spray evaporatioin

8/7/2019 spray evaporatioin

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inf. .i . Heof Mass ~ransfur.Yol 24, No. IL pp, 1913-1923, 1981. #l~-93lO~~/l2l9~3~ll SO Z.t&OPrintedin Great Britain 0 1981 PeraamonPress Ltd.

THE INF LUE NCE OF SIZE-DISTRIBUTION PARAMETE RS ONTHE EVAPORATION OF POLYDISPE RSE DILUTE SP RAYS

A. C. ALKIDASEngine Research Department, General Motors Research Laboratories, Warren, MI 48090, U.S.A.

(Received 28 January 1981 anti in reuise~~or~ 26 May 1981)Abstract - An analytical study has been ~rformed of both diffusio~~ontrolled and radiation~ontrolled(opaque droplets) evaporation of singie~omponent polydisperse sprays. The overall evaporation character-istics of sprays of different initial size distributions were found to be correlated best by their initial Sautermean diameter. In the case of diffusion-controlled evaporation, the representation of a polydisperse spray bya monodisperse spray of droplet size equal to the initial Sauter mean diameter of the polydisperse spray is agood assumption during the early stages of evaporation. When 40% of the volume of droplets hasevaporated, the error of the monodisperse model reaches 20%. On the other hand, in the case of radiation-controlled evaporation, the monodisperse spray model yields good results for most of the period ofevaporation of the spray.

~O~ENCLA~REc~~cient of size distribution, defined inequation (3);coefficient of size distribution ;transfer number ;specific heat ofgases at constant pressure ;number fraction of unevaporated drop-lets [ = n(t)/n(O)] ;volume fraction of unevaporated droplets[ = kXr)/UO)J ;number of droplets of diameter x at timer;heat of vaporization;exponent of droplet diameter in the evap-oration law ;thermal conductivity;total number of droplets at time t;total number of droplets at time t = 0;heat of reaction ;time ;temperature;volume of unevaporated droplets;diameter of droplet;linear mean diameter of spray;surface mean diameter of spray;volume mean diameter of spray:Sauter mean diameter of spray ;/Yi*;.x/.r32;mass fraction of oxidizer ;molecular weight of oxidizer ;defined in equation (33);constant of size distribution, defined inequation (1);gamma function ;constant of size distribution, defined inequation (1);

defined in equation (18);coefficient of evaporation ;density of liquid droplet ;defined in equation (18);= ~ij(t)/xij(O);dimensionless evaporation time based onXij [see equation (34)].

conditions far away from droplet;conditions on the surface of droplet.

Tut: WAPOKAIWN of fuel sprays in diesel and otherreciprocating engines strongly influences the ignitionand subsequent combustion of the fuel in theseengines. Modelling of the evaporation and combustionof polydisperse fuel sprays requires considerablenumerical effort. This problem may be simplified,however, if the polydisperse fuel spray can be repre-sented by an equivalent monodisperse spray. Thevalidity of such representation requires answers to thefollowing questions: (1) What is the droplet size of thetheoretical monod~sperse spray that best representsthe evaporation of the corresponding po~ydis~~~spray, and (2) what is theerror in the predicted amountof fuel evaporated if the monodisperse spray assump-tion is evoked?

This investigation was directed towards providinganswers to these questions by considering the evap-oration of polydisperse sprays of different size distri-butions. Some of the empirically determined sizedistributions of sprays used were: the Rosin-Rammlerdistribution [l], the Nukiyama-Tanasawa distri-bution [Z), and the Hiroyasu-Kadota distribution [3,

1913


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1914 A. C. ALKIDAS41. The Tanasawa-Toyoda distribution [S] (a limitingform of the Nukiyama-Tanasawa distribution) andthe Hiroyasu-Kadota distribution are consideredapplicable to diesel engine fuel-injection systems.

The diffusion-controlled evaporation of polydis-perse sprays has been considered by Probert [6],Tanasawa and Tesima [7], and Williams [8, 91.Probert [6] examined sprays with size distributionsgiven by the Rosin-Rammler distribution, whereasTanasawa and Tesima [7] examined a variety of sizedistributions that are generally described by equation(1) (see subsequent section). In both studies the volumeof unevaporated droplets of the spray was determinedas a function of time by graphical integration of theresultant equations. Williams [8, 93 developed anelegant theory of the evaporation of a polydispersespray in a variable-area quasi-one-dim. rocket cham-ber. The combustion efficiency was determined as afunction of chamber length for a spray conforming tothe Rosin-Rammler size distribution. It must be notedthat presently the Rosin-Rammler distribution israrely utilized. Ingebo [lo, 1 l] found this distributionto represent poorly the droplet size distributions ofsprays in turbulent air streams.Dickinson and Marshall [12] considered the in-fluence of the relative velocity of the droplets on therate of evaporation of polydisperse sprays of varioussize distributions which included log normal andsquare-root normal distributions. For the case ofnegligible relative velocity of the droplets, i.e. diffusion-controlled evaporation, they concluded that no meandiameter can adequately characterize the evaporationcharacteristics of polydisperse sprays.

The present study may be considered as an exten-sion of the Tanasawa and Tesima [7] study. In thepresent study, both diffusion- and radiation-controlledevaporation has been considered. Radiation in dieselengines is very significant [13, 141 and is expected toinfluence the evaporation rate of droplets stronglyafter combustion has been initiated. The influence ofradiation increases with an increase in the size of thedroplet, as well as with an increase in the temperatureof its surroundings. For an opaque droplet in a low-convective flow environment, the ratio of radiative toconvective heat transfer is directly proportional to thedroplet diameter. In the case of a 100 pm n-decane fueldroplet in a 2500 K combustion gas environment thisratio is about 0.6. However, in spray combustors,which are high-convective flow systems, radiation isnot expected to control the evaporation of droplets.The problem considered here is the steady-stateevaporation of polydisperse sprays of different sizedistributions in a constant volume chamber understeady environmental conditions. All the droplets in aspray are assumed to start evaporating at time t = 0.The present treatment of the evaporation of sprays stillremains idealized because the effects of (a) dropletinteractions, (b) relative velocity between the liquidand gas phases, and (c) droplet preheating are notconsidered.

DROPLET SIZE DISTRIBUTI ON IN SPRAYSThe size distribution of droplets in sprays produced

by diesel engine, gas turbine and other injectionsystems may be correlated by the general expression[8, 9, 151:

dn- = ax exp[b.u;] dx,nwhere n is the total number of droplets and dninrepresents the fraction of the number of droplets of adiameter between x and x + dx. Constants i: and ;Jareexperimentally determined parameters that depend onthe type of injector used. The coefficient a can beevaluated from the total number of droplets, i.e.

Thus,n= dn (2)

(3)where the Gamma function is defined as:

r(k) = xk- exp( - x) dx. (4)0

The coefficient b can be expressed in terms of acharacteristic mean diameter of the polydispersespray. In the field of spray evaporation and combus-tion, droplet mean diameters of primary importanceare: the linear mean diameter Xio, the surface meandiameter xZo, the volume mean diameter Xjo, and theSauter mean diameter .&. These diameters are definedby the general formula

ilj-[i:xidn:?^:xdn]llil (5)

Arbitrarily choosing the Sauter mean diameter as thecharacteristic mean diameter of the spray droplets, thecoefficient b is related to this mean diameter by theexpression :

Using equations (3) and (6), the coefficients a and b ofequation (1) are eliminated to give :dn-n

= jdy,,. (7)


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The evaporation of poiydisperse dilute sprays 1915

where yXZ = x/5?,,. In equation (7) the diameter of thedroplets was normalized by the Sauter mean diameter.Similar expressions may be obtained using one of theother mean diameters of the spray to normalize thedroplet diameter.

The droplet volume distribution can be obtainedfrom the droplet number distribution. Assumingspherical droplets, the volume of droplets of diametersbetween x and x + dx is

dV= zx3dn = f anxLf3exp[-bx] dx. (8)The total volume of the droplets in the spray is

Using equations (6) and (9) in (8), the volume fractionof droplets of a certain size range may be expressed interms of y,, as

dY32. (10)

It is apparent from the above equations that for a giveninjector (i.e. given values of c and 7) the number andvolume distributions of droplets is determined byeither the Sauter mean diameter or any other meandiameter of the spray.

The number distributions and volume dist~butionsfor various values oft: and y are shown in Figs. 1 and 2respectively.

A MODEL OF SPRAY EVAPORATION~overni~~~ equations

Letf(x, t) be the number of droplets of diameter x attime t. Assuming no interaction between particles,such as coagulation of particles, and that evaporationis the only process that the droplets undergo, then abalance of the number of particles of diameter x attimes t and t + dt gives

af k a f- - *dt= - dt c?x 0 1)Equation (11) is a limited form of the genera1 sprayequation preented in [9]. To proceed with its sol-ution, the variation of the diameter of the dropletswith respect to time must be specified. In the presentstudy, the vaporization process of individual dropletsis described by:

[x(01k+1 = [X(0)-p+ - IL (12)

2.5n

0 05 1 1.5 2 2.5Dimensionless Droplet Dmneter, ~32

FIG 1. Number distributions of spray droplets for variouscombinations of the spray constants E and y.Consequently

dx I,dt=-(k+l)Xk

and(13)

(14)where 2 is the coefficient ofevaporation and 0 5 k 5: 1.For diffusion-controlled evaporation, k = 1 [ 16, 173.This is known as the &-law because the square of thediameter of the droplet varies linearly with time [seeequation (12)]. In this case the rate of change of thediameter of the droplet is inversely proportional to itsdiameter, whereas the rate of change of the volume ofthe droplet is directly proportional to its diameter. Forradiation-controlled evaporation of opaque droplets,k = 0. In this case, the,diameter of the droplet varieslinearly with time; consequently the rate of change ofthe diameter of the droplet is constant. On the otherhand, the rate of change of the volume of the droplet isproportional to the square of the diameter.

The assumptions underlying equation (12) are :(1) The evaporation is quasi-steady.(2) The droplet consists of a single-eminent fuel.(3) The evaporation of one droplet is not influencedby other droplets.

1.5!1~~o~~~,,

0 1 2 3 4Dimensionless Droplet Diameter. ~32

FIG. 2. Volume distributions of spray droplets for variouscombinations of the spray constants c and y.


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1916 A. C. ALKIDAS(4) The environmental conditions (i.e. pressure,

temperature and gaseous species concentrations) aresteady and uniform.and

0 = [Xk+l + j.t]l.(k+l)(18)

(5) The convective processes are negligible. Using equation (18), equation (16) is transformed toFor steady conditions the evaporation constant i. isconstant. For example, for diffusion-controlled evap-

oration with constant properties, the coefficient ofevaporation i, is given by [9] :

!Y= 0,d6Hence the solution is

f(x, t) = &a) = 4([xk+ + i.t]l CL+1). (20)The unknown function $(cr) is determined from theinitial condition, i.e. at

t = 0, 4(a) = fk 01. (21)Thus the time dependence of the size distribution of anevaporating spray with an initial size distributiongiven by equation (1) is

8Ki. = --ln(1 + B7.)C,P,. (15)where the transfer number B, is given by:B,=

C,U, - T,lh /cl for evaporation withoutcombustion

C,( T II _ TL) + Q, Y,7 for evaporation withh , combustion.fsThe effects of interactions among droplets on their

combustion characteristics have been studied by nu-merous investigators [18-261. Experimental studieson suspended droplets [18, 19, 20, 211 and arrays offreely falling droplets [22,23] found that the d*-lawis still applicable; however the coefficient of evap-oration is significantly reduced. These results weretheoretically substantiated by Brzustowski et al. [24],Umemura et al. [25] and Labowsky [26]. The lastauthor concluded that, while not rigorous, the d*-law does approximate the burning history of aninteracting droplet, especially during the initial periodof combustion.

The conditions of individual burning of singledroplets versus collective or group burning of thedroplets were studied by Suzuki and Chiu [27],Labowsky and Rosner [28] and Chiu and Liu [29].These theoretical studies indicated that individualdroplet burning is encountered rarely in spray com-bustors. Qualitative experimental confirmations ofthese results were obtained by Chigier and McCreath[30], Twardus and Brzustowski [31] and Sangiovanniand Dodge [32]. The last study showed that groupcombustion is sustained even at droplet spacings of theorder of 25 droplet diameters.

Substituting equation (13) in (11) yieldsr_f 2 a!-= 0,~_____?t (k + 1)~~ dx (16)

The initial condition based on the general droplet sizedistribution of equation (1) isf&O) = an#exp[ - bx;]. (17)

SolutionEquation (16) subjected to the initial condition (17)may be solved by using the following independentvariable transformations:6 = [x+ l + (1 + I)t]k+

f(x,t) = anJx+ + jJ]r(k+l)x exp[ - b(xk+ + %t)(+ 1. (22)

Using equation (22), the variation with time of severalpertinent spray parameters, such as the number andvolume of unevaporated droplets of the spray and thevarious mean diameters of the spray, can be evaluatedas follows :

n(t) = J f(x, W, (23)0V(t) = i J x3f(x, t)dx, (24)0.iI

I, 11 (i-j)xf(x. t)dx (25)xii(t) = J9 xf(x, t)dx0The generalized integral H(a, b, c, z)Evaluation of the above spray parameters requiresthe evaluation of an integral of the form:sII= Xs[Xk+ 1 + )bt]O.+ 1)0

x exp[ - b(xk+ + &)Tak+l)] dx (26)

where s is an integer equal to 0, 1, 2 or 3.Defining

andu E b(Xk+1 + lbt)?(k+1) _ b(j.t)(k+ 1)

_, c b(Rt);!k+1)(2

equation (26) becomes

I=

7)

s-k k+i:+lY

(28)


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The evaporation of ~lyd~s~r~ dilute sprays 1917where the function H is defined* as:H(a, b, c, z) = exp( - z)

x s 1 [(u + z) - z]* (u + z)c- exp( - w)du. (29)The temporal variation of the linear mean diameter,surface mean diameter, volume mean diameter, Sautermean diameter, and the number and volume ofunevaporated droplets in the spray are then given by:

i-k k+i:+i ,r z/k+l j-k k-i-i:+1 \Pk-tlY Y lz/

-k k+i:+lY

l(i- j)

3-k k+c+ly

(32)where z may be expressed as a function of the initialSauter mean diameter of the spray, 2s2(O), as:

where

Numerical evaluation of H(a, b, c, z)For a given initial spray droplet distribution (i.e.specified values of I:and Y)and a given evaporation law(i.e. either k = 0 or k = l), the integral H(a, b, c, z) wasevaluated for various values of z and consequently thedimensionless evaporation time ~~~ The integrationwas performed using Rombergs numerical method[33]. Because the upper limit of integration is in~~ite,the integral was evaluated for successively larger upperlimits until the values of two successive integrationsagreed within a tolerance of 0.01%.

RESULTS AND DISCUSSIONTypical temporal variations of the number distri- 0 0.5 1 1.5bution and volume distribution of droplets in an Dimensionless Droplet Diameter, y32

*The definition of the integral N(a, b, C,z) differs slightlyfrom the definition given by Williams [9].

evaporating spray are shown in Figs. 3 and 4 re-spectiveiy. The constants of the initial distribution are:I: = 1.0, y = 1.0. Evaporation is assumed to obey thed*-law (i.e. k = 1.0). Figure 3 shows that as the timeof evaporation progresses, the maximum in the num-ber distribution moves towards smaller droplet dia-meters. The size of droplets of maximum number-fraction, yy2, is given by

- 32ldkt 1)

(35)

Thus, in the present case (i.e. E = 1.0, y = 1.0,k = 1.0) the mode occurs at yyz = 0 when 232 2 0.0625.The shift of the mode towards smaller diameters asevaporation progresses indicates that the diameters ofthe smaller droplets decrease faster than the diametersof the bigger droplets. This is expected, since the rate ofchange of the diameter of a droplet is inverselyproportional to its diameter [see equation (13) fork = 13.On the other hand, the maximum of the volumedistribution moves towards larger droplet sizes asevaporation progresses. This results because the rate ofchange of the volume of a droplet evaporating accord-ing to the @-law is proportional to its diameter.

The temporal variations of the number and volumeof unevaporated fuel droplets and of the variouspertinent mean diameters of the spray droplets are

-I2

Frc. 3. Temporal variations of the number distribution of aspray (c = 1.0, y = 1.0) undergoing diffusion-controlledevaporation.


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1918 A. c. ALKIDAS

0 1 2 3 4Dwnens~~nless Droplet Diameter, y32

FIG. 4. Temporal variations of the volume dis~~bution of aspray (1:= 1.0, 7 = 1.O) undergoing diffusion~ontrolledevaporation.

shown in Fig. 5 for the case of diffusion-controlledevaporation of a spray of initial droplet size distri-bution defined by: x = 1.0, y = 1.0. These parameters,which characterize the evaporating polydispersespray, are normalized by their initial values. As expected,Fig. 5 shows that the number and volume of un-evaporated fuel droplets decrease with increasingevaporation time. On the other hand, the variousmean diameters of the droplets, despite an initial smalldecrease in their vaiues (with the exception of theSauter mean diameter), increase with increasing evap-oration time.

Figure 6 shows the temporal variations of theparameters corresponding to Fig. 5 for the case ofdiffusion-controlled evaporation of a spray havinginitial droplet size distribution defined by E = 0.0,y = 1.0. In this case the various mean diameters of thedroplets increase monotonically with increasing evap-oration time.The general tendency of the mean diameters of thespray droplets to increase in value during most of theevaporation period is partly a consequence of the d-law of evaporation, which results in the smallerdroplets decreasing in size at a faster rate than thebigger droplets. The temporal variation of the mean

----0 I -__ ____0 1 2 3 4Dlme~~~~nlass Evaporation Ttme. 132FIG. 5. Temporal v;iriations of the number and volume ofunevaporated droplets and of the various mean diameters of aspray undergoing diffusion-controlled evaporation. (Initialsize distribution: F = 1.0, y = 1.0.)

diameters of the droplets also depends on the initialsize distribution. For example, in the limiting case of amonodisperse spray, the mean diameter of the dropletsdecreases monotonically with increasing evaporationtime. In this regard, Dickinson and Marshall [12]concluded that for diffusion-controlled evaporationthere is a general tendency for the average diameter ofthe remaining droplets to increase in the moderately orhighly nonuniform sprays and to decrease in the moreuniform sprays.

In the case of radiation-controlled evaporation ofsprays (k = 0), the temporal variations of their per-tinent parameters are shown in Fig. 7 for an initial sizedistribution defined by I: = 1.0, 1= 1.0.

The radiation-controlled evaporation law dictatesthat all droplets, independent of size, decrease in size atthe same rate. Consequently it is expected that ingeneral the mean diameters of the droplets decreasewith increasing evaporation time. The initial distri-bution defined by i: = 0.0, ; = 1.0 is a special case. Inthis case, at any instant during the evaporation periodthe number fraction of unevaporated droplets is equalto their volume fraction and the mean diameters of thedroplets are time-invariant.

In spray combustion, one of the most importantparameters to study is the amount of fuel evaporatedas a ftinction of time. Often the evaporation process isthe rate-determining step in the combustion of liquidfuels. In this case the rate of evaporation is equal to therate of combustion.In many processes involving polydisperse sprays,such as evaporation and combustion of sprays, it ismore desirable to work with the mean diameters of thedroplets than with the complete droplet size distri-bution. As was stated earlier, in the field ofevaporationand combustion of sprays the important mean dia-meters of the spray droplets are the linear meandiameter, the surface mean diameter. the volume meandiameter, and the Sauter mean diameter. The questionthus arises as to which of these mean diameters. if any,may adequately characterize the evaporation of thespray.

Dimensionless Evaporation Time, r32FIG. 6. Temporal variations of the number and volume ofunevaporated droplets and of the various mean diameters of aspray undergoing diffusion-controlled evaporation. (initialsii distribution: E = 0.0, y = 1.0.)


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The evaporation of polydisperse dilute sprays 1919

-~--..-_____.----------. . .._.._.__.__..___Fn ___

0 05 1 1.5 2Dlmenstonless Evaporatmn Time, T32

FIG. 7. Temporal variations of the number and volume ofunevaporated droplets and of the various mean diameters ofaspray undergoing radiation-controlled evaporation. (Initialsize distribution: c = 1.0, y = 1.0.)

To resolve this question, the evaporation rates offour sprays of different initial size distribution werecompared for four cases. In each case the initial valueof one of the pertinent mean diameters of the spraydroplets was kept the same. For example, in the firstcase the evaporation of the sprays was compared

Dimensionless Evaporation Time, rlo

Dlmenslonless Evaporation Ttme. rzo

c-0.5

02

keeping their initial-droplet linear mean diameters thesame, whereas in the second case the initial surfacemean diameters were kept the same.

These comparisons at constant mean diameters canbe accomplished by considering the volume fraction F,of unevaporated droplets of each of four sprays as afunction of dimensionless time, which is based on theinitial mean diameter. Thus, a plot of F, vs 7to for thefour sprays considered shows a comparison of theevaporation of the sprays at constant linear meandiameter 5,,. Figure 8 shows comparisons of theevaporation of sprays of various size distributions forthe same initial-droplet linear mean diameter, surfacemean diameter, volume mean diameter and Sautermean diameter respectively. These are plots of thevolume fraction F, of unevaporated fuel droplets as afunction of the corresponding normalized evaporationtime for various combinations of the size distributionconstants I: and y. In all cases shown, evaporationobeys the d-law, i.e. diffusion-controlled evap-oration prevails.

To aid with the subsequent discussion, Table 1shows the ratios of each of the initial mean diametersto the initial Sauter mean diameter for the com-binations of values of c and y considered in Fig. 8. Also

0 4 8 12 16Dimensionless Evaporation Time, r30

0 1 2 3 4Dlmenslonless Evaporation Time. 32

Yl-1 ___ _ ___1 ----__1 P__

FIG. 8. Comparisons of the diffusion-controlled evaporation of sprays of various initial size distributions.(a) Same initial linear diameters; (b) same initial surface mean diameters; (c) same initial volumemean diameters; (d) same initial Sauter mean diameters.


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Table IX10 .f20Y X30- _I- -232 X32 x32

-0.5 1.0 O&IO 0,346 0.4930.0 1.0 0.333 0.471 0.6061.0 1.0 0.500 0.612 0.721Data2&f 1.0 0.6CO 0.693 0.7830.21 0,3t 0.46

tabulated are the ~orr~s~ondi~g ratios compiled froma large number of experimental measurements onvarious sprays at various ~nv~ronmEntaI conditionse341,Comparison of Fig. 8 shows that the best correlationof the evaporation of the four different sprays isobtained by keeping their initial-droplet Sauter meandiameters the same [Fig. 8(d)]. In this case, thecalculated volume fraction of unevaporated fuel drop-lets at any given time of evaporation is not appreciablyinfluenced by the initial distribution within the spray.This is especially true during the early stages ofevaporation, where the volume fraction of unevap-orated droplets may be considered nearly ~nde~ndento~~n~tial-droplet size distribution within the spray. Theinfluence of the initial distribution on the amountevaporated is strongest during the final stage ofevaporation.

On the other hand, it is apparent from Figs. 8(a), (b)and (c)that the evaporation rates of the various spraysof the same initial-droplet linear mean diameter,surface mean diameter or volume mean diameter varyconsiderably for different values of the size distributionconstants I: and y. These combinations of the sizedistribution constants result in different refationsbetween the corresponding initial mean diameters(Z,,, .&,, x,,) and the initial Sauter mean diameterI,,, as shown in Table 1. It is apparent, furthermore,that the lower the initial Sauter mean diameter of thedroplets, the faster its evaporation rate.

It may therefore be concluded that the diffusion-controlled evaporation of a spray is primarily in-fluenced by its initial-droplet Sautcr mean diameterand to a much lesser degree by the other pertinentmean diameters of the droplets.In the case of radiation-~onteolled evaporation ofsprays, Fig. 9 shows the evaporation schedules forsprays in which one of the initial-droplet mean dia-meters of the sprays is, in turn, kept constant. Thetrends of these results are the same as in the case ofdiffusion-controlled evaporation. However, in the caseof radiation-controlled evaporation it appears thatwhen the initial Sauter mean diameters of the dropletsare the same [Fig. 9(d)], evaporation rate is practicallyindependent of initial-droplet size distribution. Thusthe radiation-controlled evaporation of sprays is solelyinfluenced by initial-droplet Sauter mean diameter. Asin the case of diffusjon-controlled evaporation ofsprays, a lower initial Sauter mean diameter causes ahigher rate of evaporation of the spray.

Consider now the representat~un of a polydispersespray by a monodisperse spray of droplet size equal tothe initial Sauter mean diameter of the polydispersespray droplets. The temporal variation of the volumeof unevaporated droplets of the monodisperse spray isgiven by:

V(r) Mt) 3vco> = X&z?)[ 1_I_ = [f -T&~? (36)

Tables 2 and 3 compare the variations of the volumefraction of unevaporated droplets with time, calcu-Jated from the above monodisperse model, with thecorresponding results of two polydisperse sprays forthe cases of diffusion-controlled and radiation-controlled evaporation respectively. The two polydis-perse sprays considered conform to theNukiyama-Tanasawa distribution (X= 2, 7 = I.0)and the Hiroyasu-Kadota distribution (i: 1=:0, y =1.0). In addition, Table 2 includes results obtainedusing Tanasawas empirical expression for burning of aspray (see ~ipinsk~ et al. [35]). The volume fractionof evaporated (unburned) droplets given byTanasawas empiricaE expression is:

Table 2 shows that the monodisperse model based onthe initial Sauter mean diameter of the spray givesgood agreement during the early stages of evap-oration This agreement degenerates, however, asevaporation proceeds. Theerror in the volume fractionof unevaporated droplets calculated by the monodis-perse model increases with increased evaporationtime. When 40% of the volume of droplets hasevaporated, the error of the monodisperse model isabout 200/,. Bracco [X] found that a monodispersespray model using an initial droplet size of 0.78 E@)gave acceptable results for the combustion of anenthanol spray in a constant-area rocket motor. Thisspray was assumed to have a Nukiyama-Tanasawa(E = 2.0, y = 1.0) initial size distribution. Note that forthis distribution, the initial Sauter mean diameter,X&O), is equal to 0.78 ~~~(0) (see Table 1).

In the case of radiation-controlled evaporation,shown in Table 3, the monodisperse model agreesmuch better with the evaporation of the poIydisper~sprays than in the d~~usion~ontroIIed case. Here theerror in the volume fraction of uneva~rat~d dropletscalculated with the monodisperse model based on theinitial Sauter mean diameter of the spray droplets doesnot reach 20% until about 70% of the volume of thedroplets is evaporated.

CONCLUSIONS(1) The rate of evaporation of a polydisperse sprayis primarily influenced by the initial-droplet Sauter

mean diameter and to a much lesser degree by theother initial-dro~~t mean diameters of the spray.(2) For disunion-~ontroIled evaporation, compa-


9/11

The evaporation of polydis~r~ dilute sprays 1921

Dlmenswnless Evaporation Time. r20

f-0.5

0

2

I I J0 1 2 3 4 5Dimenstonless Evaporarron Tune, ~3~

W

0 0.5 1 1.5 2Dimensionless Evaporatwn Time, r32

Frc;. 9. Comparisons of the radiation-controlled evaporation of sprays of various initial size distributions.(a) Same initial tinear mean diameters; (b) same initial surface mean diameters; (c) same initial volumemean diameters; (d) same initial Sauter mean diameters.

Table 2. Diffusion-controlled evaporation of polydisperse sprays using a monodisperse spray modelVolume fraction of unevaporated droplets, F,Nukiyama-Tanasawa Hiroyasu-Kadota MonodisperseEvaporation time distribution distribution Tanasawas model

532 ,: = 2, ) = 1 1;= 0, / = 1 model x(0) = GAO)0.05 0.938 0.931 0.918 0.9260.10 0.879 0.871 0.857 0.8540.20 0.770 0.769 0.783 0.7150.30 0,675 0.685 0.717 0.5860.40 0.593 0.615 0.660 0.4650.50 0.522 0.554 0.609 0.3540.60 0.46 L 0.502 0.564 0.2530.70 0.407 0.456 0.524 0.1640.80 0.361 0.416 0.486 0.0890.90 0.320 0.380 0.452 0.0321.00 0.285 0.348 0.421 0.0001.20 0.226 0.295 0.3641.40 0.181 0.251 0.3151.60 0.146 0.216 0.2721.80 0.118 0.186 0.2342.00 0.096 0.162 0.2003.00 0.006 0.084 0.074


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1922 A. C. ALKIDASTable 3. Radiation-controlled evaporation of polydisperse sprays using a monodispersespray model

Volume fraction of unevaporated droplets, F,Nukiyama-Tanasawa Hiroyasu-Kadota MonodisperseEvaporation time distribution distribution model

T32 ,: = 2, y = 1 i: = 0, y = 1 x(0) = &(O)0.050.100.200.300.400.500.600.700.800.901.001.201.401.601.802.00

0.8590.7350.5330.3820.2710.1900.1320.0910.0620.0420.0290.0130.0060.0020.0010.000

1.2.

3.

4.

5.6.

7.

8.9.

0.8610.7410.5490.4070.3010.2230.1650.1220.0910.0670.0500.0270.0150.0080.0040.002

0.8570.7290.5120.3430.2160.1250.0640.0270.0080.0010.000

rison of the evaporation rate of a polydisperse spray 10.with that of a monodisperse spray with droplet sizeequal to the initial-droplet Sauter mean diameter ofthe polydisperse spray shows reasonable agreement. 11.This agreement, which is excellent during the earlystages of evaporation, deteriorates as evaporation 12.proceeds.

(3) For radiation-controlled evaporation (opaque 13.droplets), the representation ofa polydisperse spray bya monodisperse spray of droplet size equal to the 14.initial-droplet Sauter mean diameter of the polydis-perse spray is an excellent assumption. 15.

REFERENCESP. Rosin and E. Rammler, Laws governing the fineness ofpowdered coal, J. Inst. Fuel 7, 29-36 (1933).S. Nukiyama and Y. Tanasawa, An experiment on theatomization of liquid (5th Report. The atomizationDattem of liquid by means of air stream), Trans. Sot.&ch. Engrs jopan 6(22), 5-7 (1940).H. Hirovasu and T. Kadota, Fuel droplet size distri-bution iA diesel combustion chamber, SAE Paper No.740715 (1974).H. Hiroyasu and T. Kadota, Fuel droplet size distri-bution in diesel combustion chamber, Bull. JSME19(135), 1064-1072 (1976).T. Kamimoto and S. Matsuoka. Predictions of sprayevaporation in reciprocating engines, SAE Paper-No.770413 (1977).R.P. Probert, The influence of spray particle size anddistribution in the combustion of oil droplets, Phil. Mag.37,94- 105 (1946).Y. Tanasawa and T. Tesima, On the theory of com-bustion rate of liquid fuel spray, Bull. JSME l(l), 36-41(1958).F. A. Williams, Progress in spray-combustion analysis,8th Int. Symp. on Combustion. pp. 50-69 (1962).F. A. Williams, Combustion Theory, Chapter 11, pp.250-261. Addison-Wesley (1965).

16.17.

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24.

R. D. Ingebo, Vaporization rates and drag coefficients foriso-octane sprays in turbulent air streams, NACA TN3265 (1954).R. D. Ingebo, Photomicrographic tracking of ethanoldrops in a rocket chamber burning ethanol and liquidoxygen, NASA TN D-290 (1960).D. R. Dickinson and W. R. Marshall. The rates ofevaporation of sprays, AIChG JI 14(4), g41-552 (1968).P. Flynn, An experimental determination of the m-stantaneouspotential radiant heat transfer, Ph.D. Thesis,University of Wisconsin (1971).J. C. Dent and S. L. Suliaman, Convective and radiativeheat transfer in a high-swirl direct injection diesel engine,SAE Paper No. 770407 (1977). _I. M. TishkoIT and C. K. Law. ADDlication of a class ofdistribution functions to drop:&; data by logarithmicleast-squares technique, Engineering for Power 99(4),684-688 (1977).G. A. E. Godsave, Studies of the combustion of drops in afuel spray -The burning of single drops of fuel, 4th Int.Symp. on Combustion pp. 818-828 (1953).M. Godsmith and S. S. Penner, On the burning of singledrops of fuel in an oxidizing atmosphere, Jet Propulsion24, 245-251 (1954).J. F. Rex, A. E. Fuhs and S. S. Penner, Interference effectsduring burning in air for stationary n-heptane, ethylalcohol and methyl alcohol droplets, Jet Propulsion 26,179-187 (1956).J. Kanevsky, Interference during burning in air for ninestationary fuel droplets arranged in a body-centeredcubic lattice, Jet Propulsion 26, 788 (1956).N. V. Fedoseeva, Combustion of a system of liquid fueldrops, Adv. aerosol Phys. 2, 110-l 18 (1972).N. V. Fedoseeva, Kinetics of evaporation of a dropletsystem, A& aerosol Phys. 3, 35-42 (1973).A. S. M. Nuruzzaman, A. B. Hedley and J. M. Beer.Combustion rates in self-supporting- flames on mono-sized droplet streams, J. Inst. Fuel 43, 301-310 (1970).A. S. M.Nuruzzaman, A. B. Hedley and J. G. Beer,Combustion of monosized droplet streams in stationaryself-supporting flames, 13th Int. Symp. on Combustion pp.787-798 (1971).T. A. Brzustowski, et ul. Interaction of two burning fueldroplets of arbitrary size, A.I.A.A. JI 17(11), 1234-1242(1979).


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The evaporation of polydisperse dilute sprays 192325. A. Umemura, S. Ogawa, N. Oshima and K. Shintani,Theoretical analysis of the interaction between twoburning fuel droplets, Bull. LSME 23(186), 2071-2075(1980).26. M. Labowsky, Calculation of the burning rates ofinteracting fuel droplets, Cornbust. Sci. Technol. 22,217-226 (1980).27. T. Suzuki and H. H. Chiu, Multi-droplet combustion ofliquid propellants, Proc. 9th fnt. Symp. Space Technol.

sci. pp. 145-154 (1971).28. M. Labowsky and D. E. Rosner, Groupcombustion ofdroplets in fuel clouds, 1. Quasi-steady predictions,Evaporation -Contbust io~ ofFuels, Adv ances in chemist rySeries 166, pp. 63-79 11978).

29. H . W . Chiu and T. M. Liu, Group combustion of liquiddroplets, Cornbust. Sci. Technol. 17, 127-142 (1977).30. N. A. Chigier and C. G. McCreath, Combustion ofdroplets in sprays, Acta nstronautica 1,687-710 (1974).31. E. M. Twardus and T. A. Brzustowski, An experimentalstudy of flame spread and burning in arrays of monosize

hydrocarban droplets, Cornbust. Sci. Technol. 17,215-225 (1978).32. J. J. Sangiovanni and L. J. Dodge, Observations of flamestructure in the combustion of monodispersed dropletstreams, 17th Int. Symp. on Combustion pp. 455-464(1978).33. H. S. Wilf, Advances in numerical quadrature, in Ma-

thematical Methodsfor Digital Computers, Vol. 2, (Editedby A. Ralston and J. S. Will), pp. 133-144, John Wiley(1967).34. D. R. Ballal and A. H. Lefebvre, Ignition and flamequenching of flowing heterogeneous fuel-air mixtures,Combustion and Flume 35, 1.5-168 (1979).

35. J. Shipinski, P. S. Myers and 0. A. Uyehara, A spray-droplet mode1 for diesel combustion, Proc. Itlstn me&.Ettgrs 184, 28-35 (1970).36. F. V. Bracco, Applications of steady-state spray equa-tions to combustion modeling, A.T.A.A. J/ 12,1534-1540(1974).

INFLUENCE DES PARAMETRES DE DISTRIBUTION DE TAILLE SUR LEVAPORATIONDES BROUILLARDS PQLYDISPERSESR&mm-Une ltude a~alytique Porte sur ~~vaporation controlee par la diffusion et par le rayonnement(goutteIettes opaques) de brouillards ~lydis~r~s dun composant unique. Les caract~ristiques globales del~va~ration pour des brouill~ds g di@rentes distributions initiales de taille sont mieux reprisentees par lediametre initial moyen de Sauter. Daus le cas de levaporation controlie par la diffusion, la representationdun brouillard polydisperse par un brouillard monodisperse de gouttelettes ayant un diametre igal audiametre moyen de Sauter du brouillard polydispersc est une bonne hypothese pendant les premieres etapesde ltvaporation. Lorsque le volume des gouttelettes est evapori au taux de 40x, lerreur atteint 207; pour lemodele monodisperi. Dautre part, dans le cas de livaporation controlte par le rayonnement, le modele dubrouillard monodisperse donne de bons risultats pour presque toute la periode dlvaporation du brouillard.

DER EINFLUSS VGN TROPFENGRC)SSENVERTEILUNGS-PARAMETERN AUF DIEVERDUNSTUNG POLYDISPERSER VERDUNNTER SPRUHNEBELZusam~nfas~ng-us wurde eine anaiytische Untersuchung der durch Diffusion und such der durchStrahlung (trtibe Tr~pfchen) bedingten Verdunstung von poiydis~r~n ~inkomponenten-Spr~hne~~durchgef~hrt. Es stellte sich heraus, da8 sich das Verdunstungsverha~ten von Spr~hne~~n mit unter~h~edli-then ~f~~~opf~~~~n-verteilun~ am besten mi t dem anfiinglichen Sauter-Durchmesser korrelierenl&t. Fiir den Fall der durch Diffusion bedingten Verdunstung ist die Darstellung eines polydispersen durcheinen monodispersen Spriihnebel mit einer dem anftiglichen Sauter-Durchmesser des polydispersenSprilhnebels entsprechenden Tropfengriibe eine fur die Anfangsstadien der Verdunstung gut zutreffendeAnnahme.Wenn 40% des Tropfenvolumens verdunstet sind, erreicht der Fehler des monodispersen Modells 20:~.Andererseits erzielt man im Fall der durch St&lung bedingten Verdunstung mit dem monodispersenSprtihnebel-Model1 fur den Hauptteil des Verdunstungsvorgangs gute Ergebnisse.

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