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HOT SPOTS IN ORGANIC GEL SPRAY DIFFUSION FLAMES

A. Kunin‡ , J.B. Greenberg§ and B. Natan** Technion – Israel Institute of Technology,Haifa 32000, Israel

In the current paper a model of a Burke-Schumann-like organic gel spray diffusion flame is presented. Experimental evidence using an organic-based gellant has revealed that the main characteristic of the gel fuel droplet that distinguishes it from a purely liquid fuel droplet is the onset of pulsating evaporation once the environment of the gel droplets reaches some critical temperature. This unusual phenomenon of the droplets in the gel spray is incorporated in the current model. The combined analytical/numerical solution of the governing equations shows that in relation to the spray diffusion flames obtained using an equivalent purely liquid fuel spray the use of a gel fuel spray can lead, under certain operating conditions, to (a) a reduction in flame height due to the effective reduction in the rate of vaporization, (b) a trail of hot spots of heterogeneous droplet burning downstream of the main homogeneous diffusion flame front. These two effects may be critical when considering flame extinction and highlight the fact that even though gel fuel sprays may have a distinct advantage over liquid sprays in terms of their safety features it is crucial that the correct operating conditions must be employed in order not to detract from attaining the desired combustion performance.

Nomenclature c inner duct’s half-width (normalized with respect to outer duct walls' half-width)

pc heat capacity

Da chemical Damkohler number H Heaviside function

totalfuelm total initial fuel mass fraction (vapor and liquid)

q single period of no evaporation followed by a spurt of evaporation

1q period of no evaporation

2q period of spurt of evaporation

Q heat of reaction

R outer duct's half-width T difference between local value of temperature and its value at 0=η

(normalized by totalfuelp Qm/c )

0T (=0) temperature difference at 0=η

vT temperature difference of onset of evaporation

Greek letters

T,γγ Schwab-Zeldovitch functions, Eq.(5)

‡ Graduate Student, Aerospace Engineering § Professor, Aerospace Engineering, Senior Member AIAA. ** Associate Professor, Aerospace Engineering, Senior Member AIAA.

44th AIAA Aerospace Sciences Meeting and Exhibit9 - 12 January 2006, Reno, Nevada

AIAA 2006-1440

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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γ mass fractions normalized with respect to total initial fuel mass fraction Γ normalized latent heat of vaporization δ ratio of mass fraction of liquid fuel to that of total fuel at central duct exit- referred to as liquid fuel load ∆ vaporization Damkohler number ν stoichiometric coefficient ω vaporization frequency

ηξ , transverse and axial coordinates, respectively

stepη half a period of no evaporation followed by a spurt of evaporation

Subscripts d relating to liquid fuel F relating to gaseous fuel fl. relating to value at flame front O relating to oxidant

I. Introduction here is a growing interest in the use of gel propellants in a variety of propulsion contexts due to their rather favorable high performance and safety characteristics. Examples of potential gel propellant utilization range from boost motors for micro-spacecraft to tactical missiles. However, there is a still lot of ground that must be covered before a sound understanding of the unique properties and combustion features of such propellants will be available. Gel propellants are liquid fuels and/or oxidizers whose rheological properties have been altered by the addition of gellants, such that they behave as non-Newtonian time dependent fluids. This change of the rheological behavior can prevent agglomeration, aggregation and separation of a metal solid phase from the fuel during storage. In short, these propellants are advantageous because, on the one hand, they are capable of providing full energy management and, on the other hand, have distinct safety benefits over conventional liquids and solid propellants. Their performance characteristics and operational capabilities, which are similar to liquid propellants, as well as their high density, increased combustion energy and long term storage capability, make them attractive for many applications, especially for volume-limited propulsion system applications. During the past few decades, many studies concerning different aspects of gel propulsion have been conducted. These studies were mainly experimental and focused on gel propellants preparation processes, basic rheology and flow, atomization, combustion and energetic performance, applications and technological demonstrators, material compatibility and impulse intensification by metal content for space applications. A thorough review on the state of the art was given by Natan and Rahimi [1]. In the current paper we turn our attention to an interesting experimental discovery that was made by Solomon and Natan[2]. In a study of the combustion of static, individual organic gellant-based fuel droplets it was found that their burning characteristics are quite different from those of inorganic gellant-based fuel droplets. In the former case, as heating occurs the gel structure ceases to exist and a binary mixture of liquids (fuel and gellant) with different properties is produced. The low boiling point liquid (the fuel) evaporates and an elastic layer of gellant forms around the droplet, which prevents further fuel vaporization from the droplet surface. Thus, a fuel vapor bubble is formed in the interior, the droplet swells and the outer layer is perforated eventually, releasing evaporated fuel to the surroundings. After the occurrence of this burst of fuel vapor the droplet shrinks, an elastic outer layer is reformed and the cycle repeats itself a number of times until the liquid fuel in the droplet is completely depleted. The actual sequence of mutually interacting physical mechanisms that lead to this pulsating-type of evaporation behavior is far from understood and remains to be thoroughly elucidated. Here, we extract from the experimental observations the fact that the organic gellant-based fuel droplet evaporates in a pulsating fashion upon reaching a critical temperature. This behavior is strikingly different not only from that of inorganic gellant-based fuel droplets but also from that of purely liquid droplets that are widely used in a range of combustion engineering applications. In the present paper we address the fundamental question as to how a spray of organic gellant-based fuel droplets undergoing pulsating evaporation influences the characteristics of the flame they are being used to fuel. The frequency of evaporation oscillations of the droplets is likely to be controlled by such factors as the droplet size, the make-up of the droplet (i.e. gellant versus liquid fuel content) and ambient conditions. Although comprehensive experimental data is not yet available in order to quantify such effects the fact that the frequency

T

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of the evaporation pulsations may vary will be taken as given and attention will be focused on this particular peculiarity of a gellant droplet’s behavior and its influence on spray gel flames. In laminar combustion theory an often-adopted strategy is to consider geometrically simple systems that lend themselves to a fairly straightforward mathematical/numerical analysis, thereby enabling dominant physical and chemical mechanisms at play to be isolated and categorized in a clear-cut manner, in terms of their relative importance. We implement such an approach here and present a simple model of a gellant-based fuel spray diffusion flame. In a previous work[3] we modeled the pulsating evaporation using a cosine function which gives a continuous flavor to the supply of fuel vapor. More recent, as yet unpublished experimental results [4] point to the more discontinuous nature of the pulsations of fuel vapor supply and it is this feature that is the main contribution of this investigation. In the ensuing sections we explain the assumptions underlying the model, describe the governing equations and boundary conditions, the method of solution and, in the final section, discuss computed results that highlight those circumstances in which the peculiarities of the gel spray profoundly influence the combustion field’s characteristics.

II. Problem Description We consider a Burke-Schumann gel spray flame configuration (see Figure 1) in which fuel vapor and organic gellant-based fuel droplets flow in an inner duct and air flows in an outer duct. Under appropriate operating conditions, after diffusive mixing of the two streams, a steady, laminar gel spray diffusion flame is maintained. A constant density model is taken and the velocities of the inner and outer ducts are taken to be constant and equal, as per Burke-Schumann's original gas flame analysis. The effect of relaxing this latter assumption was examined by Khosid and Greenberg [5] in the context of liquid spray flames and shown to be of quantitative rather than qualitative significance. The droplets in the spray are taken to be located in the far-field region in relation to the spray source so that on the average they are in dynamic equilibrium with their host carrier environment. It is assumed that the various transport coefficients such as thermal conductivity, diffusion coefficients, specific heat at constant temperature, latent heat of vaporization of the droplets etc. can be satisfactorily specified by representative constant values. Furthermore, the transport properties will be supposed to be determined primarily by the properties of the gaseous species. This follows from the implicit assumption that the gellant and liquid fuel volume fraction is sufficiently small. In addition, the Lewis numbers of the reactants (and products) are supposed to be unity. An overall reaction of the form oductsPrOxidantFuel →+ν is taken to describe the chemistry and we consider the fast chemistry limit, i.e ∞→Da , where Da is the chemical Damkohler number. The droplets are viewed from a far-field vantage point, i.e. their average velocity is equal to that of their host environment. The actual description of the spray is based on the sectional approach [6, 7]. In this method the "point-wise" size distribution of droplets in the spray is subdivided into a finite number of size sections each of which contains droplets of diameters that fall within a certain size bracket. The mass balance of droplets in section j accounts for (a) the influx of droplets from section j+1 which have diminished in size and have thus become eligible for membership in section j, and (b) mass loss due to evaporation of droplets in section j. Sectional mass conservation equations can then be rigorously derived for the droplets in each size section. Here, for the sake of simplicity at the current stage, we take the spray to be mono-sectional. An arbitrary poly-disperse spray can also be treated at the expense of further complexity and will be examined in a future publication.. The governing equations, written in terms of non-dimensional quantities (see [8]), are

)c(H)(H)()c(H)(H .fld.fld02

2

ξηηγη∆ξηηγ∆ξ

γηγ −−+−−+

∂∂=

∂∂

(1)

)c(H)(H)()1()(H)c(H)1( .fld.fld02T

2T ξηηγη∆Γηηξγ∆Γ

ξγ

ηγ

−−−+−−−+∂∂

=∂∂

(2)

)c(H)(H)()c(H)(H .fld.fld0d ξηηγη∆ξηηγ∆

ηγ

−−−−−−=∂∂

(3)

where

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( )��

�

��

�

�

≥��

��

−

<��

��

−

=���

��

�

�−�

�

��

−=

qq

qq,

qq

qq,0

qq

qqH

10

1

10 ηη∆

ηηηη∆η∆ (4)

where 21 qqq += is a single period during which there is a cessation of supply of fuel vapor to the surroundings for 1q and then a burst of evaporation for 2q . The use of the square brackets denotes the integer part of what is contained within them so that this ensures a period q such that

�,2,1,0nfor,qqq

nqq

nq =��

�

��

�

���

��

−=��

�

��

�

���

��

+−+ ηηηη

In equations (1) and (2) the Schwab-Zeldovitch functions γ and Tγ are defined by ( ) ( )T,, FOFT +−= γγγγγ (5) As usual, the stoichiometric coefficient is also incorporated in the denominator when normalizing the oxygen mass fraction. The vaporization Damkohler number for the organic gellant-based fuel is split into two parts (see equations (3) and (4)). The experimental evidence culled from single gel drop burning [2] indicates that pulsating type of vaporization sets in at some specific ambient temperature at which there is (in some sense) sufficient heat transfer to the droplet. The experimental results are rather sparse and it is unclear whether the onset of the oscillations occurs with or before the combustion of the single droplet. Here we make the non-essential assumption that oscillatory evaporation is initiated at the flame surface, )(.fl ξηη = . Upstream of this

surface, in the region containing no oxygen, evaporation is characterized by a constant vaporization Damkohler number, 0∆ , whereas downstream of the flame surface, in the oxygen-containing region, the vaporization Damkohler number is assigned a step-function dependence on distance from the flame surface, as described in equation (4). It is not hard to show that, under the assumption that the droplets are in dynamic equilibrium with the host carrier gas, the time pulsations of the evaporating droplet of the experiments can be translated into pulsating space-dependent behavior of the vaporization Damkohler number. Upstream of the flame surface the constant value of 0∆ is based on the 2d -law dependence of an evaporating gel droplet observed by Nachmoni

and Natan[9]. Note that a large value of 0∆ represents a highly volatile fuel and/or small droplets in the spray whereas, conversely, small values correspond to a non-volatile fuel and/or large droplets [10]. 0∆ is actually a complicated function of initial droplet diameters, the temperature differential between the droplets and the surrounding gas and the diffusivity and other properties of the fuel and its surroundings. This renders analytical solution of the problem at hand unfeasible. So, while bearing in mind the restrictive nature of such a step, 0∆ is taken as constant for some mathematical tractability. Despite this pragmatic reason for using a constant vaporization Damkohler number, there is some further justification to be found in the use of the 2d -law for a description of the vaporization coefficient. Reasonably accurate estimates of droplet size and vaporization time do provide some evidence of the validity of this law even under transient temperature conditions [11-13]. Labowsky[13] showed that the 2d -law predicts the actual vaporization history of an interacting liquid droplet, especially in the initial period of combustion. Studies by Annamalai and Ryan[14] and Elperin and Krasitov[15] (the latter for random clusters of liquid droplets) add further weight to Labowsky's findings. Adopting this law in the mono-sectional model of the spray then leads to a constant averaged value of the vaporization coefficient and, hence, of the vaporization Damkohler number, 0∆ . Finally, we quote the excellent agreement of theoretical predictions for purely liquid sprays (based on the use of a constant value of 0∆ ) with experimental measurements [16] as further support for our simplification. Eq.(3) is valid for the region 0,c0 ≥≤≤ ηξ ; elsewhere dγ is identically zero. This expresses the fact that droplets are to be found only in the region above the inner duct since, in this model, there is no mechanism by means of which they may be diverted transversely. It is important to note that the governing equations implicitly account for the different possible scenarios that are physically viable and that have been experimentally observed, viz. complete evaporation of the droplets before reaching the homogeneous diffusion flame front or pre-homogeneous flame front evaporation followed by post-homogeneous-diffusion-flame burning of individual (or clusters of) droplets that survive the main flame

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front. That both these possibilities are encaptured is readily discernible from the governing equations. First, Eq. (3) applies at all downstream stations i.e. for all 0>η , so that there is no restriction on the rate of vaporization, the latter being expressed via the specified value of the vaporization Damkohler number. Second, Eq. (1) for the Schwab-Zeldovitch function γ also reveals the comprehensive nature of the model. In the region where no oxygen is present γ is positive by definition and the source term on the right-hand side relates to the production of fuel vapor due to the evaporation of droplets in the spray in the pre-flame zone. However, when γ is negative the equation describes the behavior of the normalized mass fraction of oxygen, and the spray source term is still applicable, but now downstream of the homogeneous flame front (within the lateral bounds set by the Heaviside function). Here it serves as a sink that removes oxygen at a rate dictated by the (pulsating) rate of vaporization of the droplets. This corresponds to the post-flame individual droplet-burning situation. Which scenario is under consideration will be strongly dependent on the value of the vaporization Damkohler number, 0∆ . The boundary conditions are

δγγδγξη ==−=≤≤= d0T ,T,1:c0,0 0,T,V:1c,0 d0T ==−=≤≤= γγγξη

0:1,0,0 T =∂

∂=

∂∂=>

ξγ

ξγξη

In these conditions δ represents the ratio of the mass fraction of liquid fuel to that of the total fuel (i.e. fuel liquid + vapor) at the exit of the central duct. We shall take the total fuel mass fraction to be constant so that different values of δ correspond to different combinations of the initial liquid and vapor mass fractions. The first two initial conditions imply a specified mass fractions/temperature at the entrance to the duct in which combustion occurs. Actually, the onset of appreciable vaporization should occur when the temperature is equal to vT (i.e. the boiling temperature of the gel droplets). For ease of development we take the temperature implied by the second boundary condition to be v0 TT ≥ . The third set of conditions specifies the symmetry at 0=ξ and the fact that the outer duct's walls are impervious to heat and mass transfer.

III. Solution In order to develop a solution to the governing equations we note that the solution domain is essentially divided into two regions, upstream and downstream of the flame front. However, at this stage, the location of the flame front, given by those values of ),( ηξ for which 0=γ , is unknown. In the current study we consider over-ventilated flames only so that the flame tip is located at .fl,0 ηηξ == The flame temperature can be derived

from both γ and Tγ by noting (see Eq.(5)) that in the oxygen-free zone γγ −= TT whilst in the fuel vapor-free zone TT γ= . Integrating Eq.(3) with respect to η and applying the initial condition readily yields

)exp(),( 0d η∆δηξγ −= in c0 ≤≤ ξ , .fl0 ηη ≤≤ (6a) and

( )

( ) ( )( )[ ] qq,qexp

q,

i1i2i0id

d

1iiid

+≤≤++−−=

+≤≤

ηηηηη∆ηγγ

ηηηηγ

in c0 ≤≤ ξ , ∞<≤ ηη .fl (6b)

where iη is the value of η at the beginning of each total period of cessation of evaporation followed by a spurt of evaporation. For lack of reliable quantitative data we set step21 qq η== so that step2q η= . The value of stepη

will be pre-specified. This analytical solution for the liquid fuel distribution is substituted in the appropriate source terms in equations (1) and (2). These equations for γ and Tγ are parabolic and solved numerically using a standard explicit forward-η central-ξ finite difference method. As mentioned previously the location of the flame front is determined by the locus of points for which 0=γ . Satisfactory step sizes in theξ and η directions were

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obtained by comparing computed results using a series of increasingly refined finite difference meshes to cover the region of the solution.

IV. Results In the ensuing discussion we will focus on the effect of the frequency of pulsating evaporation of the droplets on the flame shape and the temperature field. We preface our discussion by pointing out that since our model considers parabolic equations for the gaseous mass fractions and the temperature fields we can only expect the influence of the pulsating evaporation of the gel droplets to appear downstream of the flame front, under operating conditions for which sufficient droplets survive the flame front, in their downstream movement, and become ignited upon its traversal. To bring to the fore the specific features of the gel spray flames we also compare their shapes to those of spray flames that would be obtained under identical operating conditions but with a purely liquid rather than a gel spray. For all the results to be presented the following data was used:

0T,6/1c,306.0V 0 === , 02.0=Γ . Other data used will be mentioned in the appropriate context. In Figure 2 we present two sets of data for the case when 70 =∆ and .8.0=δ The upper four figures show typical downstream evolution of the normalized mass fraction of liquid fuel in the spray's droplets. The upper most Figure on the left illustrates the usual exponential-like decay of a purely liquid fuel spray. The other three upper figures demonstrate how the liquid fuel in the gel spray's droplets is reduced for different values of the period parameter, stepη (denoted by “step” in the figures). For a value of 0.01 the on/off nature of the evaporation is hardly visible and the liquid mass fraction appears to decay exponentially as in the case of a purely liquid spray of droplets. However, as the period increases to stepη =0.04 the jagged appearance of the

curve clearly highlights the pulsating release of fuel vapor. For the upper right-most figure stepη has been

assigned the value 0.1 and the more prolonged periods of vapor supply during the spurts of its release is rather evident. Below the aforedescribed figures the homogeneous diffusion flame fronts are drawn for each case (only half the flames are shown due to axial symmetry). The purely liquid fuel spray flame provides the tallest flame. However, the larger the evaporation period stepη the shorter the gel spray flames become. This is due to the fact

that as the evaporation period increases less fuel vapor is effectively available for combustion upstream. It is well known that a deficiency in fuel vapor leads to lower homogeneous flame heights in gaseous Burke-Schumann diffusion flames (see, also, Chung and Law[17]). However, the homogeneous flame shapes do not tell the whole story of the influence of the organic gel droplets in the spray and a more dramatic, behind-the-scenes insight is furnished by examining the temperature fields. In Figure 3 the temperature contours are plotted for the three evaporation periods discussed in Figure 2. Consider, first, stepη =0.01. Beyond the homogeneous flame front (whose height, it will be recalled, is about 057.0≈η ) a street of hot spots trails downstream. In fact, the entire normalized temperature contour of about 0.17 has a wavy appearance and from 5.0≈η there is a further break-up into another street of trailing hot-spots. It should be emphasized that this is not a numerical artifact as the results are reproducible with increasingly finer mesh sizes. Rather this reflects downstream heterogeneous burning of the gel spray's droplets at a rate that is dictated by the rate of evaporation of the fuel vapor the droplets contain. Since the vapor is supplied in a pulsating fashion the combustion occurs in a similar fashion and produces the numerous hot-spots of heterogeneous combustion. Turning to stepη =0.04 the phenomenon of the hot-spots becomes really clear, and, indeed, it is not hard to

discern that their maximum temperatures are higher that that of the small upstream homogeneous diffusion flame. Furthermore, as stepη increases to 0.1 the periods during which there is no release of vapor from the gel

droplets grows so that there are greater distances between the hot-spots than exhibited for the smaller values of the period. On the other hand, the period during which vapor is released is now also greater so that the axial and transverse dimensions of the hot spots are greater than in the previous two cases shown. In Figure 4 the oxygen mass fraction contours reflect the aforementioned behavior from another angle – the blue regions indicating where there is considerable oxygen consumption due to combustion. In addition, the most left-hand side set of contours are those for the case when all the liquid fuel is supplied in non-gel droplets. The striking way in which these contours differ from those obtained with the gel spray flames highlights the remarkable influence of the organic gel droplets on the entire thermal and composition fields.

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In previous work[3] the evaporation of the organic gel droplets was modeled using a continuous cosinusoidal description, rather than the discontinuous pulsating model adopted here. A rough comparison between the predictions of the two models is possible by reading the figures shown in Figure 5 together with their counterparts in Figures 1-4. The flame height shown in Figure 5 is lower than that of the purely liquid fuel spray flame but taller than those predicted by the current model (compare to Figure 1). However, in qualitative terms, it is clear that the preponderance of downstream hot spots, as islands of intense post-homogeneous flame heterogeneous burning of gel droplets, persists and is an essential feature of these flames irrespective of which model is utilized.

V. Conclusion A model of a Burke-Schumann-like organic gel spray diffusion flame is presented in which the experimentally discovered pulsating evaporation of gel fuel droplets is incorporated. The combined analytical/numerical solution of the governing equations shows that in relation to the spray diffusion flames obtained using an equivalent purely liquid fuel spray the use of a gel fuel spray can lead, under certain operating conditions, to a reduction in the homogeneous flame height and to a trail of hot spots of heterogeneous droplet burning downstream of the main homogeneous diffusion flame front. These two effects may be critical when considering flame extinction and highlight the fact that even though gel fuel sprays may have a distinct advantage over liquid sprays in terms of their safety features it is crucial that the correct operating conditions must be employed in order not to detract from attaining the desired combustion performance. In addition, a comparison between the proposed phenomenological discontinuous spurting evaporation model and a previously used continuous model indicates that the basic detrimental underlying thermal and composition structure is a feature of these organic gel spray flames irrespective of the way in which the pulsating evaporation is modeled.

Acknowledgments J.B.Greenberg gratefully acknowledges the partial support of the Lady Davis Chair in Aerospace Engineering and the Technion Fund for the Promotion of Research.

References 1. Natan, B. and Rahimi, S., “The Status of Gel Propellants in Year 2000”, in Combustion of Energetic Materials, K.K. Kuo

and L. DeLuca, editors, CRC and Begel House, Boca Raton, 2001. 2. Solomon, Y. and Natan, B., “Experimental Investigation of the Combustion of Organic-Gellant-Based Gel Fuels,”

accepted for publication in Combustion Science and Technology, March, 2005. 3. Kunin, A., Greenberg, J.B. and Natan, B., “Influence of Organic Gel Droplets on Gel Spray Diffusion Flames”, AIAA

Paper 2005-4476, 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Tucson, July, 2005. 4. Kunin, A., “Investigation of Gel Spray Diffusion Flames” M.Sc. Thesis (in Hebrew), Faculty of Aerospace Engineering,

Technion, Israel Institute of Technology, 2005. 5. Khosid, S. and Greenberg, J.B., “The Burke-Schumann Spray Diffusion Flame in a Nonuniform Flow Field”, Combustion

and Flame, Vol. 118, 1999, pp. 13-24. 6. Greenberg, J.B., Silverman, I. and Tambour, Y., “On the Origins of Spray Sectional Conservation Equations”, Combustion

and Flame, Vol. 93, 1993, pp. 90-96. 7. Laurent, F. and Massot, M., “Multi-fluid Modeling of Laminar Polydisperse Spray Flames: Origin, Assumptions and

Comparison of Sectional and Sampling Methods”, Combustion Theory and Modelling, Vol. 5, 2001, pp. 537-572. 8. Greenberg, J.B., “The Burke-Schumann Flame Revisited – With Fuel Spray Injection”, Combustion and Flame, Vol. 77

(3&4), 1989, pp. 229-240. 9. Nachmoni, G. and Natan, B., “Combustion Characteristics of Gel Fuels”, Combustion Science and Technology, Vol.

156, 2000, pp. 139-157. 10. Greenberg, J.B. and Cohen, R., “Dynamics of a Pulsating Spray Diffusion Flame”, Journal of Engineering Mathematics,

Vol. 31, 1997, pp. 387-409. 11. Law, C.K. and Sirignano, W.A., “Unsteady Droplet Combustion with Droplet Heating-II: Conduction Limit Combustion

and Flame”, Vol. 28, 1977, pp. 175-186. 12. Law, C.K., “Adiabatic Spray Vaporization with Droplet Temperature Transient”, Combustion Science and Technology,

Vol. 15, 1977, pp. 65-74. 13. Labowsky, M., “Calculation of the Burning Rates of Interacting Fuel Droplets”, Combustion Science and Technology,

Vol. 22, 1980, pp. 217-226.

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14. Annamalai, K. and Ryan, W., “Interactive Processes in Gasification and Combustion. Part I: Liquid Drop Arrays and Clouds”, Progress in Energy Combustion Science, Vol. 18, 1992, pp. 221-295.

15. Elperin, T. and Krasovitov, B., “Analysis of Evaporation and Combustion of Random Clusters of Droplets by a Modified Method of Expansion into Irreducible Multipoles”, Atomization and Sprays, Vol. 4, 1994, pp. 79-97.

16. Golovanevsky, B., Levy, Y., Greenberg, J.B. and Matalon, M., “On Oscillatory behavior of Laminar Spray Diffusion Flames: Theory and Experiment”, Combustion and Flame, Vol. 117(1&2), 1999, 373-383.

17. Chung, S.H. and Law, C.K., ”Burke-Schumann Flame with Streamwise and Preferential Diffusion” ,Combustion Science and Technology,1984, Vol.37 ,pp. 21-46.

Figure 1: Configuration for gel spray diffusion flame formation.

Gel FuelDroplets

OxidantOxidant

ξ

η

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Figure 2: Influence of period (step) of pulsating evaporating organic gel droplets on gel spray flames: Upper figures compare dγ profile for purely liquid fuel (left hand side figure) with dγ profiles for three periods of pulsating evaporation of gel spray; Lower figures compare the resulting homogeneous diffusion flame shapes for conditions considered in the upper figures.

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Figure 3: Temperature contours for gel spray diffusion flames; vaporization Damkohler number,

8.0,70 == δ∆ , other data as in text.

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Figure 4: Oxygen mass fraction contours for gel spray diffusion flames; vaporization Damkohler number, 8.0,70 == δ∆ , other data as in text.

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Figure 5: Predictions using oscillating evaporation model [3] in which

( )( ).fl0

0 cos12

)(f)( ηηω∆η∆η∆ −+== ; πωδ∆ 12,8.0,70 === .