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A rod of length $L$ with sides fully insulated is of a material whose thermal conductivity varies with $\alpha$ temperature as $ K= \frac{\alpha }{T}$, where $\alpha$ is a constant. The ends of the rod are kept at temperature $T_1$ and $T_2$. The temperature $T$ at $x,$ where $x$ is the distance from the end whose temperature is $T_1$ is
${T_1}{\left( {\frac{{{T_2}}}{{{T_1}}}} \right)^{\frac{x}{L}}}$
$\frac{x}{L}\ln \frac{{{T_2}}}{{{T_1}}}$
${T_1}{e^{\frac{{{T_2}x}}{{{T_1}L}}}}$
${T_1} + \frac{{{T_2} - {T_1}}}{L}x$
Solution
By definition the heat flux (per unit area) is
$Q=-K \frac{d T}{d x}=-\alpha \frac{d}{d x} \ln T$
$=$ constant $=+\alpha \frac{\ln T_{1} / T_{2}}{l}$
Integrating $\quad \ln T=\frac{x}{l} \ln \frac{T_{2}}{T_{1}}+\ln T_{1}$
where $T_{1}=$ temperature at the end $x=0$
$\mathrm{So}$
$T=T_{1}\left(\frac{T_{2}}{T_{1}}\right)^{x / l}$
