Problem Description:

A one$-$meter rod is initially at $20C.$ For simplicity, assume rod length $$ rod diameter. Suddenly, the

end at $x=0[m]$ is dropped to zero $C$ while the end at $x=1[m]$ is brought to $40C.$

Two materials are considered for the same rod, steel, and copper. Compare the temperatures at

zero second $($i$.$e$.$ initial temperature distribution$)$ and after $500$ s$,1000$ s$,1500$ s$,$ and $2000$ s along the

rod for each of the materials $($steel vs$.$ copper$).$

Consider the following properties:

For steel

Conductivity $=30\frac{W}{m}-K,$ Specific heat $=400\frac{J}{k}g-K,$ Density $=8000k\frac{g}{m}3$

For copper

Conductivity $=400\frac{W}{m}-K,$ Specific heat $=400\frac{J}{k}g-K,$ Density $=8000k\frac{g}{m}3$

Limit your intervals along the rod length to $10$ intervals

Use the same time step for both cases

What time step will you use? Why?

Your report should clearly contain $AL{L}_{?}$ the following:

Which equation are you solving?

What is the analytical temperature distribution at time $t=0$ s for BOTH materials?

Find the steady state solution for the temperature distribution in the rod $($after the temperature

at both ends were changed$).$ This should be VERY simple.

Clearly indicate how you are going to discretize your equation using the finite difference

method. Write the finite difference equation that you are solving.

For simplicity, use the explicit method

Clearly indicate how you computed your time step. Use the same time step for the two rods.

Plot the temperature distribution through the entire rod at $0s,500s,1000s,1500s,$ and $2000$ s

a$.$ Compare the temperature distribution along the two rods at the above times

Plot the temperature at $x=0\mathrm{.}5[$m$]$ vs$.$ time during the time interval $0-2000s$ for the two rods.

Plot the heat flux at $x=0[m]$ against time during the time interval $0-2000s$ for the two rods.

The heat flux is defined as: $q^{\text{'}\text{'}}=-k\frac{dT}{dx}$ evaluated at the point of interest.