Problem $5$:

Consider a two$-$stage cascade refrigeration system operating between the pressure limits of $1\mathrm{.}2$ MPa and $200$ kPa with $R-407C$ as the working fluid. The refrigerant leaves the condenser as a subcool liquid of $5C$ 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 $0\mathrm{.}15k\frac{g}{s}.$ Assuming the refrigerant leaves the evaporator as a superheat vapor of $7C$ and the isentropic efficiency is $85$ percent for both compressors, using EES determine

$($a$)$ the mass flow rate of the refrigerant through the high$-$pressure compressor,

$($b$)$ the rate of heat removal from the refrigerated space,

$($c$)$ the COP of this refrigerator, and

$($d$)$ sketch the P$-$h and T$-$s diagrams of the cycle.

$($e$)$ If this refrigerator operated on a single$-$stage cycle between the same pressure limits with the same compressor efficiency and the same mass flow rate mentioned above. Determine the rate of heat removal and the COP changes, compare and discuss the results.