Wiedemann–Franz law and thermoelectric inequalities: Effective <i>ZT</i> and single-leg efficiency overestimation

We derive a thermoelectric inequality in the thermoelectric conversion between the material figure of merit (ZT) and the effective ZT of the module by combining the constant Seebeck coefficient approximation with the Wiedemann–Franz law. In a P–N leg-pair module, the effective ZT lies between the individual ZT values of the P- and N leg materials. In a single-leg module, however, the effective ZT is less than approximately one-third of the leg's ZT because an external wire is necessary to complete the circuit, introducing additional thermal and electrical losses. Multi-dimensional numerical analysis shows that although structural optimization can mitigate these losses, the system efficiency remains limited to below half of the ideal single-leg efficiency. Our findings explain the overestimation of single-leg efficiency and highlight the importance of optimizing the P–N leg-pair module structure. They also underscore the need for thermoelectric leg compatibility, particularly regarding Seebeck coefficients.