System instabilities in combustion equipment

Chugging in liquid-propellant rockets is only one of many examples of system instabilities in combustion devices [135], [136]. As may be seen from the contribution of Putnam to [137], a considerable amount of research has been performed on mechanisms of these instabilities. Interactions of processes occurring in intakes and exhausts with those occurring in the combustion chamber typically are involved, and it may or may not be necessary to consider acoustic wave propagation in one or more of these components in theoretical analyses [138]-[142]. Here we shall not address problems involving acoustic wave propagation we shall restrict our attention to bulk modes, in which spatial variations of the pressure in the combustion chamber are negligible. 

The simplest example of an oscillation in a bulk mode is the classical Helmholtz resonator. Consider a gas-filled chamber of volume V, into which is inserted a tube of length I and of cross-sectional area that is open to the atmosphere. Assume that the atmospheric pressure remains constant at the outer end of the tube at a value p, and consider slowly varying processes for which the pressure p at the inner end of the tube is equal to the spatially uniform pressure within the chamber. With friction neglected, a force balance on the gas in the tube is — p) = (a O dv/dt, where p is the average gas density in the tube and v is the average outward velocity of gas in the tube. Conservation of mass for the gas of density p in the chamber is t dp/dt — 

Applying equation (3) to the gas in the chamber, we have dp/dt = (dp/dt)/a. By substituting this relation into the one preceding it, differentiating, and then substituting the result into the first expression, we find that the pressure history is described by 

Equation (80) possesses solutions that are oscillatory in time, with frequency 

The derivation of equation (81) is invalid unless the resulting Helmholtz frequency cojj is small compared with the lowest natural frequency of acoustic waves in the chamber, since otherwise an acoustic field with a nonuniform p would have to be considered a related restriction limits the length I of the exit tube. If I is comparable with a characteristic length of the chamber, then a representative acoustic frequency is a/l, and the ratio o)fj/(ci/l) is the square root of the ratio of the volume of the tube to that of the chamber, which is small for small tubes in large chambers—the configuration envisaged in the Helmholtz model. 

Since frictional damping always is present but has been neglected in the derivation of equation (80), a source of excitation, such as a suitably 

An example of bulk modes in solid-propellant rockets is afforded by the low-frequency, or L, instability [7]. A characteristic length of importance in rocket design is the ratio of the gas volume in the chamber to the throat area of the nozzle this ratio often is denoted by L, and its ratio to a characteristic exhaust velocity provides an estimate of the residence time of a fluid element in the gas phase inside the chamber. A mass balance for the gas inside a rocket chamber with a choked nozzle is 

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