Problem$-2($Marks: $30+3$ for Cleanliness$)$

Consider a regenerative reheat steam power plant with two steam turbines and two closed feedwater heaters $($FWH$)$ as shown in the figure below. The steam enters the high$-$pressure turbine $($HPT$)$ at $14$MPa,$540C.$ The steam is reheated to $540C$ at $1\mathrm{.}4$ MPa before entering the low$-$pressure turbine $($LPT$).$ The condenser pressure is $7$ KPa $.$ The pressure of the extracted steam for the high$-$pressure FWH is $3\mathrm{.}1$ MPa and for the low$-$pressure FWH is $0\mathrm{.}30$ MPa $.$ The terminal temperature difference $($TTD$)$ for both the FWHs is $5C.$ The condensate leaves both the FWHs as saturated liquid. The highpressure heater condensate is throttled into the low$-$pressure heater, which in turns drains into the condenser through a throttling. The isentropic efficiencies are $0\mathrm{.}88$ for the HPT$,0\mathrm{.}90$ for the LPT$,$ and $0\mathrm{.}80$ for the boiler feed pump. The relevant T$-$s diagram is provided following the flow diagram.

Determine:

$($a$)$ the mass mass fractions of steam into both FWHs

$($b$)$ the net work output, in $k\frac{J}{k}g.$

$($c$)$ the thermal efficiency of the cycle.