$2$ Finite Difference Method $(28$ Points$)$

Finite Differences: Consider the square channel shown in the sketch operating under steady$-$

state conditions. The inner pipe is made of a radio$-$active material with a uniform and constant

heat source, while the outer surface is exposed to convection with a fluid at $300$ K and a

convection coefficient of $50\frac{W}{m^2}K.$ From a symmetrical element of the channel, a two$-$

dimensional grid has been constructed and the nodes labelled. The temperatures for nodes $1,3,$

$6,$ and are identified.

$T_1=427,3K$

$T_3=389,8K$

$T_5=501,8K$

$T_6=482,8K$

c$.$ Derive the finite$-$difference equations for nodes $2,4,7,8$ and $9$ and determine the

temperatures $T2,T4,T7,T8$ and $)$

d$.$ Calculate the heat loss per unit length from the channel