$10\mathrm{.}22$ A vapor$-$compression refrigeration system uses the arrangement shown in Fig. $10\mathrm{.}8$ for two$-$stage compression with intercooling between the stages. Refrigerant $134$a is the working fluid. Saturated vapor at $\backslash (-32^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ enters the first compressor stage. The flash chamber and direct contact heat exchanger operate at $5$ bar $,$ and the condenser pressure is $14$ bar. Saturated liquid streams at $14$ and $5$ bar enter the high$-$ and low$-$pressure expansion valves, respectively. If each compressor operates isentropically and the refrigerating capacity of the system is $12$ tons, determine

$($a$)$ the power input to each compressor, in kW $.$

$($b$)$ the coefficient of performance. ig$.10\mathrm{.}10$ Refrigeration cycle with two stages of compression and flash intercooling.