$(45$points$)$ A Rankine cycle with regeneration develops a net power output of $1000Bt\frac{u}{s}.$ Steam enters the first turbine stage at $1000$ psi and $1000$ F $[$state $2].$ It expands in the first turbine state to a pressure of $200$psi $[$state $3]$ where it enters a closed feedwater heater, condenses, and exits as a saturated liquid $[$state $5].$ The remaining steam is expanded feedwater heater $[$arbe stage to a final pressure of $10$ psi $[$state $4].$ The feedwater exits the closed sate $1]$ at the same temperature of the condensate $[$at state $5].$

Cooling water $[$indicated by $"$cw$"$ on the figure$]$ removes heat from the cycle and increases in temperature from $50$ F to $63$ F with a negligible pressure drop. You may treat the cooling water as an incompressible substance with a negligible pressure drop and constant specific heats $[h=CT,$ with $C=1Bt\frac{u}{l}bmR.$

The isentropic efficiency of the first and second turbine stage is $100\%.$ The isentropic efficiency of the pump is $85\%.$

Note $-$ If you can not calculate a value in part a$,$ I suggest writing the governing equation for parts $b,c,d,$ and e only in terms of $"h"$ values $[$and $"y"$ in the case of parts $c,d,$ and e if needed$].$

b$.(\frac{5}{45}pt)$ The fraction of steam bled off at state $3,y.$

c$.(\frac{9}{45}pt)$ The thermal efficiency of the cycle

d$.(\frac{7}{45}pt$ Mass flow rate of the water entering the high pressure turbine at state $2$inlb$\frac{m}{s}$ e$.(\frac{9}{45}pt)$ Mass flow rate of the cold water in lbm$/$s