PDE: Heated Sphere Homework Solution

[skrat]

Homework Statement

This is not really a school problem, it's actually something I am trying to figure out. So, we have a sphere with given radius. (Actually let's assume that all the parameters are known). The sphere has equally distributed heaters and is in the beginning at constant temperature. It also radiates as a black body. I would like to know the temperature as a function of radius and time.

Homework Equations

The Attempt at a Solution

So if I am not mistaken, what I have to solve is $$\frac{\partial T}{\partial t}-D\nabla ^2 T= \frac{q}{\rho c_p}$$ if $q$ is the density of the heaters. Since everything is symmetrical, the solution of homogeneous equation should like something like $$T(r,t)=(aj_0(kr)+bn_0(kr))e^{-k^2Dt}$$ where $j_0$ is spherical Bessel function and $n_0$ is spherical Neumann. Due to the nature of Neumann functions when $r\rightarrow 0$ the constant $b=0$.

Now in order to simplify my boundary conditions, which are: $$T(r,t=0)=T_0$$ and $$-\lambda \frac{dT(r=R)}{dr}=\sigma T(R)^4$$ I decided to add a constant to my solution. Therefore the solution should look something like $$T(r,t)=aj_0(kr)e^{-k^2Dt}+T_0$$ which also changes my boundary condition to $$T(r,t=0)=T_0=aj_0(kr)+T_0$$ meaning that $kr=\xi_n$ where $\xi_n$ is n-th zero of a Bessel function.

Now I am not sure about this part above. Really not. Is everything ok so far? :/ I am not sure because my $k$ is now actually $k(r)$. This confuses me a bit.

 

[Chestermiller]

In a problem like this, the first thing to do is to find the long-time steady state solution. This will then be the starting point for developing the transient solution. Furthermore, if you can't get the steady state solution, you will never be able to find the transient solution. So, follow the most important rule of modeling: Start Simple.

Chet