THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

14.12 Gas-Jet Noise 387said to be choked. In order to increase the velocity further, the nozzle must increasein its cross-sectional area beyond the duct station where choking occurs.In the frictionless, isentropic flow of an ideal gas from one point to another, theapplicable energy equation describing this flow ise + p ρ + u2= constant (W/kg) (14.37)2where e is the internal energy of the gas, and u is the gas flow velocity. For thereservoir, u = 0 and p = p r . Also, from thermodynamics theory for an ideal gas,e + p ρ = enthalpy = c pTwhere c p is the specific heat of the gas at constant pressure and T is the absolutetemperature. Mach number M is defined byM = u c(nondimensional characteristic)But propagation speed of sound c in an ideal gas is given byEquation (14.37) becomesc 2 = γ RT = γ p ρe + p ρ + u22 = c pT + γ M2 RT2= c ppRρ= c p T(1 + γ M2 (γ − 1)2γ(1 + γ )M2 R2c p)= constant (14.38)We apply the fact that R = c p – c v in an ideal gas, and the ratio of the specificheats c p /c v = γ . Selecting two flow stations, one at a point r inside the reservoirand the other at the point in the duct where choking occurs (Mach number M =1.0), we obtain from Equation (14.37)p rρ r= p (ch1 + γ − 1ρ ch 2)orp r= ρ (ch1 + γ − 1 )p cr ρ ch 2(14.39)In an isentropic flow, according to thermodynamic theory:pρ −γ = constantThen Equation (14.37) becomes(p r= 1 + γ − 1 ) γγ −1(14.40)p ch 2The ratio γ of specific heats is equal to 1.4 for air, so Equation (14.40) yields thecritical pressure ratio for a shock to appear:p rp ch=(1 + 1.4 − 12) 1.41.4−1= (1.2) 3.5 = 1.89