The inlet of a supersonic aircraft engine is a supersonic diffuser. Air enters at high speed ?_1?at state 1 (T₁, P₁), and exits at high pressure P₂ at much lower speed 2 . The diffuser is adiabatic and has no moving parts.

The Mach number M is defined as the ratio of the flow velocity to the speed of sound c, which for an ideal gas is c = ??RT, so M₁ = ?₁/a₁ = ?₁/??RT1, where M₁ is the flight Mach number. Starting from basic principles, and using the ideal gas equations of state, show that

Then, derive an expression for the discharge pressure P₂/P₁ in terms of M₁, PS (the entropy production per unit mass), and constants for the gas (neglect the kinetic energy at 2 compared to that at 1). From this determine the expression for the maximum possible P₂/P₁ that could be obtained, in terms of M₁ and ?. Sketch the process representation for an ideal and an actual diffuser with the same inlet conditions on a Ts diagram, showing the pertinent lines of constantP for both cases.

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VI 1 1 2