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upper R Subscript e x Baseline Over upper R Subscript i n Baseline EndFraction comma"/> the equation (1.55) becomes

(1.56) upper P equals StartFraction left-parenthesis 2 StartAbsoluteValue upper S EndAbsoluteValue upper Delta upper T right-parenthesis squared Over upper R Subscript i n Baseline EndFraction StartFraction m Over left-parenthesis 1 plus m right-parenthesis squared EndFraction period

When R _in = R _ex, the external load has the maximum power (P _max),

(1.57) upper P Subscript max Baseline equals StartFraction left-parenthesis 2 StartAbsoluteValue upper S EndAbsoluteValue upper Delta upper T right-parenthesis squared Over 4 upper R Subscript i n Baseline EndFraction period

Figure 1.23 shows the variations of the output power with the external load resistance at different temperature differences [34]. The output power increases when the temperature difference increases. However, at the optimal output power, the corresponding load resistance is independent of the temperature difference.

The equation for the efficiency (η) can be obtained as it is the ratio of the output power to the incident heat (Q _in) into the system,

(1.58) eta equals StartFraction upper P Over upper Q Subscript i n Baseline EndFraction period

Heat energy transport through this system with the two thermoelectric legs, p‐ and n‐type legs. The incident heat at the hot side is balanced by the Peltier effect at the contacts between the legs and the metal electrodes, and the heat transfers due to the temperature gradient. The heat transferring from the hot to the cold side is equivalent to the Peltier effect at the contacts and the heat out from the cold side. Thus, the equations at the two boundaries (hot and cold side) are given by

Schematic illustration of variations of the power output of a TEG with the load resistance at ΔT = 10 and 20 K.

Figure 1.23 Variations of the power output of a TEG with the load resistance at ΔT = 10 and 20 K.

Source: Madan et al. [34]. © American Chemical Society.

(1.59) StartFraction upper Q Subscript i n Baseline Over upper A EndFraction minus 2 StartFraction upper S upper T Subscript upper H Baseline upper I Over upper A EndFraction plus 2 kappa left-parenthesis StartFraction partial-differential upper T Over partial-differential x EndFraction right-parenthesis Subscript 0 Baseline equals 0 comma a t x equals 0 comma h o t s i d e comma

(1.60) minus StartFraction upper Q Subscript o u t Baseline Over upper A EndFraction plus 2 StartFraction upper S upper T Subscript upper C Baseline upper I Over upper A EndFraction minus 2 kappa left-parenthesis StartFraction partial-differential upper T Over partial-differential x EndFraction right-parenthesis Subscript upper L Baseline equals 0 comma a t x equals upper L comma c o l d s i d e period

By ignoring the Thomson effect, the equation (1.50) for one leg becomes

(1.61) kappa StartFraction partial-differential squared upper T Over partial-differential x squared EndFraction plus StartFraction upper I squared Over upper A squared sigma EndFraction equals 0 period

Using the boundary conditions of T(0) = T _H and T(L) = T _C, the following equations can be obtained,

(1.62) upper T left-parenthesis x right-parenthesis equals minus StartFraction upper I squared x squared Over 2 kappa upper A squared sigma EndFraction plus left-parenthesis StartFraction upper I squared upper L Over 2 kappa upper A squared sigma EndFraction minus StartFraction upper Delta upper T Over upper L EndFraction right-parenthesis x plus upper T Subscript normal upper H Baseline comma

(1.63) StartFraction d upper T Over d x EndFraction equals minus StartFraction upper I squared Over 2 kappa upper A squared sigma EndFraction x plus StartFraction upper I squared upper L Over 2 kappa upper A squared sigma EndFraction minus StartFraction upper Delta upper T Over upper L EndFraction period

At x = 0,

(1.64) left-parenthesis StartFraction d upper T Over d x EndFraction right-parenthesis Subscript 0 Baseline equals StartFraction upper I squared upper L Over 2 kappa upper A squared sigma EndFraction minus StartFraction upper Delta upper T Over upper L EndFraction period

Using Eq. (1.54) in Eq. (1.59), the expression for Q _in can be obtained as below,

(1.65) upper Q Subscript i n Baseline equals 2 StartAbsoluteValue upper S EndAbsoluteValue upper T Subscript upper H Baseline upper I plus upper I squared upper R Subscript l e g Baseline minus 2 kappa upper A StartFraction upper Delta upper T Over upper L EndFraction comma

where upper R Subscript l e g Baseline equals StartFraction upper L Over sigma upper A EndFraction is the intrinsic resistance of one leg.

In terms of Eqs. (1.54), (1.56), and (1.65), the conversion efficiency is given by

(1.66)

where z is the figure of merit of the thermoelectric module,

(1.67) z equals StartFraction 2 upper S squared Over kappa upper R Subscript i n Baseline EndFraction StartFraction upper L Over upper A EndFraction period

The internal resistance (R _in) includes the contact resistances and

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