A cycle that uses R$-134$a has two compression stages. The maximum pressure is $1100$ kPa. The minimum pressure is $160$ kPa. The R$-134$a exits the condenser with a quality of $0($saturated liquid$)$ and goes to the flash chamber which is at a pressure of $500$ kPa. The saturated vapor from the flash chamber mixes with the refrigerant coming from the low pressure compressor, both are at $500$ kPa. This mixture stream is then compressed by the high pressure compressor. The liquid leaving the flash chamber goes through a valve $($throttled$)$ to the pressure of the evaporator. The refrigerant leaves the evaporator as a saturated vapor. Both compressors operate isentropically$.$ The mass flow rate through the condenser is $0\mathrm{.}25$ kg$/$s$.$ Determine:

a$)$ The fraction of refrigerant that becomes saturated vapor in the flash chamber $($no units$)0\mathrm{.}21$

b$)$ The rate of heat removed from the refrigerated space $($kJ$/$s$)33\mathrm{.}1$

c$)$ The COP $($no units$)3\mathrm{.}76$

d$)$ The rate of heat sent to the warm region $($kJ$/$s$)41\mathrm{.}86$