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Unformatted text preview: Compressible Flow Compressible Flow z When density variations cannot be ignored, we are dealing with a compressible flow – An aircraft cruising faster than 30% of the speed of sound – Flow in a standard combustion reciprocating engine – Flow in rocket engines – Flow in gas or steamturbine engines – Flow of traffic on a large highway z We will examine onedimensional compressible flows, including… – Isentropic flow, Laval nozzle – Normal shock waves – Fanno flow, which includes viscous effects – Rayleigh flow, which includes heattransfer Classification Classification z With regard to compressibility, we classify flows according to Mach Number… z The various compressibleflow regimes are… z Virtually all liquid flows are incompressible…not so with gases The M 2 column is relevant because most compressibleflow algebraic expressions involve M 2 Example: Density in a lowspeed flow is Thermodynamic Properties of Air Thermodynamic Properties of Air 1 1 z We will deal mostly with air, which obeys the perfectgas law where R is the perfectgas constant… z Air is calorically perfect so that where specific heats c p and c v are constant and given by with Thermodynamic Properties of Air Thermodynamic Properties of Air 2 2 z As we found in Chapter 7, the entropy of a perfect, caloricallyperfect gas follows from Gibbs’ equation so that z Thus, for isentropic flow… z All of these thermodynamic relations are valid for temperatures as high as 2000 K Speed of Sound Speed of Sound 1 1 z Consider an acoustic wave, i.e., a weak wave moving though a fluid, where weak means z Using a Galilean transformation, it is a simple exercise to compute the speed of sound An acoustic wave moves at the speed of sound, a Speed of Sound Speed of Sound 2 2 z The flow in this reference frame is steady and we use a stationary control volume so that mass conservation is z So, for constant crosssectional area (out of the page) z Dropping the (verysmall) quadratic term yields Æ Speed of Sound Speed of Sound 3 3 z Ignoring body forces and viscous effects, the equation for xmomentum conservation simplifies to z Evaluating the integrals, a little algebra yields z Before dropping the quadratic term, mass conservation is ρ a = ( ρ + ∆ ρ )( a – ∆ u ) , so that z Simplifying, we have Æ Speed of Sound Speed of Sound 4 4 z Recall from above that so that which, in the limit ∆ ρ → , is z Pressure, like all thermodynamic state variables, is a function of two other state variables, so that this is a partial derivative z Since an acoustic wave is isentropic, we conclude that ← Subscript s means the derivative is taken with s held constant Speed of Sound Speed of Sound 5 5 z As noted when we summarized thermodynamic properties of air, p = A ρ γ for isentropic flow, so that z Therefore, the speed of sound of a perfect gas is z Example: Compute the speed of sound of air on a summer day when T = 95 o F = 554.67 o R and on a winter day when...
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This note was uploaded on 10/12/2009 for the course AME 309 at USC.
 '06
 Phares

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