Chemical Engineering 535
Fall, 1996
B. Bernstein, Instructor
Notes 1: Thermal Conduction in a Cylinder

This is a discussion of the heat equation (diffusion equation) in a cylinder. Let us start by defining the Laplacian operator, : In cartesian coordinates, x,y,z, it is defined by

 

In cylindrical coordinates, , the Laplacian is

 

The heat equation is given by

 

where t is time and where here, the thermal diffusivity, corresponds to in our book. The problem we now address is as follows:

A cylinder r<a, 0<z<L, has initial temperature distribution at time t=0. Thenceforth, the lateral face r=a is kept at temperature T=0 and the ends, z=0 and z=L are thermally isolated: i.e.  is zero at both ends. Find the temperature at times t>0.

The method for attacking this problem will be separation of variables: We start by seeking a solution of (3) in the form

 

Substitution of equation 4 into equations (2) and (3) gives

 

Dividing both sides of (5) by , we get

 

Now, since the right hand side depends only on t, and the left hand side is independent of t, they must both be equal to a constant, say . We get, then

 

Consider, now, the right hand side of the first equation (7): Its right hand side depends only on z, whereas its left hand side is independent of z. Therefore, both sides must be constant, say . We get, after a little algebra,

 

Once more, the right hand side of equation (8) depends only on and the right hand side depends only on r, so that both sides are equal to the same constant, say . We get

 

Let us now summarize where we are, we are to solve

    

We have added a couple of comments to equations (10) and (13): These comments should be clear, but they are needed.

There are several eigenvalue problems of Sturm-Liouville form which now arise. The first has to do with equation (10). We shall leave it to you to show that the constant must be zero or positive. Indeed, we get

 

We leave it to you, the reader, then, to establish that we must have , where . These values of are eigenvalues of what we call the periodic Sturm-Liouville problem and the eigenfunctions are as follows: For n=0 the eigenfunction is a constant, say . For positive n, the eigenfunctions are and , . Note that for each positive eigenvalue, the space of eigenfunctions is two dimensional ( is given by an arbitrary linear combination of two linearly independent functions, and ) unlike the situation where homogeneous end values are specified: When they are so specified, the space of eigenfunctions corresponding to each eigenvalue is one dimensional.

So, now, equation (13) becomes

 

Now equation (13) is really the statement of a Sturm-Liouville problem for R(r), except that it has two parameters. However, equation (15) has essentially fixed one of the parameters (or has, at least, incorporated the allowable values of one of the parameters), so we now have an eigenvalue problem for each value of n, where has to be an eigenvlue. Now equation (15) looks almost like Bessel's equation: Indeed, a simple transformation will transform it into Bessel's equation: Let , so that the chain rule gives

so that equation (15) becomes

 

Those solutions of (16) which are continuous at are

 

The end condition R(a)=0 is met by requiring to be one of the zeroes of the Bessel function . Let these zeroes be denoted by : i.e.  , , . So the eigenvalues are and the eigenfunctions are

 

What remains is only to put these separation of variables solutions together with unknown coefficients and to evaluate the coefficients using the initial conditions. That it is possible to do this stems from the result in Sturm-Liouville theory that one may represent a fairly arbitrary function as a linear combination of eigenfunctions, a result which one uses in determining the coefficients from the initial conditions.

We wish, now, to carry through (with your help) an example of such a solution. But we shall scale the example down so as to illustrate the notions without a very excessive amount of work.

To begin with, let us assume that the initial condition is independent of z: . Then equation the boundary value problem stated in (11) will have the solution z=1, and we can forget about dependence of T on z.

So, our solution is, then

Problems:

1. The problem involving the Bessel functions is of the Sturm-Liouville type. What are its eigenfunctions and eigenvalues. Write down a formula for evaluationg , , .
2. Consider the simplest case, namely where the initial value depends only on r and not on or z.

   

   In that case, assuming that a=1, one gets

   

   Given that g(r)=r, and that the first zeroes of are

   

   (rounded off, of course), evaluate for m=1,2,3,4,5.

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