$1$ m$2/$s of air enters at $2$ bar and $20$C through a heat exchange at a speed of $100$ m$/$s$,$ where the temperature rises to $820$C$.$ The air then enters a turbine with the same speed of $100$ m$/$s and expands until the temperature drops to $620$C$.$ As soon as it leaves the turbine, the air takes a speed of $250$ m$/$s to enter a nozzle where it expands until the temperature drops to $530$C$,$ neglecting the potential energy calculated.

a$)$ The rate of heat transfer to the air in the heat exchanger, in kW

b$)$ the power output at the turbine, in kW$,$ assuming no heat loss

c$)$ the nozzle exit velocity, in m$/$s$,$ assuming no heat loss