Consider a cogeneration system operating as shown in figure. Steam enters the first turbine stage at $6$ MPa, $540$ C$.$ Between the first and second stages, $45\%$ of the steam is extracted at $500$ kPa and diverted to a process heating load of $53108$ kJ$/$h$.$ Condensate exits the process heat exchanger at $450$ kPa with specific enthalpy of $589\mathrm{.}13$ kJ$/$kg 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 $469\mathrm{.}91$ kJ$/$kg$.$ Saturated liquid at $60$ kPa leaves the condenser. The turbine stages and the pumps operate with isentropic efficiencies of $82\%$ and $88\%,$ respectively. 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 $,$ Perform calculations to account for all outputs, losses, and destructions of this exergy. Let T$0=15$ C$,$ p$0=0\mathrm{.}1$ MPa

write the equations for the input, outputs, losses and destructions