R$134$a vapor compression system operating to cool a chilled water to $5C$ when the condenser is water cooled supplied with cooling water at $35C.$ The required cooling capacity is $20$ tons of refrigeration. Assume:

The cooling water temperature in the condenser rises by $8C$ and the chilled water temperature in the evaporator drops also by $8$ degrees.

A minimum of $5C$ temperature difference is required to allow heat transfer between the condenser and the cooling water as well as between the evaporator and the chilled water.

The cycle operates with $3-$degree superheat, $3-$degree sub$-$cool, and $80\%$ isentropic efficiency.

The compressor has $4$ cylinders that have a piston stroke length is $7$ cm and the piston diameter is $5$ cm with a clearance ratio of $5\%,$ with a polytropic index of $1\mathrm{.}3.$

Using the P$-$H diagram determine:

a$)$ Sketch a flow diagram, T$-$S$,$ P$-$H diagrams of the process, and the temperature distribution with the coil length in the evaporator and condenser $($for refrigerant and chilled water: Evaporator and for the refrigerant and cooling water: Condenser$)$

b$)$ The mass flow rate of the refrigerant circulated as well as the volumetric efficiency in this case.

c$)$ The cooling water and the chilled water flowing velocities if the cooling water and chilled water flow in a $0\mathrm{.}25-$inch inner diameter pipes.

d$)$ The compressor speed in RPM

e$)$ The power required in the compressor in horsepower.

f$)$ The refrigeration effect as well as the total heat rejected in the condenser.

g$)$ The Carnot COP, and the cycle COP of the cycle, as well as the cycle refrigeration efficiency.