P003.md

P003: Device Figure-of-Merit (ZTdev) (G.J. Snyder et al., 2017)

• G.J. Snyder, A.H. Snyder, Figure of merit zt of a thermoelectric device defined from materials properties, Energy Environ. Sci. 10 (11) (2017) 2280-2283. https://doi.org/10.1039/C7EE02007D.

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$$ (ZT)_{dev} = \left[ \frac{T_{h} - T_{c}(1-\eta)}{T_{h}(1-\eta) - T_{c}} \right] ^{2} - 1 $$

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$$ \eta = 1 - \frac{\Phi(T_{c})}{\Phi(T_{h})} $$

$$ \Phi = ST+1/u $$

$$ \frac{1}{u_{n}} = \frac{1}{u_{n-1}} \sqrt{1 - u_{n-1}^2 (\rho _{n} \kappa _{n} + \rho _{n-1} \kappa _{n-1})(T_{n} - T_{n-1})} - \left( \frac{T_{n} + T_{n-1}}{2} \right) (S_{n} - S_{n-1}) $$

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An Explanation :

• Ref: G. J. Snyder, Thermoelectric Power Generation: Efficiency and Compatibility, in Thermoelectrics Handbook: Macro to Nano, ed. D. M. Rowe, CRC Press, 2006, ch. 9.

The steady-state heat equation is given as

$$ \nabla (\kappa \nabla T) = - T \frac{dS}{dT} J \nabla T - \rho J^2 $$

Consider the relative (reduced) current density u, which is defined as

$$ u = \frac{J}{\kappa \nabla T} $$

we can rewirte the steady-state heat equation as

$$ \frac{du}{dT} = u^{2} T \frac{dS}{dT} + u^{3} \rho \kappa $$

also as

$$ \frac{d}{dT} \left( \frac{1}{u^{2}} \right) = -2 \left( \frac{T}{u} \frac{dS}{dT} + \rho \kappa \right) $$

For computation, it can be approximated by

$$ \frac{1}{u_{n}^2} - \frac{1}{u_{n-1}^2} = -2 (T_{n} - T_{n-1}) \left( \frac{T_{n} + T_{n-1}}{2} \frac{1}{u_{n}} \frac{S_{n} - S_{n-1}}{T_{n} - T_{n-1}} + \frac{\rho _{n} \kappa _{n} + \rho _{n-1} \kappa _{n-1}}{2} \right) $$

$$ \frac{1}{u_{n}^2} + \frac{1}{u_{n}} (T_{n} + T_{n-1}) (S_{n} - S_{n-1}) = \frac{1}{u_{n-1}^2} -(T_{n} - T_{n-1}) (\rho _{n} \kappa _{n} + \rho _{n-1} \kappa _{n-1}) $$

$$ \left[ \frac{1}{u_{n}} + \left( \frac{T_{n} + T_{n-1}}{2} \right) (S_{n} - S_{n-1}) \right] ^2 = \frac{1}{u_{n-1}^2} -(T_{n} - T_{n-1}) (\rho _{n} \kappa _{n} + \rho _{n-1} \kappa _{n-1}) + \left[ \left( \frac{T_{n} + T_{n-1}}{2} \right) (S_{n} - S_{n-1}) \right] ^2 $$

$$ \frac{1}{u_{n}} + \left( \frac{T_{n} + T_{n-1}}{2} \right) (S_{n} - S_{n-1}) \cong \sqrt{\frac{1}{u_{n-1}^2} -(T_{n} - T_{n-1}) (\rho _{n} \kappa _{n} + \rho _{n-1} \kappa _{n-1})} $$

$$ \frac{1}{u_{n}} = \frac{1}{u_{n-1}} \sqrt{1 - u_{n-1}^2 (\rho _{n} \kappa _{n} + \rho _{n-1} \kappa _{n-1})(T_{n} - T_{n-1})} - \left( \frac{T_{n} + T_{n-1}}{2} \right) (S_{n} - S_{n-1}) $$

When a trial u is given, we can slove the steady-state heat equation and calculate the efficiency $\eta$ of a thermoelectric device by thermoelectric potential $\Phi$,

$$ \eta = 1 - \frac{\Phi(T_{c})}{\Phi(T_{h})} $$

$$ \Phi = ST+1/u $$

As the value of u is optimized, the efficiency will reach the maximum.