Engineering & Science
Thermoelectric Figure of Merit (ZT) & Ioffe Maximum Generator Efficiency
Calculate the dimensionless thermoelectric figure of merit ZT = (S^2 * sigma * T_avg) / kappa and Abram Ioffe’s maximum solid-state heat-to-electricity conversion efficiency eta_max (%) relative to Carnot.
Balance Seebeck voltage, electrical conductivity, and phonon thermal leakage in ZT!
Deep-space Voyager/Mars radioisotope thermoelectric generators (RTGs) and automotive exhaust waste-heat harvesters convert temperature gradients (T_h to T_c, with mean temperature T_avg = (T_h + T_c) / 2) directly into DC electricity with zero moving parts! Yet good electrical conductors (high sigma, S/m) usually also conduct parasitic heat (high thermal conductivity kappa, W/(m*K)), while electrical insulators boost Seebeck thermopower S (uV/K) at the cost of huge internal resistance!
Abram Ioffe unified all three transport properties into the dimensionless Thermoelectric Figure of Merit ZT = ((S * 10^-6)^2 * sigma * T_avg) / kappa, which governs maximum possible generator efficiency: eta_max = ((T_h - T_c) / T_h) * ((SQRT(1 + ZT) - 1) / (SQRT(1 + ZT) + (T_c / T_h)))! For a nanostructured PbTe/SnSe leg at T_h = 750 K, T_c = 450 K (T_avg = 600 K, Carnot efficiency = 40.0%) with S = 220 uV/K, sigma = 100,000 S/m, and kappa = 1.80 W/(m*K), ZT = 1.613 and maximum generator efficiency reaches eta_max = 11.13% (jumping from just 5.38% for a conventional ZT ~ 0.56 alloy at kappa = 5.20 W/(m*K))!
| Thermoelectric Semiconductor (Th = 750 K, Tc = 450 K, S = 220 uV/K) | Thermal Cond. kappa (W/(m*K)) | Figure of Merit ZT | Max Generator Efficiency eta_max (%) |
|---|---|---|---|
| Nanostructured Phonon-Glass (SnSe/PbTe) | 1.80 W/(m*K) | 1.613 | 11.13% (27.8% of Carnot) |
| Conventional Bulk Alloy | 5.20 W/(m*K) | 0.558 | 5.38% |
| Invalid Temps (Th <= Tc or Tc <= 0) | Thermodynamic Guard | "Invalid TEG Inputs" | Requires Th > Tc > 0 and kappa > 0 |
Dimensionless Thermoelectric Figure of Merit ZT at Mean Temperature T_avg
Converts Seebeck coefficient B2 from uV/K to V/K (* 10^-6) at mean temperature T_avg = (T_h + T_c) / 2.
=IF(OR(B3 <= 0, B4 <= 0, B5 <= B6, B6 <= 0), "Invalid TEG Inputs", ROUND((((B2 * 0.000001) ^ 2) * B3 * ((B5 + B6) / 2)) / B4, 3))Ioffe Maximum Thermoelectric Conversion Efficiency eta_max (%)
Multiplies Carnot efficiency (T_h - T_c) / T_h by the exact Ioffe load-matched ZT factor.
=IF(OR(B3 <= 0, B4 <= 0, B5 <= B6, B6 <= 0), "Invalid TEG Inputs", ROUND(100 * ((B5 - B6) / B5) * ((SQRT(1 + ((((B2 * 0.000001) ^ 2) * B3 * ((B5 + B6) / 2)) / B4)) - 1) / (SQRT(1 + ((((B2 * 0.000001) ^ 2) * B3 * ((B5 + B6) / 2)) / B4)) + (B6 / B5))), 2))Try the example
Use the original values or change them to test your understanding.