Heat Transfer Problem in Cylindrical

[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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[danai_pa]

Anyone please help me. I don't understand this problem. Thankyou

 

[quantumdude]

1) Assume that the temperature distribution is independent of the direction
along the cylinder

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

2) Use Laplace equation in cylindrical coordinates

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

3) the temperature at the center is determined from the temperature
distribution for which r=0

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

4) The functions Sin beta(x) and Cos beta(x) have a periodicity if and only if
the values of beta are integer

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

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)]

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.

 

[danai_pa]

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

 

[quantumdude]

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