Question

A rod of length $l$ with thermally insulated lateral surface consists of material whose heat conductivity coefficient varies with temperature as k = a / T, where a is a constant. The ends of the rod are kept at temperatures $T_1$ and $T_2$. If the function T(x), where x is the distance from the end whose temperature is $T_1$ is given as $T=T_1{\left(\frac{T_2}{T_1}\right)}^{kx/l}$, then the value of k will be

Answer 1

$i=\frac{△Q}{△t}=-kA\left(\frac{dT}{dx}\right){\int }_{T_1}^{T_2}\frac{dT}{T}=\frac{-i}{aA}{\int }_o^{l}dx\Rightarrow \ln \frac{T_2}{T_1}=\frac{-il}{aA}\Rightarrow \ln \frac{T_1}{T_2}=\frac{il}{aA}\Rightarrow i=\frac{aA}{l}\ln \frac{T_1}{T_2}\left(1\right){\int }_{T_1}^{T_2}\frac{dT}{T}=\frac{-\ln \left(\frac{T_1}{T_2}\right)}{l}{\int }_o^{x}dx\Rightarrow \frac{T_1}{T_2}={\left(\frac{T_1}{T_2}\right)}^{\frac{-x}{l}}T\left(x\right)=T_1\times {\left(\frac{T_1}{T_2}\right)}^{\frac{-x}{l}}\Rightarrow T\left(x\right)=T_1{\left(\frac{T_2}{T_1}\right)}^{\frac{x}{l}}$