 Thermoelectric couple figure of merit

For a constant-property thermoelectric module consisting of two material legs, let ${\alpha }_{np}=S_p-S_n$ be their Seebeck coefficient difference, $R$ their total series electrical resistance, and $K$ their total parallel thermal conductance. The couple's thermoelectric figure of merit isIts units are ${\mathrm{K}}^{-1}$; multiply by an absolute temperature to form a dimensionless figure. This expression assumes that contact losses are absent or consistently included in $R$ and $K$.

For equal leg lengths $l$ and areas $A$, with equal electrical resistivity $\rho$ and thermal conductivity $\kappa$, one has $R=2\rho l/A$ and $K=2\kappa A/l$. Consequently $Z_{\mathrm{pair}}={\alpha }_{np}^2/(4\rho \kappa )$. The factor four follows from combining two series resistances and two parallel conductances; it disappears only if the symbols have already been defined as pair properties. With opposite equal Seebeck coefficients $S_p=-S_n=S$, the result reduces to $S^2/(\rho \kappa )$.