Electronic Supplementary Material (ESI) for Energy & Environmental Science. This journal is © The Royal Society of Chemistry 2020 1 Supporting Information for 2 System efficiency and power: the bridge between the device and 3 system of thermoelectric power generator 4 Kang Zhu1,2, Biao Deng1,2, Pengxiang Zhang1,2, Hee Seok Kim3, 5 Peng Jiang4 and Weishu Liu1,2* 6 1 Department of Materials Science and Engineering, Southern University of Science and Technology, Shenzhen, 7 Guangdong 518055, China 8 2 Shenzhen Engineering Research Center for Novel Electronic Information Materials and Devices, Southern 9 University of Science and Technology, Shenzhen, Guangdong 518055, China 10 3 Department of Mechanical Engineering, University of South Alabama, Mobile, Alabama 36688, USA 11 4 State Key Laboratory of Catalysis, CAS Center for Excellence in Nanoscience, Dalian Institute of Chemical 12 Physics, Chinese Academy of Sciences, Dalian, Liaoning 116023, China 13 *Email: [email protected] 14 15 1. Derivation of the Governing Equation of a Thermoelectric Process 16 The earlier works by Callen1 and Domenicali2 are reviewed, to present a complete derivation 17 of the differential equation which governs the temperature distribution in a thermoelectric leg in 18 steady state. The particle current density, the energy current density and the entropy current 19 density within a TE leg are respectively denoted as 퐽⃗, 푊⃗ and 푆⃗. The heat current density is by 20 definition written as 푞⃗ = 푇푆⃗. In a steady-state process the following conditions should be met, 21 ∇ ∙ 푊⃗ = 0#(푆1) 22 ∇ ∙ 퐽⃗ = 0#(푆2) 23 ∇ ∙ 푆⃗ = 푆̇ #(푆3) 24 where 푆̇ is the entropy production rate per unit volume. The energy current density consists of 25 two parts, i.e., the kinetic part and the potential part, which gives 26 푊⃗ = 푞⃗ + 휇퐽⃗ = 푇푆⃗ + 휇퐽⃗#(푆4) 휇 = 휇 + 휇 휇 휇 27 where 푐 푒 is the electrochemical potential energy per particle, with 푐 and 푒 respectively 28 representing the chemical portion and the electrical portion. Under this circumstance the chemical 29 portion is minor and is thus neglected in the following derivations. The electrical part could be 휇 = 푒휑 30 written as 푒 , with 푒 being the charge on a particle and 휑 the electrostatic potential. 31 The entropy production rate is obtained by Eqs. (S1)-(S4) as 1 휇 1 1 푆̇ = ∇ ∙ 푊⃗ ‒ ∇ ∙ 퐽⃗ = ∇ ∙ 푞⃗ ‒ ∇휇 ∙ 퐽⃗#(푆5) 32 푇 푇 푇 푇 1 1 ∇휇 ∇ 33 Taking ‒ 퐽⃗ and 푞⃗ as currents, and 푇 and 푇 as the driving forces, the relationships between 34 the currents and forces could be written as 1 1 1 ‒ 퐽⃗ = 퐿 ∇휇 + 퐿 ∇ 11푇 12 푇 1 1 #(푆6) 푞⃗ = 퐿 ∇휇 + 퐿 ∇ { 21 22 1 푇 푇 퐿 = 퐿 2 Onsager’s theorem3 states that a reciprocal relation, in the form of 12 21 in a two-flow 3 system, exists when there is no magnetic field or Coriolis force. Then Eq. (S6) could be rewritten 4 as 1 1 ‒ 퐽⃗ = 퐿 ∇휇 + 퐿 ∇ 11푇 12 푇 1 1 #(푆7) 푞⃗ = 퐿 ∇휇 + 퐿 ∇ { 12 22 5 푇 푇 퐿 퐿 퐿 6 The kinetic coefficients ( 11, 12 and 22) are involved with the physical properties (훼, 휎 and 7 휅) of the medium, and are derived as follows. First of all, the thermal conductivity is defined as the 8 heat current density per unit temperature gradient for zero particle current. Keep in mind that the 9 heat current is in the opposite direction of the temperature gradient, 푞⃗ 휅 =‒ #(푆8) 10 ∇푇 11 Based on Eq. (S7) and the zero particle current condition, i.e., 퐽⃗ = 0, the thermal conductivity 12 could be derived as 2 퐿11퐿22 ‒ 퐿12 휅 = #(푆9) 2 13 푇 퐿11 퐽⃗ = 푒퐽⃗ 14 Secondly, the electrical conductivity is defined as the electric current density ( 푒 ) per unit 15 potential gradient under an isothermal condition. Also keep in mind that the electric current is in 16 the opposite direction of the electrical potential gradient, 푒퐽⃗ 휎 =‒ #(푆10) 17 ∇휑 휇 = 푒휑 18 Again, by Eq. (S7), 푒 , and the isothermal condition, i.e., ∇푇 = 0, the electrical 19 conductivity could be derived as 2 푒 퐿11 휎 = #(푆11) 20 푇 21 Eliminating the ∇휇 term in Eq. (S7), the entropy current density could be written as 2 푞⃗ 퐿12 퐿11퐿22 ‒ 퐿12 1 푆⃗ = =‒ 퐽⃗ + ∇ #(푆12) 22 푇 푇퐿11 푇퐿11 푇 23 In Eq. (S12), the first term in the right hand side represents the entropy flow caused by the 퐿12 ‒ 푇퐿 푆 24 particle flow, hence the term 11 means the entropy flow per particle ( 퐽), which is in a 25 relationship with the Seebeck coefficient by 퐿12 푆퐽 =‒ = 푒훼#(푆13) 26 푇퐿11 27 The kinetic coefficients could be obtained by solving Eqs. (S9), (S11) and (S13), 2 푇휎 퐿11 = 푒2 푇2 푇2 퐿12 =‒ 휎푆퐽 =‒ 휎훼#(푆14) 푒2 푒 푇3 퐿 = 휎푆2 + 푇2휅 = 푇3휎훼2 + 푇2휅 { 22 2 퐽 1 푒 2 Taking the kinetic coefficients back into Eq. (S7), formulas of the particle current density and 3 heat current density are derived as 휎 휎훼 퐽⃗ =‒ ∇휇 ‒ ∇푇 푒2 푒 #(푆15) { ⃗ ⃗ 4 푞 = 푇훼푒퐽 ‒ 휅∇푇 5 The zero divergence of the energy current density (Eq.(S1)) and the particle current density 6 (Eq. (S2)) lead to 7 ∇ ∙ 푞⃗ =‒ ∇휇 ∙ 퐽⃗#(푆16) 8 Substituting Eq. (S15) into Eq. (S16) gives the governing equation as 푒2퐽2 ∇ ∙ (휅∇푇) + ‒ 푒푇∇훼 ∙ 퐽⃗ = 0#(푆17) 9 휎 퐽⃗ = 푒퐽⃗ 10 Taking the electrical current density ( 푒 ) into Eq. (S17), 퐽2 푒 ⃗ ∇ ∙ (휅∇푇) + ‒ 푇∇훼 ∙ 퐽푒 = 0#(푆18) 11 휎 12 In a homogeneous medium, Eq. (S18) could be rewritten as 퐽2 푒 ⃗ ∇ ∙ (휅∇푇) + ‒ 휏퐽푒 ∙ ∇푇 = 0#(푆19) 13 휎 푑훼 휏 = 푇 14 where 푑푇 is the Thomson coefficient. In case of temperature independent physical properties 15 and one-dimension energy flow, Eq. (S19) degrades into a simpler form as 2 2 푑 푇 퐽푒 휅 + = 0#(푆20) 2 16 푑푥 휎 17 2. Derivation of the Maximum Efficiency and Power Formula 18 The analytical solution of the governing equation and boundary conditions (Eq. (5)) leads to 19 a temperature profile of the TEG leg as 0.5퐼2푅푅 푡,푙 2 푇 =‒ 푥 + 퐶1푥 + 퐶2 퐿2 2 2 1푐1푓푐퐼 푅푅푡,푙 + 0.5푐1푐2퐼 푅푅푡,푙 + 훼푅푡,푙퐼(푓ℎ푇푐 + 푓푐푇ℎ) ‒ Δ푇 퐶1 = 퐿 푓ℎ + 푓푐 + 푐1푐2 #(푆21) 2 2 1 푓ℎ[푐1푓푐퐼 푅푅푡,푙 + 0.5푐1푐2퐼 푅푅푡,푙 + 훼푅푡,푙퐼(푓ℎ푇푐 + 푓푐푇ℎ) ‒ Δ푇] 퐶2 = + 푇ℎ 푐1{ 푓ℎ + 푓푐 + 푐1푐2 } 푐1 = 1 + 푓ℎ훼푅푡,푙퐼 { 푐 = 1 ‒ 푓 훼푅 퐼 20 2 푐 푡,푙 21 where 푅 denotes the electric resistance of the TEG leg. From the above equation the leg terminal 22 temperatures and effective temperature difference are given by 3 ' 푇ℎ = 퐶2 ' 2 푇푐 =‒ 0.5퐼 푅푅푡,푙 + 퐶1퐿 + 퐶2 2 2 #(푆22) Δ푇 ‒ 훼푅푡,푙퐼(푓ℎ푇푐 + 푓푐푇ℎ + 푓ℎ푓푐퐼 푅푅푡,푙) + 0.5퐼 푅푅푡,푙(푓ℎ ‒ 푓푐) Δ푇' = { 푓 + 푓 + 푐 푐 1 ℎ 푐 1 2 ' 2 From the equation of Δ푇 , the contributions of the external thermal resistances, the Peltier 3 effect and the Joule effect to the effective temperature difference across the TEG leg could be 4 clearly clarified. Among these factors, the external thermal resistances will definitely decrease ' 푓 + 푓 5 Δ푇 as exhibited by the division of Δ푇 by ℎ 푐. Since the Peltier effect absorbs heat from the hot 푇 ' 푇' 6 side and releases heat to the cold side, it acts to lower ℎ and raise 푐, leading to a further ' 7 reduction in Δ푇 , reflected by the negative sign before the Peltier term (the second term in the 8 numerator). With regard to the Joule effect (the third term in the numerator), it acts primarily to 9 raise the temperature level of the whole leg rather than only influence some specific regions, so ' 10 its contribution to Δ푇 is generally trivial compared to other factors. 11 According to the relationship between the electric current and the effective temperature ' 12 difference Δ푇 , i.e., 훼Δ푇' 퐼 = #(푆23) 13 푅(1 + 푚) 14 a cubic equation of the electric current is deduced as 3 2 푏0퐼 + 푏1퐼 + 푏2퐼 + 푏3 = 0 2 2 푏0 = 푓ℎ푓푐훼 푅푡,푙푅푚 푏1 = ‒ 훼푅푡,푙푅(푓ℎ ‒ 푓푐)(푚 + 0.5)#(푆24) 2 푏2 =‒ [훼 푅푡,푙(푓ℎ푇푐 + 푓푐푇ℎ) + (1 + 푓ℎ + 푓푐)(1 + 푚)푅] { 푏 = 훼Δ푇 15 3 16 where 푚 is the ratio of the load resistance to the TE leg electrical resistance. Unfortunately, even 17 though the analytical solutions of a cubic equation do exist, the complicated formation of Eq.(S24) 18 will eventually make the analytical derivations of maximum power output and conversion 19 efficiency unattainable. To make a balance between precision and concision (and also feasibility), 20 it is necessary to eliminate or simplify some terms within the temperature profile Eq.(S21) 21 carefully. Besides, the original equations of (S21), (S22) and (S24) could be packaged into a 22 computational program to generate exact numerical solutions, serving as reference data for the 23 validation of the simplified analytical model. 24 First of all, as an attempt to neglect the influence of Joule effect on the effective temperature 2 25 difference, the terms including 퐼 푅 in Eq.(S1) are eliminated, resulting in a simplified temperature 26 profile as 푇 = 퐶1푥 + 퐶2 1훼푅푡,푙퐼(푓ℎ푇푐 + 푓푐푇ℎ) ‒ Δ푇 퐶1 = 퐿 푓ℎ + 푓푐 + 푐1푐2 푓ℎ푇푐 + 푓푐푇ℎ + 푐2푇ℎ#(푆25) 퐶2 = 푓ℎ + 푓푐 + 푐1푐2 푐1 = 1 + 푓ℎ훼푅푡,푙퐼 { 푐 = 1 ‒ 푓 훼푅 퐼 27 2 푐 푡,푙 푐 푐 28 Secondly, the terms 1 and 2 are checked for any chance of simplification, since their 4 퐶 퐶 1 multiplication in the denominator of 1 and 2 will continue to generate a cubic equation of 퐼, 2 which is still difficult to solve analytically.