# Heat Transfer Problem in Cylindrical

1. Sep 6, 2005

### danai_pa

I don't understand this problem. I think it is difficult for me. Please anyone
suggestion this problem to me. Thanf you

Let us cosider steady state heat transfer problem in which laplaceT(r)=0
What is the temparature at the center of a thin disc of radius a
whose average boundary temparatue is 70 degree?

Hint:
1) Assume that the temperature distribution is independent of the direction
along the cylinder
2) Use Laplace equation in cylindrical coordinates
3) the temperature at the center is determined from the temperature
distribution for which r=0
4) The functions Sin beta(x) and Cos beta(x) have a periodicity if and only if
the values of beta are integer
5) The average boundary temperature at r=a is given by

T(average) = 1/2*Pi intregrate from 0 to 2*Pi [(T(a,seta)d(seta)]

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Last edited: Sep 6, 2005
2. Sep 12, 2005

### danai_pa

3. Sep 12, 2005

### Tom Mattson

Staff Emeritus
That means that $T$ is a function of $r$ only, and not $\theta$.

Laplace's equation is $\nabla^2T=0$. Look up the Laplacian in cylindrical coordinates and write down the equation for $T=T(r)$.

Apply this boundary condition after you get a general solution for $T(r)$.

We'll get to this after you complete #3.

Since you're solving a 2nd order Diff Eq, you need 2 pieces of information to eliminate the 2 arbitrary constants that arise. The first piece was in Hint 3, and this is the other one.

Please try the problem. If you get stuck, let us know how you started and how far you got.

Last edited: Sep 12, 2005
4. Sep 12, 2005

### danai_pa

Why this problem is not depent on seta. and h ?

5. Sep 12, 2005

### Tom Mattson

Staff Emeritus
Because the problem says so. You could achieve this by holding the cylindrical wall at a constant temperature.