A bootstrap refrigeration system used for an aircraft needs a $30$ TR load. The air from ambient is rammed upto a pressure of $0\mathrm{.}4$ bar and $260$ K $.$ The pressure at exit to main compressor is $2$ bar. The air bled off from main compressor is cooled by ram air in primary heat exchanger upto $60C$ which is then compressed in secondary compressor run by the air cooling turbine. This is cooled in the secondary heat exchanger upto $57C$ by ram air before supplied to cooling turbine where it expands upts satin pressate of I dar. The cabin is maintained at $27C.$ The power developed by cooling turbine is used to drive the secondary compressor.

Assuming the isentropic efficiency of both the compressors as $90\%$ and that of turbine as $80\%,C_p=1\mathrm{.}0k\frac{J}{k}gK$ and $=1\mathrm{.}4,$ find the delivery pressure of secondary compressor, mass flow rate of air to cabin, power input to compressor and the COP of the system.