Heat transfer on a cylinder (doubt)

[Adrian F]

Hi, there.

I remember when I was in the University (mech. engineering), I had an exam on partial differential equations about heat transfer in a cylinder. We had to determine the temperature distribution. I remember the conditions were that the cylinder was insulated in the side area, had a heat source in the bottom face at 9°C and the top face had a heat transfer coefficient that was taken from a digit in the student's ID number. For my particular case, this digit was 0, so I knew that the result was going to be that the cylinder ended up at 9º in all of its volume, or that the limit of the temperature function when t (time) tends to infinity equaled 9, independent of any other parameter.

Now, I don't remember the procedure, but I remember that I assumed that the rate of heat transfer in the radial direction or ∂u/∂r was going to be 0 because there's no heat being transfer in that direction and proceded from there. I got the result right: Lim(T) when t tends to infinity = 9 and wrote the reasoning. The teacher gave me all points in the problem because of the reasoning but said that the procedure was wrong.

My question is, was I correct in making that assumption? If anyone could maybe solve this problem here, I'd appreciated.This happened 10 years ago, but I never got the answer. It's been bugging me ever since and I forgot about D.E.

Thanks in advance

 

[Chestermiller]

This is a 1D transient heat transfer problem in the axial z direction. Do you remember the partial differential equation describing transient heat conduction in 1D?

Chet

 

[Adrian F]

Q = kA(T1-T2), right?

Edit: no, nevermind. That's not a DE. I don't remember!

Since you say this is a 1D transient heat transfer problem, I take it that my initial assumption was correct. Am I right?

 

[NukeProf]

Your approach was correct only if the cylinder is perfectly insulated on its lateral surfaces. You solve the steady heat equation in 2-D r-z cylindrical geometry, and apply the boundary condition of zero heat flow at the outer radius. This will prove that there is no temperature gradient in the radial direction at steady-state. If the top is also perfectly insulated, the cylinder will reach equilibrium with a spatially uniform temperature.

 

[Adrian F]

Yes, that was one of the boundary conditions: the cylinder was insultated on its lateral surface. And, because the digit in my ID number was 0, which corresponded to HT coefficient of the top face, the top face was also insulated, so the cylinder was completely insulated except for the bottom face where the heat source was. I'd love to see what I did, because I don't remember. I do remember just assuming that ∂u/∂r = 0.

 

[Chestermiller]

Look up "Transient Heat Conduction" on Google.