c$)$ The nozzle in a jet engine is designed to increase the kinetic energy of the air flows entering the jet turbine engine section as shown in Figure Q$2$b$.$ At the initial stage, the air entering the nozzle without any work or heat interaction with velocity of $25\frac{m}{s},$ pressure $2\mathrm{.}1$ MPa and temperature $510$ K and exit at $1\mathrm{.}3$ MPa and temperature $500$ K $.$ Then the air enters an adiabatic turbine engine stage and leaves the turbine section at $100$ kPa and temperature $127C$ through $0\mathrm{.}5m^2.$ The mass flow rate of the air is $80k\frac{g}{s}.$ Determine:

$($Note: Take $h=c_p(T)$; where $c_p$ is calculated at $T_{\text{a}\text{v}\text{g}})$

i$.$ The exit velocity of a nozzle $($in $\frac{m}{s})$

ii$.$ The specific volume at the turbine inlet and outlet $($in $\frac{m^3}{k}g)$

iii. The change in kinetic energy in the turbine $($in $\frac{m^2}{s})$ and

iv$.$ The power produce by the turbine $($in KW$)$

$m^{}=80k\frac{g}{s}$

Figure Q$2$b