ch913 - CH.9: COMPRESSIBLE FLOW It required an unhesitating...

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CH.9: COMPRESSIBLE FLOW It required an unhesitating boldness to undertake such a venture …. an almost exuberant enthusiasm…but most of all a completely unprejudiced imagination in departing so drastically from the known way. J. Van Lonkhuyzen, 1951, discussing designing Bell XS-1
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Ch. 9: Introduction to Compressible Flow • Liquids, ρ = constant for us (1% increase in ρ for every 1.6 km deep) • Air, 1% change for every 26 m deep • M = 0.3 ~ 5% = Δρ / ρ ; M = 0.3 ~ 100 m/s or 230 mph • Significant density changes imply significant compression or expansion work on the gas, which can change T, u, s, … • Compressibility: fluid acceleration because of friction, fluid deceleration in a converging duct, fluid temperature decrease with heating • Ideal Gas: p = ρ RT (simple, good approximations for our engineering applications, captures trends)
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HOW DO FLOW-THERMODYNAMIC PROPERTIES CHANGE? WHAT CHANGES THEM? AREA CHANGE ; +/- Q; FRICTION; SHOCK
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WHERE IS WALDO? Isentropic? Adiabatic? Reversible /Frictionless? Constant area? Ideal gas Constant c p and c v Quasi-1-Dimensional No shaft work p, T, ρ , u, h, s, V,… p o , T o , ρ o , u o , h o , s o , V=0 p*, T*, ρ *, u*, h*, s*, V=a
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ρ≠ 0 leads to increased complexity Consider Conservation of Mass
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Conservation of Mass ρ VA = constant so d ρ / ρ + dV/V + dA/A = 0 But if compressible: If incompressible: dV/V = -dA/A and d ρ / ρ = 0 dV/V = - (dA/A)(1 – M 2 ) -1 and d ρ / ρ = (dA/A)[M 2 /(1-M 2 )]
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GOVERNING EQUATIONS FOR NEWTONIAN FLUIDS 4 Equations: continuity and three momentum 4 Unknowns: p, u, v, w (know: μ , ρ , f j ) Continuity and Momentum INCOMPRESSIBLE COMPRESSIBLE Continuity, Momentum, Energy, State Equations Need to find ρ and T Sometime also interested in s, h, u p = ρ RT* u = c v T* h = c p T* p/ ρ k = constant + * Ideal gas and constant c v and c p; + isentropic
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Chapter 9: Compressible Flow 9.1 Introduction: Review of Thermodynamics 9.2 The Speed of Sound 9.3 Adiabatic and Isentropic Steady Flow 9.4 Isentropic Flow with Area Changes 9.5 The Normal Shock Wave 9.6 Operation of a Converging and Diverging Nozzles
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Ideal Gas p = ρ RT (always assuming that flow is in equilibrium)
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EQUATION OF STATE FOR IDEAL GAS p = ρ RT - if average distance between molecules is 10 diameters or more, then very weak attractive forces approximately point non-interacting particles assuming no interactions between gas particles
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p = ρ RT Good to 1% for air at 1 atm and temperatures > 140 K (-130 o C) or for room temperature and < 30 atm At large pressures, great departure from ideal gas equation of state.
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pV= N m R univ T p =(1/V)N m m mole {R univ / m mole }T p =(m/V){R univ /m mole }T p = ρ { R univ /m mole }T p = ρ R T
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Note: It is assumed that system always in equilibrium. Assume all gases obey ideal gas law:
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This note was uploaded on 04/26/2010 for the course MAE 101B 101B taught by Professor Rohr during the Summer '09 term at UCSD.