How Do You Solve Heat Conduction in a Sphere with Fixed Boundary Temperature?

[genius2687]

A solid sphere of radius a is immersed in a vat of fluid at a temperature T_0. Heat is conducted into the sphere according to

dT/dt = D(d^2T/dr^2) (d-> partial derivative btw)

If the temperature at the boundary is fixed at T_0 and the initial temperature of the sphere is T_1, find the temperature within the sphere as a function of time.

My reasoning

Ok. Here's my reasoning. Use a solution of the form T=X(t)R(r), and plug into the above equation to get

R''/R=X/(DX')=-k^2.

I get X(t) = C*[exp(-t/(D*k^2))]. (k^2>0 for convergence)

Then I have R'' +k^2*R = 0

so R= Acos(kr) + Bsin(kr), since k^2>0.

The Problem

Assuming that the above steps are right, we could set the boundary conditions and have Acos(ka)+Bsin(ka)=T_0. That only eliminates one variable, either A or B. I don't know where to go from there however. Should this be like a Fourier series or something?

 

[Meir Achuz]

The Laplacian in spherical coords is not d^2T/dr^2.
It is $\frac{1}{r}\left[\partial_r^2(rT)\right]$