Compressible Flow Intro_L8_1

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What are we dealing with: (gas – air) high speed flow – Mach Number 0.3 Perfect gas law applied – expression for pressure field Need some thermodynamic concepts, eg, enthalpy, entropy, etc

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Compressible Flow Intro_L8_1 BACKGROUND INFO What are we dealing with: (gas – air) high speed flow – Mach Number > 0.3 Perfect gas law applied – expression for pressure field Need some thermodynamic concepts, eg, enthalpy, entropy, etc Flow Classification: - Subsonic: Ma < 1 - Supersonic: Ma > 1 - Hypersonic: Ma > 5 Pressure always expressed as absolute pressure likewise Temperature will be in absolute unit, eg Kelvin Density becomes a flow variable and Temperature is a variable too Application: turbomachinery – turbine, compressor, airfoil, missile Mach Number: Ma = fluid velocity / sonic velocity Sonic velocity , c,speed of sound in fluid medium
Compressible Flow Intro_L8_2 Perfect Gas Law pV = mRT MR = 8314.4 J kg-1K-1 Mass of gas Mol wt of gas Another relation: p = ρRT What do we need from THERMODYNAMICS ? Few properties : P, T, u (internal energy), h (enthalpy), s (entropy) Few processes : adiabatic, isoentropic, reversible Adiabatic process: system insulated from surrounding – no heat exchange Isoentropic process: constant entropy, no change in entropy Reversible process: ideal process (most efficient); return to original state How does all these correlate? Adiabatic dQ For a reversible process, ∆s = ∫ ∆s = 0 T dQ ∆s > 0 For an irreversible process, ∆s > ∫ T Note: the terms, u, h & s comes from 1st & 2nd law of thermodynamics
Compressible Flow Intro_L8_3 Few more terms & relations (thermo) cp → specific heat at constant pressure cv → specific heat at constant volume  ∂h   ∂u  ratio of sp heats, k = cp / cv cp =   cv =    ∂T  p  ∂T  v kair ≈ 1.4 cp = cv + R h=u+p/ρ Speed of Sound (sonic velocity) sound is a measure of pressure disturbance defined as propagation of infinitesimal pressure disturbance In fluid medium process is assumed isoentropic (reversible)  ∂p   ∂p  p In gas, sonic velocity, c =   Ideal gas,   = k 0 = kRT  ∂ρ   ∂ρ   s ρ  s Static pressure (or stream pressure), p Stagnation pressure (p0)- pressure when gas brought to rest isoentropically
Compressible Flow Intro_L8_4 Note: more on stagnation pressure few slides later Sonic velocity in liquid or solid medium: c = K /ρ where K is bulk modulus Air at moderate pressure is assumed to behave as ideal gas Speed of sound in air at a pressure 101lPa becomes c = kP / ρ = 1.4 * 101000 / 1.2 = 343m / s at sea level, c = 340 m/s at 11km altitude, c = 295 m/s In water sonic speed, c = K / ρ = 2.14E09 / 10 3 = 1463m / s Inertial force Mach Number, Ma = U / c compressible force U → local fluid speed Note: local sonic speed, u is defined as propagation of a infinitesimal pressure disturbance when fluid is at rest
Compressible Flow Intro_L8_5 Shock Waves Finite pressure disturbances can cause sound propagation greater than local sonic speed Examples: bursting of a paper bag or a tire; disturbances caused by high velocity bullets, jet aircraft & rockets WAVE PROPAGATION (pressure disturbance) Pressure disturbance occuring at an interval of every ∆t S is the disturbance source c∆t c(2∆t) c(3∆t) c(2∆t) c(3∆t) c∆t S U=0 U
Compressible Flow Intro_L8_6 Shock Wave propagation: Case (c): U = c Inside cone aware c(2∆t) of sound nd α c(3∆t) ou eo e 1 a r co n 2 fs 3 aw e S S un tsid 3 2 1 3 2 1 Ou U∆t U(2∆t) Ma = 1 sonic U(3∆t) The Mach Cone all wavefronts touch Ma > 1 source S Case (d): U > c Mach (Cone) angle: α = Sin-1(1/Ma)
Compressible Flow Intro_L8_7 Isentropic Flow (1-D): Local Isentropic Stagnation properties Integration of differential equations result in an integration constant To evaluate this constant, a reference location is required This reference location is zero velocity where Ma = 0 Stagnation point is thus when fluid is brough to stagnant state (eg, reservoir) Stagnation properties can be obtained at any point in a flow field if the fluid at that point were decelerated from local conditions to zero velocity following an isentropic (frictionless, adiabatic) process Notation: pressure : p0 Temperature : T0 Density : ρ0 Fluid chosen in most cases will be air or superheated steam which can be treated as perfect gas
Compressible Flow Intro_L8_8 Conservation Equations for 1-D Isentropic Flow Conservation of Mass(Continuity) : d(ρVxA) = 0 or ρVxA = m = constant dp  Vx2  Conservation of Momentum : +d  2 =0  ρ   Conservation of Energy :  Vx2  Vx2 d h +  =0 or h+ = h 0 = cons tan t (1st Law of Thermo)  2  2 2nd Law of Thermo: s = constant Equations of State: h = h(s,p) ρ = ρ(s,p) p Isentropic Process of Ideal Gas: = cons tan t (derivation available) ρ k
Compressible Flow Intro_L8_9 1-D Analysis: CV y Starts with Stream tube (stream lines) Flow x Useful Relations: p p0 = cons tan t = k 1 ρk ρ0 ( k −1) / k ( k −1) Other Variables: T0  p 0  ρ  =  = 0  (from ideal gas relation) T  p     ρ    How can we get local Isentropic variables in terms of Mach # Pressure: p0/p = [1 + 0.5(k-1)Ma2]k/(k-1) At Sonic condition (Ma = 1) 2k (V* = c*) Temperature: T0/T = [1 + 0.5(k-1)Ma2] V =c = * * * RT0 k +1 Density: ρ 0/ρ = [1 + 0.5(k-1)Ma2]1/(k-1) p0 * = 0.5(k + 1)k /( k −1) = 1.89 ( for air ) p ρ0 T0 = 0.5(k + 1)1 /( k −1) = 1.577 ( for air ) = 0.5(k + 1) = 1.2 ( for air ) p* T*
Compressible Flow Intro_L8_10 Effects of Area Variaion on Properties of Isentropic Flow dp  Vx2  dp dV Momentum eqn: +d =0 =− x ρ  2  ρVx2 Vx   dA dV dρ From Continuity: =− x − A Vx ρ dA dp dp ρVx2 dρ Conbine the 2 equations: = − A ρVx2 ρVx2 dp ρ  Vx2  dA = dp 1 −   dp / dρ  = dp (1 − Ma 2 ) A ρVx2   ρVx2 dA A = - dVx Vx ( 1 − Ma 2 ) Above relation illustrates how area can be affected by Ma
Compressible Flow Intro_L8_11 Effects of Area Variaion on Properties of Isentropic Flow dA A = - dVx Vx ( 1 − Ma 2 ) Nozzle – Diffuser Sonic velocity reached where area is minimum (throat) Comes from the principle dA = 0 → Ma = 1 Nozzle dp>0 Diffuser dp0 dVx 1