$1$ Finite Difference Method $(16$ Points$)$

Finite Differences: Consider the triangular fin $($heat transfer from extended surfaces$)$ shown in

the sketch operating under steady$-$state conditions. The base temperature of the surface is

uniform 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, a two$-$dimensional grid has been

constructed and the nodes labelled. The temperatures for nodes $1,2,3,4,5,7,8,11,12$ and $13$

are identified.

$T_1=T_2=T_3=T_4=T_5=500K$

$T_8=T_7=433K$

$T_{11}=367K$

$T_{13}=347,7K$

$T_{12}=380K$

a$.$ Derive the finite$-$difference equations for nodes $6,10,14$ and $15$ and determine

the temperatures T$6,$ T$10,$ T$14$ and T$15($K$)$

b$.$ Calculate the heat loss per unit length from the fin