Superheated steam at $40$ bar absolute and $500C$ flows at a rate of $250k\frac{g}{m}in$ to an adiabatic turbine, where it expands to $5$ bar. The turbine develops $1500kW.$ From the turbine the steam flows to a heater, where it is reheated isobarically to its initial temperature. Neglect kinetic energy changes.

$($a$)$ Write an energy balance on the turbine and use it to determine the outlet stream temperature.

$($b$)$ Write an energy balance on the heater and use it to determine the required input $(kW)$ to the steam.

$($c$)$ Verify that an overall energy balance on the two$-$unit process is satisfied.

$($d$)$ Suppose the turbine inlet and outlet pipes both have diameters of $0\mathrm{.}5$ meter. Show that it is reasonable to neglect the change in kinetic energy for this unit.