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and ZT value. The top thick line is the Carnot efficiency."/>

      Source: Fanciulli et al. [31]. © Elsevier.

      1.2.2 Effect of Electrical and Thermal Contact Resistances On Thermoelectric Performance

      (1.35)upper R Subscript e Baseline equals upper R plus 2 upper R Subscript c Baseline comma

      where R is the resistance of the thermoelectric leg. The temperature drop (ΔT c) at the contacts lowers the actual temperature difference (ΔT) on the thermoelectric leg,

      (1.36)upper Delta upper T equals upper Delta upper T 0 minus 2 upper Delta upper T Subscript normal c Baseline comma

Schematic illustration of structure of a thermoelectric module with n-type and p-type thermoelectric legs.

      The deterioration of the thermoelectric performance by the electrical and thermal contact resistances can be modeled into their effects on the effective ZT value. If a thermoelectric leg with a length of l and figure of merit of ZT is connected to the contact layers with the conductivity of (σ con) and thermal conductivity of (κ con), the effective ZT e value will decrease,

      (1.37)upper Z upper T Subscript normal e Baseline equals StartStartFraction upper Z upper T OverOver left-parenthesis 1 plus StartFraction 2 sigma Over sigma Subscript c o n Baseline l EndFraction right-parenthesis left-parenthesis 1 plus StartFraction 2 kappa Over kappa Subscript c o n Baseline l EndFraction right-parenthesis EndEndFraction comma

      where σ and κ are the electrical conductivity and thermal conductivity of the thermoelectric material, respectively. Thus, the decrease in the ZT value is related to the electrical conductivities and thermal conductivities of the contact layer and the thermal electric leg and the length of the thermoelectric leg. An effective material for the contact layer should have high electrical conductivity and high thermal conductivity. It is important to make the thermoelectric interfaces with small electrical and thermal contact resistances. To design high thermoelectric conversion efficiency, the contact area between the electrodes and thermoelectric legs and the length of the thermoelectric legs are important parameters [32, 33].

      The energy conversion efficiency of a thermoelectric generator or module is the electrical power output divided by the heat power input. But it is difficult to accurately determine the heat power input. Thus, electrical powers instead of the efficiency values are usually provided in literature. The electrical power output depends on the temperature difference (ΔT). Because of the temperature dependences of the Seebeck coefficient, the ZT value and the resistances of thermoelectric legs and electrodes, the open‐circuit voltage and the total electrical resistance vary with ΔT as well. In addition, cracks and materials degradation can happen under thermal cycling. Thus, the evolutions of the power output, open‐circuit voltage and total electrical resistance are sometimes presented with the number of the thermal cycles.

      1.2.3 Equation of Thermoelectric Efficiency

      The equation for the thermoelectric efficiency of thermoelectric generators is derived as below. The electrical current density through a thermoelectric material includes the drift current density (J d) driven by the potential gradient and the thermal diffusion current density (J t) driven by the temperature gradient,

      where V is the potential, T is the temperature, and σ and S are the electrical conductivity and the Seebeck coefficient of the thermoelectric material, respectively.

      (1.39)upper Delta upper V Subscript upper O upper C Baseline equals minus upper S left-parenthesis upper T Subscript normal upper H Baseline minus upper T Subscript normal upper C Baseline right-parenthesis period

Schematic illustration of steady temperature and voltage profiles of a thermoelectric leg under (a) open-circuit and (b) short-circuit conditions.
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