Comparison of combustion instabilities found in various types of combustion chambers

COMPARISON OF COMBUSTION INSTABILITIES FOUND IN VARIOUS TYPES OF COMBUSTION CHAMBERS M. BARRI~RE

O~ce National d'Etudes et de Recherches Aerospatiales, Chatillon sous Bagneux, Hauls de Seine, France AND

F. A. WILLIAMS

Departme~t of the Aerospace and Mechanical EngiT~eering Sciences, University of California (San Diego), La Jolla, CaliforT~ia Combustion instabilities in motors and furnaces are contrasted under three separate headings: System instabilities involve, in an essential way, interactions between processes occurring in the combustion chamber and processes occurring in at least one other component of the system, Combustion chamber instabilities consist of unsteady phenomena that are localized within the combustion chamber. Intrinsic instabilities are inherent in the reactants themselves and would be observed if combustion were to occur in the absence of any external influences. The second category, combustion-chamber instabilities, is again divided into three subcategories: Acoustic instabilities involve, in an essential way, the propagation of acoustic waves in the combustion chamber. Shock instabilities are characterized by the presence of steep-fronted finite-amplitude shock or detonation waves in the combustion chamber. Fluiddynamic instabilities are associated with the establishment of special kinds of flow patterns, such as vortices, in the chamber. Experimental and theoretical aspects of each of these types of instabilities are discussed separately. Characteristics of each type of instability are described, and the kinds of chambers in which each has been observed are indicated. Attention is given to surface-burning systems, such as solid-propellant rockets and hybrid rocket motors, and also to volume-burningsystems, such as liquid-propellant rocket engines and both gas-fired and oil-fired industrial burners. Experimental procedures for investigating each type of instability are reviewed along with selected experimental results, and the character and state of development of the theoretical methods that have been used for describing each type of instability are compared. The review is an attempt to bring about discussions between specialists who study combustion instabilities in different systems. Classification according to the type of instability rather than the type of system can emphasize the similarities and differences that exist among combustion instability problems encountered in different systems. One resulting observation is that examples of system instabilities and of acoustic instabilities are common to all kinds of combustion systems. 1. Introduction in the past, combustion instabilities in furnaces ~md engines of different types have traditionally been treated as separate topics. Few studies have endeavored to establish the similarities between the various types of instability phenomena. One indication of this is the scarcity of contacts between the specialists on liquid-propellant systems and the experts on solid-propellant systems. ConVersations between specialists on furnaces and on laropulsion systems are practically nonexistent.

This situation is unfortunate, since many results obtained in any one of the fields could benefit another and could certainly lead to a better understanding of the phenomena in general and of the specific instability problems encountered in each type of combustion system in particular. For this reason, it is desirable that discussions begin between the specialists in the various fields, and a congress such as the present one could initiate the necessary associations. Therefore, in the following sections, we outline a comparison between the various types of insta-

169

170

COMBUSTION INSTABILITY

bilities, from both experimental and theoretical points of view, stressing the principal results already obtained and those that are expected in the future. Let us first establish a scheme of classification for the various phenomena that are encountered. This scheme will, to a certain extent, help to clarify the presentation? Few combustion chambers operate with a homogeneous mixture of fuel and oxidizer; in most of them, the combustion process is heterogeneous. This heterogeneity can arise from the geometry of the condensed phases or from the flow pattern. There are combustion chambers in which the oxidizer and the fuel are both in the same phase (gaseous, liquid or solid) and chambers in which they are in different phases, as in hybrid systems (liquid-gas, solid-gas, etc.). These various combustion chambers can experience instabilities, which we discuss next. One may define three main classes of instabilities: 1. Instabilities that are specific to the chamber (combustion-chamber instabilities). 2. Instabilities involving the entire system (system instabilities). 3. Instabilities involving only the reactants (intrinsic instabilities). 1. The first group includes the most common type of instabilities, characterized by the propagation of acoustic waves in the combustion chamber--viz., "acoustic instabilities." Any such instability is oscillatory and possesses a frequency that is determined by the chamber geometry and by the average velocity of sound in the medium contained in the chamber. Combustion oscillations involving steep-fronted waves, such as propagating shock or detonation waves, can also be observed. We designate them "shock instabilities." The flow can organize itself in such a way that combustion occurs periodically in the chamber. For example, in hybrid systems with certain injector configurations, vortices can appear, inside of which the oxidizer and the fuel require a finite time to react. At the end of the reaction time, a sudden heat release takes place, and a puff of gases propagates along the chamber, destroying the vortex. The vortex reappears quickly, and periodic combustion by puffs is established. We call phenomena of this type "fluid-dynamic instabilities"; they are related to the establishment of certain flow patterns. 2. In the second group, there is an interaction between the processes taking place in the chamber and those occurring elsewhere in the system. Lowfrequency instabilities in liquid-propellant rocket

motors are examples for which there is a coupling between the combustion phenomena, the pro. pellant feed system and also the vehicle structure by means of the thrust fluctuations that arise from pressure fluctuations in the chamber. 3. The third group, which is more difficult to demonstrate, consists of instabilities inherent to the reactants and independent of the chamber properties; they can, for instance, depend upon the combustion kinetics. This simply means that the set of equations representing the combustion kinetics exhibits unstable solutions. This three-level classification scheme, which localizes the phenomena at the level of the reactants, of the chamber, or of the entire system, is not a rigid one, in that different types of instabilities can interfere. For example, the acoustic mode, which for moderate chamber lengths (below 1 meter) involves oscillations that are localized within the chamber, can affect the entire system if the chamber dimensions are increased to a point at which the acoustic frequency reaches a value low enough for conditions in another component of the system to vary appreciably during an oscillation period. This can occur in large propulsion chambers and in industrial furnaces, for example.

2. Experimental Aspects Three attributes are common to representative instability experiments on chambers of all types: 1. A parametric study is made of the phenomenon, determining the dependence of the oscillation frequency and amplitude upon various parameters. 2. A stability domain is defined in terms of a parameter characteristic of the combustion (mixture ratio, pressure) and a parameter characteristic of the oscillation (frequency, period, chamber dimensions). 3. These results enable us to draw conclusions of a fundamental character concerning causes of the phenomenon, as well as conclusions of a practical character concerning sustaining and amplifying mechanisms and means to widen the stability domain and to design damping devices. Measurement techniques have been described elsewhere ~-4 and can be applied to all systems. The measurement devices used in motor chambers, which are smaller than usual industrial furnaces, generally have a much shorter response time. There has been so much progress in developing pressure- and temperature-measurement techniques that the experimentalist presently has few insurmountable problems, either in the labor-

COMPAIHSON OF COMBUSTION INSTABILITIES atory or in processing the information. However, the determination of local conditions is still difficult, and efforts should be made to achieve intprovements in this direction: for example, ill the case of solid propellants, it would be useful to determine accurately the temperature profile in the solid and in the gas close to the burning surface during unstable motor operation. Experimental procedures differ according to the type of instability that is to be investigated. Two general classes of instability that occur are those that arise spontaneously and those that are triggered. The latter class warrants some discussion since it bears on the sensitivity of the system to externally imposed perturbations. When the flow rate into the chamber can be modified, triggered instabilities can be investigated by modulating the flow rate sinusoidally and by analyzing the effects of this modulation on chamber operation. Alternatively, a step in flow rate can be imposed at the chamber entrance, causing perturbations whose time histories can be measured at various positions along the chamber. Another widely used technique is to pulse the chamber with a gas jet, oriented in such a way that it triggers a given type of instability. The additional flow rate, or pulse flow rate, is often characterized by the parameter I=rhlvi/gorht, the ratio of the pulse momentum to the total mass flow rate of the chamber (go is the standard acceleration of gravity). An example of the influence of I on the oscillation amplitude is shown in Fig. 1. The type of instability studied here was an acoustic tangential mode, and the pressure fluctuation i~/i5 was found to be proportional to I. Although pulse techniques have been used in certMn industrial furnaces, their

.2C -- ~ P g

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ALUMINUM

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Fro. 1, Effect of aluminum addition on stability rating (Heidmann and Povinelli, Ref. 5).

171

primary al)plication is for liquid, solid, and hybrid prol)ulsion chambers. 5-r The flow-rate perturbatious cat~ SOluctimes be replaced by pressure perturbations (sinusoidal pressure-wave generators, detonators, etc.). 2.1 Combustion Chamber Instabilities 2.1.1 Acoustic Instabilities Acoustic instability has been encountered in most chambers and appears when, for one or more of the acoustic modes of the chamber, the amount of energy fed into the mode exceeds the amount extracted. A very important factor is the location of the combustion zone, since the mechanism of amplification usually is associated with the combustion process that can vary from one type of combustion chamber to another. In chambers that use gaseous fuels, atomized liquid fuels, or pulverized solid fuels, the amplificatiou sources are distributed nearly uniformly throughout the chamber. Diffusion processes and chemical kinetics must then be taken into account. On the other hand, for solid or hybrid fuels, the amplification sources usually are concentrated close to the propellant surface, within the combustion zone. This distinction enables us to identify as a first approximation, two categories of chambers and to discuss and compare the results obtained with each. The longitudinal mode occurs most easily when the energy release due to combustion takes place in a zone where the pressure variations are large, that is, in a system of standing waves, at a pressure antinode. For this reason, chambers of the first category often experience longitudinal instabilities. This ]node can be very intense in industrial furnaces and in gaseous- or liquidpropellant propulsion chambers, as well as in solid-propellant chambers, provided that the surface combustion zone is located sufficiently close to a pressure antinode. According to Rayleigh's theory, maximum amplification occurs when the pressure fluctuations are in phase with the periodic energy release due to combustion phenomena. Radial modes that require maximum heat release near the chamber axis are seldom observed. The tangential mode, on the other hand, requires heat release close to the walls of a cylindrical chamber, since the corresponding pressure fluctuations exhibit their highest intensity there. Tangential modes have been observed particularly in surface-burning systems. In solid-propellant systems, the combustion takes place close to the propellant surface, at the periphery of the chamber. The heat release takes

172

COMBUSTION INSTABILITY

place under conditions most favorable to tangential-type instabilities. This mode can also be observed in liquid-propellant systems, where the amount of energy released per unit volume is high, and particularly when a combustion zone can develop close to the injector or close to the outer walls of the chamber. Seldom has it been observed in industrial furnaces. The oscillation frequencies observed experimentally are close to the theoretical acoustic frequencies, which can be expressed as functions of the velocity of sound and of the chamber geometry (length, diameter). A frequency spectrum reveals the modes that are most intense. Frequencies corresponding to longitudinal modes have been observed in chambers of all types. In propulsion chambers, the exhaust is generally supersonic, and the fundamental mode corresponds to a pipe closed at both ends. In industrial furnaces, the exhaust is subsonic, and the mode usually corresponds to a pipe closed at one end and open at the other. This mode has been observed in chambers exceeding 30 m in length. The oscillation amplitude/~//5 depends strongly upon the experimentM conditions and can assume values ranging from a few per cent up to levels that cause destruction of the chamber. Correlations between theoretically predicted and experimentally measured amplitudes are easier to obtain when the combustion zone is concentrated close to the surface. 8 Generally speaking, the pressure at any given point in the chamber can be written in the form P = P + P 0 exp(at)

exp(iwt),

(1)

where a is, depending on its sign, the (real) amplification or damping constant of the wave. Since a is additive, we can separate the amplification mechanisms (constant a,) from the damping mechanisms (constant ad). In a chamber of volume 9 with a burning area Ab, the real part of the acoustic admittance Y at the burning surface is Re{ Y} = --

(aa+ad)~/Ab'yp.

(2)

In order to compare theory and experiment, it is preferable to introduce a nondimensional response function y defined as

Re{y}

(~+~a)vP_ Ab'y~

(/~/~b)Re{ Y}+~-~. (3)

The left-hand side of this equation, containing

the coefScie~t a = a ~ + a a as well as the chamber volume ~, the burning area Ab, the average density ~, the ratio of the specific heats % and the average mass flux from the burning surface ~, is determined by experiment. The right-hand side is given by the theory? a A systematic study of y in a T-burner has revealed the influence of various parameters: nature of the propellant, pressure, chamber geometry. The agreement between theory and experiment is good in this case. In another experiment involving surface combustion, the amplitudes of the tangential mode are such that they modify the burning rate, and hence the amount of energy involved. An important parameter is then the one expressing the sensitivity of the burning rate r to a pressure perturbation ~3

N= (~/~)/(p/$). The increase in burning rate is due not only to the pressure fluctuations but also to variation of the gas velocity parallel to the surface (erosive burning). The amplitude of the oscillations, and the parameters that control this amplitude, are more complex to analyze when the combustion is distributed homogeneously or heterogeneously in the chamber, and few valid correlations have been obtained, because of the impossibility of defining a single admittance, the latter depending upon the spatial distribution of the combustion. The definition of stability domains depends upon the nature of the chamber. When the combustion is localized close to the surface, the stability domain is represented in a diagram of pressure (characteristic parameter of combustion) as a function of frequency (characteristic parameter of the fluctuation). The frequency is sometimes replaced by the chamber length for the longitudinal mode or by the diameter of the acoustic cavity for the tangential mode. Such a stability diagram is shown in Fig. 2, which pertains to experiments on a T-burner using a composite propellant. The unstable domain narrows at high and at low pressures. With certain propellants this domain can be closed34 In the unstable domain we show the isoadmittance or isoresponse curves; a point of maximum response usually appears. These domains, as defined on a T-burner, frequently differ from those obtained on a motor using the same propellant; the results obtained on one type of chamber may not be valid for another type, and must be applied cautiously. The longitudinal mode generally appears in a motor when the pressure in the chamber exceeds a critical value which is itself a function of the characteristic length. On the other hand, the tangential mode appears

COMPARISON OF COMBUSTION INSTABILITIES

173

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(s-~ l F~G. 2 I S ~ b i l i ~ y l i m i ~ ( N I E L ~ O N ~ R A

at low pressures and also exhibits a point of maximum response. When the combustion is distributed within the chamber volume (propulsion chambers or industrial furnaces), the stability domain is usually described on a diagram of equivalence ratio (characteristic parameter of combustion) as a function of chamber length or diameter according to the mode (parameter linked to the perturbation frequency). Here, too, one does not find a unique domain valid for all types of chambers, and various parameters such as the nature of the fuel and of the oxidizer and the operating pressure in the chamber can deform the domain. For example, in premixed gaseous propellant systems, two unstable zones appear, one on each side of stoichiometric conditions, when the chamber length exceeds a critical value; as the chamber length increases, the instability domain widens. In liquid propellant systems however, the unstable zone is centered at stoichiometric conditions and occurs for chamber lengths lying between two critical values. It differs for the fundamental mode and harmonics} ~ We mentioned at the beginning of this paragraph the possibility of the acoustic mode being sustained by a fraction of the energy released in

)l

combustion. As a first approximation, it can be assumed that instabilities can develop only if the combustion time is less than the oscillation period. This is a necessary, but not sufficient condition, since certain other factors can cause damping of the oscillation amplitude. When acoustic-type instabilities appear in a chamber, there are usually several means to control them. Damping can be achieved by baffes located inside the chamber. For example, in liquid-propellant motors, the tangential mode can be attenuated by placing radial baffles on the injector plate, thereby preventing rotation of the gases. Another approach is to place nozzles or diaphragms within the chamber, thus modifying the acoustics of the chamber. The geometry of the chamber and of the nozzle can be designed to produce damping. Homogeneous mechanisms such as viscosity, heat conduction, chemical and molecular relaxation can also act as dampers. One of the most efficient solutions consists of using propellants that yield combustion products containing condensed phases such as alumina (Al20~). The stabilizing action of condensed particles has been observed with solid, liquid, and hybrid propellants} ,5 Figure 1 shows an example of the decrease in amplitude due to the presence

174

COMBUSTION INSTABILITY

ill the combustion products of condensed particles. For a given value of the pulse-flow rate that triggers the tangential mode, the amplitude is reduced approximately by a factor of one-half. This effect has been produced with a few per cent of aluminum. 5's'~5 2.1.2 Shock Instabilities "Shock instabilities" have been observed in liquid-propellant motors and sometimes in solidpropellant motors; they appear mainly when the combustion intensity in the chamber, that is the amount of energy released per unit volume, is high. This condition precludes their appearance in industrial chambers. They are characterized by the propagation of steep-fronted pressure waves in the chamber (shock and combustion waves) resembling in certain respects detonation waves. These waves propagate at sonic velocity or slightly above, with observed frequencies close to acoustic frequencies, but these nonlinear phenomena are not as well understood as acoustic instabilities. Longitudinal propagation of these waves has been observed, as well as tangential or spiral propagation. They can develop progressively in a manner similar to detonation waves. They can sometimes be observed after an acoustic mode is established; the acoustic amplitude increases and the sinusoidal waveform progressively steepens. Shock instabilities can also be triggered by shocks or small explosions. Studies of the wave shape show that the wave divides into two elements: a shock wave followed at some distance by combustion. This distance varies during the establishment of steady propagation. The heat release can drive the wave only if proper propellant distribution is achieved, in order to concentrate the combustion behind the wave. This condition is particularly well fulfilled in liquid-propellant systems in which steep-fronted axial and tangential waves have been observed, a,1~,27 In liquid rocket chambers the amount of available energy is sufficiently high to assure rapid combustion behind the wave, even though the propellant may not be entirely vaporized. Conditions are even more favorable with gaseous propellants which can initiate a detonation that does not become fully developed. Solid-propellant chambers can also experience this type of instability if triggered by a shock. In contrast to acoustic instabilities, it is observed that these nonlinear instabilities are more severe for solid propellants with low rates of heat release and at high operating pressures. For a given geometry, a critical pressure can be defined, above which the system is nonlinearly unstable.

2.1.3 Fluid-Dynamic Instabilities This type of instability appears in many combustion systems, notably in solid and hybrid propellant rockets. Apparently, it has never been mentioned for industrial furnaces, although such systems can quite conceivably exhibit instabilities of this type. Fluid-dynamic instabilities are often characterized by the formation of vortices in the combustion chamber. In cylindrical chambers of solid-propellant systems, one or more vortices have been observed to develop from finite-amplitude acoustic tangential modes of instability. A separate vortex is sometimes confined to each star recess, and sometimes the rotation of the gas in the central port of the motor can influence the spin of the rocket vehicle. The most detailed studies of this type of instability in cylindrical motors can be found in Refs. 17, 18, and 19; a cinematographic investigation of the phenomenon was performed on a motor in which the fore-end closure was fitted with a transparent window. The average burning rate and the average pressure in the chamber increase, due to the vortex itself (erosion phenomenon) and also to a vortical blockage of the nozzle throat. Fluid-dynamic instabilities have also been observed in hybrid systems for certain injector configurations. A toroidal vortex is es-

COOL CORE

VORTEX

..

!

I

FIG. 3. Flow pattern in in]ection zone (hybrid rocket motor).

COMPARISON OF COMBUSTION INSTABILITIES tablished close to the injector, as indicated in Fig. 3. This vortex entrains the hot gases arising from combustion, oxidizer from the injector, and fuel gases from the region of the vortex in contact with the fuel surface. Initially this recirculation zone is highly stratified; it gradually becomes more homogeneous and, at a critical stage, expertences a sudden heat release that produces a local pressure rise. This "puff" then propagates downstream. The vortex forms again, and after a cerrain delay, combustion recurs. The period of this relaxation phenomenon is determined by the time required for mixing of fuel and oxidizer within the vortex and by the combustion time inside the core. The puff propagates inside the motor at a subsonic velocity and does not trigger any acoustic mode. This type of instability can be suppressed by modifying the injection system or else by using, close to the injector, a fuel with a low ablation rate so as to avoid providing the vortex with fuel. The conditions which define the onset of this type of instability are still uncertain; stable combustion has been achieved under certain operating conditions in the presence of a vortex. Coupling with the injection system and with the feed system may also occur. The frequencies are generally low, of the order of 20 cps.

2.2 Intrinsic Instabilities Intrinsic instabilities can assume different forms since they are specific to the reactants. The periodic shedding of aluminum from the surface of a burning propellant is an example of intrinsic instability. Homogeneous solid propellants burning in a bomb can conceivably exhibit local instabilities due to an interplay between the heat flux to the solid and the mass flow rate from the burning surface. Heterogeneous solid propellants show periodic local burning due to their heterogeneous structure; these oscillations are rarely coherent and are difficult to observe but may trigger a particular mode such as the acoustic mode or a system instability. An experimental technique used to study this type of instability consists of subjecting the combustion zone to periodic pressure fluctuations in an effort to define conditions under which the combustion intensity increases. This technique has been applied to solid propellants ~~and also to gaseous or liquid propellants3 ~,~ In the case of gaseous mixtures, spontaneous instabilities of the flamefront can appear at a certain frequency, the stability domain being defined by representing the parameter A=k/27rL in abscissa and W = [(Po--pb)/po](2vNe/rb ~ in ordinate. The term

175

A contains the perturbation wavelength ~,, and a characteristic length L that is related to the flame-zone thickness. W is proportional to the motion velocity of a gas element e, subjected to an acceleration of frequency N; p0 and Pb are the gas densities upstream and downstream from the combustion zone, and rb~ is the normal burning rate in the absence of perturbations. Also falling within the present category are instabilities artsing from the chemical kinetics of combustion. The combustion kinetics is defined by a set of chemical equations which in some cases may lead to a periodic formation of certain chemical species.

2.3 System Instabilities System instabilities occur both in propulsion chambers and in industrial furnaces. The fluctuations generally are periodic, and in a system which can experience various types of instabilities they exhibit the lowest frequencies. In analyzing system instabilities one must consider various components of the system and the characteristic times or transfer functions for each component It is again useful to distinguish between surfaceburning systems and systems where combustion takes place throughout the chamber volume. With surface burning, the propellant "injection system" is governed by conditions applied at the surface, such as the transient energy conservation laws, since these conditions determine the rate of gas flow into the chamber. An important parameter is the characteristic length of the chamber (ratio of the chamber volume to the nozzlethroat area), and the stability domain (Fig. 4) is defined in a diagram of the pressure as a function of characteristic length. The unstable zone is located at small values of the characteristic length and at low chamber pressures. The unstable domain widens when L* increases, so long as the nozzle is choked. This tendency reverses itself for subsonic flow. Many investigations have been devoted to the study of this phenomenon. ~-26 For volume-burning chambers, the components of the system, i.e., feed system, injectors, combustion chamber, exhaust of combustion products, play a very important role and modify the stability conditions. Consider, for example, a large industrial furnace (e.g., Cooper furnace) using gaseous reactants; the system usually will be quite tightly coupled acoustically. Then, it is necessary to make an acoustic analysis of the entire system in order to obtain the stability conditions. A practical approach is to measure the stability

176

COMBUSTION INSTABILITY 1

Prassurr 3

'

\'~,.

stable \

t,5

o

4

2

3

4-

~

6

(,.)

FIG. 4:. Stability domain at low frequency ( N I E L - - O N E R A ) . domain, taking as coordinates the lengths of the fuel and oxidizer lines; the instabilities will appear for certain lengths. For industrial furnaces, the gas-exhaust duet plays an important role. The system can behave like a Helmholtz resonator, for which the oscillation frequency r is given by the relation w=a(S/h~) 1/2, where ~d is the chamber volume, S the cross-sectional area of the outlet tube, and h the length of the outlet tube. The natural frequency r can be affected by the iniection system and by the combustion itself? s System instabilities are very common for liquid rocket motors; specialists term them "chugging." The underlying cause of the instability is a delay between iniection and propellant combustion. An analog method for analyzing this type of instability consists of simulating electrically the transfer functions of all systems and all possible couplings. These techniques use methods developed in the field of cybernetics. Methods of cybernetics or systems analysis enable one to define stability conditions for a system and to develop feedback servomechanism devices for stabilizing unstable systems. 3. Theoretical Aspects 3.1 Combustion Chamber Instabilities 3.1.1 Acoustic Instabilities In a fundamental sense, small-amplitude (linear) acoustic instabilities are understood better than any other type of instability. 1,29-zl There are three aspects to theoretical studies of linear acoustic instability: description of the

acoustic field, analysis of damping mechanisms, and analysis of amplification mechanisms. Methods for describing the acoustic field, including such complicating factors as mean flow in the chamber, 3~ rest on firm ground. Theoretical descriptions of a variety of damping mechanisms (nozzle damping, wall friction, wall-heat conduction, homogeneous viscous and heat-conduction effects, homogeneous chemical and molecular relaxation, damping by small solid particles in the gas, etc.) are well developed. Noteworthy among recent achievements is the experimental demonstration of close agreement between theory and experiment in regard to particle damping, for certain ranges of parameters. 32 The area of greatest theoretical ignorance concerns mechanisms for acoustic amplification. In spite of a great den of theoretical work on acoustic amplification by surface and homogeneous combustion processes, t,2s,33a number of unanswered questions remain, primarily because of the complexity of the equations that must be solved in obtaining proper theoretical descriptions of the phenomenon. These remarks apply for combustion chambers of all types. Since acoustic modes and damping mechanisms can be described basically in the same way for all types of chambers, we shall not discuss these topics further but instead shall contrast briefly theoretical studies on amplification mechanisms in chambers of various types. The greatest amount of attention has been devoted to studie~ of amplification by burning surfaces, such s~ occurs in solid-propellant rocket motors. A lesse$ amount of attention has been given to the distributed amplification mechanisms that can occur in liquid-propellant rocket motors. Essentially nO theoretical studies have addressed themselves to

COMPARISON OF COMBUSTION INSTABILITIES analyses of detailed combustion models for acoustic amplification in industrial burners, although some basic experimental information relevant to the excitation of linear and nonlinear acoustic vibrations in gas-fired devices has been developed. ~4 Moreover, some of the results obtained for liquid rockets are, in principle, applicable directly to oil-fired furnaces. The need for developing a theoretical understanding of the mechanisms of acoustic amplification in industrial burners has been recognized~s,35 and has been cited as stemming from increasingly stringent requirements on combustion intensity: Increased combustion intensity, which is needed for minimi~.ation of corrosion, oxide deposition and atmospheric pollution, usually enhances the severity of combustion instability problems in industrial burners. A proper discussion of theories for acoustic amplification mechanisms in surface-burning systems would occupy many pages. The theories have proceeded from simple time-lag concepts, to mechanistic models that treat an infinitesimally thick gas-phase flame standing above a condensed phase, to models that employ the one-dimensional transient equations of aerothermochemistry for describing a planar distributed gas-phase reaction 9 adjacent to a gasifying condensed phase that experiences transient heat conduetionJ An analysis in this last category, first stated clearly by Denison and Baum, 36 that may describe homogeneous solid propellants subjected to lowfrequency ( < 1000 cps) acoustic vibrations, and which has currently been adopted for use in attempting to explain a number of observed solidpropellant instability phenomena, yields a simple explicit formula for the response function y defined in Eq. (3), viz.,

y= (n/2)bk/[k 2- (1-[-a--b)k~-a],

(4)

where

a~ ( Es/R~

(1-- T~/T~),

b-~ 2E~cpTp/EIc~T~ 2,

(5)

and

~-89

(l+4i~poXJ~h~c~).~].

(6)

Here rh is the steady-state mass burning rate, n the pressure exponent of ~h, p~ the density of the condensed phase, X~ the thermal conductivity of the condensed phase, c~ the specific heat of the condensed phase, cp the specific heat of the gas, T~ the initial temperature of the propellant, T~ the average surface temperature, T / t h e average

177

flame temperature, and E8 and E/ are over-all activation energies for gasification and for gasphase combustion, respectively. The only delay process in this theory is heat conduction in the condensed phase; all other processes are treated quasisteadily. A few theories have attempted to take into account effects of structural heterogeneities of the condensed fuels, and some qualitative ideas concerning erosive phenomena (i.e., effects of a nonzero steady or oscillatory component of gas velocity parallel to the surface of the condensed phase) have been developed. However, the roles of these last two phenomena in acoustic amplification are largely unknown, 1 and the study of these topics remains an active area of current research. To develop a theoretical description of acoustic amplification that is distributed throughout the combustion chamber (e.g., amplification in liquidpropellant rocket motors) is more difficult than developing a corresponding description for surface-burning systems. A framework has been givenss within which specific combustion models can be inserted to predict whether amplification will occur. For many years, the combustion models have been restricted severely in one of two ways: Either they have been based on timelag concepts that are necessarily imprecise and are often physically unrevealing, or they have treated all aspects of the vaporization and combustion processes as strictly quasisteady, thereby limiting their applicability to very low frequencies. Recently, theoretical analyses of specific combustion processes have begun to appear in which these two restrictions are removedY-~ These studies develop detailed descriptions of unsteady droplet combustion, unsteady combustion in a stagnation-point flow field and unsteady boundary-layer combustion, all for initially nonpremixed systems. Very little effort has been invested in using the results of these analyses for predicting whether acoustic amplification will occur in specific chambers, and additional work of this kind appears to be warranted. Comparison of theory with experiment is desirable whenever possible; in particular, it is of interest to compare the results of acoustic amplification theories with experimental measurements. For surface-burning systems, a few detailed comparisons have recently been reported. In Ref. 40, the real part of Eq. (4) was compared with the experimental frequency dependence of the response function, obtained from T-burner measurements by use of Eq. (3). The results, shown in Fig. 5, demonstrated that it was possible to choose the values of a and b in such a way that reasonable agreement between theory and experiment was achieved, although this necessitated using rather abnormally large values for the

178

COMBUSTION INSTABILITY 2.4

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2.0

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1.6 0

CURVE: a=37, b=40

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1.2

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0.8

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80

120

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200

240

280

320

Oapc k c / m 2 c c

FIG. 5. Experimental response function from NOTS data on A-13 propellant with theoretical curve through data (Oberg, Ref. 40). surface-activation energies. This marginal success of amplification theories for surface-burning systems has not been emulated for systems with distributed amplification because correspondingly detailed comparisons, at the basic level of aerothermochemical theories, have not been attempted. 3.1.2 Shock Instabilities The troublesome phenomena of shock instabilities are certainly nonlinear and are not understood nearly so well as acoustic instabilities. The coordinate perturbation theory of Lighthill, Chu, and others has been applied 33,41-e to the problem of describing these steep-fronted waves. In spite of its restriction to small-amplitude disturbances, the theory has met with partial success in correlating experimental observations made on a particular gaseous-fueled laboratory burner. ~3 Were the theory shown to be of more general relevance to shock instabilities in surface-burning systems, it would engender a host of new tractable but challenging "admittance" problems (problems of first- and second-order "acoustic" response of burning surfaces) that would have to be solved in order to obtain a complete description of this type of instability. 3.1.3 Fluid-Dynamic Instabilities The stage of development of theories of fluiddynamic instabilities is so primitive that it can

hardly be said to have begun. Complicated nonlinear flow phenomena are involved here and, therefore, to obtain adequate theoretical descriptions of these instabilities is likely to be a formidable task. Since these flows sometimes develop from finite-amplitude acoustic waves, theories of acoustic streaming are sometimes relevant. Preliminary theoretical studies on certain instabilities of the vortex type have been reported. TM 3.2 Intrinsic Instabilities If the equations describing steady-state combustion for any given set of reactants can be solved, then the theoretical approach to the investigation of intrinsic instabilities for the given set of reactants is straightforward in principle. The time-dependent conservation equations are linearized about the steady-state solution, and studies are made of the existence of solutions to these linear equations which diverge from the steady-state solutions exponentially in time. If a divergent solution exists, then the steady-state combustion is intrinsically unstable. Seldom can an entirely accurate solution for the steady-state combustion be obtained. However, it is often possible to construct models for the steady-state combustion that possess tractable solutions. In principle, whenever any such model solution is presented, its intrinsic instability should be investigated. Many investigations of this type have been reported, but often the

COMPARISON OF COMBUSTION INSTABILITIES study of intrinsic instability is so complex algebraically that it is omitted entirely. Relatively few combustion models have been subjected to a detailed study of intrinsic instability. Apparently, theoretical studies on the intrinsic instability of combustion models for motors and furnaces have been reported only for surfaceburning systems. Notable among these is the analysis in Ref. 36, where the intrinsic instability of a particular model for solid-propellant combustion was investigated. It was found that, for certain combinations of values of the parameters a and b defined in Eq. (5), the combustion was intrinsically stable, while for other combinations it was intrinsically unstable. This property of the existence of intrinsic stability only for certain ranges of parameters appears to be shared by many combustion models. If a model is intrinsically unstable, then it cannot properly describe combustion of a set of reactants that burns steadily in the laboratory when isolated from resonant devices or other external influences. Theoretieians, especially those who construct very complicated models, appear too often to overlook the possibility of occurrence of such phenomena. It should be added that, while the linearized methods of analysis that we have been discussing reveal conditions under which intrinsic instabilities can occur, they do not produce descriptions of the combustion process in the unstable domain. A theoretical finding of linear intrinsic instability may imply that combustion ceases, that explosion occurs, or that a periodic mode of combustion is established. When intrinsically unstable burning occurs, it is generally a highly nonlinear phenomenon that can at best be described qualitatively by theories. An example of this is the classical description I of "ehufling" in solid-propellant rocket motors as a periodic thermal explosion phenomenon. 3.3 System Instabilities Theories have been developed for describing system instabilities in engines and furnaces, including both surface-burning and volume-burning systems. Static stability ideas can sometimes be used to discover mechanisms for system instabilities, but considerations of dynamic stability are often required. Most of the dynamic stability theories explicitly introduce a combustion time lag and do not relate this parameter to more basic combustion characteristics. The principal exceptions to this statement are theories of lowfrequency instability in solid-propellant rockets (considered as surface-burning systems). These "L* instability" theories, 44-45 that treat' systems

179

with quasisteady nozzle flow and with spatially uniform chamber pressures, employ combustion models with quasisteady heat release in the gasphase adjacent to the solid, and with timedependent heat conduction in the solid propellant. The time-dependent heat-conduction process plays the same part as the combustion time lag in the less explicit theories. Dynamical theories of system instabilities in volume-burning chambers have apparently received more attention for liquid propellant rocket motors 33'~-4~ (see literature cited in Ref. 47) than for industrial furnaces? s,4s-49 The absenco of recent research on low-frequency instability in liquid-propellant rockets, and the increasingly stringent requirements being placed on combustion intensity in industrial burners, may herald the beginning of a reversal of this situation. Time-lag theories can be useful in spite of the fact that they are less fundamental than theories of the kind that have been used for surfaceburning systems. It is worth remarking that dynamic theories of system instabilities can sometimes be generalized in such a way that they merge continuously into theories of other types of instabilities as one or more of the parameters is varied. For example, the theory for fuel-oil burners, given in Ref. 28, consists of an analysis of system instabilities that is entirely nonacoustic in certain limiting cases but contains acoustical phenomena in others. That a similar transition from system instability to acoustic instability can be embodied in a single theory for solid-propellant rocket motors may be inferred from the recent observations 1,4~176that the basic acoustic amplification mechanism underlying Eq. (4) is in essence identical to the combustion mechanism introduced in theories of L* instability. Experimental evidence that, for solidpropellant rocket motors, L* instability merges continuously into chugging (as the pressure is decreased, for example) suggests ~ that it may be possible to develop a theory for such motors that exhibits the properties of system instabilities and of intrinsic instabilities in different limiting cases. 4. Conclusion In this short presentation, we have attempted to outline the principal results obtained with each type of combustion chamber, in order to classify and to compare them. Similarities among instability problems in various systems occur in areas ranging from questions concerning the underlying theoretical description of the phenomena to questions concerning useful experimental instrumentation for observing the phe15o l n e n a .

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COMBUSTION INSTABILITY

Further comparisons of the techniques and the results seem to us desirable and useful especially in two domains, that of "acoustic instabilities" and that of "system instabilities." This study is a first attempt, certainly very incomplete, to coordinate our knowledge in this field, but it may constitute a basis for further and more exhaustive work.

ACKNO~LEDGMENT

One of us (FAW) wishes to thank the Air Force Office of Scientific Research for support under Project THEMIS, Contract F44620-68-C-0010, for his contributions to the present study.

REFERENCES 1. WILLIAMS, F. A., BARRhRE, M., AND HUANG, N. C. : Fundamental Aspects of Solid Propellant :Rockets, AGAP~Dograph 116, Teehnivision, Ltd., London, 1968. 2. JONES, H. B. AND HAARJE, D. T. : An Integrated Combustion Instability Recording and Analysis Installation, Progr. Astron. Rocketry 2, 175 (1960). 3. REISSIG, G. W. AND ANGELUS, T. A.: Instrumentation for Studies of Unstable Burning Phenomena, Progr. Astron. Rocketry 2, 201

(:[960). 4. BARR~iRE,M. AND CORBEAU, J. : Les instabilit6s de combustion darts les fusdes ~ propergol liquide, pp. 637-685, AGARD Colloquim, Pergamon, 1963. 5. I-IEIDMANN, M. F. AND POVINELLI, L. A.: An Experiment on Particulate Damping in a Two Dimensional Hydrogen Oxygen Combustion, NASA Technical Memorandum, NASA TMX 52359. 6. PICKFORD, R. S., PEOPLES, R. C., AND KRIEG, H. G.: Analytical and Experimental Scaling of Thrust Chambers, 15th Annual Meeting, ARS Preprint, 1436-60. 7. DICKINSON,L. A.: ARS J. 32, 643 (1962). 8. PRICE, E. W.: Review of the Combustion Instability Characteristics of Solid Propellants, 25th Meeting of the AGARD Combustion and Propulsion Panel, San Diego, California, 22-24 April 1965. 9. COATES, R. L., HORTON, M. D., AND RYAN, N. W.: AIAA J. 2, 1119 (1964). 10. CROCC0, L., GREY, J., AND ItAARJE, D. T.: ARS J. 30, 159 (1960). 11. CROCCO, L., IIAARJE, D. T., AND REARDON, F. H.: Transverse Comt)ustion Instability in

Liquid Propellant Rockets Motors, ARS Preprint, 1491-60, 1960. 12. OSBORN,J. R. ANDBONNELL,J. M.: Jet Propulsion 31, 482 (1961). 13. POVINELLI, L. A. AND HEIDMANN, M. F.: ExperimentM Investigation of Transverse-Mode Solid-Propellant Combustion Instability in a Vortex Burner, NASA TND-3708. 14. IRIRICU, M. M.: Stable and Unstable Regimes in Solid Propellant Combustion, 17th Congress of the International Astronautical Federation, Madrid, 1966. 15. I)ICKINSON, L. A. AND JACKSON, F.: Combustion in Solid Propellant Rocket Engines, pp. 531550, 5th AGARD Colloquium, High Temperature Phenomena, Macmillan, I963. 16. BERMAN, K. AND CHENEY, S. H.: ARS J. 28, 89 (1953). 17. GREEN, L., JR.: Jet Propulsion 28, 483 (1958). 18. SWITHENBANK, J. AND SOTTER, G.: AIAA J. 2, 1297 (1964). 19. SWITHENBANK, J. AND SOTTER, G.: AIAA J. 1, 1682 (1963). 20. NADAUD, L. AND GICQUEL, M.: Influence des ondes sonores sur la vitesse de combustion des propergols solides, La Recherche Adronautique, No. 88, p. 59, 1962. 21. LEYER, J. C.: Rev. Gen. Therm. (Septembre 1967). 22. SIJMMERFIELD,M.: ARS J. 21, (1951). 23. CRocco, L.: ARS J. 21, (1952). 24. MARBLE, F. E. AND COX, D. W.: AI~S J. 23, (1953). 25. BECI~STZAD,M. W. AND PRICE, E. W.: AIAh J. 5, 1989 (1967). 26. PRICE, E. W.: Tenth Symposium (International) on Combustion, p. 1067, The Combustion Institute, 1965. 27. LEVINE, I{. S. : Tenth Symposium (International) on Combustion, p. 1083, The Combustion Institute, 1965. 28. Mxvss, F., PERTItUIS, E., AND SALE, B.: Tenth Symposium (International) on Combustion, p. 1241, The Combustion Institute, 1965. 29. BLOKHINTSEV, D. I.: Acoustics of a Nonhomogeneous Moving Medium (translation), Chap. V, p. 1399 NACA Tech. Memo. (1956). 30. McCLURE, E. T., HART, R. W., AND CANTRELL~ R. H.: AIAA J. 1, 586 (1963). 31. WEINER, S. D. AND SMITH, P. W., JR.: The Acoustics of T-Burners, Bolt Beranek and Newman, Inc., Rept. No. 1077, Nov. 1963. 32. TEMK1N, S. AND DOBBINS, R. A.: J. Acoust, Soc. Am. 1016 (1966). 33. CRocco, L.: Tenth Symposium (International) on Combustion, p. 1101, The Combustion Institute, 1965. 34. TOONG, T. Y., SALANT, P~. F., STOPFORD,J. M., AND ANDERSON, C. Y.: Tenth Symposium

COMPARISON OF COMBUSTION INSTABILITIES (International) on Combustion, p. 1301, The Combustion Institute, 1965. 35. BlZOWN, A. ~/[. AND THRINO, ~{. W.: Tenth

Symposium (International) on Combustion, p. 1203, The Combustion Institute, 1965. 36. DENNISON, R. AND BAUM, E.: ARS J. 31, 1112 (1961). 37. STI1AHLE, W. C.: Tenth Symposium

(Inter-

national) on Combustion, p. 1315, The Combustion Institute, 1965. 38. WILLIAMS,F. A.: AIAA J. 3, 2112 (1965). 39. STRAHLE, W. C. : Eleventh Symposium (Inter-

national) on Combustion, p. 747, The Combustion Institute, 1967. 40. OB~no, C. L.: AIAA J. 6, 267 (1968). 41. SIRIGNANO,W. A. AND CROCCO, L.: AIAA J. 2, 7 (1964). 42. CRoceo, L., unpublished notes on Nonlinear Transverse Instability in a thin Annular Chamber, Princeton Univ., 1967.

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43. BOWMAN, C. T., GLASSMAN, I., AND CROCCO,

L.: AIAA J. 2, 1981 (1965). 44. SEHaAL, R. AND STRAND, L.: AIAA J. 2, 696 (1964). 45. BECKSTEAD, ~I. W., RYAN, N. W., AND BAER,

A. D.: AIAA J. J, 1622 (1966). 46. TSlEN, H. S.: Engineering Cybernetics, p. 94, McGraw-Hill, 1954. 47. Caoeco, L. AND CHENa, S. I.: Theory of Combustion Instability in Liquid Propellant Rocket Motors, AGARDograph 8, Butterworths, 1956. 48. FalTSeH, W. H.: Oelfeuer-Technik, Stuttgart, 1958. 49. THnlNa, M. W.: Seventh Symposium (International) on Combustion, p. 458, Butterworths, 1959. 50. COATES, R. L., CO~EN, N. S., AND HAaVlLL, L. R.: AIAA J. 5, 1097 (1967). 51. MARKSTEIN, G. H. AND SQUIRE, W.: J. Acoust.

Soc. Amer. 27, 416 (1955).

COMMENTS P. D. McCormack, Dartmouth College. I would appreciate a qualitative explanation of how a vortex burns. How does the oxidizer diffuse into the fuel held in the vortex? Your methods of suppression of the instability must be determined by this. I might point out that in gas and liquid rockets, pressure fluctuations at the injectors form local vortices in the jets. There probably is no connection, however. M. Barrere and F. A. Williams. In the present experiment two vortices are formed in the vicinity of the oxidizer injector; they contain gasified oxidizer. Erosion and hypergolic surface reactions gasify the adjacent solid fuel and a fraction of this fuel penetrates the oxidizer vortices by turbulent diffusion. The fuel and the oxidizer mix within these vortices and react after a certain lapse of time. This local combustion causes a rapid local-pressure increase and sends off a puff of gases that flows along the combustion chamber.

These instabilities can be eliminated first by using, close to the injector end, a slowly regressing solid fuel, which prevents large amounts of fuel from penetrating within the vortices and limits the intensity of the subsequent combustion and puff emission. A second method consists in organizing the flow in order to prevent the formation of vortices. I t is well possible that such vortices can form close to the injector end of gaseous or liquid propellant rockets and that intense localized combustion zones appear and disappear but, considering the number of injection orifices, these zones may have a random distribution in time and space. The resulting over-all pressure fluctuation is then small. In the case of hybrid systems, and especially in the case of the present experiment, there were two injection orifices and this configuration favors the onset of this instability type.