Question 3.13: Consider a bismuth telluride (Bi2Te3) p- and n-type thermoel...

Using (\dot{S}_{e,p})_{c}=-\alpha _{s}J_{e}T_{c},        \alpha _{s}=\alpha _{S.p}-\alpha _{S,n}, and from the properties in Table (a), we have

Using \frac{1}{R_{k,h-c}} =\frac{1}{(R_{k,h-c})_{p}} +\frac{1}{(R_{k,h-c})_{n}} =\left(\frac{A_{k}k}{L} \right)_{p} +\left(\frac{A_{k}k}{L} \right)_{n} ,    R_{e,h-c}= \left(\frac{\rho _{e}L}{A_{k}} \right)_{p} +\left(\frac{\rho _{e}L}{A_{k}} \right)_{n} , and from the properties in Table (a), we have

The thermoelectric figure of merit Z_{e}=\frac{\alpha _{S}^{2}}{[(k\rho _{e})_{p}^{1/2}+(k\rho _{e})_{n}^{1/2}]^{2}} for this system is Z_{e} = 0.003078 K^{−1}. For T_{h} = 35.45^{\circ } C = 308.6 K, using Q_{c}=-\sigma _{S}J_{e}T_{c}+R_{k,h-c}^{-1}(T_{h}-T_{c})+\frac{1}{2}R_{e,h-c}J^{2}_{e} , the variation of the heat removal rate from the cold junction Q_{c} is plotted in Figure, as a function of T_{h} − T_{c}, for various currents J_{e} = A_{k} j_{ e}.
The results show that as the current increases, first −Q_{c} increases and when the Joule heating contributions become significant, then −Q_{c} decreases. For the system considered, maximum in −Q_{c} occurs at J_{e} = A_{k} j_{ e} = 7.997 A, and the maximum in T_{h}-T_{c} occurs at 5.92 A. In practice, a smaller current is used.
The maximum −Q_{c} occurs for T_{h} = T_{c} (i.e., T_{h} = T_{c} = 0) and here it is Q_{c,max}(T_{h} = T_{c}) = −0.5430 W. Also, for a given current, there is a maximum value for T_{h} − T_{c} corresponding to Q_{c} = 0, i.e., no heat added at the cold junction. For this system and for J_{e} = 7.997 A, this gives (T_{h} − T_{c})_{max} = 146.3^{\circ} C.