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Knudsen aerosol evaporation
Knudsen Aerosol Evaporation RANDOLPH CHANG 1 AND E. JAMES DAVIS Department of Chemical Engineering, Clarkson College of Technology, Potsdam, New York 13676 Received August 26, 1975; accepted October 15, 1975 The evaporation kinetics of submicron aerosol droplets has been studied by measuring the size of single droplets as a function of time using laser light scattering from the droplet. By suspending a charged droplet in an electric field in a light scattering cell, precise measurements of droplet size have been made over a range of total pressures corresponding to diffusion-controlled evaporation and Knudsen aerosol evaporation. The results are compared with available theoretical and semitheoretical predictions for Knudsen aerosol transport processes. For the systems studied, dioctyl phthalate in helium and in nitrogen, the results are consistent with an evaporation coefficient of unity. Furthermore, the techniqu e can be used to determine diffusion coefficients and vapor pressures for relatively nonvolatile species.
INTRODUCTION
Fuchs (3) has reviewed the extensions of Maxwell's analysis to unsteady-state evaporaThe classical theories of droplet evaporation tion, and corrections for Stefan flow (4) have represent two limiting cases of the phenomenon, been considered by Newbold and Amundson diffusion-controlled evaporation into a conti(5) for multicomponent evaporation of droplets. nuum and free-molecule effusion into a Numerous other aspects of diffusion-controlled vacuum. Fuchs (1) has pointed out that the rates of transfer of momentum, energy, mass, evaporation have been studied and analyzed, and electric charge can be calculated relatively including the problem of the moving interface simply only for these two limiting cases, which and nonisothermal evaporation, but these are the limits of very small and very large refinements, based on continuum theory, need Knudsen numbers. The Knudsen number is not be elaborated on here. When the mean free path of escaping moledefined as the ratio of the mean free path k cules is large compared with the size of the and the droplet radius a, that is, Kn = X/a. evaporating droplet, the classical kinetic For isothermal, quasi-steady-state evaporatheory of gases can be applied to determine tion of a spherical droplet, Maxwell (2) obthe evaporation rate. This free-molecule tained for the time rate of change of the regime has been discussed thoroughly by Hidy droplet radius, da/dt, the equation and Brock (6), and in this limit the time rate of & / d t = -- (I)/ap) (C~ -- C~), ~-13 change of droplet radius is given by where D is the diffusion coefficient for the vapor molecules in the surrounding continuous phase, p is the molar density of the liquid droplet, and C, and C~ are the molar concentrations of vapor at the droplet surface and in the bulk of the surrounding gas, respectively. 1 Dr. Chang is with the Sun Oil Co.
da .
ceen.~ .
dt
.
a~p°~ ,
.
N.~4p
[-2-1
4pRT
where a~ is the evaporation coefficient, NA is Avogadro's number, and hoe/4 is the rate at which gas molecules strike the droplet surface, obtained from the kinetic theory of gases. The
352
Journal of Colloid and Interface Science, Vol. 54, No. 3, March 1976
Copyright ~ 1976 by Academic Press, Inc. All rights of reproduction in any form reserved,
353
AEROSOL E V A P O R A T I O N
average velocity of molecules, ~, is given by = (8Kr/Irm)½,
where K is the Boltzmann constant, T is the absolute temperature, and m is the mass of a molecule. The number density of gas molecules, no, is related to the partial pressure of molecules at the droplet interface (the vapor pressure, p0) by no = pO/KT.
There has been considerable controversy in the literature over the numerical value of the evaporation coefficient. In 1962, Paul (7) compiled an extensive list of data available at that time. His tabulations together with more recent data indicate that for metals and most organic compounds a~ is unity, but for inorganic compounds a~ varies widely. The lower alcohols and water are the source of the controversy, for reported values of oe, vary from 0.02 to 1.0. The jet evaporation studies of Hickman (8) and Maa (9), which avoid surface contamination by the continuous generation of fresh surface, indicate that ae is of order unity for all of the compounds they studied (carbon tetrachloride, isopropyl alcohol, toluene and water). Davis, Chang, and Pethica (10) corrected Maa's analysis of his data to take into account the varying surface temperature of the jet, finding a, to be very near unity for Maa's data. The technique used in the present paper offers a new method for determining the evaporation coefficient. Intermediate Knudsen numbers, that is, K n = 0(1), involve theoretical and experimental difficulties that have prevented a definitive treatment of transfer rates for Knudsen aerosols. It is the purpose of this study to provide precise measurements of evaporation rates in the Knudsen aerosol regime to assess the validity and accuracy of available theoretical and sernitheoretical analyses. K N U D S E N AEROSOL T R A N S P O R T PHENOMENA
Perhaps the simplest method proposed for determining transfer rates in the Knudsen
aerosol regime is the flux-matching method of Fuchs (11), for which it is assumed that transport is controlled by diffusion through the surrounding medium for distances greater than some distance & = 0(~.) with a discontinuity in diffusion in the region between the droplet surface and A. Wright (32) reviewed the analyses and experimental data for droplet evaporation in an attempt to rationalize the conflicting values of A proposed by Bradley, Evans, and Whytlaw-Gray (12) and by F~Sssling (13) and Monchick and Reiss (14). Because of the scatter in the available experimental data, however, Wright was not able to discriminate between the values of A to reach a definite decision about the best choice of A. Monchick and Reiss applied the nonequilibrium distribution function based on Enskog's development (t5) of the Boltzmann equation, and their resulting equation for the evaporation rate indicates that (da/dt) -1 should be a linear function of (a ;< _P), where a is the droplet radius and P is the total pressure of the system. We shall test this relationship below. By matching fluxes at a radial distance Aa from the droplet center, the flux-matching theory leads to the equation
y/]~ = ~/(1 + ~/~D~),
[3]
where ] is the mass flux for a Knudsen aerosol, and Je is the mass flux obtained from kinetic theory, & = ~M~¢/4,
[4]
where M is the molecular mass of the evaporating species. Although flux matching provides an adequate fit of experimental data, more rigorous analyses of transport rates have been attempted by solving the Boltzmann rate equation. Because the Boltzmann equation is nearly intractable unless simplified, linearization techniques have been applied by Wellander (16), Bhatnager el el. (17), and Brock (18, 19). These methods are valid only in the limit m~/mo ~ O, where m, and mo are the masses of vapor and gas molecules, respectively. In
Journal of Colloid and Interface Science, Vol. 54, No. 3, March 1976
354
CHANG AND DAVIS
ATOMIZER this limit the vapor molecules acquire a Maxwellian distribution; otherwise, the velocities of vapor molecules show some persistence, STORAGE ~ (~ GAS FLASK CYLINDER and their distribution after collision is not isotropic. MANOMETER Sahni (20) obtained a rigorous solution for neutron transfer to a black sphere for all Knudsen numbers assuming that all neutrons t,. MCLEOD SCATTERING GAUGE VACUUM have the same absolute velocity and a r e CELL PUMP scattered isotropically by heavy atoms filling the space arourrd the sphere, and Fuchs and FIG. I. A schematic of the experimental system. Sutugin (21) developed an interpolation formula based on Sahni's results which agrees Because of the unavailability of precise with free-molecule theory and converges on experimental data over a wide range of the Maxwell equation. One problem with the Knudsen numbers, the interpolation formulas evaporation rate equation of Monchick and have not been compared with experimental Reiss is its failure to predict the correct free- data. The results of the present investigation molecule kinetics as Kn---)O, but the semi- provide the much-needed data. empirical expression of Fuchs and Sutugin has the correct behavior in both the large and EXPERIMENTS small limits of the Knudsen number, Their The experimental technique developed by interpolation formula may be written as Davis and Chorbajian (23) was used to 3ID(C, - C~)/a measure the evaporation rates of single subJ= micron aerosol droplets. A schematic of the [1 + Kn(1.333Kn + 0.71)/(Kn + 1)]' system is shown in Fig. 1. An aerosol was [s] produced by injecting a carrier gas, either nitrogen or helium in these experiments, into where M is the molecular weight. an atomizer, shown in Fig. 2. The size distriBademosi and Liu (22) used a "flux- bution of the aerosol was adjusted by generresistance" concept to obtain general equations ating the aerosol from an emulsion. The for the rates of transfer of momentum, energy, emulsion was prepared from distilled water mass, and charge for aerosol droplets with and dioctyl phthalate by vigorous shaking arbitrary Knudsen number. Their result is without the addition of an emulsifier, which
U
U
J
47rKn
7k
1 + 4~rKn[Kn/(2.08Kn + 1.51)]
[6]
cTO LIGHT SCATTERING CELL
where it is assumed that D--/~Xe from the Maxwell-Chapmann-Enskog equation and fl is given by
/~--- ( l + ~ ) ~ - ~ \
M,
/
[7] JLSION
where ~ = 0.132 when the molecular masses of vapor and gas, M. and Mo, are extremely unequal, which is the situation of interest in this paper. Journal of Colloid and Interface Science, Vol. 54, No. 3, March 1976
1 N2 OR He
FIG, 2. T h e aerosol generator.
AEROSOL EVAPORATION would contaminate the aerosol. The organic phase was usually 10% by volume of the emulsion. By controlling the droplet size of dioctyl phthalate droplets in water, the aerosol droplet size was easily adjusted to the desired size range (0.5 /~m < a < 0.9 #m). The atomizer served two functions, (1) the generation of the aerosol, aLd (2) charging of the aerosol. Due to the highly polar nature of the water, atomization from the emulsion produced the charged droplets essential for the technique employed to measure the evaporation rates. Atomization of the pure organic generally led to weakly charged or uncharged droplets. The aerosol passed from the atomizer to a storage flask from which it was introduced into the light scattering cell of a Science Spectrum, Differential II Light Scattering Spectrophotometer. The light scattering cell was modified from the Science Spectrum design by including vacuum ports and a vacuum manifold. A cross-sectional view of the light scattering cell is shown in Fig. 3. A single aerosol droplet was captured in the electric field produced by the high voltage pin/grounded plate electrostatic system, and the remaining aerosol droplets in the aerosol cloud were allowed to settle out of the viewing area. To avoid significant evaporation of droplets during the capturing stage of the experiment, which often took from 2 to 5 min, a very dilute aerocolloidal suspension was used. That is, very few droplets were introduced into the initially vapor-free cell. Thus, the vapor concentration in the surrounding medium was maintained very near zero. It was essential for the analysis and interpretation of the evaporation rate data that the bulk concentration of organic in the light scattering cell be as close to zero as possible. Because all droplets entering the light scattering chamber evaporated during a run, the absolute number of droplets in the cell had to be kept to a minimum. Several precautions were taken to limit the number of droplets entering the cell. As indicated in Fig. l, dilution of the aerosol was carried out in a
355 AEROSOL FROM ATOMIZER
O-RING ~-~ G#%~:~x~ /HIGH VOLTAGE RAYLEIGH ~/~ ' ,~(~#¢~/" PIN HORN ~ ~ . ^ , ~ , . ,~,
± GROUND
FIG. 3. T h e light scattering cell.
storage flask. The diluted aerosol passed into the upper chamber of the light scattering cell shown in Fig. 3, and only a very small (and unknown) fraction of these droplets passed through the small hole near the high-voltage pin into the lower half of the cell. Prior to the start of an experiment the system was pumped out for several hours to eliminate residual vapor and/or aerosol droplets. The aerosol generated in the atomizer was so diluted with fresh carrier gas that at any time only a single droplet could be seen in the viewing eyepiece attached to the light scattering cell. It is not possible to determine quantitatively the droplet population density, but 50 droplets per cubic centimeter with radii of 0.5 ~*m would be sufficient to saturate the carrier gas, so it was important to provide sufficient dilution to maintain the aerosol concentration below one droplet per cubic centimeter. To prevent any significant accumulation of settled droplets in the light scattering cell, an experiment was stopped if a dropIet was not captured within 5 min after introduction of aerosol into the chamber. The system was then thoroughly flushed with pure carrier gas and pumped out before attempting another experiment. Since it was not possible strictly to control, regulate and measure the aerosol number concentration to minimize the bulk concentration of vapor in the cell, systematic procedures were developed to achieve highly reproducible results. The results reported below provide an a posteriori verification of the procedure, and it is likely that the major source of experimental error is the accuracy
Journal of Colloid and Interface Science, Vol. 54, No. 3, March 1976
356
CHANG AND DAVIS RUN4271 -
-
.....
SCAN1
TIME=0
EXPERIMENT MIE THEORY FOR o(=6,15
_z
I 50°
I 100° SCATTERING ANGLE
I 150~
Fla. 4. Intensity versus scattering angle for Run 4271, Scan l, using DOP in N2 at atmospheric pressure. with which C~, the bulk concentration of vapor, can be reduced to zero. I t should be pointed out that the large amount of dilution used also reduced the water vapor present as a result of atomization of the organic/water emulsion to a negligible concentration. Hence, the effects of water vapor on the evaporation process (by affecting the diffusion of vapor in the water vapor,/ carrier gas) are expected to be negligible. Even without dilution the maximum concentration of water vapor, corresponding to saturation of "the carrier gas, is approximateIy 4% by volume at atmospheric pressure. The total pressure in the light scattering cell was measured by means of a mercury manometer and a McLeod gauge. The introduction of aerosol caused a small fluctuation in the total pressure, but the pressure quickly stabilized and was constant during the period of evaporation. For the experiments reported below, the pressure ranged from atmospheric to 68 mbar, although the vacuum system was capable of reducing the pressure to 10-a mbar. A thermocouple was installed in the light scattering cell, and the temperature was recorded on an oscillograph. Due to heat generation of the compact electronics equipment, the cell usually operated at approximately 35°C, although the room temperature was maintained at 25°C. During the period
of a run the cell temperature remained constant because the small amount of aerosol injected into the cell at the start of a run was preheated to the cell temperature in the upper chamber of the two-part light scattering cell. To make sure that the system was thermally at steady state, the equipment was turned on and allowed to operate 3 or 4 hr before the evaporation experiments were performed. Theoretical calculations for unsteady-state heat and mass transfer from an evaporating droplet in the continuum regime by Chang and Davis (24) indicate that, for slowly evaporating droplets such as dioctyl phthalate (droplets of low volatility), a droplet introduced into a surrounding medium at different temperature very quickly attains the temperature of the surrounding medium. It can be safely assumed that the droplet temperature was the measured cell temperature during the period of evaporation. After a single droplet was captured by manually adjusting the field strength and polarity of the electric field, the instrument was set to hold the droplet automatically by means of the automatic control circuit of the Differential II module. The intensity of the light scattered by the droplet was measured by photomultiplier traverses, which were begun as soon as the droplet was stabilized in the field. Traverses were made from 40 to 150 °, where the incident laser beam was RUN 4271 SCAN 9 -
-
......
TIME = 555 S
EXPERIMENT
MIE THEORY FOR c(=5.875
.,I
/I
;"t77 '\
,rx l
V
I 50 °
/
100 ° SCATTERI NG ANGLE
__L 150°
FIC. 5. Intensity versus scattering angle for Run 4271, Scan 9 using DOP in N2 at atmospheric pressure.
Journal of Colloid and Interface Science, Vol. 54, No. 3, Marck 1976
AEROSOL EVAPORATION RUN 5031 SCAN 1 -
-
......
changed size slightly during the period of a traverse of the photomultiplier tube, and because of electronic noise and oscillation of the droplet, the matching of experimental light scattering curves to Mie theory is not trivial. It is necessary to obtain a "best fit" of the data and Mie theory to establish the droplet size from the experimental data shown in Figs. 4-7. The measured angular scattering intensities can be related to the Mie theory prediction by assuming the equation
TIME = 0
EXPERIMENT MIE THEORY FOR d=7.45
it t 50 °
100 ° SCATTERING ANGLE
357
1_ 150°
i~(O) -- b(O) = Ki,(0, ~) + e,
FIG. 6. Intensity versus scattering angle for Run 5031, Scan l, using D O P in N2 at a pressure of 195 mbar.
located at 0 °. At the highest scan speed, a scan took approximately 12 sec, and a typical run lasted more than 10 rain, so many scans were taken for each run. A run usually terminated when the droplet was lost from the field due to increasingly rapid oscillations in the electric field as the droplet became smaller. This usually occurred when the droplet radius reached about 0.2/~m. Occasionally a droplet was lost suddenly during the course of a run when it was still fairly large, possibly due to loss of charge. The laser light was vertically polarized and the scattered light was measured by photomultiplier traverses in the horizontal plane. The output of the traversing photomultiplier tube was recorded as the ordinate on a stripchart recorder, and the angle of traverse was the abscissa. Typical raw data from the instrument are shown in Figs. 4-7, which are for dioctyl phthalate (DOP) droplets evaporating into nitrogen. Superimposed on the raw data are plots of intensity versus angle obtained from Mie theory (25), which requires some elaboration.
E8~
where it is the experimental measurement of intensity as a function of angle, it is the Mie theory intensity, which is a function of angle 0 and dimensionless radius a defined by a = 2~ra/Xs. The parameter ~,L is the wavelength of the laser light source (XL = 514 nm); K is a scale factor; b is the background intensity; and e represents measurement error, which is considered to be a random variable of mean zero and constant variance. I t should be noted that in general b is a function of 0, but for the range of scattering angles used here (45 ° < 0 < 150°), b was relatively constant. The error e can be due to a number of sources including electronic noise, small-amplitude oscillatory motion of the droplet in the electric field, and the fact that the droplet size changes slightly during the period of a traverse. At the highest scan speeds, hysteresis associated with the light intensity recorder pen was noted. To minimize this hysteresis, which was most prevalent at the highest scan speeds, the smallest time-constant setting of the elecRUN 5031 SCAN 5 -
-
......
TIME = 200 S
EXPERIMENT MIE THEORY FOR m= 7.20
INTERPRETATION OF DATA The theory and application of light scattering from spheres are well established, as indicated in the treatise by Kerker (26), but because in these experiments the droplet
I 50 °
t 100° SCATTERING ANGLE
I 150"
FIO. 7. Intensity versus scattering angle for Run 5031, Scan 5, using DOP in N2 at a pressure of 195 mbar.
Journal of Colloid and Interface Science, Voh 54, No. 3, March 1976
358
CHANG AND DAVIS
tronics was used. But this adjustment prohibited damping out some of the electronic noise and oscillations of the droplet. Thus, minimization of e experimentally involved experimental compromise between the scan speed, the time constant, and the amplification of the signal from the light-sensitive photomultipliers. Figure 5 shows differences between Mie theory and experiment between 85 ° and 95 ° that are probably due to pen overshoot, and the small-ampiitude fluctuations superposed on the primary intensity-versus-angle curve are probably associated with smallamplitude oscillations of the droplet. For a given scattering diagram, an arbitrary number of points can be used for curve fitting. For N points, Eq. [-8] can be written in matrix form as
[9]
L = ~f, + E, where _?~,?t, and E are the matrices :
Ie
g
- b(01} 7.
ki~(ON) " b(O~v)J'
Fki~(bx)Jt '
e
Applying the theory of least-squares analysis, the experimental data were fitted to the theoretical values for different a by choosing = (L'L)-wL,
[1o]
where the prime_refers to the transpose of a matrix, and (I,'It) -1 is the inverse of matrix (It l It). The value of a that best fits the experimental data determines the droplet size. One criterion used for the best fit was to minimize the value of N
E
[ie(0m) -- b(O~) -- i,(O,~, ~ ) 2 .
A second criterion can be used to avoid overemphasis on the low-intensity regions, and that is the minimization of
Both of these criteria were used in processing the experimental data. To apply these optimization procedures for matching, data points were taken directly from the stripchart records at every 5 ° in the range 4 5 ° < 0 < 145 ° for most runs. For some runs (e.g., Figs. 4 and 5), data were limited to the range 6 0 ° < 0 < 140 ° either because of large peaks or because of large background corrections near the limits. Mie theory results were calculated by digital computer for each of these angles for a set of a values in a range determined by examining the positions of the peaks and troughs, using precalculated light scattering curves to provide quick estimates of size. The optimization was done by computer. The two criteria discussed above were found to agree to within 1% for the determination of droplet size, and the accuracy of the matching was generally to within 1%. For small droplets (a < 3.0) the accuracy was found to be no better than 1% because of the small amplitude of peaks, but in the range 5.0_< a _ 10.0 the method described can give an accuracy better than 1~o. This procedure is far superior to that of Phillips et al. (27), who used only the first peak and trough to characterize particle size, particularly when the particle size changes slightly during the light scattering measurement. The Mie theory curves based on the best-fit matching technique outlined above are plotted in Figs. 4-7. The agreement between theory and experiment is excellent. EVAPORATION RATES Once the light scattering curves are matched with the Mie theory to obtain the droplet size, it is possible to plot the droplet size as a function of time. Typical evaporation curves for DOP evaporating into nitrogen are plotted in Fig. 8 for various total pressures of the system. The slopes of these size-versus-time curves are seen to increase as the total pressure decreases until a nearly constant slope is reached. These results indicate t h a t the
Journal of Colloid and Interface Science, Vol. 54, No. 3, M a r c h 1976
359
AEROSOL EVAPORATION
0 6q~
l
-
l
Q%
/
[~o
•
I 'v'~.s
d
a.
o.
o....._ ° i
0.8 /
I I
""O-o-..o ~.
DOP/He
2o F
]
-t
°---o.4
0.5~b--cl--
DOP/N 2
" ~
\d
\o
0.~
F--
07 F
\u
~o,5 ~ ~ . \
c3
-}
\
c2
fi
I
\
1000 2000 TIME, S
0 3/ • 0
-
I 1000
fx.~ 2000 TINE, S
I 3000
1 I 4000
i 3000
FIG. 10. Droplet radii as a function of time for D O P in N2 and in He at atmospheric pressure.
FIG. 8. Droplet radii as a function of time for various total pressures using D O P and N2.
evaporation process progressed from the diffusion-controlled regime, for which the diffusion coefficient is inversely proportional to pressure, to nearly the free-molecule regime, for which the rate is independent of the total pressure of the system. We shall compare the results with Maxwell's equation, Eq. [-1], and with equations for the Knudsen aerosol regime below. The evaporation rate data are highly reproducible, as shown in Fig. 9, which also indicates that the evaporation proceeded at a quasisteady state; that is, the data for Run 5031 with an initial radius of 0.609 um can be superimposed on the data for Run 5012 with an initial radius of 0.421 urn. The excellent reproducibility of the data also confirms that
o4
°3F 0
,
,
;000
~ TIME, S
,
2000
rx
I
3000
Fie. 9. Superposition of R u n s 5012 and 503i at and 195 mbar.
34.4°C
the concentration of DOP vapor in the surrounding gas could be maintained effectively at zero, for it was not possible to maintain the vapor concentration at any other level in a reproducible manner because of the introduction of a dilute cloud of aerosol at the start of an experiment. The effect of the carrier gas on the evaporation rate is shown in Fig. 10, which shows typical data at atmospheric pressure for DOP in nitrogen and in helium. The higher evaporation rate in helium is consistent with a higher diffusion coefficient of DOP in helium than in nitrogen, and we must now turn our attention to the problem of predicting the diffusion coefficients. Data on the diffusion coefficient for DOP in helium and DOP in nitrogen are not readily available, but Johnstone and Eads (28) reported a value of 0.064 cm2/sec for dibutyl phthalate (DBP) in air at 135°C, and Bradley el al. (12) reported 0.031 cm2/sec for the same system at 20°C and 0.030 cm2/sec for butyl stearate in air at 20°C. For diamyl sebecate (DAS) in air, Monchick and Reiss (14) reported 0.0237 cm2/sec at 25.2°C and 0.0254 cm2/sec at 34.9°C, and for dibutyl phthalate in air they obtained 0.0340 cm2/sec at 19.9°C. Because the molecular weight of dioctyl phthalate (MDoP = 390) is greater than that of dibutyl phthalate (MD~r = 278), the diffusion coefficient for DOP in nitrogen or air can be expected to be lower than for
Journal of Colloid and Interface Science, Vol. 54, No. 3, March 1976
360
CHANG AND DAVIS
0,S
--~I-" _
I
1
1
I
[
t
j
"--.%.
0.26
0.6
TABLE I The Diffusion Coefficient for Dioctyl Phthalate in Nitrogen at Atmospheric Pressure and 36°C
3.24 E
Run number
a (um)
D (em~/sec)
Equation
4271 4272 4272 4271 4272
0.481 0.450 0.413
0.0158 0.0161 0.0158 0.0161 0.0149
Ill] Ill] [12] [12]
,~0.~ 0,22
0P/N 2
\
0.2 1000
2000 TIME, S
3000
4000
3,20
FIG. 11. A plot of a~versus t for DOP in N~ and DOP in He at atmospheric pressure. DBP. Now the diffusion coefficient for DOP in nitrogen can be calculated from Eq. [-1] using the experimental data in the diffusion-controlled regime for a and da/dt and information on the vapor pressure po, for Eq. [1] can be written as
da/dt
=
- (D/ao)(p/RT),
[,11]
where R is the ideal gas constant, T is the temperature of the system, and it has been assumed that Coo = 0. Alternatively, we can integrate Eq. [11] to give a ~ = a02 --
2Dp°/pRT(t -/0),
[,12]
where a0 is the radius at time to = 0, which is taken to be the time of the first scan of the photomultiplier tube. A plot of a 2 versus t (with t o - - 0 ) should be a straight line with slope equal to 2Dp°/pRT for data in the diffusion-controlled regime. Figure 11 shows typical results for DOP/N2 and D O P / H e experiments made at atmospheric pressure plotted as a 2 versus t. The calculation of the diffusion coefficient from the slopes requires knowledge of the vapor pressure at the temperature of the droplet. As we shall show, the vapor pressure of DOP can be obtained from the evaporation-rate data in the free-molecule regime. Thus, using p 0 = 8.07 X 10-~ mbar at 36°C and using typical rate data at atmospheric pressure for DOP in nitrogen, we obtain the results tabulated in Table I. The first three results in the table were obtained by measuring da/dt at
---
[11]
the values of a indicated, then calculating D from Eq. [11]. The last two values were obtained by applying Eq. [12] to the measured slopes of the plots of a 2 versus t. These results are not in agreement with the diffusion coefficient for DOP in N2 reported by Davis and Chorbajian, due to an unreported error in the temperature of the light scattering cell in the previous experiments. The diffusion coefficient reported in Ref. (23), DDOV/N2 = 0.0307 cm2/sec, was based on a vapor pressure of 2.92 X 10-7 mbar, which they obtained by extrapolation of available data to 25°C, which was the ambient temperature and not the light scattering cell temperature. Based on the more likely cell temperature of 33°C, their diffusion coefficient should be corrected to DDOP/N2 = 0.0169 cm2/sec at 33°C and atmospheric pressure, which is in good agreement with the data reported here. However, both values appear to be quite low compared to diffusion coefficients calculated from semitheoretical methods discussed by Davis and Chorbajian. Similar results based on DOP/helium data at atmospheric pressure give DDovza, = 0.0504 cm2/sec, and the average value for D O P / nitrogen is DDopm~ = 0.0159 cm2/sec. Now applying the Maxwell-Chapman-Enskog equation, x
8~/8 \
M~
/
we obtain X = 6.75 X 10-~ cm for the mean free path of DOP in nitrogen at 36°C and 1
Journal of Colloid and Interface Science, Vol. 54, No. 3, M a r c h 1976
AEROSOL EVAPORATION TABLE II
1.00 0.50 0.25
I - - 7
1.2
The Effect of Droplet Radius on the Vapor Pressure of DOP a (~m)
361 1--
~, ~,0 o
Oo
x 0.8 E
exp(2aV/aKT)
/8~/° ~.~-
"T" 0./.
1.009 1.018 1.037
u 0.2
06
arm. Hence, the Knudsen number for the droplets reported in Table I is of order 0.015, which indicates that the evaporation was carried out in the diffusion-controlled regime. Since the mean free path is inversely proportional to the total pressure, we can calculate the Knudsen number for the data at other pressures.
o DOP/N 2 DATA
/
- I
0.05
LEAST SQUARES FI" I
0.1 a P,,um - bar
I
0.15
0.2
F~G. 12. The reciprocalevaporation rate as a function of the product aP.
limp~o(da/dt) -1 = 2520 s/#m for DOP in nitrogen at 36°C. For Fig. 12 the raw data, which were taken at temperature ranging from 32.1 to 36.3°C, were corrected to 36°C by adjusting the vapor pressure using the ClausiusClapeyron equation, assuming da/dt to be VAPOR PRESSURE proportional to p0. For the data in the diffusionI t was indicated above that the vapor controlled regime the diffusion coefficient was pressure was calculated from results for the adjusted by assuming D ~ TI as well. These free-molecule regime. As the free-molecule adjustments were generally very small. Thus, regime is the limit as K n becomes large, it is applying Eq. [2_] and assuming that the appropriate to take the limit of the rate data evaporation coefficient of DOP is unity, we as the total pressure goes to zero to determine calculate the vapor pressure at 36°C to be the vapor pressure of DOP. Monchick and 8.07 4- 0.36 X 10-7 mbar. This result is comReiss showed that the evaporation rate follows pared with available data in Table III. the equation The data of Small el al. (30) are probably the best of the previous data, because they exp ( 2 a V / a K T ) = A(1 + B P), D4-I were able to measure vapor pressures as low (d /dO as 10.4 mbar, but all of the previous results where ~ is the surface tension, V is the molecu- require extrapolation from measurements at lar volume, A and B are constants, and P is higher temperatures. The present method is in the total pressure. The exponential term in good agreement with the extrapolated data, Eq. [14-] arises by applying the Kelvin equa- but the data reported here are probably of tion to obtain the dependence of the vapor much better accuracy because the vapor pressure on the droplet radius. For the droplets pressure is obtained without extrapolation encountered in the present study the correc- through many orders of magnitude. The tion due to the Kelvin effect was generally less reasonableness of the vapor pressure suggests than 2% as indicated in Table II. that the assumption of unity for the evaporaSince the correction for the Kelvin effect is negligible for most of the present data, we tion coefficient is valid. If a smaller value of can expect (da/dt) -1 to be a linear function of ae is used, the vapor pressure is proportionately the product of droplet radius and total pres- increased, and any significant increase of the sure. Figure 12 shows this to be valid. Ex- vapor pressure is inconsistent with the vapor trapolation to zero total pressure gives pressure data of Small el el. Journal of Colloid and Interface Science, VoL 54, No. 3, March 1976
362
CHANG AND DAVIS TABLE III Comparison of Vapor Pressure Data for DOP Investigator
po in mbar at 25°C
This work Hickman et al. (29) Small et al. (30) Kapff and Jacobs (31) Reported in Ref. (31)
1.73 4- 0.08 X 3.07 X 1.89 X 0.93 X 2.67 X
po in mbar at 36°C
10-7~ 10-v 10-7 10-7 10-7
8.07 4- 0.36 X 13.3 X 9.2 X 4.27 X 12.4 X
10-7 10-v 10-v 10-7 10-7
Extrapolated using the Clausius-Clapeyron equation with A//wp = 109 Kjoule/g-mole.
COMPARISON OF THEORY AND EXPERIMENT Knowing the vapor pressure, the diffusion coefficient, and the mean free path, it is possible to compare theoretical predictions of the evaporation rate with the experiments. Figure 13 is a comparison between the radiusversus-time results predicted from Eq. [12-] and the experimental data at higher pressures, which correspond to the diffusion-controlled regime• To obtain the diffusion coefficient at pressures below 1 arm, we have assumed that D is inversely proportional to the total pressure. The data for one atmosphere are in excellent agreement with the theory for diffusioncontrolled evaporation, but as the pressure decreases the deviations between Eq. [-12] and the data become larger, the evaporation rate being lower than that predicted from Eq. [12-] at the lower pressures. The data for the entire region of Knudsen numbers covered by the experiments are most 0.6 K
I r
_~0.5
i
\ \
w~
--~ ~
o RUN 4271 ~- RUN 4282 v RUN 5013
03L---, BDUATION(12) 0
1
Yk
1 + 1/(7.28Kn)
[15]
Using fl = 1.82 (for DOP in nitrogen) and Eq. 1,,13] to relate the diffusion coefficient and the mean free path, Fuchs's interpolation formula, Eq. [5], can be written as J
Kn
Jk 0.7SO2[l+Kn(1.333Kn-]-O.71)/(Kn+l)] 1-16]
The theoretical and semitheoretical expressions predict similar results, but the interpolation formula of Bademosi and Liu has the best agreement with experimental data. Although the data do not extend all the way to the free-molecule regime (where J / J k = 1), the Knudsen aerosol regime is reasonably well covered by the experimental data. Figure 14 ~.0L
"-, /o
\ ~0
~ O,Z
I
J
~°~ ~..o~,=1.013 bar \
a~
conveniently presented as the flux ratio J / J k versus the Knudsen number. Such results are plotted in Fig. 14 together with the results predicted using the interpolation formulas of Fuchs and of Bademosi and Liu and the fluxmatching method, Eq. [3-], written as
oo.
\o
" "O v%"
'
'
'
'
'~*~'~
I-
"<~
\,~6\~,~
V
\
i
500 1000 TIME, S
\ \l \ , --
FIG. 13. A comparison between Eq. [12-1 for diffusioncontrolled evaporation and the data for DOP in N2 at
the higher pressures used. Journal o f Colloid and Interface Science,
~ n ~- I --u001 002
I I I IL I 00501 02 0.5 1 KNUDSEN NUMBER, ?,./a
1500
l 2
Fro. 14. A comparison between the flux ratio Y / J k from theory and experiment for DOP in N~ at 36°C.
Vol. 54, No. 3, March 1976
AEROSOL EVAPORATION shows rather little scatter data over more than one in K n , again indicating that can be attained with technique.
in the experimental order of magnitude the good precision the light scattering
CONCLUSIONS The measurement of submicron aerosol droplet evaporation rates by means of light scattering offers a very precise tool for the study of Knudsen aerosol evaporation. The droplet size can be measured to within 1% in the nondimensional size range 5.0 < o~ < 10.0. For the wavelength of light used here (kL = 514 nm), this corresponds to droplet sizes in the range 0.409/~m < a /-. 0.818 #m. If the vapor pressure is known accurately, the method can be used to determine the evaporation coefficient and the diffusion coefficient b y measuring evaporation rates over a range of pressures including the diffusioncontrolled regime and the Knudsen aerosol regime. If the evaporation coefficient can be assumed to be unity or if is measured independently, the vapor pressure of relatively nonvolatile species ( p 0 < 10-6 mbar) can be measured with better accuracy than by most other methods. The data supplied here provide a critical test of the flux-matching theory of Fuchs for Knudsen evaporation and of the semitheoretical equations of Bademosi and Liu and Fuchs. The data can also be used to test theories based on the solution of the Boltzmann equation for Knudsen aerosols. ACKNOWLEDGMENT The authors are grateful to the National Science Foundation for Grant GP-33654X, to Clarkson College of Technology for equipment funds, and to M. Kerker and D. Cooke for the use of their facilities and for valuable discussions related to the experimental techniques. REFERENCES I. Fuc~Is, N. A., "Recent Progress in the Theory of Transfer Processes in Aerosols at Intermediate Values of Knudsen Numbers," Proc. 7th Int. Conf. on Condensation and Ice Nuclei, Prague and Vienna, 1969. 2. MAXWELL, J. C., "Collected Scientific Papers," Vol. 11, p. 625. Cambridge Univ. Press, Cambridge, 1880.
363
3. FucHs, N. A., "Evaporation and Droplet Growth in Gaseous Media." Pergamon Press, New York, 1959. 4. STErAN,J., Wien. Ber. 83, 943 (1881). 5. NEW:BOLD,F. R., AND AMUNDSON,N. R., A I C h E J
19, 22 (1973). 6. I-IID¥,G. M., ANDBROCK,J. B., "The Dynamics of Aerocolloidal Systems." Pergamon Press, New York, 1970. 7. PAUL, B , A R S J 3 2 , 1321 (1962). 8. HICK~AN,K., Desalination 1, 13 (1966). 9. MAA,J. R., Ind. Eng. Chem. Fundam. 6, 504 (1967). 10. DAVIS, E. J., CHANG, R., AND PETHICA, B. D., Ind. Eng. Chem. Fundam. 14, 27 (1975). 11. FlJcl~s, N. A., Phys. Z. Sowjet 6, 225 (1934). 12. BRADLEY, R. S., EVANS, N. G., AND WHYTLAWGRAY,R. W., Proc. Roy. Soc. (London) 186, 368
(1946). 13. FR6SSLInG, N., Gerlands Beitr. Geophys. 52, 195 (1938). 14. MONCHICI~,L., ANDREISS, H., J. Chem. Phys. 22, 831 (1954). 15. C~IAP~AN,S., ANDCOWLINC,T. G., "Mathematical Theory of Non-Uniform Gases." Cambridge Univ. Press, Cambridge, 1952. 16. W~LANDER,P., Ark. Fys. 7, 507 (1954). 17. BHATNAGAR,P., GROSS, E., AND KROOK,H., Phys. Rev. 94, 511 (1954). 18. BROCK,J. B., Y. Colloid Interface Sci. 22, 513 (1966). 19. BROCK,J. B., J. Colloid Interface Sci. 24, 344 (1967). 20. SAHnI,D., J. Nuel. Energy 20, 915 (1965). 21. FucHs, N. A., A:NDSUTUGIN,A. G., "Highly Dis-
22.
23. 24. 25. 26.
27. 28. 29.
persed Aerosols," Ann Arbor Science Publishers, Ann Arbor, Mich., 1970. BADEMOSI,F., ANDLILT,B. Y. H., Publications 155, 156, 157, Particle Technology Laboratory, Mech. Eng. Dept., Univ. of Minnesota, May 1971. DAVIS,E. J., ANDCHORBAJIAN,E., Ind. E~g. Chem. Fundam. 13, 272 (1974). CHANG,R., AnD DAVIS, E. J., J. Colloid Interface Sci. 47, 65 (1974). MIE, G., Ann. Phys. 25, 377 (1908). KERICER,M., "The Scattering of Light and Other Electromagnetic Radiation," Academic Press, New York, 1969. PmLLIPS,D. T., WYAtt, P. J., AnD BERIC-~An,R. M., J. Colloid Interface Sci. 34, 159 (1970). J0~InS~ONE, H. F., AnD LADS, D. K., Ind. Eng. Chem. 42, 2293 (1950). HICKMA~,K. C. D., HECK~R,J. C., AnD EM~REE, N. D., Ind. Eng. Chem. 9, 264 (1937).
30. SMALL, P. A., SMALL, K. W., AND COWLEY, P., Trans. Faraday Soe. 44, 810 (1948). 31. KAWF, S. F., AnD JACOBS, R. B., Rev. Sci. Inst.
18, 581 (1947). 32. WRIGHT,P. G., Discussions Faraday Soc. 30, 100 (1960).
Journal of Colloid and Interface Science, Vol. 54, No. 3. March 1976
INTRODUCTION
Fuchs (3) has reviewed the extensions of Maxwell's analysis to unsteady-state evaporaThe classical theories of droplet evaporation tion, and corrections for Stefan flow (4) have represent two limiting cases of the phenomenon, been considered by Newbold and Amundson diffusion-controlled evaporation into a conti(5) for multicomponent evaporation of droplets. nuum and free-molecule effusion into a Numerous other aspects of diffusion-controlled vacuum. Fuchs (1) has pointed out that the rates of transfer of momentum, energy, mass, evaporation have been studied and analyzed, and electric charge can be calculated relatively including the problem of the moving interface simply only for these two limiting cases, which and nonisothermal evaporation, but these are the limits of very small and very large refinements, based on continuum theory, need Knudsen numbers. The Knudsen number is not be elaborated on here. When the mean free path of escaping moledefined as the ratio of the mean free path k cules is large compared with the size of the and the droplet radius a, that is, Kn = X/a. evaporating droplet, the classical kinetic For isothermal, quasi-steady-state evaporatheory of gases can be applied to determine tion of a spherical droplet, Maxwell (2) obthe evaporation rate. This free-molecule tained for the time rate of change of the regime has been discussed thoroughly by Hidy droplet radius, da/dt, the equation and Brock (6), and in this limit the time rate of & / d t = -- (I)/ap) (C~ -- C~), ~-13 change of droplet radius is given by where D is the diffusion coefficient for the vapor molecules in the surrounding continuous phase, p is the molar density of the liquid droplet, and C, and C~ are the molar concentrations of vapor at the droplet surface and in the bulk of the surrounding gas, respectively. 1 Dr. Chang is with the Sun Oil Co.
da .
ceen.~ .
dt
.
a~p°~ ,
.
N.~4p
[-2-1
4pRT
where a~ is the evaporation coefficient, NA is Avogadro's number, and hoe/4 is the rate at which gas molecules strike the droplet surface, obtained from the kinetic theory of gases. The
352
Journal of Colloid and Interface Science, Vol. 54, No. 3, March 1976
Copyright ~ 1976 by Academic Press, Inc. All rights of reproduction in any form reserved,
353
AEROSOL E V A P O R A T I O N
average velocity of molecules, ~, is given by = (8Kr/Irm)½,
where K is the Boltzmann constant, T is the absolute temperature, and m is the mass of a molecule. The number density of gas molecules, no, is related to the partial pressure of molecules at the droplet interface (the vapor pressure, p0) by no = pO/KT.
There has been considerable controversy in the literature over the numerical value of the evaporation coefficient. In 1962, Paul (7) compiled an extensive list of data available at that time. His tabulations together with more recent data indicate that for metals and most organic compounds a~ is unity, but for inorganic compounds a~ varies widely. The lower alcohols and water are the source of the controversy, for reported values of oe, vary from 0.02 to 1.0. The jet evaporation studies of Hickman (8) and Maa (9), which avoid surface contamination by the continuous generation of fresh surface, indicate that ae is of order unity for all of the compounds they studied (carbon tetrachloride, isopropyl alcohol, toluene and water). Davis, Chang, and Pethica (10) corrected Maa's analysis of his data to take into account the varying surface temperature of the jet, finding a, to be very near unity for Maa's data. The technique used in the present paper offers a new method for determining the evaporation coefficient. Intermediate Knudsen numbers, that is, K n = 0(1), involve theoretical and experimental difficulties that have prevented a definitive treatment of transfer rates for Knudsen aerosols. It is the purpose of this study to provide precise measurements of evaporation rates in the Knudsen aerosol regime to assess the validity and accuracy of available theoretical and sernitheoretical analyses. K N U D S E N AEROSOL T R A N S P O R T PHENOMENA
Perhaps the simplest method proposed for determining transfer rates in the Knudsen
aerosol regime is the flux-matching method of Fuchs (11), for which it is assumed that transport is controlled by diffusion through the surrounding medium for distances greater than some distance & = 0(~.) with a discontinuity in diffusion in the region between the droplet surface and A. Wright (32) reviewed the analyses and experimental data for droplet evaporation in an attempt to rationalize the conflicting values of A proposed by Bradley, Evans, and Whytlaw-Gray (12) and by F~Sssling (13) and Monchick and Reiss (14). Because of the scatter in the available experimental data, however, Wright was not able to discriminate between the values of A to reach a definite decision about the best choice of A. Monchick and Reiss applied the nonequilibrium distribution function based on Enskog's development (t5) of the Boltzmann equation, and their resulting equation for the evaporation rate indicates that (da/dt) -1 should be a linear function of (a ;< _P), where a is the droplet radius and P is the total pressure of the system. We shall test this relationship below. By matching fluxes at a radial distance Aa from the droplet center, the flux-matching theory leads to the equation
y/]~ = ~/(1 + ~/~D~),
[3]
where ] is the mass flux for a Knudsen aerosol, and Je is the mass flux obtained from kinetic theory, & = ~M~¢/4,
[4]
where M is the molecular mass of the evaporating species. Although flux matching provides an adequate fit of experimental data, more rigorous analyses of transport rates have been attempted by solving the Boltzmann rate equation. Because the Boltzmann equation is nearly intractable unless simplified, linearization techniques have been applied by Wellander (16), Bhatnager el el. (17), and Brock (18, 19). These methods are valid only in the limit m~/mo ~ O, where m, and mo are the masses of vapor and gas molecules, respectively. In
Journal of Colloid and Interface Science, Vol. 54, No. 3, March 1976
354
CHANG AND DAVIS
ATOMIZER this limit the vapor molecules acquire a Maxwellian distribution; otherwise, the velocities of vapor molecules show some persistence, STORAGE ~ (~ GAS FLASK CYLINDER and their distribution after collision is not isotropic. MANOMETER Sahni (20) obtained a rigorous solution for neutron transfer to a black sphere for all Knudsen numbers assuming that all neutrons t,. MCLEOD SCATTERING GAUGE VACUUM have the same absolute velocity and a r e CELL PUMP scattered isotropically by heavy atoms filling the space arourrd the sphere, and Fuchs and FIG. I. A schematic of the experimental system. Sutugin (21) developed an interpolation formula based on Sahni's results which agrees Because of the unavailability of precise with free-molecule theory and converges on experimental data over a wide range of the Maxwell equation. One problem with the Knudsen numbers, the interpolation formulas evaporation rate equation of Monchick and have not been compared with experimental Reiss is its failure to predict the correct free- data. The results of the present investigation molecule kinetics as Kn---)O, but the semi- provide the much-needed data. empirical expression of Fuchs and Sutugin has the correct behavior in both the large and EXPERIMENTS small limits of the Knudsen number, Their The experimental technique developed by interpolation formula may be written as Davis and Chorbajian (23) was used to 3ID(C, - C~)/a measure the evaporation rates of single subJ= micron aerosol droplets. A schematic of the [1 + Kn(1.333Kn + 0.71)/(Kn + 1)]' system is shown in Fig. 1. An aerosol was [s] produced by injecting a carrier gas, either nitrogen or helium in these experiments, into where M is the molecular weight. an atomizer, shown in Fig. 2. The size distriBademosi and Liu (22) used a "flux- bution of the aerosol was adjusted by generresistance" concept to obtain general equations ating the aerosol from an emulsion. The for the rates of transfer of momentum, energy, emulsion was prepared from distilled water mass, and charge for aerosol droplets with and dioctyl phthalate by vigorous shaking arbitrary Knudsen number. Their result is without the addition of an emulsifier, which
U
U
J
47rKn
7k
1 + 4~rKn[Kn/(2.08Kn + 1.51)]
[6]
cTO LIGHT SCATTERING CELL
where it is assumed that D--/~Xe from the Maxwell-Chapmann-Enskog equation and fl is given by
/~--- ( l + ~ ) ~ - ~ \
M,
/
[7] JLSION
where ~ = 0.132 when the molecular masses of vapor and gas, M. and Mo, are extremely unequal, which is the situation of interest in this paper. Journal of Colloid and Interface Science, Vol. 54, No. 3, March 1976
1 N2 OR He
FIG, 2. T h e aerosol generator.
AEROSOL EVAPORATION would contaminate the aerosol. The organic phase was usually 10% by volume of the emulsion. By controlling the droplet size of dioctyl phthalate droplets in water, the aerosol droplet size was easily adjusted to the desired size range (0.5 /~m < a < 0.9 #m). The atomizer served two functions, (1) the generation of the aerosol, aLd (2) charging of the aerosol. Due to the highly polar nature of the water, atomization from the emulsion produced the charged droplets essential for the technique employed to measure the evaporation rates. Atomization of the pure organic generally led to weakly charged or uncharged droplets. The aerosol passed from the atomizer to a storage flask from which it was introduced into the light scattering cell of a Science Spectrum, Differential II Light Scattering Spectrophotometer. The light scattering cell was modified from the Science Spectrum design by including vacuum ports and a vacuum manifold. A cross-sectional view of the light scattering cell is shown in Fig. 3. A single aerosol droplet was captured in the electric field produced by the high voltage pin/grounded plate electrostatic system, and the remaining aerosol droplets in the aerosol cloud were allowed to settle out of the viewing area. To avoid significant evaporation of droplets during the capturing stage of the experiment, which often took from 2 to 5 min, a very dilute aerocolloidal suspension was used. That is, very few droplets were introduced into the initially vapor-free cell. Thus, the vapor concentration in the surrounding medium was maintained very near zero. It was essential for the analysis and interpretation of the evaporation rate data that the bulk concentration of organic in the light scattering cell be as close to zero as possible. Because all droplets entering the light scattering chamber evaporated during a run, the absolute number of droplets in the cell had to be kept to a minimum. Several precautions were taken to limit the number of droplets entering the cell. As indicated in Fig. l, dilution of the aerosol was carried out in a
355 AEROSOL FROM ATOMIZER
O-RING ~-~ G#%~:~x~ /HIGH VOLTAGE RAYLEIGH ~/~ ' ,~(~#¢~/" PIN HORN ~ ~ . ^ , ~ , . ,~,
± GROUND
FIG. 3. T h e light scattering cell.
storage flask. The diluted aerosol passed into the upper chamber of the light scattering cell shown in Fig. 3, and only a very small (and unknown) fraction of these droplets passed through the small hole near the high-voltage pin into the lower half of the cell. Prior to the start of an experiment the system was pumped out for several hours to eliminate residual vapor and/or aerosol droplets. The aerosol generated in the atomizer was so diluted with fresh carrier gas that at any time only a single droplet could be seen in the viewing eyepiece attached to the light scattering cell. It is not possible to determine quantitatively the droplet population density, but 50 droplets per cubic centimeter with radii of 0.5 ~*m would be sufficient to saturate the carrier gas, so it was important to provide sufficient dilution to maintain the aerosol concentration below one droplet per cubic centimeter. To prevent any significant accumulation of settled droplets in the light scattering cell, an experiment was stopped if a dropIet was not captured within 5 min after introduction of aerosol into the chamber. The system was then thoroughly flushed with pure carrier gas and pumped out before attempting another experiment. Since it was not possible strictly to control, regulate and measure the aerosol number concentration to minimize the bulk concentration of vapor in the cell, systematic procedures were developed to achieve highly reproducible results. The results reported below provide an a posteriori verification of the procedure, and it is likely that the major source of experimental error is the accuracy
Journal of Colloid and Interface Science, Vol. 54, No. 3, March 1976
356
CHANG AND DAVIS RUN4271 -
-
.....
SCAN1
TIME=0
EXPERIMENT MIE THEORY FOR o(=6,15
_z
I 50°
I 100° SCATTERING ANGLE
I 150~
Fla. 4. Intensity versus scattering angle for Run 4271, Scan l, using DOP in N2 at atmospheric pressure. with which C~, the bulk concentration of vapor, can be reduced to zero. I t should be pointed out that the large amount of dilution used also reduced the water vapor present as a result of atomization of the organic/water emulsion to a negligible concentration. Hence, the effects of water vapor on the evaporation process (by affecting the diffusion of vapor in the water vapor,/ carrier gas) are expected to be negligible. Even without dilution the maximum concentration of water vapor, corresponding to saturation of "the carrier gas, is approximateIy 4% by volume at atmospheric pressure. The total pressure in the light scattering cell was measured by means of a mercury manometer and a McLeod gauge. The introduction of aerosol caused a small fluctuation in the total pressure, but the pressure quickly stabilized and was constant during the period of evaporation. For the experiments reported below, the pressure ranged from atmospheric to 68 mbar, although the vacuum system was capable of reducing the pressure to 10-a mbar. A thermocouple was installed in the light scattering cell, and the temperature was recorded on an oscillograph. Due to heat generation of the compact electronics equipment, the cell usually operated at approximately 35°C, although the room temperature was maintained at 25°C. During the period
of a run the cell temperature remained constant because the small amount of aerosol injected into the cell at the start of a run was preheated to the cell temperature in the upper chamber of the two-part light scattering cell. To make sure that the system was thermally at steady state, the equipment was turned on and allowed to operate 3 or 4 hr before the evaporation experiments were performed. Theoretical calculations for unsteady-state heat and mass transfer from an evaporating droplet in the continuum regime by Chang and Davis (24) indicate that, for slowly evaporating droplets such as dioctyl phthalate (droplets of low volatility), a droplet introduced into a surrounding medium at different temperature very quickly attains the temperature of the surrounding medium. It can be safely assumed that the droplet temperature was the measured cell temperature during the period of evaporation. After a single droplet was captured by manually adjusting the field strength and polarity of the electric field, the instrument was set to hold the droplet automatically by means of the automatic control circuit of the Differential II module. The intensity of the light scattered by the droplet was measured by photomultiplier traverses, which were begun as soon as the droplet was stabilized in the field. Traverses were made from 40 to 150 °, where the incident laser beam was RUN 4271 SCAN 9 -
-
......
TIME = 555 S
EXPERIMENT
MIE THEORY FOR c(=5.875
.,I
/I
;"t77 '\
,rx l
V
I 50 °
/
100 ° SCATTERI NG ANGLE
__L 150°
FIC. 5. Intensity versus scattering angle for Run 4271, Scan 9 using DOP in N2 at atmospheric pressure.
Journal of Colloid and Interface Science, Vol. 54, No. 3, Marck 1976
AEROSOL EVAPORATION RUN 5031 SCAN 1 -
-
......
changed size slightly during the period of a traverse of the photomultiplier tube, and because of electronic noise and oscillation of the droplet, the matching of experimental light scattering curves to Mie theory is not trivial. It is necessary to obtain a "best fit" of the data and Mie theory to establish the droplet size from the experimental data shown in Figs. 4-7. The measured angular scattering intensities can be related to the Mie theory prediction by assuming the equation
TIME = 0
EXPERIMENT MIE THEORY FOR d=7.45
it t 50 °
100 ° SCATTERING ANGLE
357
1_ 150°
i~(O) -- b(O) = Ki,(0, ~) + e,
FIG. 6. Intensity versus scattering angle for Run 5031, Scan l, using D O P in N2 at a pressure of 195 mbar.
located at 0 °. At the highest scan speed, a scan took approximately 12 sec, and a typical run lasted more than 10 rain, so many scans were taken for each run. A run usually terminated when the droplet was lost from the field due to increasingly rapid oscillations in the electric field as the droplet became smaller. This usually occurred when the droplet radius reached about 0.2/~m. Occasionally a droplet was lost suddenly during the course of a run when it was still fairly large, possibly due to loss of charge. The laser light was vertically polarized and the scattered light was measured by photomultiplier traverses in the horizontal plane. The output of the traversing photomultiplier tube was recorded as the ordinate on a stripchart recorder, and the angle of traverse was the abscissa. Typical raw data from the instrument are shown in Figs. 4-7, which are for dioctyl phthalate (DOP) droplets evaporating into nitrogen. Superimposed on the raw data are plots of intensity versus angle obtained from Mie theory (25), which requires some elaboration.
E8~
where it is the experimental measurement of intensity as a function of angle, it is the Mie theory intensity, which is a function of angle 0 and dimensionless radius a defined by a = 2~ra/Xs. The parameter ~,L is the wavelength of the laser light source (XL = 514 nm); K is a scale factor; b is the background intensity; and e represents measurement error, which is considered to be a random variable of mean zero and constant variance. I t should be noted that in general b is a function of 0, but for the range of scattering angles used here (45 ° < 0 < 150°), b was relatively constant. The error e can be due to a number of sources including electronic noise, small-amplitude oscillatory motion of the droplet in the electric field, and the fact that the droplet size changes slightly during the period of a traverse. At the highest scan speeds, hysteresis associated with the light intensity recorder pen was noted. To minimize this hysteresis, which was most prevalent at the highest scan speeds, the smallest time-constant setting of the elecRUN 5031 SCAN 5 -
-
......
TIME = 200 S
EXPERIMENT MIE THEORY FOR m= 7.20
INTERPRETATION OF DATA The theory and application of light scattering from spheres are well established, as indicated in the treatise by Kerker (26), but because in these experiments the droplet
I 50 °
t 100° SCATTERING ANGLE
I 150"
FIO. 7. Intensity versus scattering angle for Run 5031, Scan 5, using DOP in N2 at a pressure of 195 mbar.
Journal of Colloid and Interface Science, Voh 54, No. 3, March 1976
358
CHANG AND DAVIS
tronics was used. But this adjustment prohibited damping out some of the electronic noise and oscillations of the droplet. Thus, minimization of e experimentally involved experimental compromise between the scan speed, the time constant, and the amplification of the signal from the light-sensitive photomultipliers. Figure 5 shows differences between Mie theory and experiment between 85 ° and 95 ° that are probably due to pen overshoot, and the small-ampiitude fluctuations superposed on the primary intensity-versus-angle curve are probably associated with smallamplitude oscillations of the droplet. For a given scattering diagram, an arbitrary number of points can be used for curve fitting. For N points, Eq. [-8] can be written in matrix form as
[9]
L = ~f, + E, where _?~,?t, and E are the matrices :
Ie
g
- b(01} 7.
ki~(ON) " b(O~v)J'
Fki~(bx)Jt '
e
Applying the theory of least-squares analysis, the experimental data were fitted to the theoretical values for different a by choosing = (L'L)-wL,
[1o]
where the prime_refers to the transpose of a matrix, and (I,'It) -1 is the inverse of matrix (It l It). The value of a that best fits the experimental data determines the droplet size. One criterion used for the best fit was to minimize the value of N
E
[ie(0m) -- b(O~) -- i,(O,~, ~ ) 2 .
A second criterion can be used to avoid overemphasis on the low-intensity regions, and that is the minimization of
Both of these criteria were used in processing the experimental data. To apply these optimization procedures for matching, data points were taken directly from the stripchart records at every 5 ° in the range 4 5 ° < 0 < 145 ° for most runs. For some runs (e.g., Figs. 4 and 5), data were limited to the range 6 0 ° < 0 < 140 ° either because of large peaks or because of large background corrections near the limits. Mie theory results were calculated by digital computer for each of these angles for a set of a values in a range determined by examining the positions of the peaks and troughs, using precalculated light scattering curves to provide quick estimates of size. The optimization was done by computer. The two criteria discussed above were found to agree to within 1% for the determination of droplet size, and the accuracy of the matching was generally to within 1%. For small droplets (a < 3.0) the accuracy was found to be no better than 1% because of the small amplitude of peaks, but in the range 5.0_< a _ 10.0 the method described can give an accuracy better than 1~o. This procedure is far superior to that of Phillips et al. (27), who used only the first peak and trough to characterize particle size, particularly when the particle size changes slightly during the light scattering measurement. The Mie theory curves based on the best-fit matching technique outlined above are plotted in Figs. 4-7. The agreement between theory and experiment is excellent. EVAPORATION RATES Once the light scattering curves are matched with the Mie theory to obtain the droplet size, it is possible to plot the droplet size as a function of time. Typical evaporation curves for DOP evaporating into nitrogen are plotted in Fig. 8 for various total pressures of the system. The slopes of these size-versus-time curves are seen to increase as the total pressure decreases until a nearly constant slope is reached. These results indicate t h a t the
Journal of Colloid and Interface Science, Vol. 54, No. 3, M a r c h 1976
359
AEROSOL EVAPORATION
0 6q~
l
-
l
Q%
/
[~o
•
I 'v'~.s
d
a.
o.
o....._ ° i
0.8 /
I I
""O-o-..o ~.
DOP/He
2o F
]
-t
°---o.4
0.5~b--cl--
DOP/N 2
" ~
\d
\o
0.~
F--
07 F
\u
~o,5 ~ ~ . \
c3
-}
\
c2
fi
I
\
1000 2000 TIME, S
0 3/ • 0
-
I 1000
fx.~ 2000 TINE, S
I 3000
1 I 4000
i 3000
FIG. 10. Droplet radii as a function of time for D O P in N2 and in He at atmospheric pressure.
FIG. 8. Droplet radii as a function of time for various total pressures using D O P and N2.
evaporation process progressed from the diffusion-controlled regime, for which the diffusion coefficient is inversely proportional to pressure, to nearly the free-molecule regime, for which the rate is independent of the total pressure of the system. We shall compare the results with Maxwell's equation, Eq. [-1], and with equations for the Knudsen aerosol regime below. The evaporation rate data are highly reproducible, as shown in Fig. 9, which also indicates that the evaporation proceeded at a quasisteady state; that is, the data for Run 5031 with an initial radius of 0.609 um can be superimposed on the data for Run 5012 with an initial radius of 0.421 urn. The excellent reproducibility of the data also confirms that
o4
°3F 0
,
,
;000
~ TIME, S
,
2000
rx
I
3000
Fie. 9. Superposition of R u n s 5012 and 503i at and 195 mbar.
34.4°C
the concentration of DOP vapor in the surrounding gas could be maintained effectively at zero, for it was not possible to maintain the vapor concentration at any other level in a reproducible manner because of the introduction of a dilute cloud of aerosol at the start of an experiment. The effect of the carrier gas on the evaporation rate is shown in Fig. 10, which shows typical data at atmospheric pressure for DOP in nitrogen and in helium. The higher evaporation rate in helium is consistent with a higher diffusion coefficient of DOP in helium than in nitrogen, and we must now turn our attention to the problem of predicting the diffusion coefficients. Data on the diffusion coefficient for DOP in helium and DOP in nitrogen are not readily available, but Johnstone and Eads (28) reported a value of 0.064 cm2/sec for dibutyl phthalate (DBP) in air at 135°C, and Bradley el al. (12) reported 0.031 cm2/sec for the same system at 20°C and 0.030 cm2/sec for butyl stearate in air at 20°C. For diamyl sebecate (DAS) in air, Monchick and Reiss (14) reported 0.0237 cm2/sec at 25.2°C and 0.0254 cm2/sec at 34.9°C, and for dibutyl phthalate in air they obtained 0.0340 cm2/sec at 19.9°C. Because the molecular weight of dioctyl phthalate (MDoP = 390) is greater than that of dibutyl phthalate (MD~r = 278), the diffusion coefficient for DOP in nitrogen or air can be expected to be lower than for
Journal of Colloid and Interface Science, Vol. 54, No. 3, March 1976
360
CHANG AND DAVIS
0,S
--~I-" _
I
1
1
I
[
t
j
"--.%.
0.26
0.6
TABLE I The Diffusion Coefficient for Dioctyl Phthalate in Nitrogen at Atmospheric Pressure and 36°C
3.24 E
Run number
a (um)
D (em~/sec)
Equation
4271 4272 4272 4271 4272
0.481 0.450 0.413
0.0158 0.0161 0.0158 0.0161 0.0149
Ill] Ill] [12] [12]
,~0.~ 0,22
0P/N 2
\
0.2 1000
2000 TIME, S
3000
4000
3,20
FIG. 11. A plot of a~versus t for DOP in N~ and DOP in He at atmospheric pressure. DBP. Now the diffusion coefficient for DOP in nitrogen can be calculated from Eq. [-1] using the experimental data in the diffusion-controlled regime for a and da/dt and information on the vapor pressure po, for Eq. [1] can be written as
da/dt
=
- (D/ao)(p/RT),
[,11]
where R is the ideal gas constant, T is the temperature of the system, and it has been assumed that Coo = 0. Alternatively, we can integrate Eq. [11] to give a ~ = a02 --
2Dp°/pRT(t -/0),
[,12]
where a0 is the radius at time to = 0, which is taken to be the time of the first scan of the photomultiplier tube. A plot of a 2 versus t (with t o - - 0 ) should be a straight line with slope equal to 2Dp°/pRT for data in the diffusion-controlled regime. Figure 11 shows typical results for DOP/N2 and D O P / H e experiments made at atmospheric pressure plotted as a 2 versus t. The calculation of the diffusion coefficient from the slopes requires knowledge of the vapor pressure at the temperature of the droplet. As we shall show, the vapor pressure of DOP can be obtained from the evaporation-rate data in the free-molecule regime. Thus, using p 0 = 8.07 X 10-~ mbar at 36°C and using typical rate data at atmospheric pressure for DOP in nitrogen, we obtain the results tabulated in Table I. The first three results in the table were obtained by measuring da/dt at
---
[11]
the values of a indicated, then calculating D from Eq. [11]. The last two values were obtained by applying Eq. [12] to the measured slopes of the plots of a 2 versus t. These results are not in agreement with the diffusion coefficient for DOP in N2 reported by Davis and Chorbajian, due to an unreported error in the temperature of the light scattering cell in the previous experiments. The diffusion coefficient reported in Ref. (23), DDOV/N2 = 0.0307 cm2/sec, was based on a vapor pressure of 2.92 X 10-7 mbar, which they obtained by extrapolation of available data to 25°C, which was the ambient temperature and not the light scattering cell temperature. Based on the more likely cell temperature of 33°C, their diffusion coefficient should be corrected to DDOP/N2 = 0.0169 cm2/sec at 33°C and atmospheric pressure, which is in good agreement with the data reported here. However, both values appear to be quite low compared to diffusion coefficients calculated from semitheoretical methods discussed by Davis and Chorbajian. Similar results based on DOP/helium data at atmospheric pressure give DDovza, = 0.0504 cm2/sec, and the average value for D O P / nitrogen is DDopm~ = 0.0159 cm2/sec. Now applying the Maxwell-Chapman-Enskog equation, x
8~/8 \
M~
/
we obtain X = 6.75 X 10-~ cm for the mean free path of DOP in nitrogen at 36°C and 1
Journal of Colloid and Interface Science, Vol. 54, No. 3, M a r c h 1976
AEROSOL EVAPORATION TABLE II
1.00 0.50 0.25
I - - 7
1.2
The Effect of Droplet Radius on the Vapor Pressure of DOP a (~m)
361 1--
~, ~,0 o
Oo
x 0.8 E
exp(2aV/aKT)
/8~/° ~.~-
"T" 0./.
1.009 1.018 1.037
u 0.2
06
arm. Hence, the Knudsen number for the droplets reported in Table I is of order 0.015, which indicates that the evaporation was carried out in the diffusion-controlled regime. Since the mean free path is inversely proportional to the total pressure, we can calculate the Knudsen number for the data at other pressures.
o DOP/N 2 DATA
/
- I
0.05
LEAST SQUARES FI" I
0.1 a P,,um - bar
I
0.15
0.2
F~G. 12. The reciprocalevaporation rate as a function of the product aP.
limp~o(da/dt) -1 = 2520 s/#m for DOP in nitrogen at 36°C. For Fig. 12 the raw data, which were taken at temperature ranging from 32.1 to 36.3°C, were corrected to 36°C by adjusting the vapor pressure using the ClausiusClapeyron equation, assuming da/dt to be VAPOR PRESSURE proportional to p0. For the data in the diffusionI t was indicated above that the vapor controlled regime the diffusion coefficient was pressure was calculated from results for the adjusted by assuming D ~ TI as well. These free-molecule regime. As the free-molecule adjustments were generally very small. Thus, regime is the limit as K n becomes large, it is applying Eq. [2_] and assuming that the appropriate to take the limit of the rate data evaporation coefficient of DOP is unity, we as the total pressure goes to zero to determine calculate the vapor pressure at 36°C to be the vapor pressure of DOP. Monchick and 8.07 4- 0.36 X 10-7 mbar. This result is comReiss showed that the evaporation rate follows pared with available data in Table III. the equation The data of Small el al. (30) are probably the best of the previous data, because they exp ( 2 a V / a K T ) = A(1 + B P), D4-I were able to measure vapor pressures as low (d /dO as 10.4 mbar, but all of the previous results where ~ is the surface tension, V is the molecu- require extrapolation from measurements at lar volume, A and B are constants, and P is higher temperatures. The present method is in the total pressure. The exponential term in good agreement with the extrapolated data, Eq. [14-] arises by applying the Kelvin equa- but the data reported here are probably of tion to obtain the dependence of the vapor much better accuracy because the vapor pressure on the droplet radius. For the droplets pressure is obtained without extrapolation encountered in the present study the correc- through many orders of magnitude. The tion due to the Kelvin effect was generally less reasonableness of the vapor pressure suggests than 2% as indicated in Table II. that the assumption of unity for the evaporaSince the correction for the Kelvin effect is negligible for most of the present data, we tion coefficient is valid. If a smaller value of can expect (da/dt) -1 to be a linear function of ae is used, the vapor pressure is proportionately the product of droplet radius and total pres- increased, and any significant increase of the sure. Figure 12 shows this to be valid. Ex- vapor pressure is inconsistent with the vapor trapolation to zero total pressure gives pressure data of Small el el. Journal of Colloid and Interface Science, VoL 54, No. 3, March 1976
362
CHANG AND DAVIS TABLE III Comparison of Vapor Pressure Data for DOP Investigator
po in mbar at 25°C
This work Hickman et al. (29) Small et al. (30) Kapff and Jacobs (31) Reported in Ref. (31)
1.73 4- 0.08 X 3.07 X 1.89 X 0.93 X 2.67 X
po in mbar at 36°C
10-7~ 10-v 10-7 10-7 10-7
8.07 4- 0.36 X 13.3 X 9.2 X 4.27 X 12.4 X
10-7 10-v 10-v 10-7 10-7
Extrapolated using the Clausius-Clapeyron equation with A//wp = 109 Kjoule/g-mole.
COMPARISON OF THEORY AND EXPERIMENT Knowing the vapor pressure, the diffusion coefficient, and the mean free path, it is possible to compare theoretical predictions of the evaporation rate with the experiments. Figure 13 is a comparison between the radiusversus-time results predicted from Eq. [12-] and the experimental data at higher pressures, which correspond to the diffusion-controlled regime• To obtain the diffusion coefficient at pressures below 1 arm, we have assumed that D is inversely proportional to the total pressure. The data for one atmosphere are in excellent agreement with the theory for diffusioncontrolled evaporation, but as the pressure decreases the deviations between Eq. [-12] and the data become larger, the evaporation rate being lower than that predicted from Eq. [12-] at the lower pressures. The data for the entire region of Knudsen numbers covered by the experiments are most 0.6 K
I r
_~0.5
i
\ \
w~
--~ ~
o RUN 4271 ~- RUN 4282 v RUN 5013
03L---, BDUATION(12) 0
1
Yk
1 + 1/(7.28Kn)
[15]
Using fl = 1.82 (for DOP in nitrogen) and Eq. 1,,13] to relate the diffusion coefficient and the mean free path, Fuchs's interpolation formula, Eq. [5], can be written as J
Kn
Jk 0.7SO2[l+Kn(1.333Kn-]-O.71)/(Kn+l)] 1-16]
The theoretical and semitheoretical expressions predict similar results, but the interpolation formula of Bademosi and Liu has the best agreement with experimental data. Although the data do not extend all the way to the free-molecule regime (where J / J k = 1), the Knudsen aerosol regime is reasonably well covered by the experimental data. Figure 14 ~.0L
"-, /o
\ ~0
~ O,Z
I
J
~°~ ~..o~,=1.013 bar \
a~
conveniently presented as the flux ratio J / J k versus the Knudsen number. Such results are plotted in Fig. 14 together with the results predicted using the interpolation formulas of Fuchs and of Bademosi and Liu and the fluxmatching method, Eq. [3-], written as
oo.
\o
" "O v%"
'
'
'
'
'~*~'~
I-
"<~
\,~6\~,~
V
\
i
500 1000 TIME, S
\ \l \ , --
FIG. 13. A comparison between Eq. [12-1 for diffusioncontrolled evaporation and the data for DOP in N2 at
the higher pressures used. Journal o f Colloid and Interface Science,
~ n ~- I --u001 002
I I I IL I 00501 02 0.5 1 KNUDSEN NUMBER, ?,./a
1500
l 2
Fro. 14. A comparison between the flux ratio Y / J k from theory and experiment for DOP in N~ at 36°C.
Vol. 54, No. 3, March 1976
AEROSOL EVAPORATION shows rather little scatter data over more than one in K n , again indicating that can be attained with technique.
in the experimental order of magnitude the good precision the light scattering
CONCLUSIONS The measurement of submicron aerosol droplet evaporation rates by means of light scattering offers a very precise tool for the study of Knudsen aerosol evaporation. The droplet size can be measured to within 1% in the nondimensional size range 5.0 < o~ < 10.0. For the wavelength of light used here (kL = 514 nm), this corresponds to droplet sizes in the range 0.409/~m < a /-. 0.818 #m. If the vapor pressure is known accurately, the method can be used to determine the evaporation coefficient and the diffusion coefficient b y measuring evaporation rates over a range of pressures including the diffusioncontrolled regime and the Knudsen aerosol regime. If the evaporation coefficient can be assumed to be unity or if is measured independently, the vapor pressure of relatively nonvolatile species ( p 0 < 10-6 mbar) can be measured with better accuracy than by most other methods. The data supplied here provide a critical test of the flux-matching theory of Fuchs for Knudsen evaporation and of the semitheoretical equations of Bademosi and Liu and Fuchs. The data can also be used to test theories based on the solution of the Boltzmann equation for Knudsen aerosols. ACKNOWLEDGMENT The authors are grateful to the National Science Foundation for Grant GP-33654X, to Clarkson College of Technology for equipment funds, and to M. Kerker and D. Cooke for the use of their facilities and for valuable discussions related to the experimental techniques. REFERENCES I. Fuc~Is, N. A., "Recent Progress in the Theory of Transfer Processes in Aerosols at Intermediate Values of Knudsen Numbers," Proc. 7th Int. Conf. on Condensation and Ice Nuclei, Prague and Vienna, 1969. 2. MAXWELL, J. C., "Collected Scientific Papers," Vol. 11, p. 625. Cambridge Univ. Press, Cambridge, 1880.
363
3. FucHs, N. A., "Evaporation and Droplet Growth in Gaseous Media." Pergamon Press, New York, 1959. 4. STErAN,J., Wien. Ber. 83, 943 (1881). 5. NEW:BOLD,F. R., AND AMUNDSON,N. R., A I C h E J
19, 22 (1973). 6. I-IID¥,G. M., ANDBROCK,J. B., "The Dynamics of Aerocolloidal Systems." Pergamon Press, New York, 1970. 7. PAUL, B , A R S J 3 2 , 1321 (1962). 8. HICK~AN,K., Desalination 1, 13 (1966). 9. MAA,J. R., Ind. Eng. Chem. Fundam. 6, 504 (1967). 10. DAVIS, E. J., CHANG, R., AND PETHICA, B. D., Ind. Eng. Chem. Fundam. 14, 27 (1975). 11. FlJcl~s, N. A., Phys. Z. Sowjet 6, 225 (1934). 12. BRADLEY, R. S., EVANS, N. G., AND WHYTLAWGRAY,R. W., Proc. Roy. Soc. (London) 186, 368
(1946). 13. FR6SSLInG, N., Gerlands Beitr. Geophys. 52, 195 (1938). 14. MONCHICI~,L., ANDREISS, H., J. Chem. Phys. 22, 831 (1954). 15. C~IAP~AN,S., ANDCOWLINC,T. G., "Mathematical Theory of Non-Uniform Gases." Cambridge Univ. Press, Cambridge, 1952. 16. W~LANDER,P., Ark. Fys. 7, 507 (1954). 17. BHATNAGAR,P., GROSS, E., AND KROOK,H., Phys. Rev. 94, 511 (1954). 18. BROCK,J. B., Y. Colloid Interface Sci. 22, 513 (1966). 19. BROCK,J. B., J. Colloid Interface Sci. 24, 344 (1967). 20. SAHnI,D., J. Nuel. Energy 20, 915 (1965). 21. FucHs, N. A., A:NDSUTUGIN,A. G., "Highly Dis-
22.
23. 24. 25. 26.
27. 28. 29.
persed Aerosols," Ann Arbor Science Publishers, Ann Arbor, Mich., 1970. BADEMOSI,F., ANDLILT,B. Y. H., Publications 155, 156, 157, Particle Technology Laboratory, Mech. Eng. Dept., Univ. of Minnesota, May 1971. DAVIS,E. J., ANDCHORBAJIAN,E., Ind. E~g. Chem. Fundam. 13, 272 (1974). CHANG,R., AnD DAVIS, E. J., J. Colloid Interface Sci. 47, 65 (1974). MIE, G., Ann. Phys. 25, 377 (1908). KERICER,M., "The Scattering of Light and Other Electromagnetic Radiation," Academic Press, New York, 1969. PmLLIPS,D. T., WYAtt, P. J., AnD BERIC-~An,R. M., J. Colloid Interface Sci. 34, 159 (1970). J0~InS~ONE, H. F., AnD LADS, D. K., Ind. Eng. Chem. 42, 2293 (1950). HICKMA~,K. C. D., HECK~R,J. C., AnD EM~REE, N. D., Ind. Eng. Chem. 9, 264 (1937).
30. SMALL, P. A., SMALL, K. W., AND COWLEY, P., Trans. Faraday Soe. 44, 810 (1948). 31. KAWF, S. F., AnD JACOBS, R. B., Rev. Sci. Inst.
18, 581 (1947). 32. WRIGHT,P. G., Discussions Faraday Soc. 30, 100 (1960).
Journal of Colloid and Interface Science, Vol. 54, No. 3. March 1976