(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

show ## \rho c_m \frac{\partial T}{\partial t} = \kappa \frac{\partial^2T}{\partial x^2} -\frac{2}{a}R(T)##

where ##R(T)=A(T-T_0) ##

a) Obtain an expression for T as a function of x for the case of an infinitely long rod whose hot end has temperature ##T_m##

b) Show that the heat that can be transported away by a long rod of radius a is proportional to ##a^{\frac{3}{2}}##, provided A is independent of a.

2. Relevant equations

3. The attempt at a solution

so for part a) I got

##T=T_0 +(T_m-T_0)e^{-\sqrt{\frac{2A}{a}}x}##

Then for b) I thought that the rate of heat transfer will be ##2 \pi a R(T)## but this gives something that is proportional to a not ##a^{\frac{3}{2}}##

Many thanks in advance

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# Homework Help: Solve the heat equation

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