Derivation of heat transfer equation for spherical coordinates

[eddysd]

Homework Statement

where λ= thermal conductivity
\dot{q}= dissipation rate per volume

Homework Equations

q_x=-kA\frac{dT}{dx}

The Attempt at a Solution

I don't know where to start from to be honest, so any help would be greatly appreciated

 

[rude man]

I've had to think on this one for some time; I hope what I write is correct:

Start with the fundamental equation for heat transfer:

dQ/dt = λAΔT/Δr
where
dQ/dt = Qdot = rate of heat flow across area A;
λ = conductivity;
ΔT = temperature difference across volume element AΔr.

What is ΔT/Δr in the limit as Δr → 0?

Then: what is the volume element AΔr in spherical coordinates? (Heat flows thru the volume element from one side of area A to the other side, also of area A, the two sides separated by Δr. )

Now for the big step: realize that Qdot need not be constant along Δr. In other words, Qdot can be different for the two end-sides of your elemental volume. So in the limit the derivative d(Qdot)/dr can be finite. So your last equation is to equate how Qdot changes along Δr to what the problem calls the "dissipation rate per volume".

 

[eddysd]

OK so this is what I got:

-λ4r²\frac{dT}{dr} + \dot{q}4∏r²dr = ρc4∏r²\frac{dT}{dτ}dr -4∏r²(λ\frac{dT}{dr} + \frac{d}{dr}(λ\frac{dT}{dr})dr)

Is this correct?

Since the flow is steady the time derivative \frac{dT}{dτ}=0

But then when I rearrange everything I get:

r²\frac{d}{dr}(λ\frac{dT}{dr}) + \dot{q}r² = 0

can I just take the r² inside the differential bracket?

EDIT: missed out a dr in the rearranged equation:

r²\frac{d}{dr}(λ\frac{dT}{dr})dr + \dot{q}r² = 0

 

[rude man]

Your (edited) equation has incompatible terms: the first is infinitesimal, the second isn't. Plus, the terms' dimensions don't agree: the first one's are (using SI) J/sec whereas the second one's are J/(sec-m).

Ironically, your unedited equation has matching dimensions but you can't smuggle the r² into the d/dr bracket as you wondered. (That's just basic calculus: for example, r²d/dr(r²) = 2r³ whereas d/dr(r⁴) = 4r³.)

Going back to my "first principles" equation , Q_dot = λAΔT/Δr, you seem to have correctly determined that, in spherical coordinates, A = 4πr² and, of course, ΔT/Δr → dT/dr. So your remaining task, and it does take some thinking, is to somehow get rid of Q_dot and substitute for it an expression containing q_dot. (Sorry, I haven't learned the itex thing yet). So that you wind up with
-d/dr{λr(dT/dr)} = r²q_dot. That is really the hard part about this problem.