A vapour compression refrigerator uses Freon$-12$ as the working fluid and has refrigerating effect of $5$ kW $.$ The compressor is a twin$-$cylinder, single$-$acting machine running at $500re\frac{v}{m}in.$ The cylinder bore diameter and the piston stroke are equal and the volumetric efficiency is $0\mathrm{.}7.$ Compression is adiabatic, with an isentropic efficiency of $0\mathrm{.}9.$ The condenser and evaporator

$45$

saturation temperatures are $40C$ and $10C$ respectively. The refrigerant leaves the condenser as liquid at $35C$ and the evaporator as saturated vapour. Calculate :$-$

$($a$)$ the coefficient of performance of the refrigerator,

$($b$)$ the power to compress the refrigerant,

$($c$)$ the dimensions of the compressor cylinders.