derive the heat conduction equation in spherical coordinates.
Answers
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".