Consider the square channel shown in the sketch operating under steady state condition. The inner surface of the channel is at a uniform temperature of $600K$ and the outer surface is at a uniform temperature of $300K.$ From a symmetrical elemental of the channel, a two $-$ dimensional grid has been constructed as in the right figure below. The points are spaced by equal distance.

$($a$)$ The heat transfer from inside to outside is only by conduction across the channel wall. Beginning with properly defined control volumes, derive the finite difference equations for locations $(1)(2)(3).$ You can also use $(n,m$ to represent row and column. For example, location $(1)$ is $(3,3),$ location $(2)$ is $(3,1),$ and location $(3)$ is $(3,5).($hint: I have already put a control volume around this locations with dashed boarder.$)$

$($b$)$ Please use excel to construct the tables of temperatures and finite difference. Solve for the temperatures of each locations. Print out the tables in the spread sheet.

$($hint: the red dots here is at constant temperature, $300$ or $600K.$ The black and blue ones are the ones to be solved. Black ones are edge ones, among which the ones on right is slightly different.$)$