A refrigeration system operates on a vapor$-$compression cycle with R$-134$a as the working fluid. The system provides $18$ tons of refrigeration $(1$ton$=12,000BT\frac{U}{h}).$ The refrigerant enters the compressor as a saturated vapor at $15F$ and leaves the condenser as a saturated liquid at $110F.$ The evaporator and condenser pressures are $30$ psia and $150$ psia, respectively. The isentropic efficiency of the compressor is $80\%.$ The refrigerant is subcooled by $15F$ before entering the expansion valve, and the superheated vapor exiting the evaporator is $15F$ above the saturation temperature. Additionally, the condenser has a $5F$ temperature difference between the refrigerant and the cooling water inlet.

Determine the following:

a$.$ The mass flow rate of the refrigerant $(lb\frac{m}{h}).$

b$.$ The power input to the compressor $($ kW $).$

c$.$ The coefficient of performance $($COP$)$ of the cycle.

d$.$ The rate of heat rejection in the condenser $($BTU$/$h$).$

e$.$ The exergy destruction in the compressor $($BTU$/$h$),$ assuming the dead state conditions as $14\mathrm{.}7$ psia and $70F.$