- #1
Physgeek64
- 247
- 11
Homework Statement
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.
Homework Equations
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