Lean flame dynamics through a 2D lattice of alkane droplets in air

Transcription

1 ean lame dynamics through a 2D lattice o alkane droplets in air Colette Nicoli, Bruno Denet, Pierre Haldenwang o cite this ersion: Colette Nicoli, Bruno Denet, Pierre Haldenwang. ean lame dynamics through a 2D lattice o alkane droplets in air. Combustion Science and echnology, aylor Francis, 2014, 186 (2), pp <hal > HA Id: hal Submitted on 15 Oct 2015 HA is a multi-disciplinary open access archie or the deposit and dissemination o scientiic research documents, whether they are published or not. he documents may come rom teaching and research institutions in France or abroad, or rom public or priate research centers. archie ouerte pluridisciplinaire HA, est destinée au dépôt et à la diusion de documents scientiiques de nieau recherche, publiés ou non, émanant des établissements d enseignement et de recherche rançais ou étrangers, des laboratoires publics ou priés.

2 ean Flame Dynamics through a 2D-attice o Alkane Droplets in Air C. Nicoli 1, B. Denet 2, P. Haldenwang 1 1 M2P2, 2 IRPHE Uniersité d Aix-Marseille / CNRS / Ecole Centrale Marseille; France Abstract Flame propagation along a 1-D array or through a 2D-lattice o uel droplets has long been suggested to schematize spray-lames spreading in a two-phase premixture. he present numerical work considers the resh aerosol as a system o indiidual alkane droplets initially located at the nodes o a ace-centred 2D-lattice, surrounded by a ariable mixture o alkane and air, in which the droplets can moe. he main parameters o the study are s, the lattice path and ϕ, the liquid loading, which are both aried, whereas ϕ, the oerall equialence ratio, is maintained lean ( ϕ = 0. 85). Main results are: a) or large lattice path (or when the droplets are large enough), spreading occurs as resulting rom two stages : a short time o combustion ollowed by a long time lag o aporization; a classical triple lame (with a ery short rich wing) spreads around the droplets; b) spray-lame speed decreases as liquid loading increases; c) an elementary model inoking both propagation stages allows us to interpret lame speed as a unction o the sole parameter s ϕ ; d) when the lattice path shortens, the spray-lame exhibits a pattern that continuously goes rom this situation to the plane lame ront; Keywords : spray-lame; two-phase combustion; heterogeneous combustion; aporizationdiusion ront; droplet combustion Nomenclature ( ϕ ) F heat o reaction depending on equialence ratio e i ewis number o species i in the mixture R droplet radius s lattice path adiabatic lame temperature or stoichiometric gaseous mixture b U adiabatic speed or single-phase premixed lame U adiabatic lame speed or the stoichiometric gaseous mixture U spray-lame speed SF Z mixture raction Ze Zeldoich number or stoichiometric gaseous mixture φ local equialence ratio ϕ liquid equialence ratio o the resh spray (liquid loading) ϕ gaseous equialence ratio o the resh spray ϕ oerall equialence ratio o the resh spray θ reduced temperature

3 Ψ i reduced mass raction o specie i δ adiabatic lame thickness or the stoichiometric gas mixture τ characteristic time o aporization τ characteristic time o combustion c Introduction Combustion spreading through a spray is an important concern in a large number o applications, such as diesel engines or gas turbines. On the one hand, the state o the art has clearly identiied dierent regimes o propagation: group combustion o droplets, aporization controlled propagation, pulsating spray-lames (Umemura and akamori 2005, Mikami et al. 2006, Suard et al. 2004). Recent micrograity experiments hae brought improements o our understanding o the dierent mechanisms inoled (Pichard et al , Nunome et al 2002, Nomura et al 2000, Nomura et al 2007). hose well-controlled contributions hae been particularly useul in order to proide inormation on droplet size inluence and liquid loading eects on spray aporization and spray-lame speed promotion. On the other hand, the deelopment o monodisperse droplet generator has led to a number o recent works on lame propagation in an array o droplets under dierent conigurations: 1D array (Kikuchi et al (2005), Mikami et al. (2005), Chen and in (2012), Nomura et al. 2013) regular 2D array (Wu and Sirignano 2011) or random conigurations (Oyagi et al 2009). In the paper, we present a set o numerical studies o combustion spreading through a acecentred 2D-lattice o droplets. he simulations are carried out using a simpliied chemistry with a global exothermic reaction, and we are particularly interested in the combustion o a globally lean spray, a regime which has not much beneited rom numerous studies, although this corresponds to a general trend in the applications. Combustion spreading through this system is here aimed at proiding new insights into the problem o spray-lame propagation through a mist composed o droplets, the aporization time o which is not negligible in comparison with the chemical times. Note urthermore that the lattice o droplets is an initial condition only: when the lame propagates, the droplets are allowed to moe. hey are drien by two cooperatie eatures: droplet aporization and thermal expansion o the mixture. Although the oerall spray composition is supposed lean (the oerall equialence ratio is set to ϕ = 0.85 ), the local equialence ratio (ϕ ) can be ound rich close to droplets. o study the combustion in such a heterogeneous medium, we hae implemented two dierent chemical schemes, which are both o one-step reaction type. he irst scheme is the classical one-step irreersible reaction model, which is known as acceptable in lean pre-mixture. Borrowed rom the recent literature (arrido-opez and Sarkar, 2005), the second one introduces a progress ariable that allows us to adapt heat release to resh composition, in order to get satisactory results on the rich side, too. heir comparison howeer shows that the spreading eatures reported here do not markedly depend on those chemical models. Non-dimensioning is carried out thanks to time and length scales o the stoichiometric premixed lame. Four lattice paths hae been inestigated: s = 1.5, s = 3, s = 6 and s = 12 in units o the latter lame thickness, as well as our dierent liquid loadings : ϕ = 0. 85, ϕ = 0. 65, ϕ = and ϕ = 0. 25, where ϕ is the equialence ratio 2

4 linked to the uel initially under liquid phase. For each set o these parameters, the droplet size is assessed to keep the oerall equialence ratio ixed to ϕ = Modelling the spray lames At low pressure, lame thickness oten appears large in comparison with droplet interspacing. In the recent years, this allowed us to resort to homogenization or deeloping an appropriate numerical modelling. In such an approach which also neglects droplet inertia, liquid uel appears as an additional species subjected to enter into the chemical scheme ater a aporization step. Seeral spreading regimes hae been predicted [Suard et al. 2004], in particular an intrinsic oscillatory regime [Hanai et al. 1998, Atzler et al. 2001] occurring as a Hop biurcation. he existence o this regime does not require the implication o dierential diusiity eects [Nicoli et al & 2007]. At moderate and high pressure, spray-lame thickness can no longer be large enough -in comparison with droplet interspacing- to allow any process o homogenization. In such a system, spray-lame tends to be controlled by aporization, the chemical heat release permitting the aporization o the droplets one ater another. hereore, spray-lame propagates within a heterogeneous mixture with large droplets, as it is the case o the n-decane experiment by Nunome et al his is also the coniguration o the experiments by Nomura et al. 2000, which was concerned with globally lean mixtures o ethanol and air (with an equialence ratio about 0.8): with droplet interspacing typically o the same size as spray-lame thickness. he aim o the present work is hence to proide a numerical analysis o such lean conigurations where the lame eels heterogeneities at the droplet length scale. We ix the oerall equialence ratio to 0.85 or the sprays we study. he mist structure is schematized by a ace-centred 2D-lattice o alkane droplets in a leaner pre-mixture alkane-air. On the one hand, the present numerical modeling considers the usual set o conseration laws: mass, momenta, energy and species. Since the accurate chemical schemes or alkane are too complex or eicient numerical simulations, the standard approach searches ater a simpliied chemical kinetics. As this work is deoted to spray with droplet inter-distance not small in comparison with the characteristic combustion scales (at least, o the same order), our spray-lames will propagate through a medium with arying chemical composition. he simplest manner consists o choosing an irreersible 1-step reaction, the parameters o which are adjusted to mimic the lame dynamics. It is known that the classical one-step Arrhenius law largely oerestimates the adiabatic lame temperature on the rich side. Although the calculations presented here concern lean spraylames, we use two dierent manners o computing the heat release: either the classical (constant) heat o reaction, or a heat o reaction as a linear unction o the resh gases equialence ratio (Nicoli and Haldenwang 2010). he second model correctly mimics the characteristics o the premixed single-phase lame (adiabatic lame temperature and lame speed). he use o this model in a heterogeneous mixture is carried out by introducing the mixture raction as proposed in arrido- opez and Sarkar (2005) and Fernandez-arrazo et al (2006) to suitably adapt heat release to the local composition o the unburnt spray. (see below). he latter point seems, a priori, to be important, since the resulting chemical scheme is able to take account o the heterogeneities in the mixture composition (it must be recalled that lame dynamics depends on the manner the mixture is perormed, in the close icinity o the lame especially). 3

5 Non-dimensioning Non-dimensional orm o the conseration laws is perormed with the use o the theoretical data related the stoichiometric (gaseous) premixed lame, as deried in the theoretical papers by Joulin and Mitani (1981), or arcia-ybarra et al (1984). We deine the stoichiometric lame temperature, as, gien by b b ( Y ) p u Q = u + (1) C ν M emperature and species mass ractions are handled under the reduced orms u ( ) θ = ( ) ; (i= or the considered alkane and i=o or oxygen) As or the time and length scales, we deine them with the use o b u i ψ = Y Y (2) i i,u D th,b, the thermal diusiity coeicient o the burnt gases, and U, the stoichiometric (single-phase) lame speed, gien by ( U ) ρ b λb = 3 Ze ν ρu ρbc p 2 [( ρb ) M Y Bb ] exp( A / ) o, u b (3) his allows us to establish the scalar conseration laws as ollows θ 1 + V. θ = di( λ θ ) + F( ϕ) W ( ρ, ψ i, θ ), (4) t ρc p ψ i 1 + V. ψ i = di( ρdi ψ i) ν im iw ( ρ, ψ i, ), (5) t ρ where the reaction rate is now deined by Ze W( ρ, ψ i, θ ) = 4 with ( ) 2 Ze = A( b u ) b and γ = ) ρ u ρ θ 1 ψ ψ exp Ze (6) o ρb ρb 1+ γ ( θ 1) ( b u b he oerall equialence ratio o the spray is gien by the ratio o the total amount o uel to the total amount o oxygen in the whole lattice: ν om ρ d o esh lattice ϕ =, (7) ν M ρ d resh lattice o 4

6 while the quantity ϕ [resp. ϕ ] only takes account o the uel density under liquid phase [resp. gas phase]. We obiously hae ϕ = ϕ + ϕ. Combustion models in a heterogeneous medium As mentioned beore, the irst model denoted by " Q 0 " supposes that the heat o reaction is independent o the local composition. his assumption is only satisying or the lean side o the composition. On the other hand, we searched ater an improement useul or the rich side, because rich combustion can aect the close icinity o the droplets. As well-known, combustion on the rich side is characterized by the production o metastable species, the enthalpy o which reduces the lame temperature and thereore the lame speed. A simple analysis o the enthalpy budget at equilibrium and the resulting lame speed has been carried out or seeral alkanes in (Nicoli and Haldenwang 2010). his approach allowed us to adapt heat o reaction to resh gas composition. It turned out that the procedure led us to a good agreement with the experimental data on gaseous premixed lame speed. he result took the orm o F(ϕ ), a multiplying actor aecting the heat production term in the energy conseration law. F(ϕ ) depends on the equialence ratio o the resh gases, linearly as [ 1 ( )] F( ϕ ) = ϕ 1 (8) where α is a coeicient depending on the considered uel [reer to a orthcoming paper by the same authors]. Next, we hae to leae the concept o homogeneous resh mixture or considering a medium o ariable composition. We now ollow the lamelet spirit deeloped in arrido-opez and Sarkar (2005) (see also Fernandez-arrazo et al (2006)). It is well known that mixture raction is a quantity presered when crossing a premixed lame, as long as ewis number is close to unity. As or the opposite case o a diusion lame, the mixture raction only results rom the transport-diusie processes and allows us to ind the lame positioning. For the triple lame (the intermediate situation), it permits to predict the correct mixing in the resh gases just in ront o the lame. In other words, mixture raction allows us to assess the mixture composition that the lame burns. More precisely, we compute at any point o the domain Z ( ψ + r) / ( r) = νψ ν (9) o + where ν = 24 / 7 is the stoichiometric oxidizer-alkane mass ratio or a heay alkane and r, the oxidizer proportion in air with r = ( ψ ) /( ψ + ψ ) We hence hae Z 1 in the O inj O N inj = droplets (pure uel), and Z 0 in pure air. he dotted line, in igure 1, represents the mixture raction or any resh mixture equialence ratio. It can be obsered that the whole lammability domain corresponds to ery small mixture ractions: rom Z or the lean limit ( ϕ = 0. 5 ) up to Z at the rich limit ( ϕ = 1. 6 ). In other words, the locus where an alkane spray-lame can propagate occupies a limited area (in Z) during mixing. o sum up, the method considers Z as a continuous marker o the equialence ratio in the resh mixture right beore the lame location. 5

7 Furthermore, or the diusion lame, a classical result leads to linearly link mass raction o reactants with mixture raction as ollows : ( ψ ) Z ψ = and = ( ψ ) 1 ) injection ψ ( Z o o injection (10) Note these hypotheses are consistent with the deinition o Z. Consequently, in this context, the equialence ratio that results rom a diusion problem now reads [arrido-opez and Sarkar (2005)] ϕ = νψ ψ = ν ψ ( 1 Z ) ψ = ν Z r 1 Z (11) o ( ) ( ) ( ) inj Hence, the preious adjustable heat o reaction introduced in the equation o energy now reads F( ϕ ) = [ 1 α( νz / r(1 Z) 1) ] i ν Z r( 1 Z ) [ 0.5,2] W ( ϕ) = i Z r( 1 Z ) [ 0.5,2] or 0 o inj ν. (12) For the sake o illustrating the eiciency o the adjustment, the homogeneous laminar lame speed o octane-air premixture obtained or α = is plotted in Fig. 1. (Ze being set to 8) and compared with experimental data. Figure 1: Octane-air lame speed s. equialence ratio: comparison between numerical premixed gaseous lame speeds computed with α = and experimental data. astly, as in the conseration laws appears the elocity ield V, the reaction-diusion system is coupled with the Naier-Stokes equations. he oerall numerical scheme considers periodic boundary conditions in the (y-) direction transerse to lame propagation, open boundary conditions in the downstream (x-) direction, and closed in the x-direction upstream rom the lattice. he numerical approach o low Mach number type has preiously been described in Denet and Haldenwang (1995). In contrast with this preious paper which used constant diusion coeicients, the present numerical approach uses diusion coeicients arying with temperature. At low temperature the diusion coeicient is ery small, at high temperature we use the diusion 6

8 coeicient o an ideal gas, and the ormula used interpolates between these two regimes. he result is that diusion is eectiely rozen at low temperature. et us also note that we use a high ewis number or the uel (1.9), which also tends to reduce droplet diusion. he resulting model is denoted by " Q ( z)" Numerical experiments he uel droplets are positioned at the nodes o the ace-centered lattice or a gien liquid loading. Since the oerall equialence ratio is set to ϕ = 0. 85, the droplet radius will only depend on lattice path s and liquid loading ϕ Four lattice paths hae been inestigated: s =1. 5, s = 3, s = 6 and s = 12 in units o the stoichiometric premixed lame thickness. Four dierent liquid loadings hae been inestigated: ϕ = 0. 85, ϕ = 0. 65, ϕ = and ϕ = 0. 25, where ϕ is the equialence ratio linked to the amount o uel initially under liquid phase. At the open (downstream) end, the temperature ield is initiated with the proile o a premixed lame, that allows us to ignite the irst droplet o the lattice. o ollow the combustion spread, we compute < > (x), the mean temperature aeraged in the periodic (y-)direction. hen, we decide to consider as the lame ront, x, the position where < > y ( x ) = his deinition can sometimes be misleading because x is not by deinition a monotonic unction o time: when droplet aporization is long (in comparison with reaction time), the temperature proile can latten and x can seemingly admit a regressing position. y Figure 2: initial ields; a) uel mass raction, b) temperature, c) oxygen mass raction. 7

9 o summarize the conditions o the numerical experiments, in Figure 2 we proide the initial conditions gien to the dierent scalar ields. Figure 2 a recalls the pattern o the initial uel supply. Figure 2.b illustrates the temperature proile which starts to ignite the lattice, whereas Figure 2.c displays the oxygen initial mass raction ield. wo-stage spreading he plot o x s. time presents arious shapes as shown in Figure 3, where the lame location is drawn ersus time. Combustion spreading deelops rom the right to the let in the x-direction. his is why the oerall slope o the lame position s. time, i.e. the lame speed, is negatie. On the one hand, or large liquid loading, or equialently or large droplets, the lame ront does not propagate monotonously. here are in act two dierent stages during the propagation rom a large droplet to another: the irst one corresponds to aporization; it is characterized by an apparent ront regression, due to the act that the temperature proile becomes less sharp during the preheating o air and uel aporization; thereore, the locus where > ( x ) = 0. 5 can appear as regressing. his eect is < y increased by the act that gas phase is produced and heated, and the low that results rom the gas expansion pushes this locus back. Figure 3: lame location s. time or lattice path s= 6 ( ϕ = 0. 85) and arious liquid loading. he second stage is much shorter in time and corresponds to the reaction stage, which is itsel decomposed into two sub-steps: a ery rapid propagation o a triple lame-like ront which goes around the droplet, ollowed by the propagation o a slower lame that burns the mixture between the current droplet and the next one. his two-stage propagation remains noticeable as ar as the droplet radius is larger than 0.15 (i.e. a small raction o the stoichiometric gaseous lame thickness). Indeed, this two- 8

10 stage propagation also disappears when the initial pre-mixture increases (i.e or anishing ϕ ), as illustrated by the lowest two cures in Fig.3. hose cures correspond to a lammable pre-mixture (their slope is always negatie), or which during the aporization time simultaneously occurs the propagation o a ery lean (and thereore slow) lame. On the other hand, when the droplet radius diminishes (simultaneously with liquid loading), aporization time becomes shorter, leading us to a lame propagation through a nearly homogeneous medium; two cooperatie arguments are adanced or explaining this behaiour: as liquid loading decreases, the pre-mixture increases (since ϕ = is gien), whereas the aporisation time becomes negligible or smaller and smaller droplets. For the sake o illustrating the propagation through a large droplet system, we hae plotted heat release at arious times ater ignition in Figure 4. he succession o eents can be told as ollows. Figure 4.a) proides us with the irst instant ater ignition. he lame is nearly lat with a ery weak heat production, except in the close icinity o the droplet where the reaction is deeloping. Figure 4.b) illustrates that the lame skirts round the droplet that aporizes in the same time. Note the lame is not a strict triple lame, since the rich wing is hardly isible (likely due to high gradients on the rich side). Figure 4.c) shows the diusion lame that remains behind the ront. Combustion spread is now ery slow since a weak ront o reaction is sustained beore the burning droplet. In Figure 4.d) this reaction ront becomes een weaker. In the meantime, the next droplet aporizes. Figure 4.e) illustrates a new stage o intense heat release that corresponds to the combustion o the next droplet, similarly with Figure 4.b). astly, Figure 4.) displays the last instants o lie o the diusion lame, that surrounds the current burning droplet. Note additionally that the diusion lame occupies a large olume. his is due to the large stoichiometric coeicients o heay alkanes. he aboe sequence o pictures -with their corresponding instants- corroborates the preious interpretation o Figure 3 about the dierent stages that characterize the combustion spreading through the lattice. Moreoer, the monotonic combustion spreading rom the right to the let conirms that there is no ront regression (as could hae been a possible wrong interpretation o Figure 3). When the lattice path becomes smaller, the aporizing uel pocket (as schematized in Figure 4.b) still remains isible. But, since the lame thickness becomes o the same size as the lattice path, the aporization o the next droplet is quickly actiated. In other words, combustion spread appears rather as a continuous process, close to the propagation in a premixed gaseous medium. he transitional process is inestigated in the last paragraph (see Fig. 8). Note also that the droplets can moe during the calculation. As an illustration, the comparison o the heat release presented in Fig.4.b and Fig.4.c (or in Fig.4.e and Fig.4.) gies a good idea o the droplet motion. he elocity ield that dries the droplets results rom two cooperatie eatures: uel droplet aporization and gas thermal expansion due to lame propagation, the latter eature being classical in any premixed lame propagation. 9

11 a) t = 0 b) t =12. 5 c) t = d) t = 25 e) t = ) t = Figure 4: Snapshots o heat release at arious times ater ignition ( s = 6, r = 0. 25, ϕ = 0. 40, ϕ = 0. 45). d Eect o liquid loading For ixed oerall equialence ratio ( ϕ = 0. 85) and lattice path s, increasing the liquid loading corresponds to an enhancement o the droplet radius. Consequently, the aporization characteristic time being a priori scaled by the square o the radius, the time lag deoted to aporization is expected to increase. his is illustrated in Figure 5 where the ront history is plotted or a large lattice path and arious liquid loading (or arious corresponding apour press 10

12 Figure 5: Front location s. time; or s = 12 and arious apour pressures. s, lattice path ϕ R τ τ c τ 2 R τ s ϕ 2 τ c s able 1: Characteristic times o aporization and combustion, and their corresponding scalings, or arious liquid loadings and lattice paths (with the Z-corrected 1-step model). For ϕ = 0 (or ϕ = 0. 85), the lame has to utterly create the mixture permitting its propagation. his is noticeable that the necessary time becomes ery large. he dierent time lags deoted either or aporization or or reaction are analysed in able 1, where we report on the 11

13 characteristic times as indicated in Figure 5. et us deine as τ, the characteristic time or aporization. As mentioned in Figure 5, its alue reads τ = t 2 t1. As or τ c, the combustion time, we deine it as τ c = t 3 t2. In able 1, we only report on s 6, because or s 3 the dierent characteristic instants (i.e. t 1, t2 and t3 ) appear less markedly. Accordingly with able 1, the characteristic time o the combustion stage only depends on the lattice path (see the ith column). he combustion step occurs ater a aporization step which also depends on the actual droplet size. O course, the scaling o the latter time is more or less ound in direct ratio to the square o the radius, or equialently proportional to the product s 2 ϕ. o analyse the part played by liquid loading, it must be recalled that combustion must stand ar rom the droplet, since we are concerned with heay alkanes. Vaporization must thereore be carried out enough to ill the sphere surrounded by the lame. In other words or globally lean spray, combustion deelops intensiely only when aporization is rather complete. his is corroborated by the act that τ c, the combustion time, does not depend on liquid loading (see the last column o able 1). On the other hand, or a gien lattice path the liquid loading modiies the droplet radius and the initial surrounding gas composition. hereore, τ, the aporization time depends on liquid loading. Accordingly with the penultimate column o able 1, we ound τ proportional to liquid loading. Although we also possess a large set o results or the cases s = 3 and s = 1. 5, let us obsere that able.1 does not contain any data rom those cases. he reason is the ollowing: as the droplet radius decreases, the determination o the two stages scaled by τ and τ c more and more becomes arbitrary. So that, to derie the scalings o τ and τ c, we hae decided to exploit the data rom large droplets only. Spray-lame speed We hae obsered that combustion spreads through the lattice with an oerall elocity gien by the mean slope o the cures drawn in Figures 3 & 5. In what ollows, this slope is called spray-lame speed. As mentioned in Introduction, we perormed the numerical simulations with the use o two dierent chemical models. Hereater, the model that uses Z, the mixture raction, is called Zcorrected 1-step (and labelled by Q (z) ), while the non-corrected (classical) model is called Standard 1-step (and labelled by Q 0 ),. In Figure 6, we plot the numerical spray-lame speed, normalized by the single-phase premixed lame at ϕ = ϕ = 0. 85, with respect to droplet radius. In this igure, both chemical models are compared or each case o liquid loading. For a gien symbol, the dashed line expresses the results or the standard 1-step model, while the plain line represents the Zcorrected 1-step model. In the cases drawn here, we obsere that the discrepancy remains weak. Next, we want to use the preious analysis in terms o characteristic aporization and combustion time scales, in order to interpret the data shown in Figure 6. I we consider a combustion spread through a distance corresponding to the lattice path s. As the lattice is ace-centred, we meet 2 droplets per lattice period. hereore, according to Figure 5, we need to spend two aporization times and two combustion times. hus, the spray-lame speed reads: 12

14 U SF s = (13) 2 τ + 2τ c Figure 6: Spray-lame speed (i.e. the combustion spread through the lattice) ersus droplet radius or arious liquid loading, accordingly with both chemical models Now, we turn towards able 1, where the scales or aporisation and combustion are analysed. 2 aking the mean alue o the concerned column, we can set τ 1.5ϕ s and τ c 2s. Incorporating those quantities within the spray-lame speed, we simply obtain U SF ϕ s his expression being in terms o the stoichiometric gaseous premixed lame, we are interested in its normalization with respect to the gaseous lame o the same equialence ratio, denoted by U ( ϕ = 0.85), i.e. or ϕ ϕ = We then obtain = U U SF 13 ( ϕ = 0.85) 8 + 6ϕ s (14) 13

15 Next, we compare the results o this elementary theory with the data obtained rom the numerical simulation. his is carried out in Figure 7, where we plot the same elocity data as those o Figure 6 (again normalized with 0.153, the single-phase lame speed at ϕ ϕ = 0. 85) as a unction o the actor ϕ s. = Figure 7: Spray-lame speed (reduced by the single-phase lame speed in a gas mixture o the same oerall equialence ratio) ersus the actor ϕ. In Figure 7, we perorm the plot or the data resulting rom both kinetics models. We again obsere that the discrepancy between both models remains eeble. hereore, we conclude that the rough theoretical expression (14) is alid or both chemical models. Note that the cases s=1.5 and s=3 correspond to the smallest alues o the actor ϕ in igure 7. It is thereore not surprising that the agreement o the numerical data with our simple model [i.e. ormula (14)] is weak in the domain o the small actors ϕ. O course, a better agreement would be obtained with the ollowing it cure s s s U SF /U = 1 /( ϕ s) (15) which corresponds to the dashed line in Figure 7, while the plain line is related to equation (14). Note additionally that -or a gien liquid loading- droplet size are in direct ratio to lattice path. In other words, or large droplets, both equations (14) or (15) retriee the classical result characterizing 1 the spray combustion regime controlled by aporization, which behaes as R (as obsered by Ballal & eebre 1981 or predicted in Suard et al. 2001). 14

16 Eect o diminishing the lattice path 2 As the lattice path shortens, the characteristic time or aporization (i.e. τ s ) diminishes aster than the characteristic combustion time ( τ c s ). Hence, or small alues o ϕ s (say, ϕ s 1), total droplet aporization and some degree o oxygen diusion (inside the apour pu resulting rom the droplet aporization) are carried out beore lame spreading. hereore, when a critical alue o s is reached (rom aboe), a rich premixed lame starts to slowly cross the apour uel pocket. hen, i the lattice path still decreases (and oxygen diusion increases), this premixed lame crosses the apour pocket aster,. Behind the rich premixed lame, a diusion lame takes place around the pu o uel apour. his process or small s is described in Figure 8, where heat release and the corresponding uel mass raction are presented at three consecutie times (with a time interal o t = 0. 5 ). Figure 8. Spreading through a small lattice (s= 3 ; φ = 0.25): heat release (top) and uel mass raction (bottom) at three times separated by t = In Figure 8, the pu o aporized uel (added with oxygen diused rom its icinity) is swept by a rich lame, leaing behind a diusion lame surrounding a hot gas pocket, which still contains an amount o uel. In the present situation, no clear triple lame occurs since the lame thickness is large enough to aporize the droplet in the pre-heating zone and to allow diusion o oxygen towards the uel pu. he transition to classical (single-phase) premixed lame occurs when this uel pocket appears negligible or smaller lattice path. Conclusion he present numerical study on lean spray combustion uses a lattice to schematize the droplet deposition. Our results concern the inluence o the spray characteristic parameters ( lattice path and liquid loading or equialently droplet radius and liquid loading ) on lame dynamics. A particular attention has been paid to ocus on large lattice path in comparison with the lame thickness. Additionally, we inestigate the transition to premixed lame when the lattice path becomes small. wo simple chemical schemes with a 1-step exothermic reaction hae been used. he irst one is associated with the standard Arrhenius law with a reaction heat independent o the equialence ratio o the mixture to burn. he second one adapts the heat o reaction to the equialence ratio 15

17 deduced rom the ield o mixture raction, with the introduction o a correction applied particularly to rich mixtures. For both cases o chemical models, we hae obtained similar qualitatie results in lame structure, as well as quantitatie alues o spray-lame speed. his indicates that corrections o the triple lame structure on the rich side are probably not ery important or lame propagating in a globally lean spray. Our simulations include a preaporized raction o the uel, which allows the lame propagation rom one droplet to the next one through a ariable lean pre-mixture. Een or ϕ = ϕ = 0.85 (and ϕ = 0 ), no lame extinction has been obsered (although combustion spreading was ound ery slow). We hae shown that the combustion spreading or relatiely large droplets (compared to the premixed lame thickness) is characterized with two stages: a slow phase corresponding to the partial aporization o the next droplet until a lammable mixture is carried out, and a aster phase corresponding to the propagation o a triple lame around the droplet. On the other hand, or smaller droplets, the lame propagates in a regime much closer to a premixed lame, but a premixed lame with a heterogeneous equialence ratio inside the lame thickness. his regime should desere a urther study because we did not obtain (as isible in Figures 6 and 7) any maximum o the lame speed at non-zero droplet radius. his maximum has, howeer, been obsered in experiments on lean spray-lames in micrograity by Nomura et al Further studies will turn our attention towards other situations o oerall equialence ratio. Concerning een leaner sprays, we enisage the eects o a random distribution o droplets to explain unexpected lammability o spray, as obsered in the experiments o Nunome et al. [2002]. With respect to rich sprays, the experimental contributions are more numerous. he corresponding amount o data should help us to urther check the dierent reduced chemical schemes we hae presently used.. Acknowledgements he present work has receied the support o the Research Program Micropesanteur Fondamentale et Appliquée DR n 2799 CNRS/CNES under the contract CNES/ Reerences Atzler F., Demoulin F. X., awes M., ee Y., Marquez N., [2006] Burning rates and lame oscillations in globally homogeneous two-phase mixtures (lame speed oscillations in droplet cloud lames), Combustion Science and echnology, Vol. 178, pp Ballal D.R., eebre A.H., Proc. Combust. Inst., Vol. 18 : (1981) Chen C K, in H,. [2012], Streamwise interaction o burning drops, Combustion and Flame, 159, Dais S.. and aw C.K. [1998], inear lame speeds and oxidation kinetics o iso-octane-air and n- heptane-air lames, Proc. Combust. Inst., Vol. 22, pp Denet B., Haldenwang P. [1995], A numerical study o premixed lames Darrieus-andau instability. Combust. Sci. echn., Vol. 104, pp Fernandez-arrazo E., Sanchez A.., inan A., Williams F.A [2006], A simple one step chemistry model or partially premixed hydrocarbon, Combustion and Flame,. 147, arcia-ybarra P., Nicoli C., Clain P. [1984], Soret and Dilution Eects on Premixed Flames, Combust. Sci. ech. 42, pp

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