Problem $\backslash$#$1(10$ pts$)$

Consider water in a Rankine Cycle used for a nuclear powerplant with a turbine, a condenser, a pump, and a boiler. A nuclear reactor supplies heat to the boiler. Water enters the turbine as a superheated vapor at $520$ C and $8$ MPa $.$ It exits the turbine at $100$ kPa $.$ The turbine has an isentropic efficiency of $\backslash (80\backslash \%\backslash ).$ The pump has an isentropic efficiency of $\backslash (70\backslash \%\backslash ).$ Water enters the pump as a saturated liquid at $100$ kPa $,$ and exits at the same pressure as it enters the turbine $(8$ MPa $).$ The cycle is required to have a net power output of $50$ MW $.$

A$.$ Determine the work$/$mass from the turbine.

B$.$ Determine the work$/$mass into the pump $($report as a positive value; you may assume the isentropic case is isothermal and incompressible$)$

C$.$ Determine the mass flow rate of the cycle

D$.$ Determine overall efficiency of the cycle $(\backslash (\backslash$eta$=\backslash$dot$\{$W$\}\_\{$n e t$\}/\backslash$dot$\{$Q$\}\_\{$H$\}\backslash ))$

E$.$ If the boundary temperature on the hot side is $520$ C and the boundary temperature on the cold side is $50$ C $,$ what is the rate of entropy production $(\backslash (\backslash$dot$\{\backslash$sigma$\}\backslash ))$ for the entire cycle?

F$.$ Draw the process on a T$-$s diagram; label all $4$ states $(1$ into turbine, $2$ into condenser, $3$ into pump, $4$ into boiler$)$

G$.$ Problem $\backslash$#$2(10$ pts$)$

The Rankine Cycle from problem $2$ is modified to add reheat $($this adds two state points$).$ The initial temperature and pressure into the first turbine $($state $1)$ remain the same as problem $1.$ The first turbine drops to a pressure of $1$ MPa $.$ From State $2$ to State $3,$ it is heated isobarically until it has a temperature of $360$ C $.$ In the second turbine, it drops to $100$kPa, so the pressure into the condenser $($and into the pump$),$ and the pressure out of the pump $($and into the boiler and first turbine$)$ are the same as Problem $2.$ The turbine and pump efficiencies are the same as Problem $2.$

Note: this means that the pump has the same enthalpy in and out $($and work$/$mass$)$ as problem $2,$ so you do not need to repeat that work; $\backslash ($ h$\_\{6\}-$h$\_\{5\}\backslash )$ for this problem is the same as $\backslash ($ h$\_\{4\}-$h$\_\{3\}\backslash )$ in problem $2.$

The net power output $(\backslash (\backslash$dot$\{$W$\}\backslash ))$ remains at $50$ MW $.$

A$.$ Determine the work$/$mass from state $1$ to state $2.$

B$.$ Determine the work$/$mass from state $3$ to state $4.$

C$.$ Determine the mass flow rate of the cycle

D$.$ Determine overall efficiency of the cycle

E$.$ Determine how much less heat is required from the nuclear reactor; report as a positive value in kW$.$

F$.$ Suppose that the If the cost to add the extra turbine is $\backslash (\backslash$$ $100\backslash )$ million, the energy is worth $\backslash (\backslash$$ $0\mathrm{.}15/\backslash$mathrm$\{$kW$\}\backslash )-$hr$,$ and that the power plant will last longer if there is less heat required due to the extra turbine. How much longer $($in years$)$ will the power plant need to last to pay off the cost? Will it last this this long if the formula for how much longer it will last is ExtraYears $\backslash (=40\backslash )$ years $\backslash (*\backslash$frac$\{\backslash$dot$\{$Q$\}\_\{\backslash$text $\{$saved $\}\}\}\{\backslash$dot$\{$Q$\}\_\{$H$,1\}\}\backslash )$ where $\backslash (\backslash$dot$\{$Q$\}\_\{\backslash$text $\{$saved $\}\}\backslash )$ is your answer to part e and $\backslash (\backslash$dot$\{$Q$\}\_\{$H$,1\}\backslash )$ is the heat required in problem $1?$

G$.$ Draw the process on a T$-$s diagram; label all $6$ states $(1$ into first turbine, $2$ into reheater, $3$ into second turbine, $4$ into condenser, $5$ into pump, $6$ into boiler

Please just solve Problem #$2$ A$-$G