$12-61$ Consider a two$-$stage cascade refrigeration cycle with a flash chamber as shown in the flgure with refrigerant$-134$a as the working fluid. The evaporator temperature is $\backslash (-10^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and the condenser pressure is $1600$ kPa $.$ The refrigerant leaves the condenser as a saturated liquid and is throttled to a flash chamber operating at $0\mathrm{.}45$ MPa $.$ Part of the refrigerant evaporates during this flashing process, and this vapor is mixed with the refrigerant leaving the low$-$pressure compressor. The mixture is then compressed to the condenser pressure by the high$-$pressure compressor. The liquid in the flash chamber is throttled to the evaporator pressure and cools the refrigerated space as it vaporizes in the evaporator. The mass flow rate of the refrigerant through the low$-$pressure compressor is $\backslash (0\mathrm{.}11\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}\backslash ).$ Assuming the refrigerant leaves the evaporator as a saturated vapor and the isentropic efficiency is $36$ percent for both compressors, determine $($a$)$ the mass flow rate of the refrigerant through the high$-$pressure compressor, $($b$)$ the rate of refrigeration supplied by the system, and $($c$)$ the COP of this refrigerator. Also. determine $($d$)$ the rate of refrigeration and the COP if this refrigerator operated on a single$-$stage vapor$-$compression cycle between the same evaporating temperature and condenser pressure with the same compressor efficiency and the same flow rate as calculated in part $\backslash ($ a $\backslash ).$