Frequency Response of Blower/Duct/Plenum Fluid Systems

Abstract

An analytical and experimental study has been made of the aerodynamic frequency response of the basic blower/duct/plenum system, which is one of the elements of the lifting air cushion system of surface-effect ships. An analysis has been developed that includes unstalled blower dynamics, distributed-duct effects, and plenum flexible-wall effects. Theoretical predictions may be made of pressure amplitude ratios at several system locations for variable-frequency sinusoidal modulation of the plenum exit area, of the duct discharge area, or of the blower's inlet guide-vanes. Experimental frequency-response data have been obtained in a 40 HP blower/duct/plenum laboratory system for three blower speeds, over a range of frequencies up to 50 Hz, using rotating butterfly valves at the plenum exit or at the duct discharge. These data have compared well with the analytical predictions up to -25 Hz and thus the theoretical method may be considered as validated for the lower frequencies.Nomenclature A d = duct area, ft 2 A e = exit area, ft 2 A p = plenum effective wall area, ft 2 Ai (A 2 ) = amplitudes of area variation defined in Eqs.(21) and (22), ft 2 c G = velocity of sound in air (fps) C -plenum capacitance, ft 5 /lb) d = fan impeller diameter, ft f d = duct resonant frequency, Hz f p = frequency of peak response Hz / = duct fluid inertance, Ib-sec 2 /ft 5 k = polytropic constant for air, 1.4 for an adiabatic process K p = effective plenum wall stiffness, Ib/ft L d = duct length, ft n = fan speed, rps N -number of blades in fan P = pressure, psf P u = atmospheric pressure, psf P d = pressure in duct, psf P f = fan outlet pressure, psf P p = plenum pressure, psf Q -volume flow, ft 3 /sec Q d = flow in duct, ft 3 /sec Q e = flow exiting from plenum, ft 3 /sec Q f = fan flow, ft 3 /sec Qo = nominal flow through system, ft 3 /sec R d = duct resistance, lb-sec/ft 5 R e = exit resistance, lb-sec/ft 5 R f = fan resistance, lb-sec/ft 5 S = La Place transform variable, I/sec / = time, sec T e = duct delay time, sec u = fan impeller tip speed, fps V = plenum volume, ft 3 x = positive along duct, ft y -plenum wall position, ft Z £ . = duct surge impedance, lb-sec/ft 5 a d -duct valve modulation gain, fps a. e = exit valve modulation gain, fps A ( ) = increment of quantity () above reference value X = fan blade angle, rad co = angular frequency, rad/sec <t > -fan flow coefficient 4/ -fan pressure coefficient p = fluid density, slug/ft 3 T = a system time constant, sec TV = fan time constant, sec T P = plenum time constant, sec ( ) o = nominal value of quantity ()

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