Analytical Solutions for 1D Transient Heat Conduction in Cylindrical Coordinates

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Hi, I have written a numerical code to solve the 1D heat equation in cyclindrical coordinates:

$$\frac{\partial T}{\partial t}=\kappa\left(\frac{\partial^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}\right)$$

The problem I'm considering is a hollow cylinder in an infinite boundary, i.e. an underground tunnel surrounded by earth. All I want to do is verify that my code is working correctly so to do this I want to find the simplest analytical solution for 1D transient heat conduction using the simplest boundary condition, i.e. constant values. Now if this was in cartesian coordinates I would simply compare it with the solution for a semi infinite approximation. Unfortunately from my research I haven't found an equivalent for cyclindrical coordinates.

I may be missing some here so does anybody have ideas for a simple analytical solution? Thanks.

 

[xlines]

Hi, I have written a numerical code to solve the 1D heat equation in cyclindrical coordinates:

$$\frac{\partial T}{\partial t}=\kappa\left(\frac{\partial^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}\right)$$

The problem I'm considering is a hollow cylinder in an infinite boundary, i.e. an underground tunnel surrounded by earth. All I want to do is verify that my code is working correctly so to do this I want to find the simplest analytical solution for 1D transient heat conduction using the simplest boundary condition, i.e. constant values. Now if this was in cartesian coordinates I would simply compare it with the solution for a semi infinite approximation. Unfortunately from my research I haven't found an equivalent for cyclindrical coordinates.

I may be missing some here so does anybody have ideas for a simple analytical solution? Thanks.

I'm not sure you got the equation right: if you are considering hollow cylinder, than r isn't variable of temperature, but a fixed quantity (radius of your pipe) . In an angle-independent case, you should have equation for T (z, t) rather than T (r, t) . Setting situation like that, hollow cylinder shouldn't be unlike standard, exactly solvable 1D case.

 

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Sorry, maybe my description isn't quite clear enough. Essentially I'm considering an infinitely long cyclinder, such that I'm only considering temperature changes in the radial direction which is assumed independent of $$\theta$$ and z. Therefore the 1D governing equation still holds, but the boundary conditions change. It's this bit that I am having trouble with.

In the end I have found a solution in Carslaw and Jaegar but I'm not sure whether it is correct.

 

[xlines]

Sorry, maybe my description isn't quite clear enough. Essentially I'm considering an infinitely long cyclinder, such that I'm only considering temperature changes in the radial direction which is assumed independent of $$\theta$$ and z. Therefore the 1D governing equation still holds, but the boundary conditions change. It's this bit that I am having trouble with.

In the end I have found a solution in Carslaw and Jaegar but I'm not sure whether it is correct.

Ok than, you confused me mentioning hollow cylinder in opening post - obviously cylinder needs to be solid to consider radial heat conduction. You did get your equation right and general solutions would be linear combination of modified Bessel function of 0th order (because it is angle independent) and Neumann modified functions, it seems. Considering your boundary conditions, where temperature at r = 0 being finite, Neumann's function need not to be considered (they diverge at r = 0) , so all you need is to scale Bessel's functions. But take this cum grano salis, I'm not sure my memory is good on that one :) Good luck!

 

[kseng]

See book solution for 1-D transient.

http://highered.mcgraw-hill.com/sites/dl/free/0073129305/314124/cen29305_ch04.pdf

I was taught similar methods in college.

I am working on a 2-D problem - see thread in maths/diff eqns section. Any advice?