To link to this article:

Transcription

1 This article was downloaded by: [The UC Irvine Libraries] On: 31 July 2013, At: 19:38 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Combustion Theory and Modelling Publication details, including instructions for authors and subscription information: Transient convective burning of interactive fuel droplets in single-layer arrays Guang Wu a & William A. Sirignano a a Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, , USA Published online: 10 Feb To cite this article: Guang Wu & William A. Sirignano (2011) Transient convective burning of interactive fuel droplets in single-layer arrays, Combustion Theory and Modelling, 15:2, 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 ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Combustion Theory and Modelling Vol. 15, No. 2, 2011, Transient convective burning of interactive fuel droplets in single-layer arrays Guang Wu and William A. Sirignano Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA , USA (Received 5 May 2010; final version received 22 October 2010) The transient convective burning of n-octane droplets interacting within single-layer arrays in a hot gas flow perpendicular to the layer is studied numerically, with considerations of droplet surface regression, deceleration due to the drag of the droplets, internal liquid motion, variable properties, non-uniform liquid temperature and surface tension. Infinite periodic arrays, semi-infinite periodic arrays with one row of droplets (linear array) or two rows of droplets, and finite arrays with nine droplets with centers in a are investigated. All arrays are aligned orthogonal to the free stream direction. This paper compares the behavior of semi-infinite periodic arrays and finite arrays with the behavior of previously studied infinite periodic arrays. Furthermore, it identifies the critical values of the initial Damköhler number for bifurcations in flame behavior at various initial droplet spacing for all these arrays. The initial flame shape is either an envelope flame or a wake flame as determined by the initial Damköhler number, the array configuration and the initial droplet spacing. The critical initial Damköhler number separating initial wake flames from initial envelope flames decreases with increasing interaction amongst droplets at intermediate droplet spacing (when the number of rows in the array increases or the initial droplet spacing decreases for a specific number of rows in the array). In the transient process, an initial wake flame has a tendency to develop from a wake flame to an envelope flame, with the moment of wake-to-envelope transition advanced for the increasing interaction amongst droplets at intermediate droplet spacing. For the array with nine droplets with centers in a, the droplets at different types of positions have different critical initial Damköhler number and different wake-to-envelope transition time for initial wake flame. Keywords: transient burning; single-layer arrays; droplet interactions; convective droplet vaporization; flame transition Nomenclature Latin B H B M C D C L c p d Spalding number for heat tranfer Spalding number for mass tranfer drag coefficient lift coefficient constant pressure specific heat droplet diameter Corresponding author. gwu1@uci.edu ISSN: print / online C 2011 Taylor & Francis DOI: /

3 228 G. Wu and W. A. Sirignano D Da E a h L M ṁ Nu p Pr q Q r R Re S Sc Sh sp t T u U d U We Y diffusion coefficient Damköhler number activation energy specific enthalpy latent heat of vaporization molecular weight mass burning rate Nusselt number pressure Prandtl number heat of combustion heat flow rate radial coordinate droplet radius Reynolds number stoichiometric number Schmidt number Sherwood number droplet spacing time temperature velocity velocity of the droplet relative velocity between the air stream and the droplet Weber number mass fraction Greek θ,φ spherical coordinates ξ normalized radial coordinate in the liquid phase ρ density α thermal diffusivity λ thermal conductivity µ kinetic viscosity ν fuel-to-oxygen mass stoichiometric ratio σ surface tension τ shear stress tensor ω chemical reaction rate Subscripts ambient value avg average value 0 initial value F fuel vapor film film conditions (average of ambient and surface conditions) g gas phase i the ith species

4 Combustion Theory and Modelling 229 l s liquid phase surface value Superscripts bar dimensionless quantities o reference value 1. Introduction Great interest has been shown in the study of vaporization and burning of droplets in a spray environment over the past two decades because of their important applications in many combustors. Due to the complexities of this problem and various limitations for experiments, numerical simulation is a desirable approach in the relevant studies. An accurate numerical simulation should include as many features of real physics (such as forced convection, droplet interactions and 3D configurations) as possible under currently available computing resources. Some studies have been made in the literature on the convective vaporization of axisymmetric droplets analytically [1] and computationally [2, 3] with the assumption of constant thermophysical properties. Other studies, by Chiang et al. [4 6], considered variable properties in the numerical simulation, with the conclusion that the thermal dependence of physical properties must be considered for high-temperature calculation. Convective burning of axisymmetric droplets has also been studied experimentally [7] and computationally [8 13]. Dwyer et al. [9, 10] and Wu and Sirignano [11] studied the effects of surface tension and found that the surface tension had a significant influence on the liquid motion inside the burning droplet. Wu and Sirignano [11] and Pope et al. [13] identified some of the transient behavior of an isolated convecting burning droplet, with considerations of droplet surface regression, deceleration due to the drag of the droplet, internal circulation inside the droplet, and variable properties. The numerical calculations for three-dimensional configurations have been made for non-vaporizing spheres by Kim, Elghobashi and Sirignano [14], and for vaporizing and burning interactive droplets by Stapf, Dwyer and Maly [15,16] without the consideration of internal circulation in the liquid phase. They found that the interactions inside the droplet arrays had a strong influence on the flow field and the physical-chemical processes. Wu and Sirignano [17] numerically studied the transient convective burning of fuel droplets interacting within an infinite periodic array with considerations of internal circulation inside the droplets and non-uniform surface temperature. The spacing amongst droplets was found to influence the burning rate by affecting the droplet surface temperature and interactions amongst droplets. A more complete review of droplet vaporization and burning is given by Sirignano [18]. The task of this numerical study is to simulate convecting, burning and interactive droplets in several single-layer arrays, by solving the Navier Stokes, energy and species equations. Droplet surface regression, deceleration of the stream flow due to the drag of the droplets, internal circulation, variable properties, non-uniform surface temperature, and surface tension are considered. The array configurations examined in this study include infinite periodic arrays, semi-infinite periodic arrays with one row or two rows of droplets, and finite arrays with nine droplets with centers in a. The transient flame shape, surface temperature, burning rate, and dimensionless numbers are studied for different initial droplet spacing, initial Reynolds number and initial Damköhler number. Particularly,

5 230 G. Wu and W. A. Sirignano the critical parameters for the determination of the initial flame shapes, the flame transition (from a wake flame to an envelope flame) time and its influence on the burning rate are determined. The model arrays studied are qualitatively characteristic of real spray situations in which there is a very large number of droplets interacting in arrays. These models do allow substantial savings of computational resources. A real spray is of course not a single-layer array along the flow direction. However, based on the similarity between an isolated droplet and the upstream droplet in a tandem arrangement of droplets reported in [5, 6], there is good reason to expect that the behavior of a single layer of droplets is also similar to the behavior of the most upstream droplets in a real spray. The various array configurations considered in this study account for some possible situations in real spray systems: the infinite periodic arrays are close to the situation when the spray is well formed and uniformly distributed, and the semi-infinite periodic arrays and finite arrays represent other situations when all or some of the droplets are less interactive than a droplet in the infinite periodic arrays. Although the results for droplets in single-layer arrays are most relevant for the most upstream droplets in a real spray, the important issues such as interactions amongst droplets, the flame configurations, and their influence on the burning rate as addressed in this study also provide some useful insight for the understanding of the fundamental aspects of a whole real spray. This paper compares the behavior of semi-infinite periodic arrays and finite arrays with the behavior of previously studied infinite periodic arrays [17]. Furthermore, it identifies the critical values of the initial Damköhler number for bifurcations in flame behavior at various initial droplet spacing for all these arrays. The dimensionless numbers are also studied at various initial droplet spacing and initial Reynolds number for these arrays. These results are first reported in this paper even for the infinite periodic arrays. 2. Problem formulation Figure 1(a) shows the configuration of an infinite periodic array in a hot gas stream, which has a larger number of single-component fuel droplets with the same initial size, uniform distributioninthex-y, and perpendicular flow direction to this (in the z- direction). The infinite periodic array can be considered as a periodic array with an infinite number of rows of droplets. Thereby, it is periodic in two directions. The finite array with nine droplets with centers in a is shown in Figure 1(b). The semi-infinite periodic arrays with one row or two rows of droplets have an infinite number of droplets only in one direction, as shown in Figure 1(c,d). For an infinite periodic array, the three-dimensional flows are periodic along the two directions in the, and thus only one droplet (with the domain of ABCD in Figure 1a) is considered in the calculation using four symmetry s. In fact, the actual domain for the calculation can be further reduced to OECF with only a quarter of the droplets due to symmetries. Similarly, the reduced number of droplets by symmetries is a quarter for semi-infinite periodic arrays with one row of droplets, a half for semi-infinite periodic arrays with two rows of droplets, and it becomes one full droplet, two half-droplets, and a quarter-droplet for finite arrays with nine droplets with centers in a. The free-stream air flow has velocity U, pressure p, and temperature T. The initial droplet temperature T s,0 is uniform and low compared to the boiling point. The droplets are first heated and vaporized, and then autoignited by the hot free stream and burned. Internal circulation is caused by the shear stress at the gas-side droplet surface and the nonuniform distribution of surface tension around the droplet surface. Although the droplets have a time-varying velocity U d, we consider that they are not moving by instantaneously

6 Combustion Theory and Modelling 231 y B y-z symmetry x-z symmetry z E C y y-z symmetry x-z symmetry O x F droplet surface y-z symmetry 3 1 y-z symmetry x-z symmetry 4 2 x A x-z symmetry D z y-z symmetry z (a) y y-z symmetry x-z symmetry (c) x y-z symmetry y-z symmetry z (b) y y-z symmetry x x-z symmetry (d) y-z symmetry Figure 1. (a) The infinite periodic array for the droplets (the flow direction is perpendicular to the x-y in which the droplet centers are located), and the gridding scheme for the three-dimensional multi-droplet calculation (in the left lower quarter). (b) The finite array with nine droplets with centers in the x-y and flow in the z-direction (1: droplet at the center, 2, 3: droplet at the edge center, 4: droplet at the corner). (c) The semi-infinite periodic array with one row of droplets, with centers in the x-y and flow in the z-direction. (d) The semi-infinite periodic array with two rows of droplets, with centers in the x-y and flow in the z-direction. having an inertial frame of reference moving at the droplet velocity. The relative velocity becomes U = U U d. This is justified for an infinite periodic array and a semi-infinite periodic array with one row of droplets with no relative motion amongst the droplets, and for a semi-infinite periodic array with two rows of droplets with no relative motion in the flow direction and also negligible relative motion in the other two directions because of small side forces. However, for a finite array with nine droplets with centers in a, not all the droplets are subject to the same drag and have the same velocity, but we still assume that the relative motion amongst the droplets can be neglected and U d is the same for all the droplets for simplicity. The consideration of the relative motion amongst droplets in the nine-droplet array might be included in our future studies. As the droplets are slowed by the drag, and vaporization occurs, the relative velocity and droplet radius are updated continuously. The following assumptions are made: (1) the Mach number is much less than unity and the dissipation terms are neglected; (2) there is no natural convection and other gravity effects; (3) the droplets remain spherical; (4) the gas mixture is an ideal gas; (5) variation in the properties of the liquid phase is neglected; and (6) radiation is neglected.

7 232 G. Wu and W. A. Sirignano The variables have been non-dimensionalized and are listed as follows: radial position r = r/d 0, time t = tu,0 /d 0, velocity vector or components ū = u/u,0, pressure p = p/(ρ U,0 2 ), density ρ = ρ/ρ, enthalpy h = h/[c p (T T s,0 )], temperature T = (T T s,0 )/(T T s,0 ), molecular weight M i = M i /M F, specific heat c p = c p /c p, kinetic viscosity µ = µ/µ, thermal conductivity λ = λ/λ, thermal diffusivity ᾱ = α/α, mass diffusivity D i = D i /D i,, shear stress tensor τ = τd 0 /(µ U,0 ), reaction rate ω = ω/ ω o, surface tension σ = σ/σ 0, where d 0 and U,0 denote the initial droplet diameter and the initial relative stream velocity. The superscript o denotes the reference value, and the subscripts F, i, and 0 denote the fuel vapor, the ith species, the ambient value and the initial value, respectively. There are certain dimensionless numbers generated: initial Reynolds number Re 0 = ρ U,0 d 0/µ, Prandtl number Pr = µ /(α α ), Schmidt number Sc i, = µ /(ρ D i, ), reference Spalding number for heat transfer BH o = c p (T T s,0 )/L, initial Weber number We 0 = ρ U,0 2 d 0/σ 0, and initial Damköhler number Da 0 = d 0 /U,0 /[ρ YF o/( ωo M F )] (where YF o is the reference mass fraction for the fuel vapor). The governing equations for both gas and liquid phases are: Continuity equation: Momentum equation: ρ u t ρ t + ( ρ u) = 0. (1) + ( ρ u u) = p + 1 Re 0 ( τ), (2) in which τ is the shear stress tensor given by τ = µ( u + u T ) 2 µ( u)1,intheform 3 for a Newtonian compressible fluid. The divergence of u becomes 0 for the incompressible liquid phase. Energy equation: ρ h t + ( ρ u h) = 1 Re 0 Pr ( ρᾱ h) + S h, (3) in which S h =.( N i=1 ρ D i h i Y i ) (Re 0 Sc i, ).( ρᾱ N i=1 h i Y i ) (Re 0 Pr ) + q ωy o F Da 0 M F c p (T T s,0 ) for the gas phase, and S h = 0 for the liquid phase. q is the heat of combustion. The reaction rate is given by ω = Ae E a/(r u T ) [Fuel] a [Oxidizer] b mol cm 3 s 1, with A = , E a = , a = 0.25, b = 1.5 for n-octane, from Westbrook and Dryer [19]. The reference value of the reaction rate ω o is calculated based on T, ρ, Y O2,, and the stoichiometric mass fraction for the fuel vapor Y o F. The values of ωo and the initial Damköhler number Da 0 will be very sensitive to the choice of the reference temperature, and T is a good choice for high ambient temperature where autoignition becomes possible. The temperature is obtained from the enthalpy.

8 Gas-phase species equation: Combustion Theory and Modelling 233 ρy i t + ( ρ uy i ) = 1 Re 0 Sc ( ρ D i Y i ) + ωs i M i Y o F Da 0, (4) in which M i and S i are the normalized molecular weight, and stoichiometric number for the ith species which represents moles of this species produced (+) or consumed ( ) for each mole of fuel consumed. The gas-phase continuity, momentum, energy and species equations and liquid-phase continuity, momentum and energy equations are coupled and solved simultaneously. There are in total N = 5 species including the fuel vapor, oxygen, water vapor, carbon dioxide, and nitrogen considered in the calculation. The species equations are applied to the first four species, while the concentration of nitrogen is obtained from the relation that the mass fraction for all the species sums to unity. The computational domain for the gas phase in Figure 1 will be divided into N d (N d is the number of droplets) spherical domains, with one of these domains around each droplet, and one Cartesian domain for the rest. For the liquid phase, each droplet has one separate spherical domain. The detailed form of the momentum equation (2) in spherical coordinates and Cartesian coordinates is provided in [20]. The droplet surface regresses during vaporization, and the droplet radius is a function of time. Therefore, the liquid-phase domains shrink while the gas-phase domains expand. To consider this, the adjustments of radial coordinate or mesh are needed for the spherical domains. Details are provided by Wu and Sirignano [11]. The outer boundaries of the Cartesian domain are the inflow and outflow boundaries of the gas stream, and the symmetry s. Matching boundary conditions at the droplet surface need to be specified for both the gas and liquid phase based on a balance of force, and heat and mass transfer. Boundary conditions at the ends of θ- and φ-directions for the spherical domains can be derived based on continuity and the features of the flow. Boundary conditions at the interfaces of gas-phase spherical domains and the Cartesian domain are given by the continuity of the calculated quantities and their first derivatives. More details for the boundary conditions are provided by Wu and Sirignano [17]. The forces on any of the droplets (z-direction drag C D, x-direction lift C Lx, and y- direction lift C Ly ) are composed of three types of forces: force due to pressure difference, thrust force, and friction force. Pressure force: C D,p C Lx,p = 2 2π π cos θ p πū 2 s sin θ cos φ sin θdθdφ. (5) 0 0 sin θ sin φ C Ly,p Thrust force: C D,t C Lx,t = 2 2π π cos θ sin θ ū C πū 2 g,r sin θ cos φ ū g,θ cos θ cos φ Ly,t 0 0 sin θ sin φ cos θ sin φ + ū g,φ 0 sin φ cos φ ρ g,s ū g,r,s sin θdθdφ. (6) s

9 234 G. Wu and W. A. Sirignano Friction force: C D,f C Lx,f C Ly,f = 2 2π π sin θ cos θ τ πre 0 Ū 2 rθ cos θ cos φ + τ rr sin θ cos φ 0 0 cos θ sin φ sin θ sin φ 0 + τ rφ sin φ cos φ sin θdθdφ. (7) s The total force is the sum of the three types of forces: C D C D,p + C D,t + C D,f C Lx = C Lx,p + C Lx,t + C Lx,f. (8) C Ly C Ly,p + C Ly,t + C Ly,f The droplets are slowed down by the drag in the transient process, and the instantaneous velocity of any of the droplets is determined by dū d d t = dū d t = Ū 2 ρ l R C D. (9) The Spalding transfer numbers, Nusselt number, and Sherwood number are defined as: B H = c p,f ilm(t T s,avg ) + νy O q L eff, (10) B M = Y F s,avg + νy O, 1 Y F s,avg (11) R 2π π ( ) T g Nu = λ 2π(1 T s,avg + νy O g,s sin θdθdφ, q 0 0 r s (12) Sh = c p,f ilm (T T 0 ) ) R 2π π 2π(Y F Y F s,avg ) 0 0 ρ g,s D g,s ( YF r ) s sin θdθdφ, (13) where ν is the fuel-to-oxygen mass stoichiometric ratio (0.285 for n-octane), c p,f ilm = 1 3 (c p + 2c ps,avg ), and L eff = Q s,g /ṁ. The modified gas-phase Reynolds number is defined as Re m = ρ U d/µ film, with µ film = 1 3 (µ + 2µ s,avg ). 3. Solution procedure We consider droplets with single component of n-octane. The thermophysical properties for the gas mixture are calculated by polynomials and semi-empirical equations [21 23]. The one-step oxidation kinetics in [19] is used for the gas-phase reaction. The number of droplets N d during the calculation is 1 for infinite periodic arrays and semi-infinite periodic arrays with one or two rows of droplets, because all the droplets have identical positions in these arrays. For finite arrays with nine droplets with centers

10 Combustion Theory and Modelling 235 in a, N d is 3 because there are three types of droplet positions: at the center, at the edge center, and at the corner. A hybrid gridding scheme is used for the threedimensional calculation, as shown in Figure 1(a). More details regarding the gridding scheme are provided in [17]. The computation is paralleled to reduce the computational time. It requires a total of (N d + 1) processors in the parallel computation, because each droplet (including both gas-phase and liquid-phase spherical domains) uses one processor and the Cartesian domain uses one more. The governing equations for each droplet and the Cartesian domain are thus solved simultaneously at each time step. The Semi-Implicit Method for Pressure Linked Equations (SIMPLE) is used to solve the coupled Navier Stokes, energy and species equations for both gas and liquid phases. Staggered grids are used. Forward time and a hybrid differencing scheme are applied in the discretization. For the iterations at each time step, TDMA (Tri-Diagonal Matrix Algorithm) is used to solve over nodes in each row along the θ- orz-direction, and it sweeps forward and backward in the other two directions. The mesh sizes for the spherical domains and Cartesian domain are similar to those described in [17]. For the Cartesian domain, different array configurations have different domain sizes in the x- and y-directions and thus different number of mesh points in the two directions. The time step size is fixed and it takes about 10 5 time steps for 99% of the droplet volume to be vaporized. Each calculation requires hours of time on Pentium-based computers depending on the array configurations. The finite arrays with nine droplets with centers in a require most computational time because they have the biggest Cartesian domain and thus the most Cartesian mesh points. The codes have been validated by Wu and Sirignano [17] using an isolated burning fuel droplet moving through a gas stream. 4. Results and discussion We investigated infinite periodic arrays, semi-infinite periodic arrays with one row or two rows of droplets, and finite arrays with nine droplets with centers in a. Infinite periodic arrays have been examined previously [17] but some additional results will be included in this paper. The ambient pressure is p = 20 atm and the ambient temperature is T = 1500 K. The initial droplet temperature is T s,0 = 300 K, and the initial droplet radius is 25 µm, typical of a droplet size in the spray. The initial relative stream velocity U,0 and the initial spacing amongst droplets sp 0 are varied to study the effects of forced convection and droplet interaction. The change in the initial relative stream velocity changes the initial Reynolds number Re 0 and Damköhler number Da 0. The calculations are made for a few cases until most of the volume (more than 99%) has been vaporized, and the flame is stabilized well and asymptotic behavior is found near the end of the lifetime. So, we made calculations only for a major part of the lifetime (with about 80% of the volume vaporized) for all the other cases since the asymptotic behavior, as shown in Figures 2 and 3, had been reached. Comparisons are made for three different types of arrays: infinite periodic array, and semi-infinite periodic array with one row or two rows of droplets. The interaction amongst droplets increases as the number of rows in the array increases. Figures 2(a,b) show the instantaneous average surface temperature and normalized mass burning rate for the three types of arrays with the same initial droplet spacing sp 0 = 2.4d 0, at the initial Reynolds number Re 0 = 5.5 and 45, respectively. The mass burning rate is defined based on the surface regression rate or droplet vaporization rate, which might have a minor deviation from the fuel consumption rate due to fuel accumulation in the region between droplet and flame in the unsteady process. For Re 0 = 5.5 (Da 0 = 2.4) with an envelope flame

11 236 G. Wu and W. A. Sirignano (a) Re 0 =5.5 (Da 0 =2.4) (b) Re 0 =45(Da 0 =0.3) Figure 2. Comparison of the instantaneous average surface temperature and normalized mass burning rate for three different types of arrays (with the same initial droplet spacing sp 0 = 2.4d 0 ), at the initial Reynolds number Re 0 = 5.5 and 45, respectively. during the lifetime, the increase of the surface temperature for the semi-infinite periodic array with one row of droplets is higher than the other two arrays due to a higher flame temperature resulting from less interaction amongst droplets and greater chemical reaction rate. For Re 0 = 45 (Da 0 = 0.3) with an initial wake flame, the wake-to-envelope transition during the lifetime is indicated by the sharp increase of the average surface temperature and the mass burning rate in Figure 2(b). It was found in [17] that the moment of waketo-envelope transition is advanced as the initial droplet spacing is decreased if the initial droplet spacing is not too small, due to an increase in the velocity-decrease effect as the interaction amongst droplets increases. A velocity-decrease effect means that the gas velocity between droplets is decreased as the spacing deceases because the flow is cooled and density increases. In a similar behavior, the moment of wake-to-envelope transition is earliest for the infinite periodic array with the most interaction amongst droplets, and latest for the semi-infinite periodic array with one row of droplets with the least interaction amongst droplets. The mass burning rate for the three types of arrays is varied because of the (a) Re 0 =5.5 (Da 0 =2.4) (b) Re 0 =45(Da 0 =0.3) Figure 3. Comparison of the instantaneous average surface temperature and normalized mass burning rate for droplets at three different types of positions in a finite array with nine droplets with centers in a (with the initial droplet spacing sp 0 = 2.4d 0 ), at the initial Reynolds number Re 0 = 5.5 and 45, respectively.

12 Combustion Theory and Modelling 237 Figure 4. The values of the critical initial Damköhler number (for the determination of whether the flame is initially a wake or envelope flame) at different initial droplet spacing, for an infinite periodic array and a semi-infinite periodic array with one row of droplets, respectively. different interaction of fuel vapor amongst droplets and different time of wake-to-envelope transition for an initial wake flame. At Re 0 = 5.5 with an envelope flame all the time, the mass burning rate is greater for the arrays with smaller interaction of fuel vapor amongst droplets. At Re 0 = 45 for an initial wake flame, the mass burning rate is also influenced by the wake-to-envelope transition which elevates the surface temperature, and the array with more interaction amongst droplets (e.g. the infinite periodic array) may have greater mass burning rate for some period during the lifetime due to an earlier wake-to-envelope transition. It was reported in [11] that, for an isolated droplet burning, there exists a critical value of the initial Damköhler number which determines whether the flame is initially a wake or envelope flame regardless of the initial Reynolds number; the value is 1.02 for n-octane under the ambient conditions of T = 1500 K and p = 20 atm. Similar results are also found for multi-droplet burning. Figure 4 shows the values of the critical initial Damköhler number at different initial droplet spacing, for an infinite periodic array and a semi-infinite periodic array with one row of droplets, respectively. From the discussions for the moment of wake-to-envelope transition in [17], the preference for an envelope flame is increased as the initial droplet spacing is decreased, but this trend is hindered as the initial droplet spacing becomes too small. Therefore, the critical initial Damköhler number becomes smaller (implying preference for an envelope flame) for decreasing initial droplet spacing when it is intermediate but has no substantial change when the initial droplet spacing becomes too small. The critical initial Damköhler number for a semi-infinite periodic array is generally greater than that for an infinite periodic array at the same initial droplet spacing, because the effective droplet spacing is actually greater for a semi-infinite periodic array

13 238 G. Wu and W. A. Sirignano Figure 5. The values of the critical initial Reynolds number (for the determination of whether the flame is initially separated or connected) at different initial Damköhler number under the same initial droplet spacing (sp 0 = 2.4d 0 ), for an infinite periodic array and a semi-infinite periodic array with one row of droplets, respectively. with one row of droplets in which there are no droplet interactions in the other direction. We also investigated the initial flame shape as either a separated or connected flame (group flame) determined by both the initial Reynolds number and the initial Damköhler number. Critical values of the initial Reynolds number (for the determination of an initial separated or connected flame) can be found for given initial Damköhler number and initial droplet spacing. Figure 5 shows the critical initial Reynolds number at different initial Damköhler number at the initial droplet spacing sp 0 = 2.4d 0, for the two array configurations. For each array configuration, the solid curve of the critical Re 0 and the dashed vertical line of the critical Da 0 (for the determination of an initial wake or envelope flame) divide the into four domains, with the initial flame wake-separated for the left-upper corner, wake-connected for the left-lower corner, envelope-separated for the right-upper corner, and envelope-connected for the right-lower corner. It is found that the critical Re 0 increases as the flame is changed from an initial wake flame to an initial envelope flame, and does not change much in the initial envelope flame zone. This implies that an initial envelope flame can be an initial connected flame over a wider range of initial Reynolds number than an initial wake flame, as shown in Figure 5. The instantaneous dimensionless numbers C D, Nu, Sh, B H, and B M are examined for infinite periodic arrays and semi-infinite periodic arrays with one row or two rows of droplets. The lift coefficient is examined only for the y-direction component C Ly for semi-infinite periodic arrays with two rows of droplets, because there are no net side forces amongst droplets in x-direction for semi-infinite periodic arrays with two rows of droplets and no net side forces in both the x- and y-directions for infinite periodic arrays and

14 Combustion Theory and Modelling 239 (a) Drag coefficient (b) Nusselt number (c) Sherwood number (d) Spalding number for heat tranfer (e) Spalding number for mass tranfer (f) Lift coefficient Figure 6. Comparison of the dimensionless numbers (C D, Nu, Sh, B H, B M,andC Ly ) as functions of the instantaneous modified Reynolds number Re m, for cases with different initial Re m, or different array configurations or initial droplet spacing (inf 1.8, 2.4, or 5.9: infinite periodic array with sp 0 = 1.8d 0,2.4d 0,or5.9d 0 ; one 2.4: semi-infinite periodic array with one row of droplets with sp 0 = 2.4d 0 ; two 1.8 or 2.4: semi-infinite periodic array with two rows of droplets with sp 0 = 1.8d 0 or 2.4d 0 ). semi-infinite periodic arrays with one row of droplets. Figures 6(a f) plot these dimensionless numbers as functions of the modified Reynolds number Re m, which decreases over time, for cases with different initial modified Reynolds number Re m,0, array configurations or initial droplet spacing. Re m,0 = 30.6, 38.3, 73.6, and correspond to Re 0 = 16, 22,

15 240 G. Wu and W. A. Sirignano 45, and 67, respectively. All the dimensionless numbers become close for various cases after a period of relaxation, when the liquid heating slows down and the relative spacing amongst droplets is increased. The cases for greater initial Reynolds number generally have greater B H and B M at the same Re m, because the same Re m at greater initial Reynolds number corresponds to a later moment during the lifetime which has greater surface temperature and less liquid heating. However, the cases for greater initial Reynolds number have smaller C D, Nu, and Sh at the same Re m, because C D, Nu, and Sh are inversely correlated to Spalding numbers as predicted by Chiang et al. [4] for an isolated droplet vaporization. The differences of the instantaneous Nuand B M for different array configurations or initial droplet spacing are similar to the differences in the instantaneous average surface temperature (greatly influenced by the moment of wake-to-envelope transition for an initial wake flame). The differences of other dimensionless numbers for different array configurations or initial droplet spacing are more complicated and do not follow an obvious pattern. For a finite array with nine droplets with centers in a, there are three different types of positions for a droplet: at the center, at the edge center, and at the corner. Figures 3(a,b) make comparisons of the instantaneous average surface temperature and normalized mass burning rate for the three different types of positions for a droplet in the nine-droplet array with the initial droplet spacing sp 0 = 2.4d 0, at the initial Reynolds number Re 0 = 5.5 and 45, respectively. For Re 0 = 5.5 (Da 0 = 2.4), there is an envelope group flame around the nine droplets, and the droplet at the center is farthest from the flame and is also subject to the strongest interaction amongst droplets. So, the droplet at the center has the slowest increase of the surface temperature and smallest mass burning rate. For Re 0 = 45 (Da 0 = 0.3), there are initially nine separated wake flames, which are transitioned into nine separated envelope flames later during the lifetime. Based on the previous discussions, the droplet at the center subject to the strongest interaction should have the earliest moment of wake-to-envelope transition. So, the droplet at the center has the earliest sharp increase in the surface temperature which results in a greatest mass burning rate, although the strongest interaction of the fuel vapor amongst droplets tends to decrease the mass burning rate for the droplet at the center. The critical initial Damköhler number for the nine-droplet arrays is also found for the determination of whether the flame is initially a wake or envelope flame. At the initial droplet spacing sp 0 = 2.4d 0, the critical initial Damköhler number is 0.84, 0.90, and 0.96 for the droplet at the center (droplet 1 in Figure 1b), the droplet at the edge center (droplet 2 or 3 ), and the droplet at the corner (droplet 4 ), respectively. The droplet at the center with the strongest droplet interaction has the smallest critical initial Damköhler number, which is consistent with the earliest wake-to-envelope transition for an initial wake flame shown above. However, the critical initial Damköhler number for the droplet at the center is greater than the value (0.78) for a droplet in an infinite periodic array with the same initial droplet spacing, because the droplet in an infinite periodic array has even stronger droplet interaction. As the critical initial Damköhler number is different for the three types of droplet positions in a nine-droplet array, the initial flame shape as either a wake or envelope flame might be different for the three types of droplet positions. Figures 7(a d) show the initial flame shapes (represented by the contours of the chemical reaction rate at three cross-sections) for the nine-droplet arrays (with sp 0 = 2.4d 0 ) at different initial Reynolds number and initial Damköhler number. The initial flames are separated for the former three cases, while the initial flames are connected as a group flame for the last case with a low initial Reynolds number. At Da 0 = 0.78 which is smaller than the critical initial Damköhler number Da 0,cr for all the three types of droplet positions, the initial flames are wake flames for all the droplets. At Da 0 = 1.02 which is greater than Da 0,cr for all the

16 Combustion Theory and Modelling 241 Figure 7. The initial flame shapes (represented by the contours of chemical reaction rate at three cross-sections) for the nine-droplet arrays (with sp 0 = 2.4d 0 ) at different initial Reynolds number and initial Damköhler number. three types of droplet positions, the initial flames are envelope flames for all the droplets. At Da 0 = 0.90, the droplet at the center with Da 0,cr < 0.90 has an initial envelope flame, while the droplet at the corner with Da 0,cr > 0.90 has an initial wake flame. 5. Concluding remarks The transient convective burning of n-octane droplets within several single-layer arrays in a hot air stream is simulated with considerations of droplet surface regression, deceleration of the stream flow, liquid motion, variable properties, non-uniform surface temperature and surface tension. We investigated semi-infinite periodic arrays with one row or two rows of droplets, and finite arrays with nine droplets with centers in a, and also compared the results of these arrays with the infinite periodic arrays which we had examined previously [17]. The transient flame shape, surface temperature, and burning rate are investigated under different initial parameters. The critical parameters for the determination of the initial flame shapes and the moment of wake-to-envelope transition for an initial wake flame are determined for all these arrays.

17 242 G. Wu and W. A. Sirignano The stability of the studied configurations remains an open question. Based on some previous studies with a few droplets, there is some promise of stability. The attractive or repulsive side forces determined in [14] and a much lower drag for the downstream droplet observed in [5] give good reasons to expect linear stability for lateral movement of a droplet toward a neighboring droplet, and fast restoration to a single-layer array if one droplet advances or falls behind. For a given array configuration under specific initial droplet spacing and ambient conditions, there exists a critical initial Damköhler number below which the flame is initially a wake flame and above which the flame is initially an envelope flame. For the initial Damköhler number below the critical value, the initial wake flame will be transitioned into an envelope flame later during the lifetime. The interaction amongst droplets at intermediate droplet spacing increases as the number of rows in the array increases. The interaction also increases as the initial droplet spacing decreases for a specific number of rows in the array. Consequently, the critical initial Damköhler number decreases and the moment of wake-toenvelope transition from an initial wake flame is advanced, both implying a preference for an envelope flame. The initial flame shape as either separated flames or a connected flame (group flame) is determined by the initial Reynolds number, initial Damköhler number and initial droplet spacing. An initial envelope flame can be an initial connected flame over a wider range of initial Reynolds number than an initial wake flame. The mass burning rate is influenced by the Reynolds number, interaction of fuel vapor amongst droplets, and the moment of wake-to-envelope transition for an initial wake flame. The array with more interaction amongst droplets may have greater mass burning rate for some period during the lifetime due to an earlier wake-to-envelope transition. Lower ambient pressure causes a later wake-to-envelope transition and smaller mass burning rate because of slower reaction kinetics. The instantaneous dimensionless numbers C D, Nu, Sh, B H, and B M become close for various cases after a period of relaxation, when the liquid heating slows down and the relative spacing amongst droplets is increased. The cases for greater initial Reynolds number generally have greater B H and B M, but smaller C D, Nu, and Sh at the same Re m. This indicates that C D, Nu, and Sh are inversely correlated to Spalding numbers, which is consistent with the results for isolated droplet vaporization in the literature. The changes of the instantaneous Nu and B M for various cases are characterized by the changes of the surface temperature. For the array with nine droplets with centers in a (nine-droplet array), there are three different types of droplet positions: at the center, at the edge center, and at the corner. The droplet at the center has the lowest vaporization rate for cases with a group envelope flame, but may have the greatest vaporization rate for cases with nine separated wake flames initially due to the earliest wake-to-envelope transition. The droplet at the center has the smallest critical initial Damköhler number, and the droplet at the corner has the greatest. The critical initial Damköhler number for the droplet at the center in a nine-droplet array is greater than that for a droplet in an infinite periodic array at the same initial droplet spacing. The initial flame shape as either a wake or envelope flame might be different for the three types of droplet positions in a nine-droplet array, due to the difference in the critical initial Damköhler number for the three types of droplet positions. References [1] K. Asano, I. Taniguchi, and T. Kawahara, Numerical and experimental approaches to simultaneous evaporation of two adjacent volatile drops,inproc. 4th Int. Conf. on Liquid Atomization

18 Combustion Theory and Modelling 243 and Spray Systems, The Institute for Liquid Atomization and Spray Systems, Sendai, Japan, 1988, pp [2] G. Patnaik, A numerical solution of droplet vaporization with convection, PhD dissertation, Carnegie-Mellon University. Department of Mechanical Engineering, [3] M.S. Raju and W.A. Sirignano, Interaction between two vaporizing droplets in an intermediate- Reynolds-number flow, Phys. Fluids 2 (1990), pp [4] C.H. Chiang, M.S. Raju, and W.A. Sirignano, Numerical analysis of convecting, vaporizing fuel droplet with variable properties, Int. J. Heat Mass Transf. 35 (1992), pp [5] C.H. Chiang and W.A. Sirignano, Interacting, convecting, vaporizing fuel droplets with variable properties, Int. J. Heat Mass Transf. 36 (1993), pp [6] C.H. Chiang and W.A. Sirignano, Axisymmetric calculations of three-droplet interactions, Atomiz. Sprays 3 (1993), pp [7] V. Raghavan, V. Babu, T. Sundararajan, and R. Natarajan, Flame shapes and burning rates of spherical fuel particles in a mixed convective environment, Int. J. Heat Mass Transf. 48 (2005), pp [8] Y. Aouina and U. Maas, Mathematical modeling of dropet heating, vaporization, and ignition including detailed chemistry, Combust. Sci. Technol. 173 (2001), pp [9] H.A. Dwyer, I. Aharon, B.D. Shaw, and H. Niazmand, Surface tension influences on methanol droplet vaporization in the presence of water, Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh PA (1996), pp [10] H.A. Dwyer, B.D. Shaw, and H. Niazmand, Droplet/Flame interactions including surface tension influences, Twenty-Seventh Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA (1998), pp [11] G. Wu and W.A. Sirignano, Transient burning of a convective fuel droplet, Combust. Flame157 (2010), pp [12] D.N. Pope and G. Gogos, Numerical simulation of fuel droplet extinction due to forced convection, Combust. Flame 142 (2005), pp [13] D.N. Pope, D. Howard, K. Lu, and G. Gogos, Combustion of moving droplets and suspended droplets: transient numerical results, AIAA J. Themophys. Heat Transf. 19 (2005), pp [14] I. Kim, S. Elghobashi, and W.A. Sirignano, Three-dimensional flow over two spheres placed side by side, J. Fluid Mech. 246 (1993), pp [15] P. Stapf, H.A. Dwyer, and R.R. Maly, A group combustion model for treating reactive sprays in I.C. engines, Twenty-Seventh Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA (1998), pp [16] H.A. Dwyer, P. Stapf, and R.R. Maly, Unsteady vaporization and ignition of a three-dimensional droplet array, Combust. Flame 121 (2000), pp [17] G. Wu and W.A. Sirignano, Transient convective burning of a periodic fuel-droplet array, Proc. Combust. Inst. doi: /j.proci [18] W.A. Sirignano, Fluid Dynamics and Transport of Droplets and Sprays, 2nd edn, Cambridge University Press, Cambridge, [19] C.K. Westbrook and F.L. Dryer, Chemical kinetic modeling of hydrocarbon combustion, Prog. Energy Combust. Sci. 10 (1984), pp [20] R.B.Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, John Wiley and Sons, New York, [21] C.L. Yaws, Chemical Properties Handbook, McGraw Hill, New York, [22] S. Bretsznajder, Prediction of Transport and Other Physical Properties of Fluids, Pergamon Press, Oxford, [23] R.C. Reid, J.M. Prausnitz, and T.K. Sherwood, The Properties of Gases and Liquids, 3rd edn, McGraw-Hill, New York, 1977.