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Jan21-08, 02:33 PM
P: 279
Quote Quote by phioder View Post

The temperature in the cylinder is given by the above equation, but is dimensionless, what about if one would like to calculate the temperature of cylinders of different materials?

Is there a way to add to that equation, the density, thermal conductivity or other variables that would describe more physically and thermally the cylinder in question? Some type of constant? or the equation needs to be solved completely different?

Any hint or book would be kindly appreciated

Very Best Regards
The solution is not exactly dimensionless as such, it can very easily made dimensionless by rewriting it as:

[tex]\frac{u(r,z)}{u_0}= \sum_{n=1}^{\infty}\frac{sinh(\lambda_n z) \cdot J_0(\lambda_n r)}{\lambda_n \cdot sinh(4 \lambda_n) \cdot J_1(2 \lambda_n)}[/tex]

The unit of [tex]\lambda_n[/tex] is [tex]m^{-1}[/tex]. The unit of [tex]u[/tex] is [tex]K[/tex]. Now to answer the question of different materials, it does not change anything in case you are only considering the steady-state solution. This means that no time dependency is introduced into your solution. To see this consider the general heat equation (without generated heat):

[tex]\frac{\partial u}{\partial t} = \frac{k}{c_p \rho} \cdot \left(\frac{\partial^2 u}{\partial r^2}+\frac{1}{r} \cdot \frac{\partial u}{\partial r}+\frac{1}{r^2} \cdot \frac{\partial^2 u}{\partial \varphi ^2}+ \frac{\partial^2 u}{\partial z^2} \right)[/tex]

If you have a time independent solution as in the original post, this is equal to 0. And therefore the material properties vanish. Also the [tex]\varphi[/tex] is left out due to symmetry. This can be seen in a somewhat physical way as well. Indeed if we have established a "steady-state" state, the complete temperature profile in the volume has reached it's final value. Also the heat flow is steady and one can "feel" that this is not influenced by the material properties anymore. Everything is now steady-state. A good book, mmm, there are several books on this subject. It depends a bit on what exactly you are looking for phioder. For the mathematics the one I use sometimes is: "Fourier Analysis with applications to boundary value problems" by Murray R. Spiegel, Schaum's theory and problem solver. I know that people or like or dislike this series very much, I have always found it a nice background for additional learning through exercises. I adore doing calculations and exercises :-) For a more theoretical study of partial differential equations, I can't really advice one, my original text is from university. I can recommend if you are also interested in ordinary differential equations (also here opinions vary): "Elementary Differential Equations and Boundary Value Problems" by Boyce and DiPrima, John Wiley & Sons. If you want to know more about heat transfer, I can recommend the following work: I use this one sometimes at work.

Hope this helps you ahead.