(adsbygoogle = window.adsbygoogle || []).push({}); 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(d-> partial derivative btw)

dT/dt = D(d^2T/dr^2)

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?

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# Homework Help: Heat conduction problem: sphere

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