Problem $\backslash$#$2(10$ points$)$

$\backslash (10\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}\backslash )$ of water enters a turbine at $4$ MPa and $\backslash (500^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )($state $1)$ and leaves the turbine at $10$ kPa $($state $2).$ After leaving the turbine, the water is condensed isobarically in a condenser $($heat exchanger$)$ to $\backslash (40^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )($state $3).$ The energy from the condensing steam is absorbed by liquid cooling water that enters the condenser at $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )($state $5).$

a$)$ If the power output of the turbine is $30$ MW $,$ what is the state of the water at the outlet of the turbine? If it is a saturated mixture, calculate its quality.

b$)$ What is the rate of heat transfer from the process water $($the water used to generate power$)$ to the cooling water?

c$)$ What is the mass flow rate of the cooling water? You may treat the cooling water as having a constant specific heat at an appropriate average temperature. There is negligible pressure drop on the cooling water side of the heat exchanger. Show that you can make this calculation $3$ different ways:

i$.$ by considering just the cooling water side of the condenser as your CV$,$

ii$.$ by considering the entire condenser as your CV$,$ and

iii. by considering a CV that includes both the turbine and the condenser.