Matlab or similar math software package must be used. (hints: heat equation finite difference method)

A short description of the problem (5 points)

A sketch of the problem (2 points)

Derivation of difference equation (D.E.) (5 points)

Results in tabular and plot forms (15 points)

Discussions and conclusions (10 points)

Computer program and its output (3 points)

Make any necessary assumptions.

Project No. 1

A laterally insulated bar of length 10 cm and constant cross-sectional area 1 cm², of density 10.6 gm/cm³, thermal conductivity 1.04 cal/cm sec C, and specific heat 0.056 cal/gm C has an initial temperature distribution of f(x) = 50 sin 0.1?x and is kept at 0 C at the ends.

a) Using finite difference method and explicit scheme, derive the Difference Equation for the bar using h = 1 cm and Dt = 0.25 s.

b) Solve the temperature at all unknown nodes from t = 0 to 20 s.

c) Plot the temperature distributions from t = 0 to 20s for every 4s interval.

d) Determine the temperature values at x = 2 and 4 cm at t = 20 s using analytical solution.

e) Compare the results of analytical and numerical solutions at x = 2 and 4 cm at t = 20 s.

f) Plot the temperature distributions from both numerical and analytical solutions from t = 0 to 20s for every 4s interval.