Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 1 / Section B / 12 / iii

For the thermoelectric couple figure of merit, the coefficient ${\alpha }_{np}$ denotes a pair difference, whereas the table gives one value each for resistivity and thermal conductivity without saying whether they are per-leg or effective pair values. To keep the factors consistent, first take both legs to have the listed $\rho$ and $\kappa$, and equal $A,L$ as stated. ThenThis is identical to treating each symmetric leg as having Seebeck-coefficient magnitude ${\alpha }_{np}/2$. The mean temperature is $\overline{T}=(523.15+323.15)/2=423.15\mathrm{K}$, giving $Z_{\mathrm{pair}}\overline{T}=0.162$.

If the examination instead intends its $\rho$ and $\kappa$ as already combined effective pair coefficients in $R=\rho L/A$ and $K=\kappa A/L$, the alternative convention gives $Z=1.53\times 10^{-3}{\mathrm{K}}^{-1}$, with $Z\overline{T}=0.649$. That convention must be declared; using the full pair Seebeck difference with individual-leg $\rho ,\kappa$ while omitting both factors of two would overestimate the device figure of merit by four.

Use absolute temperatures $T_H=523.15\mathrm{K}$, $T_C=323.15\mathrm{K}$ and $\overline{T}=423.15\mathrm{K}$. With the explicit identical-per-leg interpretation from the preceding part,and the stated ideal efficiency expression givesWith the alternative already-combined pair convention, $Z\overline{T}=0.6488$ instead gives ${\eta }_{\max }\simeq 5.71\%$. Both lie below the Carnot efficiency $1-T_C/T_H=38.2\%$. These are constant-property ideal conversion efficiencies, neglecting contact resistance, parasitic heat paths and temperature dependence. They are not the efficiency specifically at the maximum-power load.

The temperature difference is $200\mathrm{K}$, so the pair open-circuit voltage isFor two identical legs with the stated per-leg geometry,At matched load $R_L=R$,The load voltage is $0.0336\mathrm{V}$, and multiplying it by the current reproduces the power. If $\rho$ instead denotes an already combined pair resistivity, $R=0.01196\Omega$, giving $I=2.81\mathrm{A}$ and $P_{\max }=0.0944\mathrm{W}$. These alternatives follow the same convention distinction as the figure-of-merit calculation; they should not be mixed.

For $N$ identical couples connected electrically in series, the open-circuit voltage is $N\mathcal{E}$ and the internal resistance is $NR$. At the module's matched load $R_{L,\mathrm{module}}=NR$,Maintaining the stated temperature difference across every couple therefore requiresunder the identical-per-leg interpretation. Sixty-four ideal couples supply about $3.02\mathrm{W}$; sixty-three supply about $2.97\mathrm{W}$. The alternative already-combined pair-resistance convention gives 32 couples, each producing approximately $0.0944\mathrm{W}$.

The module is thermally parallel: adding couples increases the heat flow that must be supplied and rejected. The count assumes that source and sink keep the actual leg junctions at the stated temperatures and that interconnect losses are negligible. Connecting the couples does not multiply the maintained temperature difference.