Archived Thermal Diffusion with Heat Source

[rabbit44]

Homework Statement

A cylindrical wire of thermal conductivity k, radius a and resistivity p uniformly
carries a current I. The temperature of its surface is fixed at T0 using water cooling.
Show that the temperature T(r) inside the wire at radius r is given by
T(r) = T0 + p(I^2)(a^2 - r^2)/4pi^2a^4k

Homework Equations

del squared(T) = (c/k)dT/dt - H/k

Where H is heat generated per unit volume

The Attempt at a Solution

I took the system to be in a steady state as the temperature is fixed. So:

d2T/dr2 = -H/k

I took H = I^2 (pl/A)(1/lA)
= I^2p/pi^2a^4

Subbing this into my differential equation and integrating twice wrt r, I get:

T = -(I^2.p.r^2)/(2pi^2.a^4.k) + ba + c

Where b and c are integration constants. I think b must be zero but am not sure why. Then putting in T=T0 at r=a I would get a close answer but with a 2 on the bottom of the second term instead of a 4.

Where did I go wrong?!?

Thanks

 

[mrb132]

remember to take the ∇²T n cylindrical coordinates not Cartesian ie it is not simply ∂²/∂r² this will give the factor of two and the term multiplying b will be ln(r) this is clearly not physical as it blows up at the origin, so must be zero.

https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates