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Research ArticleOn Laminar Rich Premixed Polydisperse Spray FlamePropagation with Heat Loss

G. Kats and J. B. Greenberg

Faculty of Aerospace Engineering, Technion-Israel Institute of Technology, 32000 Haifa, Israel

Correspondence should be addressed to J. B. Greenberg; [email protected]

Received 22 November 2015; Revised 4 February 2016; Accepted 7 February 2016

Academic Editor: Ashwani K. Gupta

Copyright © 2016 G. Kats and J. B. Greenberg. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

A mathematical analysis of laminar premixed spray flame propagation with heat loss is presented. The analysis makes use of adistributed approximation of the Arrhenius exponential term in the reaction rate expression and leads to an implicit expression forthe laminar burning velocity dependent on the spray-related parameters for the fuel, gas-related parameters and the intensity ofthe heat losses. It is shown that the initial droplet load, the value of the evaporation coefficient, and the initial size distribution arethe spray-related parameters which exert an influence on the onset of extinction.The combination of these parameters governs themanner in which the spray heat loss is distributed spatially and it is this feature that is the main factor, when taken together withvolumetric heat loss, which determines the spray’s impact on flame propagation and extinction.

1. Introduction

Spalding [1] was the first to treat the problem of a laminargas flame propagating through a combustible premixture inthe presence of heat losses, for example, due to heat loss byconduction to the walls of the combustion chamber or radi-ation. In keeping with experimental evidence it was foundthat for a given heat loss there exist two possible burningvelocities, one stable and the other not. Extinction occurs atthe point of traversal from the stable to the unstable mode ofpropagation. Essentially, this happens when the heat loss istoo great. The theory agreed well with experiments for flamepropagation and extinction in tubes. Subsequent work [2, 3]also dealt with similar problems of one-dimensional flamepropagation and examined different aspects of extinction ofthese flames. Buckmaster [4] reexamined the aforementionedproblem using asymptotic tools and was able to construct theslow and fast waves as well as to predict a simple explicitquenching criterion. Joulin and Clavin [5] considered thestability of laminar premixed flames subject to linear heat lossand found a variety of instabilities for the different regimes(slow and fast waves) examined by previous researchers.Nicoli and Clavin [6] considered the effect of variable heat

loss intensities on the dynamics of a premixed flame. Clavinand Nicoli [7] investigated heat loss effects on stability limitsof downward propagating premixed flames.

In the context of mathematical analysis of one-dimensional premixed spray flames, some attention wasdirected to the influence of heat losses [8, 9], although whenthe stability of such flames was considered heat losses werenot accounted for [10–13]. However, in [8, 9] the linearvolumetric heat losses were taken as being of order 𝜀, where𝜀 is inversely proportional to the activation energy of theassumed global chemical reaction, and were only appliedin the region between the onset of droplet evaporation andthe flame front. In addition, the sprays were taken to bemonodisperse.

As pointed out by Sirignano [14] radiation impacts onindividual droplet heating and evaporation in several ways.Primarily, droplets may be heated by radiation from hightemperature gases. Or, alternatively, radiation may decreasethe flame temperature so that radiative (and conductive)heat transfer to droplets will be diminished. In modelingthe behavior of single droplets the radiative heating effect isexpressed via a modification to the latent heat of evaporation.However, Sazhin [15], in discussing single droplets, argues

Hindawi Publishing CorporationJournal of CombustionVolume 2016, Article ID 1069873, 14 pageshttp://dx.doi.org/10.1155/2016/1069873

2 Journal of Combustion

x∗→ −∞x∗ = x∗e x

∗ = 0 x∗ = x∗�

x∗→ +∞

x∗

R4 = {−∞ < x∗ ≤ x∗e } R3 = {x

∗e ≤ x

∗ ≤ 0} R2 = {0 ≤ x∗ ≤ x∗� } }R1 = {x

∗� ≤ x

∗ < ∞

Figure 1: Schematic of the subregions considered for the analysis of a one-dimensional planar premixed spray flame in the presence ofvolumetric/radiative heat losses.

that taking them as grey opaque bodies “overlooks the factthat droplet radiative heating takes place not at their surface(as in the case of convective heating) but via the absorptionof thermal radiation penetrating inside the droplets.” Hetherefore assumes for modelling purposes that the droplet issemitransparent.

In the current paper we investigate the propagation of anoff-stoichiometric rich laminar premixed polydisperse sprayflame in the presence of heat loss, for the first time. Inthis paper we restrict our attention exclusively to the stablebranch of propagation down to conditions of extinction. Ourultimate aim is to explore spray flame ignition since duringthe first moments of application of the igniter the role of heatlosses can be rather dominant. It is a well-established factthat modern combustors in aircraft need to satisfy a largenumber of requirements. Of particular interest is the fact thatunder extreme conditions they must reignite following flameextinction without any problems and without any externalhelp. The possibility of extinction also exists in cold and wetconditions (e.g., in a hailstorm) as well as at high altitudesdue to oxygen starvation. The presence of liquid fuel in theform of a multisized spray of droplets that must first producea sufficient amount of fuel vapor for successful ignitionincreases the difficulties.The currentwork is a prelude to suchan ignition study that will be reported in the future.

In a previous publication [16] we modified a nonasymp-totic mathematical approach [17, 18] to analyzing gas flamepropagation and successfully applied it to examine the prop-agation of liquid fuel spray flames and double spray flames(i.e., both fuel and oxidizer supplied as a spray of droplets).For propagation studies this approach seems to be a viablealternative to an asymptotic approach. Here we adopt thesame methodology.

The structure of the paper is as follows. We present thegoverning equations and the assumptions upon which theyare based. We then explain how they are solved and presenttheir solution. Finally, we examine how the combinationof volumetric and spray-related heat losses influences thepropagation and extinction conditions of the spray flames.

2. Governing Equations andProblem Definition

2.1. Assumptions. We consider a laminar one-dimensionalpremixed flame propagating into an off-stoichiometric freshhomogeneous mixture of fuel vapor, liquid fuel droplets,oxygen, and an inert gas. A schematic of the situation

considered is shown in Figure 1. The flame is taken topropagate from left to right. The droplets are viewed from afar-field vantage point; that is, their average velocity is equalto that of their host environment. For qualitative purposesthis approach has been demonstrated to be quite valid [19].The spray is taken to be polydisperse; that is, at any pointin space and time there is a distinct size distribution of thespray’s droplets.The temperature of the droplets is taken to bethat of the surroundings; essentially the droplets heat-up timeis small compared to the characteristic time associated withtheir motion. Droplet evaporation is assumed negligible untila prescribed reference temperature 𝑇V (such as the boilingtemperature of the liquid fuel) is attained.

The stoichiometry of the gas mixture that the flame frontmeets is taken to be fuel rich, so that the limiting reactantconsumed by chemical reaction is oxygen. It is assumedthat the various transport coefficients, such as thermalconductivity, diffusion coefficient, specific heat at constantpressure, and latent heat of vaporization of the droplets, canbe satisfactorily specified by representative constant values.An overall reaction of the form ]

𝐹fuel + ]

𝑂oxidant →

products is taken to describe the chemistry.As the velocity of propagation of the flame is much less

than the velocity of sound, dynamic compressibility effects inthe mixture can be neglected.Thus, the density becomes onlya function of the temperature through the gas law.

The polydisperse spray is described using the sectionalmethod [20] in which the droplet size-distribution is dividedinto sections (or bins) and conservation equations are derivedfor the liquid fuel in each section allowing for dropletevaporation from a given section, for example, 𝑗, and additionto that section as droplets evaporate in the next section up andbecome eligible for membership in 𝑗.

The spatial region from 𝑥∗ → +∞ to 𝑥∗ → −∞ can bedivided into four distinct regions:

(a) A prevaporization regionwhere the system’s tempera-ture has not yet reached the critical temperature of thefuel at which significant evaporation begins to occur.This region is denoted by 𝑅

1= {𝑥∗

V ≤ 𝑥∗< ∞}.

(b) A preflame region in which chemical reaction hasnot yet begun because the temperature is less thanthe temperature required to initiate it. This is region𝑅2= {0 ≤ 𝑥

∗≤ 𝑥∗

V }.

(c) The flame region where reaction occurs, denoted by𝑅3= {𝑥∗

𝑒≤ 𝑥∗≤ 0}.

Journal of Combustion 3

(d) A postflame region where the reaction has essentiallyceased because, due to heat loss, the temperaturedrops below the temperature that sustains the flame,𝑅4= {−∞ < 𝑥

∗≤ 𝑥∗

𝑒}.

Note that Figure 1 is simply a schematic drawing of the fouraforementioned regions and does not reflect their actualscales which will be determined by the spray and gasenvironment parameters.

2.2. Governing Equations. In the presence of linear heat lossthe governing conservation equations can be shown to be

𝜌𝑐𝑝

𝜕𝑇

𝜕𝑡∗= 𝜆

𝜕2𝑇

𝜕𝑥∗2

+ 𝜌𝑞𝑌𝑂𝐴 exp(− 𝐸

𝑅𝑇

) − �̂� (𝑇 − 𝑇0)

− 𝐿𝑆V,

(1)

𝜌

𝜕𝑌𝑂

𝜕𝑡∗

= 𝜌𝐷𝑂

𝜕2𝑌𝑂

𝜕𝑥∗2

− 𝜌𝑌𝑂𝐴 exp(− 𝐸

𝑅𝑇

) , (2)

𝜌

𝜕𝑌𝑑,𝑗

𝜕𝑡∗

= −𝑆V,𝑗, 𝑗 = 1, 2, 3, . . . , 𝑁𝑆, (3)

where𝑆V𝑗 = Δ 𝑗𝑌𝑑,𝑗 − Ψ𝑗𝑌𝑑,𝑗+1 (4)

is the source term for production of liquid fuel in droplets ofsection 𝑗, and the sectional vaporization coefficients are (see[20])

(Δ𝑗, Ψ𝑗) = 1.5�̂�V (

3𝑑𝑢𝑗

− 2𝑑𝑙𝑗

𝑑3

𝑢𝑗− 𝑑3

𝑙𝑗

,

𝑑𝑙𝑗+1

𝑑3

𝑢𝑗+1− 𝑑3

𝑙𝑗+1

) ,

𝑗 = 1, 2, . . . , 𝑁𝑆, Ψ𝑁𝑆

= 0,

(5)

where �̂�V is the liquid fuel’s evaporation coefficient and𝑑𝑙𝑗 and𝑑𝑢𝑗are the lower and upper diameters, respectively, defining

droplet size section 𝑗.

𝑆V =

𝑁𝑆

∑

𝑗=1

𝑆V𝑗 (6)

𝑆V is the source term for the production of fuel vapor byevaporating droplets in all sections.The sectional Damkohlernumbers are based on 𝑑2 law, which was confirmed byprevious studies [21] to predict the actual vaporization historyof an interacting droplet, especially in the initial period ofcombustion.

Note that the terms resulting from the chemical reactionare linear in the deficient reactant, which, for the rich off-stoichiometric case herein discussed, is oxidant. (The near-stoichiometric case, in which the product of the oxidantand fuel vapor concentrations is present, is not consideredhere since the approach to be adopted was reported [18]as yielding considerably less satisfactory results under suchcircumstances.)

The boundary conditions for this set of equations are

𝑥∗→ ∞: 𝑌

𝑂= 𝑚𝑂𝑢

, 𝑇 = 𝑇0, 𝑌𝑑,𝑗

= 𝛿𝑗,

𝑥∗→ −∞: 𝑌

𝑂= 0, 𝑇 = 𝑇

0, 𝑌𝑑= 0.

(7)

2.3. Solution Approach. Following [18, 22] the nonlinearnature of the chemical source terms can be alleviatedby replacing the exponential temperature-dependent term𝐴 exp(−𝐸/𝑅𝑇) by the step function 𝑘(𝑇

𝑚)𝐻(𝑇−𝑇

∗) in which

𝑘 (𝑇𝑚) = 𝐴 exp(− 𝐸

𝑅𝑇𝑚

) . (8)

𝑇∗is found so that there is equality of the integrals with

respect to 𝑇 of both the Arrhenius exponential and the stepfunction over the entire range of temperatures (from the coldmixture temperature, 𝑇

0, to the highest temperature attained

in the system, 𝑇𝑚):

∫

𝑇𝑚

𝑇0

𝑘 (𝑇𝑚)𝐻 (𝑇 − 𝑇

∗) 𝑑𝑇 = ∫

𝑇𝑚

𝑇0

𝐴 exp(−𝐸𝑅𝑇

)𝑑𝑇. (9)

It is not hard to show that this yields

∫

𝑇𝑚

𝑇0

𝑘 (𝑇𝑚)𝐻 (𝑇 − 𝑇

∗) 𝑑𝑇 = 𝑘 (𝑇

𝑚) (𝑇𝑚− 𝑇∗) ;

∫

𝑇𝑚

𝑇0

𝐴 exp(−𝐸𝑅𝑇

)𝑑𝑇 ≈

𝑅𝑇2

𝑚

𝐸

𝑘 (𝑇𝑚)

(10)

and we obtain

𝑇∗= 𝑇𝑚 (

1 − 𝜀) , 𝜀 =

𝑅𝑇𝑚

𝐸

≪ 1 (11)

so that in (1) and (2) use is made of

�̂�𝑂 (

𝑇) = 𝐴 exp(− 𝐸𝑅𝑇𝑚

)𝐻 (𝑇 − 𝑇∗)

= 𝑘 (𝑇𝑚)𝐻 (𝑇 − 𝑇

∗) .

(12)

For a spray flame the heat loss due to droplet evaporationalso plays a role in the heat balance and it is thereforeintroduced via 𝑇

𝑚, the maximum temperature attained in

the system. In this way the step function specifically reflectsthe maximum value of the Arrhenius exponential functionthereby capturing the essence of the physical meaning of thatfunction.

The above governing equations (with their appropriateboundary conditions) describe the spray flame propagationfrom the perspective of laboratory coordinates. However, inorder to extract a solution it proves more straightforward torewrite the equations in coordinates attached to the flamefront. Assuming the flame is moving at a constant speed 𝑢 inthe positive 𝑥-direction (i.e., propagation from left to right)we can define the new coordinate 𝜉 = 𝑥∗ − 𝑢 ⋅ 𝑡∗, wherebythe governing set of equations reduces to the followingordinary differential equations:

𝐷𝑇

𝜕2𝑇

𝜕𝜉2+ 𝑢

𝜕𝑇

𝜕𝜉

+

𝑞

𝐶𝑝

𝑌𝑂�̂�𝑂 (

𝑇) −

𝐿

𝐶𝑝

𝑆V�̂�𝑑 (𝑇)

− 𝛼 (𝑇 − 𝑇0) = 0,

(13)

𝐷𝑂

𝜕2𝑌𝑂

𝜕𝜉2

+ 𝑢

𝜕𝑌𝑂

𝜕𝜉

− 𝑌𝑂�̂�𝑂 (

𝑇) = 0, (14)

𝑢

𝜕𝑌𝑑,𝑗

𝜕𝜉

− 𝑆V𝑗�̂�𝑑 (𝑇) = 0, 𝑗 = 1, 2, 3, . . . , 𝑁𝑆, (15)


where

�̂�𝑑 (

𝑇) = 𝐻 (𝑇 − 𝑇V) (16)

with 𝐻 being the Heaviside function used to ensure thatdroplet evaporation only occurs significantly once the tem-perature of the fuel droplets reaches the fuel’s boiling point,𝐷𝑇

= 𝜆/𝜌𝑐𝑝

is the thermal diffusion coefficient, and

𝛼 = �̂�/𝜌𝑐𝑝is the heat loss coefficient. Henceforth, for

convenience, we assume a unity Lewis number (𝐷𝑇= 𝐷𝑂).

Note that the chemical source terms in (13) and (14)are applicable only for 𝑇 ≥ 𝑇

∗but due to our coordinate

transformation we can replace 𝐻(𝑇 − 𝑇∗) by 𝐻(−𝜉) as the

system is invariant under spatial translation and we can takethe location of the interface between the second and thirdregions as = 0.

Accordingly, the boundary and matching conditions thatare applicable are

𝜉 = 𝜉V: [𝑌𝑂] = [𝑌

𝑂] = 0, [𝑌

𝑑,𝑗] = 0, 𝑇 = 𝑇V, [𝑇] = [𝑇

] = 0, (17a)

𝜉 = 0: [𝑌𝑂] = [𝑌

𝑂] = 0, [𝑌

𝑑,𝑗] = 0, 𝑇 = 𝑇

∗, [𝑇] = [𝑇

] = 0, (17b)

𝜉 = 𝜉𝑒: [𝑌

𝑂] = [𝑌

𝑂] = 0, [𝑌

𝑑,𝑗] = 0, 𝑇 = 𝑇

∗, [𝑇] = [𝑇

] = 0, (17c)

𝜉 = 𝜉𝑚: [𝑌

𝑂] = [𝑌

𝑂] = 0, [𝑌

𝑑,𝑗] = 0, 𝑇 = 𝑇

𝑚, [𝑇] = [𝑇

] = 0, 𝑇

(𝜉𝑚) = 0, (17d)

where derivatives with respect to 𝜉 are denoted by and 𝜉V, 𝜉𝑒,and 𝜉𝑚are, respectively, the interface locations where finite-

rate vaporization begins and the interface between the regionwhere the chemical reaction takes place and the region whereit ceases and a spatial point where themaximum temperature𝑇𝑚is attained.Physically, 𝜉

𝑚must be located in the middle of the third

region 𝜉𝑒

< 𝜉𝑚

< 0, where the reaction is adding heatto the system but the volumetric/radiative heat loss is ofconsiderable competitive importance.

3. Solution

The solution of the governing equations is found in everyregion separately with the matching conditions connectingthe solutions. We present the results for the four relevantregions:

In 𝑅1= {𝜉V ≤ 𝜉 < ∞}

𝑇 (𝜉) = 𝑇0+ (𝑇V − 𝑇0) exp [𝜆2 (𝜉 − 𝜉V)] , (18a)

𝑌𝑂 (

𝜉) = 𝑚𝑂𝑢(1 +

𝜇1

𝜇2

exp(− 𝑢𝐷𝑂

𝜉)) , (18b)

𝑌𝑑,𝑗

= 𝛿𝑗, 𝑗 = 1, 2, 3, . . . , 𝑁

𝑆. (18c)

In 𝑅2= {0 ≤ 𝜉 ≤ 𝜉V}

𝑇 (𝜉) = 𝑇0+ 𝜔1exp [𝜆

1(𝜉 − 𝜉V)]

+ 𝜔2exp [𝜆

2(𝜉 − 𝜉V)] +

𝐿

𝐶𝑝

𝑃 (𝑆V (𝜉)) ,(19a)

𝑌𝑂 (

𝜉) = 𝑚𝑂𝑢(1 +

𝜇1

𝜇2


𝜉)) , (19b)

𝑌𝑑,𝑗 (

𝜉) =

𝑁𝑆

∑

𝑖=𝑗

Ω𝑗𝑖exp [

Δ𝑖

𝑢

(𝜉 − 𝜉V)] ,

𝑗 = 1, 2, 3, . . . , 𝑁𝑆.

(19c)

In 𝑅3= {𝜉𝑒≤ 𝜉 ≤ 0}

𝑇 (𝜉) = 𝑇0+ 𝜔1exp [𝜆

1(𝜉 − 𝜉V)] − Γ𝑚𝑂𝑢 (

𝜇1

𝜇2

+ 1)

⋅ [

𝜆2− 𝜇1

𝜆1− 𝜆2

exp (𝜆1𝜉) + exp (𝜇

1𝜉)] +

𝐿

𝐶𝑝

⋅ 𝑃 (𝑆V (𝜉)) ,

(20a)

𝑌𝑂 (

𝜉) = 𝑚𝑂𝑢(1 +

𝜇1

𝜇2

) exp (𝜇1𝜉) , (20b)

𝑌𝑑,𝑗 (

𝜉) =

𝑁𝑆

∑

𝑖=𝑗

Ω𝑗𝑖exp [

Δ𝑖

𝑢

(𝜉 − 𝜉V)] ,

𝑗 = 1, 2, 3, . . . , 𝑁𝑆.

(20c)

In 𝑅4= {−∞ < 𝜉 ≤ 𝜉

𝑒}

𝑇 (𝜉)

= 𝑇0

+ {𝑇∗− 𝑇0−

𝐿

𝐶𝑝

𝑃 (𝑆V (𝜉𝑒))} exp [𝜆1 (𝜉 − 𝜉𝑒)]

+

𝐿

𝐶𝑝

𝑃 (𝑆V (𝜉)) ,

(21a)


𝑌𝑂 (

𝜉) = 0, (21b)

𝑌𝑑,𝑗 (

𝜉) =

𝑁𝑆

∑

𝑖=𝑗

Ω𝑗𝑖exp [

Δ𝑖

𝑢

(𝜉 − 𝜉V)] ,

𝑗 = 1, 2, 3, . . . , 𝑁𝑆,

(21c)

where

𝑃 (𝑆V (𝜉)) =

𝑁𝑆

∑

𝑗=1

{

{

{

Δ𝑗

𝑁𝑆

∑

𝑖=𝑗

Ω𝑗𝑖exp [(Δ

𝑖/𝑢) (𝜉 − 𝜉V)]

𝐷 (Δ𝑖/𝑢)2+ 𝑢 (Δ

𝑖/𝑢) − 𝛼

− Ψ𝑗

𝑁𝑆

∑

𝑖=𝑗+1

Ω𝑗+1,𝑖

exp [(Δ𝑖/𝑢) (𝜉 − 𝜉V)]


𝑖/𝑢) − 𝛼

}

}

}

,

(22a)

𝑃(𝑆V (𝜉))

=

𝑁𝑆

∑

𝑗=1

{

{

{

Δ𝑗

𝑁𝑆

∑

𝑖=𝑗

Ω𝑗𝑖(Δ𝑖/𝑢) exp [(Δ

𝑖/𝑢) (𝜉 − 𝜉V)]


𝑖/𝑢) − 𝛼

− Ψ𝑗

𝑁𝑆

∑

𝑖=𝑗+1

Ω𝑗+1,𝑖

(Δ𝑖/𝑢) exp [(Δ

𝑖/𝑢) (𝜉 − 𝜉V)]


𝑖/𝑢) − 𝛼

}

}

}

,

(22b)

Ω𝑗𝑖=

Ω𝑗+1,𝑖

Ψ𝑗

Δ𝑗− Δ𝑖

, 𝑖 ̸= 𝑗, (22c)

Ω𝑗𝑖= 𝛿𝑗−

𝑁𝑆

∑

𝑖=𝑗+1

Ω𝑗𝑖, 𝑖 = 𝑗, (22d)

𝜔1= −

𝐿

𝐶𝑝

𝑃(𝑆V (𝜉V)) − 𝜆2𝑃 (𝑆V (𝜉V))

𝜆1− 𝜆2

, (22e)

𝜔2= 𝑇V − 𝑇0 +

𝐿

𝐶𝑝

𝑃(𝑆V (𝜉V)) − 𝜆1𝑃 (𝑆V (𝜉V))

𝜆1− 𝜆2

, (22f)

Γ =

𝑞𝑘 (𝑇𝑚)

𝐶𝑝[𝑘 (𝑇𝑚) − 𝛼]

, (23a)

𝜆1=

𝑢

2𝐷𝑇

(𝑑𝑇− 1) , (> 0) , (23b)

𝜆2= −

𝑢

2𝐷𝑇

(𝑑𝑇+ 1) , (< 0) , (23c)

𝜙 =

𝛼𝐷𝑇

𝑢2

, (23d)

𝑑𝑇= √1 + 4𝜙, (> 1) , (23e)

𝜇1=

𝑢

2𝐷𝑂

(𝑑𝑂− 1) , (> 0) , (23f)

𝜇2= −

𝑢

2𝐷𝑂

(𝑑𝑂+ 1) , (< 0) , (23g)

𝑑𝑂= √1 + 4𝐵, (23h)

𝐵 =

𝑘 (𝑇𝑚)𝐷𝑂

𝑢2

. (23i)

An explicit formula for the burning velocity cannot beextracted; however, by applying matching and boundaryconditions a set of coupled implicit algebraic equations areobtained through which 𝑢, 𝑇

𝑚, 𝜉𝑚, 𝜉𝑒, 𝜉V can be found by

using a numerical iterative method. The solution is based onthe assumption that exp(𝜇

1𝜉𝑒) → 0 which can be readily

verified.In the limit of infinite vaporization coefficient (�̂�V → ∞)

when the spray of liquid fuel droplets evaporates in a singlevaporization front, some simplification of the afore-describedsolutions is achieved (see Appendix for details).

4. Results and Discussion

Use was made of the analytical solution in the previoussection to examine the effect of heat loss and fuel sprayparameters on conditions for spray flame propagation andextinction. The data used for the calculations was as follows(unless otherwise specified):

𝑞 = 1.279 × 107 J/kg,

𝐿 = 0.04Q,

𝜆 = 0.02512Wm/K,

𝐴 = 1010 s−1,

𝐸 = 2 × 108 J/kmol,

𝑐𝑝= 1255.92 J/kgK,

𝑇0= 300K,

𝑇V = 400K,

�̂�V = 1.4524 ⋅ 10−15m2/s.

(24)

The chemical kinetic scheme employed concerns the burningof n-decane and relevant thermochemical data was takenfrom [23, 24]. By specifying the initial fraction of liquidfuel to the total fuel (vapor + liquid) in the fresh mixture,∑𝑁𝑆

𝑗=1𝛿𝑗/(𝑚𝐹𝑢

+ 𝑚𝑑𝑢) = 𝛿, it can be shown that the mass

fractions in the fresh mixture are given by the followingexpressions:

(𝑚𝑂𝑢

, 𝑚𝑑𝑢, 𝑚𝐹𝑢)

= (𝑠 {1 − 𝑚𝑑𝑢

− 𝑚𝐹𝑢} ,

𝑠𝛿

(1 + 𝛼𝑂𝐹

/𝜑)

,𝑚𝑑𝑢

(1 − 𝛿)

𝛿

)

(25)

unless 𝛿 = 0 for which

𝑚𝐹𝑢

=

𝑠

(𝑠 + 𝛼𝑂𝐹

/𝜑)

, (26)

where 𝑠 is the mole fraction of oxygen in the fresh mixture,𝛼𝑂𝐹

is the stoichiometric coefficient, and 𝜑 is the equivalenceratio. Here 𝜑 is taken as 2 and 𝛼

𝑂𝐹= 3.5.


𝛿 = 0

𝛿 = 0.5

𝛿 = 1

0.080.09

0.10.110.120.130.140.150.16

u (m

/s)

0 10 2520 305 15𝛼 (s−1)

Figure 2: Influence of heat loss parameter on spray flame propa-gation velocity for different initial droplet loads—evaporation frontcase.

In order to extract and highlight the various factors atwork in the fuel rich spray flames under consideration herewe focus on three cases: (a) evaporation of the droplets in asingle vaporization front (the solution for this case is givenin Appendix), (b) finite rate evaporation but with all dropletssubsumed into a single section (monosectional descriptionof the spray with solution derivable by setting 𝑁

𝑆= 1 in the

analysis of the previous section), and (c) the afore-detailedfull polydisperse case. Case (a) is applicable when the liquidfuel is highly volatile. Case (b) applies to a less volatile fueland is modeled to capture only the gross features of the sprayimpact on the combustion. Case (c) applies to a less volatilefuel but with the details of the spray size structure accountedfor.

(a) Evaporation in a Front. We begin our discussion of thepredictions of the theory by considering the case for whichthe liquid droplets evaporate in a sharp front. In Figure 2a plot is presented of the spray flame’s propagation velocityfor different initial liquid fuel loads, 𝛿, as a function of theheat loss parameter, 𝛼. For the purely gaseous case, 𝛿 = 0,the behavior of the velocity follows the classical behavior(see, e.g., [4]), with a decrease in the velocity resulting fromincreasing the heat loss. Of course, this is to be expected asa result of the competing exothermic-endothermic mecha-nisms at play. Eventually, flame extinction occurs at somecritical value of𝛼.With the fuel supplied as a liquid spray only,that is, 𝛿 = 1, it is readily observed that the flame propagationvelocity is less than that of its gaseous counterpart for anygiven value of 𝛼. This is not surprising since, even withoutvolumetric/radiative heat loss, the droplets themselves mustabsorb heat for evaporation, thereby automatically lower-ing the flame temperature and, hence, the flame velocity(see, also, [10, 16]). As the droplet evaporation takes placein a front this source of heat loss is quite concentratedleading to a notable influence on the flame velocity. Inaddition, extinction of the spray flame occurs at a lowervalue of the heat loss parameter 𝛼. This is understandableas both the distributed heat loss from the surroundings and

𝛿 = 0

𝛿 = 0.5

𝛿 = 1

5 10 15 20 25 300𝛼 (s−1)

1900

1950

2000T m(K

) 2050

2100

2150

Figure 3: Influence of heat loss parameter on spray flame tempera-ture for different initial droplet loads—evaporation front case.

the liquid droplets heat loss combine to overcome theexothermic chemical reaction.

The case in which 50% of fuel is supplied as liquid and50% as vapor is also illustrated in Figure 2. It can be seenthat, not surprisingly, the relevant curve lies in between thelimiting curves which we have discussed.

In Figure 3 the flame temperature is drawn as a functionof the heat loss parameter. The curves clearly reflect thediscussion of Figure 2 and show the rather drastic effectof the heat loss particularly in the proximity of extinctionconditions.

It is known from the theory of gas flame propagation[4] that the extinction velocity when linear heat loss ispresent is 𝑒−1/2 times the adiabatic flame velocity. For sprayflames the two sources of heat loss play a role, namely,radiative/volumetric heat loss andheat loss, due to absorptionof heat by the liquid droplets for evaporation. In Figure 4we examine the relative importance of these heat losses byplotting the log of the ratio of the flame velocity at extinctionto the adiabatic gas flame velocity as a function of the initialliquid spray load, 𝛿. The green line with circles is the classicalresult when only radiative/volumetric heat loss is accountedfor and it is readily seen to be constant at the value of 𝑒−1/2.The blue continuous unmarked line shows the decrease inthe spray flame burning velocity predicted when only dropletheat loss is accounted for. In this case the heat absorbed bythe droplets for evaporation is not sufficient to extinguishthe flame and the velocity ratio is always larger than 𝑒−1/2.When both heat losses are included in the model the red linewith boxes is obtained. (Note that since the flame velocity atextinction is used to construct this figure the value of the heatloss parameter, 𝛼, varies along the curve (cf. Figure 2).) Theratio of velocities is now evidently dependent on the initialliquid droplet load, and a factor of about 𝑒−0.66 is found when𝛿 = 1.

In view of these findings and the underlying rationale itwould seem that the latent heat of vaporization of the liquidfuel should be an important factor in determining spray flamevelocity and extinction conditions.This influence is shown in


DHLRHLDHL + RHL

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

ln(u

/uad

)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10𝛿

Figure 4: Relative importance of sources of heat loss in determiningthe flame velocity of laminar spray flames as function of the initialliquid fuel load, evaporation front case. DHL = droplet heat lossonly, RHL = radiative/volumetric heat loss only, and DHL + RHL= combined droplet and radiative/volumetric heat loss.

0 5 10 15 20 25

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

u (m

/s)

𝛼 (s−1)

L/q = 0.008

L/q = 0.04

L/q = 0.08

Figure 5: Influence of latent heat of vaporization on spray flamepropagation velocity—evaporation front case.

Figure 5 in which the flame velocity is drawn as a function ofthe volumetric/radiative heat parameter for different values ofthe ratio of the latent heat to the heat of reaction.The effect isquite striking with the value of the flame velocity decreasingby a factor of 50% (for a fixed value of 𝛼) as the latent heatincreases by a factor of 10. Moreover, the critical value of 𝛼for extinction decreases by a factor of 67% as the latent heatof vaporization increases tenfold.

(b) Monosectional Spray. The graphs presented so far relatedto the case in which the spray of fuel droplets evaporatesin a sharp front. Attention will now be turned to themonosectional case with a finite evaporation rate. In Figure 6

300 400 500 600 700 800 900 1000 1100 1200C (s−1)

𝛿 = 0

𝛿 = 0.2

𝛿 = 0.4

𝛿 = 0.6

𝛿 = 0.8

𝛿 = 1

0.0820.0840.0860.088

0.090.0920.0940.096

uD

HL+

RHL(

m/s

)

Figure 6: Influence of vaporization coefficient on spray flamepropagation velocity at critical values of volumetric/radiative heatloss parameter for different initial droplet loads, monosectionalspray.

the spray flame propagation velocity is drawn as a functionof the evaporation coefficient, 𝐶 = 1.5�̂�V(3𝑑𝑢 − 2𝑑𝑙)/(𝑑

3

𝑢−

𝑑3

𝑙) (see (5)), at the values of the relevant critical volumet-

ric/radiative heat loss parameter, and for different values ofthe initial fuel droplet load. For this case increasing the valueof𝐶 is equivalent to using a more volatile fuel (i.e., increasing�̂�V) and/or using smaller droplets in the spray. For any givenload it is clear that the velocity decreases as the evaporationcoefficient increases. In fact as 𝐶 → ∞ the velocity levelsoff at the value appropriate to the evaporation front case, asanticipated. In addition, as the initial droplet load increasesthe critical velocity decreases, irrespective of the value ofthe vaporization coefficient. This, too, is in keeping with theprediction of the evaporation front case (see, e.g., Figure 2).

In Figure 7 we examine the influence of the vaporizationcoefficient in determining the flame velocity as a functionof the initial liquid fuel load by plotting the log of the ratioof the flame velocity at extinction to the adiabatic gas flamevelocity as a function of the vaporization coefficient. It is clearthat as 𝐶 increases, for all values of 𝛿, the logarithm of thevelocity ratio decreases below the value of −0.5 with the effectbeing greatest when the fuel is supplied in liquid form, that is,𝛿 = 1. In addition, it can be observed that for large valuesof the evaporation coefficient the relevant curve overlapswith that of Figure 4 for the case of the evaporation front.Whereas the heat loss due to volumetric/radiative heat lossis distributed throughout the entire field, the more focusedthe heat loss due to droplet evaporation the greater the effecton critical conditions for extinction. Once again it is evidentthat the influence of the spray parameter, this time in theform of the evaporation coefficient, combines with that of thevolumetric/radiative heat loss so that the classical 𝑒−1/2-factoris modified. The maximum modification corresponds to thecase of an infinitely large rate of evaporation.

(c) Polydisperse Spray. Having established the importantcharacteristic features of a fuel spray and their role in flame


Table 1: The sectional diameters 𝑑𝑗(𝜇m) and initial droplet size distributions.

Section number 1 2 3 4 5 6 7 8 9Section diameters 1–5 5–10 10–20 20–30 30–40 40–50 50–70 70–90 90–110Distribution 1 0 0 0 0 0 1 0 0 0Distribution 2 0 0 0.207 0 0 0 0 0 0.793Distribution 3 0.0005 0.0005 0.0141 0.0793 0.1662 0.2464 0.2349 0.1547 0.1034

C = 300C = 600C = 900

C = 1200C = 1500

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.10𝛿

−0.66−0.64−0.62

−0.6−0.58−0.56−0.54−0.52

−0.5

ln(u

DH

L+RH

L/uad)

Figure 7: Influence of the vaporization coefficient in determiningthe flame velocity of laminar spray flames as function of the initialliquid fuel load, monosectional spray.

propagation in the presence of heat losses without recourse tothe actual droplet sizes within the spray we turn our attentionto the impact of the initial spray droplet size distribution.To this end we make use of three quite different initialspray distributions that are listed in Table 1, together with thedefinitions of the size sections utilized.

The numbers in each distribution’s columns representthe fraction of liquid fuel (in the relevant size section)in the total liquid fuel supplied. Distribution 1 is initiallymonosectional but will become multisize once evaporationoccurs and droplets become ineligible for membership insection 6. Distribution 2 is bidisperse, initially having almost80% liquid fuel in the highest section, number 9, and therest in much smaller droplets in section 3. Finally, the thirddistribution has droplets initially well spread out throughoutall size sections and is probably closer to what exists in reallife. These three different size distributions have a commonSauter Mean Diameter (SMD) of 44.8 𝜇m. The SMD is theratio of the volume of droplets in the spray to their surfacearea and is often used to characterize a polydisperse spray byan equivalent spray of single size droplets all of which have adiameter equal to the SMD. However, there is evidence thatthis characterization may sometimes be misleading [11, 25,26].

In Figures 8, 9, and 10 the profiles of the sectional fuelmass fractions for the three initial size distributions aredrawn, for comparison.𝑚

𝑑𝑢was taken as 0.1.

Recall that the flame is propagating from left to right withthe close vicinity of the point 𝜉 = 0 marking the intense

0

0.02

0.04

0.06

0.08

0.1

Sect. 1 Sect. 2 Sect. 3



2 3 4 5×10−4

×10−4

×10−4

yd

3.5 4 4.5 5 5.53𝜉 (m)

02468

Figure 8: Liquid fuel sectional mass fractions profiles in one-dimensional laminar spray flame propagation, initial size distribu-tion 1.

00.010.020.030.040.050.060.070.08

−10 −8 −6 −4 −2 0 2 413579

×10−4

×10−4

×10−4

yd

Sect. 1Sect. 2Sect. 3

Sect. 7Sect. 8Sect. 9Sect. 4

Sect. 5

Sect. 1Sect. 2Sect. 4

Sect. 5

Sect. 6

Sect. 6

−4 −2 0 2 4 6 8−6𝜉 (m)

Figure 9: Liquid fuel sectional mass fractions profiles in one-dimensional laminar spray flame propagation, initial size distribu-tion 2.

flame reaction zone. The redistribution of liquid dropletsin all three cases as they evaporate and migrate down thesize sections is apparent. For example, for distribution 1 alldroplets initially occupy section 6. As they evaporate thereis a transfer to the initially unoccupied section 5 in whicha build of liquid fuel is readily observable. Subsequently, therelocation to lower sections from section 5 occurs as dropletsevaporate and become ineligible for membership in section5. Similarly, behavior is found for the lower sections, too.


−5 0 5 10

×10−4

×10−4

×10−5

Sect. 1Sect. 2

Sect. 1Sect. 2

Sect. 3Sect. 7Sect. 8Sect. 9Sect. 4

Sect. 5

Sect. 6

0

0.005

0.01

0.015

0.02

0.025

yd

−4 −2 0 2 4 6−6𝜉 (m)

1357

Figure 10: Liquid fuel sectional mass fractions profiles in one-dimensional laminar spray flame propagation; initial size distribu-tion 3.

However, what should also be noted, for the data utilized here,is the spatial distribution of the liquid fuel. For distribution1 virtually all the liquid fuel evaporates before reaching thereaction zone, whereas for distributions 2 and 3 this is notthe case. Although for these latter distributions much vaporis released upstream of the flame region noticeable continuedevaporation occurs downstream of the flame front as thedroplets pass through the flame into the hot products region.This will lead to different spatially distributed heat releasebehaviors depending on the internal spray structure (i.e., sizedistribution) and its evolution as the droplets absorb heat forevaporation.

Consider Figure 11 in which the velocity at extinction isdrawn as a function of the initial liquid load for all threeinitial droplet size distributions. It is clear that for 𝛿 ̸= 0the initial droplet distribution has a noticeable influenceover the velocity at extinction. Distribution 1 leads to thelowest velocity; distribution 2 leads to the highest velocitywith distribution 3 in between. At most the discrepancyis about 9% (comparing distributions 1 and 2) and about2% (comparing distributions 1 and 3). This ordering is alsoreflected in Figure 12 in which the influence of the volumetricheat loss on the flame velocity is charted for three initialliquid fuel loads and for all three initial size distributions.Theunderlying rationale for this behavior can be deduced fromFigure 13 where the flame temperatures associated with thevelocities of Figure 11 are drawn. First we isolate the effectof the initial total liquid load. For any value of 𝛼 for whicha flame exists and for any initial droplet size distributionthe flame temperature drops as 𝛿 increases. As we have seenbefore this is due to the increased heat loss sustained due tothe droplets heat absorption for evaporation. This, in turn,influences the speed of propagation and lowers it accordingly.

Now isolate the effect of the droplet size distribution for afixed value of 𝛿 and𝛼. It is clear that size distribution 1 leads tothe lowest flame temperature followed by distribution 3 withthe highest flame temperature supplied when distribution 2

0.084

0.086

0.088

0.09

0.092

0.094

0.096

0.098

1st dist.2nd dist.3rd dist.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10𝛿

uex

tinct

ion

(m/s

)

Figure 11: Flame velocity at extinction versus initial liquid dropletload—influence of initial droplet distribution.

𝛿 = 0, 1st dist.𝛿 = 0.5, 1st dist.𝛿 = 1, 1st dist.𝛿 = 0, 2nd dist.𝛿 = 0.5, 2nd dist.

𝛿 = 1, 2nd dist.𝛿 = 0, 3rd dist.𝛿 = 0.5, 3rd dist.𝛿 = 1, 3rd dist.

0.080.09

0.10.110.120.130.140.150.16

u (m

/s)

5 10 15 20 25 300𝛼 (s−1)

Figure 12: Influence of heat loss parameter on spray flame propaga-tion velocity for different initial droplet loads—polydisperse case.

is used. Evidently, the initially monosectional distributionproduces the most focused heat loss due to the initial con-centration of droplets in section 6. At the other extreme theleast focused heat loss results from the bidisperse distribution2. Although the smaller droplets in section 3 evaporate fairlyrapidly thereby lowering the flame temperature somewhat,it is the large fraction of large droplets initially in section 9which evaporate gradually that dominates the droplet heatloss mechanism thereby lessening the effect of heat loss whencompared to that of distribution 1. Distribution 3, which hasdroplets spread throughout the size range involved, yields asituation between that generated by the other two distribu-tions, as can be readily observed in Figures 11 and 12. Thus,it is the way in which heat loss due to droplet evaporation isspatially distributed that determines the flame temperatureand velocity. This sharpens our observations for the afore-discussed cases of the evaporation front and monosectionalspray for which the details of the initial size distribution weredealt with in an integral fashion.


Table 2: The sectional diameters 𝑑𝑗(𝜇m) and initial droplet size distributions for examining the influence of droplet size on spray flame

propagation.

Section number 1 2 3 4 5 6 7 8 9Section diameters 1–5 5–10 10–20 20–30 30–40 40–50 50–70 70–90 90–110Distribution 1 0 1 0 0 0 0 0 0 0Distribution 2 0 0 0 1 0 0 0 0 0Distribution 3 0 0 0 0 0 1 0 0 0Distribution 4 0 0 0 0 0 0 0 1 0

𝛿 = 0, 1st dist.𝛿 = 0.5, 1st dist.𝛿 = 1, 1st dist.𝛿 = 0, 2nd dist.𝛿 = 0.5, 2nd dist.

𝛿 = 1, 2nd dist.𝛿 = 0, 3rd dist.𝛿 = 0.5, 3rd dist.𝛿 = 1, 3rd dist.

5 10 15 20 25 300𝛼 (s−1)

1900

1950

2000

2050

2100

2150

T (K

)

Figure 13: Influence of heat loss parameter on spray flame temper-ature for different initial droplet loads—polydisperse case.

In Figure 14we examine the influence of the initial dropletsize distributions on the ratio of the velocity at extinctionto the adiabatic flame velocity. The ordering of the curvesfollows the pattern we have explained and impacts on theclassical factor of 𝑒−1/2 in accordance with that ordering andthe initial droplet loading.

The impact of the initial droplet size distributions on thetemperature distribution is illustrated in Figure 15. Althoughthe profiles are similar, the differences reflect the differentrates of heat loss due to droplet evaporation. As mentionedpreviously in connection with Figures 8, 9, and 10, initialsize distribution 1 provides the most concentrated heat lossso that the associated temperature profile lies below thoseof the other two size distributions. Distribution 2 providesthe most protracted droplet heat loss behavior thereby lead-ing to greater temperatures than associated with the otherdistributions. Due to this disparity in droplet-related heatloss, differences of several tens of degrees occur at the peaktemperature.

We now turn to examine more explicitly the influenceof droplet size on the spray flame behavior described before.For this purpose we make use of the four initial droplet sizedistributions listed in Table 2.

Note that these four distributions are initially monosec-tional. The difference between them is the size of droplets inthe fresh mixture. The SMDs for these four distributions are


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10𝛿

−0.7

−0.65

−0.6

−0.55

−0.5

−0.45

ln(u

/u(𝛿=0,𝛼=0)

)

Figure 14: Influence of the initial droplet size distribution indetermining the flame velocity of laminar spray flames as functionof the initial liquid fuel load, polydisperse spray.

−8 −6 −4 −2165017001750180018501900

400600800

10001200140016001800

Tem

pera

ture

(K)

×10−3

×10−4

0−16 −14 −12 −10 −8 −6 −4 −2−18−20𝜉 (m)


Figure 15: Effect of initial droplet size distribution on thermalprofile in one-dimensional laminar spray flame propagation.

7.213, 24.663, 44.814, and 79.582 𝜇m, respectively.The solutionin (18a)–(21c) is, of course, applicable as the polydispersedevelopment of the spray is independent of the initial sizedistribution.

In Figure 16 the spray flame velocity at extinction isplotted for all four cases of Table 2 as a function of the initialliquid fuel load, 𝛿. As anticipated and explained previously,


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10𝛿

0.075

0.08

0.085

0.09

0.095

0.1

uex

tinct

ion

(m/s

)

1st dist.2nd dist.

3rd dist.4th dist.

Figure 16: Influence of initial droplet size on extinction velocity inpolydisperse spray flames.

for all four initial distributions, the critical velocity decreasesas the initial liquid fuel load increases. The largest effect isnoted for the case of smallest droplets for which adecrease ofabout 26%occurs.The initially larger droplets of distributions2–4 of Table 2 lead to more modest maximum decreases ofbetween 14% and 11%, with the largest droplets supplying thesmallest increase.This is entirely in keeping with the previousassertion that it is the way in which heat loss due to dropletevaporation is spatially distributed that determines the flamevelocity. This is further confirmed by Figure 17 in which therelative importance of the volumetric and spray-related heatlosses is plotted, for the initial droplet sizes of Table 2. Thecompound effect of both sources of heat loss is apparent.Once again it is clearly evident that the smaller the initialdroplet size the more focused the droplet related heat loss,and, hence, its impact on the flame propagation velocity. Thepreviouslymentioned classical factor of 𝑒−1/2 at extinction fora gas flame can become as small as 𝑒−0.8 for size distribution 1(of Table 2) and an initial liquid fuel load 𝛿 = 1.

5. Conclusions

The role of both volumetric/radiative heat loss and that ofheat absorbed by a spray of evaporating droplets in deter-mining premixed spray flame propagation and extinctionwasinvestigated analytically using a nonasymptotic solution ofthe governing equations. This is the first analytical treatmentof laminar premixed polydisperse spray flame propagationwith volumetric/radiation heat loss that we are aware of.Such analysis is important in terms of the physical insightsit is able to supply and its potential use as a benchmark forcomputational studies.

Calculated results indicate that the presence of the fueldroplets in the premixture reduces both the critical value ofboth the flame velocity prior to extinction and the heat loss,primarily due to the aforementioned heat absorption by thedroplets. In addition to the initial droplet load, the value of theevaporation coefficient and the initial size distribution are the

DHL, 1st dist.DHL, 2nd dist.DHL, 3rd dist.DHL, 4th dist.RHL, 1st dist.RHL, 2nd dist.

RHL, 3rd dist.RHL, 4th dist.RDHL, 1st dist.RDHL, 2nd dist.RDHL, 3rd dist.RDHL, 4th dist.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10𝛿

−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

0

ln(u

/u(𝛿=0,𝛼=0)

)

Figure 17: Influence of the initial droplet size in determining theflame velocity of laminar polydisperse spray flames as function ofthe initial liquid fuel load.

other spray-related parameters which exert an influence onthe onset of extinction.The combination of these parametersgoverns themanner in which the spray heat loss is distributedspatially and it is this which is the main factor, when takentogether with the volumetric heat loss, which determines thespray’s impact on flame propagation and extinction.

In addition, the analysis demonstrates that use of a SauterMeanDiameter to characterize the behavior of a polydispersespray flame may lead to erroneous conclusions. In fact, theactual polydispersity of the flame must be considered.

Finally, the results clearly show how the presence of thespray and volumetric heat losses lead to a reduction of theburning velocity in comparison with a single phase gas flame,due to the total heat loss, which leads to a lowering of theburned gas temperature. It should be mentioned that, inter-estingly, this result does not match some experimental datafor laminar premixed spray flames for which flame velocityincreases were found for rich spray flames [27–29]. However,this discrepancy is due to the fact that in the current steadystate, one-dimensional theory the inherently two- or threedimensional phenomenon of flame cellularization clearlycannot be captured. This necessitates a stability analysis forthe two- or three dimensional perturbed flame front, whichis beyond the scope of the current work.

Appendix

Solution of Governing Equations When theSpray of Droplets Evaporates in a Sharp Front

We apply a matching condition at 𝜉 = 𝜉V that reflects thepresence of vaporization front:


𝜉 = 𝜉V:

[𝑌𝑂] = [𝑌

𝑂] = 0, 𝑇 = 𝑇V, [𝑇

] =

𝐿𝑚𝑑𝑢𝑢

𝐷𝑇𝑐𝑝

.

(A.1)

The solution can then be readily found:

In 𝑅1= {𝜉V ≤ 𝜉 < ∞},

𝑇 (𝜉) = 𝑇0+ (𝑇V − 𝑇0) exp [𝜆2 (𝜉 − 𝜉V)] ,

𝑌𝑂 (

𝜉) = 𝑚𝑂𝑢[1 +

𝜇1

𝜇2


𝜉)] .

(A.2)

In 𝑅2= {0 ≤ 𝜉 ≤ 𝜉V}

𝑇 (𝜉) = 𝑇0− Λ exp [𝜆

1(𝜉 − 𝜉V)]

+ [𝑇V − 𝑇0 + Λ] exp [𝜆2 (𝜉 − 𝜉V)] ,

𝑌𝑂 (

𝜉) = 𝑚𝑂𝑢[1 +

𝜇1

𝜇2


𝜉)] .

(A.3)

In 𝑅3= {𝜉𝑒≤ 𝜉 ≤ 0}

𝑇 (𝜉) = 𝑇0

+ [

Γ𝑚𝑂𝑢

[𝜆2− (𝜇1/𝜇2) 𝜆1]

𝜆2− 𝜆1

− Λ exp (−𝜆1𝜉V)]

⋅ exp (𝜆1𝜉) − 𝑚

𝑂𝑢(1 +

𝜇1

𝜇2

)Γ exp (𝜇1𝜉) ,

𝑌𝑂 (

𝜉) = 𝑚𝑂𝑢(1 +

𝜇1

𝜇2

) exp (𝜇1𝜉) .

(A.4)

In 𝑅4= {−∞ < 𝜉 ≤ 𝜉

𝑒}

𝑇 (𝜉) = 𝑇0+ (𝑇∗− 𝑇0) exp [𝜆

1(𝜉 − 𝜉

𝑒)] ,

𝑌𝑂 (

𝜉) = 0,

(A.5)

where

Λ ≡

𝐿𝑚𝑑𝑢

𝑑𝑇𝐶𝑃

. (A.6)

In this vaporization front case, it can be shown that explicitexpressions are obtained for 𝜉

𝑚, 𝜉𝑒, 𝜉V:

𝜉V = −ln [Γ𝑚

𝑂𝑢[𝑑𝑂− 𝑑𝑇] /𝑑𝑇(1 + 𝑑

𝑂) (𝑇V − 𝑇0 + Λ)]

𝜆2

,

𝜉𝑒=

ln {(𝑇∗− 𝑇0) / (−Δ

2exp (−𝜆

1𝜉V) + Γ𝑚𝑂𝑢 (𝑑𝑇 + 𝑑𝑂) /𝑑𝑇 (1 + 𝑑𝑂))}

𝜆1

,

𝜉𝑚

=

ln (𝜆1[−Δ2exp (−𝜆

1𝜉V) + Γ𝑚𝑂𝑢 (𝑑𝑇 + 𝑑𝑂) /𝑑𝑇 (1 + 𝑑𝑂)] / (2Γ𝑚𝑂𝑢𝜇1/ (1 + 𝑑𝑂)))

𝜇1− 𝜆1

.

(A.7)

Nomenclature

𝐴: Preexponential constant𝐵: Parameter in solution (Equation (23i))𝑐𝑝: Specific heat

𝑑𝑙𝑗, 𝑑𝑢𝑗: Lower and upper diameters of size section

𝑗

𝑑𝑂, 𝑑𝑇: Parameters in solution ((23e) and (23h))

𝐷𝑇: Thermal diffusion coefficient

𝐷𝑂: Oxygen mass diffusion coefficient

𝐸: Activation energy�̂�V: Evaporation coefficient𝐻: Heaviside function𝑘,𝐾𝑑, �̂�𝑂: Functions defined in (8), (16), and (12),respectively

𝐿: Latent heat of vaporization𝑚𝑂𝑢: Initial mass fraction of gaseous oxygen

𝑚𝑑𝑢: Initial mass fraction of liquid fuel

𝑁𝑆: Number of size sections

𝑃: Spray-related function ((22a) and (22b))𝑞: Heat of reaction

𝑅: Universal gas constant𝑅1, 𝑅2, 𝑅3, 𝑅4: Solution subdomains

𝑠: Mole fraction of oxygen in the freshmixture

𝑆V: Total rate of droplet evaporation𝑆V,𝑗: Rate of evaporation of droplets in section 𝑗𝑡∗: Time𝑇: Temperature𝑇0: Ambient temperature

𝑢: Velocity𝑥∗: Spatial coordinate

𝑌: Mass fraction�̂�, 𝛼: Radiative heat loss coefficients𝛼𝑂𝐹: Stoichiometric coefficient

Γ: Parameter in solution (Equation (23a))𝛿𝑗: Mass fraction of liquid fuel in section 𝑗

𝛿: Initial ratio of mass fraction of liquid fuelto total fuel

Δ𝑗, Ψ𝑗: Vaporization Damkohler numbers for sec-

tion 𝑗𝜀: Small parameter (Equation (11))


𝜙: Parameter in solution (Equation (23d))𝜑: Equivalence ratio𝜆: Thermal conductivity𝜆1, 𝜆2: Parameters in solution (Equation (23b) and(23c))

𝜇1, 𝜇2: Parameters in solution (Equations (23f)and (23g))

𝜌: Density𝜔1, 𝜔2: Parameters in solution ((22e) and (22f))

Ω𝑗𝑖: Coefficients defined in (22c) and (22d)

𝜉: Flame front coordinate.

Subscripts

𝑑, 𝑗: Relating to droplets in size section 𝑗𝑒: Reaction extinction point value𝑗: Relating to size section 𝑗𝐹: Fuel𝑚: Maximum temperature point value𝑂: Oxygen𝑢: Unburnt valueV: Vaporization front value∗: Value at 𝜉 = 0.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

J. B. Greenberg wishes to acknowledge the partial support ofthe Lady Davis Chair in Aerospace Engineering.

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