On the oscillatory behavior of laminar spray diffusion flames: experiment and theory

On the Oscillatory Behavior of Laminar Spray Diffusion Flames: Experiment and Theory B. GOLOVANEVSKY, Y. LEVY, J. B. GREENBERG*

Faculty of Aerospace Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israel

M. MATALON

McCormick School of Engineering and Applied Science, Northwestern University, Evanston, Illinois 60208, USA In this study an experimental, well-controlled parametric investigation of the behavior of an oscillating Burke-Schumann type spray diffusion flame is described. It is demonstrated that these unique spray-related oscillations occur at frequencies in the range 1–5 Hz and it is established that their genesis is due to a heat and mass transfer mechanism resulting from the presence of the droplets in the system. A complementary stability analysis of a one-dimensional model problem, containing the essential features of the experimental conditions, is performed. It is found that the correct order of magnitude of oscillations is predicted by the analysis, confirming a posteriori that the model is a satisfactory paradigm for examining the observed phenomenon. © 1998 by The Combustion Institute

INTRODUCTION The classical Burke-Schumann diffusion flame configuration [1] has served as a convenient platform for studying the behavior of coflowing laminar combustion. On the purely theoretical side, research has involved attempts to relax the simplifying assumptions of the original analysis to try to provide a more realistic description of the physics. On the experimental side, research has been performed both to examine the detailed hydrodynamical/chemical structures of these flames and for the purpose of comparison with the predictions of theoretical/numerical models. However, the aforementioned works all considered only gaseous fuel and oxidizer. In many practical settings, diffusion flames are formed when the fuel is introduced as a spray of droplets. A confined laminar spray diffusion flame with iso-octane as the fuel was investigated experimentally and theoretically by Moore and Moore [2]. A fixed fraction of the fuel vapor was assumed to diffuse, and the liquid droplets were taken to be moving with the local host gas velocity. The flame was assumed to be located at the surface where the flow had a fixed equivalence ratio. Computed results were obtained using the approach of Patankar and Spalding [3] and agreement was generated by altering the *Corresponding author. COMBUSTION AND FLAME 117:373–383 (1999) © 1998 by The Combustion Institute Published by Elsevier Science Inc.

fixed fuel fraction parameter. Although the flame was supposed to be laminar, a closer examination reveals that it was probably transitional. In addition, the large diameter of the base part of the flame implies that large amounts of oxygen were entrained into the core so that the diffusional nature of the flame is questionable. Kim and Sichel [4] presented an analytical study of a liquid jet coflowing with oxidizing gas. In their model the liquid surface maintains the same level even though evaporation occurs. In addition, the effect of a spray droplet size distribution is not accounted for. Their results appear to be in qualitative agreement with experimental evidence. Greenberg and coworkers [5–9] have studied both theoretically and numerically a series of Burke-Schumann type spray flames for both quasi-monodisperse and polydisperse sprays and have clearly demonstrated the manner in which both spray and host– carrier gas properties can individually and/or jointly influence the characteristics of these flames. Motivated by some of these theoretical predictions, Levy and Bulzan [10] carried out a series of experiments on spray flames using the coflow configuration with fuel droplets supplied in an inner stream while air flowed in the outer stream, both at approximately the same velocity. They discovered that under certain operating conditions a unique mode of flame oscillations occurs, that had never been observed previously. Moreover, this 0010-2180/99/$–see front matter PII S0010-2180(98)00073-X

374 phenomenon is completely different from the flame flicker phenomenon that is well known from experimental and theoretical research on gas flames [11–13]. The latter occurs at the upper part of the diffusion flame at frequencies in the range of 15–20 Hz and is primarily associated with buoyancy effects. The spray flame oscillations, however, were found to have a frequency of about 1–5 Hz; Levy and Bulzan [10] gave a preliminary qualitative description of the process, including arguments based on the presence of buoyancy forces. Mawid and Aggarwal [14] attempted to simulate this experimentally observed spray flame phenomenon using the KIVA-II code but were unsuccessful in reproducing the flame oscillations— only a stable laminar spray diffusion flame was obtained. They attributed this failure to the neglect of buoyancy forces in their model. Later work by Park et al. [15] did include the effect of gravity in a numerical simulation of an unsteady spray diffusion flame. Although results indicated an interaction between the spray and the vortical structures usually associated with flame flicker, flame oscillations of the aforementioned frequency range were not reported. Further experimental research on Burke-Schumann spray diffusion flames was performed by Chen and Gomez [16, 17]. They considered monodisperse spray flames (the sprays were produced using an electrostatic technique). The flames they obtained were completely stable, presumably because of the large velocity difference between the droplet fuel stream and the oxidant stream (the ratio of the former to the latter was of the order of 5). In this paper an expanded parametric experimental and theoretical study of the spray flame oscillations is presented in an attempt to pinpoint the underlying prime mechanism responsible for them. We begin with a brief description of the experimental apparatus and measuring procedures. Experimental data and conclusions are then given. A simplified one-dimensional spray diffusion flame model problem capturing the essential physical features of the experimental setup is then formulated. A linear stability analysis about steady-state solutions produces flame oscillations in a given range of mass flow rates through the flame, with a frequency obtained as an eigenvalue of a system of equa-

B. GOLOVANEVSKY ET AL.

Fig. 1. Schematic drawing of combustor.

tions. Agreement between experimentally measured and theoretically predicted frequencies is demonstrated. The role of the droplets on the onset of the flame oscillations is discussed. EXPERIMENTAL APPARATUS AND PROCEDURES The combustor used in the experiments is illustrated in Fig. 1, and is similar to the one described by Levy and Bulzan [10]. It consisted of two coaxial tubes. The outer quartz tube had an inner diameter of 71 mm with a wall thickness of 2 mm, while the inner stainless steel tube had an inner diameter of 12.90 mm and an outer diameter of 14.80 mm. The height of the outer tube was 30 cm above the nozzle. The combustion air entered through four radial holes and passed through a series of fine mesh screens and flow straighteners. The upstream distance from the screens to the combustor (to the tip of the

SPRAY FLAME OSCILLATIONS inner tube) was about 25 cm. The total length of the inner tube (from the atomizer to the tip) was 50 cm. For later reference, this particular configuration will be termed the standard configuration. Heptane fuel was introduced into the combustor through the inner tube in the form of a fine spray of droplets. A slightly modified Sonotek ultrasonic nozzle (# 8700-60 MS) was used to generate the droplets. The droplets were injected upwards to form a narrow spray and were carried upstream by a nitrogen gas stream. The nitrogen was injected through a tube which surrounded the “horn” of the ultrasonic atomizer. As the droplets traveled upwards through the inner tube they partially evaporated and interacted with each other, causing changes in their initial size and momentum. The flow of the nitrogen carrier gas was adjusted, with the aid of a flowmeter mounted upstream, so that its velocity was equal to that of the air at the entrance to the combustor. This created conditions similar to that inherent in the analysis of the model of Burke and Schumann [1] for gaseous diffusion flames. However, some flow differences in the two streams existed due to the parabolic nature of their velocity profiles. A 110-Watt electric heater was attached to the outside of the inner tube, just above the atomizer. The heater was activated to control the amount of prevaporization of the fuel droplets before their arrival at the entrance to the combustor. Thus, the flow in the inner tube contained a homogeneous mixture of nitrogen gas and fuel vapor and droplets. Henceforth, this stream will be referred to as the fuel stream. Experimental measurements were obtained using two techniques. A phase doppler anemometer (PDA) simultaneously measured the size and velocity of individual droplets. Largescale observations of the two-phase flow and the flame dynamics employed a CCD camera coupled with laser light-sheet illumination. A schematic illustration of the measurement system, incorporating the PDA components, is shown in Fig. 2. During the course of the experiments the effect of geometrical changes to the apparatus was examined. In the top left-hand corner of Fig. 2 variations of the outer tube are sketched. The properties of the droplets were measured

375 at the exit of the inner tube. For average nitrogen and air velocities of about 50 mm/sec at the combustor entrance a typical droplet size distribution at the center of the inner tube is shown in Fig. 3a. Figure 3b illustrates the total velocity histogram, while Fig. 3c shows the size–velocity correlation. It can be seen that the velocity histogram has almost a Gaussian distribution which extends to small negative values. As can be deduced from Fig. 3c, the latter negative velocity values are associated with the larger droplets that are apparently too heavy to be swept upwards by the carrier nitrogen gas. The small droplets, having diameters of less than about 15 mm, were found to be appropriate for representing the velocity of their carrier gas. Experimental uncertainty in the measurements is estimated to be 65% for the droplet sizes and 60.5% for their velocities. These values are based on calibration tests using a vibrating orifice type of droplet generator and a rotating fiber at known velocities. The flame oscillations that were observed occurred at frequencies that were recovered using a fast Fourier transform (FFT) routine over measured instantaneous gas velocities. The latter, represented by the velocities of small droplets in the size range 2–15 mm, were recorded as a function of time over more than 100 flame oscillation cycles by the PDA system in the combustion zone. EXPERIMENTAL RESULTS Levy and Bulzan [10] have already noted that the oscillatory flame dynamics that occur in this Burke-Schumann type of spray flame are unique inasmuch as the entire volume of the flame region behaves in an expansion– contraction fashion. A sequence of stills from an interlaced video recording (50 fields per second) of an oscillating spray flame is displayed in Fig. 4. This set of pictures is intended to qualitatively convey the manner in which the spray flame grows and collapses during a typical cycle, and the essentially longitudinal nature of the oscillations. The length of the flame from its visible base to tip varies from between about 5 to 100 mm. The oscillating velocities (from which the frequency of oscillations was deduced) were

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Fig. 2. Schematic description of measurement system.

measured along the centerline in the fuel evaporation region between the tip of the fuel nozzle and the homogeneous flame reaction zone. Results are summarized in Figs. 5a and b. In Fig. 5a the variation of the mean vertical velocity and its peak-to-peak values are given as a function of height. Figure 5b displays the recorded frequency of oscillation along the centerline. It is readily seen that both the average and the peak-to-peak velocities increase with height. The average velocity increases by a factor of about 10, whereas the peak-to-peak values are augmented from zero to 0.5 m/s within a height of 15 mm. The acceleration of the flow is mainly due to heat addition from the reaction zone and the associated expansion of reaction products. However, in this region the frequency of oscillation remains almost constant

at a value just below 4 Hz. The importance of these results is that they strongly indicate that the oscillatory motion is not induced by the fuel or air supply lines. Rather, the oscillations are self-induced downstream of the fuel nozzle, in which region they build up and grow in amplitude. In addition, the fact that the frequency remains unchanged implies that the whole of the flame region is oscillating. (It should be noted that the absence of frequency values in Fig. 5b at heights above 12 mm is due to insufficient data.) The effect of the geometry on the oscillating behavior was analyzed by making geometrical changes to the apparatus. A set of outer tubes having inner diameters ranging from 40 to 71 mm were employed, but the flow velocities were maintained constant. Hence, the ratio of the

SPRAY FLAME OSCILLATIONS

Fig. 3. Droplet characteristics at exit of inner (fuel supply) tube/entrance to the combustor: (a) size distribution, (b) velocity histogram, (c) size–velocity correlation.

377 diameters of the outer to inner tubes was varied between 5.5 to 3.2 with an attendant change in the fuel to air ratio by a factor of three. However, the operating conditions all involved overall ratios below stoichiometric. Additionally, the height of the combustor was altered from 1 m down to zero (no outer wall) for the outer tubes having an inner diameter of 71 mm. The frequency of the flame oscillations was found to be completely insensitive to all the aforementioned geometrical variations. This result clearly eliminates the possibility that the oscillations are induced by acoustic coupling. The effect of the average flow velocity on the frequency of oscillation is demonstrated in Fig. 6 for the standard configuration. With the exception of the low velocity point (V 5 30 mm/s), it is observed that the frequency of oscillations decreases monotonically as the velocity increases. The measurement was repeated 10 times for each point, each of which measurement appears in the figure. The high precision of repeatability, which is clear from Fig. 6, indicates the permanence of the phenomenon. It is interesting to note that the flame oscillation exists within a quite specific large velocity range (about 30 to 120 mm/s) for the operating conditions considered here. However, it is remarkable that the range of values of the oscillating frequency lies within a relatively narrow range of about 2–5 Hz. At velocities outside the aforementioned range, both on the low and high sides, stable nonoscillating flames were established. It was clearly observed that in the low velocity regime only fuel vapor was transported to the combustion zone, thus rendering the situation identical to that of the familiar stable gaseous BurkeSchumann diffusion flame. In the high-velocity regime conditions were no longer laminar at the fuel inlet, and the flames that were established were located between the inner tube and its droplet flow with the latter functioning as a bluff body stabilizer. Clearly, these latter flames have a completely different nature to the oscillatory ones. The responsibility of the fuel droplets for the oscillatory flame behavior was demonstrated in the following manner. By activating the electric heater (see Fig. 1) the inner tube was preheated. The longer the tube was heated the greater

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Fig. 4. Oscillatory behavior of a laminar spray diffusion flame—sequence of stills from interlaced video recording (50 fields per second).

became the relative mass of fuel vapor to liquid fuel, supplied to the combustor at the inner tube exit. Under all circumstances the fuel flow rate was maintained constant. It took about 40 minutes from the time the heater was turned on until all the fuel exuding from the inner tube was in the vapor phase. It was found that the frequency of flame oscillation decreases monotonically as the amount of liquid fuel decreases, and that a completely stable nonoscillating flame was established when all the fuel was supplied as vapor. However, once again, the range of measured nonzero frequencies (obtained when fuel droplets were present at the inner tube’s exit) lay within a rather narrow bandwidth of about 1–3.5 Hz. These measurements provide evidence that the cause of the flame oscillations is rooted in the presence of the fuel droplets in the system. Moreover, more compelling evidence of the role of the droplets is afforded by a series of experiments with butane gas and nitrogen supplied in the fuel stream. A completely stable gaseous diffusion flame was established. However, when droplets of water were injected into the aforementioned fuel stream, flame oscillations having a frequency of 3 Hz were recorded. The phenomenon repeated itself for velocities in the range of 50 to 100 mm/s. These experimental findings clearly point to the dominant role of heat and

Fig. 5. Variation of flow properties along centerline between fuel nozzle and flame core: (a) average centerline velocity and peak-to-peak velocity values, (b) frequency of flame oscillations.

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Fig. 6. Variation of frequency of oscillation of spray flames with feed velocity; in the low-velocity regime the feed stream contains only fuel vapor while in the high-velocity regime turbulent flow conditions exist.

mass transfer in these spray flame instabilities, so that when modeling such flames it is imperative to account for the latent heat of vaporization—an important factor that has sometimes been neglected [2]. THEORY The two major features of the oscillating spray diffusion flames disclosed by these controlled experiments are (i) the narrow range of frequencies of oscillation, between about 1–5 Hz, and (ii) the crucial function of the heat and mass transfer resulting from the presence of the droplets in the fuel stream. In order to attempt to grasp the essential physics of the system and to reproduce, at least in some general sense, its oscillating nature, it seems reasonable to consider a geometrically simpler model that contains the main components of the experiments. We examine an idealized one-dimensional combustion chamber, shown in Fig. 7. A stream containing fuel droplets and vapor is fed at a constant rate, m ˙ , from the bottom of the chamber ( ˆx 5 0), that permits the passage of the feed stream only, and oxidizer is supplied from the top ( ˆx 5 L). The combustion products are transported away by a sufficiently fast flowing stream across the exit, from which the oxidizer diffuses inwards. The chamber is considered to be sufficiently narrow so that the flow field

Fig. 7. Schematic diagram of idealized combustion chamber used in theoretical study.

remains uniform throughout the chamber and variations of the physical properties occur only along its length. The walls are taken to be noncatalytic and adiabatic. Under appropriate conditions a diffusion flame will be established within the combustion chamber, fueled by the vapor and the oxygen. This model contains the main characteristics of the real system and is sufficient for testing whether the experimentally conclusive evidence can be theoretically substantiated. Indeed, a close look at a video movie of the oscillating flames (Levy and Bulzan [10], Golovanevsky [18]) shows that such a onedimensional situation with oscillations normal to the flame provides a fairly realistic replica of the spray flame (see also Fig. 4). The spray is considered to be dilute, so that its volume fraction is relatively small, and quasimonodisperse, i.e. containing droplets of approximately the same size. Following the experimental situation we ensure that the droplets in the spray completely evaporate upstream of the flame front. The homogeneous chemical reaction that occurs between the fuel vapor and the oxygen is modeled by a global one-step mechanism of the form

n FF 1 n O 3 P

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where F stands for the fuel, O for the oxidant, and P for the products, and where n F , n O are the corresponding stoichiometric coefficients. The reaction rate is described by the usual Arrhenius type law with activation energy E and a preexponential factor B. The conservation equations relevant to our analysis are written below using the following dimensionless variables (the accent ˆ denotes the dimensional quantities):

S

D

l ˆx m ˙ ,M5 , t 5 ˆt 2 , x 5 r 0c pL L l /c pL r5

ˆ ˆr T ,T5ˆ , rˆ 0 Tc

ˆF ˆ Y Y w ˆj ˆO 5 O YF 5 ˆ , wj 5 ˆ , Y ˆO YF YF Y c

u5

c

c

E Lv ,b5 , ˆ Q/ n FW F RT c

ˆO , T ˆ c are ˆF , Y where the characteristic values Y c c given by: ˆF 5 Y ˆF 1 w ˆO 5 Y ˆ 0, Y c 0 c

n OW O ˆ , Y n FW F Fc

ˆ F n FW F ˆ c 5 ~Q/c p!Y T c The subscript “0” denotes the value of a given property in the feed stream, at ˆx 5 0. The fluid density, which is approximately equal to the gas density (the mass of gas per unit volume of space available to the gas) is denoted by rˆ. The total (gas and condensed phase) mass-flow rate per unit area is m ˙ , and l and c p are the thermal conductivity and specific heat of the whole mixture. The mass fraction of the fuel vapor is ˆ F , of the fuel droplets by w denoted by Y ˆ , and of ˆ O . The molecular masses of the the oxidizer by Y fuel and oxidizer are W F and W O , respectively. ˆ; The temperature of the mixture is denoted by T R is the gas constant, Q the total chemical heat release, and L v the latent heat of vaporization. In order to simplify matters we restrict attention to a constant density model. The governing equations are then

r

­T ­ 2T ­T 1M 2 5 r 2DY FY Oe 2u/T 2 b F ­t ­x ­ x2 (1)

r

­Y F ­Y F ­ 2Y F 1M 2 L 21 F ­t ­x ­ x2 5 2r 2DY FY Oe 2u/T 1 F

r

r

(2)

­Y O ­Y O ­ 2Y O 1M 2 L 21 O ­t ­x ­ x2 5 2r 2DY FY Oe 2u/T

(3)

­f ­f 1M 5 2C r F ­t ­x

(4)

Equation 1 is the energy equation. Equations 2 to 4 describe the conservation of fuel vapor, oxygen, and liquid fuel, respectively. The parameters L F 5 l / r c pD F, L O 5 l / r c pD O, D5

S D

ˆ O c r 2L 2 n FY p 0 c B WO l

are, respectively, the Lewis numbers representing the ratio of thermal to molecular diffusivities (D F , D O denoting the fuel and oxidant molecular diffusivities) and the Damko ¨hler number representing the ratio of the flow to chemical reaction times. The source term F is given by: F 5 rCf

(5)

The coefficient C reflects the fuel’s volatility and/or the size of droplets. Typically, it depends on the temperature and is of the form C , 1 1 b~T 2 T*! a

(6)

where T* is a reference temperature and a and b are constants. The boundary conditions for the problem are at x 5 0: MY F 2

­Y F 5 M~1 2 a ! ­x

MY O 2

­Y O 5 0, ­x

­T MT 2 5 MT 0 ­x

w5a where

(7)

SPRAY FLAME OSCILLATIONS

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ˆF 1 w a5w ˆ 0/~Y ˆ 0! 0 is the droplet loading parameter, representing the fraction of liquid fuel contained in the droplets from the total fuel in the feed stream. The uniform conditions at x 5 1 are Y F 5 0, Y O 5 Y O1, T 5 T 1

(8)

For the gas phase the conditions at x 5 0 result from material and energy balances at the membrane. The mass fractional flux of liquid fuel is also specified at x 5 0. The conditions at x 5 1 are specified temperature and concentrations. For future reference, a measure of the combustion inefficiency is given by the fraction of the entering fuel that escapes unburned from the chamber. The parameter h

U

1 ­Y F h52 M ­x

,

(9)

x51

which represents the fractional mass flux of the fuel vapor that remains unburned, takes on values between zero and unity. LINEAR STABILITY ANALYSIS AND RESULTS Steady-state solutions are constructed numerically as described in [19]. In general it is found that burning states, with h Þ 1, exist for a restricted range of the mass flux M. Of two possible steady states the one corresponding to the larger value of h is unstable and therefore physically unattainable. The other one is shown in Fig. 8. Note that the range of M for which steady burning is possible depends on the physicochemical parameters and on the droplet loading parameter a. By increasing, or decreasing, M past, or below, its upper/lower limit the flame will be extinguished. The extinction that occurs at the higher value of M results from incomplete combustion because the mixing time is too short compared with the relatively high rate at which fuel is supplied. The extinction limit that corresponds to the lower value of M results because fuel is supplied at a smaller rate than the diffusion time so that the reactants are so dilute that heat cannot be generated at the appropriate rate to sustain steady combustion. Indeed, it has been shown [19] that by increas-

Fig. 8. Combustion inefficiency vs fuel mass flux.

ing M the diffusion flame moves further downstream. Thus, the low velocity extinction occurs when the flame is essentially attached to the porous plug at x 5 0. In Fig. 8 we illustrate typical curves showing the parameter h as a function of the total mass flux, M. The two curves correspond to the two cases of purely gaseous flame (with no liquid fuel present in the incoming flow) and spray flame with 80% droplet loading (a 5 0.8). The addition of droplets to the system reduces the “flammability” range considerably (the range of allowable values of M) as well as the burning intensity (testified to by the fact that the curve for a Þ 0 lies entirely above the curve corresponding to a 5 0). We now subject the aforementioned burning states to an arbitrary small disturbance. To this end any dependent variable, C, is written as C~ x, t! 5 C s~ x! 1 C9~ x!e Vt

(10)

where Cs ( x) is the steady state and the perturbation is assumed to depend exponentially on time with C9( x) satisfying iC9i ,, iCs i; here i z i denotes an appropriately defined norm of the specified function. Substituting in the governing equations and boundary conditions, and linearizing about the steady state leads to a set of linear ordinary differential equations and boundary conditions for the perturbed functions in which V appears as an eigenvalue. The eigenvalue V is, in general, a complex number.

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Fig. 9. Computed eigenvalues in complex plane showing shift (left to right) from stable to unstable solutions via neutral stability of steady states corresponding to the case a 5 0.8 of Fig. 8. Key: Marked points correspond to the following values of M: 1 M 5 0.4125; 3 M 5 0.4150; o M 5 0.4178.

A steady state is stable to small disturbances if the real part of V is always negative; for instability the real part of V must be positive. In order to identify the conditions for the onset of an instability, it is sufficient to seek solutions for which the real part of V is zero. If, at the instability threshold, the imaginary part of V is nonzero the unstable states that emerge beyond criticality have an oscillatory behavior. In the present study the eigenvalue problem was solved numerically by recasting the equations in finite difference form and determining the eigenvalues of the coefficient matrix, P, of the resulting system of algebraic equations. This system assumes the form ~P 2 VI!C9 5 0

(11)

where I is the unit matrix and C9 is now the vector of the perturbed dependent variable functions, at the finite difference mesh points. The eigenvalues were computed using a standard routine DVGLRG from the IMSL library of scientific subroutines. The coefficients of the linearized stability problem, and, hence, the elements of the matrix P, contain information about the steady states illustrated in Fig. 8. Thus, for a given state the eigenvalue system (11) is well defined, and is used to test the stability of this state. The system (11) leads to a sequence of eigenvalues (as large as the number of mesh points). Clearly the eigenvalue with the largest (in magnitude) real part characterizes the stability/instability of that state. Figure 9 shows the characteristic eigen-

value, corresponding to burning states for a 5 0.8 illustrated in Fig. 8, in the complex V plane; the arrows indicate the direction of increasing M for the range 0.4125 , M , 0.4178. The solutions on the solid parts of the curve of Fig. 8 are stable to small disturbances as is evident from the eigenvalues being all in the left halfplane of Fig. 9. The nonzero imaginary part suggests that arbitrary small disturbances superimposed on a steady state exhibit damped oscillations in approaching the steady solution. Similarly, our results indicate that the whole curve in Fig. 8 corresponding to a 5 0 is stable, Re(V) , 0. For points on the dashed part of the curve with a 5 0.8 in Fig. 8, there is at least one eigenvalue with a positive real part; these states are therefore unstable. The onset of instability occurs at M approximately equal to 0.4150 and the imaginary part of V there is nearly 3.28. Although disturbances for M beyond criticality perform growing oscillations, nonlinearities are likely to limit the growth giving rise to purely oscillatory states at least in the vicinity of the stability threshold. The frequency of oscillations is therefore typified by the value 3.28. In order to relate our findings to the experimental conditions we must first identify the length scale L. An appropriate estimate is the width of the flame, proportional to the inner tube’s radius, representing the diffusion length. With L 5 6.45 mm, a mass diffusivity D F 5 105 mm2/sec corresponding to relevant binary diffusion coefficient at about 1000 K [20], and representative values of L F 5 L O 5 1.32 a frequency of oscillations for the spray flame of our model is found to be 1.7 Hz. Further calculations for different droplet loadings revealed similar results with frequencies of the same order of magnitude, in the same range as the measured frequencies. Note that, for the same mass flux, the steady solution for the purely gaseous diffusion flame is absolutely stable while the spray flame may be oscillatory. This, too, is in keeping with the experimental findings. CONCLUSIONS An experimental well-controlled parametric investigation of the behavior of the oscillating Burke-Schumann type spray diffusion flame has

SPRAY FLAME OSCILLATIONS been carried out. It has been demonstrated that these unique spray-related oscillations occur at frequencies in the range 1–5 Hz and it has been established that they are brought on by a heat and mass transfer mechanism resulting from the presence of the droplets in the system. A complementary stability analysis of a one-dimensional model problem, containing the essential features of the experimental conditions, has been carried out. It was found that the correct order of magnitude of oscillations was reproduced by the analysis, confirming a posteriori that the model was indeed a satisfactory paradigm for examining the observed phenomenon. J.B.G. wishes to acknowledge the partial support of the Technion Fund for the Promotion of Research. M.M. acknowledges the partial support of NASA’s Microgravity Combustion Program and the National Science Foundation. J.B.G. and M.M. thank the U.S.–Israel Binational Science Foundation (Grant 94-00012/3) for its support of this research, and Mr. Ron Odes for his dedicated technical assistance. REFERENCES Burke, S. P., and Schumann, T. E. W., Indust. Eng. Chem. 20:998 –1004 (1928). 2. Moore, J. G., and Moore, J., Sixteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1976. 3. Patankar, S. V., and Spalding, D. B., Heat and Mass Transfer in Boundary Layers, Intertext Books, London 1970. 4. Kim, J. B., and Sichel, M., Fall Technical Meeting, Eastern Section of the Combustion Institute Annual 1.

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Received 10 September 1997; accepted 25 April 1998