Problem #$1$: $(50$ points$)$

A mass flow of $6k\frac{g}{s}$ of water vapor at $20$ bar and $500C$ flows steadily through the inlet of an insulated turbine. The pressure of the water is reduced to $1\mathrm{.}0$ bar and an internal energy of $2329\mathrm{.}25K\frac{J}{k}g$ at the turbine outlet. Neglecting kinetic and potential energy effects, determine:

a$)$ The power output in $kW$ and the outlet temperature. $12,5$ points

b$)$ The inlet Area if the Velocity at the inlet is $25\frac{m}{s}\mathrm{.}12\mathrm{.}5$ points

c$)$ The outlet velocity to keep the same pipe area at the outlet. $12\mathrm{.}5$ points

d$)$ Sketch the process in a neat and labelled $P-v$ and $T-v$ diagrams. $12\mathrm{.}5$ points