Project Description

Conduction within relatively complex

geometries can sometimes be evaluated easily

by using the finite$-$difference methods that are

applied to subdomains and patched together.

Consider the two$-$dimensional domain formed

by rectangular and cylindrical subdomains

patched at the common, dashed control

surface. Note that, along with the dashed

control surface, temperatures in the two

subdomains are identical and local conduction

heat fluxes to the cylindrical subdomain are

identical to local conduction heat fluxes from

the rectangular subdomain. The base of the

rectangular subdomain is held at T$\_($h$)=100\backslash$deg C$,$

while the vertical surface of the cylindrical

subdomain and the surface at outer radius ro

are at T$\_($c$)=30\backslash$deg C$.$ The remaining surfaces are adiabatic, and the thermal conductivity is

k$=30($W$)/($m$)-$K$.$ In this project you are supposed to do the following:

$($a$)$ Develop the nodal network by using using $\backslash$Delta x$=\backslash$Delta

u $=\backslash$Delta r$=10$mm and $\backslash$Delta $\backslash$phi $=(\backslash$pi $)/(8).$

$($b$)$ Develop finite$-$difference equations for all the nodes,

$($c$)$ Calculate the nodal temperatures by writing a computer program based on

Gauss$-$Seidel iterative procedure, and finally

$($d$)$ Calculate the heat transfer per unit depth into the page, q$^(\text{'})($W$)/($m$)$