a comparison of vaporization models in spray...

1448 AIAA JOURNAL VOL. 22, NO. 10

A Comparison of Vaporization Models in Spray Calculations

S. K. Aggarwal,* A. Y. Tong,t and W. A. SirignanoJCarnegie-Mellon University, Pittsburgh, Pennsylvania

The effects of different gas- and liquid-phase models on the vaporization behavior of a single-componentisolated droplet are studied for both stagnant and convection situations in a high-temperature gas environment.In conjunction with four different liquid-phase models, namely, d2 law, infinite conductivity, diffusion limit,and internal vortex circulation, the different gas-phase models include a spherically symmetric model in thestagnant case and Ranz-Marshall correlation plus two other axisymmetric models in the convective case. Acritical comparison of all these models is made. The use of these models in a spray situation is examined. Atransient one-dimensional flow of an air-fuel droplet mixture is considered. It is shown that the fuel vapor massfraction can be very sensitive to the particular liquid- and gas-phase models. The spherically symmetric con-duction or diffusion limit model is recommended when the droplet Reynolds number is negligible compared tounity, while the simplified vortex model accounting for internal circulation is suggested when the dropletReynolds number is large compared to unity.

f(Bk)H'

L'

M}Mlm'k

TT'kt'tr

V'eV

Ax7

Nomenclature^constants in Eq. (19); bj = 1.458x 10-5g/cm/s/

K°-5,&2 = 110.4K= specific heat at constant pressure, cal/gm/K= DDC, gas diffusivity, cmVs= Blasius function used in Eqs. (15) and (A5)= HC'p T'c, heat from the gas phase given to the liquid

phase per unit mass of fuel vaporized, cal/g= length of the tube, cm= LC'C T'c, heat vaporization, cal/g= Lc/r'0, ratio of gas-phase characteristic length and

the initial drop size= MyMc', molecular weight of fuel= M0MC', molecular weight of oxidizer= mkiJif

cr'0, vaporization rate, g/s= nk/L2

c9 number of droplets per unit cross-sectionalarea, I/cm2

-pprc, gas pressure, atm

= rkrf), droplet radius, cm= gas temperature at the tube entrance, K= TT'C, gas temperature= Tk TC, droplet surface temperature= ttc, time= tc/D'c/L2

c, ratio of convection time to diffusiontime in the gas phase

- gas velocity at the tube entrance, cm/s= We, gas velocity, cm/s-VkV'c, droplet velocity= xLc, axial coordinate, cm= XkLc, droplet location= fuel vapor mass fraction at the droplet surface= fuel vapor mass fraction= oxidizer mass fraction= neutral species mass fraction= spatial grid size= ratio of specific heats= /ijitc, gas viscosity, g/cm/s

Received April 25, 1983; revision received Dec. 5, 1983. Copyright© American Institute of Aeronautics and Astronautics, Inc., 1984.All rights reserved.

* Research Engineer and Lecturer, Department of MechanicalEngineering. Member AIAA.

tGraduate Student, Department of Mechanical Engineering.|G. T. Ladd Professor and Head, Department of Mechanical

Engineering. Member AIAA.

</>' = 00C', variable defined in Eq. (11)p' = pp'c, gas densityp'k = liquid fuel densitypr =p'c/pk, ratio of initial gas-phase density and the

liquid fuel density

I. Introduction

SPRAY vaporization and combustion studies are ofprimary importance in predicting and improving the

performance of systems utilizing spray injection. Com-bustors, fire suppression, spray drying, and various forms ofchemical power plants are typical examples of these systems.Often the vaporization of a droplet in the spray is affected byneighboring droplets. However, in spray combustion com-putations, it is assumed that the overall spray behavior can beobtained by summing behavior of individual isolated dropletssurrounded by a gas phase that itself has varying properties.Even when the assumption that droplets behave as if theywere isolated from each other is not satisfactory, the behaviorof a single isolated droplet in an oxidizing environment willprovide a fundamental input to the overall spray analysis.

Several models exist to represent the quasisteady behaviorof the gas film or boundary layer surrounding the droplet.Also, various representations have been made of the transientbehavior of the liquid phase. The purpose of this paper is topresent two studies that are strongly coupled. First, an in-vestigation is made of the impact of the various gas- andliquid-phase models on isolated droplet vaporization rates in ahigh-temperature environment. Second, the variation of thefuel vapor concentration in a fuel-air spray due to dropletmodel differences is evaluated. The accurate prediction offuel-air mixture ratio, of course, is critical in a combustionsituation. No chemical reaction is considered in the analysesbut a high-temperature gas is considered; this is adequate toevaluate vaporization models for isolated droplets and spraysin combustion situations. Both steady and unsteady sprays areof interest in combustion. In this paper the more generalunsteady case is studied.

In Sec. II, the effects of d2 law, infinite conductivity,conduction limit, and vortex models on the vaporizationbehavior of a single-component droplet are studied. Both thestagnant and convective environments are considered. For theconvective case, several models of the gas-phase boundarylayer are studied. All these gas- and liquid-phase models are

OCTOBER 1984 VAPORIZATION MODELS IN SPRAY CALCULATIONS 1449

examined for three hydrocarbon fuels, namely, w-hexane, n-decane, and Ai-hexadecane. The use of different liquid-phasemodels in a spray vaporization situation is described in Sec.Ill, where a transient one-dimensional flow of an air-fueldroplet mixture in an open tube is considered. The effects ofdifferent droplet models on the fuel vapor mass fractiondistribution in the tube are discussed in detail. An un-steadiness in the gas-phase properties, which is entirely due tothe discrete droplet group locations, is also discussed.Conclusions are stated in Sec. IV.

II. Single-Droplet VaporizationThe basic droplet vaporization/combustion model for an

isolated single-component droplet in a stagnant environmentwas given by Godsave1 and Spalding.2 Since then this modelhas been studied extensively both experimentally andtheoretically. These studies have been reviewed by Williams,3Faeth,4 and Law.5 More sophisticated studies have also beenreported. These consider the effects of relaxing the restric-tions of the basic model. For example, the studies on thetransient droplet heating models include the infinite con-ductivity model6 and the conduction-limit model.7 Law andLaw8 considered the variable gas-phase transport propertyeffects. Hubbard et al.9 considered the transient effects forthe spherically symmetric case and concluded that thequasisteady gas-phase approximation yields small errors, atleast for pressures below 10 atm. Their comparison with theconstant transport property models using reference conditionsindicated that the one-third rule gives the best agreement.

Basically, the existing literature on single-dropletvaporization can be classified into two major categories:spherically symmetric and axisymmetric. The differentmodels in these two categories are discussed below.

Spherically Symmetric Modelsd2 Law Model

The most notable earlier work on droplet vaporization is byGodsave. * In that study, a quasisteady spherically symmetricmodel was used for both liquid and gas phases. The droplettemperature was assumed to be uniform and remainedconstant at its wet-bulb value. The properties in both the gasand liquid phases were assumed to be constant, together withLewis number equal to unity. At the gas-liquid interface, itwas assumed that the fuel vapor mass fraction was a functionof the surface temperature given by some equilibrium vaporpressure equation such as the Clausius-Clapeyron relation.This theory gives the classic d2 law and is the simplest possiblemodel describing droplet vaporization. It should be noted thatthis model neglects the liquid-phase heat and mass transferand is basically a gas-phase model. It does not consider someof the important physics and yields only a crude estimate ofthe droplet vaporization rate.

Some of the resulting relationships are given below.

d)

B = -*fs

(2)

where m is the fuel mass vaporization rate, B the Spaldingtransfer number, L the latent heat of fuel, and Yfs the fuelvapor mass fraction at the interface. Subscripts g and s denotegas phase and droplet surface, respectively.

Note that Yfs = Yj-s(Ts) is a function of surface tem-perature only. Therefore Ts and B can be determined by Eq.(2) and subsequently can be obtained through Eq. (1). For agiven ambient condition, Ts and B are fixed in this model, Eq.(1) can be written as

which says that the radius squared (proportional to thedroplet surface area) decreases linearly in time and hence thismodel is referred to as d2 law model in the literature.

Infinite Conductivity ModelIn a combustor the droplet is initially cold and heats up

with time. Law6 studied droplet combustion with rapid in-ternal mixing where droplet temperature is spatially uniformbut varying with time. It was found that droplet heating is asignificant cause of the unsteadiness of droplet combustionand should be taken into account in any realistic analysis ofunsteady droplet combustion phenomena. Basically, thismodel is the same as the d2 law model except that the constantliquid-phase temperature assumption is relaxed and replacedby a uniform but time-varying temperature inside the droplet.The gas-phase model remains spherically symmetric andquasisteady.

Some authors believed that this uniform temperature limitwas related to the rapid internal liquid circulation limit and,hence, is referred to as rapid mixing model. Sirignano10

showed that even in the limit of high vortex strength, theinternal liquid circulation can only reduce the characteristiclength scale for diffusion by an order of magnitude. The rapidmixing limit can never exist. Rather, it would be conceptuallymore correct to think that the uniform temperature limitresults from the infinite conductivity limit. Hence, it is moreappropriate to call it an infinite conductivity model.

Equations (1) and (2) remain the same except that L isreplaced by

The additional term accounts for the transient liquid dropletheating. Subscript (denotes the liquid phase.

Conduction Limit ModelWhen the internal liquid motion is not significant, heat

transfer inside the droplet will be controlled by thermaldiffusion only. This will be a reasonable model for thestagnant case and represents the slowest heat-transfer limit.

The liquid-phase temperature variation is governed by thewell-known heat diffusion equation,

(4)dt r2 dr V a/-

with the initial and boundary conditions

BT((r,t)dr

d T ( ( r , t )' dr

= 0 atr=0

a t r=r 5 ( f ) (5)

dr2 ,—— = k=constdt (3)

where T 0 ( r ) is the initial temperature distribution and q ( ( t )is the liquid-phase heat flux at the droplet surface.

Equations (1) and (2) remain essentially the same exceptthat L is replaced by

-Lm

The additional term accounts for the liquid-phase heat flux atthe surface.

Since the droplet is vaporizing, the diffusion field has amoving boundary. Obviously this model is more complicatedthan the infinite conductivity limit model.

1450 AGGARWAL, TONG, AND SIRIGNANO AIAA JOURNAL

Droplet Vaporization with Convective EffectsIn many practical situations, the droplet vaporizes in a

convective gas field. The gas-phase convection influences thevaporization process in two ways. First, it increases thegasification rate as well as the heat-transfer rate between thephases. Second, it generates liquid circulation inside thedroplet which increases the liquid-transfer rate. Note that atvery high relative velocity, droplet deformation and shatteringcan occur. This situation is not considered in this paper.Semiempirical correlations11 exist which account for the gas-phase convection by expressing the vaporization and in-terphase heat-transfer rates as a modification of thespherically symmetric case. Sirignano10 analyzed the con-vective case through a combination of stagnation-point andflat-plate analyses and concluded that the convective caseshould not be treated by a correction on the sphericallysymmetric case. Prakash and Sirignano12 analyzed the gas-and liquid-phase flowfields for a single droplet in a convectivegas field. They considered a gas-phase boundary layer outsidethe droplet and a Hill's vortex in the droplet core with thinviscous and thermal boundary layers near the droplet surfaceand an inviscid internal wake near the axis of symmetry. Theirresults show that the infinite conductivity case is neverrealized and the characteristic liquid-phase heat diffusion timereduces (from the diffusion-limit case) by an order ofmagnitude. Their analysis, however, is too complicated andcomputer-time consuming to be included in spray calcu-lations. Tong and Sirignano13 simplified that analysis sub-stantially by neglecting the thin boundary layer inside thedroplet surface. The results of the simplified model are inclose agreement with those obtained from the more exactanalysis.12

Ram-Marshall ModelMany investigators4'5 suggested empirical correlations for

vaporization rate in a convective field as a correction to thespherical symmetric case. The typical form of correlation is^convection =™sPhericai/(^^V")» where f(Re,Pr) is the cor-rection factor. In the present study, the Ranz-Marshallcorrelation is examined, for which the factor f(Re,Pr) is givenby

f(Re,Pr) =1 + 0.3 Re1/2Pr1/3 (6)

Although this type of correlations is very simple, there is< really very little theoretical justification for them. Moreover,these correlations are based on experiments conducted underquasisteady conditions. As mentioned previously, there is atransient droplet heating stage and during that period thequasisteadiness assumption for the liquid phase is invalid.Indeed, Sirignano10 showed, through theoretical analysis,

LIQUID-GASINTEFACE

FREE-STREAM(REGION I)

LIQUID STREAMLINE

GASSTREAMLINE

AXIS OF SYMMETRY

GAS BOUNDARYLAYER (REGION 13)

LIQUID BOUNDARYLAYER (REGION HI)

Fig. 1 Flow regions outside and within a fuel droplet at highReynolds number.

that a correlation of Ranz-Marshall type cannot predict thevaporization rate satisfactorily.

Prakash and Sirignano's Axisymmetric ModelRecently, Prakash and Sirignano12'14 studied the problem

of transient liquid droplet vaporization in a hot convectiveenvironment. They first14 studied liquid internal circulationand droplet heating and later12 developed a gas-phaseboundary-layer analysis and coupled it to the previous liquid-phase analysis. Basically, they used a two-dimensionalaxisymmetric model (valid for a droplet Reynolds numberlarge compared to unity) and divided the problem into severalphysical regions as shown in Fig. 1. These regions were outerinviscid gas flow, gas-phase boundary layer, liquid-phaseboundary layer, internal liquid wake, and inviscid liquid core.Because of the relatively high Reynolds number and highPeclet number, the boundary-layer approximation was usedin their analysis for both the momentum and the energytransfers in both the gas and liquid phases. The outer inviscidgas flow was treated as steady potential flow around a spherewith no flow separation. The gas and liquid-phase boundarylayers were both treated as quasisteady for both momentumand energy transfer while, for the liquid core, the momentumtransfer was treated as quasisteady and the energy transferwas considered transient. This model, named as Prakash andSirignano's model herein, has been included in the single-droplet study. It should be mentioned that this model,although quite detailed, uses an algorithm that is too cum-bersome to be included in a complete combustion analysis.Simplifications, taking into account the important physics,are needed.

In Prakash and Sirignano's analysis, there is a thin thermalboundary layer near the droplet surface which is coupled tothe thermal core in the matching region. The thermalboundary layer which allows for the heat flux to adjust fromtwo-dimensional behavior along the droplet surface to one-dimensional behavior along the boundary layer and thermalcore matching region was treated as quasisteady. The im-portance of the thermal boundary layer and thequasisteadiness assumption have been reviewed recently byTong and Sirignano.13 The results of that study show that,unless the thermal boundary layer is very thin (very largePeclet numbers), the thermal inertia term is important andquasisteadiness assumption for that layer is invalid. Theresults also tend to suggest that the elimination of the thermalboundary-layer representation and the assumption that thethermal core solution is valid up to the droplet surface maystill give solutions with an acceptable degree of accuracy. Itshould be noted that while the thermal boundary-layer for-mulation is two dimensional, the thermal core formulation isone dimensional. Hence, the removal of the thermal bound-ary-layer will simplify the problem tremendously. With theelimination of the thermal boundary layer and the assumptionthat the thermal core solution is valid up to the droplet sur-face, Tong and Sirignano13 subsequently simplified thethermal core solution. Their simplified liquid-phase model isincluded in this study.

Tong and Sirignano's Axisymmetric ModelTong and Sirignano15 further developed a one-dimensional

gas-phase model. They simplified the axisymmetric convectiveanalysis in the gas phase by representing the heat- and mass-transfer rates by an optimum average for the stagnation pointregion and the shoulder region of a droplet. The results fromcoupling this simplified gas-phase model to their simplifiedliquid-phase model which was obtained earlier are inreasonably good agreement with the results of the moredetailed model of Prakash and Sirignano. The gas-phasemodel has been employed in the spray calculations reported inSec. III.


Results and Discussions on Single-Droplet VaporizationLiquid fuel droplets initially at 300 K vaporizing in 1000 K,

10 atm pressure fuel-free environment is used to study thevarious gas- and liquid-phase models. Hexane, decane, andhexadecane droplets of initial radius of 0.00476 cm areconsidered. The physical properties are the same as those inthe Prakash and Sirignano's analysis.12

The results for the stagnant case are given in Figs. 2 and 3.The gas-phase model is quasisteady and spherically sym-metric. Three different liquid-phase models, namely, the d2

law, the conduction limit, and the infinite conductivity limitare compared. Since the existence of a vortex in a stagnantenvironment cannot be realized conceptually, it is excludedfor comparison. One would think that if the vortex modelwere included, it would be in between the infinite conductivityand conduction limits. This indeed can be shown to be true.Figures 2 and 3 show the temporal variation in the surfacetemperature and surface area, respectively. The physical timet is nondimensionalized by using the liquid thermal diffusivityoi( and the initial droplet radius r0 as T=t/(r$/af). For the d2

law model the surface temperature is at the wet-bulb tem-perature and remains constant while the surface area regresseslinearly in time. The d2 law serves as an asymptotic limit forthe other two models. This is reflected in Fig. 2 where thesurface temperature of the other two models approaches thewet-bulb temperature, and in Fig. 3 where the curves becomemore linear.

Since the d2 law neglects the initial transient dropletheating, the droplet vaporizes much faster than the other twomodels. The difference is bigger for the heavier fuels whichhave higher boiling temperatures and, hence, a longertransient heating period. Conversely, the conduction limit andthe infinite conductivity limit have about the same dropletlifetime. Note that these two limits intersect each other. Sincethe droplet temperature is uniform in the infinite conductivitylimit, the surface temperature increase per unit of energyabsorption is less. Consequently, the fuel vapor mass fractionat the droplet surface is lower, which leads to lowervaporization rate. On the other hand, the difference betweenthe ambient and surface temperatures, which relates directlyto the heat-transfer rate to the liquid, is higher. This even-tually results in a higher surface temperature and faster

vaporization rate during the latter portion of the lifetime forthe infinite conductivity model. Note that the wet-bulbtemperatures in these calculations are considerably lower thantheir corresponding boiling temperatures of the fuels. Theeffect of the different liquid-phase models on the interiorliquid temperature is quite significant and, therefore, theconduction limit model, which is considered to be exact,should be used if detailed temperature distribution inside thedroplet is needed. The infinite conductivity model may beuseful in the low ambient temperature case when the dropletlifetime is long. The d2 law model, which gives pooragreement with the conduction limit model, is oversimplifiedand should be discarded.

i.oo

0.90

a 0.80UJ(T

0.60

0.50

0.40

0.30

z 0.20OZ

0.10

0.00 I

, GAS PHASE : SPHERICALLY -<<V SYMMETRIC

LIQUID PHASE:——— CONDUCTION LIMIT MODEL——— INFINITE CONDUCTIVITY MODEL——— D SQUARE LAW MODEL

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80NON-DIMENSIONAL T I M E

Fig. 3 Nondimensional radius square vs time comparisons.

600

GAS PHASE: SPHERICALLY-SYMMETRICLIQUID PHASE :

—— CONDUCTION LIMIT MODEL— -INFINITE CONDUCTIVITY MODEL—— D SQUARE LAW MODEL

3000.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

NON-DIMENSIONAL TIME

Fig. 2 Surface temperature vs time comparisons.

OBO

600

570

540

510

480

450

420

390

360

330

300

GAS PHASE: TONG-SIRIGNANO'S MODELLIQUID PHASE:CONDUCTION LIMIT MODELINFINITE CONDUCTIVITY MODELVORTEX MODEL

0.00 0.04 0.08 0.12NON-DIMENSIONAL TIME


0.16


The results for the convective case are given in Figs. 4-7. Inthese calculations, the Reynolds number (based on dropletdiameter) is initially 200 and decreases with the diameter asthe droplet vaporizes. The relative velocity is 2500 cm/s and isassumed to be constant throughout. In reality the relativevelocity will be reduced by the drag force; this is consideredlater in the spray vaporization case.

Although Prakash and Sirignano's model is the mostdetailed and should be considered the most exact, it is quitecumbersome to be included in a complete spray analysis.Instead, Tong and Sirignano's model and the simplifiedvortex model, which have been shown15 to give closeagreement with Prakash and Sirignano's results, are used asreference model in the figures.

Figures 4 and 5 show the results of the various liquid-phasemodels. The d2 law does not exist in this convective case.Again, the conduction and infinite conductivity limits in-tersect due to the same reasoning as in the stagnant case. Thevortex model lies in between the other two models as expected.Note that the droplet lifetime is considerably shorter than thatin the stagnant case. In the convective case, (r/r0)3'2regresses asymptotically linearly with time and is used in Figs.5 and 7. Figure 5 shows that, for the heavier fuel, the per-centage variation in droplet lifetime can depend quitesignificantly on the liquid-phase model.

Figures 6 and 7 show the effects of the different gas-phasemodels. The Ranz-Mar shall correlation over predicts thevaporization rate and underestimates (for decane) the surfacetemperature. As indicated by Sirignano,10 this type of corre-lation has very little theoretical justification and cannot givesatisfactory results. The present study supports Sirignano'squalitative analysis. Tong and Sirignano's model appears tobe in good agreement with the more detailed model ofPrakash and Sirignano.

The simplified vortex model applies to a situation where 1)the streamfunction patterns are described reasonably well by aHill's spherical vortex pattern, and 2) the circulation time isvery short compared to other characteristic times. Hill'sspherical vortex can be obtained when the nonlinear inertialterms in the momentum equation can be neglected comparedto other terms. This can occur at low Reynolds number via anorder-of-magnitude argument. Fortuitously, in similarfashion to Couette flow or Poisseuille flow, the inertial terms

go identically to zero so that the streamfunctions are valideven at higher Reynolds number. The very short circulationtime or very high vortex strength implies a very high Pecletnumber for the liquid (upon comparison of diffusion time tocirculation time). Since Prandtl number is of order ten, a highReynolds number is implied. For low Reynolds number, theheat conduction will occur in a two-dimensional fashion, bothnormal and tangent to the streamsurfaces. This complicatesthe application of a simplified model to a low Reynoldssituation. The gas-phase models are also limited to highReynolds numbers, since thin boundary layers are considered.Therefore, for several reasons, additional modeling isrequired if the low Reynolds number case is to be considered.

The results indicate that the droplet heating andvaporization are essentially unsteady for most of theirlifetimes, particularly for the heavier fuel. For a detailedanalysis, Prakash and Sirignano's model is recommended, but

UJOCe>

o2DCO

540

510

480

450

420

390

360

330

300 I

HEXANE

LIQUID PHASE: VORTEX MODELGAS PHASE:

—— TONG-SIRIGNANO MODEL—— RANZ-MARSHALL MODEL——— PRAKASH- SIRIGNANO MODEL

I_____I_____I_____I_____I_____I

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16NON-DIMENSIONAL TIME


1.00

1 0.20 -

0.10-

0.00

GAS PHASE: TONG-SIRIGNANO'S MODELLIQUID PHASE:CONDUCTION LIMIT MODELINFINITE CONDUCTIVITY MODELVORTEX MODEL

J_____I_____I_____I_____I_____

l.00r

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16NON-DIMENSIONAL TIME

Fig. 5 (R/R0)3'2 vs time comparisons.

\ \

LIQUID PHASE'• VORTEX MODELGAS PHASE:

—— TONG-SIRIGNANO MODEL— - RANZ-MARSHALL MODEL——— PRAKASH-SIRIGNANO MODEL

0.000.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

NON- DIMENSIONAL TIME

Fig. 7 (R/R0)3/2 vs time comparisons.


for more practical application, Tong and Sirignano's modelwith the simplified vortex model would be useful.

III. Spray Vaporization StudyIn the spray vaporization problem, a one-dimensional

transient flow of air and fuel droplets in an open tube isconsidered, where the motion and vaporization of amonodispersed spray in a laminar, hot gas stream are studied.The purpose is to examine the effects of different liquid-phasemodels on the bulk vaporization characteristics as well as onthe gas medium. The different liquid-phase models consideredare the infinite conductivity, the conduction limit, and thevortex models. The different convective models for the gasphase are the Ranz-Marshall correlation and the model ofTong and Sirignano. The physical situation consists of acontinuous laminar flow of hot air in an open tube. The gas-phase properties initially in the tube and later at the tubeentrance are prescribed. The injection of fuel droplets isintermittent. One group of droplets is injected every giventime interval. The number of droplets in a group (or with acharacteristic) represents the number of droplets per unitcross-sectional area. That is, the spacing between dropletgroups is precisely the spacing (in the flow direction) betweenindividual droplets. The grouping of the droplets eliminatesonly the effect of droplet spacing in the transverse direction.The constancy of the frequency of injected droplets is, ofcourse, artificial, but will not affect any of the qualitativeresults of this paper. The frequency of injection and initialdroplet velocity depends on the overall fuel-to-air ratio in thetube, the mass flow of air, droplet spacing, and dropletdiameter at the tube entrance. The prescription of the overallfuel-to-air ratio, the droplet diameter, and the droplet velocityat the entrance yields the value of the droplet number density.The droplet spacing in the axial direction and the number ofdroplets per unit cross-sectional area are so adjusted as toprovide initially an isotropic droplet spacing in a unit cube.Initially a high relative velocity is provided between the twophases. As a droplet group moves in the hot gas stream itaccelerates. At the same time, the droplets heating andvaporization is taking place. The spray processes influence thestate of the gas, i.e., the gas phase is continuously retarded,cooled, and enriched with fuel vapor.The gas-phaseproperties are also being influenced by the upstream con-vection. All of these gas- and liquid-phase processes are

Table 1 Values used in the computation

Parameter Value

FuelLiquid density, p'kInitial drop radius, r'0Initial drop temperature, T'kNumber of drops per unit

cross-sectional area, N'kDrop spacing in the axial direction, d'sGas velocity at the tube entrance, V'cDrop velocity at the tube entrance, V'kTube length, LcPressure, p'cGas temperature at the entrance, T'cTemporal step size, A/'Spatial step size, Ax'Specific heat at constant pressure, C'pMolecular weight of fuel, Mj-Molecular weight of oxidizer, M'0Boiling temperature of the

fuel at 1 atm, T'bnHeat of vaporization, L'Gas constant, /?gas ________

N-decane0.77350 /im300 K

400cm'20.05 cm

1000 cm/s0.1 V'c10cm10 atm1000 K10~5s0.1 cm

0.25 cal/g/K142.3 g/mole

32.0

447.3 K86.5 cal/g

2.8393 atm cm3/g/K

modeled by a system of unsteady, one-dimensional equations.The gas phase is represented in Eulerian coordinates, whereasthe liquid phase is represented in Lagrangian coordinates. Thegoverning equations in nondimensional form are given asfollows.

Gas-Phase Equations

dp d-W+Tx

3Y dY d2YTt+vTx^D^

Y=YrY0.<j>

= 1.0-Yf-Y0

Liquid-Phase Equations

sk

for Ranz-Marshall correlation model

dSk

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

.f(Bk)

for Tong-Sirignano model

_trLr

where

(16)

(17)

(18)

(19)

(20)

The source terms in the gas-phase equations result from thecoupling between the two phases. These are given in theAppendix. It should be noted that a transformation, as givenby Eq. (11), has been employed. This transformation is usefulfor a constant-volume situation.16 However, it has beenretained here (a constant-pressure case) for the sake ofgenerality.

In nondimensionalizing the gas-phase equations, the lengthscale is the length of the tube, and the velocity scale is the gasvelocity at the tube entrance. The time scale is determined bythese two scales. The gas-phase properties are non-


dimensionalized by using the respective properties at theentrance. For the liquid-phase equations, the droplet location,velocity, and surface temperature are nondimensionalized bythe gas-phase length, velocity, and temperature scales,respectively. The droplet radius is nondimensionalized by theinitial droplet radius. The above nondimensionalization givesrise to three additional dimensionless groups, tr, Lr, and pr;tr is the ratio of convective time to diffusion time in the gasphase, Lr the ratio of gas-phase length scale and initial dropradius, and pr the ratio of initial gas-phase density and liquiddensity.

The important assumptions made in writing these equationsare that the gas pressure is constant, radiative heat transfer isnegligible, the species diffusion follows Pick's law with equalmass diffusivities for each pair, the specific heat at constantpressure is constant, and the gas-phase Lewis and Schmidtnumbers are unity. In addition, the product pD (= /x ) isassumed constant. It is noteworthy, however, that in thecalculation of liquid-phase properties /x is considered afunction of temperature as given by Sutherland correlation.17

As indicated in Eq. (19), a reference temperature is used forthe calculation of /z and the reference temperature is obtainedby the one-third rule as recommended by Sparrow andGregg.18

The drag coefficient is evaluated by using an expression asproposed by Putnam.19 The evaluation of Reynolds numberfor the drag coefficient is based on the freestream density andviscosity evaluated at the one-third reference state. Thisfollows the recommendation of Yuen and Chen.20 In thecalculations the Reynolds number for each droplet varies withtime due to changes in size, relative velocity, and local con-ditions. It should also be noted that the effect of relativedroplet-gas velocity on the vaporization rate [see Eq. (15)]has been treated by a semiempirical correlation,11 as well asby an axisymmetric model.15

The Solution ProcedureA hybrid Eulerian-Lagrangian numerical scheme is used to

calculate the gas- and liquid-phase properties at the («+ l)thtime level from the values known at the nth time level. First, asecond-order accurate scheme is employed to interpolate thegas-phase properties from the Eulerian locations (fixed-grid

points) to the Lagrangian (or droplet) locations. The schemeuses the gas-phase properties at two grid points xf and xi+1and gives the corresponding properties at a Lagrangianlocation Xk, where Xk is between xt and xi+ / . Using these gas-phase properties, the droplet surface temperature is thencalculated. For the infinite conductivity model it involves thesolution of an ordinary differential equation for each dropletgroup. For the conduction limit and vortex models a partialdifferential equation needs to be solved. The details have beendiscussed in Sec. II. Knowing Tk for each droplet group,other droplet properties (Xk,Vk>rk) can be obtained bysolving Eqs. (13-16). A second-order Runge-Kutta scheme isused for this purpose. It should be noted, however, that thereis some loss of accuracy due to the fact that the gas-phaseproperties are not updated in the Runge-Kutta scheme. Usingthe new liquid-phase properties, the source terms as given inthe Appendix can be evaluated at the Lagrangian locations.Then a second-order accurate scheme is used to distributethese source terms from a Lagrangian location to the twoneighboring gas-phase locations. Using these source terms,the gas-phase properties (Yf, Y0, and </>) at the (AI+ l)th timelevel are obtained by solving Eq. (8). An explicit finite dif-ference method is used for this purpose. The gas temperatureand gas density are then obtained by using Eqs. (11) and (12).Finally, the gas velocity is obtained from an integral form ofthe continuity equation (7). It is worth emphasizing that, inthe present case, the gas-phase convection term dominates thediffusion term. Therefore, an upwind difference scheme21 isemployed for the convection term. It also should be noted thatan explicit method was found to be the most efficient forreacting two-phase flow.21 However, an implicit method maybe more efficient for the present vaporizing case withoutcombustion since the equations may not be as stiff.

Discussion of ResultsThe effects of using different liquid-phase models on the

bulk spray and gas-phase properties are now discussed. Thevarious values used in the computation are listed in Table 1.The criterion used in selecting these values was to consider aspray vaporization situation with a moderate gas-phaseconvection for the droplets. The selected values provide aninitial Reynolds number (based on the relative velocity and

0.75— CONDUCTION LIMIT MODEL

— VORTEX MODEL— INFINITE CONDUCTIVITY

MODEL

0.000.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75 10.00

DISTANCE (CM)

Fig. 8 Fuel vapor mass fraction vs distance at various times fordifferent liquid-phase models. x

0.75

0.60

o °-45

——— CONDUCTION LIMIT MODEL

——— VORTEX MODEL

— — INFINITE CONDUCTIVITYMODEL

COco

0.30troQ.

0.15

0.000.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75 10.00

DISTANCE (CM)

Fig. 9 Fuel vapor mass fraction vs distance at 16 ms for differentliquid-phase models.


Table 2 Properties of the first droplet group for different liquid-phase models

, ms X^, cm

Conduction limit model

02468

1012

02468

1012

00.8852.2723.9165.6917.5409.433

00.8892.2573.8695.6157.4459.328

1.0000.9610.9020.8420.7810.7160.648

1.0000.8870.7340.5970.4760.3670.272

Infinite conductivity model

1.0000.9940.9480.8670.7790.6920.604

1.0000.9820.8520.6520.4730.3310.220

Vortex model

300.0445.0453.3458.1462.2465.9469.1

300.0412.5459.4471.8474.5475.0475.1

100.0590.0773.1861.2909.2937.2954.5

100.0587.6758.2845.0897.1930.3951.6

02468

1012

00.8852.2643.8955.6587.4969.382

1.0000.9730.9200.8580.7910.7210.648

1.0000.9210.7790.6320.4950.3750.272

300.0438.1452.6460.6465.6468.8470.9

100.0588.0767.5854.7903.4932.7951.3

drop diameter) of 113 and a droplet residence time of about12.5 ms. During this time 80% of the droplet mass isvaporized. The effects of different liquid-phase models on thebulk gas-phase properties are presented in Figs. 8-10. Figures8 and 9 indicate that the choice of a particular droplet heatingmodel can influence the fuel vapor distribution in the tubesignificantly, especially in the early vaporization period. Thiscan have a profound effect on the subsequent combustionprocess in a practical situation. The maximum differencebetween the fuel vapor mass fraction values is as much as100% between the conduction limit and the infinite con-ductivity models. As Fig. 9 indicates, the conduction modelinitially (up to a distance of 5 cm) predicts the highest fuelvapor mass fraction, whereas the infinite conductivity modelpredicts the lowest. At later times the situation is reversed.The vortex model results are always in between those of theother two models. The difference in the fuel vapordistribution is a direct consequence of the difference in thedroplet surface temperature values for the three models. Asdiscussed in Sec. II, the conduction limit model initiallypredicts the highest droplet surface temperature and,therefore, the fastest vaporization rate, whereas the infiniteconductivity model predicts the lowest surface temperatureand the slowest vaporization rate. As also discussed in Sec. II,this behavior is reversed at later times.

The fuel vapor mass fraction profiles in Figs. 8 and 9exhibit an oscillatory and undulating behavior. This is due tothe periodic nature of the droplet injection process whichcauses a finite droplet spacing in the streamwjse direction.Thus the period of this oscillation is the same as the timeinterval between two subsequent injections. This is clearlyillustrated in Fig. 10. As this figure indicates, the profilesoverlap after every 0.5 ms, which is the time interval betweentwo subsequent injections. This unsteady gas-phase behavior,which is entirely due to the spray discretization, should bekept in mind in spray modeling. A continuous drop in the gastemperature occurs in the downstream direction and is in-dicative of the continuous cooling of the gas phase due to

0.60o*zoo 0.45

0.30<roa.

0.15

0.00

I 1 i I I i

I _

(5) 10.00 MILLISECONDS(4) 10.25(3)10.50(2)10.75

1 ) 1 1 . 0 0

0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75 10.00DISTANCE (CM)

Fig. 10 Fuel vapor mass fraction vs distance at various times for thevortex model.

droplet heating and vaporization. The difference in the gastemperature distribution due to the different liquid-phasemodels is not as significant as that in the fuel vapordistribution. This is due to the fact that the spray heating andvaporization constitute only a small heat loss in the overallgas-phase heat budget. An oscillatory behavior in the gastemperature profiles occurs again due to the intermittentdroplet injection process. This intermittency is prescribedexactly in our idealized calculation. In a practical situation itmay not be so well defined and perhaps should be taken asrandom. That is, the distance between droplets as they flowinto the domain of interest will not be fixed as they are given


in this calculation but rather will be distributed about someaverage value.

If random injection rather than orderly injection had beenchosen, the oscillatory behavior in the fuel-vapor con-centrations would not have been eliminated. Droplet spacingwould vary from droplet pair to droplet pair so that thewavelength or period would no longer be constant in a ran-dom injection case but the fundamental oscillatoryphenomenon would still be present.

The effect of different liquid heating models on the dropletproperties is illustrated in Table 2. The properties of the firstdroplet groups, i.e., the location, the nondimensional radius,the surface temperature, and the velocity, are given as func-tions of time for the three models. The difference in thesurface temperature, droplet radius, and droplet volumevalues for the three models is very similar to that discussed inSec. II. The conduction limit model predicts the highestdroplet velocity, whereas the infinite conductivity modelpredicts the lowest value. This occurs since the conductionlimit model initially predicts the fastest vaporization rate, thesmallest drop size, and, consequently, the highest dropletacceleration. It should be noted, however, that the differencein the droplet velocities for the three models is quite small.

The effect of different convective models is presented inFig. 11. It should be noted that the model of Tong andSirignano is valid only when the Reynolds number is largecompared to unity; as indicated by Eq. (15), this modelpredicts a vaporization rate, which is proportional to thesquare root of the Reynolds number. For this reason, initialdroplet radius of 100 /xm is used for these calculations. Thisgives a value of 226 for the initial Reynolds number. Figure 11shows that the Ranz-Marshall correlation overpredicts thevaporization rate as compared to that given by the axisym-metric model. This behavior is consistent with the one ob-served for the single-droplet case in the previous section. Sincethe Ranz-Marshall correlation predicts a higher vaporizationrate and, therefore, a higher heat loss from the gas phase, itgives lower gas temperatures than those given by thecorrelation. It is also important to note that at very smallReynolds number, our model predicts an unacceptably smallvaporization rate. Consequently, the model should have thecapability to switch from the Tong and Sirignano model tosome valid low Reynolds number model as the Reynoldsnumber becomes small. This is presently under investigation.

IV. ConclusionsThe most common vaporization models have been com-

pared for fuels of varying volatility in a high-temperatureenvironment. Both isolated droplet and spray vaporizationhave been studied. The use of the d2 law or the infiniteconductivity model (sometimes named the rapid mixingmodel) has been shown to be very inadequate. For sphericallysymmetric vaporization (no relative gas-droplet motion), the

zo

0.30

0.15

2 0.00

—— RANZ-MARSHALL CORRELLATION-— TONG-S IR IGNANO MODEL

0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75 10.00DISTANCE (CM)

Fig. 11 Fuel vapor mass fraction vs distance at 16 ms for differentgas-phase models.

conduction limit model for transient heating is recommended.In practical situations where a relative gas-droplet motionexists, the simplified vortex model of Tong and Sirignano isrecommended when the droplet Reynolds number based onrelative velocity is high compared to unity. That modelcompares well with the more detailed model of Prakash andSirignano and predicts well the effects of the laminar gas-phase boundary layer over the droplet and the internal cir-culation in the droplet.

The spray calculations indicate an inherent unsteadinessdue to the intermittent character of a spray. This unsteadinesswill be interesting whenever we wish to resolve structure onthe scale of the average distance between droplets.

AppendixThe source and sink terms in gas-phase equations (7) and

(8) are given as follows:

_ *' l V> ~ 7~ 77- 2-/ m*«*

SYf=-Sp(l-Yf)

Sy0 = --spr0

H=

(Al)

(A2)

(A3)

(A4)

for the Ranz-Marshall correlation

[-/te*j "/(**)

for the Tong-Sirignano model (A5)

Bk = T-Tk/H (A6)

(/-y /5)(r-rj(A7)

(A8)

In the above relations, SYf and SYo are the source and sinkterms for the fuel vapor and oxidizer mass fractions, xfs themole fraction at the droplet surface, M'a the molecular weightof the gas phase (excluding fuel vapor), T'bn the liquid fuelboiling temperature at the normal pressure p'n, p theprevailing pressure, and k represents the summation over alldroplet groups which happen to be in a given gas-phase mesh,of size Ax. A phase equilibrium assumption has been made atthe droplet surface and the Clausius-Clapeyron relation [Eq.(A9)] has been used. The different methods of computing thedroplet surface temperature Tk distinguish the various liquid-phase models examined in the present spray problem. Thesehave been discussed in Sec. II.

References1 Godsave, G.A.E., "Studies of the Combustion of Drops in a Fuel

Spray: The Burning of Single Drops of Fuel," Fourth Symposium(International) on Combustion, Williams and Wilkins, Baltimore,Md., 1953, pp. 818-830.

2Spalding, D. B., "The Combustion of Liquid Fuels," FourthSymposium (International) on Combustion, Williams and Wilkins,Baltimore, Md., 1953, pp. 847-864.


3Williams, A., "Combustion of Droplets of Liquid Fuels: AReview," Combustion and Flame, Vol. 21, 1973, pp. 1-31.

4Faeth, G. M., "Current Status of Droplet and Liquid Com-bustion," Proceedings of Energy Combustion Science, Vol. 3, 1977,pp.191-224.

5 Law, C. K., "Recent Advances in Droplet Vaporization andCombustion," Proceedings of Energy Combustion Science, Vol. 8,1982, pp. 171-201.

6Law, C. K., "Unsteady Droplet Vaporization with DropletHeating," Combustion and Flame, Vol. 26, Feb. 1976, pp. 17-22.

7Law, C. K. and Sirignano, W. A., "Unsteady Droplet Com-bustion with Droplet Heating II, Conduction Limit," Combustionand Flame, Vol. 29, March 1977, pp. 175-186.

8Law, C. K. and Law, H. K., "Theory of Quasi-Steady One-Dimensional Diffusional Combustion with Variable Properties In-cluding Distinct Binary Diffusion-Coefficients," Combustion andFlame, Vol. 29, Nov. 1977, pp. 269-275.

9Hubbard, G. L., Danny, V. E., and Mills, A. F., "DropletVaporization: Effects of Transients and Variable Properties," In-ternational Journal of Heat and Mass Transfer, Vol. 18, Sept. 1975,pp. 1003-1008.

10Sirignano, W. A., "Theory of Multi-Component Fuel DropletVaporization," Archives of Thermodynamics and Combustion, Vol.9, No. 2, 1978, pp. 231-247.

HRanz, W. E. and Marshall, W. R., "Evaporation from Drops,"Chemical Engineering Progress, Vol. 48, 1952, pp. 141-146, and 173-180.

12Prakash, S. and Sirignano, W. A., "Theory of ConvectiveDroplet Vaporization With Unsteady Heat Transfer in the CirculatingLiquid Phase," International Journal of Heat and Mass Transfer,Vol. 23, No. 3, March 1980, pp. 253-268.

13Tong, A. and Sirignano, W. A., "Analytical Solution for Dif-fusion and Circulation in a Vaporizing Droplet," 19th (In-

ternational) Symposium on Combustion, The Combustion Institute,Pittsburgh, Pa., 1982, pp. 1007-1020.

14Prakash, S. and Sirigano, W. A., "Liquid Fuel Droplet Heatingwith Internal Circulation," International Journal of Heat and MassTransfer, Vol. 21, July 1978, pp. 885-895.

15Tong, A. Y. and Sirignano, W. A., "Analysis of VaporizingDroplet with Slip, Internal Circulation and Unsteady Liquid PhaseHeat Transfer," JSME-ASME Thermal Engineering Joint Con-ference, Honolulu, Hawaii, March 1983.

16Seth, B., Aggarwal, S. K., and Sirignano, W. A., "FlamePropagation Through an Air-Fuel Spray Mixture with TransientDroplet Vaporization," Combustion and Flame, Vol. 39, Oct. 1980,pp. 149-168.

17 Fox, R. W. and McDonald, A. T., Introduction to FluidDynamics, John Wiley and Sons, New York, 1978, pp. 613-616.

18Sparrow, E. M. and Gregg, J. L., "The Variable Fluid-PropertyProblem in Free Convection," Transactions ofASME, Vol. 80, 1958,pp. 879-886.

19Putnam, A., "Integratable Form of Droplet Drag Coefficient,"American Rocket Society Journal, Vol. 31, Oct. 1961, pp. 1467-1468.

20Yuen, M. C. and Chen, L. W., "On Drag of EvaporatingDroplets," Combustion Science and Technology, Vol. 14, Nov. 1976,pp. 147-154.

21 Aggarwal, S. K. and Sirignano, W. A., "Effect of NumericalModeling on One-Dimensional Enclosed Homogeneous andHeterogeneous Deflagrations," Paper 81-WA/HT-46, ASME WinterAnnual Meeting 1981.

22Sirignano, W. A., "Fuel Vaporization and Spray CombustionTheory," Progress in Energy and Combustion Science, Vol. 9, No. 4,1984.

From theAIAA Progress in Astronautics and Aeronautics Series

AERO-OPTICAL PHENOMENA—v. 80Edited by Keith G. Gilbert and Leonard J. Often, Air Force Weapons Laboratory

This volume is devoted to a systematic examination of the scientific and practical problems that can arise inadapting the new technology of laser beam transmission within the atmosphere to such uses as laser radar, laserbeam communications, laser weaponry, and the developing fields of meteorological probing and laser energytransmission, among others. The articles in this book were prepared by specialists in universities, industry, andgovernment laboratories, both military and civilian, and represent an up-to-date survey of the field.

The physical problems encountered in such seemingly straightforward applications of laser beam transmissionhave turned out to be unusually complex. A high intensity radiation beam traversing the atmosphere causes heat-up and breakdown of the air, changing its optical properties along the path, so that the process becomes anonsteady interactive one. Should the path of the beam include atmospheric turbulence, the resulting nonsteadydegradation obviously would affect its reception adversely. An airborne laser system unavoidably requires thebeam to traverse a boundary layer or a wake, with complex consequences. These and other effects are examinedtheoretically and experimentally in this volume.

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412pp., 6x9, illus., $30.00Mem., $45.00List

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a comparison of vaporization models in spray...

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