Effect of Inter Particle Interaction

7/29/2019 Effect of Inter Particle Interaction

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Aerosol Science 34 (2003) 837857

www.elsevier.com/locate/jaerosci

Eect of inter-particle interactions on evaporation of dropletsin a linear array

Venkat Devarakonda, Asit K. Ray

Department of Chemical Engineering, University of Kentucky, Lexington, KY 40506-0045, USA

Received 29 May 2002; received in revised form 14 October 2002; accepted 14 October 2002

Abstract

Eects of interparticle interactions on evaporation have been examined on droplets in a linear array generated

by a modied vibrating orice aerosol generator (VOAG). The size as well as the spacing between the droplets

was varied by modulating the frequency of the generator, and the liquid ow rate through the orice. Ethanol

and methanol droplets of diering initial sizes and spacings were studied, and the instantaneous evaporation

rates as well as droplet temperatures were measured using a resonance-based light scattering technique. The

results show that at a xed dimensionless spacing between the droplets (i.e., the ratio of the spacing to

the droplet radius, l=a), the size and temperature versus time data are highly reproducible. The measured

evaporation rates were normalized by the evaporation rates of an isolated droplet under identical conditionsto obtain the interaction parameter () as a function of the dimensionless spacing between the droplets. The

value of was observed to decrease with a decrease in l=a as predicted by the theoretical models, and as

expected, found to be the same for ethanol and methanol droplets.

? 2003 Elsevier Ltd. All rights reserved.

Keywords: Multi-droplet evaporation; VOAG; Optical resonance

1. Introduction

Various industrial and atmospheric processes, such as evaporation and growth of cloud droplets,

spray drying and combustion, entail heat and mass transfer from assemblages of droplets. It is well

known that interactions among member droplets of an assemblage retard heat and mass transfer rates

between the droplets and their surrounding medium. Heat and mass transfer processes associated with

single isolated droplets have been the subjects of extensive examination in many theoretical ( Fuchs,

1959; Williams, 1975; Crespo & Linan, 1975; Chang & Davis, 1988; Law, 1982) and experimental

Corresponding author. Tel.:+1-859-257-7990; fax: +1-859-323-1929.

E-mail address: [email protected] (A.K. Ray).

0021-8502/03/$ - see front matter? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0021-8502(03)00065-X
mailto:[email protected]:[email protected]


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838 V. Devarakonda, A.K. Ray / Aerosol Science 34 (2003) 837 857

studies. In recent years light scattering techniques have been used on single droplets suspended in

electrodynamic balances to obtain high precision data on evaporation of isolated droplets in controlled

environments (Richardson, Hightower, & Pogg, 1986; Ray, Souyri, Davis, & Allen, 1991; Davis,

1997; Tu & Ray, 2001). Due to the complicated nature of inter-droplet interactions the resultsfrom single droplet studies cannot be used directly to model processes involving assemblages of

droplets. Therefore, a knowledge of interparticle interactions is essential for understanding a number

of processes involving droplet sprays and clouds.

A number of investigators have theoretically examined the eects of interparticle interactions

on evaporation and combustion of droplets. Umemura (1994) has reviewed the analytical models

available in the literature for predicting the extent of interactions in quasi-steady state processes,

while Annamalai and Ryan (1992) and Annamalai (1995) have provided extensive reviews of the

experimental and theoretical advances made in the study of interaction eects on droplet dynamics,

heat and mass transfer, ignition and combustion of droplet arrays. Most theoretical studies are limited

to two interacting particles (Brzustowski, Twardus, Wojcicki, & Sobiesiak, 1979; Umemura, Ogawa,& Oshima, 1981; Rajuand & Sirignano, 1990). Labowsky (1978, 1980) has developed a more general

treatment based on the method of images for analysis of quasi-steady evaporation and burning rates

of interacting droplets in an arbitrary array. Later, Ray and Davis (1980, 1981) and Marberry,

Ray, and Leung (1984) have presented a rigorous treatment of heat and mass transport between an

assemblage of particles and a surrounding medium. The results of theoretical studies show that the

extent of interparticle interactions depends on the geometrical parameters, such as the shape and

dimensions of the array, as well as on the size distribution of the droplets.

A number of investigators (Nuruzzaman, Hedley, & Beer, 1971; Sangiovanni & Kesten, 1977;

Sangiovanni & Labowsky, 1982; Silverman & Dunn-Rankin, 1994) have experimentally examined

the eects of interactions on evaporation, combustion and ignition of linear arrays of droplets. They

measured changes in size using photographic imaging techniques, and found that the combustion ratesof droplets in a linear array may decrease by as much as 50% compared to the rate of an isolated

droplet. Sangiovanni and Labowsky (1982) have observed that the extent of interactions increases

as the separation distance between droplets decreases, and applied the method of images technique

to estimate the interaction parameter for a linear droplet array. Konig, Anders, and Frohn (1987)

have used a more accurate technique, based on light scattering intensity distribution along rainbow

angles, to estimate the size and temperature changes of linear arrays of droplets. Even though this

technique is more accurate than the imaging method (i.e., size and refractive index are accurate

to about 2% and 4%, respectively), the applicability of this technique is restricted only to cases

where signicant changes in size and refractive index occur over a small period of time, such as in

high-temperature combustion. Chen, Mazumder, Chang, Swindal, and Acker (1996) have investigatedthe evaporation rates of ethanol droplets in single as well as closely spaced multiple linear streams

generated using a vibrating orice aerosol generator (VOAG). They have used a sizing technique

based on positions of output resonances observed in inelastic scattered intensity versus wave-number

shift spectra. Although the temperature of the droplets varied with time, they assumed a constant

refractive index (i.e., constant temperature) in the sizing technique, and compared the evaporation

rate of droplets in a linear array with the evaporation rate of an isolated droplet at the wet-bulb

temperature.

In addition to the geometrical parameters and droplet size, the evaporation rate of a droplet in

an array depends on the temperature and the concentration of vapor in the surrounding medium.


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V. Devarakonda, A.K. Ray / Aerosol Science 34 (2003) 837 857 839

In most of the prior studies on linear arrays of droplets, the heat and mass transfer processes have

been treated as quasi-steady, even though for typical time scales involved in the studies (i.e., under

a millisecond), the processes were unsteady (Devarakonda, Ray, Kaiser, & Schweiger, 1998). The

time-dependent droplet temperature has not been measured or taken into consideration in the previousstudies, although light scattering techniques can be applied for in-situ measurement of time-dependent

temperature (Vehring & Schweiger, 1992; Devarakonda & Ray, 2000) In addition to the temperature,

scant attention was given to the environment surrounding the droplets. If the vapor molecules result-

ing from evaporation of droplets are not removed instantaneously from the surrounding gas phase,

the measured evaporation rates may decrease considerably due to the partial saturation. Furthermore,

most studies examined interactions occurring at high separation distances between relatively large

droplets ( 200 m) that undergo shape distortions at high velocities involved, thus aecting heat

and mass transfer rates.

Recently, we have demonstrated that fast processes associated with microdroplets can be studied

in a linear stream of highly monodisperse droplets that traverse an identical path in a well-denedtime-invariant gas phase (Devarakonda et al., 1998; Devarakonda & Ray, 2000). We have used a

resonance-based light scattering technique to determine accurately the size and refractive index of

the droplets with high temporal resolution. The purpose of this study is to use the technique to

examine the eect of interparticle interactions on heat and mass transfer rates between droplets in a

linear array and the surrounding medium. Specically, we focus on the eects of interactions on heat

and mass transfer rates at low separation distances between droplets. We have generated linear array

of highly monodisperse droplets using a modied vibrating orice aerosol generator (VOAG) ( Lin,

Eversole, & Campillo, 1990; Devarakonda et al., 1998). For the time scale involved in this study (i.e.,

1 ms), the droplets evaporate under unsteady conditions, and thus, both the size and temperature

change simultaneously with residence time. We have determined the size and the temperature of the

droplets as functions of residence time from the resonances observed in elastic and Raman scattering,and calculated interaction parameters for both heat and mass transfer processes from the data. The

interaction parameters for a linear array of droplets depend on the droplet size and the centercenter

separation distance between successive droplets. We varied both these quantities by manipulating

the ow rate of liquid through the orice and the orice vibration frequency used for generation of

droplets.

2. Theory

The extent of interparticle interactions in a process involving momentum, heat or mass transfer ismeasured in terms of an interaction parameter, , which is dened as

=RarrayRiso

; (1)

where Rarray and Riso are, respectively, the transfer rate associated with a droplet in an array and

with a single isolated droplet of identical size and composition exposed to identical environmental

conditions. Even though prior experiments on droplets in linear arrays were conducted under un-

steady conditions on relatively fast moving droplets, most investigators have expressed the interaction


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parameter in terms of the quasi-steady evaporation rate of an isolated stationary droplet which is

given by the following relation when the Lewis number (Le) is equal to 1.0 (Marberry et al., 1984)

miso = 4Dgag ln

1 + Cpg(T

Tw)L

: (2)

Here Dg and L are the gas-phase diusivity and the latent heat of vaporization of the evaporating

component, g and Cpg are the density and the heat capacity of the gas phase, a is the droplet

radius, and T and Tw are the surrounding gas phase and the wet bulb temperatures, respectively.

The above relation assumes that the Lewis number is equal to 1.0. When Eq. (2) is used, Eq. (1)

may yield an interaction parameter, , value greater than unity for unsteady evaporation. To decipher

the eect of the presence of other droplets in the neighborhood of a droplet, one should compare

the evaporation rate of a droplet in an array with the evaporation rate of an identical isolated droplet

under identical conditions. The experiments of the present study involve unsteady evaporation of

ethanol and methanol droplets, and we shall compare our experimental results with predictions fromthe following model for unsteady evaporation of a droplet in an array.

Consider a droplet in a linear stream of monodisperse droplets that are composed of compo-

nent A. The droplet stream is exposed to a gas phase that is maintained at a temperature T.

Initially, the droplet has a radius a0, and a temperature T0. In the formulation of the problem we

assume that the gas phase convective processes can be described by heat and mass transfer coe-

cients. The temperature prole in the droplet is given by the following partial dierential equation

(Ray, Huckaby, & Shah, 1987)

LCpL@TL

@t=

kL

r2@

@r

r2

@TL

@r

; 06 r6 a(t); (3)

where L; CpL, and kL are the density, the heat capacity and the thermal conductivity of the droplet,

respectively, and TL is the droplet temperature at the radial position r, at time t. The following

boundary conditions apply:

@TL

@r(0; t) = 0; (4)

TL(a; t) = Ts(t): (5)

An interfacial mass balance gives the droplet evaporation rate as

2Ladadt

=mShDgMA

P0A(Ts)

RTs

pA

RT

; (6)

where Sh is the Sherwood number, MA is the molecular weight of component A; P0

A(Ts) is the vapor

pressure of A at the droplet surface temperature Ts, and pA is the partial pressure of A in the

surrounding gas phase. Similarly, an interfacial energy balance yields the following relation for the

droplet surface temperature:

2akL@TL

@r

r=a

= hNu kg(T Ts) + 2LLada

dt; (7)


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0 0.2 0.4 0.6 0.8 1

Radial position,r/a

276

280

284

288

292

Drop

lettemperature,TL

(K)

t=0.48 ms

t=0.95 ms

t=1.43 ms

t=2.39 ms

t=3.81 ms

t=6.0 ms

Parameter values:

a0=10m, T0=T=298 K, Sh=4.24, Nu=3.86

Fig. 1. Temperature distributions inside an isolated ethanol droplet at various times.

where Nu is the Nusselt number, and kg is the gas-phase thermal conductivity. In the present study,

the relation provided by Davis and co-workers (Zhang & Davis, 1987; Tain & Davis, 1987) is

employed for the estimation of Sh and Nu. The interaction parameters, m and h, are used in

Eqs. (6) and (7) to describe the reduction of gas-phase heat and mass transfer rates due to the

presence of other particles. Eq. (6) shows that when the instantaneous evaporation rate and the

surface temperature of a droplet are measured simultaneously in an experiment, m can be directly

calculated from the experimental data, and then h can be estimated from the size and temperaturehistory of the droplets. In the absence of droplet surface temperature data, the measured droplet size

versus time data must be matched with the values calculated from Eqs. (3) to (7) to obtain the

interaction parameter, , by assuming m = h = .

To elucidate the salient characteristics of unsteady evaporation, in Fig. 1, we have plotted tem-

perature proles inside an isolated droplet (i.e., h = m = 1:0) at various times. The results were

computed for an ethanol droplet with typical values involved in our experiments (i.e., a0 = 10 m,

Sh = 4:2, and Nu = 3:9). The results show that at short residence times, the temperature prole in

the droplet exhibits a sharp gradient near the surface. After about 4 ms, the temperature gradient

disappears, and the droplet attains a nearly uniform temperature. Finally, the droplet temperature


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SPEX

1403

MONO

CHRO

MATO

R

PMT 1

PMT 2

MOTORIZED VERTICALTRANSLATION STAGE

BEAMTRAP

DROPLETCHAIN

CONSTANTPRESSURE

TANK

LIQUIDRESERVOIR

FREQUENCYSYNTHESIZER

ELECTRONICS

VOAGAr-ion LASER BEAM

E

Dry Air

DRY AIR

FLOW CONTROLLER

Fig. 2. Schematic of the experimental system for generating a linear stream of monodisperse droplets.

reaches the steady-state wet-bulb temperature after about 6 ms. The time taken for a droplet to attain

a steady state increases as the droplet size increases.

Most of the experimental investigations on interactions in linear arrays were conducted on droplets

larger than 50 m, and the time needed for such droplets to attain a steady state is about 150 ms, thatis considerably longer than the short residence times involved in the experiments. The droplet surface

temperature decreases with time from the initial maximum value due to evaporative cooling, and as

a result, the evaporation rate decreases with time. When the droplet attains a steady-state temperature

(i.e., wet-bulb temperature Tw), the evaporation rate attains a constant value which can be calculated

from Eq. (2) for Le = 1. During the unsteady period, the evaporation rate of an interacting droplet

may exceed the steady-state evaporation rate of an isolated droplet, thus yielding an interaction

parameter greater than unity when the rate is normalized by the steady-state evaporation rate. It

should be noted that the surface of an isolated evaporating ethanol droplet initially may be cooler

by as much as 3C compared to the droplet core, but inter-particle interactions signicantly retard

heat and mass transfer rates associated with a droplet in a linear array. Thus, the dierence betweenthe core and surface temperatures of an interacting droplet is expected to be less than that of an

isolated single droplet.

3. Experiments

Linear streams of highly monodisperse ethanol and methanol droplets were produced with a

modied vibrating orice aerosol generator. A schematic of the experimental system is shown in

Fig. 2. Devarakonda et al. (1998) and Devarakonda and Ray (2000) have described the experimental


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system in detail. We shall recapitulate some of the salient features here. The droplet generation sys-

tem is housed inside an environmentally controlled cylindrical chamber through which a gas stream

of desired composition ows past the droplets that descend along the axis. The droplet stream is

illuminated by a polarized Ar-ion laser beam at a xed position of the chamber. Light scattered elas-tically by the droplets is detected by two photo-multiplier tubes, located along polarization angles of

= 00 (PMT1) and = 450 (PMT2), and Raman scattered light is collected into a double mono-

chromator. The orice assembly is mounted on a programmable motorized vertical translation stage,

and can be positioned accurately to any location from 1 to 10 mm distance from the laser line. Each

droplet experiences the same history, and the axial distance from the generation point is uniquely

related to the residence of the droplets in the chamber environment. Therefore, by examining droplets

at various distances the variation of a droplet with time can be deciphered.

The size of the droplets generated by a modied vibrating orice aerosol generator is given by

a = 3Q

4f1=3

; (8)

where Q is the volumetric ow rate of liquid through the orice vibrating at a frequency f. The

droplet generation system is highly stable; for example, under typical operating conditions, the in-

stantaneous uctuations in the droplet size is about 1 part in 105, and the fractional long-term drift

in the droplet size is about 105 per min. The main objective of the present study is to examine the

eect of the nondimensional spacing, l=a, between the droplets on the reduction in the evaporation

rate. The parameters that inuence the size (i.e., a) and spacing (i.e., l) between the droplets gener-

ated by a VOAG are the liquid ow rate Q, and the vibration frequency of the orice f. The ow

rate of liquid through an orice depends on the pressure dierence across the orice. In a modied

VOAG, there is only a range of backing pressures over which stable droplets can be generated. In

the present study, backing pressures of 1.65 bar (24 psi), 1.93 bar (28 psi) and 2.35 bar (34 psi)were used to generate ethanol droplets, and a backing pressure of 1.17 bar (18 psi) was used for

methanol droplets.

The velocity v of droplets generated by the VOAG is related to the separation distance l, between

two successive droplets through the relation v = lf. An increase in the backing pressure has two

eects: the droplet size (a) increases due to an increase in the ow rate (Q); and the distance

between successive droplets increases due to an increase in the velocity of the droplets. It was

found experimentally that the velocity of the droplets remains almost constant over a long range of

frequencies. Since the liquid ow rate remains constant at a given backing pressure, an increase in

the frequency results in a decrease in both the droplet size and the spacing between the droplets.

Various combinations of frequencies and backing pressures were used to vary the nondimensionalspacing, l=a.

In a typical experiment, the ow rate of liquid through the orice was maintained constant by

keeping the backing pressure constant. To prevent the build-up of vapor arising from evaporation

of droplets (i.e., to maintain pA = 0 in Eq. (6)) a dry air stream at a rate of 1000 ml=min was

owed past the droplets. It was found that the air stream ow rate above 500 ml=min has no eect

on the evaporation rate of the droplets, but a ow rate below 400 ml =min causes a reduction in the

evaporation rate due to the partial saturation of the surrounding air. To examine the evaporation rate

of droplets as a function of the residence time the droplet generator was moved in steps from 1

to 7 mm distances from the laser beam. The velocity of the droplets was measured as a function


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of downstream distance from the droplet generator head using a diraction technique (Devarakonda

et al., 1998). The residence time of the droplets was estimated from velocity versus distance data,

and varied from 90 to about 650 s as the droplet generator was moved from 1 to 7 mm. At a

given position of the generator, the droplet size was varied by scanning the vibration frequency ofthe orice over a range of either 60 kHz (i.e., for long-frequency scans) or 20 kHz (for short scans)

in steps of 10 Hz in every 0:10 s. Elastic scattering intensities detected by PMT1 and PMT2 and

Raman scattering intensity at 2900 cm1 collected by the spectrometer were recorded simultaneously

as functions of the droplet generation frequency. The Raman shift of 2900 cm1 corresponds to the

CH bond stretch of ethanol and methanol molecules. The light scattering data from the long scans

were used to determine the absolute droplet size and the refractive index, and the short scans were

used to establish droplet size and refractive index changes with respect to the position where a long

scan was recorded.

4. Light scattering

Scattered intensity versus frequency data were analyzed using LorenzMie theory (van de Hulst,

1981; Kerker, 1983; Bohren & Human, 1983) to obtain droplet size and refractive index.

Devarakonda et al. (1998) and Huckaby, Ray, and Das (1994) have described the theoretical basis

as well as the data analysis procedure in detail. Here we provide a synopsis of the analysis proce-

dure. Under the illumination of monochromatic light of wavelength , the intensity of light scattered

elastically at a point by a homogeneous spherical particle of radius a, depends on the refractive

index of the particle m, and the size parameter, which is dened as x = 2a=. In a dielectric or

nearly dielectric spherical microparticle electromagnetic waves can propagate nearly loss-less, and at

specic size parameters values resonances are excited due to the trapping of electromagnetic waveby almost total internal reection near the surface. At resonances the intensity of light inside the

particle rises dramatically (i.e., the average internal intensity can exceed the incident light intensity

by a few order of magnitudes). This gives rise to sharp peaks in elastic and inelastic (e.g., Raman)

light scattering. The value of x, at which a resonance appears depends on m, and can be computed

precisely from the theory. An experimental scattering intensity versus size parameter spectrum shows

a number of resonance peaks. By identifying the observed resonances and by aligning them to com-

puted resonances the size and refractive index of the scattering particle can be obtained with high

precision (Huckaby et al., 1994; Devarakonda et al., 1998; Tu & Ray, 2001).

For a given ow rate of liquid through the orice, the relation in Eq. ( 8) can be expressed in

terms of the size parameter as follows:x = cf1=3; (9)

where c is a constant. The variation of frequency causes the size parameter of the droplets to

change according to the above relation, and as a result, experimental scattering intensity versus

frequency spectra exhibit a number of resonance peaks. Fig. 3 shows typical experimental elastic

and Raman scattering spectra obtained from ethanol droplets at 2.35 bar backing pressure. It should

be noted that the elastic scattering spectrum from PMT1 (i.e., top spectrum) contains only transverse

magnetic (TM) mode resonances, while the spectrum from PMT2 (i.e., middle spectrum) exhibits

both transverse electric (TE) and TM mode resonances. In addition, the Raman scattering spectrum


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200 210 220 230 240 250 260

Frequency, kHz

Scatteringintensity

Output from PMT 1

Output from PMT 2

Raman scattered intensity at 2900 cm-1 shift

Fig. 3. Raw experimental elastic and Raman scattering spectra obtained by varying droplet generation frequency.

reveals more resonance peaks than those appear in the elastic scattering spectra. This is because low

order (i.e., sharp peaks) resonances only appear in Raman scattering, but are suppressed in elastic

scattering due to a variety of factors.

An experimental intensity spectrum provides the frequencies at which resonances appear. Res-

onating frequencies observed in the Raman scattering spectrum from a long scan are aligned with

theoretical resonance positions (i.e., size parameter values) computed from LorenzMie theory to

obtain the operating constant c in Eq. (9), and the refractive index m of the droplets at the position

of the long scan. Devarakonda et al. (1998) and Devarakonda and Ray (2000) have described the

alignment procedure in detail. In a nutshell, for an assumed value of m, each observed peak at fre-

quency fi, is aligned with a calculated resonance at xi. We have considered all possible theoreticalresonances. The data, xi versus fi, thus obtained from the alignments are regressed to Eq. ( 9), and

the alignment errors are calculated from the following relation:

S(m) =

Npti=1

(xi cf1=3i )

2: (10)

The values of c and m at the global minimum of S are chosen as the best estimates. A visual match

between the experimental spectrum obtained from PMT1 and the theoretical spectrum computed

from Mie theory using the values of c and m obtained from the alignment procedure is used to


10/21


132 134 136 138 140 142

Size parameter,x

ElasticscatteringintensityfromPMT1

Theory

m= 1.3645, =92.14

0

Exp. data at t=0 min;c=8321



Fig. 4. Three experimental elastic scattering spectra taken over a period of 100 min along with the theoretical scattering

spectrum computed from the optimum parameter values obtained from the analysis of observed resonance peaks.

validate the results. The alignment procedure permits determination of the operating constant c, and

the refractive index m, of the droplets with precisions of 2 parts in 104.

5. Results and discussion

5.1. Reproducibility of droplet size

At each backing pressure, light scattering data from the droplet stream were recorded by positioning

the droplet generator at various distances from the laser beam. A typical experiment at a given

pressure was conducted over a period of about 2 h. The reliability of such data requires a long-term

stability in the droplet generator. To verify the reproducibility of droplet size we periodically repeatedlong-frequency scans at a reference position. Fig. 4 shows examples of three transverse magnetic

(TM) mode elastic scattering spectra from ethanol droplets obtained from scans that were separated

by a time interval of 100 min. Experimental spectra are indistinguishable from one another, indicating

a long-term stability of the droplet generator. The analysis of the resonance peaks observed in

spectra, using the peak alignment procedure outlined before, yields the same operating constant of

c =8321, and the refractive index of m = 1:3645, for all three scans. In Fig. 4, we have compared the

experimental spectra with the theoretical spectrum computed from Mie theory using the optimum

values of c and m obtained from the analysis. The excellent match between the theoretical and

experimental spectra validates light scattering results. Additionally, the light scattering results indicate


11/21


that at a given generation frequency the droplet size is highly reproducible (i.e., uctuates less than

1 part in 104).

5.2. Eect of backing pressure

In this study we have altered the droplet size and the separation distance between the droplets in

a linear array by varying the backing pressure used to generate droplets. An increase in the backing

pressure causes an increase in the ow rate, and thus the droplet size, as can be seen from Eq. (8),

increases at any given frequency. In addition, the separation distance between successive droplets

increases as the velocity of the droplets increases. Since the liquid ow rate through the orice is

proportional to the square root of the backing pressure, the operating constants (or the frequencies

at which a given size droplets is generated) at two dierent backing pressures are related through

the following expression that can be derived from Eqs. (8) and (9)

c

c0=

f

f0

1=3

=

P

P0

1=6

: (11)

The above relation shows that the operating constant c, or the frequency f, at which a given size

droplets appears at any backing pressure P, can be calculated from the known values of the operating

constant c0, and the frequency f0, at a reference pressure P0. We have tested the relation with data

obtained at various backing pressures. Fig. 5 shows TM mode elastic scattering intensity versus

frequency spectra from ethanol droplets obtained from PMT1 at the same position of the droplet

generator but at two dierent backing pressures of 1.65 and 1.93 bar. The spectra show that the

resonance peaks labeled as A to G in the bottom spectrum obtained at 1.65 bar also appear in the top

spectrum obtained from droplets generated at 1.93 bar backing pressure, but each of the resonance

peaks shifts to a higher frequency. Each of the labeled resonances has nearly the same value of sizeparameter x (i.e., neglecting the eect of the small refractive index dierence between the droplets

generated at two dierent backing pressures). Therefore, we can predict the frequency of a peak

at the higher backing pressure from its observed value at the lower backing pressure. We nd that

the frequency shift of about 20 kHz calculated from Eq. (11) for the increase of backing pressure

from 1.65 to 1.93 bar is in excellent agreement with the frequency shift observed between the

spectra.

The application of the peak alignment procedure to the resonance peaks observed in the Raman

scattering spectra yields c = 8320:8 and m = 1:3645, for the experiment at 1.65 bar, and c = 8580:5

and m = 1:3643 for the experiment at 1.93 bar, respectively. The higher value of the operating

constant c, implies that the size of droplets generated at a given frequency is larger at 1.93 barpressure than the size obtained at 1.65 bar pressure. The value of c = 8552 at 1.93 bar predicted

from Eq. (11) using the value at 1.65 bar, matches closely with the value (i.e., c =8580:5) obtained

from the analysis of resonance peaks. The dierence of about 0:30% between the values of c can

be attributed to inaccuracies in pressure measurements.

It should be noted that there is a small dierence in the refractive index (i.e., about 0.0002)

between the droplets generated at two backing pressures. This is because the droplets generated at

the higher pressure travel faster than the droplets at the lower pressure, and thus, the residence time

(i.e., the time taken by droplets to travel from 0 to 4 mm) of the droplets generated at the higher

pressure is shorter compared to the droplets at the lower pressure. Since the evaporation process


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200 210 220 230 240 250 260Frequency (kHz)

ScatteringintensityfromPMT1

A

A

B

C

D

E

F

G

H

I

JL

K

M

N

B

C

D

E

F

G

H

I

J

Pback=1.65 bar, z= 4 mm, c= 8320.8, m= 1.3645

Pback=1.93 bar, z= 4 mm, c= 8320.8, m= 1.3643

Fig. 5. Elastic scattering intensity versus frequency spectra from ethanol droplets generated at two dierent backing

pressures. As the pressure increases the resonance peaks shift towards higher frequencies, indicating an increase in droplet

size.

is unsteady for the residence times involved in the experiments, the temperature of the droplets

decreases with increasing residence time due to the evaporative cooling. Therefore, at a distance of

4 mm from the generator the temperature of droplets generated at the lower pressure drops more (by

about 0:5C) than the droplets generated at the higher pressure, and the small temperature dierence

is reected through the dierence in the refractive index.

5.3. Evaporation rates

To determine evaporation rates of droplets at a given backing pressure we obtained elastic and

Raman scattering spectra by scanning the frequency synthesizer over a 20-kHz range (i.e., short

scans) at various positions of the droplet generator. Fig. 6 shows Raman spectra from ethanol droplets

generated at 1.65 bar obtained by positioning the droplet generator at various distances from the laser

beam. As the droplet generator moves away from the laser beam (i.e., as the residence time of the


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230 235 240 245 250Frequency (kHz)

Ramanscatteringintensityat2900cm-1s

hift

A

A

A

A

A

A

A

z=1 mm

z=2 mm

z=3 mm

z=4 mm

z=5 mm

z=6 mm

z=7 mm

Pback=1.65 bar

Fig. 6. Raman intensity versus droplet generation frequency spectra at various positions of the droplet generator. Resonance

positions shift to lower frequencies as the droplet generator moves away from the laser beam, indicating a decrease in

droplet size due to evaporation.

droplets increases), the resonance peaks shift to lower frequencies (e.g., see peaks labeled A). This

indicates that at a given frequency, the droplet size decreases with increasing residence time due to

evaporation. In a typical experiment, the short scans were repeated a number of times to examine

the reproducibility of experimental data. From the observed frequency shifts in the resonance peaks,we have determined the changes in the operating constant c, and the refractive index m, with respect

to the reference position values, using an alignment procedure described by Devarakonda et al.

(1998) and Devarakonda and Ray (2000). The optimum values of c and m obtained from analysis

of resonance peaks observed in a number of experiments on ethanol droplets are summarized in

Table 1. The results show that at any position the maximum dierence among the operating constant

values is only about 1.5, and among the refractive index values is 0.0001. The maximum dierence

in c corresponds to a drift of about 1 A for a 11:0 m droplet. The high degree of reproducibility

attests to the stability of the experimental conditions, in addition to the precision of the light scattering

measurements.


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Table 1

Results obtained from ethanol droplets generated at 1.65 bar backing pressure

Position Experiment 1 Experiment 2 Experiment 3 Experiment 4

z c Refractive c Refractive c Refractive c Refractive

(mm) (Hz)1=3 index, m (Hz)1=3 index, m (Hz)1=3 index, m (Hz)1=3 index, m

7 8302.2 1.3656 8301.4 1.3657 8300.6 1.3657 8300.7 1.3657

6 8306.3 1.3654 8306.3 1.3654

5 8313.8 1.3649 8313.4 1.3650 8312.9 1.3650 8312.9 1.3650

4 8321.7 1.3644 8320.8 1.3645 8320.3 1.3645 8320.4 1.3645

3 8329.7 1.3639 8328.6 1.3640 8328.8 1.3640 8328.1 1.3640

2 8337.8 1.3634 8336.2 1.3635 8336.5 1.3635 8336.5 1.3635

1.5 8343.2 1.3631

1 8348.4 1.3627 8347.6 1.3627 8346.9 1.3627 8347.5 1.3627

The operating constant c, and refractive index m, values at various positions of the droplet gener-

ator were translated into droplet size and temperature versus residence time data. It should be noted

that the size of droplets generated at a given frequency is proportional to the operating constant c,

and therefore, the size of droplets generated at that frequency can be calculated at various positions

from the known value of size at a given position. The droplet size at a given backing pressure

varies according to Eq. (8) during the frequency scans of the generator. We have translated c and

m, versus z data with respect to a reference size of 11:0 m at z= 3 mm. We have converted the

droplet generator position data to residence times using the diraction fringe measurement technique(Devarakonda et al., 1998). In Fig. 7, we have plotted droplet size change versus residence time

results for ethanol and methanol droplets generated at various backing pressures. The experimental

data are highly reproducible, that is, at a given backing pressure the maximum dierence between

the data from two sets of experiments is less than 3%. The results show that the rate of size change

decreases from the initial value as the time progresses, indicating the usual unsteady state evaporation

characteristics for the residence times involved in this study. A comparison between the data from

ethanol droplets generated at two dierent pressures reveals that over the same residence time the

size of droplets generated at 1.93 bar pressure decreases more than droplets at 1.65 bar. This can

be attributed to the eects of interparticle interactions. The evaporation rate of droplets generated

at the lower pressure is retarded more because of the lower interparticle separation distance. Theresults also show that more volatile methanol droplets evaporate signicantly faster than ethanol

droplets.

The refractive indices of both ethanol and methanol decreases linearly by 0:0004 for each 1C

increase in the temperature (Weast, 1985). We have used this information to convert refractive

index values at various positions to droplet temperatures. The changes in the droplet temperature with

respect to time are shown in Fig. 8. Again, the results are highly reproducible. The temperature values

of one set of experiments are duplicated in another set of experiments under identical conditions.

Temperature data exhibit characteristics similar to size data, that is, initially, the temperature changes

rapidly, and then the rate of change decreases as the residence time increases.


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0 100 200 300 400 500

Residence time (s)

-80

-60

-40

-20

0

Dr

opletsizechangea(nm)

Ethanol droplets at 1.65 bar




Methanol droplets at 1.17 bar


Polynomial fit

Fig. 7. Droplet size change versus residence time data for ethanol and methanol droplets generated at various backing

pressures.

5.4. Interaction parameter

To examine the eect of interparticle interactions on evaporation of droplets we have varied the

dimensionless spacing (i.e., l=a) between droplets through manipulation of backing pressure and

frequency used for droplet generation. We have used backing pressures of 1.65, 1.93 and 2.35 bar

to generate ethanol droplets, and calculated the separation distance between two successive droplets,

using the relation l = v=f.As mentioned before, at a given frequency an increase in the backing pressure increases the

separation distance l, between droplets, while at a given backing pressure an increase in the frequency

decreases the separation distance. Within each set of experiments (i.e., for a given backing pressure

and frequency range), the spacing between two successive droplets decreases with increasing distance

from the orice due to the unsteady state momentum transfer (i.e., drag eects). To minimize

the eect of the variation in the separation distance, we considered only data obtained between

4 and 7 mm distance from the orice, and in this range separation distance, l, varies less than

10%. We shall present results on the basis of the average value of l=a in the range of distance

considered.


16/21


0 100 200 300 400 500

Residence time (s)

-10

-8

-6

-4

-2

0

Dro

plettemperatureechangeT(C)

Ethanol droplets at 1.65 barEthanol droplets at 1.65 bar





Polynomial fit

Fig. 8. Droplet temperature change versus residence time data for ethanol and methanol droplets generated at various

backing pressures.

We calculated interaction parameters for the experimental droplet streams from the size and tem-

perature versus residence time data. It should be pointed out here that the surface temperatures of the

experimental droplets, as mentioned before, were cooler than the center temperatures because of the

high evaporation rates involved in the experiments. The temperature variation in a droplet introduces

a variation in the refractive index. The technique, based on Raman scattering resonances, used in

this study estimates refractive index of a droplet that is averaged over a narrow region near the sur-

face of the droplet (Lin & Campillo, 1995; Devarakonda & Ray, 2000). Therefore, the temperature

obtained from the refractive index corresponds closely to the surface temperature of the droplet. We

have used the temperatures estimated from the refractive index as the surface temperatures, Ts, in

the following expression that was used to calculate experimental values of the interaction parameterfor mass transfer

m =L

ShDgMA

da2

dt

RTs

P0A(Ts): (12)

It should be noted that the above relation has been obtained by rearranging Eq. (6) with pA=0, for

the experimental conditions of this study. Since all the parameters on the right-hand side of Eq. ( 12)

are either known or can be estimated, we can directly calculate m, from the experimental size

and temperature versus residence time data. To obtain the interaction parameter, h, we numerically

solved the partial dierential equation describing the temperature prole in a droplet (i.e., Eq. (3))


17/21


3 5 6

Dimensionless spacing, l/a

0.2

0.3

0.4

0.5

Interactionparameter,

mo

rh

Experimental m

Experimental h

Polynomial fit

4

Fig. 9. Interaction parameter values obtained at various dimensionless separation distances.

using the value of m obtained from the above procedure in the boundary condition, Eq. (6), forthe interfacial mass transfer. We varied the value of h, in the boundary condition, Eq. (7), and the

value of h for which the overall changes in the size (i.e., a) and the surface temperature matched

with the experimental changes was used chosen as the best estimate for the interaction parameter

for heat transfer for the experimental data.

The values of interaction parameters for heat and mass transfers obtained using the procedure

outlined above are plotted in Fig. 9 for various experimental values of l=a. The results indicate that

the interaction parameters for both heat and mass transfer are almost identical, that is, m =h =, and

the interaction parameter, as expected, increases as the dimensionless separation distance between

the droplets increases. We regressed experimental versus l=a data to a second-order polynomial to

obtain the following expression:

= 0:0611 + 0:0505

l

a

+ 0:0029

l

a

2for 36

l

a6 6: (13)

Theoretically, the interaction parameter for a given geometry is independent of the physical proper-

ties of droplets, and this has been conrmed experimentally by Sangiovanni and Labowsky (1982).

Therefore, the relation between and l=a, Eq. (13), obtained from experiments on ethanol droplets

can be applied to estimate the interaction parameter of linear arrays of droplets containing any

components. In order to validate the results obtained from the experiments on ethanol droplets, we

have computed the evaporation rates of linear arrays of methanol droplets for m = h = = 0:27


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0 100 200 300 400 500

Residence time (s)

10.90

10.92

10.94

10.96

10.98

11.00

11.02

DropletRadius(m)

284

286

288

290

292

294

Dropletsurfacetemperature

Experimental radius

Computed radius

Experimental temperatureComputed temperature

Fig. 10. Comparison of experimental droplet size and temperature versus residence time data for methanol droplets with the

values predicted by a mathematical model using the interaction parameter value obtained from ethanol droplet experiments.

obtained from Eq. (13) for the experimental separation distance of 3:5. The droplet size and sur-

face temperature computed as functions of the residence time from the model (i.e., Eqs. (3)(7))

are compared with experimental data in Fig. 10. The results show excellent agreement between the

observed and computed droplet size and surface temperature versus residence time data. The max-

imum deviation in the droplet size between the observed and computed values is less than 3 nm,

and this is within the accuracy of the light scattering technique for size determination. The droplet

temperatures computed from the model mostly deviate from the experimental values by about 0 :5 K,

and this corresponds to the accuracy of our temperature determination from the refractive index

data. The maximum deviation occurs initially because of the inconsistency between the initial and

boundary conditions in the model. The experimental temperature values are consistently higher than

the computed values. Since the evaporation process is unsteady, there is a sharp temperature gradient

near the droplet surface, and the droplet core is slightly warmer than the surface. The temperature

estimated from the refractive index provides a temperature that is averaged over a narrow regionnear the surface of the droplet, and thus, the refractive index may overestimate surface temperature

by about 0:5 K.

6. Conclusions

We have examined evaporation of highly monodisperse droplets in linear arrays. Droplets were

generated using a modied VOAG. We have varied the residence time (i.e., exposure time in the

gas phase since generation) of the droplets by varying the distance between the droplet generator


19/21


and the laser beam used for light scattering. We have determined the size and refractive index of

the droplets at various residence times using a light scattering technique based on the resonances

observed in the elastic and Raman scattering spectra. The technique permits determination absolute

size and refractive index and their changes with relative errors of about 2 parts and 1 part in 10 4,respectively. We have interpreted refractive index data to obtain droplet surface temperature as a

function of the residence time. We have demonstrated that the size and temperature versus residence

time data under given experimental conditions are highly reproducible.

Because of the inter-droplet separation distances involved the eects of interparticle interactions

were inherently present in the experiments. To examine the eects of interparticle interactions we

have varied the non-dimensional spacing (i.e., l=a) between the droplets by manipulating the fre-

quency and the backing pressure used for droplet generation. We have normalized the measured

evaporation rates by the rates of single isolated droplets evaporating under identical conditions to

obtain the interaction parameter, , as a function of the dimensionless spacing between the droplets.

For the residence times involved in this study the evaporation process was transient, and thus, theevaporation rate as well as droplet temperature changed with time. We have directly estimated in-

teraction parameters for mass transfer from the instantaneous evaporation rate as well as droplet

temperature data obtained from the analysis of light scattering data. We have established interaction

parameters for heat transfer by comparing the experimental droplet surface temperature versus resi-

dence time data with values computed from an unsteady state evaporation model. The results show

that the interaction parameter values obtained at various dimensionless spacings are identical for

both heat and mass transfer processes, and decrease with decreasing dimensionless spacing between

the droplets.

To examine the validity and the applicability of the experimental results we have compared the

size and the temperature versus residence time data obtained from methanol droplets with values

computed from the unsteady state evaporation model using the interaction parameter value obtainedfrom ethanol droplet experiments. The experimental size and temperature data of methanol droplets

agree well with the computed results. This validates the interaction parameter results obtained in

this study, and conrms that the interaction parameters are independent of the nature of substances

involved and depend only on the geometrical parameters of the droplet array.

Acknowledgements

The authors are grateful to the National Science Foundation for their support (grant # CTS-

0130778).

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Effect of Inter Particle Interaction

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