$10\mathrm{.}21$ A vapor$-$compression refrigeration system operates with the cascade arrangement of Fig. $10\mathrm{.}7.$ Refrigerant $22$ is the working fluid in the high$-$temperature cycle and Refrigerant $134$a is used in the low$-$temperature cycle. For the Refrigerant $134$a cycle, the working fluid enters the compressor as saturated vapor at $-34C$ and is compressed isentropically to $345$ kPa $.$ Saturated liquid leaves the intermediate heat exchanger at $345$ kPa and enters the expansion valve. For the Refrigerant $22$ cycle, the working fluid enters the compressor as saturated vapor at a temperature $-15C$ below that of the condensing temperature of the Refrigerant $134$ a in the intermediate heat exchanger. The Refrigerant $22$ is compressed isentropically to $1724$ kPa $.$ Saturated liquid then enters the expansion valve at $1724$ kPa $.$ The refrigerating capacity of the cascade system is $20$ tons. Determine

$($a$)$ the power input to each compressor, in $k\frac{J}{s}.$

$($b$)$ the overall coefficient of performance of the cascade cycle.

$($c$)$ the rate of exergy destruction in the intermediate heat exchanger, in kJ$/$s$.$ Let $T_0=27C,p_0=100$kPa.