Heat transfer: Temperature distribution inside a sphere submerged in a fluid

[happyparticle]

Homework Statement

What is the temperature distribution inside a sphere and fluid

Relevant Equations

$$\kappa \nabla^2 T = 0$$

$$T(r,\theta) = \sum_{m=0}^{\infty} = (A_m r^m + \frac{B_m}{r^{m+1}}) P_m(cos \theta)$$

Consider a sphere of thermal diffusivity $\kappa_2$ is submerged in an incompressible and stationary fluid of thermal diffusivity $\kappa_1$.

The fluid is held between 2 large plates ( at T_0 for the top plate and T_1 for the bottom plate).

What is the stationary temperature distribution inside the sphere and in the fluid ?

Since $\frac{\partial}{\partial t} = 0$ and we have symmetry in $\phi$ coordinates, the heat conduction equation becomes:


$\kappa \nabla^2 T = 0$

With the following solution in spherical coordinates :

$$T(r,\theta) = \sum_{m=0}^{\infty} = (A_m r^m + \frac{B_m}{r^{m+1}}) P_m(cos \theta)$$.

Also, since $T \rightarrow \infty$ at $r \rightarrow 0$

B must be 0.

This is as far as I can go. I'm not sure what are the boundary conditions to find the value of A and m.

 

[Chestermiller]

Is the temperature distribution axisymmetric (a function of r only)? Please provide a diagram of the situation. From your description, I am unable to visualize the system.

 

[pasmith]

You need one solution inside the sphere and a separate solution outside the sphere. Since $T$ is specified on each of the plates, you must start by finding the outer solution. Having done so, the inner solution is found from the requirement that $\kappa\frac{\partial T}{\partial r}$ must be continuous on the surface of the sphere.

For the outer solution, if the plates are at $z = r \cos \theta = \pm h$, then we have the conditions $$\begin{split} \sum_{m=0}^\infty B_m \frac{1}{r^{m+1}}P_m\left(\frac{h}{r}\right) &= T_0 \\ \sum_{m=0}^\infty B_m \frac{1}{r^{m+1}}P_m\left(-\frac{h}{r}\right) &= T_1.\end{split}$$ Exploiting the parity of the $P_m$, we can add and subtract these to get one condition for even $m$
and one condition for odd $m$: $$\begin{split} \sum_{m=0}^\infty \frac{B_{2m}}{r^{2m+1}} P_{2m}\left( \frac hr \right) = \frac{T_0 + T_1}{2} \\ \sum_{m=0}^\infty \frac{B_{2m+1}}{r^{2m+2}} P_{2m+1}\left( \frac hr \right) = \frac{T_0 - T_1}{2} \end{split}$$ The left hand sides are power series in $r^{-1}$, and by comparing coefficients of powers of $r^{-1}$ it should be possible to determine the $B_m$.

EDIT: I' not sure if it's $T$ or $\kappa \frac{\partial T}{\partial r}$ which is continuous across the surface of the sphere. Also it may be necessary to include $P_0$ and $rP_1$ in the outer solution, since far from the sphere the temperature should approximate $C + Dr\cos \theta$.

 

[PhDeezNutz]

In order to relate $A$ to $B$ and solve the system we really need to know about the boundary conditions across the sphere. Both in terms of $T$ and $\frac{\partial T}{\partial R}$. I have no domain experience for this type of thermodynamics problem BUT it is very similar to electrostatics (Laplace's equation shows up everywhere doesn't it?).

I'm going to assume $T$ is continuous but $\frac{\partial T}{\partial r}$ is not continuous and it's related to $\kappa$ of each medium across the boundary.

I don't know what that relationship is but if we did this problem would be a lot easier to solve.