Acoustic admittance

The most important parameter in the analysis of pressure-coupled combustion instability is the acoustic admittance Y, which is the ratio of the amplitude of the acoustic velocity V to the amplitude of the acoustic pressure amplitude of the acoustic velocity V to the amplitude of the acoustic pressure P ... 

Thus, the exponential growth constant of the pressure oscillation is directly related to the acoustic admittance of the propellant. Hence, the acoustic admittance can be evaluated directly from the growth rate of the pressure amplitude. Ryan (R5) has also desired this espression on the basis of acoustic-energy considerations. 

Hart and McClure (H2) have considered the combustion aspects of the problem and have shown the acoustic admittance may be written in the form... 

When the propellant burning rate is espressed by Eq. (5b), the parameter p /e can be considered as the transient sensitivity of the burning rate to pressure. This parameter depends on the transient combustion characteristics, and its evaluation depends on the particular model of the combustion process. Thus, the acoustic admittance provides the link between experimental observation and theoretical prediction. 

These studies have indicated that the independent parameters controlling the postulated solid-phase reactions significantly affect the resulting acoustic admittance of the combustion zone, even though these reactions were assumed to be independent of the pressure in the combustion zone. In this combustion model, the pressure oscillations cause the flame zone to move with respect to the solid surface. This effect, in turn, causes oscillations in the rate of heat transfer from the gaseous-combustion zone back to the solid surface, and hence produces oscillations in the temperature of the solid surface. The solid-phase reactions respond to these temperature oscillations, producing significant contributions to the acoustical response of the combustion zone. 

Friedly (F4) expanded the theoretical analysis of Hart and McClure and included second-order perturbation terms. His analysis shows that the linear response of the combustion zone (i.e., the acoustic admittance) is not sign-ficantly altered by the incorporation of second-order perturbation terms. However, the second-order perturbation terms predict changes in the propellant burning rate (i.e., transition from the linear to nonlinear behavior) consistent with experimental observation. The analysis including second-order terms also shows that second-harmonic frequency oscillations of the combustion chamber can become important. 

Horton and Price (H11) have obtained acoustic-admittance data for a series of double-base and composite propellants with different burning-rate characteristics. They examined the effects of pressure at various frequencies... 

Horton (H9, H10) has obtained additional acoustic-admittance data for a series of composite propellants. At a given frequency, decreasing the mean oxidizer particle size increases the acoustic admittance and thereby the tendency for instability. Horton also investigated the effects on the acoustic admittance of the incorporation of traces of copper chromite, a known catalyst, for the decomposition of ammonium perchlorate, lithium fluoride (a burning-rate depressant), and changes in binder these data are difficult to analyze because of experimental errors. 

Y Acoustic admittance Yk Mole fraction of /fth species y Distance along propellant surface... 

Hillman et al. measured the quartz crystal impedance to determine changes in rigidity, swelling and ionic exchange in conducting polymer films [57, 58] and have also used dynamic quartz crystal impedance of modified electrodes during film growth and redox conversion [57] by qualitative analysis of the acoustic admittance-frequency peak width. 

It explains qualitatively many observations of singing and vibrating flames (and vibrations in tubes containing heated elements as well), described beautifully by Tyndall [53] and dating back at least to an observation made by Dr. B. Higgins in 1777. Of course, the amplification at a burning propellant surface is considerably more complex than the phenomena described by equation (51), since molecular transport processes are involved in an essential way for example (as was seen for the steady-state combustion in Chapter 7). However, the pistonlike mechanism implied by equation (51) when the heat release is localized and the result is expressed in terms of an acoustic admittance [as in equation (29)] is tied closely to the amplification by propellant combustion. 

---