Problem $3(30$ pts$).$ Consider a power plant with cogeneration and regeneration. Steam enters the turbine at $9$ MPa and $\backslash (400\{\}^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )($State $6)$ and expands to a pressure of $1\mathrm{.}6$ MPa $($State $7).$ At this pressure, $35$ percent of the steam is extracted out of the turbine, and the remainder expands to $10$ kPa $($State $8).$ Part of the extracted steam is used to heat the feedwater in an open feedwater heater, and the rest of the extracted steam is used for process heating and leaves the process heater as a saturated liquid at $1\mathrm{.}6$ MPa $($State $9).$ It is subsequently mixed with the feedwater leaving the feedwater heater $($State $3$ as saturated liquid$),$ and the mixture $($State $4$ as saturated liquid$)$ is pumped to $9$ MPa and enters the boiler $($State $5).$ Assuming the turbines and the pumps to be isentropic$.$

$(1)$ Show the cycle on a T$-$s diagram with respect to saturation lines $(5$ pts$),$

$(2)$ Determine the specific enthalpy h $($unit: $\backslash (\backslash$mathrm$\{$kJ$\}/\backslash$mathrm$\{$kg$\}\backslash ))$ for each state $(10$ pts $),$

$(3)$ Determine the net work output per unit mass of steam for the cycle, in unit of $\backslash (\backslash$mathrm$\{$kJ$\}/\backslash$mathrm$\{$kg$\}\backslash )(10$ pts $),$

$(4)$ Determine the steam mass flow rate through boiler $($unit: $\backslash (\backslash$mathrm$\{$kg$\}/\backslash$mathrm$\{$s$\}\backslash ))$ if a net power output is $25$ MW $(5$ pts $).$