$3.)(10$ points$)$ A dual stage refrigeration cycle uses two stages running at steady state to cool a freezer to ultralow temperatures $($below $-80$C$),$ which is required for storage of a new mRNA based therapeutic. The lower stage of this cycle uses a vapor compression refrigeration cycle $($with R$508$B as the working fluid. This cycle consists of a fluid running through an isoenthalpic expansion valve, an isobaric evaporator, a reversible and adiabatic compressor, and then finally an isobaric condenser to complete the cycle. For this problem, you only need to consider the lower temperature stage which transfers heat to the higher temperature stage through its condenser. The upper stage would be similar to the one described in problem $2($but would take in the heat from the lower stage$).$

Given:

Saturated Liquid exits the lower temperature Condenser at $-23C$

Saturated Vapor exits the lower temperature Evaporator at $-90C$

Mass flow rate in the system is: $m^{}=0\mathrm{.}25k\frac{g}{m}in$

a$.)$ Calculate the necessary heat flow rate into the evaporator of this lower stage.

b$.)$ Calculate the amount of energy required by the lower stage compressor.

c$.)$ Calculate the heat flow rate for this lower stage condenser.

d$.)$ What is the coefficient of performance for just the lower stage of the freezer?