solmid135s10 - The shock is formed at A/A =3.0 so the Mach...

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MAE135 SOLUTIONS TO MIDTERM Spring 2010 Problem 1 (5 points) C and D are wrong: C is the isentropic relation, which does not hold through the shock, because the shock is non-isentropic; D is the incompressible Bernoulli equation, which fails here because the Fow is compressible and non-isentropic. Problem 2 (5 points) Since the process is isentropic, the temperatures and pressures are linked via the isentropic relation T 2 T 1 = p p 2 p 1 P ( γ - 1) = 0 . 5 0 . 286 = 0 . 820 ±rom conservation of total enthalpy, c p T 1 + 1 2 V 2 1 = c p T 2 + 1 2 V 2 2 V 2 = r V 2 1 + 2 c p T 1 (1 - T 2 /T 1 ) = 664 m/s Problem 3 (10 points) (a) There is no heat transfer. The Fow decelerates but remains supersonic, therefore a normal shock is excluded. The only option is friction which, as we know, decelerates supersonic Fow. (b) Since Fow is adiabatic, total temperature is conserved: T 1 p 1 + γ - 1 2 M 2 1 P = T 2 p 1 + γ - 1 2 M 2 2 P T 2 T 1 = 1 . 55 (c) Continuity gives ρu = constant, which can be rewritten as pM/ T = constant. Hence p 2 p 1 = M 1 M 2 R T 2 T 1 = 1 . 87 Problem 4 (10 points)
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Unformatted text preview: The shock is formed at A/A * =3.0, so the Mach number upstream of the shock is M 1 = 2 . 65 and the total pressure upstream of the shock is p 1 = 500 kPa. The shock tables give p 2 /p 1 = 0.44, therefore the total pressure downstream of the shock is p 2 =220 kPa. In case (a), the probe is placed downstream of the shock, where the velocity is subsonic. Since the probe is in a subsonic Fow, it cannot have any shocks on it so it measures the local value of the total pressure p 2 = 220 kPa. In case (b), the probe tip is ahead if the shock, so it is in supersonic Fow. The tip must have a shock wave formed around it. The strength of this shock is identical to the strength of the shock in the test section (since they both have the same upstream Mach number M 1 = 2 . 65). The probe measures again p 2 = 220 kPa. 1...
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This note was uploaded on 08/31/2010 for the course ENGRMAE 135 taught by Professor Staff during the Spring '08 term at UC Irvine.

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