Problem $1(20$ points$)$ Consider a cogeneration system operating as shown in Fig. P$8\mathrm{.}80.$ Steam enters the first turbine stage at $6$MPa,$540C.$ Between the first and second stages, $45\%$ of the steam is extracted at $500$ kPa and diverted to a process heating load of ${5310}^{8}k\frac{J}{h}.$ Condensate exits the process heat exchanger at $450$ kPa with specific enthalpy of $589\mathrm{.}13k\frac{J}{k}g$ and is mixed with liquid exiting the lowerpressure pump at $450$ kPa $.$ The entire flow is then pumped to the steam generator pressure. At the inlet to the steam generator the specific enthalpy is $469\mathrm{.}91k\frac{J}{k}g.$ Saturated liquid at $60$ kPa leaves the condenser. The turbine stages and the pumps operate with isentropic efficiencies of $82\%$ and $88\%,$ respectively. Determine

$($a$)$ the mass flow rate of steam entering the first turbine stage, in $k\frac{g}{s}.$

$($b$)$ the net power developed by the cycle, in MW$.$

$($c$)$ the rate of entropy production in the turbine, in kW$/$K$.$

Determine the rate of exergy transfer in kJ per kg of steam entering the first$-$stage turbine, to the working fluid passing through the steam generator in Problem $8\mathrm{.}80$ Perform calculations to account for all outputs, losses, and destructions of this exergy. Let $T0=15C,p0=0\mathrm{.}1$MPa