A two$-$stage cascade refrigeration system is used for a low temperature refrigeration, as shown in the figure below. The upper stage $($A$)$ uses $R-22$ as a refrigerant and the lower stage $($B$)$ uses $R-290($propane$).$ For this application, the refrigerated space must be cooled to $-38C.$ To achieve this goal, the lower stage system must be designed such that:

The evaporator temperature must be $19\mathrm{.}3C$ lower than the refrigerated space.

The mass flow rate through the lower stage must be $0\mathrm{.}06k\frac{g}{s}.$

The lower stage condensing temperature must be $-5\mathrm{.}46C.$

Notice that for this system, the upper stage evaporator exchanges energy with the lower stage condenser, which is done in a heat exchanger that requires a temperature difference between the two refrigerants of $14C.$

The upper stage condenser rejects heat to the ambient air, which is at a temperature of $25C.$ The condensing temperature must be $11\mathrm{.}3C$ higher than the ambient air temperature. Both stages require a superheat of $12C$ Both stages require a subcooling of $14C.$

Using property tables, determine:

a$.$ The compressor power input, in kW $,$ for both stages.

b$.$ The refrigeration capacity, in kW and in tons of refrigeration.

c$.$ The rate of heat transfer from the upper stage condenser, in kW $,$

d$.$ The rate of heat transfer between the two stages, in kW $.$

e$.$ The mass flow rate required for the upper stage, in $k\frac{g}{s}.$

f$.$ The refrigeration system coefficient of performance $($COP$).$

g$.$ The COP of each stage.