Numerical Methods, Transient & 2D Conduction

1. Numerical Methods: Computation

A 2-dimensional rectangular plate is subjected to

prescribed temperature boundary conditions on 3 sides

and a uniform heat flux into the plate at the top surface.

Derive an expression for the temperature distribution in

the plate. Assume Twall = 0, L = 1 cm, the material is

copper, and qs" = 1000 W/m²

a) Simplify the heat diffusion equation to describe T₁-

conduction within the plate

b)

Provide the temperature conditions along each face

and for a central node to solve this problem with Finite

Difference methods

W

9"

↓↓↓↓

(please submit your code with the HW).

L

T₁

-T₁

c) Write down the matrix calculation for solving

temperature profile assuming there are 4 nodes along

each face.

d) Break the volume up into elements with equal numbers along each face. Compute and

graph the temperature profile at each node, using a contour map.

e) How many nodes total does it take to converge to the example in class?

f) What is the maximum temperature in the solid?

g) Plot the solution from the lecture slides on SOV alongside. Pick sufficient terms for

reasonable convergence (e.g. 200).

h) Compare simulation times and simulation difficulty between c and d, the SOV and

Numerical cases.

x