Problem $-1$: Consider a regenerative vapor power cycle with two feedwater heaters, a closed one and an open one, as shown in Figure $1.$ Steam enters the first turbine stage at $12$MPa,$480C,$ and expands to $2$ MPa $.$ Some steam is extracted at $2$ MPa and fed to the closed feedwater heater. The remainder expands through the second$-$stage turbine to $0\mathrm{.}3$ MPa $,$ where an additional amount is extracted and fed into the open feedwater heater operating at $0\mathrm{.}3$ MPa $.$ The steam expanding through the third$-$stage turbine exits at the condenser pressure of $6$ kPa $.$ Feedwater leaves the closed heater at $210C,12$MPa, and condensate exiting as saturated liquid at $2$ MPa is trapped into the open feedwater heater. Saturated liquid at $0\mathrm{.}3$ MPa leaves the open feedwater heater. Assume all pumps and turbine stages operate isentropically$.$ Determine for the cycle

$($a$)$ the heat transfer to the working fluid passing through the steam generator, in kJ per kg of steam entering the first stage turbine.

$($b$)$ the thermal efficiency.

$($c$)$ the heat transfer from the working fluid passing through the condenser to the cooling water, in kJ per kg of steam entering the first$-$stage turbine.