The good news is that you aren't doing any
programming this time. The bad news is:

Part A

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:

T_{fluid} = 400K

and

The rod is divided into the following set of finite volumes:

Construct the finite volume approximation to the conduction equation
for:

- a volume in the inner ring;
- a volume in the outer ring;
- any other (interior) volume.

Clearly define any notation that you introduce. Clearly show all steps and define additional equations (e.g. flux matching, special boundary conditions) needed to obtain the same number of equations as unknown temperatures. How many unknown temperatures have you introduced for the specific cylindrical grid above?

**Part B**

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.*