The extinction behavior of small interacting droplets in cross-flow

The Extinction Behavior of Small Interacting Dropletsin Cross-Flow

STEFANO RUSSO and ALESSANDRO GOMEZ*Yale Center for Combustion Studies, Department of Mechanical Engineering, Yale University,

New Haven, CT, USA

Experiments were conducted on monodisperse streams of small (�100 �m) methanol and ethanol droplets,injected perpendicularly to the axis of an axisymmetric methane diffusion flame. After crossing the methaneflame, the droplets ignited and continued to burn outside of the “domain of influence” of the gaseous flame tillthey eventually extinguished. The extinction behavior was studied as a function of droplet spacing, oxidizerconcentration and velocity, and droplet velocity by measuring droplet size and velocity at the time of extinctionusing a commercial Phase Doppler Anemometer. The onset of droplet interaction effects on extinction wasfound at interdroplet distances well above the limits reported in the literature. The interaction is attributed tothe vitiated atmosphere left behind by the leading droplet that affects the extinction conditions of the trailingone. The ratio between interdroplet time and convection time normal to the droplet direction of travel wasfound to be the controlling parameter. Results were correlated in terms of a critical Damkohler number thatwas found to scale as dY o,�

no /v, where d is the droplet diameter, Yo,� is the oxidizer mass fraction, v is the dropletvelocity, and no is 0.5 and 0.75 for methanol and ethanol, respectively. © 2002 by The Combustion Institute

INTRODUCTION

In many practical combustion systems such asoil burners, gas turbines, incinerators, diesel,and rocket engines, the liquid fuel is introducedin the combustion chamber as a spray of drop-lets. Different modes of burning can take placeincluding external group combustion, in which adiffusion flame surrounds all vaporizing drop-lets inside the spray core, and internal groupcombustion, in which the primary flame is lo-cated inside the spray boundary, and somedroplets may burn, individually or as a group, onthe oxidizer side of the primary flame [1, 2]. Thetransition from external group combustion tointernal group combustion and, ultimately, toindividual droplet burning can be promoted, forexample, solely by increasing the oxygen con-centration in the oxidizer stream [3]. The result-ing change in the overall stoichiometry causes,in fact, an inward displacement of the primarydiffusion flame and may result in the establish-ment of an internal group combustion regime, ifthe vaporization time is longer than the resi-dence time available on the fuel side of theprimary flame. In such a case, a droplet will passthrough the reacting layer and burn isolated on

the outside of the diffusion flame. In the case ofturbulent environments, it is even possible thatdroplets with relatively short lifetimes mightescape through the reacting layer following un-usual time-temperature pathways [4].

Regardless of the mechanism by which iso-lated droplet burning occurs in a spray flame,these droplets are known not to burn to com-pletion, but extinguish when they reach a criticaldiameter. As a result, emission of unburnedhydrocarbons may ensue, which would be prob-lematic, especially in the case of hazardouswaste combustion, in which a fuel destructionefficiency in excess of 99.99% is required [5].

The extinction of a flame surrounding eithera burning droplet or a porous sphere in aconvective flow was studied both experimentally[6] and theoretically [7, 8]. Spalding [6, 7] wasthe first to show the linear behavior betweenvelocity and size at extinction for large (3–6mm) porous spheres. More recently, the appli-cation of activation energy asymptotics [8] high-lighted the existence of a critical Damkohlernumber for extinction. Computational work ondroplet extinction also led to an interpretationof the results in terms of characteristic timescales and Damkohler number [9].

The objective of the present work is to exam-ine interaction effects on droplet extinction inconvective environments. To that end, monodis-

*Corresponding author. E-mail: [email protected]

COMBUSTION AND FLAME 130:215–224 (2002)© 2002 by The Combustion Institute 0010-2180/02/$–see front matterPublished by Elsevier Science Inc. PII S0010-2180(02)00375-9

perse streams of small (�100 �m in diameter)alcohol droplets were injected perpendicularlyto the axis of an axisymmetric methane diffusionflame. They ignited as they emerged from theflame, burned and subsequently extinguished.The array configuration provided a convenientand well-controlled way to carry out these inter-action experiments and was widely used in thepast to study flame structure [10] [11] [12], fluiddynamics interactions [13], ignition properties[14], and droplet burning time [15]. In thepresent study we identify the controlling param-eter for the onset of interactive effects on thedroplet extinction phenomenon and analyze theeffect of the oxidizer mass fraction on theextinction of alcohol droplets.

EXPERIMENTAL APPARATUS

A sketch of the experimental apparatus isshown in Fig. 1. A co-flow laminar flame ofmethane and nitrogen is established at themouth of a pipe measuring 11 mm in diameterby feeding 2.3 slpm of nitrogen and 0.16 slpm ofmethane in an oxidizer stream consisting ofoxygen and nitrogen in various concentrationssupplied on the outside of the pipe. A system of

screens, packed glass beads, and honeycombguarantees the laminarity of the oxidizer co-flow. A monodisperse stream of either metha-nol or ethanol droplets of controlled size andspacing is injected normally to the methaneflame. The droplets enter the flame, travelthrough it and are ignited at the exit by thereacting layer. The burning droplets continue totravel in the high temperature boundary layer ofthe gaseous flame for �4 mm before reachingthe region unaffected by the flame. When flameextinction occurs in this region, size and velocityof the droplet at the instant of extinction aremeasured using a commercial Phase DopplerAnemometer (PDA) (Dantec Electronik). Theextinction point is located by positioning thePDA probe volume at the point where chemi-luminescence from the burning droplets ceasesto be visible. An example of the measureddroplet size and velocity distributions at extinc-tion is shown in Fig. 2a and 2b. Because oftrajectory effects that broaden artificially thesize distribution, it is likely that the true sizedistribution is even narrower than in Fig. 2a[16]. The PDA, in the LDV mode, also providesthe velocity of the oxidizer flow from Al2O3,seed particles of 1.5-�m nominal diameter. Dis-

Fig. 1. Experimental apparatus.

216 S. RUSSO AND A. GOMEZ

crimination between droplet and particle veloc-ities is obtained from the bimodality of thevelocity distribution. Specifically, dropletspresent a large radial component and a verysmall axial component, whereas the oppositeholds for the seed particles.

The method to generate the monodispersedroplet stream used in the present investigationis similar to the one implemented by Sangio-vanni and Kesten [14]. A homemade vibratingorifice atomizer [17] provides a collimated,monodisperse stream of methanol droplets. Adigitized function generator (Global Specialties105-2003) provides the 25 V, 50 kHz squarewave needed to vibrate the 50-�m orifice of theatomizer. The typical diameter of the generateddroplets is calculated from the liquid flow rate

and the applied frequency at 70 �m, with drop-lets as large as either 90 �m or 114 �m gener-ated in a few cases. The rate at which dropletsare injected into the flame is controlled by anelectrostatic charging/deflection mechanismthat periodically selects a droplet from the mainstream. Differently from [14], the charging ofthe droplets is obtained by induction using aplate positioned at the jet break-up. The fre-quency at which droplets are extracted from themain stream varies from 50 to 500 Hz. Toachieve larger droplets and higher velocities, insome cases two consecutive droplets are se-lected from the main stream to enable them tocoalesce and form a larger droplet with suffi-cient inertia to reach the measurement point athigher speed. The occurrence of the phenome-non has been verified using the PDA in “cold”measurements.

The relative position of the injection systemand the flame is adjustable to achieve a broadrange of extinction velocities for one setting ofthe droplet generation system. Atomizer, elec-trostatic system and burner are all mounted ona tri-dimensional translational stage to focusproperly the PDA at the extinction point.

Table 1 lists the ranges of values of thefollowing relevant parameters: the extinctiondiameter, the effective droplet generation fre-quency after the application of the deflectionscheme, the extinction velocity, the oxidizervelocity and the oxygen mass fraction in theoxidizer stream. All data are for the experi-ments with methanol droplets. It should benoted that extinction diameters and velocities incolumns 2 and 4 represent extinction “bound-aries” originated from significantly different ini-tial conditions.

Chemiluminescence and laser light scatteringimages of the extinguishing droplets are ob-tained using an intensified CCD camera (SantaBarbara Instruments ST-6B)

Lastly, the temperature field induced by thediffusion flame is measured using a coated

Fig. 2. a) Typical diameter histogram of extinguishing drop-lets; b)- Typical velocity histogram of extinguishing droplets.

TABLE 1

Ranges of Values of Key Variables

Variable dext[�m] f [Hz] vext[m/s] u [m/s] Y0,�

Range investigated 40–90 50–1000 0.6–4.2 0.15–0.50 0.62–1.0

217EXTINCTION BEHAVIOR OF INTERACTING DROPLETS

Pt/Pt-10%Rh thermocouple with a bead of 200�m.

RESULTS AND DISCUSSION

As a droplet approaches the flame from the fuelside and passes through, if the conditions arefavorable, it ignites, burns and eventually extin-guishes. Figure 3 captures the last part of thissequence. It shows a long time exposure photo-graph of the streaks of two successive dropletstraveling right to left with the droplets illumi-nated by a He-Ne laser sheet collinear with thedroplet trajectory. The droplet in the middle ofthe picture is caught as it undergoes extinction,as can be inferred by the presence of a chemi-luminescent “halo” in the right half of thestreak. Clearly visible is the extinction point,when the luminosity from the flame ceases andthe narrower cylindrical shape develops duesolely to laser light scattering. The left half ofthat droplet image and the entire streak of thepreceding droplet at the left of the image aremade visible only by light scattering from theliquid phase, with no evidence of chemilumines-cence.

In Fig. 4, we show normalized temperatureand oxidizer concentration (left ordinate) ver-sus time (lower abscissa) and space (upperabscissa), with the methane gaseous flame, as-sumed to be at the origin of both abscissae,burning in pure oxygen. The measurement lo-cation is 8 mm above the tip of the burner. Themeasured peak temperature is 1720 K. Nocorrections for radiative losses were introduced,since the data are used primarily to show thequalitative behavior. The oxidizer concentrationoutside the reaction layer is estimated from the

temperature profile, assuming Le � 1 and theBurke–Schumann condition of Yo � 0 in theinfinitely thin reaction layer.

Also plotted in Fig. 4 is the evolution of thesquare of the diameter of a sample droplet(right ordinate) as a function of time. Thedroplet, in this particular case 86 �m in initialdiameter and with a velocity of 3.2 m/s, after asmall increase in size in the high temperatureregion of the gaseous flame, which can beattributed to the increasing liquid temperature,starts evaporating. After passing through thereacting layer, it burns surrounded by an enve-lope flame. It proceeds through a domain ofinfluence of the methane gaseous flame, extend-ing up to 4 to 5 mm away from the gaseousflame reacting layer, within which the ignitingflame affects the conditions at “infinity” for theburning droplet, and, consequently its burningcharacteristics. The thickness of this domain ofinfluence is regulated by a balance of convectionalong the z-axis and diffusion in the radialdirection, as represented by the Peclet number,a non-dimensional parameter defined in termsof the oxidizer convective velocity, a reactantspecies diffusivity and a mixing layer thickness,Pe � u�/D. In the ensuing discussion, even if thevalue of u is changed, for all reported measure-ments extinction will always occur outside themethane flame domain of influence.

Fig. 3. Long exposure image of two consecutive droplets.Exposure time 400 �s. Oxidizer concentration at infinityYo,� � 0.71. Droplet repetition rate f � 900 Hz.

Fig. 4. Normalized temperature and oxidizer concentration(left ordinate) induced by the gaseous diffusion flame as afunction of distance from the flame front (upper abscissa).Tf � 1720 K, T� � 290 K. Square of the droplet diameter(right ordinate) as a function of the residence time afterpassing through the flame front (lower abscissa) or distancefrom the flame front (upper abscissa). Droplet diameter �86 �m; velocity v � 3.2 m/s in the reacting layer.


Beyond the methane flame domain of influ-ence, steady conditions of temperature andoxidizer concentration are encountered, asshown in Fig. 4. After a trajectory of �7 mm thediameter of the droplet, now down to 65 �m,stops decreasing marking the extinction of theenvelope flame, at a velocity of �2.4 m/s.

We observe that after 1 ms the d2-law seemsto be established for the burning droplet withvaporization constant that, in this particularcase, is 1.75 mm2/s. The extinction point ischaracterized by a change of slope indicatingthe abruptness of the extinction phenomenon,although a small amount of vaporization fromresidual heat in the droplet and gas-phase con-tinues even after extinction, as previously re-marked by Cho et al. [18]. The location wherethe diameter slope changes coincides with theend of visible flame chemiluminescence.

In the ensuing discussion we shall focus onthose droplets which, after igniting in the react-ing layer, travel in the flame influence regionand extinguish outside such a region, that is,where temperature and oxidizer concentrationat “infinity” have recovered the undisturbedvalue.

In a diffusive environment �/d, the interdro-plet distance non-dimensionalized with respectto the droplet diameter, appears to be thecritical parameter controlling droplet interac-tion. Even for arrays in convective conditions,this parameter has been used as a measure ofdroplet to droplet interaction [14, 15]. Charac-teristic limiting values of � for interaction tooccur have been measured approximately to 20to 30 droplet diameters [15]. However, dropletclouds are considered to be more effective inpreventing single droplet burning, with values ofinterdroplet distance reaching 50 diameters inlaminar sprays of monodisperse droplets [3].

We can postulate that in a convective envi-ronment the situation is sufficiently differentfrom a diffusive situation, so that this geometricparameter should not be able to explain fullydroplet interaction effects and departures fromthe single droplet behavior. For example, if wefocus on the extinction point in the presentexperiments, an approaching burning dropletwill find a vitiated environment left by thepreceding extinguishing droplet, which will in-fluence its extinction behavior unless enough

time is given to the oxidizer cross-flow, at veloc-ity u, to sweep out the degraded layer andregenerate the oxidizing atmosphere at the ox-idizer feed stream composition and tempera-ture. Of course, whether this regenerative effectis due mostly to convection or a combination ofdiffusion and convection depends on the spe-cific experimental conditions.

A simple criterion to understand when thereis interaction between droplets can be estab-lished by comparing two characteristic times.The first, the interdroplet time, tint, is the transittime between two successive droplets, and canbe defined as the inverse of the droplet repeti-

tion rate tint �1f

. The second is the washout

time, two, that is, the time needed for theoxidizer flow to wash out the vitiated atmo-sphere left behind by the leading droplet, underthe tacit assumption to be corroborated by theexperiments that convection dominates. Thelatter can be defined as the ratio between theflame size, estimated at 5 times the dropletdiameter for the present methanol/oxygen stoi-

chiometry, and the oxidizer velocity: two �5du

.

It is a measure of the convective time in thedirection normal to the droplet trajectory.

The ratio of these two times is � �tint

two

�u

5df. We can postulate that there will be a

critical value �c, such that if � � �c the leadingdroplet will influence the trailing one and drop-let interaction effects should set in. Conversely,if � � �c single droplet behavior should berecovered, that is, each droplet burns sur-rounded by its own flame and without anyinfluence on the surrounding environment fromthe preceding droplet in the stream. It is worth-while remarking that � can be rewritten as

� �15

�

duv, where v is the droplet velocity. Thus,

the controlling parameter is the non-dimen-sional interdroplet distance multiplied by a ve-locity correction factor accounting for convec-tive effects. In light of these premises, theexperimental results will be presented as thesethree parameters, droplet size, oxidizer velocityand droplet frequency, are varied within the


ranges specified in Table 1 for fixed oxidizerconcentration.

Figure 5 shows the change in extinction ve-locity (left ordinate) as a function of the dropletfrequency for droplets extinguishing with a di-ameter of 55 �m in a 0.17 m/s cross-flow of pureoxygen. We note that by extinction velocity andextinction diameter we refer to the values ofthese variables when the flame surrounding thedroplet goes out. We change conditions up-stream at will, ensuring that the droplets wouldcross the methane flame, ignite, and extinguishoutside the gaseous flame domain of influence.If these conditions are fulfilled, all the data arecorrelated independently of the initial condi-tions. The extinction velocity is initially constantat around 1.6 m/s and independent of frequencyup to approximately f � 200 Hz. For higherfrequencies, it varies linearly with frequency.This latter behavior can be explained in terms ofthe dimensionless parameter governing the ex-tinction of a single droplet, the Damkohlernumber, representing the ratio between a char-acteristic flow time and a chemical time. For adroplet in a convective environment, the flowtime is on the order of tm � d/v, where d and vrepresents diameter and velocity, respectively.Such a time plays the role of the inverse of thestrain rate in the canonical counter-flow diffu-sion flame theory. At extinction, this time is onthe same order of the chemical time. Theincrease of extinction velocity with frequency inFig. 5 suggests the capability of the droplet to

withstand higher strain rates before extinguish-ing. In effect, it can be interpreted as a reduc-tion of the extinction Damkohler number sincethe preceding droplet in the trail “assists” theburning of the subsequent one. For example,the leading and extinguishing droplet may con-tribute to the combustion of the trailing onesome residual heat in the gas-phase and somefuel vapor. In a regime of vigorous burningbringing droplets close to one another mayhinder combustion, which has a simple mass-transport interpretation, as the droplets com-pete for the deficient reactant, the oxidizer,leading to the onset of group combustion. Incontrast, here we examine conditions of “groupextinction” in which the deficient reactant is thefuel and competition for oxygen is no longercontrolling. Rather, starvation of fuel in thegaseous phase controls, and extinction is de-layed by any process that helps increasing thepresence of fuel in the vapor phase, such as thepuff of fuel vapor from the preceding droplet inthe array after it has extinguished.

These results are consistent with the antici-pated role of �, suggesting that for f � 200 Hz,or, equivalently, � � �c, single droplet burningis observed, whereas in the region of largerfrequencies (� � �c) droplet interaction affectsthe extinction behavior. In the same figure wealso plot (right ordinate) the interdroplet dis-tance, non-dimensionalized with respect to thedroplet diameter, l/d. The spacing between thedroplets is much larger than the one usuallyconsidered maximal for interactions to takeplace (�30d), which indeed confirms the signif-icant role of forced convection in the presentsituation. In fact, the velocity correction factor,u/v, in the controlling parameter � takes valuesranging between 0.1 to 0.2

The presence of a critical value of � can beinferred by examining Fig. 6, showing thechange in extinction velocity as function of theparameter � for three different values of thedroplet extinction diameter. For each dropletsize, � was varied by increasing the oxygenco-flow normal to the droplet trajectory. In allthree cases, it appears that a critical value of �c� 3.1 is obtained beyond which there is nochange in the extinction velocity and we canassume that the trailing droplet is not influ-enced anymore by the leading one.

Fig. 5. Extinction velocity (left ordinate) and non-dimen-sional inter-droplet distance (right ordinate) for dropletswith extinction diameter of 55 �m droplets as a function ofthe droplet repetition rate at constant oxidizer cross-flowvelocity.


Let us now focus our attention on the diam-eter component of the parameter �. In Fig. 7 weshow the behavior of extinction velocity vs.droplet extinction diameter for different fre-quencies and oxidizer velocities. For fixed con-ditions of frequency and oxidizer velocity thereis a critical diameter below which no interactionis expected (� � �c). For example, when f �100 Hz and u � 0.17 m/s, if we assume �c � 3.1,the critical value of the diameter is dc

�u

5�c f� 110�m, well outside the area

currently under investigation. As a result, theopen diamonds represent the behavior of non-interacting droplets. Therefore, we can assumethe straight line for f � 100 Hz and u � 0.17 m/sto be representative of the extinction velocity vs.extinction diameter for non-interacting droplets

in the range 30 to 90 �m. Vice versa, if weexamine the case f � 500 Hz and u � 0.17 m/s,

we obtain dc �u

5�c f� 22�m. This value is

well below the range in Fig. 7. So we can assumethat for these conditions the leading dropletalways influences the trailing one. As a result,the open circles in Fig. 7 pertain to extinctionbehavior vitiated by droplet interaction.

The anticipated behavior is even confirmed atintermediate conditions, when f � 200 Hz andu � 0.17 m/s (full triangles), which yield

dc �u

5�c f� 55�m. The extinction velocity in

this case seems to follow the behavior of thenon-interacting line until approximately d � 55�m, that is, the data overlap the line of the opendiamonds. At this point, however, droplet inter-action effects start, the curve diverges and ex-tinction is observed at higher velocities as com-pared to the non-interacting line. A similarbehavior is obtained if we increase frequencyand oxidizer velocity by the same factor, that is,if we keep dc constant. This is the case shownwith open squares, corresponding to f � 500 Hzand u � 0.43 m/s. As expected also in this casethe critical diameter appears to be around d �55 �m.

It should be remarked that the “cold” Reyn-olds number for this range of size and velocitiesis comprised approximately between 2 and 40,which is not sufficiently large for droplets toexperience a transition from envelope to wakeflame [9, 19], which would have compromisedthe linear relation between diameter and veloc-ity. Chemiluminescence images taken with anintensified CCD camera, and not reported here,confirmed this hypothesis.

The correlation in Fig. 7 clearly points to adominant role of convection over diffusion, aswe initially postulated. This result can be ratio-nalized as follows.

First, an upper estimate of the diffusion ve-locity is given by

� D ��YF

YF�

D �

5dYF

YF�

1.5 � 105

5 � 60.106 � .05m/s

that is, the diffusivity over the size of the fuelregion left behind by the extinguished droplet,of the order of the flame volume, where we have

Fig. 6. Extinction velocity as a function of the � parameterfor three different extinction diameters and constant dropletrepetition rate.

Fig. 7. Extinction velocity as a function of droplet diameterfor different conditions of oxidizer velocity u, and dropletrepetition rate f.


assumed YF/YF to be of order unity initially.This value should be compared with a typicalconvective velocity of the cross flow of 0.17 m/sin the experiments. YF/YF remains of orderunity when there is a droplet in the core of thepuff that continuously provide a source of fuelvapor. Not so after extinction, when the dropletand the existing fuel puff will separate becauseof their difference in mass (inertia). Moreover,the droplet leaves fuel vapor behind only at thebeginning. It stops vaporizing after a brief ther-mal transient during which it cools down. So,the extinct droplet keeps barreling along whilethe fuel vapor puff is left behind to face thesweeping cross-flow u. Because the total mass ofthe fuel puff must stay constant, once it isseparated from the source, the diffusion velocitydecays in time for the diminishing concentrationgradients resulting from the coupled effects ofdiminished YF/YF and increasing length scale.On the other hand, convection sweeps awayleftovers cleanly and abruptly. So, althoughdiffusion may not be always negligible, convec-tion appears to dominate, as our correlationsuggests.

Second, if the interaction between two drop-lets can be interpreted thermally, as an energyinput into the trailing droplets from the com-bustion products left behind from the precedingand extinct droplet, the characteristic lengthscale is a multiple of the droplet diametergreater than the estimated stand-off ratio, sincehot combustion products are present also on theoutside of the individual flame.

Third, the length scale 5d is based on thetheoretical flame stand-off ratio from the clas-sical purely diffusive theory. From pictures ofthe chemiluminescent region, the size of thevisible flame appeared close to 0.5 mm, that is,closer to a 10d. Replacing such a lengthscale inthe expression of the diffusion velocity abovewould halve its value and make the convectiveargument even more compelling.

All the results presented so far pertained to afixed unity oxidizer mass fraction. We can nowexamine the effect of the oxidizer mass fractionon the extinction behavior. The results arecorrelated in Fig. 8, showing the extinctionvelocity as a function of the product of thedroplet diameter at extinction times the squareroot of the oxygen mass fraction. The data

collapse around two lines, one obtained for f �100 Hz and u � 0.17 m/s, which, as discussed inconnection with Fig. 7, relates to single dropletbehavior, the other, obtained for f � 500 Hz andu � 0.17 m/s, relating to droplet interactivebehavior (see also Fig. 6). The data coverdroplet sizes in the range 40 to 90 �m withvelocities between 0.7 m/s and 4.5 m/s andoxygen mass fraction spanning the range 0.62through 1.00. Limitations in the droplet gener-ation capabilities of the atomizer and in theradial dimension of the oxidizer stream pre-vented us from exploring an even broaderrange.

The success of this correlation prompts us tointerpret the results in terms of a criticalDamkohler number at extinction. If the chemi-cal time is defined as

tch1 � ��CH3OH�nF�O2�noexp�� Ea

RT�and fuel and oxidizer concentrations are rein-terpreted in terms of mass fractions, theDamkohler number at extinction, Dac, is:

Dac �tm

tch

dv

YF,dnf YO,�

no1

TnF � noexp�� Ea

RT�Where v is a rescaled velocity v � v v0, and v0� 2.5 m/s is the extrapolated value of thevelocity at d � 0.

The correlation in Fig. 8 is consistent with theconcept of a constant extinction Damkohler

Fig. 8. Extinction velocity for methanol droplets at twodifferent repetition rates as function of the product ofdroplet diameter times the square root of the oxidizer massfraction. The oxidizer velocity is held constant at u � 0.17m/s.


number, provided that nO � 0.5 and the tem-perature dependent term is approximately con-stant. Note that the exponent of the oxidizermass fraction remains at 0.5 even in the case inwhich droplet interaction effects are present(upper curve). This value, perhaps fortuitously,is in agreement with the oxidizer exponentfound in shock tube experiments [20].

Similar studies have been carried also for theextinction of small Ethyl Alcohol droplets. InFig. 9 we report extinction velocity as functionof d � Yo

0.75. The correlation for different oxi-dizer concentrations is excellent, confirmingthat the overall kinetics of ethyl alcohol isapproximately of order 0.75 with respect to theoxidizer concentration. Even in this case thisvalue agrees with the oxidizer exponent found inshock tube experiments [21].

If the correlations in Figs. 8 and 9, are to beinterpreted in terms of a critical Damkohler, theimplication of the constancy of the tempera-ture-dependent term is rather surprising, sincethe change in oxidizer concentration of thoseexperiments should result in higher flame tem-peratures. A plausible explanation for the con-stancy of the temperature-dependent term maybe offered by the following consideration. Forliquid fuels the flame temperature is affected bythe presence of the droplet that acts as aconstant temperature sink for the heat gener-ated in the flame. The higher the oxygen massfraction, the closer the flame gets to the dropletincreasing the heat flux into the droplet. This

droplet sink effect can further contribute tooffset the anticipated increase in flame temper-ature subsequent to an increase in oxidizerconcentration. This effect may be particularlysignificant for alcohols because of the high valueof the ratio of the enthalpy of vaporization tothe enthalpy of combustion (50�103 comparedwith, say, 6.7�103 for heptane), which results insteeper temperature gradients on the dropletside of the flame and larger heat fluxes ascompared to hydrocarbons [22]. Unfortunately,we could not test this point experimentally Thecharging device can work either with polarliquids (e.g., alcohols) or with liquids with finiteelectric conductivity, unlike hydrocarbons. Dop-ing hydrocarbons with a suitable additive, whichwe tried, does not work, since it requires con-centration levels that ultimately lead to micro-explosions, thereby affecting the sensitive ex-tinction condition.

A potentially important role of water vaporabsorption in the extinction of alcohol dropletshas been proposed in [18, 23, 24]. We could notverify it experimentally. However, it is likelythat, because burning occurred in a very shorttime in view of the small size of the droplets andthe large oxygen concentration, water vaporabsorption plays a minor role. In fact, the effectappears to be significant mostly for large drop-lets and long vaporization time, since theabsorbed water vapor at the surface of thedroplets is required to diffuse into the interior[23]. Because Peclet numbers for the liquidphase have values on the order of 100, thiseffect is not significant. Nor could recirculationin the droplet interior help. In fact, the ratio ofa diffusion time in the liquid droplet to ahypothetical turnover time of a vortex within adroplet, if such a circulation existed, can beshown to be equal to the droplet Reynoldsnumber times the ratio of the kinematic viscos-ity of the gas to the kinematic viscosity of theliquid. At the prevailing Reynolds number, inthe range 2 to 40, the ratio of these two times isof order one and recirculation could not changethese estimates dramatically. Nor could a pos-sible recirculation because of the droplet gen-eration scheme linger in the combustion region,since it would be attenuated by viscous effectsover characteristic times shorter than the tran-sient time across part of the cross flow oxygen

Fig. 9. Extinction velocity for ethanol droplets at two dif-ferent repetition rates as function of the product of dropletdiameter times the oxidizer mass fraction to the 0.75 power.Oxidizer velocity and repetition rate are held constant atu � 0.17 m/s and 100 Hz, respectively.


stream and the gaseous methane flame beforedroplet ignition. We conclude that water vaporabsorption should not be relevant in the presentexperimental conditions.

CONCLUSIONS

This study demonstrates that in arrays of mono-disperse droplets, interaction effects on extinc-tion in convective flows, appear at interdropletdistances well above the limits reported in liter-ature. The effect is attributed to the vitiatedatmosphere left behind by the leading dropletthat affects the extinction conditions of thetrailing one. Droplet interaction effects set inwhen the ratio between interdroplet time andconvection time, �, attains a critical value. � isequivalent to the widely used non-dimensionalinterdroplet distance multiplied by a factor rep-resenting the ratio between droplet velocity andoxidizer cross-flow velocity. When � reaches acritical value, measured to be �3.1, dropletinteraction effects cease to exist and singledroplet behavior is recovered. The measuredextinction velocities have been correlated interms of a critical extinction Damkohler num-ber that was found to scale as dYo,�

no /v, withexponent equal to 0.5 and 0.75 for methanol andethanol droplets, respectively.

The support of NSF, Grant # CTS-9904296(Dr. Farley Fisher, Contract Monitor), is gratefullyacknowledged. The authors wish to thank Profes-sor Amable Linan for helpful discussions.

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Received 16 November 2001; revised 4 March 2002; accepted25 March 2002


The extinction behavior of small interacting droplets in cross-flow

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