(Solved): 4. (35\%) Please consider an infinitely long aluminum slab with a width of 0.1m. The temperatures ...

4. (35\%) Please consider an infinitely long aluminum slab with a width of $0.1\mathrm{m}$. The temperatures are $50^{\circ }\mathrm{C}$ and $150^{\circ }\mathrm{C}$ on the left and right boundaries, respectively. However, the thermal conductivity of aluminum varies with the temperature by the equation, $\mathrm{k}=k_0(1+\beta \mathrm{T})$, in which $\beta$ and $k_0$ are $-3\times 10^{-4}(1/\mathrm{K})$ and $277\mathrm{W}/\mathrm{mK}$; respectively. Please apply the average thermal conductivity to obtain the approximate heat flux first. Then, apply Kirchhoff's transformation to determine the correct heat flux through the aluminum slab.