Simplified model for droplet combustion in a slow convective flow

Combustion and Flame 143 (2005) 599–612 www.elsevier.com/locate/combustflame

Simplified model for droplet combustion in a slow convective flow M. Ackerman, F.A. Williams ∗ Center for Energy Research, Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA Received 15 November 2004; received in revised form 28 August 2005; accepted 31 August 2005 Available online 3 October 2005

Abstract An idealized model for droplet vaporization or combustion in the Burke–Schumann reaction-sheet approximation is analyzed in terms of a Peclet number based on the Stefan velocity, taken to be of order unity, for Lewis numbers of unity and for small values of a parameter ε, defined as the ratio of the convective velocity far from the droplet to the Stefan velocity at its surface. Asymptotic solutions for the velocity, pressure, and mixture-fraction fields are obtained through second order in ε. The results are employed to calculate the effects of convection on the burning rate and on the flame shape. The prediction that the burning-rate constant increases linearly with ε for small values of ε is shown to be consistent with available experimental data. It is demonstrated that reasonable values of diffusivities provide approximate agreement of predicted burning rates and flame shapes with results of measurements.  2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Droplet combustion; Slow flow; Asymptotic analysis

1. Introduction A favorite topic of debate with Bob Bilger is the physical relevance of asymptotic analysis. We use it often, and he generally considers it hogwash. This paper, therefore, seems appropriate for an issue celebrating his many important contributions. Although the limit is highly idealized, the results will be seen to be in reasonable agreement with microgravity experiments. Droplet combustion in forced convection is of interest in connection with burning rates of liquid fuels. Mathematical analyses of such processes are com* Corresponding author.

E-mail address: [email protected] (F.A. Williams).

plicated by the fact that imposition of convection removes spherical symmetry. Addressing the limit of small convective velocities facilitates the study of the influences of convection. Except for very small droplets, however, natural convection prevents this limit from applying to experiments performed in Earth gravity. Microgravity measurements of droplet combustion in forced convective flow have been made in the Space Shuttle [1]. Although these experiments are not in the limit of small convective velocities, experiments in this limit are planned for the International Space Station, in preparation for which some preliminary ground-based experiments have been performed. Theoretical analyses of the limit of lowvelocity convection are helpful for comparison with the results of these experiments. The present study is

0010-2180/$ – see front matter  2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2005.08.025

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motivated by interest in ascertaining how simple such analyses can be for making the needed comparisons with experiment. Building on the work of Proudman and Pearson [2], Fendell et al. [3,4] addressed the combustion of a fuel droplet in a slow viscous flow by means of an asymptotic expansion for small Reynolds numbers of the forced convection. Fendell’s model contained many simplifying approximations, but more recent and more extensive investigations by Ayyaswamy and co-workers [5–8] removed many of these approximations. Similar analyses of the combustion of spherical carbon particles at low convection Reynolds numbers have been performed by Blake and Libby [9] and Blake [10]. Although these results, in principle, allow accurate comparisons with experiments to be made, the expansions are so complex that they would be exceedingly time consuming to apply, and they have not been used, except in recent comparisons with experiments for carbon spheres by Blake [10]. A need therefore exists for simplified models that can be more readily used for droplet combustion. A recent simplified analysis is that of Wichman and Baum [11], who made use of a decomposition of the equations into solenoidal and irrotational parts. While quite useful, this approach does not make clear the ordering in the asymptotic expansion—many higher order terms are in effect included at each order, making it difficult to evaluate exactly to what order the expansion has been carried. To complement these results, the present analysis therefore proceeds in a much more systematic and pedestrian way to derive the asymptotic expansion for small convective velocities, following more closely the methods of the works cited earlier. The results, then, while remaining both algebraically simpler through first order and functionally less complex than those of Ayyaswamy and colleagues, bear little resemblance in form to those of Wichman and Baum. All earlier studies cited in this paper treated the Reynolds number of convection as the small parameter. The present analysis is different in principle because a different small parameter is selected, namely the ratio, ε, of the convection velocity of the gas at infinity to the velocity of the gas at the droplet surface without nonspherical convection. The mathematical development therefore is different and precludes addressing the limit of a nonvaporizing droplet. This selection was made with a view toward the problem intended to be studied. Despite this difference, the resulting expansions contain many of the elements that are present in most of the earlier analyses, some of which [5] explicitly refer to “large radial velocity” and become equivalent mathematically to the present development if appropriate restrictions are introduced. The present expansions thus share much

of the range of applicability of earlier analyses. The resulting alternative development given here, however, can help to clarify possible differences associated with different expansion procedures.

2. Formulation Consider a fuel droplet of radius R burning steadily in an oxidizing atmosphere. Attention is focused on the gas phase, and the ratio of the liquid density to the gas density is assumed to be large, so that the time rate of change of R can be neglected in analyzing the gas phase, thereby rendering the droplet-burning problem equivalent to a poroussphere problem in which the fuel-wetted surface of the sphere has a constant radius, independent of time. Let V0 denote the radial velocity of the gas at the surface of the droplet (the Stefan velocity) in the absence of natural and forced convection, under which conditions the steady flow is spherically symmetrical, dependent only on the radial coordinate. Buoyancy is assumed negligible, as would be true in the absence of gravity, eliminating natural convection. Forced convection at a constant uniform velocity U∞ with respect to the droplet is imposed at an infinite distance from the droplet. A spherical coordinate system is selected, with its origin at the center of the droplet and its polar angle θ chosen such that the imposed convective velocity is in the direction θ = 0◦ . By symmetry there is no dependence on the azimuthal angle, and in steady flow there are then only two independent variables, θ and the radial coordinate. The small parameter of expansion is selected to be ε = U∞ /V0 .

(1)

A parameter that appears in the analysis and that is taken to be of order unity is a Peclet number based on the Stefan velocity, a = V0 R/D∞ ,

(2)

where D∞ is the diffusion coefficient for the mixture fraction evaluated in the oxidizing gas far from the droplet. The problem is formulated in terms of the velocity and mixture-fraction fields, assumptions being introduced to enable all other quantities of interest to be related to these variables. Different sets of assumptions can achieve this result. The set adopted here is that Mach numbers are small, that radiant energy fluxes are negligible, as are bulk viscosity and Soret and Dufour effects, that all Lewis numbers are unity (thermal and species diffusivities being equal to the diffusion coefficient of the mixture fraction), that specific heats are constant, and that the Burke– Schumann reaction-sheet approximation applies [12].

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To further simplify the present analysis, gas density and diffusion coefficients are taken as constants; although inaccurate, these approximations retain the correct general form of the problems while removing variable-property terms that would necessitate numerical integrations in most cases. The formulation is nondimensionalized with R as the unit of length, V0 as the unit of velocity, and ρ∞ V02 as the unit of the pressure deviation (the difference between the local pressure and that of the ambient atmosphere), where ρ∞ is the density of the ambient oxidizing gas. With these nondimensionalizations, r denotes the radial coordinate, v and u the r and θ components of velocity, respectively, p the pressure deviation (simply termed pressure), and Z the mixture fraction. By definition, the variable Z assumes the value 1 in the interior of the fuel droplet and 0 in the ambient oxidizing atmosphere. The axisymmetric problem is defined by the four partial differential equations for conservation of mass, two components of momentum, and mixture fraction, along with their associated boundary conditions at infinity and at the surface of the droplet. In these nondimensional variables the differential equations are

remains irrotational to all orders considered. Contrary to the earliest studies, therefore, the viscous Stokes flow is not part of the present solution, that viscous region having been “blown” away by the relatively strong vaporization velocities. The analysis thus applies only when b is of order unity or larger but breaks down if b becomes too small. The Reynolds number b is treated as being of order unity, corresponding to a Schmidt number, a/b, of order unity and a Reynolds number based on the convective velocity, U∞ R/ν∞ = bε, being small. In this sense the expansion for small ε is equivalent to expansions for small convective Reynolds numbers employed in previous work. Conditions at the droplet surface are identified by the subscript  and those in the ambient oxidizing atmosphere by the subscript ∞. The boundary conditions at r = ∞ are

1 ∂ 1 ∂ (vr 2 ) + (u sin θ ) = 0, (3) r ∂r sin θ ∂θ ∂p ∂v u ∂v u2 + − =− v ∂r r ∂θ r ∂r
    1 ∂v 1 1 ∂ ∂v ∂ 2 r + 2 sin θ + b r 2 ∂r ∂r ∂θ r sin θ ∂θ  2 ∂u 2u cot θ 2v − , − 2 − 2 (4) r r ∂θ r2 ∂u u ∂u vu 1 ∂p v + + =− ∂r r ∂θ r r ∂θ
    1 ∂u 1 1 ∂ ∂u ∂ 2 r + sin θ + b r 2 ∂r ∂r ∂θ r 2 sin θ ∂θ  u 2 ∂v − + 2 (5) , r ∂θ r 2 sin2 θ
  1 ∂Z 1 ∂ 2 r vZ − r ∂r a ∂r
  1 ∂Z 1 ∂ sin θ uZ − = 0. + (6) sin θ ∂θ ar ∂θ

v = v (θ ),

Here the Reynolds number based on the Stefan velocity is b = V0 R/ν∞ , where ν∞ denotes the kinematic viscosity in the ambient oxidizing atmosphere. Although these equations indicate that a and b are both parameters in the solution, it will be found that, with the assumptions to be made in the analysis, to the order to which the expansion is carried, the sums of the viscous terms in momentum conservation, those multiplying 1/b in Eqs. (4) and (5), vanish identically, removing b from the problem. Consequently, the flow

v∞ = ε cos θ, p∞ = 0,

u∞ = −ε sin θ, Z∞ = 0.

(7)

Those at r = 1 are u = 0,

(∂Z/∂r) = −av (1 − Z ).

(8)

The last of these conditions, which arises from species conservation for fuel across the interface, equates the sum of the convective and diffusive flux in the gas to the convective flux in the liquid. The condition preceding that, which applies to a rigid porous sphere or to a high-viscosity liquid droplet, excludes tangential motion at the droplet surface; it is selected largely for the simplicity of avoiding the necessity of introducing additional parameters, such as the liquid viscosity, under the realization that much of the earlier work [5–8] has addressed various influences involving surface motion and that the other parameter-free approximation, (∂u/∂r) = 0, the opposite limit, is less realistic. In the first condition given in Eq. (8), v (θ ) = 1 to leading order in ε, by definition. The condition u = 0 requires v (θ ) to depend on θ at higher order. The nature of this dependence and the variation of v (θ ) with ε physically involve coupling with the Z field in droplet burning, for example, through imposition of a condition of equilibrium vaporization, and is best discussed later, in the context of the solutions. Here it is sufficient to treat v (θ ) as specified, subject to certain restrictions, and to observe that the seventh-order system defined by Eqs. (3)–(6), along with the seven boundary conditions given in Eqs. (7) and (8), may then be expected to possess a unique solution.

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3. Asymptotic expansion Expansions of the solutions are sought in the form v = v0 + εv1 + ε2 v2 + · · · , u = εu1 + ε2 u2 + · · · , p = p0 + εp1 + ε2 p2 + · · · , Z = Z0 + εZ1 + ε2 Z2 + · · · .

(9)

Equations (3)–(8) involve ε only in the first two boundary conditions at infinity, demonstrating that the perturbation arises only from the imposed convective velocity. The radial velocity is large compared with this convective velocity when r is of order unity and becomes comparable with it only at large values of r, where the radial velocity becomes small, approaching zero in the absence of convection. This suggests the existence of a distinguished outer region where the radial velocity at leading order is of the same order of magnitude as the imposed convective velocity, so that convection is not a small perturbation there. Rescaling r in the outer region leads to an alternative specification of the problem, which can also be expressed as an expansion in ε. Solving the equations in the inner and outer regions at each order and imposing appropriate matching conditions leads to a solution by matched asymptotic expansions. Variables in the outer region will be identified by a caret. Thus rˆ = εr,

(10)

with rˆ of order unity in the outer region, and if vˆ = v, uˆ = u, pˆ = p, and Zˆ = Z, then the outer expansions vˆ = εvˆ1 + ε2 vˆ2 + · · · , uˆ = εuˆ 1 + ε2 uˆ 2 + · · · , pˆ = ε2 pˆ 1 + ε3 pˆ 2 + · · · , Zˆ = ε Zˆ 1 + ε2 Zˆ 2 + · · ·

(11)

are appropriate. In outer variables, Eqs. (3)–(6) become 1 ∂ 1 ∂ (vˆ rˆ 2 ) + (uˆ sin θ ) = 0, (12) rˆ ∂r sin θ ∂θ uˆ ∂ vˆ uˆ 2 ∂ pˆ ∂ vˆ + − =− vˆ ∂ rˆ rˆ ∂θ rˆ ∂ rˆ
    1 ∂ vˆ ε 1 ∂ ∂ v ˆ ∂ 2 rˆ + 2 sin θ + b rˆ 2 ∂ rˆ ∂ rˆ ∂θ rˆ sin θ ∂θ  2uˆ cot θ 2 ∂ uˆ 2vˆ − , − 2 − 2 (13) rˆ rˆ ∂θ rˆ 2 uˆ ∂ uˆ vˆ uˆ 1 ∂ pˆ ∂ uˆ + + =− vˆ ∂ rˆ rˆ ∂θ rˆ rˆ ∂θ
    1 ∂ uˆ ε 1 ∂ ∂ uˆ ∂ 2 r ˆ + sin θ + b rˆ 2 ∂ rˆ ∂ rˆ ∂θ rˆ 2 sin θ ∂θ

 2 ∂ vˆ uˆ + 2 − , rˆ ∂θ rˆ 2 sin2 θ
 ˆ  1 ∂ 2 vˆ Z 1 ∂ Zˆ rˆ − rˆ ∂r ε a ∂ rˆ
 ˆ  1 ∂ Zˆ  1 ∂ uˆ Z sin θ − = 0. + sin θ ∂θ ε a rˆ ∂θ

(14)

(15)

The boundary conditions, Eq. (7), in outer variables are vˆ∞ = ε cos θ, pˆ ∞ = 0,

uˆ ∞ = −ε sin θ, Zˆ ∞ = 0,

(16)

while the conditions at r = 1 are irrelevant, being replaced by matching conditions. It is well known that, instead of dealing with three differential equations for v, u, and p, a stream function ψ can be introduced, so that only one differential equation (in addition to that for Z) needs to be considered. The stream function is defined such that 1 ∂ψ 1 ∂ψ , u=− , v= 2 (17) r sin θ ∂r r sin θ ∂θ while recovering p in spherical coordinates necessitates integrating momentum conservation. The partial differential equation for ψ and the corresponding momentum equations for recovery of p are rather complex in spherical coordinates and are given in Appendix A. The boundary conditions for ψ are simply ψ = − cos θ,

ψ∞ = ε(r 2 /2) sin2 θ.

(18)

The inner and outer expansions for ψ are ψ = ψ0 + εψ1 + ε2 ψ2 + · · · , ψˆ = ε2 ψ = εψˆ 1 + ε2 ψˆ 2 + · · · .

(19)

It is evident from Eq. (17) that, unlike the other relations between inner and outer variables, there is an ε2 factor between ψ and ψˆ . Although the solutions can be obtained by working only with ψ and Z, the analysis here is instead developed in terms of the original variables v, u, p, and Z. For completeness, however, resulting expansions for ψ are given as well.

4. The inner solution at leading order At leading order the inner solution is the classical steady spherically symmetrical droplet-burning solution. This can be seen formally by substituting the expansions in Eq. (9) into Eqs. (3)–(8) and retaining only the leading-order terms. The differential equations d(v0 r 2 )/dr = 0, (20)
   dp 1 1 d 2v dv dv r 2 0 − 20 , (21) v0 0 = − 0 + dr dr b r 2 dr dr r

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  d 2 1 dZ0 r v0 Z 0 − =0 dr a dr

(22)

are thereby obtained subject to v0 = 1, (dZ0 /dr) = −av0 (1 − Z0 ), v0∞ = 0, p0∞ = 0, and Z0∞ = 0. The solutions are 1 1 p0 = − 4 , v0 = 2 , r 2r Z0 = 1 − e−a/r , ψ0 = − cos θ,

(23)

the last of which is the stream function that corresponds to the velocity listed first; see Eq. (17). This leading-order solution is the basic solution about which the expansion is performed. It may be noted that, although neither of the two viscous terms in Eq. (21) vanishes, their sum does. The solution for the mixture-fraction field has Z0 = 1 − e−a < 1, a positive value dependent on the Peclet number a. Classical droplet-burning theory determines a from the Clausius–Clapeyron equation applied at the droplet surface, along with energy conservation across the interface [12]. For brevity, the resulting well-known formula for the droplet burning rate, conveniently derived through a temperature-oxidizer coupling function that is proportional to Z, is not reproduced here.

603

Zˆ 1∞ = 0; Eq. (16) implies that all subsequent boundary conditions at infinity for all variables are zero. Matching conditions as rˆ → 0, obtained from Eq. (23) by expressing those solutions in outer variables and then expanding in ε, are vˆ1o = 0, uˆ 1o = 0, Zˆ 1o = a/ˆr . The subscript o will identify the behavior of the outer solutions as rˆ → 0 throughout this paper. The solutions to Eqs. (24)–(27), subject to these boundary and matching conditions, are vˆ1 = cos θ, pˆ 1 = 0,

uˆ 1 = − sin θ, ψˆ 1 = (ˆr 2 /2) sin2 θ,

(28)

and   Zˆ 1 = (a/ˆr ) exp −a rˆ (1 − cos θ)/2 .

(29)

Equation (28) simply describes the imposed uniform flow at infinity, for which the sums of the viscous terms in Eqs. (25) and (26) vanish. Equation (29) shows how the mixture-fraction field emanating from the spherical burning droplet responds to this uniform flow in the outer region. This nontrivial nonspherical order-ε solution for Z in the outer zone has been seen previously [6,7,11] and is significant in inducing perturbations in Z in both the inner and the outer zones at higher order.

5. The first-order outer solution The terms of order ε, found when the expansions given in Eq. (11) are substituted into Eqs. (12)–(15), provide the differential equations 1 ∂ 1 ∂ (vˆ1 rˆ 2 ) + (uˆ 1 sin θ ) = 0, rˆ ∂ rˆ sin θ ∂θ ∂ pˆ uˆ 1 ∂ vˆ1 uˆ 21 − =− 1 rˆ ∂θ rˆ ∂ rˆ
  ∂ vˆ1 2vˆ 1 1 ∂ sin θ − 21 + b rˆ 2 sin θ ∂θ ∂θ rˆ  2 ∂ uˆ 1 2uˆ 1 cot θ − , − 2 rˆ ∂θ rˆ 2 1 ∂ pˆ 1 uˆ 1 ∂ uˆ 1 vˆ1 uˆ 1 + =− rˆ ∂θ rˆ rˆ ∂θ
  ∂ uˆ 1 1 1 ∂ sin θ + b rˆ 2 sin θ ∂θ ∂θ  uˆ 1 2 ∂ vˆ1 − + 2 , rˆ ∂θ rˆ 2 sin2 θ
  1 ∂ Zˆ 1 1 ∂ 2 rˆ vˆ1 Zˆ 1 − rˆ ∂ rˆ a ∂ rˆ
  1 ∂ Zˆ 1 1 ∂ sin θ uˆ 1 Zˆ 1 − = 0. + sin θ ∂θ a rˆ ∂θ

(24)

6. The first-order inner solution Since there are no order-ε perturbations in Eqs. (3)–(6) and (8), it is only through the matching to the solutions given in Eqs. (28) and (29) that first-order perturbations arise in the inner zone. Expansion of the solutions in Eqs. (28) and (29) for small values of rˆ and expression of the results in inner variables leads to the matching conditions v1∞ = cos θ,

(25)

p1∞ = 0,

u1∞ = − sin θ, Z1∞ = −(a 2 /2)(1 − cos θ).

(30)

Here and later the subscript ∞ on inner variables identifies their asymptotic behavior as r → ∞. Collection of terms of order ε from the expansions, Eq. (9), in Eqs. (3)–(6) gives (26)

(27)

The boundary conditions at infinity for these equations are vˆ1∞ = cos θ , uˆ 1∞ = − sin θ , pˆ 1∞ = 0,

1 ∂ 1 ∂ (v1 r 2 ) + (u1 sin θ) = 0, (31) r ∂r sin θ ∂θ ∂p ∂v ∂v v0 1 + v 1 0 = − 1 ∂r ∂r ∂r
    1 1 ∂ 1 ∂v1 ∂v ∂ 2 1 + + r sin θ b r 2 ∂r ∂r ∂θ r 2 sin θ ∂θ  2u cot θ 2v 2 ∂u − 21 − 2 1 − 1 2 (32) , r r ∂θ r

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1 ∂p1 ∂u1 v0 u1 + =− ∂r r r ∂θ
    1 ∂u1 1 1 ∂ ∂u ∂ 1 2 r + 2 sin θ + b r 2 ∂r ∂r ∂θ r sin θ ∂θ  u1 2 ∂v1 − + 2 (33) , r ∂θ r 2 sin2 θ
  1 ∂Z1 1 ∂ 2 r v0 Z 1 + v 1 Z 0 − r ∂r a ∂r
  1 ∂Z1 1 ∂ sin θ u1 Z0 − = 0. (34) + sin θ ∂θ ar ∂θ

v0

The ε terms in the corresponding expansions of the boundary conditions given in Eq. (8) at r = 1 give v1 = v1 (θ ),

u1 = 0,

(∂Z1 /∂r) = −a(v1 − v1 Z0 − v0 Z1 ),

(35)

(θ ) + ε2 v

where v (θ ) = 1 + εv1 2 (θ ) + · · ·. Equations (30)–(35) lead to the solutions   A 2 v1 = 1 + 3 cos θ + 21 , r r   1 u1 = − 1 − 3 sin θ, r   A 2 1 p1 = − 2 + 5 cos θ − 41 , r r r   2 1 r + sin2 θ − A1 cos θ, ψ1 = (36) 2 r where A1 is a constant discussed below, and     a 2 −a/r A1 a a 2 −a/r a − e e + + Z1 = r 2 2 r2
   2r 2r −a/r a2 1− + 1+ e cos θ, + C a a C = (a 2 − 2a + 2)ea − 2
0.

(37)

With these solutions, the sums of viscous terms in Eqs. (33) and (34) vanish identically, and from the first expression in Eq. (36), v1 (θ ) = 3 cos θ + A1 . There are a number of points to note concerning these solutions. First, because of the no-slip condition u1 = 0, a θ dependence is necessarily present in v1 . It is possible to eliminate the angular dependence of the vaporization rate and retain irrotationality by relaxing the no-slip condition, resulting in u1 = −[1 + 1/(2r 3 )] sin θ in place of the tangential velocity function appearing in Eq. (36), with corresponding modifications to the other three functions there. These complications underscore the intricacies of the interactions between the flow-field and heattransfer phenomena at the droplet surface. The local vaporization rate at any angle θ depends on fluid-flow, heat-transfer, and thermodynamic phenomena. If, for example, the surface temperature has

a nearly fixed value, practically independent of θ , which could occur as a consequence of a high heat of vaporization forcing the surface temperature to be essentially equal to a constant equilibrium boiling temperature under the high thermal input from a flame surface at which the heat release is large, then a strong no-slip condition (infinite liquid viscosity, say) forces heat conduction to occur in the interior of the droplet, in such a way that the effective heat of vaporization (the sum of the heat of vaporization and the conductive loss to the interior) varies with θ , thereby permitting the vaporization velocity (determined locally by the energy balance at the surface) to vary correspondingly. Mass and momentum conservation in both the gas and the liquid, as well as energy conservation, the thermodynamics of equilibrium vaporization, possible flow driven by gradients of surface tension, etc, all influence the θ dependence of v . To avoid introducing the many additional parameters that would arise (liquid-to-gas viscosity and thermal-conductivity ratios, ratio of heat of vaporization to heat of combustion, equilibrium vapor pressure, etc.) the θ variation of v dictated by the no-slip condition is simply accepted in the present study. Because of the proportionality to cos θ , the average over the entire droplet surface vanishes, so that there is no influence of the θ variation on the net burning rate of the droplet. An additional reason that it is of interest to investigate no-slip consequences, as is done here, is that this approximation represents a limiting case that maximizes the θ variation of v . Introducing slip would reduce the magnitude of the variation of v with θ . It is worthwhile to see how large the variation of v with θ may be. Since the θ dependence of v1 (θ ) averages to zero, the constant A1 measures a correction of order ε to the net burning rate. Exactly what this correction will be again depends on the conditions that are imposed at the interface. With Z proportional to a temperatureoxidizer coupling function and no oxidizer inside the flame sheet, as in the usual Burke–Schumann development [12], the surface temperature is determined by Z . One limiting boundary condition that can be applied is that the average surface temperature remains constant, unaffected by convection. This condition requires that the part of Z1 not proportional to cos θ , obtained from Eq. (37), must vanish. The results of this approximation is A1 = a/2,

Z1 = (a 4 cos θ)/(2C).

(38)

Since the average of Z may depend on ε under other conditions, the constant A1 will be retained in the solutions, as will a corresponding constant at the next order. For most purposes, however, the value in Eq. (38) may be considered to apply.

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The θ -dependent Z1 in Eq. (38) is bounded, since a 4 /(2C) varies from 3a/2 as a → 0 to (a 2 /2)e−a as a → ∞. The variation thus vanishes in the limits of both small and large Peclet numbers and is largest for a Peclet number of order unity. The associated functional form of Z1 for r = 1 in Eq. (37) is slightly messy but is needed for calculating flame shapes, which are defined by setting Z equal to its stoichiometric value Zc . Flame shapes are addressed in a later section. It is possible to eliminate the θ dependence of Z1 by introducing a rotational component to the velocity field, achieving a condition that corresponds best to evaporative equilibrium with a large heat of vaporization. This is not done here because it complicates the solutions, introducing exponential integrals and eventually involving numerical integrations. The resulting modifications to the predicted flame shapes will be small for the small values of Zc encountered in practice.

7. The second-order outer solution Since all boundary conditions at infinity vanish at second order, the second-order outer solution is driven only by matching and first-order source terms in the field equations. Substitution of the expansions appearing in Eq. (11) into Eqs. (12)–(15) and collection of terms of order ε 2 yield 1 ∂ 1 ∂ (vˆ2 rˆ 2 ) + (uˆ 2 sin θ ) = 0, rˆ ∂ rˆ sin θ ∂θ uˆ ∂ vˆ uˆ ∂ vˆ vˆ uˆ ∂ vˆ vˆ1 2 + 1 2 + 2 1 − 2 2 1 ∂ rˆ rˆ ∂θ rˆ ∂θ rˆ
  1 1 ∂ ∂ pˆ 2 ∂ vˆ2 2 + rˆ =− ∂ rˆ b rˆ 2 ∂ rˆ ∂ rˆ   ∂ vˆ2 2vˆ 1 ∂ sin θ − 22 + 2 ∂θ rˆ sin θ ∂θ rˆ  2 ∂ uˆ 2 2uˆ 2 cot θ − , − 2 rˆ ∂θ rˆ 2 uˆ ∂ uˆ uˆ ∂ uˆ vˆ uˆ vˆ uˆ ∂ uˆ vˆ1 2 + 1 2 + 2 1 + 1 2 + 2 1 ∂ rˆ rˆ ∂θ rˆ ∂θ rˆ rˆ
  1 1 ∂ 1 ∂ pˆ 2 ∂ uˆ 2 2 + rˆ =− rˆ ∂θ b rˆ 2 ∂ rˆ ∂ rˆ   ∂ uˆ 2 1 ∂ sin θ + 2 ∂θ rˆ sin θ ∂θ  uˆ 2 2 ∂ vˆ + 2 2− , rˆ ∂θ rˆ 2 sin2 θ
  1 ∂ Zˆ 2 1 ∂ 2 rˆ vˆ1 Zˆ 2 + vˆ2 Zˆ 1 − rˆ ∂ rˆ a ∂ rˆ
  1 ∂ Zˆ 2 1 ∂ sin θ uˆ 1 Zˆ 2 + uˆ 2 Zˆ 1 − + sin θ ∂θ a rˆ ∂θ = 0. (39)

605

Expansion of the two-term inner solutions, v0 + εv1 , etc. for large r, from Eqs. (23), (36), and (37), writing the results in outer variables, and collecting terms of order ε2 eventually provide the matching conditions vˆr2o = 1/ˆr 2 ,  a ˆ A1 + Z2o = rˆ

uˆ 2o = 0,

 a a2 − − cos θ . 2 2ˆr 2

a2

(40)

This expansion also gives pˆ = − cos θ/ˆr 2 , but this is not an independent matching condition and, therefore, is not listed. The solutions to Eq. (39) subject to Eq. (40) are vˆ2 = 1/ˆr 2 ,

uˆ 2 = 0, 2 ψˆ 2 = − cos θ, pˆ 2 = − cos θ/ˆr ,   3 a2 a3 A1 a a + − 2− cos θ Zˆ 2 = rˆ

4ˆr 4ˆr 2ˆr 
a rˆ × exp − (1 − cos θ) . 2

(41)

These results indicate a source-flow contribution in the outer zone at second order, perturbing the uniform flow, again with the sums of the viscous terms vanishing through irrotationality, and an associated perturbation to the mixture-fraction field.

8. The second-order inner solution The second-order solution in the inner zone is driven by matching conditions for the mixture-fraction field and source terms in the expanded field equations. The irrotational flow field at this order is not complicated; it only involves a possible further perturbed source flow. The mixture-fraction field, however, is much more algebraically complex than at previous order because of the appearance of source terms such as v1 Z1 . The matching conditions are v2∞ = 0,

u2∞ = 0,

p2∞ = 0,

A a2 Z2∞ = − 1 (1 − cos θ) 2  3  3a 4 a r + − (1 − cos θ)2 . 8 16

(42)

The boundary conditions from the expansion of Eq. (8) are v2 = v2 (θ ),

u2 = 0,

(∂Z2 /∂r) = −a(v2 − v2 Z0 − v1 Z1 − v0 Z2 ). The expansions of Eqs. (3)–(6) are 1 ∂ 1 ∂ (v2 r 2 ) + (u2 sin θ) = 0, r ∂r sin θ ∂θ

(43)

606

v0

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u ∂v ∂v2 ∂v ∂v + v1 1 + v2 0 + 1 1 − ∂r ∂r ∂r r ∂θ
  1 1 ∂ ∂p2 ∂v 2 2 + r =− ∂r b r 2 ∂r ∂r   ∂v 2v 1 ∂ sin θ 2 − 22 + 2 ∂θ r sin θ ∂θ r  2u cot θ 2 ∂u , − 2 2− 22 r ∂θ r

u21

9. Composite expansions

r Composite expansions that afford uniformly valid solutions are constructed from these results by adding the inner and outer solutions and subtracting the matched, common part. Through order ε2 , the composite expansion for the stream function can be written as

u ∂u v u v u ∂u2 ∂u + v1 1 + 1 1 + 0 2 + 1 1 ∂r ∂r r ∂θ r r
  1 1 ∂ 1 ∂p2 ∂u + r2 2 =− r ∂θ b r 2 ∂r ∂r   ∂u 1 ∂ sin θ 2 + 2 ∂θ ∂θ r sin θ  u2 2 ∂v + 2 2− , r ∂θ r 2 sin2 θ
  1 ∂Z2 1 ∂ 2 r v0 Z 2 + v 1 Z 1 + v 2 Z 0 − r ∂r a ∂r
  1 ∂ 1 ∂Z2 sin θ u1 Z1 + u2 Z0 − + sin θ ∂θ ar ∂θ

v0

= 0.

(44)

The solutions become v2 = A2 /r 2 ,

u2 = 0,

ψ2 = −A2 cos θ,   A2 1 A 1 2 1 p2 = 3 − 42 − 41 − 6 − A1 2 + 5 cos θ r r r 2r r r   1 1 − 3 3 + 6 cos2 θ ; (45) r 2r Z2 is given in Appendix A because of its size. There is no θ dependence of the vaporization rate in this order for this flow, and v2 (θ ) = A2 . The constant A2 thus defines the perturbation of order ε2 to the vaporization rate. Imposition of the previously discussed condition that the average of Z2 must vanish gives the value of A2 listed in Appendix A. As in the first-order inner solution the cos θ term in Z2 is bounded; the coefficient of cos θ in this term, given in Appendix A, varies from −A1 /64 as a → 0 to (a 4 /8 − a 3 A1 /2)e−a as a → ∞. Thus, the variation vanishes in the limits of both small and large Peclet numbers and is largest for a Peclet number of order unity just as found at first order. In contrast, the coefficient of the cos2 θ term vanishes only as a → 0 and increases with increasing Peclet number. The second-order results, therefore, may be less representative of what would be obtained by strict imposition of a limit of high heat of gasification.

ψ = −(1 + A1 ε + A2 ε2 ) cos θ   2 1 r + sin2 θ, +ε 2 r

(46)

and through order ε, the composite expansion for the mixture fraction is  a Z = 1 − e−a/r − 1 − e−εar(1−cos θ)/2 r  A1 a −a/r a 2 e + (1 − e−a/r ) +ε r 2
a2 a + cos θ 2 e−a/r − (1 − e−a/r ) 2 r   2r 2r −a/r a2 −a/r 1− +e . (47) + e + C a a This last composite expansion, which could be extended to order ε2 at the expense of adding a large number of terms, is sufficient for obtaining the first perturbation to the flame shape caused by the imposed convection.

10. The burning rate The net burning rate of the droplet, v¯ = 1+A1 ε + A2 ε2 , through terms of order ε is simply v¯ = 1 + (a/2)ε

(48)

as may be seen from Eqs. (35)–(38). This formula shows that the convection increases the burning rate by the factor 1 + U∞ R/2D∞ , where use has been made of Eqs. (1) and (2). This increase is proportional to the Peclet number based on the convection velocity. This result, of course, applies only when the average droplet surface temperature is unaffected by the imposed convection velocity. Comparisons with experimentally measured effects of convection on the burning rate can be made using this result.

11. The flame shape The flame shape is obtained from Eq. (47) by putting Z = Zc . Although the calculation, in general, needs to be done numerically, an analytical expansion can be derived for small ε if Zc is not small. For

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607

flame contraction. They differ significantly from the composite solutions, demonstrating the importance of using the composite expansion to accurately calculate flame shapes.

12. Comparison with experiment

Fig. 1. Comparison of theoretical flame shapes.

ε = 0, the flame is spherical, and Eq. (47) shows that the nondimensional flame radius rc , denoted by rc0 in that limit, is   rc0 = a/ ln 1/(1 − Zc ) . (49) If Zc is large compared with aε, then the flame is in the inner zone, and by putting rc = rc0 + εrc1 in Eq. (37) and expanding to first order in ε it is found that  
ar 2 r a 2A1 + 1 + c0 rc1 = − c0 rc0 − 2 a 2   2 arc0 2r Zc 2 − Zc − c0 cos θ. + C(1 − Zc ) a (50) If Zc is small compared with aε, then the flame is in the other zone, and Eq. (29) gives rc = (a/Zc )e−εarc (1−cos θ)/2 ,

(51)

which is a transcendental equation to be solved for rc as a function of θ . For intermediate values of Zc , nearly always encountered in practice, Eq. (47) must be used directly with Z = Zc and r = rc , providing a more complicated transcendental equation to be solved for rc as a function of θ . A Newton iteration was employed to solve the transcendental equations. Fig. 1 shows illustrative flame-shape solutions for two different values of ε, namely ε = 0.1 in the upper half and ε = 0.5 in the lower half, with the prediction for combustion with no flow, ε = 0. For the former value of ε, with the selected values of Zc and a, the flame is more nearly in the inner region, and, therefore, the solution obtained using Eq. (50) is shown as well as the first-order composite solution. For the latter value of ε, the flame is more nearly in the outer zone, and, therefore, the solution according to Eq. (51) is shown along with the first-order composite solution. The solutions show flame elongation in the downstream direction and contraction laterally and especially upstream, caused by the convective velocity. The inner and outer solutions alone are seen to overestimate the extents of

Comparison of predicted burning rates and flame shapes with experimental measurements is of interest. Since the analysis is based on an expansion for small values of ε, such comparisons are best made with experiments in which ε is small. Because typical values of V0 are quite small, on the order of 0.1 m/s, there exist very few experiments in which ε is small. Some preliminary microgravity drop-tower experiments were preformed [13] with values of ε less than unity in preparation for experiments planned for the International Space Station. Tests in air with both a sooting fuel, n-heptane, and a nonsooting fuel, methanol, were completed to investigate possible influences of soot on the combustion process. Attempts are made here to compare the present theoretical results with those of these preliminary experiments. The experiments were performed in the 2.2- and 5-s drop towers at the NASA Glenn Research Center [13]. The 2.2-s experimental apparatus used a pressurized gas bottle and regulator to develop a uniform, forced flow in a chamber; the flow was allowed to develop before the rig was dropped. The 5-s apparatus induced a flow by translating a wire supporting the droplet along a track perpendicular to the wire. Both chambers contained a quartz fiber 100 µm in diameter with a quartz bead 200 µm in diameter that was used to tether the droplet. Just after the apparatus was released to microgravity, the droplet was ignited by a retractable hot-wire igniter. The timing of the droplet formation, ignition, and flow was controlled by an on-board microprocessor. Images of the flame and droplet were captured with a color CCD camera at 30 frames per second; these images were used to determine the droplet diameter and flame shape.

13. Burning rate The burning-rate constant, K, the slope of the graph of the square of the droplet diameter as a function of time, was determined for each test run. According to the present theory, in which K = 8R × V0 (ρg /ρ )v¯ , where (ρg /ρ ) is the ratio of the gas to liquid density at the droplet surface, the ratio of the burning-rate constant with convection, K, to that without convection, K0 , depends on the Peclet number based on the convective velocity, aε, as given by Eq. (48) at first order. It is more conventional,

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Fig. 2. Fractional increase of the burning rate versus Reynolds number bε.

however, to employ a Reynolds number rather than a Peclet number in burning-rate correlations. The Peclet number aε can be converted to the Reynolds number bε by dividing by the Schmidt number, ν∞ /D∞ . When considering the burning rate, the most relevant Schmidt number pertains to the diffusion of oxygen outside the flame [14], which is between 0.7 and 1.0; consequently, it seems reasonable, within the accuracy of the present analysis, to set this Schmidt number equal to unity, thereby equating aε to bε. Usually experimental results are exhibited [13] by plotting K/K0 as a function of the square root of a Reynolds number because a square-root correlation, the Frössling correlation, is known to be approximately linear over a relatively wide range of Reynolds number. Since the present analysis instead indicates a linear dependence on bε for small values of ε, Fig. 2 shows the data plotted in that manner (Reynolds number here being based on radius rather than diameter, which is more conventional and was used previously [13]). In Fig. 2 the points represent experimental data and the line represents the theoretical prediction, K/K0 − 1 = bε/2. The correlation is seen to be reasonable for both fuels; agreement is somewhat better than that in a previous plot [13], as might be expected from the fact that ε < 1. There are a number of points to be made concerning the agreement shown in Fig. 2. First, for both fuels, the value K0 = 0.75 mm2 /s was employed. The original paper [13], which considered only heptane, employed this value, although a somewhat higher value (probably 0.8 mm2 /s, within experimental accuracy) would be more appropriate for this fuel, as suggested theoretically and by the data point [13] at bε = 0 in the figure. The original value was retained for consistency of comparisons, and this value is also appropriate for these small methanol droplets, which do not experience the significant burning-rate reduction through water absorption during their histories, which has been observed for large methanol droplets [1]. Second, in calculating the Reynolds

number the kinematic viscosity was taken to be that of air at a mean temperature of 850 K, again for consistency with the original selection [13], although a somewhat higher mean temperature (perhaps as high as 1200 K) may be more appropriate [14], which would move the data points in the figure to smaller values of bε. Since the present theory does not account for variable properties, it is essential in comparisons to make an adjustment of this type; if the theoretical formula is used with ν∞ evaluated at the ambient temperature of about 300 K, then the slope of the theoretical line is much too steep. With the prescription adopted and a Schmidt number of unity, the first-order theoretical prediction shown in Fig. 2, bε/2, is remarkably close to the least-squares fit of the data, which is 0.49bε. In view of the uncertainties in K0 and in the manner of evaluating mean properties, it is best to anticipate differences between theory and experiment on the order of a factor of 2 and to consider the agreement that is seen here as being only qualitative.

14. Flame shapes Nayagam et al. [13] give some data on flame shapes at different droplet diameters and different values of imposed flow velocity, as obtained from measurements with the CCD camera; a great deal of additional data from these experiments was reduced for the purposes of the present comparisons [15]. The flame shapes were characterized by three different measurements, namely, standoff radius, S, flame height, H , and flame width, W , as depicted in Fig. 3. While the values of S and W are relatively well defined experimentally, the flame tip is somewhat jagged, so that the value of H depends on the flame luminosity and, for heptane, is likely to exceed the height of the stoichiometric contour because of soot penetration and burnout beyond this distance [16]. To avoid consideration of chemical kinetics, however, and for simplicity of calculation, when making comparisons with the present theory the stoichiometric surface was taken to define the flame shape. Thus, H was interpreted as the radius of the stoichiometric surface in the downstream direction, W as the diameter of the stoichiometric surface measured through the droplet center in a direction perpendicular to the flow, and S as the upstream radius of the stoichiometric surface. In comparing the flame measurements with the theory, for heptane the unperturbed burning-rate constant, K0 , was set, for all tests at 0.8 mm2 /s, the average of the values found by Nayagam et al. [13], and also that determined by Ackerman et al. [17] for droplets of this size. For methanol, on the other hand,

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609

Fig. 4. Diffusion coefficient calculated based on flame width. Fig. 3. Definition of flame dimensions.

the literature value of 0.6 mm2 /s was employed. The stoichiometric mixture fraction, Zc , was fixed at a value of 0.062 for heptane and 0.135 for methanol, values which correspond to an ambient oxygen mass fraction of 0.23, that of air. In order to calculate ε from Eq. (1), the unperturbed velocity at the droplet surface was determined from the droplet radius according to V0 = (ρ /ρg )K0 /8R. With ε, Zc , and a flame dimension thus found from the experimental data, Eq. (47) was employed to determine the Peclet number, a, by a Newton iteration. In view of Eq. (2), a then determines the value of D∞ needed to fit the data. Within the framework of this constant-property theory, the resulting value of D∞ should lie somewhere between that at ambient temperature and that at flame temperature [14]. Different values of D∞ are, of course, obtained depending on whether S, H , or W is employed in the above procedure. Fig. 4 illustrates some results obtained using W , for heptane experiments, with three different convection velocities. The points shown in Fig. 4 are a representative subset of experimental data from several tests. The constant value of 2 cm2 /s provides reasonable agreement in this figure, likely within experimental error. There is no clearly discernible dependence on U∞ , but there is a tendency for the resulting D∞ to increase with increasing droplet diameter. This might be associated with a decreasing flame temperature (through finite-rate chemistry) as the droplet diameter decreases, approaching extinction; a lower flame temperature would imply a lower average temperature for evaluating the diffusion coefficient and therefore a lower value of the average diffusion coefficient. Similar results were obtained in figures (not shown) based on S and H rather than W . All values of D∞ for heptane were between 1 and 3 cm2 /s. The values obtained with H tended to be somewhat lower than the others, which is expected from the fact that the experimental H is larger, the flame generally extending beyond the stoichio-

metric contour downstream (see the sketch in Fig. 3). For methanol, on the other hand, the experimental H is smaller, so that results for D∞ obtained using H tend to be larger than those found from S or W ; this is caused by the dimness of the downstream flame for this nonsooting fuel, making it difficult to resolve the flame there experimentally and possibly indicating partial radiative extinction. Aside from this difference, the methanol trends are similar to those for heptane. Values of D∞ for methanol ranged from 0.5 to 1.5 cm2 /s and averaged about 1 cm2 /s. One way to assess the values obtained for D∞ is to ascertain at what temperature suitable representative binary diffusion coefficients take on these values. According to information in the literature [18], for the heptane D∞ (2 cm2 /s) these temperatures are 850 K for H2 O into air, 1360 K for CO2 into air, and 1160 K for O2 into N2 . For methanol (1 cm2 /s) they are, correspondingly, 610, 780, and 720 K. Although somewhat lower than the suggested [14] mean of the flame temperature and ambient temperature, these values lie within the necessary bounds and are consistent with the average temperature for methanol being less than that for heptane as a consequence of the lower heat release of methanol. The comparisons may, therefore, be considered reasonable. After determining a suitable average diffusion coefficient in the manner described above, it is of interest to use this constant value to calculate S, H , and W from the theory for comparison with experimental values. Fig. 5 exhibits one such representative comparison for the same experimental points shown in Fig. 4. The predicted trends of increasing flame dimensions with increasing droplet diameter and (to a lesser extent) with decreasing convection velocity are seen to agree with the experiment. Results shown here are in reasonable agreement with those obtained from other such plots presented and discussed elsewhere [15]. Fig. 6 compares representative complete flame shapes calculated theoretically and measured from the

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Fig. 5. A comparison of theoretical and experimental flame widths.

Fig. 6. Theoretical and experimental flame shapes.

experiments for heptane and methanol. Heptane results at a convection velocity of 6 cm/s are shown in the top half of the graph, and methanol results at 1.5 cm/s are shown in the bottom half. The solid lines in the graph represent the experimental flame shapes and the dashed line represents the theoretical prediction. Excellent agreements upstream may be noted for both fuels, but there are differences downstream, in opposite directions for the two fuels, for the reasons suggested previously. The comparisons clearly indicate that, even after optimizing the selection of the value of the diffusion coefficient, flame-shape differences between theory and experiment of at least 30% should be anticipated.

15. Conclusions The droplet-combustion model developed here, for small ratios of the imposed convection velocity to the radial Stefan velocity at the droplet surface, is much simpler, at least conceptually, than other models available in the literature. Even the present model, however, tends to become algebraically complicated at second order in the asymptotic expansion, as do other approaches if written explicitly, as is done here. Among the main limitations of this model are the

assumptions that the Lewis number is unity and the transport properties are constant. These assumptions necessitate judicious selection of the value of the transport coefficient when making numerical predictions and testing against experimental data. Nevertheless, reasonable selections can be made which yield better agreement with experiment for burning rates and flame shapes than would be initially expected. The restriction to irrotational flow, however, prevents the limit of a nonvaporizing droplet from being addressed. For most droplet combustion experiments in convective flow, the convection velocities are too large for the expansion procedure employed here to apply. There are, however, some experiments for which the expansion should be good. Some of the results of these experiments were compared with the theoretical predictions based on carrying the expansion to first order. Comparisons were also made with the expansions carried to second order, but those results are poor and are not shown here; they also involve appreciably more complicated calculations. One may conclude that, as often occurs with asymptotic expansions, it is best to restrict the use of the present theory to first order. To address larger values of the expansion parameter, it would be better to consider the opposite limit, in which, for example, the burning-rate perturbation is believed to increase as the square root of the Reynolds number instead of linearly in Reynolds number. In the comparisons that were made, somewhat different values of transport coefficients were needed for flame dimensions than for burning rates to best fit the measurements. Such differences are to be expected for such a theory because different transport coefficients control different aspects of droplet combustion. Since comparisons were performed only for heptane– air and methanol–air systems, it would be of interest to obtain experimental data and make comparisons for other fuels and other ambient atmospheres, to obtain a better idea of how values of transport properties might best be selected under other circumstances. Although best-fit transport coefficients for flame dimensions were taken here to be independent of droplet diameter, there is an indication from some comparisons that the values should decrease with decreasing droplet diameter, possibly because of effects of finiterate chemistry, which are not included in the theory; it would be of interest to have experimental results over much wider ranges of droplet diameters, to help ascertain whether this observed effect persists. Predicted qualitative trends are supported by experiment. The prediction that the burning rate increases linearly with Reynolds number for small Reynolds numbers is consistent with experiment, although it definitely would be desirable to have more

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accurate experimental results to obtain more stringent tests of this prediction. The predicted general elongation of the flame in the flow direction and its narrowing transversely are consistent with experiment, but for heptane these variations appear to be somewhat more pronounced experimentally, likely because of effects of variable properties and finiterate chemistry, especially of soot. For methanol they seem less pronounced experimentally, possible because of downstream flame weakening and perhaps extinction thorough radiative energy loss. It may thus be concluded that qualitative trends may be obtained from the theory, but refinements (introducing greater complexity) are needed for improved quantitative predictions. Burning rates and flame dimensions at low convective Reynolds numbers can confidently be calculated numerically within about a factor of 2 from the model presented here.

Acknowledgments We thank Vedha Nayagam and Amable Liñán for many important discussions concerning this work, which was supported by the NASA Microgravity Combustion Science Program.

Appendix A Equations deemed too long for inclusion in the main text are listed here. The equation to calculate the stream function is


∂ 3ψ ∂ 2ψ ∂ψ 2 ∂ 3 ψ −1 + − 2r 2 cot θ 2 r r 2 3 r sin θ ∂r ∂r ∂θ ∂θ ∂r  ∂ 2ψ ∂ψ − 3 cot θ 2 + 3(cot2 θ + 1) ∂θ ∂θ
3 3 ∂ ψ ∂ 2ψ ∂ψ 3 ∂ ψ r + r − r cot θ − ∂θ ∂r∂θ ∂r 3 ∂r∂θ 2  2 2 ∂ ψ ∂ ψ ∂ψ ∂ψ − 2r 2 2 − 4 2 + 2r + 4 cot θ ∂r ∂θ ∂r ∂θ
4 4 4 ∂ ψ ∂ ψ 1 4∂ ψ r + 2r 2 2 2 + = 4 b ∂r ∂r ∂θ ∂θ 4 ∂ 3ψ ∂ 3ψ ∂ 3ψ − 4r − 2 cot θ ∂r 2 ∂θ ∂r∂θ 2 ∂θ 3 2 2 ∂ ψ ∂ ψ + 4r cot θ + (3 cot2 θ + 8) 2 ∂r∂θ ∂θ  ∂ψ . − 3 cot θ (cot2 θ + 3) (A.1) ∂θ − 2r 2 cot θ

Equations for recovering the r and θ dependences of the pressure from the stream function are

611


 2  1 ∂ψ ∂ ψ ∂ψ r −2 r sin θ ∂θ ∂r∂θ ∂θ   ∂ψ ∂ψ ∂ 2 ψ ∂ψ − r −r − cot θ ∂r ∂θ 2 ∂θ ∂r ∂ 3ψ ∂ 3ψ ∂p + r2 2 + ∂r ∂r ∂θ ∂θ 3 2 ∂ ψ ∂ψ − cot θ 2 + (cot2 θ + 1) ∂θ ∂θ

= −r 4 sin θ

(A.2)

and 
 ∂ψ ∂ 2ψ 1 −r 2 r sin θ ∂θ ∂r   ∂ψ ∂ 2ψ ∂ψ −r −r + r cot θ ∂r ∂r∂θ ∂r ∂ 3ψ ∂p ∂ 3ψ + r3 3 − r ∂θ ∂r ∂r∂θ 2 2 2 ∂ ψ ∂ψ ∂ ψ + r cot θ + 2 2 − 2 cot θ . ∂r∂θ ∂θ ∂θ

= −r 2 sin θ

(A.3)

The second-order inner solution for the mixture fraction is  
1 a A A2 a + A1 a 2 − + − 12 Z2 = r 2 2r 2r   3 4 2 3 a r a a a a2 + − − − 2 − 4 e−a/r 6 12 3r 6r 6r
 2 2 a a2r ar a ar + 3 − + e−a/r + + − 6C 2 r 2 2  3 2 2a a a − − 2 −3+ 4 r 2r   3  a4 a3 a r + + 2 + cos θ e−a/r − 4 8 2r   a 1 1 − − 3 + A1 a 2 e−a/r 2 2r r
  1 3 a 2r A a 2 2r − + e−a/r − − − + 1 2C a 2 a 2 2r
4 a −a A1 a −a e + e + − 2C 8C  A a2 − 1 2 (2 − a − 2e−a − ae−a ) C  
1 1 r r −a/r − −e + × a 2 a 2   a4 1 3a 2 r 2 3a 3 r 2 + − + cos θ − − 3 4 8 16  2 2  a3 r a2 a3 3a r a2 + + − 2− 4 + e−a/r 4 2 2r 2r 2r
2 2 3 4 2 5a r 3a r a a 1 − + − + + 2C 4 4 8 2r

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+ e−a/r



a2 a3 5a 2 r 2 a 3 r + − − 2 4 2 2r 2r


3 9a a2 − + B 2 8  3a 2 1 − + + 2C 4



3a 2 a 3 + 2 16  a3  2 12r − 6ar + a 2 8  −a/r 2 2 −e (12r + 6ar + a ) , (A.4)

+

−

where C = ea (a 2 − 2a + 2) − 2 and B = ea (a 3 − 6a 2 + 18a − 24) + 6a + 24. If the average surface temperature remains constant, unaffected by convection, then from Eq. (A.4)     3a a 3 1 a A1 A2 = A 1 a − + + + 2 2 2 8 12 

 2 3 a 1 9a a 3 a a − + + 3+ − + ea 8 16 96 6C 8 4  2 3  3a a 15a + − − ea 3 − 8 8 16  
a 3 9a 3a 2 a 3 − + − + 2B 2 8 2 16  
3 a a2 a 3 a − −1 − − + BC 4 8 2 12   2 a a . + ea 1 − + (A.5) 2 12

References [1] D.L. Dietrich, J.B. Haggard Jr., F.L. Dryer, V. Nayagam, B.D. Shaw, F.A. Williams, Proc. Combust. Inst. 26 (1996) 1201–1207. [2] I. Proudman, J.R.A. Pearson, J. Fluid Mech. 2 (1957) 237–262.

[3] F.E. Fendell, M.L. Sprankle, D.S. Dodson, J. Fluid Mech. 26 (1966) 267–280. [4] F.E. Fendell, D.F. Coats, E.B. Smith, AIAA J. 6 (10) (1968) 1953–1960. [5] S.S. Sadhal, P.S. Ayyaswamy, J. Fluid Mech. 133 (1983) 65–81. [6] G. Gogos, S.S. Sadhal, P.S. Ayyaswamy, T. Sundararajan, J. Fluid Mech. 171 (1986) 121–144. [7] G. Gogos, P.S. Ayyaswamy, Combust. Flame 74 (2) (1988) 111–129. [8] M.A. Jog, P.S. Ayyaswamy, I.M. Cohen, J. Fluid Mech. 307 (1996) 135–165. [9] T.R. Blake, P.A. Libby, Combust. Flame 86 (1/2) (1991) 147–161. [10] T.R. Blake, Combust. Flame 129 (1/2) (2002) 87–111. [11] I.S. Wichman, H.R. Baum, in: C.T. Avedisian, V.A. Arpaci (Eds.), Heat Transfer in Microgravity, American Society of Mechanical Engineers, New York, 1993, p. 111. [12] F.A. Williams, Combustion Theory, Addison-Wesley, Redwood City, CA, 1985, pp. 2–4, 52–78. [13] V. Nayagam, M.C. Hicks, N. Kaib, M. Ackerman, J.B. Haggard Jr., F.A. Williams, Droplet Combustion in a Slow Convective Flow, in: Sixth International Microgravity Combustion Workshop Proceedings, NASA CP 2001-210826, NASA, 2001. [14] C.K. Law, F.A. Williams, Combust. Flame 19 (1972) 393–405. [15] M.D. Ackerman, Ph.D. thesis, University of California, San Diego, 2003. [16] R.E. Mitchell, A.F. Sarofim, L.A. Clomburg, Combust. Flame 37 (3) (1980) 227–244. [17] M.D. Ackerman, R.O. Colantonio, R.K. Crouch, F.L. Dryer, J.B. Haggard, G.T. Linteris, A.J. Marchese, V. Nayagam, J.E. Voss, F.A. Williams, B.L. Zhang, A Treatment of Measurements of Heptane Droplet Combustion Aboard MSL-1, NASA TM 2003-212553, NASA, 2003. [18] N.B. Vargaftik, Handbook of Physical Properties of Liquids and Gases, second ed., Hemisphere, Washington, 1975.