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1 This article was ownloae by: [ ] On: 12 December 214, At: 18:48 Publisher: Taylor & Francis Informa Lt Registere in Englan an Wales Registere Number: Registere office: Mortimer House, Mortimer Street, Lonon W1T 3JH, UK Combustion Science an Technology Publication etails, incluing instructions for authors an subscription information: Regimes of Spray Vaporization an Combustion in Counterflow Configurations Amable Liñán a, Daniel Martínez-Ruiz b, Antonio L. Sánchez b & Javier Urzay c a E. T. S. I. Aeronáuticos, Mari, Spain b Grupo e Mecánica e Fluios, Universia Carlos III e Mari, Leganés, Spain c Center for Turbulence Research, Stanfor University, Stanfor, California, USA Publishe online: 1 Dec 214. To cite this article: Amable Liñán, Daniel Martínez-Ruiz, Antonio L. Sánchez & Javier Urzay (215) Regimes of Spray Vaporization an Combustion in Counterflow Configurations, Combustion Science an Technology, 187:1-2, , DOI: 1.18/ To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) containe in the publications on our platform. However, Taylor & Francis, our agents, an our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions an views expresse in this publication are the opinions an views of the authors, an are not the views of or enorse by Taylor & Francis. The accuracy of the Content shoul not be relie upon an shoul be inepenently verifie with primary sources of information. Taylor an Francis shall not be liable for any losses, actions, claims, proceeings, emans, costs, expenses, amages, an other liabilities whatsoever or howsoever cause arising irectly or inirectly in connection with, in relation to or arising out of the use of the Content. This article may be use for research, teaching, an private stuy purposes. Any substantial or systematic reprouction, reistribution, reselling, loan, sub-licensing, systematic supply, or istribution in any form to anyone is expressly forbien. Terms &

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3 Combust. Sci. Technol., 187: , 215 Copyright Taylor & Francis Group, LLC ISSN: print / X online DOI: 1.18/ REGIMES OF SPRAY VAPORIZATION AND COMBUSTION IN COUNTERFLOW CONFIGURATIONS Amable Liñán, 1 Daniel Martínez-Ruiz, 2 Antonio L. Sánchez, 2 an Javier Urzay 3 1 E. T. S. I. Aeronáuticos, Mari, Spain 2 Grupo e Mecánica e Fluios, Universia Carlos III e Mari, Leganés, Spain 3 Center for Turbulence Research, Stanfor University, Stanfor, California, USA This article aresses the problem of spray vaporization an combustion in axisymmetric oppose-jet configurations involving a stream of hot air counterflowing against a stream of nitrogen carrying a spray of fuel roplets. The Reynols numbers of the jets are assume to be large, so that mixing of the two streams is restricte to a thin mixing layer that separates the counterflowing streams. The evolution of the roplets in their fee stream from the injection location is seen to epen funamentally on the value of the roplet Stokes number, St, efine as the ratio of the roplet acceleration time to the mixing-layer strain time close to the stagnation point. Two ifferent regimes of spray vaporization an combustion can be ientifie epening on the value of St. For values of St below a critical value, equal to 1/4 for ilute sprays with small values of the spray liqui mass-loaing ratio, the roplets ecelerate to approach the gas stagnation plane with a vanishing axial velocity. In this case, the roplets locate initially near the axis reach the mixing layer, where they can vaporize ue to the heat receive from the hot air, proucing fuel vapor that can burn with the oxygen in a iffusion flame locate on the air sie of the mixing layer. The character of the spray combustion is ifferent for values of St of orer unity, because the roplets cross the stagnation plane an move into the opposing air stream, reaching istances that are much larger than the mixing-layer thickness before they turn aroun. The vaporization of these crossing roplets, an also the combustion of the fuel vapor generate by them, occur in the hot air stream, without significant effects of molecular iffusion, generating a vaporization-assiste nonpremixe flame that stans on the air sie outsie the mixing layer. Separate formulations will be given below for these two regimes of combustion, with attention restricte to the near-stagnation-point region, where the solution is self-similar an all variables are only epenent on the istance to the stagnation plane. The resulting formulations isplay a reuce number of controlling parameters that effectively emboy epenences of the structure of the spray flame on spray ilution, roplet inertia, an fuel preferential iffusion. Sample solutions are given for the limiting cases of pure vaporization an of infinitely fast chemistry, with the latter limit formulate in terms of chemistry-free coupling functions that allow for general nonunity Lewis numbers of the fuel vapor. Receive 15 August 214; revise 29 September 214; accepte 29 September 214. Publishe as part of the Special Issue in Honor of Professor Forman A. Williams on the Occasion of His 8th Birthay with Guest Eitors Chung K. Law an Vigor Yang. Aress corresponence to Antonio L. Sánchez, Area e Mecanica e Fluios, Universia Carlos III e Mari, Leganes 28911, Spain. asanchez@ing.uc3m.es Color versions of one or more of the figures in the article can be foun online at gcst. 13

4 14 A. LIÑÁN ET AL. Keywors: Counterflow spray flames; Coupling functions; Diffusion-controlle reaction; Spray mixture fraction INTRODUCTION For the high Reynols numbers typically encountere in combustion applications the flow is turbulent an the flames appear embee in thin mixing layers that are locally istorte an straine by the turbulent motion (Peters, 2). In applications involving spray combustion, the interactions of the flame with the flow are also epenent on the presence of the fuel roplets (Sirignano, 21). These interactions can be investigate by consieration of simple laminar problems, an example being the counterflow mixing layer investigate here, which has been wiely use as a cartoon to represent local flow conitions in straine mixing layers (Peters, 2). Counterflow structures that move with the mean velocity can be abstracte from the interface ynamics of shear an mixing layers (Corcos an Sherman, 1976). Local counterflow spray configurations are encountere in typical combustion chambers aroun the stagnation point that forms near the injector exit as a result of vortex breakown of the swirling air-fee stream (see, for example, Ewars an Ruoff, 199). Counterflow configurations have been employe in previous experimental analyses of spray iffusion flames, with numerous funamental contributions originating from the combustion laboratories at UCSD (Li, 1997; Li an Williams, 2; Li et al., 1993; Puri an Libby, 1989) an at Yale university (Chen an Gomez, 1992; Gao et al., 1996; Massot et al., 1998; Santoro an Gomez, 22; Santoro et al., 22). Numerical analyses were evelope in parallel efforts. Continillo an Sirignano (199) provie for the first time a two-continua formulation for spray flames in counterflow mixing layers an the conitions neee for the solution to remain self-similar in the vicinity of the stagnation point, where flui properties are functions of the istance to the stagnation plane. The two-continua escription applies to the ilute spray conitions typically foun in the main vaporization an combustion region of practical liqui-fuele combustion evices (Sirignano, 21), when the interroplet istances are significantly larger than the roplet iameter an, for the counterflow configuration, smaller than the mixing-layer thickness. Then each roplet moves an vaporizes iniviually in the gas environment provie collectively by the roplets, which inclues the statistically smoothe effect of the wakes of the neighboring roplets, where the exchanges of fuel, energy, an momentum with the gas have been umpe. This allows us to use a homogenize treatment of the isperse phase, in which the roplets appear as istribute point sources, resulting in source terms in the gas-phase equations that are proportional to the number of roplets per unit volume. The two-continua formulation, terme multicontinua formulation when use for the analysis of polyisperse sprays by incorporation of several roplet classes in the computation, has been use to explore ifferent aspects of counterflow spray iffusion flames. The computation is simplifie when the roplets are sufficiently small that they vaporize completely before crossing the stagnation plane (Kee et al., 211; Laurent an Massot, 21; Schlotz an Gutheil, 2; Wang et al. 213; Zhu et al., 212). However, as note by Puri an Libby (1989) an Chen et al. (1992), sufficiently large roplets cross the stagnation plane an even unergo oscillatory trajectories, a general complicating characteristic of particle-laen stagnation-point flows (Fernánez e la Mora an Riesco-Chueca, 1988;

5 SPRAY VAPORIZATION AND COMBUSTION 15 U s INERT +SPRAY z r δ m 2H AIR U A Figure 1 Schematic view of the typical experimental arrangement employe in experimental stuies of counterflow spray flames. Robinson, 1956). As shown by Gutheil an Sirignano (1998), this can be successfully hanle in the self-similar counterflow formulation by consieration of ifferent sheets of solutions, thereby enabling computations that may account for oscillatory roplet trajectories (Gutheil, 21; Hollmann an Gutheil, 1998; Olguin an Gutheil, 214). The multicontinua formulation can be extene to the treatment of realistic roplet-size istributions by consieration of a large number of roplet classes (or sectionals ). A ifferent sectional approach is followe by other authors (Gao et al., 1996; Massot et al., 1998), who use as starting point the spray equation originally erive by Williams (1985). THE COUNTERFLOW PROBLEM In this article, we shall analyze the vaporization an combustion of sprays in axisymmetric counterflow arrangements involving two high-reynols-number opposing streams, one of air an the other containing a polyisperse fuel spray carrie by nitrogen. Figure 1 represents the typical setup use in experimental stuies, which involves two opposing nozzles of raius R whose exits are locate a istance 2H apart. The resulting axisymmetric coaxial counterflowing jets are separate by a laminar stagnation-point mixing layer, to be escribe in terms of the raial an axial coorinates r an z measure from the stagnation point. The Reynols number Re = U s R/ν s, base on the characteristic injection velocity U s an kinematic viscosity ν s of the spray-carrier gas, an the accompanying Reynols number of the hotter air stream are moerately large in typical applications. Uner those conitions, the flow of the counterflowing streams is nearly invisci an inclues a potential region near the stagnation point where the gas velocity v = (u, v) is etermine by the uniform strain rate foun on each sie of the stagnation plane. On the spray sie the flow is given by R u = A s z an v = A s r/2 (1)

6 16 A. LIÑÁN ET AL. of the counterflow mixing layer, measures the coupling of the roplets with the gas flow, whereas the ratio α/st measures the coupling of the gas phase with the roplets. In vaporizing sprays, effective two-way coupling occurs in the ouble istinguishe limit St = O(1) an α = O(1). The coupling is more pronounce in the presence of combustion, because the heat release by burning the fuel vapor is enough to lea to flame temperatures several times larger than the spray fee temperatures. In analyzing the interphase coupling in burning sprays one shoul bear in min that in the combustion of typical hyrocarbon fuels the air-to-fuel stoichiometric ratio S (i.e., the mass of air neee to burn the unit mass of fuel) is a large quantity of orer S 15. As a result, very ilute sprays with relatively small values of α S 1 1 may generate iffusion-flame temperatures of the orer of the stoichiometric aiabatic flame temperature, thereby proucing a strong effect on the gas flow through the associate gas expansion. The analysis in this article will focus on values of the Stokes number of orer unity an values of the liqui mass-loaing ratio α of orer S 1. Since α 1, we fin one-way coupling of the roplets in the spray stream, but strong two-way coupling in regions affecte by the fuel-vapor combustion if the gas-phase reaction has been ignite. For these ilute sprays, the computation of the roplet motion ownstream from the injection plane, given in the Appenix, reveals ifferent behaviors epening on the value of St. For St < 1/4, the roplets are seen to approach the stagnation plane with a vanishing transverse velocity, whereas for St > 1/4 they cross the stagnation plane an move into the opposing air stream. These two behaviors lea to two istinct regimes of spray vaporization an combustion, which are analyze separately below. For St < 1/4, we fin that the roplets are trappe in the mixing layer, where roplet vaporization an gas-phase chemical reactions occur. For St > 1/4, on the other han, the roplets traverse the stagnation plane with a crossing velocity that is much larger than the transverse gas velocity in the mixing layer, penetrating large istances of the orer of the initial injection istance into the counterflowing stream before they turn aroun. Droplet vaporization occurs in this case on the air sie, with the in terms of the spray-sie strain rate A s, a quantity of orer U s /R. The corresponing strain rate foun on the air sie is, in general, ifferent, with a value A A = A s ρs /ρ A ictate in terms of the inert-to-air ensity ratio by the conition of negligible pressure variation across the mixing layer. Because of the prevailing large Reynols number flow, mixing between both streams occurs only in a thin layer at the separating surface, whose characteristic thickness is δ m (ν s /A s ) 1/2 R/Re 1/2 R. In the vicinity of the central stagnation point, the mixing layer exhibits a self-similar structure in terms of the strain rate A s in which v/r an the other flui variables are a function of the istance z to the stagnation plane. Typically in experiments the roplets are injecte at a istance z I from the stagnation plane much larger than the mixing-layer thickness. The initial temperature of the roplets an of the inert gas are often sufficiently lower than the boiling temperature of the liqui fuel for roplet vaporization in the spray stream to be negligible. The escription of the motion of the nonvaporizing roplets in the nearly-invisci inert stream is given in the Appenix. Because of their iverging raial motion, only the roplets initially locate near the axis, where r R, eventually enter the self-similar region of the mixing layer aroun the stagnation point. Two important parameters, epenent on the roplet size, govern the coupling between the liqui an gas phases in vaporizing sprays, namely, the Stokes number St, efine in (9), an the ratio α of the liqui mass to the mass of gas per unit volume, efine in (1) (Sánchez et al., in press). The Stokes number, which for the counterflow is the ratio of the roplet acceleration time (which is of the orer of its vaporization time) to the characteristic strain time A 1 s

7 SPRAY VAPORIZATION AND COMBUSTION 17 inertial roplets istributing the fuel vapor over transverse istances much larger than the mixing-layer thickness. Corresponingly, when this fuel vapor reacts with the oxygen of the air, the iffusion flame that forms stans away from the mixing layer, with a structure markely ifferent from that foun for St < 1/4. SPRAY VAPORIZATION AND COMBUSTION IN THE COUNTERFLOW MIXING LAYER The roplet velocity v = (u, v ) an the roplet number ensity foun near the stagnation plane outsie the mixing layer are etermine by the evolution of the nearaxis roplets as they move from z = z I until they finally reach the stagnation plane z =. As shown in the Appenix, for ilute sprays with small values of the liqui mass-loaing ratio, the roplets with Stokes number St < 1/4 approach the stagnation-point region with axial an raial velocity components an u = 1 1 4St A s z (2) 2St 2St v = A s r/2, (3) St inepenent of the injection conitions. Because of their vanishing axial velocity, instea of crossing to the air sie, these roplets remain in the mixing layer, corresponing to small axial istances z of the orer of the mixing-layer thickness δ m, where they vaporize when encountering the hot air. In this section we give the multi-continua formulation for spray vaporization an combustion in the counterflow mixing layer, the relevant regime for roplets with St < 1/4. Attention is restricte to the near-stagnation-point region, where the flow has a selfsimilar structure etermine by the strain rate A s, in which the gas phase is escribe in terms of the raial an axial velocity components v = A(z) r / 2 an u(z), temperature an ensity T (z) an ρ(z), an mass fractions Y i (z). A polyisperse spray with N c ifferent roplet classes is consiere. For each roplet class j, the continuum solution is given in terms of the roplet number ensity n j (z), roplet raial an axial velocity components v j = A j (z) r/ 2 an u j (z), an roplet raius a j (z) an temperature T j (z), the latter assume to be uniform insie the roplet, a vali approximation when the thermal conuctivity of the liqui fuel is much larger than that of the gas surrouning the roplet (Law an Sirignano, 1977). We begin by giving the expressions for the exchange rates of momentum, energy, an mass between the two phases, followe by the equations an bounary conitions for the liqui an gas phases. The formulation inclues in the bounary conitions for the liqui phase the roplet velocity istributions given in (2) an (3) an the accompanying roplet number ensity given in (A1), which hol at intermeiate istances δ m z R. Together with the case of pure spray vaporization, specific consieration will be given below to the limit of infinitely fast reaction an its formulation in terms of coupling functions (Arrieta- Sanagustín et al., 213; Sánchez et al., in press).

8 18 A. LIÑÁN ET AL. Droplet Submoels The rag force f j acting on the iniviual roplet of each class, its rate of vaporization ṁ j, an heating rate q j, which epen in general on the roplet-gas slip motion, are evaluate below for the case of roplet Reynols numbers small compare with unity, leaing to a set of compact expressions. Effects of near-roplet convection associate with the slip velocity introuce corrections to the exchange rates that, surprisingly, remain moerately small as the slip-flow Reynols number increases to values of orer unity, so that the escription given in (4) (8) provies sufficient accuracy uner most conitions of interest. More complete roplet moels, incorporating epenences on roplet Reynols number as well as influences of aitional effects not contemplate in the erivation given below are available (Abramzon an Sirignano, 1989) an coul be incorporate in the counterflow formulation. The expressions given below result from the quasi-steay analysis of the flow fiel near the iniviual roplet, using the local gas-phase values for the outer conitions. They inclue the familiar Stokes law for the force of the gas on the iniviual roplet f j = 6πμa j( v v j ) [ ( ) = 6πμa j u u j, A A j r / ] 2 where μ is the viscosity of the gas surrouning the roplet. The rate of vaporization an the rate of heating of the iniviual roplet an ṁ j = ( 4πκa j/ c p ) λ j ( ) j q j T T = 4πκa j e λ j 1 L v λ j (6) c p are expresse in terms of the imensionless vaporization rate λ j, an eigenvalue of the problem, representing a Stefan-flow Peclet number base on the mean raial gas velocity at the roplet surface. Here, κ an c p are the thermal conuctivity an the specific heat at constant pressure of the gas an L v is the latent heat of vaporization of the fuel. The value of λ j is foun to be given by (4) (5) λ j = 1 Le F ln ( ) 1 Y F 1 Y j F,S (7) in terms of the fuel-vapor Lewis number Le F an the values of its mass fraction in the atmosphere surrouning the roplet Y F an at the liqui surface Y j F,S ; the latter etermine by the Clasius Clapeyron relation in terms of the roplet temperature M j S M F Y j F,S = exp ( ) L v L v R F T B R F T j (8)

9 SPRAY VAPORIZATION AND COMBUSTION 19 Here, M F an M j S are the molecular mass of the fuel an the mean molecular mass of the gas at the roplet surface, R F = R o/ M F is the fuel gas constant, an T B is the boiling temperature of the fuel at the chamber pressure. The computation is simplifie here by employing / ) j the expression M F M S = Y j FS (1 + Y j / FS M F MN2, an approximation that accounts for the large ifferences of the molecular masses of the fuel vapor an N 2, while taking the molecular mass of all other species equal to that of nitrogen. In that case, Eq. (8) can be use to etermine Y j j F,S as a function of T, while (7) gives explicitly λ j in terms of Y j F,S an Y F. For most liqui fuels, the latent heat of vaporization is sufficiently large that the conition L v R F T B is satisfie. Accoring to (8), the fuel-vapor mass fraction on the roplet surface Y F,S remains exponentially small as long as the roplet temperature T j stays ( sufficiently below T B, i.e., its value is such that T B T) j /TB [ / L v (RF T B ) ] 1.Asa result, when the roplets are injecte in a col carrier gas, the initial rate of vaporization becomes negligibly small, as can be seen from (7) with Y F = an Y j F,S 1. In this case, significant vaporization is seen to occur only after the roplets enter in contact with the hot air in the mixing layer; an changes in the roplet raius can be neglecte altogether when stuying the roplet evolution in the outer stream, as one in the Appenix. Dimensionless Formulation The spray-sie value of the strain rate A s an the associate characteristic mixinglayer thickness δ m = ( / ) 1 /2 D Ts As, where DTs is the thermal iffusivity of the unperturbe carrier gas, will be use as scales in efining the imensionless variables z = z / δ m, Ã = A / A s, ũ = u / (A s δ m ), Ã j = A / j As, an ũ j = u / j (As δ m ). Similarly, the unperturbe ensity ρ s an temperature T s of the carrier gas will be use to scale ρ = ρ / ρ s, T = T / T s, an T j = T / j Ts. The initial raius of each roplet class at the injection location a j I will be use to efine the imensionless value of the roplet raius ã j = a j/ a j I. For counterflow configurations with large Reynols numbers Re, the analysis given in the Appenix reveals that n j, the number of roplets per unit volume, has a characteristic value in the mixing layer nm j much larger than the value at the injection plane n j I accoring to n / m j j n I = B ( R / ) C j δ m 2St ) Re C j/2 /, where C j = 1 2( j (1 ) 1 4St j an B is a constant of orer unity. Hence, to investigate the solution in the mixing layer, we use nm j to scale the number ensity accoring to ñ j = n j/ nm j. For each roplet class, the roplet raius at injection a j I an the characteristic number ensity nm j will be seen to appearin the resulting formulation through the Stokes number St j = 2 9 A s ( a j I) 2 ρl /μ s (9) an the liqui mass-loaing ratio α j = ( 3 (4π/3) ai) j n j m ρ l (1) ρ s

10 11 A. LIÑÁN ET AL. where ρ l is the ensity of the liqui fuel. For simplicity, the tile enoting nonimensional quantities is remove in the remainer of the article. Given the gas-phase istributions of temperature an velocity, the evolution of each roplet class j requires integration of the equations following the roplet trajectories ( ) 2 A j 2 u j u j z = 1 T σ ( ) St j ( ) u u j a j 2 + u j A j z = 1 St j T σ ( ) ( ) A A j a j 2 (11) (12) u j ( a j) 3 = 2 z 3PrSt j a j T σ λ j (13) u j T j z = 2c ( ) p/c l T σ j T T 3PrSt j ( ) a j 2 e λ j 1 L v λ j (14) c p T s ( ) n j u j + n j A j z = (15) supplemente with the expressions (7) an (8), neee to compute the imensionless vaporization rate λ j. For roplets with St < 1/4, the initial conitions consistent with the solution foun at intermeiate istances δ m z R, given in (2), (3), an (A1), are a j 1 = u j St j 2St j z = A j 2St j St j = T j 1 = n j z C j = asz yieling a convenient escription inepenent of the specific injection conitions. In writing (16), roplet vaporization prior to entering the mixing layer has been neglecte along with ifferences of the roplet temperature from that of the carrier gas. To complete the formulation we give now, using the nonimensional variables efine above, the gas-phase conservation equations, beginning with the continuity an raial momentum equations (16) 2 (ρu) + ρa = z 3Pr N c j=1 α j St j n j a j T σ λ j (17) ρa ρua z = Pr z ( T σ A ) z N c + j=1 α j ( ) ( St j n j a j T σ A j A ) 3Pr λ j (18)

11 SPRAY VAPORIZATION AND COMBUSTION 111 If the chemical reaction between the oxygen of the air an the fuel vapor is assume to occur accoring to the global irreversible step F + so 2 (1 + s) P + q, where s an q are the mass of oxygen consume an the amount of heat release per unit mass of fuel burne, then the equations for energy an reactants become z (ρut) + ρat = ( T σ T ) z z + q S ( SωF ρ s A s ) + 2 3Pr N c j=1 ( α j St j n j a j T σ λ j T j T T ) (19) j e λ j 1 z (ρuy F) + ρay F = 1 ( T σ Y ) F 1 ( ) SωF + 2 Le F z z S ρ s A s 3Pr ( ) ) (ρuŷ O + ρaŷ O = T σ Ŷ ( ) O SωF z z z ρ s A s N c j=1 α j St j n j a j T σ λ j (2) where q = q / ( c p T s ) is a imensionless combustion heat per unit mass of fuel, an the factor S = s/y O2 A in (21) represents the amount of air neee to burn the unit mass of fuel vapor, a fairly large quantity for most fuels of practical interest (e.g., S 15 for oecane). Here, ω F is the mass of fuel consume per unit volume per unit time an Ŷ O = Y O2 /Y O2 A is the mass fraction of oxygen scale with its value on the air sie Y O2 A.232. A Fickian escription is aopte for the species iffusion velocities, with Le F enoting the Lewis number of the fuel vapor an a unity value assume for that of O 2. The gas Prantl number Pr appearing in (19) an (2) is assume to be Pr =.7. A simple power-law T σ with exponent σ =.7 has been assume for the temperature epenence of the ifferent transport coefficients. The chemical-reaction terms appear written in (19) (21) in terms of the imensionless oxygen-consumption rate (Sω F ) / (ρ s A s ), which when important shoul result in changes of orer unity in Ŷ O, as can be inferre from (21). The same imensionless rate is multiplie by q/s in (19), thereby introucing changes in the imensionless temperature T of orer q/s, an by S 1 in (2), generating changes in Y F of orer S 1. This fuel mass fraction will be provie by the last term in (2) if α j is of orer S 1,asitisinthe istinguishe regime α j S 1 consiere below. The above equations (17) (21) are to be integrate with the bounary conitions (21) { u + z = A 1 = T 1 = YF = Ŷ O = as z + A T A = T T A = Y F = Ŷ O 1 = as z. (22) Differences in molecular weight between the two fee streams have been neglecte in writing the bounary conition for the strain rate on the air sie, so that the value A A = A s ρs /ρ A simplifies to T A when expresse in imensionless form. Note that an arbitrary zero isplacement of the spray stream is assume in writing the bounary conition u + z = asz +. The location z = z of the stagnation plane, where u =,

12 112 A. LIÑÁN ET AL. is obtaine as part of the integration. The above equations must be supplemente with the equation of state written in the nonimensional form ρt = [ 1 Y F ( 1 MN2 /M F )] 1 (23) To complete the formulation we shoul give the finite rate of fuel consumption ω F.Inthis article, we shall limit the escription to the two extreme limiting cases of negligible an infinitely fast reaction rate. Governing Parameters The imensionless formulation given above serves to ientify the parameters that control the structure of spray iffusion flames. Some of the parameters are relate to the properties of the fuel, incluing its specific heat c l an molecular mass M F, which appear through the ratios c p /c l an M N2 /M F in (14) an (23), respectively, the latent heat of vaporization L v, which appears in imensionless form in (8) an (14), the fuel Lewis number Le F, present in (7) an (2), an the boiling temperature T B, which enters in the Clasius Clapeyron relation (8). The main thermochemical parameters involve in the chemical reaction, i.e., the mass S of air neee to burn the unit mass of fuel vapor an the imensionless heat of reaction q = q / ( ) c p T s, are also fuel epenent, although the ifferences are only small between fuels that share the same molecular structure, such as saturate hyrocarbons. For instance, for heptane an oecane S (15.2, 15) an q =(45, 44.5) kj/g, giving a characteristic imensionless temperature increase q / S = q /( ) Sc p T s =(8.22, 8.24) when evaluate at the normal temperature Ts = 3 K with the average specific heat c p = 12 J / (kg K). For each roplet class, the inertia of the roplets an the ilution of the spray are characterize by the Stokes number St j an the liqui mass-loaing ratio α j given in (9) an (1), respectively. It is of interest that, since the characteristic times for roplet vaporization an roplet heating are comparable to the roplet acceleration time (Sánchez et al., in press), the Stokes number St j characterizes not only the coupling of the roplet motion with the gas flow in (11) an (12) but also their vaporization an heating, as can be seen in (13) an (14). As previously anticipate, α j/ St j measures in (17) (2) the coupling of the gas flow with the roplets. Since for all liqui fuels the mass of air S neee to burn the unit mass of fuel is always a large quantity, fairly small values of α 1 are sufficient to generate a robust spray flame. For these ilute conitions, the irect effects of roplet vaporization, heating, an acceleration on the gas motion are negligible, as can be inferre from observation of the roplet source terms in (17) (2), although significant interphase coupling still exists associate with the strong exothermicity of the chemical reaction. The bounary conitions (22) introuce only one aitional parameter in the escription, namely, the free-stream temperature ratio T A. An attractive characteristic of the formulation given here is that the bounary conitions for the liqui phase, given in (16), are inepenent of the injection conitions, whose effects are reflecte mainly on α j through the value of the apparent number ensity nm j. The Burke Schumann Formulation of Counterflow Spray Flames The above formulation can be use to compute reacting sprays an also purely vaporizing sprays, the latter given by ω F = in (19) (21). Reactive solutions epen on the

13 SPRAY VAPORIZATION AND COMBUSTION 113 competition of the chemical reaction rate with the transport rates of heat of species an also with the interphase exchange rates. The solution can be simplifie in the Burke Schumann limit of infinitely fast reaction rate, when the chemical reaction is seen to occur in a flame sheet locate at z = z f, separating a region for z > z f where Ŷ O = from a region for z < z f where Y F =, whereas at the flame both reactant concentrations are simultaneously zero. As inicate elsewhere (Arrieta-Sanagustín et al., 213; Sánchez et al., in press), to hanle the Dirac-elta character of the reaction term associate with the limit of infinitely fast reaction one may follow the general proceure suggeste by Shvab (1948) an Zelovich (1949) for gaseous iffusion flames, appropriately extene to account for the nonunity Lewis number of the fuel vapor (Liñán, 1991; Liñán an Williams, 1993; Liñán et al., 215). Thus, subtracting (21) from (2) times S leas to [ )] ) ρu (SY F Ŷ O + ρa (SY F Ŷ O = [ T σ / ) (SY ] F LeF Ŷ O z z z + 2S 3Pr N c j=1 α j which can be written in the alternative form / S LeF + 1 (ρuz) + ρaz = z S + 1 z St j n j a j T σ λ j (24) ( T σ Z z involving a iffusion-weighte mixture-fraction variable, ) Z = SY / F LeF Ŷ O + 1 S / Le F + 1 in aition to the classical mixture-fraction variable, Z = SY F Ŷ O + 1 S Pr N c j=1 α j St j n j a j T σ λ j (25) (26) (27) A similar manipulation of (19) an (21) yiels z (ρuh) + ρah = ( T σ H ) + 2 z z 3Pr N c j=1 ( α j St j n j a j T σ λ j H j T T j e λ j 1 ) (28) for the excess enthalpy, ) H = T T A +(ŶO 1 q / S (29)

14 114 A. LIÑÁN ET AL. In the sum over roplet classes, H j = T j T A q / S represents the excess-enthalpy value for the vaporizing fuel vapor of each roplet class. In this case, since the Lewis number of oxygen is assume to be unity, the coupling functions emerging in the iffusion an convective terms in (28) are ientical, thereby simplifying the formulation. The bounary conitions for (25) an (28) are given by Z Z st = Z Z st = an H = T s T A q / S as z (3) Z = Z = an H = as z (31) where Z st = 1 / (1 + S) an Z st = 1 /( 1 + S / Le F ). In the escription of the limit of infinitely fast reaction, the three conservation equations for the energy an the reactants (19) (21) are replace with the chemistry-free equations (25) an (28), together with the conition, Y F Ŷ O = (32) of noncoexistence of the reactants. The flame is locate where both the vapor fuel Y F an the oxygen Ŷ O are simultaneously zero, corresponing to values of the mixture fraction Z = Z st or Z = Z st.forz Z st, we fin Ŷ O = an whereas for Z Z st, Y F = an Y F = Z Z st 1 Z st = Z Z st 1 Z st an T = T A + H + q S (33) Ŷ O = 1 Z = 1 Z an T = T A + H + q Z (34) Z st Z st S Z st These relationships link the values of Z, Z, an H an provie the mass fractions of reactants an the temperature in terms of the coupling functions across the mixing layer. If neee, source-free conservation equations that etermine the prouct concentrations can be obtaine from linear combinations accounting for nonunity Lewis numbers of CO 2 an H 2 O (Arrieta-Sanagustín et al., 213). Sample Numerical Results The above formulation can be use to investigate ifferent aspects of straine spray iffusion flames for the two limiting regimes of zero an / infinitely fast reaction / rates. In the sample /( integrations ) shown below, / the values c p cl =.543, M N2 MF =.165, Le F = 2.62, L v cp T s = 1.5, TB Ts = 1.63, an q = are employe, as correspons to oecane with T s = 3 K an with a constant mean value c p = 12 J / (kg K) assume for the specific heat of the gas mixture. Also, since the air is often preheate in fuel-spray applications, an elevate air-to-inert temperature ratio T A = 2 is consiere. We begin by investigating solutions corresponing to chemically frozen flow, obtaine by removing the chemical source terms in (19) (21). Sample profiles obtaine for

15 SPRAY VAPORIZATION AND COMBUSTION 115 T 2n 3 2 Y F, Yˆ O2 1 u a 1 T, u, u, a, n.5 Ŷ O2 Y F /α u z - z Figure 2 Structure of a vaporizing monoisperse oecane spray in a counterflow mixing layer for α =.2 an St =.2. a monoisperse oecane spray with α =.2 an St =.2 are shown in Figure 2, where the axial istance is measure with respect to the stagnation plane, which was foun to lie at z =.69. Due to their inertia, the roplets are seen to accumulate, as can be seen in the profile of n. The roplet raius remains constant until the surrouning gas temperature increases to values sufficiently close to the boiling temperature as the roplets approach the stagnation plane. The large resience time associate with the limite axial velocities foun as the roplets approach z = z facilitates roplet vaporization, so that the raius a is seen to ecrease rapily across a thin vaporization region ajacent to the stagnation plane. Rapi roplet vaporization generates fuel vapor that accumulates near z = z an then iffuses to both sies of the mixing layer, mixing with the oxygen of the air. The limit of infinitely fast reaction is consiere in Figure 3, with all parameters being ientical to those of Figure 2. The computation makes use of (25) an (28) as a replacement for (19) (21). The profiles of Z, Z, an H, scale with their characteristic values Z Z α an H q/s, are given in the lower plot, an the associate profiles of Y F, Ŷ O, an T, calculate from (33) an (34), are shown in the upper plot, along with the profiles of u, u, a, an n. As can be seen, in the fast-reaction limit the graients of temperature an mass fractions have jumps at the flame sheet, while the graients of Z an H are continuous. The graient of the classical mixture fraction Z also jumps at the flame, as correspons to a localize chemical source. The comparison of Figures 2 an 3 clearly shows how thermal expansion moifies significantly the velocity fiel in the presence of combustion, as can be seen by observation of the profile of axial velocity. As a result, the stagnation plane, locate at z =.69 for chemically frozen flow, is isplace to z = 2.75 for infinitely fast reaction. The roplet behavior is also ifferent when a spray iffusion flame is present, because the temperature increase associate with the chemical heat release enhances roplet vaporization, with the result that the roplets isappear far from the stagnation plane at a relatively thin vaporization layer where the fuel vapor is seen to accumulate, giving a peak value of Y F of orer α. The fuel vapor iffuses both upstream, against the incoming flow, an also ownstream, to

16 116 A. LIÑÁN ET AL. Y F, Yˆ O2 u T a 2n T, u, u a, n Ŷ O2 Y F /α u z - z Z st /α z - z Z/α Z/α HS/q Figure 3 Structure of a monoisperse oecane spray flame in a counterflow mixing layer for α =.2 an St =.2. reach the iffusion flame an react with the oxygen of the air arriving there by iffusion, with fluxes in stoichiometric proportions. The external sheath combustion regime shown in Figure 3, with the spray vaporizing at a istance from the flame, is the configuration encountere in most spray counterflow iffusion flames; this was verifie in numerical integrations by varying the ifferent controlling parameters. For larger values of α, the flame tens to move into the air sie of the mixing layer. To enable the assessment of preferential iffusion effects, Figure 4 exhibits the results obtaine when the fuel-vapor Lewis number is set equal to unity in the integrations. Changing the fuel-vapor iffusivity moifies its transport rate across the mixing layer an also the solution for the local fuel-vapor profile in the vaporization region aroun the roplets. The latter moification has an impact on the spray flow through the perturbe roplet vaporization rate, as can be seen in (7), with λ being proportional to the reciprocal of Le F. The two separate phenomena have counteracting effects on the amount of fuel vapor present in the vaporization region. Thus, ecreasing the Lewis number from Le F = 2.62 to Le F = 1 is expecte to increase irectly the prouction rate of fuel vapor as ictate by (7), an therefore the associate local value of Y F. However, a smaller Lewis number promotes also the rate of fuel-vapor iffusion from the vaporization region, thereby ecreasing

17 SPRAY VAPORIZATION AND COMBUSTION 117 Y F, Yˆ O u Ŷ O2 Y F Le F = 1. Le F = a T u 2n T, u, u, a, n z - z Figure 4 Structure of a monoisperse oecane spray flame in a counterflow mixing layer for α =.2 an St =.2. Besies the results obtaine with the Lewis number of oecane (i.e., Le F = 2.62), shown in soli curves, the figure represents in ashe curves results obtaine by setting the fuel Lewis number equal to unity. The black curves represent the liqui-phase properties a, n,anu in the region where roplets exist. the resulting peak value of Y F there. As can be seen, both effects are approximately in balance for the case consiere in Figure 4, with the result that the peak value of Y F is almost the same for both computations. The larger iffusivity of the fuel vapor for Le F = 1 results in an increase transport rate from the vaporization region, leaing to a wier Y F profile an to a iffusion flame that lies farther into the air stream. The local balance between the rate of heat loss from, an the rate of fuel iffusion into, the reaction sheet etermines largely the peak temperature achieve at the flame. A ecrease in Le F results in a reuction of the rate of heat loss relative to that of fuel iffusion, an therefore causes an increase of the flame temperature, a well-known ifferential-iffusion effect observe for instance in hyrogen combustion (Sánchez an Williams, 214). This reasoning, base on the local molecular-transport balance at the flame, explains the results shown in Figure 4, where the peak temperature foun for Le F = 1 is consierably larger than that corresponing to the heptane iffusivity. Evaluation of Extinction Conitions The reaction layer shown in Figures 3 an 4 (a sheet in the infinitely fast reaction limit use here) is not affecte irectly by the presence of the roplets. Corresponingly, its internal structure, etermine by a balance between the chemical reaction an the iffusive transport of heat an chemical species, woul be ientical to that foun in gaseous combustion. Computation of finite-rate effects, incluing critical extinction conitions, coul be therefore investigate a posteriori by consiering the gaseous reacting layer locate at Z = Z st. If a chemistry moel with a one-step Arrhenius reaction is aopte, then the extinction regime involves, as shown by Liñán (1974), small eviations from the Burke Schumann solution. The analysis has been generalize to account for preferential iffusion effects associate with nonunity values of the fuel Lewis number (see the etaile extinction

18 118 A. LIÑÁN ET AL. analysis given in the online supplemental appenix of Liñán et al., 215). The structure of the reacting layer is seen to epen on the flame-sheet temperature T f of the Burke Schumann solution, on the flame-sheet value of the scalar issipation rate χ f, an on the fraction of the chemical heat release at the flame that is conucte towars the oxiizer sie, γ ; compute with use mae of the graient of excess enthalpy (H/z) f. Thus, values of T f, χ f, an γ obtaine in the limit of infinitely fast reaction with the Burke Schumann formulation presente previously coul be combine with the analysis of the reaction-iffusion layer of gaseous flames to etermine critical extinction conitions for spray iffusion flames. Note that, in this nonequiiffusional case, the scalar issipation rate at the flame sheet must be evaluate in terms of the graient of the moifie mixture fraction Z, asχ = D T ( Z/z ) 2. This has a value that, contrary to the scalar issipation rate base on the stanar mixture fraction Z, oes not jump across the flame sheet when Le F = Le O2. It is also worth mentioning that, since for all fuels the chemical reaction rate is strongly epenent on the temperature, the extinction conitions are very sensitive to variations of the peak temperature. Therefore, influences of spray ilution, roplet inertia, an fuel-vapor iffusivity on flame extinction coul be easily assesse from the results of the Burke Schumann integrations by investigating how variations of St, α, an Le F affect the resulting values of T f. AIR-SIDE VAPORIZATION AND COMBUSTION OF INERTIAL SPRAYS The evolution of the roplets ownstream from their injection location in high- Reynols-number oppose-jet configurations, investigate in the Appenix, inicates that, when the Stokes number is sufficiently large (i.e., St > 1/4 for ilute sprays of nonvaporizing roplets), the roplets cross the stagnation plane to reach values of z of orer z I into the opposing air stream. The vaporization of the roplets an the reaction of the resulting fuel vapor with the oxygen of the air occur mainly, after crossing the mixing layer, in the air stream, without significant iffusion effects. The escription will be simplifie by consiering that roplet injection occurs in the near-stagnation-point region, i.e., at istances z I much larger than δ m for the Reynols number ( z I / δm ) 2 to be large, but small enough compare with R for the gas-phase solution (1) to apply. The resulting formulation, which employs the length an velocity scales z I an A s z I associate with the injection istance, is elineate below an use to generate some illustrative results for the limiting cases of purely vaporizing sprays an infinitely fast chemistry. Conservation Equations an Bounary Conitions For the analysis, the conservation equations for the liqui an gas phases, given in (11) (15) an in (17) (21), respectively, must be rewritten by introucing the rescale transverse coorinate z / z I along with the rescale variables u / (A s z I ), u/ j (As z I ), an n j/ n j I, while the remaining nonimensional variables are those employe earlier in the mixinglayer analysis, i.e., A / A s, A j / As, T / T s, ρ / ρ s, a j/ a j I, an T / j Ts. The resulting equations for the liqui phase can be seen to be equal to (11) (15), but the bounary conitions (16) use in the mixing-layer analysis must be replace now by a j 1 = u j + u I/ j (As z I ) = A j A j / I As = T j T j / I Ts = n j 1 = at z / z I = 1 (35)

19 SPRAY VAPORIZATION AND COMBUSTION 119 involving the nonimensional injection velocity components u j I /(A sz I ) an A j I /A s an the nonimensional injection temperature T j I /T s. Since the scales for the problem are base on the injection istance z I, in the nonimensional equations for the gas flow the Reynols number ( / ) 2 z I δm appears iviing the molecular transport terms in (18) (21) (an also in Eqs. (25) an (28) for the coupling functions of the fast-reaction limit). In the limit z I δ m, therefore, the equations reuce to the Euler equations. The integration for the spray sie z > must employ as bounary conitions u = atz = an A 1 = T 1 = Y F = Ŷ O = asz ; whereas for z < we must use u = atz = an A T A = T T A = Y F = Ŷ O 1 = asz.the solution must allow for a iscontinuity at the stagnation plane z =, with orer-unity jumps in temperature, strain rate, an composition that are smoothe across the thin mixing layer, which is not escribe in the simplifie iffusionless analysis given here. The numerical computation with the multicontinua formulation requires the couple solution of the gas an liqui phases in an iterative scheme that may start by solving the Euler form of the gas-phase equations (17) (21) in the two separate omains z > an z <, with an aequate starting guess use for the roplet properties. The resulting profiles of velocity, temperature, an reactant mass fractions are next use in computing for each roplet class the istributions of a, T, u, A, an n by integrating (11) (15) from z = z I. The proceure is followe iteratively until convergence is achieve. For ilute sprays with small values of the liqui mass-loaing ratio α (now efine in terms of the roplet number ensity at injection n I ) of orer α S 1, there exists one-way coupling of the roplets with the gas flow in the spray stream z >, where we fin in the gas only small epartures, of orer α, from the unperturbe properties u + z / z I = A 1 = T 1 = Y F = Ŷ O =. For these ilute sprays, strong two-way coupling may appear on the air sie if combustion occurs there. If the spray-carrier temperature T s an the roplet injection temperature ( T I are sufficiently smaller than the boiling temperature T B for the conition T B T) j /TB [ / L v (RF T B ) ] 1 to hol everywhere on the spray sie of the counterflow, then roplet vaporization is entirely negligible on the spray stream. That is the case consiere in the sample computations in Figures 5 an 6 (to be iscusse later), which correspon to oecane sprays injecte at normal atmospheric temperature. Treatment of Reversing Droplets For St > 1/4, the roplets are seen to cross the stagnation plane an penetrate into the air sie, a characteristic of sprays in counterflows note in early work (Chen et al., 1992; Puri an Libby, 1989). In the presence of reverse roplet motion the solution for a given roplet class is no longer uniquely efine in terms of the istance to the stagnation plane, because we may fin avancing roplets an returning roplets at the same location z but with ifferent values of a, T, u, an A. In the Eulerian escription of the roplet ynamics, which is convenient for the self-similar analysis of the spray counterflow, this can be accounte for in the integrations, as propose by Gutheil an Sirignano (1998), by introucing ifferent sheets of solutions or, equivalently, by consiering the avancing an returning roplets as belonging to ifferent classes; so that an aitional inepenent roplet class is ae to the escription when the roplets reverse their motion (Sánchez et al., in press). The implementation of the integration proceure for the turning roplets must account for the local escription of the flow near the turning plane z = z t, where u j =.

20 12 A. LIÑÁN ET AL. 4 n T 4 n T z/z I z/z I 1 u a 1 u a 1 u 1 u z/z I z/z I 2 1 Ŷ O2 Y F /α z/z I 2 1 Ŷ O2 Y F /α z/z I Figure 5 Structure of a vaporizing oecane spray in a counterflow for α =.5 an St = 1. with u I / (As z I ) = 1,A I / As = 1, T I / Ts = 1, an T A / Ts = 3. The profiles in the left-han-sie panel correspon to iffusionless results for (z I /δ m ) 2 1, whereas the right-han-sie plots are obtaine with the complete equations of the mixing-layer formulation for z I / δm = 85. There, the integration of the avancing roplets provies nonzero finite values of a j = a j A j = A j t, an T j = T j t. On the other han, the local axial-velocity istribution u j = 2Tσ t u t ( St a j t ) 2 1/2 (z z t ) 1 /2 t, (36) obtaine from (11) in terms of the local values, at z = z t, of the gas temperature T t an gas velocity u t (with the minus an plus signs corresponing to avancing an returning roplets, respectively), can be use in (15) to show that the roplet number ensity iverges at the turning plane in the form n j = C (z z t ) 1 /2 (37) where the constant C is etermine numerically. To avoi the existence of multivalue functions within a given roplet class, the roplets that have turne are assigne to a newly create roplet class, whose raius, velocity, an temperature are etermine by integrating (11) (14) for increasing z with initial conitions a j = a j t, A j = A j t, u j =, an T j = T j t