The cooling system for a small private jet airplane consists of the hardware shown here.

High-pressure air from the aircraft engine compressor is fed to a small turbine at state 1 (temperature T₁). The turbine extracts energy from the air, lowering the air temperature to state 2. This cold air is then passed through a heat exchanger, where it picks up energy transferred as heat from the warmer cabin air, thereby cooling the cabin air. The air at state 3 is then fed to a compressor, which is driven by the turbine. The air emerges from the compressor at state 4, and is fed back into the jet engine at temperature T₄. The mass flow rate of engine air through the cooling system is Ṁ. Using the perfect gas equation of state with constant specific heats, derive an expression for the heat transfer rate Q̇ in terms of the mass flow rate Ṁ, the inlet and discharge temperatures T₁ and T₄, and constants for the gas. Treat the turbine and compressor as adiabatic, and neglect kinetic and potential energy of 3 and 4.

Data From A.3

Transcribed Image Text:

Turbine Compressor Cabin air 3