7. Accounting for Variation of k in Steady Heat Transfer

\[\begin{aligned} Q_{\text{flat}} &= k_{\text{avg}}A\frac{T_1-T_2}{L} = \frac{A}{L}\int_{T_2}^{T_1}k(T)\,dT \\\\ Q_{\text{cylinder}} &= 2\pi k_{\text{avg}}L\frac{T_1-T_2}{\ln(r_1/r_1)} = \frac{2\pi L}{\ln(r_2/r_1)}\int_{T_2}^{T_1}k(T)\,dT \\\\ Q_{\text{sphere}} &= 4\pi k_{\text{avg}}r_1r_2\frac{T_1-T_2}{r_2-r_1} = \frac{4\pi r_1r_2}{r_2-r_1}\int_{T_2}^{T_1}k(T)\,dT \end{aligned}\]