Interesting Heat Diffusion Problem

[heritage972]

We solved a simple looking heat diffusion problem because it describes an apparatus used in an NIH sponsored research project. It resembles problems discussed in textbooks and in many papers on the web. A textbook method solves all those problems but a straight forward application of that method fails for our problem. (The textbook approach is to isolate the heat source, if any, and expand the rest in well known eigenfunctions.)

Has anyone seen this problem in print, textbook or other source. Two days of Google searches come up empty and it is not in our library; e.g. Morse and Feshbach.

A homogeneous sphere of radius 1 satisfies the heat diffusion equation with a time independent heat source Q(r). The boundary condition is grad (T)=0 on the boundary. The initial distribution T(r,0)=f(r) is known. What is T(r,t)?

Our first surprise was that blindly applying the textbook method fails and our second surprise was that such a simple, but interesting, problem doesn't seem to appear in print. We are trying to decide on distributing our solution. Thanks

 

[Chestermiller]

You might try Carslaw and Jaeger, Conduction of Heat is Solids.

I've seen this kind of thing before, but not this exact problem.

The first thing to do is to solve for the temperature distribution at long times, which is equal to the average value of f(r) plus t times a function of r. Then you represent T(r,t) as the long-time solution plus an initial transient u(r,t). Solving for the initial transient is simple to do. Is this how you approached the problem?

Chet

 

[heritage972]

Thanks and Yes (almost). The long term solution uses the average of the heat source which I called Q(r). Not nit-picking - just don't want to confuse the next reader.
I figured that this had to be discussed somewhere. Will probably put your reference for the next paper (hopefully there is one).

In terms of the whole project, this was pretty minor but it really bugged me.

You don't know of a solution using Laplace Transforms do you?

I found what I think is the Green's function for this geometry and Boundary Condition that is definitely NOT the ways to go.

 

[Chestermiller]

Thanks and Yes (almost). The long term solution uses the average of the heat source which I called Q(r). Not nit-picking - just don't want to confuse the next reader.

I think you would be much happier with what you got if you did the long term solution with the actual Q(r), rather than the average. This would eliminate the heat generation term from the equation for u(r,t).

Chet

 

[heritage972]

I appreciate your reference very much and you clearly understand the problem. I certainly don't want to nit-pick or argue.
I hope you don't mind if I disagree with this technical point.

Pretty sure you have to separate out the long term solution with the average heat source and solve for the difference (your transient?) with a heat source whose volume average is zero.

Trying to figure out how to post an equation. Bear with me. Not getting anywhere and Latex will take me forever.
hopefully the forum will pass hyperlinkhttp://meiere.com/ForumReply1.docx

http://meiere.com/ForumReply1.docx

 

[Chestermiller]

I take back what I said in my previous post. What you did using the average generation rate is preferable. Doing this, you get:

$T(r,t)=\bar{f}+\frac{ \bar{Q} }{ρC}t+g(r)+u(r,t)$

where g(r) satisfies the steady state heat conduction equation with generation rate $Q(r)-\bar{Q}$.

Chet

 

[heritage972]

Thanks again. Impressed by anyone who knew the reference off the top of your head.
Currently driving our librarian crazy trying to find the book. May just buy the paperback.

 

[Chestermiller]

Thanks again. Impressed by anyone who knew the reference off the top of your head.
Currently driving our librarian crazy trying to find the book. May just buy the paperback.

It's a classic.

 

[heritage972]

Close but no Cigar,
Carslaw and Jaeger is a very nice book and discusses an incredible number of examples. Just not this one.
Closest they come is in section 9.2 (VI) and 9.8 (IX) which are the same. Their terminology is weird "radiation at the surface into a medium at zero" apparently means grad(T) + hT=0. Presumably, as h -->0 this is grad(T)=0. Unfortunately, in their solution there is no term linear in t. They assume a steady state solution which is the usual one for T=0 on the boundary plus a term which goes like b/h (b=sphere radius). This blows up as h --> 0, reflecting the fact that there is no steady state solution.
Nice book though. I like the discussion of Integral Equations since they were a big part of my thesis long ago (high energy, not diffusion).

Still looking for a reference. Will probably submit the solution (follows the procedure described above by Chestermiller) at which point my guess is that someone will point out how bad I am at searches.

I really should learn Latex for this forum. Sorry.