A double$-$flash steam GPS$,$ as shown in the figure below, is used for power production by using a

HP and LP turbines. In this plant, $140k\frac{g}{s}$ gaturated liquid geothermal water at a temperature and

pressure of $218C$ and $2316$ kPa $,$ respectively, is isenthalpically flashed in the Flash Chamber $1$

at a pressure ratio of $2,$ and separated in the saturated form in the vapor$-$liquid Separator $1.$ The

fraction of vapor at the flash chamber outlet is $0\mathrm{.}07623,$ and then the produced vapor is used to

generate power in the HP turbine. The quality and pressure of working fluid exiting from the HP

turbine are $0\mathrm{.}9605$ and P$3/2.$ The liquid working fluid enters Flash Chamber $2$ and is

isenthalpically flashed in this flash chamber at the same pressure level of Flow $6.$ and separated

in the saturated phase in the vapor$-$liquid Separator $2.$ The fraction of vapor at this flash chamber

outlet is $0\mathrm{.}0607,$ and the produced vapor is used to generate power in the LP turbine. The

discharged geothermal working fluid from the steam turbine with a quality of $0\mathrm{.}8392$ is at $10$ kPa $,$

which is then condensed and reinjected to the reinjection well. The saturated liquid working fluid

exiting Separator $2$ is at the same pressure level of Flow $6.$ Calculate the: $($a$)$ the work production

rate by the HP and LP turbine, and the total work production rate; $($b$)$ the thermal efficiency of

the double$-$flash steam GPS; and $($c$)$ the variation of the produced power of the double$-$flash GPS

when the geothermal water mass flow rate increases from $100$ to $180k\frac{g}{s}.$