$(15$ points$)$ A house is maintained at $22C$ on a hot day where the outdoor temperature is $35C.$ Insulation is placed in the house at a volumetric rate of $1000\frac{W}{m^3}.$ The walls have a thickness of $70cm$ and a total surface area of $88m^2.$ The temperatures of the inner and outer surfaces of the wall are $25C$ and $30C,$ respectively.

$($a$)(5$ points$)$ Draw a diagram of the system assuming the inner and outer surfaces of the wall are at the boundary. Show all exergy flows coming into and out of the wall, and the respective boundary temperatures.

$($b$)(5$ points$)$ Calculate the rate of exergy destruction in the wall.

$($c$)$ points$)$ Now calculate the total rate of exergy destruction of the heat transfer process taking both the wall and its surroundings as the system.

$($d$)(10$ points$)$ Determine the minimum possible amount of power required to drive the refrigerator responsible for keeping this room cool. HINT: Think carefully about this. Remember the refrigeration cycle and the refrigerated space itself can be treated as two different systems. You should have everything that you need.