The figure below shows a turbine$-$driven pump that provides water to a mixing chamber located $dz=45m$ higher than the pump, where $m^{}=50k\frac{g}{s}.$ Steady$-$state operating data for the turbine and pump are labeled on the figure. Heat transfer from the water to its surroundings occurs at a rate of $2$ kW $.$ For the turbine, heat transfer with the surroundings and potential energy effects are negligible. Kinetic energy effects at all numbered states can be ignored.

Determine:

$($a$)$ the magnitude of the pump power, in kW$.$

$($b$)$ the mass flow rate of steam, in $k\frac{g}{s},$ that flows through the turbine.