P001.md

P001: Engineering Thermoelectric Model (ZTeng) (H.S. Kim et al., 2015)

• H.S. Kim, W. Liu, G. Chen, C. Chu, Z. Ren, Relationship between thermoelectric figure of merit and energy conversion efficiency, Proceedings of the National Academy of Sciences 112 (27) (2015) 8205-8210. https://doi.org/10.1073/pnas.1510231112

$$ $$

$$ +------------------------------------------+ $$

$$ (PF)_{eng} = \frac{\left[ {\int _{T_{c}}^{T_{h}} {S(T)} dT} \right] ^{2}}{\int _{T_{c}}^{T_{h}} {\rho (T)} dT} $$

$$ (ZT)_{eng} = \frac{\left[ {\int _{T_{c}}^{T_{h}} {S(T)} dT} \right] ^{2}}{{\int _{T_{c}}^{T_{h}} {\rho (T)} dT}{\int _{T_{c}}^{T_{h}} {\kappa (T)} dT}} \Delta T $$

$$ P_{d,max} = \frac{(PF)_{eng}}{4 L} \Delta T $$

$$ \eta _{max} = \eta _{c} \frac{\sqrt{1 + (ZT)_{eng} \alpha _{1} \eta _{c}^{-1}} - 1}{\alpha _{0} \sqrt{1 + (ZT)_{eng} \alpha _{1} \eta _c^{-1}} + \alpha _{2}} $$

$$ \eta _{c} = \frac{\Delta T}{T_{h}} , \tau (T) = T \frac{dS(T)}{dT} $$

$$ \alpha _{i} = \frac{S(T_{h}) \Delta T}{\int _{T_{c}}^{T_{h}} {S(T)} dT} - \frac{\int _{T_{c}}^{T_{h}} {\tau (T)} dT}{\int _{T_{c}}^{T_{h}} {S(T)} dT} W_{T} \eta _{c} - i W_{J} \eta _{c} $$

$$ W_{T} = \frac{\int _{T_{c}}^{T_{h}} {dT} {\int _{T}^{T_{h}} {\tau (T) dT}} }{\Delta T \int _{T_{c}}^{T_{h}} {\tau (T)} dT} , W_{J} = \frac{\int _{T_{c}}^{T_{h}} {dT} \int _{T}^{T_{h}} {\rho (T) dT}}{\Delta T \int _{T_{c}}^{T_{h}} {\rho (T)} dT} $$

$$ +------------------------------------------+ $$

$$ \frac{d}{dx} \left( \kappa (T) \frac{dT}{dx} \right) + J^{2} \rho (T) - J \tau (T) \frac{dT}{dx} = 0 $$

$$ Q_{h} = \frac{A}{L} \int _{T_{c}}^{T_{h}} {\kappa (T)} dT + I T_{h} S(T_{h}) - W_{J} I^{2} R - W_{T} I \int _{T_{c}}^{T_{h}} {\tau (T)} dT $$

$$ P_{out} = I \int _{T_{c}}^{T_{h}} {S(T)} dT - I^{2} R $$

$$ R = \frac{1}{\Delta T} \frac{L}{A} \int _{T_{c}}^{T_{h}} {\rho (T)} dT $$

$$ I = \frac{\int _{T_{c}}^{T_{h}} {S(T)} dT}{R (1+m)}, m = \frac{R_L}{R} $$

$$ m_{opt, P_{d}} = 1 $$

$$ m_{opt, \eta} = \sqrt{1 + (ZT)_{eng} \alpha _{1} \eta _{c}^{-1}} $$

$$ +------------------------------------------+ $$