Numerical Prediction of Dispersion and Evaporation of Liquid Sprays in Gases Flowing at all Speeds

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1 Numerical Heat Transfer, Part B: Funamentals ISSN: (Print) (Online) Journal homepage: Numerical Preiction of Dispersion an Evaporation of Liqui Sprays in Gases Flowing at all Spees F. Moukalle & M. Darwish To cite this article: F. Moukalle & M. Darwish (2008) Numerical Preiction of Dispersion an Evaporation of Liqui Sprays in Gases Flowing at all Spees, Numerical Heat Transfer, Part B: Funamentals, 54:3, , DOI: / To link to this article: Publishe online: 29 Jul Submit your article to this journal Article views: 96 View relate articles Citing articles: 2 View citing articles Full Terms & Conitions of access an use can be foun at Downloa by: [American University of Beirut] Date: 05 July 2017, At: 07:31

2 Numerical Heat Transfer, Part B, 54: , 2008 Copyright # Taylor & Francis Group, LLC ISSN: print= online DOI: / NUMERICAL PREDICTION OF DISPERSION AND EVAPORATION OF LIQUID SPRAYS IN GASES FLOWING AT ALL SPEEDS F. Moukalle an M. Darwish Faculty of Engineering an Architecture, Mechanical Engineering Department, American University of Beirut, Beirut, Lebanon This work is concerne with the formulation, implementation, an testing of an all-spee numerical proceure for the simulation of turbulent ispersion an evaporation of roplets. The pressure-base metho is formulate, for both the iscrete an continuous phases, within a Eulerian framework following a finite-volume approach an is equally applicable in the subsonic an supersonic regimes. The two-equation k e turbulence moel is use to estimate turbulence in the gas phase with moifications to account for compressibility at high spees, while an algebraic moel is employe to preict turbulence in the iscrete phase. Two configurations involving streamwise an cross-stream injection are investigate, an solutions for evaporation an mixing of roplets spraye into subsonic an supersonic streams are generate over a wie range of operating conitions. Results, isplaye in the form of velocity vector fiels an contour plots, reveal the egree of penetration of the injecte roplet into the gas phase, an the rate of evaporation as a function of inlet gas temperature, inlet roplet temperature, an=or length of the omain. INTRODUCTION Recently there has been a revive interest in the injection of liquis in supersonic streams, particularly with respect to fuel injection techniques for hypersonic flights. These esigns require air-breathing engines capable of supersonic combustion. The scramjet (supersonic combustion ramjet) appears at present to be a practical engine for these types of applications. Its concept is fairly ol, an was the subject of stuies throughout the 1960s an again in the 1980s. However, its coming to fruition epens on, among other things, the evelopment of numerical tools for the simulation of its supersonic combustion process an relate phenomena. More specifically, effective penetration an enhance mixing of hyrocarbon fuels in a gas flowing at supersonic spee are crucial ingreients for the success of any scramjet esign [1]. Three key issues govern the performance of the liqui injection process in the scramjet engine: the penetration of the fuel into the free stream, the atomization Receive 2 November 2007; accepte 30 April The support provie by the Lebanese National Council for Scientific Research (LNCSR) through Grants an is gratefully acknowlege. Aress corresponence to F. Moukalle, Mechanical Engineering Department, Faculty of Engineering an Architecture, American University of Beirut, P.O. Box , Ria El Solh, Beirut, , Lebanon. memouk@aub.eu.lb 185

3 186 F. MOUKALLED AND M. DARWISH NOMENCLATURE A ðkþ P ;... B ðkþ P c P C D Cq ðkþ D ðkþ P ½/ðkÞ Š F B F D h h cor; H H P ½/ ðkþ Š H P ½u ðkþ Š k m cor; _m _M p P k Pr Pr t Q ðkþ R ðkþ Re S S f coefficients in the iscretize equation for / ðkþ source term in the iscretize equation for / ðkþ specific heat at constant pressure rag coefficient coefficient equals to 1=R ðkþ T ðkþ roplet iameter the matrix D operator boy force rag force static enthalpy correction coefficient for heat transport in roplet evaporation moel total enthalpy the H operator the vector form of the H operator turbulence kinetic energy correction coefficient for mass transport in roplet evaporation moel mass rate of roplet evaporation volumetric mass rate of roplet evaporation pressure prouction term in k an e equations laminar Prantl number of flui=phase k turbulent Prantl number of flui=phase k general source term of flui=phase k gas constant for flui=phase k Reynols number base on the roplet iameter source term surface vector Sc Schmit number t time T temperature of flui=phase k u; v velocity components in x an y irections, respectively U f interface flux velocity ðv ðkþ f S f Þ v velocity vector Y vapor mass fraction a volume fraction b ðkþ thermal expansion coefficient for phase=flui k C iffusion coefficient t time step Dh v latent heat e turbulence issipation rate g Kolmogorov microscale k conuctivity coefficient m; m t ; m eff laminar, turbulent, an effective viscosities of flui=phase k q ensity s the stress tensor / general scalar quantity X cell volume Subscripts refers to the roplet phase eff refers to effective values f refers to interface g refers to the gas phase i refers to size group i k refers to phase k nb refers to the east, west,..., face of a control volume NB refers to the East, West,..., neighbors of the main gri point P refers to the P gri point s refers to the roplet surface conition sat refers to the saturation conition vap, g refers to the vapor species in the gas phase of the injecte fuel rops, an the level of fuel=air mixing [2]. It is important for the fuel to penetrate effectively into the free stream so that the combustion process prouces an even temperature istribution; otherwise it will mostly occur along the surface of the combustor, causing inefficient combustor operation an increase cooling problems. Rapi atomization of the fuel is also require for efficient combustion, as it results in increase fuel=air mixing, allowing a higher percentage of the fuel to be burne in the short time before the entire mixture passes out of the combustor

4 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 187 (generally, the flow resience time is of the orer of a few millisecons [3]). This article is aime at eveloping a numerical metho capable of preicting the spreaing an evaporation of liqui roplets injecte in gases flowing at all spees. The complex multiphase flow phenomena governing liqui injection applications involve a continuous gas phase usually compose of air an the evaporating vapor species an one or more isperse liqui phases, each compose of either a single component or a multicomponent fuel. In the case of a single-component fuel, the evaporation rate of the roplet will be uniform, since only one species of spatially uniform properties is present. It shoul be note, however, that it is possible to have several evaporating but chemically istinct species evaporating, i.e., several kins of single-component fuel roplet evaporating, where each is treate as a separate flui interacting with the gas phase. Moreover, each type will have its own evaporation rate. On the other han, a multicomponent fuel [4] consists of a blen of several species of hyrocarbons containe in the same roplet. These hyrocarbons generally have ifferent volatilities, an the high-volatility components will evaporate early in the process while the lower-volatility components will be retaine until later in the process. Thus the molar weight of the multicomponent fuel will vary uring evaporation, which will affect all thermophysical properties of the fuel. Singlecomponent fuel is of interest in this work. Approaches for the simulation of roplet transport an evaporation in combustion systems can be classifie uner two categories, the Lagrangian an Eulerian methos. In both techniques, the gaseous phase is calculate by solving the Navier- Stokes equations with a stanar iscretization metho such as the finite-volume metho. In the Lagrangian approach [5 7], the spray is represente by iscrete roplets, which are avecte explicitly through the computational omain while accounting for evaporation an other phenomena. Due to the large number of roplets in a spray, each iscrete computational roplet is mae to represent a number of physical roplets averaging their characteristics. The equations of motion of each roplet are a set of orinary ifferential equations (ODE) which are solve using an ODE solver, a numerical proceure ifferent from that of the continuous phase. To account for the interaction between the gaseous phase an the spray, several iterations of alternating solutions of the gaseous phase an the spray have to be conucte. In the Eulerian approach [6 9], the evaporating spray is treate as an interacting an interpenetrating continuum. In analogy to the continuum approach of single-phase flows, each phase is escribe by a set of transport equations for mass, momentum, an energy extene by interfacial exchange terms. This escription allows the gaseous phase an the spray to be iscretize by the same metho, an therefore to be solve by the same numerical proceure. Because of the presence of multiple phases, a multiphase algorithm is use. Several investigations ealing with spray moeling have been reporte. Nmira et al. [10] use a Eulerian-Eulerian two-phase approach to stuy thermoplastic fire suppression by water sprays. Raju [11] employe the Monte Carlo probability ensity function metho to moel turbulent spray flames on unstructure gris. Chow [12] stuie numerically the interaction of a water spray with a smoke by subiviing the spray into several classes base on the roplet istribution function.

5 188 F. MOUKALLED AND M. DARWISH Tolpai et al. [13] evelope a quasi-steay roplet vaporization moel in which roplet heating an vaporization take place simultaneously. Kim et al. [14] employe a Eulerian-Lagrangian approach to stuy the initiation an propagation of etonation waves in an air fuel spray mixture. Raju [15] integrate the Monte Carlo probability ensity function, a Lagrangian-base ilute spray moel, an an Eulerian solver to moel turbulent spray flames using parallel computing. Liu an Reitz [16] evelope a mixe laminar an turbulent moel of heat transfer for escribing impinging fuel sprays in irect injection iesel engines. Chen an Pereira [17] use a Eulerian-Lagrangian stochastic moel to investigate a confine evaporating isopropyl alcohol spray issuing into a co-flowing, heate turbulent air stream. Jicha et al. [18] aopte a Eulerian-Lagragian approach to stuy a turbulent gas liqui roplet flow in a two-imensional plane channel. In this work, a numerical metho for the simulation of roplet evaporation an scattering in a stream flowing at any spee is evelope. This is achieve through a multiflui, all-spee, pressure-base finite-volume flow solver in which a roplet evaporation moel is implemente. The moel as it stans oes not take into consierations roplet breakup or coalescence, but ifferent roplet sizes are accounte for. The use of a Eulerian approach has many avantages: the same valiate numerical proceure use for all phases, ease of implementation of acceleration techniques (such as multigri), an improvements to coe can be carrie over to all phases. In the remainer of this article, the governing equations for roplet transport an evaporation are first presente, followe by a brief escription of the iscretization metho an solution proceure. Then, results obtaine for two physical configurations are iscusse. THE GOVERNING EQUATIONS The conservation equations neee to solve for the interacting flows of interest can best be unerstoo by referring to Figure 1a. A gas moving at subsonic= supersonic spee enters a omain with liqui roplets being injecte into the gas while flowing. The roplets move with the gas, evaporate, an ecrease in size. The equations require to solve for this multiphase flow are those representing the conservation of mass, momentum, an energy for both the gas an roplet phases. Moreover, equations to track the mass fraction of the evaporating liqui in the gas phase an to compute the size of the roplets for each roplet phase are neee. Furthermore, for turbulent flows, aitional equations to compute the turbulent viscosity or Reynols stresses are necessary. The number of these equations epens on the turbulence moel use. In this work the stanar k e moel [19, 20] is employe for the gaseous phase, while an algebraic moel base on a Boussinesq approach [8] approximates the turbulence terms in the roplet-phase transport equations. Neglecting interaction between roplets, the flow fiels are escribe by the transport equations presente next. Droplet Evaporation Moel Evaporation is accounte for in the various conservation equations via source terms that are erive following the uniform temperature moel [21 23]. This

6 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 189 Figure 1. (a) A schematic epiction of the physical situation consiere. (b) Control volume. computationally effective roplet moel is base on the assumption of a homogeneous internal temperature istribution in the roplet an phase equilibrium conitions at the surface. The analytical erivation of this moel oes not consier contributions to heat an mass transport through force convection by the gas flow aroun the roplet. Force convection is taken into account by means of two empirical correction factors, m ðkþ is commonly expresse as cor; an hðkþ cor; [24, 25]. The evaporation rate from a roplet m t ¼ _m ð1þ where m is the mass of the liqui roplet an _m is the mass flux, correcte using the Frössling correlation an base on the classical roplet vaporization moel [24, 25]. Using reference values for variable flui properties base on the 1=3 rule of Sparrow an Gregg [26], an integration of the raially symmetric ifferential equations yiels an expression for the transport fluxes, which is given by _m ¼ m cor; _m ¼ m cor; 2p q g;ref C g;ref ln 1 Y vap;g;1 ð2þ 1 Y vap;g;s In Eq. (2) an the equations to follow, the subscript ref inicates that the variable is evaluate at the reference temperature an mass fraction, which are efine as T ref ¼ 1 3 T vap;g;1 þ 2 3 T vap;g;s Y ref ¼ 1 3 Y vap;g;1 þ 2 3 Y vap;g;s ð3þ where T vap;g;s an Y vap;g;s are the temperature an mass fraction of the vapor at the surface of the roplet. Thus q g;ref an C g;ref are the gas ensity an vapor iffusion coefficient evaluate at the reference temperature an mass fraction. Since the

7 190 F. MOUKALLED AND M. DARWISH uniform temperature moel [21 23] is use, the temperature at the roplet surface is basically equal to that of the roplet (i.e., T vap;g;s ¼ T Þ. On the other han, the vapor concentration on the surface of the roplet is foun using the exponential law of Cox-Antoine [21] as X vap;g;s ð p sat Þ¼ p sat p ð4þ with the saturation pressure p sat for a roplet at temperature T obtaine from P sat ðt Þ¼e AþB=ðT þcþ ð5þ where A, B, an C are specific values for the liqui uner consieration. Thus X vap;g;s MW vap Y vap;s ¼ X vap;g;s MW vap þð1 X vap;g;s Þ MW air ð6þ where MW is the molecular weight. The roplet temperature increases ue to heat transfer from the hotter gas phase. Once enough energy has been transferre to overcome the latent heat of evaporation, evaporation is initiate. This can be expresse mathematically as h vap;g;s h ;s ¼ Dh vap ðt ;s Þ ð7þ The heat balance equation for the roplet can be written as m ðc p T Þ t ¼ _ Q evap;s þ _ Q conv;s ð8þ where h is the static enthalpy an _ Q conv;s an _ Q evap;s are the convection an evaporation heat transfer rates, respectively, given by _Q conv;s ¼ p 2 b ðt g;s T Þ ð9þ where b is the correcte convective heat transfer coefficient given by b ¼ h cor; _m c p;vap;ref; =p 2 expð _m c p;vap;ref; =2p k g;ref Þ 1 ð10þ an _Q evap;s ¼ _m Dh v ð11þ The correction factors m cor; an h cor; account for convective mass an heat transport an are compute from [24, 25] m cor; ¼ 1 þ 0:276 Re 1=2 Sc 1=3 h cor; ¼ 1 þ 0:276 Re 1=2 Pr 1=3 ð12þ

8 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 191 where Re, Sc, an Pr are the Reynols, Schmit, an Prantl numbers, respectively, efine as Re ¼ q gkv v g k m g;ref Sc ¼ m g;ref q g;ref C g;ref Pr ¼ m g;refc p;g;ref k g;ref ð13þ From the above, it follows that the energy equation for the roplet can be written as ðm h Þ t ¼ _m ðdh v þ h Þþp 2 b ðt T g Þ ð14þ The right-han sies of Eqs. (2) an (14) represent the mass an energy sources ue to evaporation from the roplet. Gas Balance Equations The continuity, momentum, energy, turbulence kinetic energy, an turbulence issipation rate equations for the gas phase, which is compose of two species, air an vapor, in aition to the mass fraction equation of the fuel vapor in the gaseous phase, are given by q qt ða m t;g gq g Þþr a g q g v g ¼r ra g X _M ðkþ ð15þ Sc t;g k6¼g q qt ða gq g v g Þþrða g q g v g v g Þ¼ a g rp þrs g þ F B g þ FD g X k6¼g _M ðkþ vðkþ ð16þ q qt ða gq g k g Þþrða g q g v g k g Þ¼rða g m eff;g rk g Þþa g ðp k q g e g ÞþS k; q qt ða gq g e g Þþrða g q g v g e g Þ¼rða g m eff;e;g re g Þþa g ð17þ! e g e 2 g C e1 P k C e2 q k g þ S e; g k g q qt ða m t;g gq g H g Þþrða g q g v g H g Þ¼rða g k g rt g Þþr a g rh g Pr t ð18þ þ a g q g g v g þ q qt ða gpþþrða g v g s g þ m t;g rk g Þ þ X p ðkþ 2 ðkþ ðkþ X b T T g _M ðkþ ðkþ Dh v; þ hðkþ k6¼g k6¼g q qt ða gq g Y vap;g Þþrða g q g v g Y vap;g Þ¼rða g C eff ry vap;g Þ ð1 Y vap;g Þ X k6¼g _M ðkþ ð19þ ð20þ

9 192 F. MOUKALLED AND M. DARWISH _M ðkþ The evaporate liqui ð _m ðkþ Þ, given by Eq. (2), as X k6¼g _M ðkþ appearing in the above equations is calculate from ¼ _M vap;g ¼ X k6¼g 6a ðkþ pð ðkþ Þ 3 ð _mðkþ Þ ð21þ Further, the terms F B g an FD g in Eq. (16) represent the boy an rag forces, respectively. For the gas phase, the boy force ðf B g ¼ a gq g gþ can be neglecte; while the rag force ue to liqui roplets is written as F D g ¼ X k6¼g 3 C ðkþ 4 aðkþ D q g ðkþ v ðkþ v g ðvg v ðkþ Þ ð22þ where ðkþ is the roplet iameter of the kth phase an the aeroynamic rag coefficient is given by [24] C ðkþ D 24 5:48 ¼ 0:36 þ þ ðkþ Re ½Re ðkþ Š 0:573 ð23þ In Eqs. (16) an (19), s g is the stress tensor given by s g ¼ a g m eff rv g þrv T g 2 3 ðr v gþi where m eff ¼ m g þ m t;g ð24þ The total enthalpy H g in the energy equation is given in terms of the static enthalpy h g by H g ¼ h g þ 1 2 v g v g þ k g where h g ¼ Y air h air;g þ Y vap h vap;g ð25þ In aition, the two terms on the left-han sie of the energy equation [Eq. (19)] escribe the rate of increase of H g an the rate at which H g is transporte into an out of the control volume by convection. Further, the terms on the right-han sie of Eq. (19) represent, respectively, the rate of energy transfer into the control volume by conuction, the turbulent flux, the rate of work one by boy forces, the pressure work, the viscous work, the heat transfer by convection between the liqui at temperature from T ðkþ an the gas at temperature T g, an the heat ae to the gas phase ue to evaporation of the liqui roplets. Moreover a g, v g,anq g are, respectively, the volume fraction, velocity, an ensity of the gas phase. The gas ensity may either be compute from the air an vapor ensities or the ieal gas relation as 1 ¼ Y ðairþ ðvapþ Y þ or q q g qðairþ q ðvapþ g ¼ p p ¼ R g T g R 0 ðy ðairþ =MW ðairþ ÞþðY ðvapþ =MW ðvaporþ Þ T g where Y an q represent the mass fraction an ensity, R g is the gas constant, an R 0 is the universal gas constant. ð26þ

10 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 193 Droplet Balance Equations The mass, momentum, roplet iameter, an energy conservation equations for a roplet phase are given by q qt ðaðkþ qðkþ ÞþrðaðkÞ vðkþ qðkþ Þ¼r m ðkþ turb; ra ðkþ Sc ðkþ turb;! þ _M ðkþ ð27þ q qt ðaðkþ qðkþ vðkþ ÞþrðaðkÞ qðkþ vðkþ vðkþ Þ¼ aðkþ rp þ FB;ðkÞ þ F D;ðkÞ þ _M ðkþ vðkþ ð28þ q qt ðaðkþ qðkþ ðkþ ÞþrðaðkÞ qðkþ vðkþ ðkþ Þ¼r aðkþ m ðkþ t; Pr ðkþ t; r ðkþ! þ 4 3 ðkþ _M ðkþ ð29þ q qt ðaðkþ qðkþ hðkþ ÞþrðaðkÞ qðkþ vðkþ hðkþ Þ¼r aðkþ þp ðkþ m ðkþ t; Pr ðkþ t; 2ðb ðkþ rh ðkþ! þa ðkþ Þ ðt g T ðkþ qðkþ gvðkþ Þþ _M ðkþ Þ ð30þ ðdhðkþ v; þ hðkþ The meaning of the various terms in the continuity, momentum, an energy equations of the roplet phase is as escribe for the gas-phase equations. The aitional roplet iameter equation (29), which is erive from the mass conservation equation, is solve for every roplet phase to track the variation of the roplet iameter as evaporation occurs. The roplet iameter fiel is use to calculate the evaporation rate an the rag term. As mentione earlier, roplet roplet interactions through breakup an coalescence are not accounte for in this work, an only the liqui mass of each roplet size is transforme to vapor. This effect is currently uner investigation an will form the subject of future stuies. In aition to the above equations, the volume fractions of the various phases have to satisfy a compatibility equation, which for an n-phase flow is given by X n k¼1 a ðkþ ¼ 1 ð31þ The turbulent viscosity of the isperse (roplet) phase, m ðkþ t;, is moele using the approach of Melville an Bray [27], accoring to which m ðkþ t; is given by m ðkþ t; ¼ m t;g q ðkþ k ðkþ q g k g ð32þ

11 194 F. MOUKALLED AND M. DARWISH The ratio of the turbulent kinetic energies of the kth isperse () phase an gas (g) phase is calculate following the approach in [8, 28] as k ðkþ k g ¼ 1 where k ðkþ 1 þðx ðkþ Þ2 ðs ðkþ Þ2 ¼ 1 2 v0ðkþ v 0ðkÞ ð33þ Since roplets o not generally follow the motion of the surrouning flui from one point to another, the ratio k ðkþ =k g is ifferent from unity an varies with particle relaxation time t an local turbulence quantities. Krämer [28] recommens the following equation for the frequency of the particle response: 0qffiffiffiffiffiffiffi x ðkþ ¼ k 11=4 s ðkþ A s ðkþ s ðkþ ¼ 1 q ðkþ ð ðkþ Þ 2 1 L x 18 q g n g 1 þ 0:133ðRe ðkþ with a characteristic macroscopic length scale of turbulence given by Þ0:687 ð34þ L x ¼ðc m Þ 3=4 ðk g Þ 3=2 e g ð35þ For the turbulent Schmit number of the roplet phase, Sc ðkþ t;,krämer [28] suggests a value of 0.3. However, in a more recent work [8], it was foun to be particle size-epenent, an a value of 0.7 is use in this work (Sc ðkþ t; ¼ 0:7). For the turbulent Prantl number, a value of 0.85 was chosen (Pr ðkþ t; ¼ 0:85). DISCRETIZATION PROCEDURE A review of the above ifferential equations reveals that they are similar in structure. If a typical representative variable associate with phase (k) is enote by / ðkþ, the general fluiic ifferential equation may be written as qða ðkþ q ðkþ / ðkþ Þ þrða ðkþ q ðkþ u ðkþ / ðkþ Þ¼rða ðkþ C ðkþ r/ ðkþ Þþa ðkþ Q ðkþ qt ð36þ where the expressions for C ðkþ an Q ðkþ can be euce from the parent equations. The general conservation equation (36) is integrate over a finite volume (Figure 1b) to yiel ZZ X qða ðkþ q ðkþ / ðkþ ZZ Þ X þ rða ðkþ q ðkþ u ðkþ / ðkþ ÞX qt X ZZ ZZ ¼ rða ðkþ C ðkþ r/ ðkþ ÞX þ a ðkþ Q ðkþ X X X ð37þ where X is the volume of the control cell. Using the ivergence theorem to transform a volume integral into a surface integral, replacing the surface integrals by a summation of the fluxes over the sies of the control volume, an then iscretizing

12 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 195 these fluxes using suitable interpolation profiles [29 31], the following algebraic equation results: A ðkþ P /ðkþ P ¼ X A ðkþ NB /ðkþ NB þ BðkÞ P NB ð38þ In compact form, the above equation can be written as / ðkþ P ¼ H P½/ ðkþ Š¼ PNB AðkÞ NB /ðkþ NB þ BðkÞ P A ðkþ P An equation similar to Eq. (38) or (39) is obtaine at each gri point in the omain, an the collection of these equations forms a system that is solve iteratively. The iscretization proceure for the momentum equation yiels an algebraic equation of the form u ðkþ P ¼ H P½u ðkþ Š a ðkþ D ðkþ P r PðPÞ Furthermore, the phasic mass conservation equation can be viewe as a phasic volume fraction equation or as a phasic continuity equation, which can be use in eriving the pressure-correction equation. Its iscretize form is given by ð39þ ð40þ ða ðkþ P qðkþ P Þ ðaðkþ P qðkþ t P ÞOl X P þ X f ¼nbðPÞ a ðkþ f q ðkþ f u ðkþ f S f ¼ B ðkþ P ð41þ PRESSURE-CORRECTION EQUATION To erive the pressure-correction equation, the mass conservation equations of the various fluis are ae to yiel the global mass conservation equation given by 8 9 X < ða ðkþ P qðkþ P Þ ðaðkþ P qðkþ P ÞOl X þ X = a ðkþ f q ðkþ f u ðkþ f S : f t ; ¼ 0 ð42þ k f ¼nbðPÞ Denoting the corrections for pressure, ensity, an velocity by P 0, u ðkþ0, an q ðkþ0, respectively, the correcte fiels are written as P ¼ P þ P 0 ; u ðkþ ¼ u ðkþ þ u ðkþ0 ; q ðkþ ¼ q ðkþ þ q ðkþ0 ð43þ Combining Eqs. (40), (42), an (43), the final form of the pressure-correction equation is obtaine as [32] 8 9 X< X t aðkþ P CðkÞ q P0 P þ X ða ðkþ U ðkþ Cq ðkþ P0 Þ f X = ½a ðkþ q ðkþ ða ðkþ D ðkþ rp 0 ÞSŠ : f ; k f ¼nbðPÞ f ¼nbðPÞ 8 9 ¼ X < a ðkþ P qðkþ P ða ðkþ P qðkþ P Þol X þ X = ða ðkþ q ðkþ U ðkþ Þ : t f ð44þ ; k f ¼nbðPÞ

13 196 F. MOUKALLED AND M. DARWISH The corrections are then applie to the velocity, ensity, an pressure fiels using the following equations: u ðkþ P ¼ u ðkþ P a ðkþ D ðkþ P r PP 0 ; P ¼ P þ P 0 ; q ðkþ ¼ q ðkþ þ C ðkþ q P0 ð45þ SOLUTION PROCEDURE The overall solution proceure is an extension of the single-phase SIMPLE algorithm [33, 34] into multiphase flows [32]. The sequence of events in the multiphase algorithm is as follows. 1. Solve the fluiic momentum equations for velocities. 2. Solve the pressure-correction equation base on global mass conservation. 3. Correct velocities, ensities, an pressure. 4. Solve the fluiic mass conservation equations for volume fractions. 5. Solve the fluiic scalar equations (k; e; T; Y;, etc.). 6. Return to the first step an repeat until convergence. NUMERICAL VALIDATION The above-escribe solution algorithm is verifie by numerically reproucing measurements in an isopropyl alcohol turbulent evaporating spray [35]. Several workers [17, 36] have use this problem to valiate their numerical methos. The experimental setup consists of a cylinrical test section of 194-mm inner iameter into which isopropyl alcohol with a temperature of 313 K is injecte from a 20-mm-outer-iameter nozzle locate along its axis of symmetry. The co-flowing air is simultaneously blown with a temperature of 373 K through a concentric annulus of 40-mm an 60-mm inner an outer iameters, respectively. The inlet mass flow rates of air an isopropyl alcohol are 28.3 an g=s, respectively. Detaile measurements at various axial positions are available for valiating the numerical preictions. Raial profiles at x ¼ 3 mm are use to escribe the inlet conitions to the omain, while profiles at x ¼ 25, 50, 100, an 200 mm are employe for comparison. In the numerical solution obtaine, the physical omain, consiere to be axisymmetric of length 1 m an raius m, is iscretize using nonuniform gris with enser clustering near the nozzle. Droplets are ivie accoring to size into five phases, with the iameter of roplets in the first roplet phase set to 10 mm an the increment to 10 mm [i.e., roplets of iameters between 10 (roplet phase 1) an 50 mm (roplet phase 5) are consiere, the range suggeste by experimental ata [35] within which the bulk of the roplet sizes fall]. The volume fraction profiles of the various phases at the inlet are euce from available experimental ata. The outflow conition is impose at the exit from the omain, an bounary values are extrapolate from the interior solution. At the walls, a no-slip conition is applie for the momentum equations, while a zero flux conition is use for the volume fraction an mass fraction equations. For the energy equation, the available experimental wall temperature profile is employe. In Figure 2 comparisons of the numerically preicte raial profiles of the mean axial gas velocity (Figures 2a 2), the mean axial roplet velocity (average

14 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 197 Figure 2. Comparison of measure an compute raial profiles for (a) () the gas mean axial velocity, (e) (h) the roplet mean axial velocity, an (i) (l) the liqui-phase mass flow rate.

15 198 F. MOUKALLED AND M. DARWISH over the phases, Figures 2e 2h), an liqui mass flux (Figures 2i 2l) against experimental ata are presente. As shown, numerical preictions at the four axial locations (x ¼ 25, 50, 100, an 200 mm) are in goo agreement with experimental profiles, valiating the numerical implementation of the solution algorithm. RESULTS AND DISCUSSION The suggeste solution algorithm is use to preict, for the configurations epicte in Figures 3a an 3b, mixing an evaporation of roplets in gas streams flowing at subsonic an supersonic spees. Figure 3a represents a rectangular uct in which air enters with a uniform free-stream velocity U, while kerosene roplets (C 12 H 23, [37]) mixe with air are injecte through an opening 1 mm in with in the streamwise irection. For the base case, the length of the omain is L (L ¼ 1 m) an its with is W (W ¼ 0.25 m). Figure 3b iffers from Figure 3a in that fuel is spraye in the cross-stream irection through two openings, locate at 10 cm from the uct inlet, each 1 mm wie. For the base case, the length of the omain is L (L ¼ 1.1 m) an its with is W (W ¼ 0.25 m). An illustrative gri network use is isplaye in Figure 3c. For all results presente, five roplet sizes are use of iameters equally space an varying between 75 an 150 mm. Therefore all computations were performe using a total of six phases [one gas phase (phase 1) an five roplet phases (phases 2 to 6)]. For all cases presente, the volume fraction values of the roplet phases at inlet are set to a ð2þ ;in ¼ 0:1a ;in, a ð3þ ;in ¼ 0:2a ;in, a ð4þ ;in ¼ 0:4a ;in, a ð5þ ;in ¼ 0:1a ;in, an a ð6þ ;in ¼ 0:2a ;in, where a ;in ¼ P 6 k¼2 aðkþ ;in. At the walls, a no-slip Figure 3. Physical omain for (a) streamwise injection in a rectangular uct, an (b) cross-stream injection in a rectangular uct; (c) an illustrative gri.

16 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 199 conition is applie for the momentum equations, while a zero flux conition is use for the volume fraction, mass fraction, an energy equations. For subsonic flow, values for all variables except pressure are specifie at the inlet, while at the exit, pressure is the only variable with a prescribe value. For the supersonic cases, values for all variables are impose at the inlet to the omain, while values are not set for any variable at the exit section. To investigate the sensitivity of the solution to the gri use, numerical experiments were carrie out with ifferent sizes of nonuniform gris. An example of these experiments involving cross-stream injection in a subsonic flow fiel is presente in Figure 4. Results isplaye in the figure were compute on three gri systems with sizes of , , an cells. The comparisons of the axial gas velocity (Figure 4a), gas temperature (Figure 4b), gas ensity (Figure 4c), vapor mass fraction (Figure 4), gas volume fraction (Figure 4e), an gas turbulent kinetic energy (Figure 4f) profiles presente at three axial stations (x ¼ 0.25 m, 0.5 m, an 0.75 m) an generate using the various gris inicate that they are nearly coinciing. Since the purpose is to test a metho, the gri with size of cells is selecte in subsequent computations involving cross-stream injection. For streamwise injection, a nonuniform gri with size of control volumes is use. For both configurations, roplets are injecte through 12 uniformly istribute control volumes (each of with 1=12 mm). Case 1: Streamwise Injection in a Subsonic Flow Fiel For the configuration isplaye in Figure 3a, air enters the omain at a Mach number of 0.2 (subsonic flow fiel) an a temperature of 700 K. Moreover, the kerosene air mixture is injecte with a velocity of magnitue 30 m=s at angles varying uniformly between 60 an 60, a temperature of 400 K, an a liqui volume fraction of 0.1, resulting in a fuel injection rate of 2.34 kg=s=m. Results are presente in Figures 5 an 6. Figure 5 isplays the material velocity fiels (au) for the gas phase (Figure 5a) an for roplet phases 1 (75 mm in iameter, Figure 5b), 2 (93.75 mm in iameter, Figure 5c), 3 (112.5 mm in iameter, Figure 5), 4 ( mm in iameter, Figure 5e), an 5 (150 mm in iameter, Figure 5f). The effect of the spray on the gas fiel is clearly reveale by the velocity vectors presente in Figure 5a. As shown, a eceleration of the gas phase occurs in the central portion of the omain at the location where the fuel is spraye. The rate of eceleration ecreases in the streamwise irection, but its effect spreas over a wier cross-sectional area ue to the ispersion of the injecte fuel. Vector fiels presente in Figures 5b through 5f reveal a larger roplet spreaing (or cross-penetration) with increasing roplet iameter, which is physically correct because larger particles possess higher inertia an are more capable of penetrating into the omain as compare to smaller ones, which align faster with the flow fiel. In Figure 6 contour maps of several variables are isplaye. The volume fraction of the gas fiel epicte in Figure 6a inicates that because of the higher air velocity, the spreaing of injecte fuel is low an roplets quickly align with the air velocity. The istribution of kerosene vapor in the gas phase is isplaye in Figure 6b. As shown, the amount of fuel vapor in the gas phase increases as the mixture moves ownstream in the channel, ue to the increase in the evaporate

17 200 F. MOUKALLED AND M. DARWISH Figure 4. Comparison of the (a) gas u-velocity, (b) gas temperature, (c) gas ensity, () vapor mass fraction, (e) gas volume fraction, an (f) gas turbulent kinetic energy profiles at three axial stations generate using three ifferent gri systems.

18 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 201 Figure 5. Velocity fiels for the gas phase (a) an the roplet phases (b) (f) in increasing roplet size for streamwise injection in a subsonic flow fiel (M in ¼ 0.2). amount with istance, which is physically plausible. The pressure fiel is epicte in Figure 6c an inicates larger changes in the spray region where the highest roplet volume fraction exists as a result of the gas-phase eceleration cause by the rag of the injecte roplets. As expecte, the gas temperature (Figure 6) ecreases in the core region of the omain because of roplet evaporation. The gas turbulent viscosity map shown in Figure 6e inicates that the highest values are in the regions of the omain where the roplets are present an high liqui gas interaction occurs. Case 2: Streamwise Injection in a Supersonic Flow Fiel For the same physical situation epicte in Figure 3a, the air Mach number is set at 2 (supersonic flow fiel) an the temperature at 700 K. Moreover, the

19 202 F. MOUKALLED AND M. DARWISH Figure 6. Comparison of (a) gas volume fraction, (b) vapor mass fraction, (c) pressure, () gas temperature, an (e) gas turbulent viscosity contours for streamwise injection in a subsonic flow fiel (M in ¼ 0.2). kerosene air mixture is injecte with a velocity of magnitue 200 m=s at angles varying uniformly between 60 an 60, a temperature of 400 K, an a liqui volume fraction of 0.015, resulting in a fuel injection rate of 2.34 kg=s=m. Results generate are presente in Figures 7 an 8. Figure 7 isplays the material velocity fiels (au) for the gas phase (Figure 7a) an for roplet phases 1 through 5 (Figures 7b 7f). The effect of the spray on the gas fiel (Figure 7a) is similar to the subsonic case (Figure 5a) but it is not as strong because for the same injecte amount of fuel, higher velocities are involve, resulting

20 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 203 Figure 7. Velocity fiels for the gas phase (a) an the roplet phases (b) (f) in increasing roplet size for streamwise injection in a supersonic flow fiel (M in ¼ 2). in lower volume fraction values. Droplet velocity vectors reveal that the egree of liqui spreaing (cross-penetration) increases with increasing roplet iameter. This is expecte, since larger particles possess higher inertia an are more capable of penetrating into the omain. At this supersonic spee, the high egree of roplet penetration obtaine is ue to the high injection velocity (200 m=s). Droplet velocity fiels (not reporte here) obtaine with low injection velocities in supersonic flow fiels resulte in very little spreaing of the roplets an remaine confine to a narrow region aroun the centerline of the omain. In Figure 8, contour maps of the gas volume fraction fiel (Figure 8a), the fuel vapor in the gas-phase fiel (Figure 8b), the pressure fiel (Figure 8c), the gas temperature fiel (Figure 8), an the gas turbulent viscosity fiel (Figure 8e) are presente. The volume fraction of the roplets ecreases in the streamwise irection

21 204 F. MOUKALLED AND M. DARWISH Figure 8. Comparison of (a) gas volume fraction, (b) vapor mass fraction, (c) pressure, () gas temperature, an (e) gas turbulent viscosity contours for streamwise injection in a supersonic flow fiel (M in ¼ 2). (i.e., an increase in the gas volume fraction is obtaine), while the mass fraction of the fuel vapor in the gas phase increases in the streamwise irection as more kerosene evaporates. To be notice is the increase in pressure values in the streamwise irection ue to the ecrease in the flow velocity cause by rag. At supersonic spees the ecrease in velocity, the turbulent fluctuations, an the viscous issipation increase the gas temperature. This statement can be clarifie by consiering the ecrease in velocity of the gas phase as an example. Numerical results reveal a ecrease in the

22 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 205 gas-phase velocity from nearly 1,060 m=s to 1,035 m=s over a few control volumes close to the nozzle. This ecrease in velocity alone causes the gas temperature to increase by about 26 C, which supports the previously state statement. On the other han, roplet evaporation ecreases the gas temperature. The relative strength of these factors ictates the gas temperature istribution over the omain, which is seen to increase slightly over the inlet value in this case. Finally, the largest gas turbulent viscosity values occur along the roplet trajectories, because of liqui gas interactions. Case 3: Cross-Stream Injection in a Subsonic Flow Fiel For the configuration isplaye in Figure 3b, air enters the omain at a Mach number of 0.2 (subsonic flow fiel) an a temperature of 700 K. Moreover, the kerosene air mixture is injecte with a velocity of magnitue 30 m=s at an angle of 30 to the irection of the gas flow, a temperature of 400 K, an a liqui volume fraction of 0.08, resulting in a total fuel injection rate, from both nozzles, of kg=s=m. Results obtaine using the above-escribe solution proceures are presente in Figures 9 an 10. Figures 9a 9e isplay the material velocity fiels (au) for the gas an roplet phases. The effect of the spray on the gas fiel can be inferre from the velocity vectors presente in Figure 9a. This effect is seen to be strong in the region close to the injector an to weaken as the spraye jet scatters. Moreover, roplet velocity vectors presente in Figures 9b 9f inicate larger roplet spreaing (or cross-penetration) with increasing roplet iameter, with the smallest roplets flowing close to the walls an the largest roplets penetrating into the core of the omain, which is physically correct. In Figure 10, contours of the gas volume fraction fiel (Figure 10a), the fuel vapor fiel in the gas phase (Figure 10b), the pressure fiel (Figure 10c), the gas temperature fiel (Figure 10), an the gas turbulent viscosity fiel (Figure 10e) are presente. As epicte, variations in these quantities are similar to those reporte in Figure 6, with the gas volume fraction fiel mimicking the roplet velocity fiels, the fuel vapor in the gas phase increasing in the streamwise irection as more liqui evaporates, the largest variation in pressure occurring in the spray region, the gas temperature ecreasing as evaporation takes place, an the gas turbulent viscosity maximizing along the roplet trajectories where high liqui gas interaction occurs. Case 4: Cross-Stream Injection in a Supersonic Flow Fiel For the same configuration epicte in Figure 3b, the air Mach number is set at 2 (supersonic flow fiel) an the temperature at 700 K. Moreover, the kerosene air mixture is injecte with a velocity of magnitue 200 m=s at an angle of 30 to the irection of the gas flow, a temperature of 400 K, an a liqui volume fraction of 0.012, resulting in a total fuel injection rate, from both nozzles, of kg=s=m. Generate results are isplaye in Figures 11 an 12. Due to the lower volume fractions involve, the effect of the spray on the supersonic gas phase (Figure 11a) is weaker than in the subsonic case. However, the roplet material velocity vectors isplaye in Figure 11 inicate similar behavior

23 206 F. MOUKALLED AND M. DARWISH Figure 9. Velocity fiels for the gas phase (a) an the roplet phases (b) (f) in increasing roplet size for cross-stream injection in a subsonic flow fiel (M in ¼ 0.2). to the cases presente earlier with larger particles penetrating eeper into the inner omain (compare Figures 11b 11f for phases 1 5, with phase 5 having the largest roplet iameter). In Figure 12, contour maps of the volume fraction fiel (Figure 12a), the fuel vapor in the gas-phase fiel (Figure 12b), the pressure fiel (Figure 12c), the gas temperature fiel (Figure 12), an the gas turbulent viscosity fiel (Figure 12e) are presente. The general tren in the variation of these variables resembles that presente in case 2, i.e., the volume fraction of the particles ecreases in the streamwise irection, the mass fraction of the liqui vapor in the gas phase increases in the streamwise irection as more fuel evaporates, the pressure increases in the streamwise irection with the largest variations occurring close to the nozzle exits, the gas temperature increases slightly for the same reasons state earlier, an the largest gas turbulent viscosity values occurs at locations where high fuel air mixing occurs.

24 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 207 Figure 10. Comparison of (a) gas volume fraction, (b) vapor mass fraction, (c) pressure, () gas temperature, an (e) gas turbulent viscosity contours for cross-stream injection in a subsonic flow fiel (M in ¼ 0.2). Parametric Stuy A parametric stuy was also unertaken to investigate the effects of varying the inlet gas temperature, inlet roplet temperature, an uct length on the percentage of the injecte fuel that evaporates into the gas phase in all configurations, an results are isplaye in Figure 13. In generating results, only the parameter uner investigation is varie in the range shown on the plot an, epening on the case, the remaining parameters are assigne the values presente earlier.

25 208 F. MOUKALLED AND M. DARWISH Figure 11. Velocity fiels for the gas phase (a) an the roplet phases (b) (f) in increasing roplet size for cross-stream injection in a supersonic flow fiel (M in ¼ 2). As expecte, the amount of evaporating liqui increases with increasing inlet gas temperature (Figure 13a), increasing inlet roplet temperature (Figure 13b), an increasing channel length (Figure 13c). Moreover, it can be inferre from Figures 13a 13c that the fraction that evaporates ecreases as the gas velocity increases. Furthermore, the ifference in the evaporating fractions between streamwise an cross-stream injection is insignificant at subsonic spee, with the percentage being marginally higher for cross-stream injection. The same tren is notice at supersonic spees, with the ifference being larger. The higher evaporating percentages at supersonic spees for cross-stream injection are attribute to the increase in temperature close to the wall, which enhances evaporation.

26 DISPERSION AND EVAPORATION OF LIQUID SPRAYS 209 Figure 12. Comparison of (a) gas volume fraction, (b) vapor mass fraction, (c) pressure, () gas temperature, an (e) gas turbulent viscosity contours for cross-stream injection in a supersonic flow fiel (M in ¼ 2). CLOSING REMARKS A Eulerian moel involving iscrete an continuous phases for the simulation of roplet evaporation an mixing at all spees was formulate an implemente.

27 210 F. MOUKALLED AND M. DARWISH Figure 13. Comparison of evaporation rate for the various configurations as a function of; (a) inlet gas temperature; (b) inlet roplet temperature; (c) channel length. The moel allows for continuous roplet size changes without recourse to a stochastic approach. The numerical proceures follow on a pressure-base multiflui finite-volume metho an form a soli base for the future inclusion of other moes of interactions such as roplet coalescence an breakup. Turbulence was moele using the two-equation k e turbulence moel for the continuous gas phase, with moifications to account for gas compressibility at high spees, couple with an algebraic moel for the iscrete phase. The metho was teste by solving for evaporation an mixing in two physical configurations involving streamwise an cross-stream injections, in the subsonic an supersonic regimes, over a wie range of operating conitions. Reporte results inicate an increase in the rate of evaporation with increasing inlet gas temperature, inlet roplet temperature, an=or length of the omain, which is physically correct.

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