MAE305_pset04s - M&AE 3050 Problem Set 4 Solution Fa...

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Problem Set 4 Fa 2009 Solution 1. A large vacuum tank, held at 66,400 Pa, sucks sea-level Standard Air through a con- verging nozzle with a throat diameter of 4 cm. Estimate the following: (a) The Mach number at the throat. (b) The temperature of the gas at the throat. (c) The mass flow rate through the nozzle. Solution (a) We are given the pressure at the throat (66.4 kPa) and so we can take advantage of the relationship for pressure in an isentropic process P 0 P = ± 1 + ( γ - 1) 2 M 2 ² γ/ ( γ - 1) Solving this expression for M 2 yields M 2 = 2 ( γ - 1) " ³ P 0 P ´ ( γ - 1) - 1 # Substituting numbers (for air setting γ = 1 . 4) M 2 = 2 (0 . 4) " ³ 101 , 325 66 , 400 ´ 0 . 4 / 1 . 4 - 1 # = 0 . 64 Taking the square root yields M = 0 . 8 (b) The temperature at the throat can be calculated from the relation T 0 T = 1 + ( γ - 1) 2 M 2 = 1 + (0 . 2) (0 . 64) = 1 . 13 Solving for the temperature yields T = 288 . 16 1 . 13 = 255 . 4 K (c) The mass flow rate through the nozzle is by definition ˙ m = ρuA 1
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where ρ , u and A are the density, velocity and area at the nozzle throat. Since we know P and T at the throat, we can compute the density from the ideal gas law ρ = P RT = 66 , 400 (287)(255 . 4) = 0 . 91kg / m 3 The velocity can be determined from the known Mach number and sound speed u = Ma = M p γRT = 0 . 8 p (1 . 4)(287)(255 . 4) = 256 . 3 m / s The mass flow rate is ˙ m = (0 . 91)(256 . 3) π (0 . 04) 2 4 = 0 . 29 kg / s 2. A converging/diverging nozzle has a throat area A * = 0.25 m 2 and is designed to carry a mass flow rate of 25 kg/s. The temperature at the throat is 195 K and the flow is choked (i.e., sonic). (a) Determine the pressure, density and velocity at the throat (
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MAE305_pset04s - M&AE 3050 Problem Set 4 Solution Fa...

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