Problem$-1($Marks: $30+3$ for Cleanliness$)$

Consider a cogeneration system operating as shown in the figure below. Water 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 $5{10}^{8}k\frac{J}{h}.$ 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 lower$-$pressure 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.

Draw the $T-$s diagram of the cycle.

Determine:

$($a$)$ the mass flow rate of steam entering the first turbine stage, in kg$/$s$.$

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