Consider the $2-$D configuration in which two square are nested $($a plate with a square hole in the middle$).$ The surfaces of the squares are at constant temperature. A steady$-$state heat transfer occurs between these two surfaces due to the temperature difference.

a$)$ Divide the area between these two surfaces into grids. Use a minimum of $5$ grids for distance $\backslash ($ L $\backslash ).$

b$)$ Derive formulations via energy balance method. Make necessary simplifications. You may use any physical values for the configuration $($Temperatures$,$ dimensions etc.$).$

c$)$ Solve the equations with a suitable method and obtain the temperature distributions for different A$/$a dimensions. Calculate the heat transfer between two surfaces per unit meter. Use these calculated values to obtain a shape factor for this geometry:

$\backslash [$

$\backslash$dot$\{$Q$\}=$S k $\backslash$Delta T

$\backslash ]$

The shape factor should be in terms of $\backslash ($ A $/$ a $\backslash )$ ratio. Consider at least six ratios $($you can consider more if you wish$).$

At least two ratios for $\backslash (1$