A vapor compression refrigeration unit with R$-12$ as a refrigerant has a capacity of $17$ TR operating between temperature limits of $-30$C and $20$C$,$ respectively. The refrigerant is sub$-$cooled by $5$C before the throttle valve and superheated by $6$C before leaving the evaporator. The refrigeration unit has a $6-$cylinder, single$-$acting, compressor with L$/$D $=1\mathrm{.}25.$ Calculate the following. $($i$)$ Theoretical power required $($ii$)$ COP $($ii$)$ Bore, and stroke of the cylinder

if volumetric efficiency is $87\%$ and speed of the compressor is $1000$ rpm$.$