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An analytical study of droplet combustionunder microgravity: Quasi-steady transient approach

S. Ulzama *, E. Specht

Institute of Fluid Dynamics and Thermodynamics, Otto-von-Guericke-University, Magdeburg, Magdeburg 39016, Germany

Abstract

An analytical model based on an assumption of combined quasi-steady and transient behavior of theprocess is presented to exemplify the unsteady, sphero-symmetric single droplet combustion under micro-gravity. The model used in the present study includes an alternative approach of describing the dropletcombustion as a process where the diffusion of fuel vapor residing inside the region between the dropletsurface and the flame interface experiences quasi-steadiness while the diffusion of oxidizer inside the regionbetween the flame interface and the ambient surrounding experiences unsteadiness. The modeling approachespecially focuses on predicting; the variations of droplet and flame diameters with burning time, the effectof vaporization enthalpy on burning behavior, the average burning rates and the effect of change in ambi-

ent oxygen concentration on flame structure. The modeling results are compared with a wide range ofexperimental data available in the literature. It is shown that this simplified quasi-steady transientapproach towards droplet combustion yields behavior similar to the classical droplet theory. 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords:Droplet combustion; Microgravity; Flame structure; Burning rates

1. Introduction

One of the main objectives in combustion

research is the development of comprehensivecomputer models to give a better understandingof spray combustion in many practical applica-tions e.g. gasturbine engine, diesel engine, oilfired boilers, process heater, etc. An isolated drop-let combustion study under microgravity condi-tions serves as an ideal platform in providing abasis for enhancing the existing understanding ofburning process, and gives proper explication ofthe process which is important for economical

use of fuels and for reducing the production ofpollutants. Microgravity condition is necessarynot only for the sphero-symmetric droplet com-

bustion in quiescent atmosphere, but also for theresulting one dimensional solution approach ofcombustion.

A great number of modeling studies for betterunderstanding of vaporization and combustion ofa fuel droplet under microgravity conditions havebeen reported for nearly five decades. Godsave[1]and Spalding [2] derived the classical d2-law,which yields relatively good estimates of the gasi-fication rate. Kumagai et al.[35]successfully per-formed the first droplet combustion experimentsin microgravity conditions to validate d2-law.

They showed that droplet gasification rate wasconstant over time which is one of the mostimportant features ofd2-law. Most of the existing

1540-7489/$ - see front matter 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved.doi:10.1016/j.proci.2006.07.134

* Corresponding author. Fax: +49 391 67 12762.E-mail address: [email protected]

burg.de (S. Ulzama).

Proceedings of the Combustion Institute 31 (2007) 23012308

www.elsevier.com/locate/proci

Proceedingsof the

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models are based on the assumption of processdynamics: models taking into account the quasi-steady nature of the process, and models whichare based on transient assumption. The quasi-steady character of sphero-symmetric combustionof a droplet has been extensively studied, analyti-

cally as well as numerically[610]. Most of thesemodels were reported taking into account the tem-perature dependence of transport properties,kinetics effects and the transport mechanisms.Puri and Libby [9] proposed a numerical modelfor steady state droplet combustion with a properdescription of gas-phase transport mechanism.Model predictions for gasification rate and flamelocation showed a good agreement with experi-mental data. Fachini[10]presented an analytical,steady state, droplet combustion model with con-siderations of temperature dependence of trans-port coefficients and non-unity Lewis number.Although the model considers temperature depen-dence of transport coefficients, the results do nothave good agreement with the experimentalresults.

Based on several experimental studies [4,5,1113], it was found that the predictions of d2-law for flame stand-off ratio are not in accordancewith the experimental observations. Experimentshave shown that the flame stand-off ratio contin-ues to increase while the gasification rate followsa steady state behavior shortly after the ignitionperiod. However, a better explanation of pure

liquid droplet combustion can be given by consid-ering unsteady effects as well. Theoretical studiesregarding the unsteadiness of the droplet combus-tion has been described in detail elsewhere[1417].Law [18] and Faeth [19] presented their reviewpapers for detail discussion of fuel droplet com-bustion. Recently, King[20] briefly reviewed theprevious transient droplet combustion literature.The complete modeling of droplet combustion isquite complicated because of the involvement oflow temperature auto ignition, radiative heattransfer, complex reaction kinetics, and of non-

linear transport/thermophysical properties. As aresult, droplet combustion modeling deals witheither quasi-steady approach or the transientapproach which is more complex and requires alot of numerical computations.

In this paper, a mathematical model is present-ed for single fuel droplet combustion under micro-gravity conditions. The present mathematicalanalysis is based on an alternative approach,according to which the simplicity in describingthe droplet combustion is based on the fact thatthis process is controlled by both the quasi-steady

behavior for the region between the droplet sur-face and the flame interface, and the transientbehavior for the region between the flame inter-face and the ambient surrounding. The main pur-pose of this work is to demonstrate that evensimplified quasi-steady transient approach

towards droplet combustion yields behavior simi-lar to the classical droplet combustion. The mod-eling results of variations of flame diameter anddroplet diameter-squared are compared against avariety of experimental data available in the liter-ature for isolated droplet combustion.

2. Model formulation

The mathematical model used to depict thecombustion phenomenon of isolated pure fueldroplet under microgravity condition is brieflydescribed here.

2.1. Model description

Consider an isolated spherical droplet of purefuel, initially at temperature T0, immersed in aquiescent environment at temperature T

1

(Fig. 1). The liquid droplet is surrounded by fuelvapor that diffuses outward from the droplet sur-face to the flame interface (region-I) while oxidizerdiffuses radially inward from the ambiencetowards the flame interface (region-II). Modelingwas performed using an alternative approach thatthe diffusive transport of oxygen towards theflame interface is unsteady. In general, it is foundthat a typical value of air demand for completecombustion of a light oil droplet is 14 (kg of air/kg of oil). On this basis, we calculate the diameter

of spherical volume of air associated to the fueldroplet with the density of air estimated at anaverage temperature of1200 K which is moreoften in the vicinity of the flame interface(region-II). This comes about 30 times of thedroplet diameter, which is less than the distanceneeded for steady-state profile for the oxygen dif-fusion. As a consequence, the stored amount ofoxygen in this range can not be neglected againstthe diffusive mass transport. With such a hugeamount of oxidizer associated with fuel droplet,the assumption of quasi-steadiness for disappear-

ance of oxygen cannot be taken into considerationfor the droplet combustion. Thus, a betterdescription of oxygen diffusion in region-II canbe accomplished only with an assumption ofunsteadiness. The unsteady-state diffusion of oxy-gen in region-II is similar to the case of diffusioninside semi-infinite bodies, as shown in Fig. 1.Moreover, the total amount of liquid residingwithin region-I is much more than the amountof fuel vapor accumulated in the same region.Therefore, the condition of quasi steadiness fordiffusive transport of fuel vapor from the droplet

surface to the flame interface exists for region-I.This model deals with many assumptionsincluding a few from the classical quasi-steadydroplet combustion model, which is described indetail elsewhere [1,2]. The main assumptions areas follows:

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(1) Lewis number in the gas phase is equal tounity;

(2) all thermophysical properties including theheat of vaporization remain constant withtemperature and their average values are tak-en for computation;

(3) the combustion products do not affect theprocess;

(4) the reaction zone at flame interface is restrict-ed only to a narrow region i.e. infinite-ratekinetics;

(5) heat loss due to radiation is negligible;(6) initially, the droplet is assumed to be at room

temperature.

The droplet is considered to be in a quiescentatmosphere, so that all processes in the gas phasewill have spherical symmetry. Note that on thebasis of different studies [21], it has been foundthat the Lewis number is not constant and chang-es during the process of vaporization. Further-more, because of the existence of hightemperature difference between the droplet and

the flame interface, the use of average values ofthe thermophysical properties may cause smallerrors. Hence, we have run a number of test sim-ulations to estimate the effect of change in thermo-physical properties and found 10% variations inthe simulation results, even when the actual valueof a property e.g. heat of vaporization, specificheat, diffusivity, etc. is increased by a factor of2. Despite the violation of the assumptions ofthe classical model in many cases, it is widely usedin comprehensive modeling of evaporation andcombustion process of sprays[22,23].

2.2. Droplet combustion time

The mechanism of heat transfer in region-I isquite complicated as it involves the preheating ofthe droplet, the fuel evaporation and the heating

of the fuel vapor diffusing from the droplet surfaceto the flame interface. The analytical modelsdescribed earlier did not take into account theheat absorbed by the diffusing fuel vapor. Sincethe preheating of the droplet depends on time,the combustion process of a droplet initiallyinvolves more preheating rather than vaporiza-tion. Due to high temperature difference betweenthe flame interface and droplet surface e.g. the val-ue is 1844 K for n-heptane, the heat consumed byvapor is much more than the heat required forevaporation and the heat necessary for preheatingof the droplet. Consequently, an increase ordecrease in the value of heat required for preheat-ing of droplet does not influence the process.Therefore, the effect of change ofT0 is neglectedin the analysis. For analytical solution of theproblem, the amount of heat required for preheat-ing of the droplet is taken to be constant. Basedon the assumption that heat loss due to radiationis zero, the total amount of heat transferred fromthe flame interface to the droplet surface is usedonly for these three different kinds of heat

consumption.The total amount of heat transferred due to

gas phase conduction between the flame interfaceand the droplet surface can be calculated analyti-cally by

q 4pkg Tf Ts1=rs 1=rf ; 1a

whereqis the amount of heat transferred from theflame interface to the droplet, Tf and Ts are thetemperatures at the flame interface and the drop-let surface respectively, and rsandrfare the values

of radius of the droplet and the flame, respective-ly. The value of gas phase thermal conductivity kgis averaged between the flame interface and thedroplet surface. The heat gained by the vaporizingdroplet can be calculated by the followingequation,

Fig. 1. Schematic diagram of a droplet combustion process.

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q Mv cpl Ts T0 DHv cpg Tf Ts

;

1bwhereMvis the mass flow rate of the vapor, DHvthe heat of vaporization, T0 the initial droplettemperature, and cpl and cpgthe specific heats of

the liquid phase and the gas phase respectively.Eq. (1b)takes into account the amount of heat re-quired for droplet preheating, vaporization, andheating up the fuel vapor.

The time variant radius of the fuel dropletundergoing combustion can be found from theequation,

Mv dmddt

ql4pr2sdrsdt

2

here md is the mass of the droplet and ql is thedensity of the liquid fuel. Total time taken for

complete combustion of the droplet can easilybe calculated by integrating Eq. (2). The valueof vapor mass flow rate can be obtained byconsidering Eqs. (1a) and (1b). While calculat-ing combustion time using Eq. (2), the valueof the ratio of rs/rf is excluded from the expres-sion because its value being very less as com-pared to unity. Further, the integration of Eq.(2) yields,

d2t d20 Kt; 3

wheret is the time,d0the initial droplet diameter,dtthe time-dependent droplet diameter, andKthegasification rate. Eq.(3)corresponds to behaviorsimilar to the classical d2-law with a value of gas-ification rate given by

K 8 kgql

Tf Ts cpl Ts T0 DHv cpg Tf Ts :

4

2.3. Flame dynamics

The flame dynamics of sphero-symmetricdroplet combustion involves a set of conservationequations of species and energy in region-I and II.At liquidgas interface, the vapor and liquid areassumed to be in equilibrium. The continuousdroplet evaporation rate can be calculated byapplying Ficks law of diffusion through a hollowsphere (region-I),

Nv 4pDf;gPRTdf 1=rs 1=rf ln 1

Pv;s

P

; 5

Pv;s A expB=Ts ; 6where Pv,s denotes the vapor pressure of pure li-quid, P the total pressure, R the universal gasconstant, Tdf the average temperature betweenthe droplet surface and the flame interface, Df,g

the diffusion coefficient of the fuel vapor, andNvthe molar flow rate of the fuel vapor. The val-ues of constants Aand Bused to calculate vaporpressure of pure liquid depend on the kind offuel.

The unsteady mass transfer of oxygen in

region-II can be determined by the followingequations taken from Carslaw and Jaegar[24],

NO28pD

3=2O2C1ffiffiffi

pp Nf

ffiffit

p exp g2f

; 7

gf rf

2ffiffiffiffiffiffiffiffiDO2

p t1=2; 8

Nf eg2

f

gf

!

ffiffiffip

p erf gf 9

wheret is the time, C1

the concentration of oxy-gen at infinite distance, DO2 the diffusion coeffi-cient of oxygen, and NO2 the molar flow rate ofoxygen.

In the region-II, oxygen is diffusing radiallyinward towards the flame interface while the fuelvapor (region-I) is transported radially outwardfrom the droplet surface to the flame interface.It is considered that for each unit of fuel con-sumed, t units of oxygen are used up. At theflame interface, the stoichiometric relationshipbetween oxygen and fuel can be explained as fol-lows [25],

NO2 t Nv; 10where t is the stoichiometric ratio. The molarflow rates of the fuel vapor (Mv/m*) and oxygenat the flame interface can be calculated usingEqs. (1a), (1b) and (7); wherem* is the molecularweight of the fuel. But the solution of these equa-tions involves the values of the droplet surfacetemperature and the flame interface temperature,which are required to calculate the values of

physical parameters e.g. diffusivity, vapor pres-sure, etc. These values of temperature can be cal-culated by comparing Eqs. (1a), (1b) and (5).Taking into account our assumption of Lewisnumber equal to unity, we can get the followingequation,

ln1 Pv;s=PcplTs T0 DHv cpgTf Ts

cpgTf Ts :

11It can be seen from the above equation that the

value ofTs remains constant during the burning.The unknown value of flame radius can be calcu-lated by using Eq.(10). Thus, this analytical mod-el is also capable of estimating the variations ofboth the droplet radius and the flame radius withtime via Eqs.(3) and (10), respectively.


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3. Results and discussion

The model described in the previous sectionwas applied to study the spherically symmetriccombustion process of a single pure fuel dropletin quiescent environment. Simulation were carried

out to generate a set of data consisting of, thevariations of droplet diameter and flame diameterwith time, the gasification rate, the effect of vapor-ization enthalpy on burning behavior, the varia-tions of flame stand-off ratio with time and theeffect of the ambient oxygen concentration onflame structure. A time interval of 0.05 s was usedto solve the system of nonlinear algebraic equa-tions of the model on discrete basis. The physicalproperties of the liquid and gas are taken fromReid et al.[26], and Perry and Green[27]. Simula-tion predictions for two fuels i.e. n-heptane andethanol, are compared with experimental dataavailable in several literature sources.

3.1. Droplet and flame structure characteristics

Figure 2compares the model predictions withthe experimental data of Kumagai et al. [4,11]for n-heptane droplets of different diameters.The model predicts a value of 356 K forn-heptanedroplet surface temperature. The model predic-tions are in good agreement with experimentalmeasurements for both droplet and flame diame-ters as functions of burning time, even though

the model slightly overestimates the values offlame diameter during the early period of burning,and the flame diameter decreases substantiallyover droplet burning time henceforth. It is readilyseen from these plots that as the droplet diameterincreases, the flame diameter also increases sincethe flame diameter depends primarily on the evap-orated mass of fuel droplet, and secondarily onthe diffusion process. Results shown in Fig. 2neglect the influence of radiation because thesmall droplet sizes with respect to volume of gasesresult in small view factor so the influence of radi-

ative heat loss can be neglected. Although themodel predictions yield the total time of completedroplet burning that is less than the experimentalmeasurements for droplets with diameters of0.836 and 0.92 mm (Fig. 2), it predicts well forhigher diameter droplet of 0.98 mm (Fig. 2). Thisbehavior is believed to be caused by the constanthigher values of gasification rate, which is inde-pendent of initial droplet diameters. Nonetheless,the model appears to give a good description ofthe data in terms of the general trend of d2tcurve.

3.2. Estimation of gasification rate

Figure 3 shows comparison of modelpredictions with the experimental results obtainedby Kumagai et al. [5,11] for the evolution of

gasification rate for n-heptane and ethanoldroplets of different sizes. A constant value0.84 mm2 s1 for gasification rate of then-heptanedroplets was found. As shown inFig. 3, the possi-ble reason for the discrepancy of the model pre-dictions from the experimental measurements isthat the model predicts the gasification rates withan assumption that the flame interface tempera-ture equals the adiabatic flame temperature. How-ever, under real experimental conditions anamount of heat transferred from flame interface

to the ambient surroundings might cause theflame temperature to attain a value lesser thanthe adiabatic flame temperature. Results ofmodel predictions of the gasification rate for theethanol droplets are in good agreement withexperimental data. Model predicts a constant

Fig. 2. Comparison between experimental[4,11](points)and predicted (lines) data of the droplet diameter andthe flame diameter variations with time. Initial condi-tions: n-heptane; drop diameters, (a) 0.836 mm, (b)0.92 mm, (c) 0.98 mm; ambient temperature, 298 K;atmosphere, air at 1 atm pressure.


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value 0.58 mm2 s1 for the gasification rate for theethanol droplets of different sizes. The compatibil-ity of simulation results with experimental mea-surements also depends on the factors affectedby the experimental conditions e.g. the heat lossdue to radiation. Figure 3also shows the predic-tions of the models of Puri and Libby [9] andFachini [10]for the ethanol droplets. Model pre-dictions match with those predicted by Puri andLibby[9].

3.3. Influence of vaporization enthalpy on burningbehavior

Figure 4 presents the variation of dropletdiameter-squared with time for the n-heptaneand ethanol droplets burned in air at pressure of1 atm. The heat of evaporation of ethanol is morethan the vaporation enthalpy of n-heptane. Forthe 0.93 mm initial diameter ethanol droplet the

model predictions are in good agreement withthe experiments while in the case of n-heptanewith an initial diameter of 0.92 mm the model pre-dicts a complete burnout of the droplet earlierthan experimental observation, as discussed previ-ously in Section3.1. On the basis of comparison

between the model predictions for these two liq-uids, it should be noted that the results show theeffect of change of vaporization enthalpy overburning behavior.

3.4. Flame stand-off ratio

Figure 5 compares the model predictions forflame stand-off ratio with experimental data ofKumagai et al.[11]for then-heptane droplets withinitial diameters of 0.836 and 0.92 mm. Althoughthe flame extinction occurs earlier than the exper-imental observations, the model predictions showqualitatively similar behavior to the experiments.However, the flame stand-off ratio increases con-tinuously for both diameters until burn-out andthere is no evidence of its constant values forany finite time interval during burning. Thus,the model predictions support the unsteadinessof the droplet burning. In the beginning of thecombustion process, however, the model predictshigher values of the flame diameter and afteralmost 50% burning of droplet, the values ofdroplet diameter are lower than the experimentalmeasurements. As shown inFig. 5, therefore, the

values of flame stand-off ratio are higher thanthe experimental results.

3.5. Influence of ambient oxygen concentration onflame structure

Figure 6 describes the variations of flamediameter with burning time for different ambientoxygen concentrations for the n-heptane dropletshaving initial droplet diameter of 0.836 mm. The

Fig. 3. Comparison of calculated gasification rate (solidlines) with the experimental results (points) of Kumagai[5,11]and the model predictions (dotted lines) of Puri [9]and Fachini [10]. Points: solid points for n-heptane;

empty points for ethanol.

Fig. 4. Calculated and measured droplet diameter-squared [11] for n-heptane and ethanol droplets in anair at 1 atm. Initial droplet diameter: n-heptane,0.92 mm; ethanol, 0.93 mm.

Fig. 5. Variation in flame stand-off ratio for then-heptane droplets with time. Comparison betweenexperimental [11] (points) and predicted (lines) data forn-heptane droplets burning in atmospheric pressure air.


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value of flame diameter decreases from its maxi-mum value of 11.98 to 4.33 mm with an increasein oxygen concentration from a value of 5 to30%. Model Eq.(7) accounts for this reason withinvolvement of ambient oxygen concentrationC1. The classical d

2-law also demonstrates theinfluence of ambient oxygen concentration onthe flame diameter. For low concentration ofambient oxygen, the large quantity of fuel vaporaccumulated near the droplet surface will createa flame front at a distance far away from the drop-let surface. However, an increase in oxygen con-

centration reduces the flame front locationsignificantly.

4. Conclusions

An analytical, sphero-symmetric model of anisolated droplet in microgravity, taking intoaccount both the quasi-steady and the transientbehavior of droplet combustion, has beendescribed here. In this study, the considerationof unsteady behavior of oxidizer diffusion in addi-

tion to quasi steadiness for fuel vapor diffusionyields good estimations for various droplet com-bustion characteristics such as droplet diameter-squared, flame diameter, flame stand-off ratio,gasification rate and influence of ambient oxygenconcentration on flame structure. The analyticalformulae are derived for heat and mass fluxes inthe vicinity of evaporating droplet. The compari-sons of modeling results with experimental dataavailable in literature demonstrate the validity ofthe model. Although the model predicts the littlebit higher values of flame diameter for n-heptane,

the classical trend of flame diameter to increaseand decrease from its maximum value with burn-ing time is observed. Furthermore, the behavior ofd2t curve is similar with experimental observa-tions for both n-heptane and ethanol. Althoughthe model calculates 7% higher value of gasifica-

tion rate for n-heptane, it is shown that the pre-dicted burning rates for both fuels are consistentwith the reported measurements for small dropletsizes with no radiation effect. Finally, the effect ofambient oxygen concentration on flame structureis well described by the model. The presented ana-

lytical quasi-steady transient model is sufficientenough to describe the fundamental characteris-tics of single droplet combustion. However, theassumption of quasi-steady behavior for fuelvapor diffusion and transient behavior for oxygendiffusion serves as a basis for subsequent develop-ment of analytical models to accommodate theeffects of radiation, non-unity Lewis number andpossibility of different chemical reactions duringthe combustion process. Further aspects of thiswork will be published in due course.

References

[1] G.A.E. Godsave,Proc. Combust. Inst.4 (1953) 818830.

[2] D.B. Spalding, Proc. Combust. Inst. 4 (1953) 847864.

[3] S. Kumagai, H. Isoda, Proc. Combust. Inst.6 (1957)726731.

[4] S. Kumagai, T. Sakai, S. Okajima, Proc. Combust.Inst.13 (1971) 779785.

[5] S. Okajima, S. Kumagai, Proc. Combust. Inst. 15(1975) 401407.

[6] M. Godsmith, S.S. Penner, Jet Propulsion24 (1954)245251.

[7] D.R. Kassoy, F.A. Williams, AIAA 6 (1968) 19611965.

[8] F.A. Williams, Combustion Theory, Second ed.,Addison Wesley, Menlo Park, CA, 1985, p. 5269.

[9] I.K. Puri, P.A. Libby, Combust. Sci. Technol. 76(1991) 6780.

[10] F. Fachini, Combust. Flame 116 (1999) 302306.[11] H. Hara, S. Kumagai, Proc. Combust. Inst. 23

(1990) 16051610.[12] M.Y. Choi, F.L. Dryer, J.B. Haggard, Proc. Com-

bust. Inst. 23 (1990) 15971604.[13] J.C. Yang, C.T. Avedisian,Proc. Combust. Inst. 22

(1988) 20372044.[14] C.K. Law, Combust. Flame 26 (1976) 1722.[15] S.Y. Cho, R.A. Yetter, F.L. Dryer, J. Comput.

Phys.102 (1992) 160179.[16] A.J. Marchese, F.L. Dryer, V. Nayagam, Combust.

Flame116 (1999) 432459.[17] S.Y. Cho, F.L. Dryer, Combust. Theory Model. 3

(1999) 267280.[18] C.K. Law, Prog. Energy Combust. Sci. 8 (1982)

171201.[19] G.M. Faeth,Prog. Energy Combust. Sci.9 (1983) 1

76.[20] M.K. King, Proc. Combust. Inst. 26 (1996) 1961

1967.[21] B. Abramzon, W.A. Sirignano, Int. J. Heat Mass

Transfer32 (9) (1989) 16051618.[22] W.A. Sirignano, J. Heat Transfer 108 (3) (1986)

633639.[23] G.M. Faeth, Prog. Energy Combustion. Sci. 13

(1987) 293345.

Fig. 6. Calculated variations in flame diameter with timefor various oxygen concentrations for 0.836 mmn-heptane droplets.


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[24] H.S. Carslaw, J.C. Jaeger, Conduction of Heat inSolids, Second ed., Oxford University Press, NewYork, 1959.

[25] S.R. Turns, An Introduction to Combustion: Con-cepts and Applications, McGraw-Hill Series, USA,1996, p. 319.

[26] R.C. Reid, J.M. Prausnitz, B.E. Poling, The Prop-erties of Gases and Liquids, fourth ed., McGraw-Hill, New York, 1995.

[27] R.H. Perry, D.W. Green, Perrys Chemical Engi-neers Handbook, Seventh ed., McGraw-Hill, NewYork, 1997.

Comment

Merrill K. King, NASA/ Headquarters, USA. Tenyears ago, at the 1996 Combustion Symposium[1], I pre-sented a paper on prediction of droplet burning rateswhich incorporated all the approximations (infinitekinetics, constant properties, Le= 1, no radiation ef-fects) which you have in your presentation, except forthe fact that I used a partial-differential-equation ap-proach, which allowed me to calculate the developmentof profiles with time (relaxing the quasi-steady approxi-mation used by Godsave and Spalding). You did not ref-erence my paper though you did reference GS and theconsiderably more complex models of Marahese et al.My best recollection is that I predicted much of the sameresults as you presented. I would appreciate it if youwould revise your paper to, at the very least, referencemine.

Reply. We agree that the work[1] is excellent with anassumption of both the quasi-steady behavior at the sur-face of the droplet and the transient behavior for the gasphase; however, the present analytical work doesnt

accept the transient behavior for the entire gas phase.According to our quasi-steady-transient assumption,the diffusive process of the fuel vapor in the regionbetween the droplet surface and the flame interfaceexperiences the quasi-steady behavior, while the regionbetween the flame interface and the ambience experienc-es the transient mode of disappearance of oxygen.Furthermore, the gasification rate (Eq. 4 in paper) hasits own definition with involvement of vapor preheating(in the region between the droplet surface and the flameinterface) unlike the classical quasi-steady model ([1,2]inthe paper). The model is capable to predict analyticallythe absolute values of flame diameter. Moreover, theagreement between the predictions and measurementsfor flame diameter histories are excellent (as outlinedin the paper).

Reference

[1] M.K. King, Proc. Combust. Inst. 26 (1996) 12271234.

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