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Combustion Science and Technology

ISSN: 0010-2202 (Print) 1563-521X (Online) Journal homepage: http://www.tandfonline.com/loi/gcst20

On the Physics of Jet Diffusion Flames

EMMANUEL VILLERMAUX & DANIEL DUROX

To cite this article: EMMANUEL VILLERMAUX & DANIEL DUROX (1992) On the Physics of JetDiffusion Flames, Combustion Science and Technology, 84:1-6, 279-294

To link to this article: http://dx.doi.org/10.1080/00102209208951857

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Combust, Sci. and Tech., 1992, Vol. 84, pp. 279-294Photocopying permitted by license only

© Gordon and Breach Science Publishers S.A.Printed in United Kingdom

On the Physics of Jet Diffusion Flames

EMMANUEL VILLERMAUX· and DANIEL DUROX Laboratoired'Aerothermique. CNRS, 4ter Route des Gardes 92190 Meudon. France

(Received January 4,1991; infinalform August 5,1991)

Abstract-A physical model for the laminar jet diffusion flame including chemical kinetics effectsis provided.We produce the appropriate set of dimensionless groups needed to discuss the role of buoyancy and jet outletvelocity Uo on such flames. We are particularly interested in flame lengths L; the behaviour of the flameis shown to split in two regimes. The regime of short flames is buoyancy controlled and the flame lengthscales with the intensity of the gravity g as L - g-I/3 in the reaction sheet limit. The regime of long flamesis governed by the outlet velocity as L '" Uo. The predictions compare well with published experimentaldata and the extension of our model to the candle flame problem reveals that the flux of consumplion ofthe candle is decreasing with increasing gravity. A possible mechanism leading to this result is proposed.

NOTATION

CoCFc,Do;dEFrGg

g'

HLLemN(x)nn'qRRerr,T.Vu,w

Concentration of oxidizer in the environment surrounding the flameConcentration of the pure fuelSpecific heat per unit of mass at constant pressureDiffusivity of the oxidizer in the burnt gasesThermal diffusivity in the pure fuelThickness of the flameActivation energyJet Froude numberRation of the actual gravity intensity to earth gravityIntensity of the gravityReduced gravity: g Po - Pb

PbElevation of the flame on the wick of the candleFlame length or heightLewis number: D,h/DStoichiometric coefficientLocal flux of oxidizer incorporated to the flame at the elevation x: D(C%')Order of the reaction with respect to the fuelOrder of the reaction with respect to the oxidizerHeat of reactionBurner radius; Perfect gas constantJet Reynolds numberChemical consumption rate (mol.jrn's)Temperature of burnt gasesTemperature of pure fuelLocal ascending velocity of the burned gasesInlet velocity of fuelCollision rate (S-I)

·Present address: Institut de Mecanique de Grenoble, U.M.R tol, BP 53 X, 38041 Grenoble Cedex, France.

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280 E. VILLERMAUX AND D. DUROX

Greek Symbolsex Constant defined by Eq. (2.2.3)fJ Zel'dovich number: (EIRTf) (T; - T,)(j Penetration depth of oxidizer in abscence of reaction (Eq.(2.3»(j' Effective penetration depth (Eq.(2.4)$0 Total flux of oxidizer consumed by the flame$.. Outlet flowrate of fuely Surface tensionAB Buoyant lengthscale = (DlRlg')'/4AI' Convective lengthscale = (DRIVo)'/2,\ Thermal conductivity\' Exponent defined by Eq.(5.3)P DensityPh Density of burnt gasesPo Density of the surrounding environmentr Chemical time: Colrr" Diffusive penetration time: (jlIDr, Convection time: x] V

Mean free path

INTRODUCTION

The knowledge of the incidence of gravity on combustion processes is of a greatpractical importance in aerospatial applications. From a fundamental viewpoint, it isnow generally admitted that gravity induced buoyancy is one of the basic elements ofmany combustion phenomena.Microgravity experiments permit to get rid of buoyancy effects and to study refined

mechanisms of the combustion process, but it is also profitable to vary the intensityof buoyancy compared with the other ingredients in the flame to determine in detailthe possible relations between gravity and chemical consumption, diffusion of reactantsor forced convection for instance. More generally, considering the fact that gravity ismost of the time a permanent element of the flame environment (Edelman et al.,1973). it is important to know whether it influences notably the burning process andhow.In this paper, we will be concerned with the work of Altenkirch et al. (1976) who

investigated the role of gravity on the shape ofjet diffusion flames. These authors useda centrifuge to simulate a gravity up to a hundred times the earth normal gravity andrecorded flame lengths for a wide range of the jet Froude number.We first reexamine these data with the aid of a detailed model of the diffusion flame,

and then we attempt to extrapolate our model to analyze our original data concerningthe candle flame.

2 TIME AND LENGTHSCALES IN DIFFUSIVE COMBUSTION

A diffusion flame may commonly be defined as the position in space where fuel (F)and oxidizer (usually Oxygen) meet and react according to

F + mOl = Prod ucts.

The reaction occurs with a high activation energy (EIRTb I) and a strong heatrelease.

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L

X III

(a)

0'

R

JET DIFFUSION FLAMES

--!'----co

'---;".-----+-- 0

d

(b)

281

FIGURE I (a) The jet diffusion flame viewed as a vertical cylinder of radius R and height L. J' is thepenetration depth of the surrounding oxidizer in the ascending burnt gas column. (b) The detailed modelof the flame. The reaction zone has a thickness tilfl,d being the thermal thickness of the flame. The oxidizerpenetrates the burnt gas column over a depth ii', which is the diffusive depth reduced by consumption andflame thicknesseffects.

By contrast with premixed combustion, where the feed stream of premixed fuel andoxidizer is separated from products by a narrow zone which is the flame, diffusionflames necessarily involve three phases, namely the pure fuel, the pure oxidizer andthe products, In a steady configuration, the products form a layer contiguous to theflame and must be evacuated in order to permit the oxidizer to go on reaching theflame (Figure I). The process of evacuation of the hot burnt gases depends on thegeometry of the flame: this can be forced convection as in the confined Burke-Schumann problem, or a buoyancy enhanced convective process in free boundaryconfigurations.

2, I Picture of the flameWe want to sketch the times and lengthscales ruling an isothermal and isobaricdiffusion flame, whose burnt gases ascend freely in a cold still environment. Weassume that the flame can locally be described as shown in Figure I, and that thisscheme applies to jet diffusion flames. We call CF the concentrations of pure fuel andCo the concentration of oxidizer in the ambient air. Let r(z) be the rate of reaction(moles of oxidizer converted per unit of volume and time), z being the directionperpendicular to the flame edge

r(z) = (z) C;(z) e- E/RT• (2.1.1 )

The characteristic chemical time t = Colrmax is defined from the concentration Ctof oxidizer in the vicinity of the flame as

(2.1,2)

The cold incoming fuel is heated from T; to T; by means of heat conduction fromthe reaction zone. If d is the distance over which the temperature of the fuel T isessentially variable, it is readily shown from activation energy asymptotics that thedistance over which the reaction occurs at an appreciable rate is of the order of diP,

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282 E. VILLERMAUX AND D. DUROX

- (2.1.3)

where fJ = E(Tb - Tul/RT; is the Zel'dovich number (Williams 1985). Indeed, asnoticed by Clavin (1985), if the reaction rate r bears an Arrhenius temperaturedependence r(T) - e- b1RT, and if T - T, - (T; - Tu ) z/d near the flame, the ratior(T)/r(Tb)) - e-P'/d is then essentially non-zero over a distance d/fJ. For a Lewisnumber close to unity, the distance over which the concentration C,,(z) of the fuel goesfrom C" to zero is d also, meaning that C,,(z) - C,,(z/d) at the flame edge. From (2.1)and (2.2) we get

r(z) = e-Pz/d.

Let us compute r' = So'" r(z)dz which represents the flux of oxidizer consumed perunit of flame length

(2.1.4)

where

r(n + I) = r x"e-xdx.

Given the heat of reaction q = Cp(Th - Tu ) , the flame thickness d is derived byequating the total heat release per unit of flame length q(r'/CO)Pb to the flux conductedupstream J.(Th - T.)/d. Introducing the thermal diffusivity Drh = ),/PhCp, the thick-ness of the flame is given by the classical result

d =fJn+ IDrh,

r(n + I)'(2. J.5)

For the derivation of (2.1.5), we do not consider a possible heat transport along theflame. The cancellation of longitudinal heat transport in the vicinity of the flame issupported by the fact that streamlines are strongly deflected due to the jump oftemperature when passing through the flame; the main direction of heat transport isthus perpendicular to the flame.Now, on the other side of the flame, the burnt gases at temperature T; exhaust into

the cold environment (air) rich in oxidizer (0,). Let U be the ascending velocity of thegases at the elevation x above the burner exit (U can have a buoyant or a forcedconvection origin, we will discuss this point later), the penetration depth of theoxidizer in the ascending column is given by equating the convection time" = x] Uto the penetration diffusive time 'd = 152 / D giving

(2.1.6)

The above calculated 15 holds in absence of chemical consumption of the oxidizer;if this effect is taken into account, the amount of oxidizer that has disappeared in thereaction during the time r must be deduced from 15. The actual, or effective pene-tration depth of the oxidizer is then the result of its ability to diffuse in the burnt gasestowards the flame measured by 15, reduced by the efficiency of the reaction to consumeit in the reaction zone

15' (2.1.7)

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o ---:--;r

(2.1.1 )

JET DIFFUSION FLAMES

dIl

(2.1.2)

o----:----'

d:-.-dIl

(2.2)

283

FIGURE 2 Two limit situations of the flame structure. 2.1. Reaction sheet limit, and distributed slowchemistry: (r(n + I)!fJ"+ I) 1£( t,!t) <li t. 2.1.1. The reaction sheet limit characteristic of high Zel'dovichnumber p: the thickness of the reacting part of the flame is small compared to the diffusive depth b. 2.1.2.The distributed reaction zone with large r and small p. The thickness over which the consumption rate isnon zero compares to b, but the chemistry is slow (large r] such that the profile ofoxidizer through the burntgas column is unperturbed with respect to the diffusive profile. 2.2. Fast ehemistry distributed on a widereaction zone: (r(n + I)!po+ I) 1£(t,!t) I. The reaction zone is distributed over a width that compares10 b, and the chemistry is fast enough to make the oxidizer profile essentially variable over b'; smallcompared to b.

Giving

J' s _ d r(n + 1) r,u tor r. < r,fJ"+1 r ' (2.1.8.1 )

" __ '_ d r(n + I)u u tor L;" r.[3n+1 ' (2.1.8.2)

The set of Eqs (2.1.8.1), (2.1.8.2) is derived from the equation of reaction-diffusionin the appendix and can be rewritten as follows

J' = J (I - I) Le for r, < r,

J' = 0 (I - 1) Le]"') for r, < r.

(2.1.9.1)

(2.1.9.2)

The profile of oxidizer in the vicinity of the flame is unperturbed by consumptionor reaction zone thickness effects when J if, that is

r(n + I) L r, 1[3n+1 e r (2.1.10)

1n view of (2.1.10), this condition is identically fullfilled in two very differentphysical situations (Figure 2): either the width of the reaction zone is infinitely small([3 co), this is the "flame sheet" limit or approximation (Williams, 1985; Liiian,1974; Bilger, 1976); or the chemical consumption is weak compared with the transporteffect (r r.), The latter case would correspond to a slow chemistry distributed ona large reaction zone (moderate (3). Indeed, combustion phenomena characterized bya strong heat release and large hydrodynamic timescales may be more closely relatedto the first limit situation (Clavin and Joulin, 1989).

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284 E. VILLERMAUX AND D. DUROX

(2.2.1 )

2.2 Chemical versus hydrodynamical timescalesWe now turn to the evaluation of the ratio r,./r for usual diffusion flames. The reactiontime r in Eqn (2.1.2) may be written with respect to the collision rate between reactivespecies \I' and the Arrhenius efficiency factor as

I -EIRT,- = HI e .r

If is the mean free path of the species, the thermal diffusivity writes D'h = andthe velocity of sound in the medium is c =. (Williams, 1985). Thus the ratio D'hlris of the same order of magnitude as c'e- lil 7,. For T; Tu ' the Zel'dovich numberfJ = E(Th - T,,)IRTi; is of the order of EIRTh and the characteristic chemical time rcan be approximated by (Clavin, 1985)

(2.2.2)

With the typical values Drh = 10- 4 m2 Is, c = 750mls (air at 2000 K) and fJ = 10,one finds that r is of the order of 4 . 10-6 s. The convection time r, = x] V introducedin the previous section is most of the time much larger than r in combustion problemswhen it is considered at the scale of the flame height (x = L): for instance, theresidence times r, = LIVo reported by Edelman et al. (1973) for a methane flame areabout 0.01 s. A 4cm high candle flame has a transit time (see next section) r, '- (0.04123)'1' = 0.04 s. The transient regime described in the appendix during which r,/r < Ithus only takes place on a small distance x above the burner exit compared to theflame height L. The major part of the flame is in the regime r r > r and the effectivepenetration depth J' is simply the diffusive penetration depth J minus a constant valueproportional to the thickness of the reactive part of the flame which is wanishinglysmall in the reaction sheet limit: dr(n + 1)/{i"+I. Setting IX = [(r(n + 1)/fJn+1) Le]'I',we rewrite Eq. (2.1.9.2) as

J' = J(I - IX). (2.2.3)

In accordance with the conclusions of the previous section, IX is zero in the reactionsheet limit and IX ,,:;; I in the distributed reaction zone regine (Figure 2). A firstorder reaction with respect to the fuel gives, with P= 10 and Le = I,IX = y'f(2)/100 = 0.1.

2.3 Hydrodynamic regimesIn order to make this picture of the flame more precise, let us explicate r,.. Thisconvection time is based on the ascending velocity U. Two limit cases may occur: thefirst one is the limit situation where the outlet velocity Vo of the fuel at the exit of theburner is small compared to the buoyant velocity acquired by the hot gases at,typically, the flame height L. The flame is then ruled by buoyant convection and whenthe viscous effects can be neglected, it is easily shown (Turner, 1973, Taylor, 1960) thatthe ascending velocity is simply the "free fall" velocity

where

g' Po - Phg Ph

(2.3.1 )

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JET DIFFUSION FLAMES 285

Po and Ph being respectively the density of the cold environment and of the burnt gases,g being the gravitational intensity. This expression for U appears in many othercontexts (see Turner, 1973 and also Castaing et al., 1989) and is correct when theaverage ascending velocity is not sensitive to the effect of viscosity (although itdetermines the shape of the velocity profile). In the case of a laminar plume, thisoccurs as long as L < g' R4[v', Note that g' is quite sensitive to the flame temperature:considering an environment at 20°C, a temperature for the burnt gases T; = 2000 Kgives g' = 57m/s-2 and To = 1000K givesg' = 23.5m/s-2• So, for g'L (wewill discuss later to what kind of flames this condition is related to), the convectiontime is

x (X )1/2r, = U(x) - "i .

From (2.2.3) and (2.1.6), one gets after rearrangement

(D2 )1/4

0' = g'X (I - IX),

(2.3.2)

(2.3.3)

which exhibits a natural lengthscale A B = (D2R/g,Y/4The second limit situation corresponds to flames for which g'L, In this case,

the ascending velocity having mainly a forced convection origin is simply Uo, then

xr, = U

o'

Again from (2.1.6), one gets

{)' = r (I - IX),

(2.3.4)

(2.3.5)

with the characteristic lengthscale AF = (DR/UO) ' /2It follows from these remarks that the response of the flame shape to a variation

of Uoor g will depend on the regime ruling the flame. A single parameter, for instancethe Froude number, cannot be used alone to describe the flame behaviour over a widerange of g or Uo, precisely because g is not relevant in the forced convection regime.One may rather expect a dependence of the form

(2.3.6)

in the two regimes of convection discussed above, where A B and A F are respectivelythe pertinent controlling characteristic lengthscales.

3 FLAME LENGTH

We start with the simplistic pattern proposed by Altenkirch et al. who represent thejet flame surrounded by the burnt gas ascending column as a cylinder of radius Randheight L. Although the radius of the flame can reach seven times the size of the burnerradius, we will assume, without loss of generality provided that the flame radius is

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286 E. VILLERMAUX AND D. DUROX

weakly varying or independent of Uoand g (Ross et at. 1991), R as equal to the burnerradius.Following these authors, we make the assumption that the flame adjusts its length

in such manner that the net flux ofoxidizer incorporated into the flame exactly equals,according to the stoichiometry of the reaction, the total outlet fuel flux :r,

The local flux of oxidizer is given in terms of a linearized Fick law

CoN(x) = Db"

.,(3.1)

".(3.2)

where D is the diffusivity of the oxidizer, Co its concentration in ambient air and (j'is the thickness of the oxidizer penetration depth discussed in section 2.Then the total flux of oxidizer over the flame of height L is

<Ilo = 2nR f:- N(x)dx. (3.3)

The flowrate of fuel exhausting from the burner at velocity Uowith a concentrationC, is

The basic assumption (3.1) yields:

2Co J'L dxmCp 0 (j'(x)·

(3.4)

, (3.5)

This equation has to be solved for the two different regimes of convection discussedabove. and will provide an implicit relation between L and the physical quantitiesinvolved in this problem, particularly As (or g') and Ap (or Uo).

3.1 The pure buoyant regime

As mentioned in the previous section. this limit concerns low velocities Uo and highlevels of g.The penetration depth (j' is given by (2.3.3) and the basic equality of the oxidizer

and fuel flowrate (3.5) is written as the implicit relation for L

givmg

m(1 - a)R C, rL dxgd/4 D'12 Co o, = 2 Jo x'/4' (3.1.1 )

LR

C,.)4)1/3D2g' m(1 - a) Co • (3.1.2)

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or

JET DIFFUSION FLAMES 287

(3.1.3)

The tendency derived in (3.1.3) is consistent with the fact that buoyancy increasesthe supply rate of oxidizer by decreasing ij' and then increasing N(x) (Eq, (3.2))causing the flame length to shorten. In this regime, the dependence ofL on the velocityUo is: L Ut/3•

3.2 The forced convection regimeIn the high outlet velocity Uo limit and for low levels of g, the penetration depth ij'has been found to be given by (2.3.5) and the computation of (3.5) provides:

II

CF (Uo)'/' rL dxmR(1 - a) Co IS = 2 Jo xii" (3.2.1)

The integration giving:

or

LR (3.2.2)

L (R)'R I\.F that is L Uo· (3.2.3)

(3.2.4)

The dependence L Uo is confirmed by many experimental data on jet laminardiffusion flames (van Tiegellen, 1968), especially microgravity experiments (Cochran,1972).The transition between these two hydrodynamical regimes occurs when UJ is of the

order of g'L, thus with (3.1.2) or (3.2.3) when

g'R'

Flames for which Uo g'R' /D (or I\.FR) are dominated by buoyancy andflames for which I\.FR are in the forced convection regime. When applied toEqs. (3.1.1) and (3.2.2), the criterion (3.2.4) reveals that buoyant flames are shorterthan those ruled by forced convection. Figure 3 sums up these results on a qualitativeLog-Log plot. As long as I\.MI\.FR < I (i.e., (I\.BR/M)4/3 < (R/I\. r)'), one expectsthat L/R (I\.BR/I\.;.)4/3 and for > I, one expects L/R (R/I\.,.)'.

4 COMPARISON WITH EXPERIMENT

Altenkirch et al. had suspected and analysed the L g,-I/1 dependence in thebuoyant regime and plotted their data in the L/R(Re'Fr)'/3 vs. Fr = UJ/gR plane, thejet Froude number Fr ranging from 0.1 to 104• They observe, for ethane and propaneflames that L/R(Re' Fr)'/3 becomes constant for Fr < 4 roughly, showing that theasymptotic regime L g'-I/J is just reached. For higher values of Fr, that is forhigher outlet velocities Uo or/and lower levels of g, they argue that the generalbehaviour can be smoothed by a Fr- 1/4 dependence.

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288 E. VILLERMAUX AND D. DUROX

2

RA,

2

RA,

RLog-AF

FIGURE 3 Flame length L scaling in the two convection regimes: Buoyant regime <t A}R; Forcedconvection regime AfR.

As shown on Figure 4, a close inspection of the data reveals in fact that theirrecordings lie in a transitional regime between the asymptotic behaviours discussedabove.Low Froude numbers correspond to low Uo and high g, and we established in

section 2 that the flame length is ruled by the characteristic length As. In order tocompare the predictions ofour model in the system ofcoordinates chosen by Altenkirchet al., we plot LI R(RAsIA;.)4/3 VS. for (As/A,,)4 < 0.8 (Fr < 4 roughly), the agreementwith experimental data is quite good, even if the asymptotic L - is not widelyreached by experiments. For > A"R, the physics is no more ruled by buoyancy,and the forced convection takes over. In the coordinates ofAltenkirch et al., we expecta [(AsIA,.)4]-116 law since L - A"F 2 • Again, the agreement is quite reasonable.

5 THE CANDLE FLAME

The effects of an artificially elevated gravity on a common candle flame were investi-gated on a 6 meter diameter centrifuge. The candle was a blend of 75% in parafineand 25% in stearine. Its diameter was 2.1 em and the wick (1.5 millimeter in diameter)was made of interlaced fibers of cotton. The candle was placed in a large combustionchamber (I meter high and 30cm in diameter) filled with a still ambient air safe fromdraughts and the flame was recorded by an in-board CCD camera. The shoulder ofthe candle was cut in a conical shape in order to leave the flame always clearly visible.The gravity was varied up to seven earth g, the value above which the flame died

out. The general characteristic of the flame was very similar to those reported byAltenkirch et al. onjet flames. At earth gravity, the bottom of the flame was blue andits equilibrium length reached about 4cm a few minutes after the ignition. Increasing

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LIR

JET DIFFUSION FLAMES

00 BI

00 a '0' = 'ba a0""'1, Ba

00

r--

FIGURE 4 Comparison with Altcnkirch et al. data.

289

gravity caused the flame to decrease in length, to become less luminous, more blue andtransparent. The bottom of the flame tends to descend on the wick, and to draw nearerto the shoulder of the candle. The recorded evolutions of the flame length L (measuredfrom base to tip) and of the bottom H of the flame (measured from the shoulder ofthe candle to the base) as a function of gravity intensity are shown on Figure 5. Thecharacteristics of the flame where measured from the luminous flame zone; this mightlead to an over-estimation of the absolute flame lengths, although weak according totheir order of magnitude (a few centimeters), but does not alter the scaling law of L(or H) with g.In his 1848 book, Faraday recognized the fact that the phenomena involved in the

candle are numerous and complex. If it is true that the effects, we argue that the

L(em)

10,----------------,(a)

10G

H(cm)

10,---------------,(b)

10G

FIGURE 5 (a) Candle flame height vs.G, the apparent gravity normalized by earth gravity. (b) Locationof the bottom of the candle flame on the wick vs. apparent gravity.

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290 E. VILLERMAUX AND D. DUROX

simplified model ofjet diffusion flame that we have derived above might enlighten atleast a rough characteristic of this flame, as its length dependence on gravity intensity.The main problem that arises concerns the way the fuel (gaseous hydrocarbons

produced by the evaporation and the decomposition of the blend of paraphine andstearine) is supplied to the flame. Indeed, the flame might with no possible doubt beconsidered as a buoyancy-controlled flame since the fuel, merging from the wick,bears no initial ascending velocity. In this regime, we have shown in section 3 that thestrongest possible dependence of L on g is L - g-I/3 reached in the reaction sheetlimit. Now, the apparent power law of Lon g for the candle flame (Figure 5(a» is

(5. [)

(We have no experimental nor theoretical evidence for this dependence to be apower law, this is only an apparent law). Then it becomes clear that the flowrate offuel <1>" provided to the flame is not constant and is itself a function of g: we haveestablished (Eq. (3.1.1» that, in the asymptotic limit and omitting the unessentialfactors

(5.2)

yielding L - g-I/J with constant <1>". Now (5.1) can only be recovered if we set

(5.3)

The apparent dependence (5.1) implies v - 0.25 suggesting that <1>" must be adecreasing function ofg. How can this decrease be understood? The fuel is essentiallyprovided to the flame in its liquid phase via a capillary driven creeping flow throughthe wick. It is well known from Jurin's Law (Batchelor, [967) that the ascensionheight H ofa wetting liquid of surface tension y, density p in a capillary tube of radiusr IS

H = 2ypgr

(5.4)

The elevation H is decreasing with increasing g as H - g-I. The structure of thewick is nearer to a porous media than to a simple tube, but the physical processinvolving a balance between gravitational and capillary forces is essentially the same.Thus one might expect a decrease of the equilibrium elevation height H of the liquidfuel in the wick with increasing g, consistent with the fact that the flame descents onthe wick with increasing g as shown on Figure 5(b). This fact can also be noticed onthe visualisations of Ross et al. (1991) in the context of microgravity experiments. Inthe limit of high g, liquid stearine does not elevate in the wick to provide the flameany more, which dies out, accordingly to our observations. So, one might conjecturethat the available flowrate <1>" is linked to the elevation H

(5.5)

This conjecture is surprisingly confirmed by our measurements since on Figure 5(b)the dependence H - g-O.25 is quite clear. Our conjecture is subordinated to furtherexperiments which should directly measure the relation between <1>" and g (Forinstance, one could have a direct visual measure of <1>" and of its dependence to g by

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JET DIFFUSION FLAMES 291

recording the rate of decrease of the heigh of the solid part of a thin candle as afunction of time for different levels of g)

6 CONCLUSIONS

The inspection of the different time and lengthscales ruling the physics of the jetdiffusion flame has produced the appropriate groups AF and AB needed to discuss therole of buoyancy and outlet velocity in these flames. The results are summed up inFigure 3. At high levels of g, for AFR, the length L of the flame scales asL that is to say L s:". The effect of an increase ofg is to enhancethe flux of oxidizer incorporated to the flame by decreasing the diffusional layerthickness of oxidizer. The volume of space needed to consume a fixed amount ofoxidizer (typically measured by L) is thus decreased. This regime concerns shortflames and has not been widely investigated by experiments. The transition to theconvective regime takes place at = I and is characterised by the scalingL A,-:2 that is to say L Uo'The occurrence of heat transfer along the flame has not been investigated, and the

thermal thickness of the flame d is assumed to be independent of x. Nevertheless, apossible increase of d with the elevation x such that d r;/2 if a diffusive process isinvolved does not alter the shape of the dependence ofLon AB or AF since the relativeimportance of these effects are measured by ()( which is found to be much less thanunity (section 2.2).In the present paper, our model has been tested for the influence of gravity and

outlet velocity on diffusion flames and the predictions can be considered as good. Itwould be interesting to test its predictions for the sensitivity of the flame length to thepressure P of the reactants. With a constant flowrate of fuel and since the diffusivityD varies as r:' for perfect gases, we predict that L should scale as p-2/J in the buoyantregime and should be independent of P in the convective regime (Eq. (3.1.2) and(3.2.2).We hope that forthcoming works will address these questions and will also con-

tribute to confirm at least our results concerning the candle flame, if not our conjecturelI>F g-025; this is a kind of expected conclusion.

ACKNOWLEDGEMENTS

This work was supported by a CNES grant under contract 90/229. We wish to thank J. B. Marquette for anuseful indication about a particular point discussed in this paper.

APPENDIXWe consider a semi-infinite homogeneous medium extending in the (0, z) directionexposed at t = 0 and z = 0 to a step of concentration C* of a scalar liable to undergoa chemical reaction. We want to derive the expression for the transient effectivepenetration depth of the scalar in the medium for a zero and first order reaction rate.

(a) Order zeroThe equation of diffusion-reaction writes

aeat

where r is the constant reaction rate.

scD--raz2 , (I a)

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292 E. VILLERMAUX AND D. DUROX

Going to Laplace transforms C = So'" C(r)e-" dt, (I a) writes

The initial concentration profile is a step function

o. (2a)

C(I = 0) = 0, z > 0; C(I) = C·, z = O.

The spatial integration of (2a) provides

The inversion of (3a) in time space gives

(3a)

(5a)

= erfc CJot) - (I - f>rfc )dl'). (4a)

The diffusion profile C(z, I) erfc (zI2Jl5i) is not affected by chemical consumptionat short times and it can be seen from (3a) by setting s 00 or from (4a) with 1 0that the first "chemical" correction to this profile is

C(z, I) ( Z) r f C.Ic* = errc 2Jl5i - C. 1 or I:;:; r.

The pure diffusive penetration depth is 15 Jl5i; the expansion of (5a) allows thecomputation of the first order correction to 15 due to the presence of chemicalconsumption. The series expansion of the error function erf (u) is

2 '" (- l)"u2n+ !erf (u) r::: e- u2 2: '(2 I) .v 7C n=O n. n +

Then, as erfc (u) = I - erf (u), and with u = (zI2Jl5i) (zIJ), (5a) writes as thesimple linearised form

C(z, I) z r- J - C. I. (6'1)

The chemical effects can then be embedded in an effective penetration depth 15' suchas C(z, I)/C· I - zlJ'

with

15' 15 (I - for I:;:; T,

C·r

(7a)

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JET DIFFUSION FLAMES 293

The previous analysis holds for times t shorter than the chemical time r; for t ;;;,. r,then 15' = O. Now, if r is essentially non-zero in a finite band ofwidth diP near z = 15(see section 2), and if t relates to a convective process as for the jet flame, t = r, = x] U(the equation of the problem becoming: U(ac;ax) D(a'c;az') - r), (7a) yields

15' 15 _ dr,pr

and

15' - 15 - P

(Sa.l)

(Sa.2)

which are Eqs. (2.I.S.I) and (2. I.S.2) with n = O.

(b) First order

The equation of diffusion-reaction can be written

acat

Similar calculations as in (a) provide

s c cD---az' r(I b)

C(z r) ( z) I 1" (Z)--'- = e- II' erfc -- + - e- I I, erfc --- dt'c* 2,fi5i r 0 2JD('

which again leads to (7a) for t .;;; r and give

(2b)

and

15' 15 d- /3'

(3b.l)

(3b.2)

which are Eqs. (2.1.8.1) and (2.I.S.2) with n I.

REFERENCES

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Batchelor, G. K. (1967). An introduction to fluid dynamics. Cambridge University Press.Bilger. R. W. (1976). The structure of diffusion flames. Combustion Science and Technology 13, 155-170.Castaing, 8.. Gunaratne, Goo Heslot, F., Kadanoff, L, Libchabert, A.• Thomae, S., Wu, X., Zaleski, S,

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294 E. VILLERMAUX AND D. DUROX

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Faraday, M. (1988). The Chemical History of a Candle. Chicago Rewiew Press.Landau, L. and Lifchitz, E. (1971). Mecanique des fluides. Ed. Mir.Lifian, A. (1974). The assyrnptotic structure of counterflow diffusion flames for large activation energies.

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