The good news is that you aren't doing any
programming this time. The bad news is:
You are attempting to solve a conduction problem involving a rod with internal heat source and cooling at the surface by a fluid bath. Within the rod the conductivity (k) can be assumed constant and the conduction equation is:
Heat flux at the surface from the metal rod to the fluid is given by the expression:
where for example we might have:
Tfluid = 400K
The rod is divided into the following set of finite volumes:
Construct the finite volume approximation to the conduction equation for:
For the same conduction problem use Taylor series expansions to construct a difference approximation at the interior nodal point circled below, that is at least third order accurate. Remember that for this approach we are only evaluating temperatures at the point where the grid lines cross.
Show the steps in your derivation. Clearly define any notation that you introduce and mark on a figure the location of any temperatures that you use. Give the expression for the error term, showing only the terms involving the lowest powers of mesh spacing.
Note: You should use Mathmatica, Maple, or MathCad to work through this problem. If you do, provide sufficient commented output from the program to show and explain all of the steps that you went through in generating the solution. PDF output from these programs is best for me. However, I can read any native Mathematica file and MathCad .mcd files, saved in MathCad 2001 or older format.