The Effect of Gravitational Settling on the Polydispersity of Coagulating Aerosols Chung H. Ahn and James W. Gentry* Department of Chemical Engineering, Cniversity of Maryland, College Park, M d . 20742
Numerical calculations were carried out for an approximate solution of the coagulation equation. The particle distribution was assumed io be closely approximated by a log-normal distribution. The calculations accounted for removal of particles by gravitational settling and included a correction for departure from continuum mechanics for very small particles. These calculations indicate that the noncontinuum correction is important for small particles. They indicate that the standard deviation of the distribution undergoes considerable variation, in contrast to most self-preserving models. This method provides a practical calculation procedure for following changes in the size distribution function during coagulation of aerosols.
T h e aerosol particle size distribution is in a continuous state of flux, as the smaller particles grow by coagulation and larger particles are removed by sedimentation. The first theory of coagulation was suggested by Smoluchowski (1916, 1918). His theory employs a n ordinary differential equation for discrete particle sizes. He assumed a constant coagulation coefficient (accounting only for thermal diffusion) with a unit collision probability for the particles; Le., if tn-o particles collide, they have a unit probability of combination. However, for a highly polydisperse aerosol, discrete spectrum theory as employed by Smoluchowski is invalid and a continuous spectrum model has to be developed. For this case one must employ a nonlinear integro-diff erential equation. The appropriate equations for coagulation have been derived by Miiller (1928b). The process of coagulation in polydisperse aerosols has been studied in detail using the Muller equation by Zebel (1958). He solved the Muller equation numerically for a number of initial conditions. Martynov and Bakanov (1963) have suggested that a serial solution to integro-differential equation can be obtained in the form of a power series. Their results suggest that a universal asymptotic behavior for the total number of particles exists. However, a unique size distribution cannot be expected, for many different initial distributions can produce the same limiting behavior for the total number of particles. This power solution appears to be applicable to the initial stage of coagulation. Schumann (1940) has suggested that, for a constant collision frequency factor, the functions
where
with n(v,t) being the number of particles having a volume between v and v dv a t time t are particular solutions to the
+
rate equation of a continuous spectrum model. He found that the size distribution for initially monodisperse systems asymptotically approaches the same exponential form as time approaches infinity, and concluded that the exponential form might be the asymptotic solution to any initial size distribution. Todes (1949) found that, if the collision frequency factor was a homogeneous function of its argument, the integrodifferential equation can be transformed into a n ordinary equation by the substitution D
(4) where z = Qv/t, with P and Q being constants. However, as Friedlander and K a n g (1966) point out, this transformation does not give reasonable results for the limiting case of very small times. Friedlander (1960, 1961) and Swift and Friedlander (1964) have developed a similarity theory for the polydisperse system. Substitution of the similarity transformation (eq 5), where g ( t ) and v + are functions of 15) time and the ratio v/v+ is a reduced volume where ti+ is the mean volume, into the nonlinear integro-differential equation yields an ordinary integro-differential equation, with 71 as an independent variable. The reduced spectrum function $1(7~)is said to be self-preserving since it depends only on the reduced volume and not explicitly on time. Hidy (1965) solved the rate equation of coagulation numerically for a discrete spectrum with several initial distributions, using a digital computer. Recently, Castleman, et al. (1969), and Lindauer and Castleman (1970, 1971a, b) carried out extensive numerical calculations using the Muller equation and the digital computer. Their experiments show that the assumption of a specific functional form with one time-dependent parameter yields erroneous results. Their calculations seem to imply that a two-parameter calculation (allowing both the mean volume and the standard deviation to be time dependent) holds considerable promise. Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
483
I n this paper the rate equation of coagulation, including the gravitational sedimentation term, was solved numerically utilizing the assumption that the particle size distribution may be closely approximated by a log-normal distribution. The model differs from the Lindauer-Castleman model in that the standard deviation, as well as the mean volume, is time dependent. It differs from the direct solution of the Muller equation in that it is an approximate solution. It should be stressed that only an approach of this type offers any practical possibility of including the effect of electrostatic attraction in the rate of coagulation. Theory
The basic equation expressing the time-dependent size distribution of aerosols owing to Brownian motion (thermal coagulation), gravitational settling, and gravitational coagulation is the Muller equation. With the assumption that the particles can be described as spheres, this equation can be written as (Keller and Schikarski, 1970; Zebel, 1957) r’=r/i/Z
bt
=
$
+ B(T‘,T‘’)} x
{Ko“r.’,r‘’)
r’=O
n(r’,t)“~’’,t)(;,)’
dr‘
- n(r,t)$^ {KoA(r,?’)+ B(T,T’)]x 0
where
T”
=
-
(T’)~]’’~.
The contribution due to thermal coagulation, KOA( T ‘ ” ’ ’ ) , given by
is
The coagulation constant K Ois given by
KO
=
4k T
-
3s and the Fuchs correction for noncontinuum effects, G(T’,T”), is given by 1
where 0 -
2 1 3 r’ + r”
1’
pkT
(r’
+
T”)3
The Cunningham correction
is used to account for deviations from Stokes’ law for very small particles. X is the mean free path of the gas molecules. The second term, B(T’,T”),serves to account for gravitational coagulation. It is described by the equation
(r’)2[1
+ C(r’)l - (r”)2[1 + C ( r ” ) ] }
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
-l
4 3
-*pp
2 9
-
g(T’)5 ~
hs
(Pp
-
Po) [l
+ C(r‘) ]n(r’,t)dr’
(12)
The basic idea in this calculation is that the particle size distribution can be characterized by a particular distribution. I n a sense the assumption is the same as the Lindauer and Castleman model. It differs in that the distribution is not restricted to the case where the distribution can be characterized by one parameter. [In actual fact this model is characterized by two parameters, the total number of particles and the mean volume, but the equation of continuity may be used to eliminate one of the parameters, since the equation of continuity may be used to express the rate of change of the total number of particles in terms of the rate of change of the mean volume. Specifically, we can express dN(t)/dt = f(r,t)(da/dt). Substitution of dN(t)/dt into the coagulation equation results in a differential equation in terms of a (the mean volume) .] The inclusion of more parameters requires that the Muller equation be evaluated a t more than one radius. I n essence, this method relaxes the assumption that coagulation be self-preserving and replaces it with the weaker assumption that the distribution can always be approximated with sufficient preciseness that the errors are not significant. Of critical importance is the choice of the distribution. The distribution that was chosen for this work is the lognormal distribution, which gives the number of particles with radii between r and r dr as
+
h‘(t)f(r,a(t),P(t)l dr (11)
There is considerable controversy over the proper value of c(r’,r’’), the capture efficiency of the particle. A more extensive discussion of this term is given subsequently. 484
The last term is the rate of gravitational settling. The rate of settling is equal to the product of the number of particles and the residence time of the particles in the system. The residence time was given by the quotient of the terminal velocity of the particles and the physical dimensions of the system h. By using large values of h, the effect of gravitational settling is minimized. By using very small values of h, the effect of extensive settling can be observed. The first integral term on the right-hand side of eq 6 accounts for the increase of particles of mass (4/3rp,ra) by the coagulation of two smaller particles whose combined mass is the same as the larger particle. That particles collide to form another spherical particle is strictly true only for liquids. The inclusion of shape into the coagulation equation has not as yet been successfully accomplished. The limit of integration, r’ = r / q i , corresponds to the combination of two equally sized particles. This limit guarantees that the particle combinations are counted only once. The second integral accounts for the loss of particles of size T owing to their collisions with a particle of any other size. Again the two mechanisms of gravitational and thermal coagulation are used. A second important equation is the conservation of mass. The approach discussed here is not restricted to systems where the total mass of the system is conserved. The mass balance is expressed by
The total number of particles, N ( t ) ,is given by
N(t)
=
km
n(r,t)dr = N ( t ) l mf[ln r , a ( t ) , P ( t ) ] dr
(13)
assumes that a t any time t the difference between the correct solution and the trial solution
+*
= {(Ck,X) k = 1, 2, . . , n
(22)
can be made vanishingly small. The choice of the trial function { and, in consequence, the number of undetermined functions Ckare restricted somewhat by the requirement that the function shows agreement with experiment. There are also quite formidable computational problems which impose restrictions on the number of undetermined functions. The method is now exactly analogous to the least-squares method employed in the weighted residual method. The residual, defined as R = L($*) - f(x), is minimized using the appropriate equations (23)
The nonlinear integro-differential equations of eq 6 are solved by this least-squares method. This method differs from the Lindauer and Castleman model in that the standard derivative d@/dt is a time-dependent function as well as the mean radius. But whereas for their model the equation can be solved directly for d a l d t , here it is necessary to employ an approximate treatment. It should be stressed that for their model, by choosing different radii to determine dcrldt, it is possible to obtain different results. The method which was chosen for this paper is to choose dcr/dt and d/3/dt such that the square of the residual is a minimum. If the method were an exact rather than an approximate solution, the values of da/dt and d@/dt would be the same for each pair of radii. If coagulation were described exactly by a log-normal distribution, then dp/dt would be identically zero, since the standard deviation of the distribution is constant. Unfortunately neither of these conditions is met. For this study, the Muller equation, corrected to include the continuity equation, may be expressed as
where
- Dz-dcr - D3-d/3 = R
s
dt
r/=r/Y\j/2
D1 = N(t)
dt
{KoA(r’,r”)
+ B(r’,r’’)} X
r’=O
(:,)2f(r’)f(r’’)dr’
- N(t)f(r)
f(r’) dr’
L=
+
{KoA(r,r‘) B(r,r‘)] x
- 2pgrzfo { 1 + C ( r ) } + 9hs
lm + Lrn (r’)bf(r’){l
f@)-
2Pg 9hrl
486 Ind.
C(r’)} dr‘
0
f(r‘)(r‘)3 dr’
Eng. Chem. Fundom., Vol. 1 1 , No,
4, 1972
and
where
A I R a d x = O
D1
and R is the residual. The values of da/dt and dp/dt are chosen such that
(25)
N 1 Dij = i;= xWk(Gauss-Hermite) 1 Di(tk)DJ(tk)r
(30)
Wk(Gauss-Hermite) and fk are the weights and abscissas for Gauss-Hermite integration; tk is related to the particle radii bytk = - [ ( l n r - a)/(d21n/3)1. It is very fortunate that the distribution function can be cast into the appropriate form for Gauss-Hermite quadrature. Although general Gauss-type quadrature equations may be developed using a general method (Rundell and Gentry, unpublished), there is the serious limitation that the quadrature formulas (weights and abscissas) are different for each change in the parameters. This method makes the inclusion of these undetermined coefficients an almost insurmountable problem and the inclusion of higher-order terms totally impractical. Gauss-Hermite quadrature is a method of approximating integrals of the type .v J
e-t*f(t) dt by -a
f(t,)?V, k=l
Only n points are sufficient to integrate exactly a (2% 1) polynomial. This accuracy in the calculations makes the method exceedingly attractive in that it is possible to acquire with only 5 or 7 points the same degree of accuracy that would be obtained by 50 to 100 points with more conventional quadrature. This degree of accuracy in the number of calculations is important in that each coefficient Di, requires the solution of several multiple integrations. Indeed, it is only by the use of Gauss-Hermite quadrature that calculations of this scale are obtainable and that more sophisticated calculations, including the effects of electrical charges, are even feasible. The major uncertainty in the rate of gravitational coagulation is the capture efficiency, a function of absolute and relative particle sizes. Findersen (1939) first developed the theory of gravitational coagulation without considering the curvature of the trajectories of the fine particles, assigning [ (R T ) / RIZ as the capture efficiency, where R is the radius of the coarser particle and T is the radius of the finer particle. His experiments checked his calculation only when the coagulation of droplets was of similar sizes ; otherwise his calculated values were too high. Langmuir (1948) a l l o ~ e dfor the curved trajectories, assuming the capture efficiency to be a function of the Stokes number but without accounting for the interception effect due to the finite size of the particles. His calculated capture efficiencies were zero for particles less than 15 I.(.
+
N.
& 0 A
o
IO’ P A R T I C L E S I C C 1.5 r..001 MICRON
r.=o
I
r.10329
T I M E (SECONDS)
Figure 1 . The effects of gravitational settling on the particle coagulation rate
REDUCED
NUMBER
DENSITY (N/N.)
Figure 2. Mean aerosol size as a function of time and initial size
Considerable work has been done in attempting to relate the capture efficiency to the Stokes number. For particles less than 0.1 p the Stokes number is less than 10-6. For these low values a zero collection efficiency is in accord with the theory of Langmuir and Blodgett (1948), for both viscous and potential flow, as well as the experiments of Ranz and Wong (1952) and Walton and Woolcock (1960). Because of the uncertainty in the theory (the experimental data do not agree with either viscous or potential flow theory) and because of the small particle size in our calculations the collection efficiency was assigned a value of zero in the work below. I n the calculation below, i t is assumed t h a t the particle size distribution is spatially homogeneous and that gradient coagulation can be neglected. Gradient coagulation, which is proportional to the cube of the particle diameters, is significant for small particles (a < 0.1 p) only in the immediate vicinity of the system boundaries. If gradient coagulation were included, it would be mandatory to account for spatial inhomogeneity in the particle size distributions. Furthermore, the greater part of this coagulation occurs a t the boundary where the particles are removed. Consequently, it is reasonable to neglect spatial variation in the particle size distribution when short time periods and small particles are of primary interest. On the other hand, calculations which neglect spatial variation of the particle size distribution show increasing error with time (Keller and Schikarski, 1970; Levich, 1962; Zebel, 1958). Results
Xccording to the coagulation theory of Smoluchowski (1916, 1918), the reciprocal of the particle number density is a linear relation of time. Even though this derivation is restricted to monodisperse aerosols and constant coagulation coefficient, this relationship has been confirmed by experiment (Derjaguin and Vlasenko, 1948). The derivation is also restrictive in t h a t i t only includes thermal coagulation. I n Figure 1 the reciprocal of the number density was plotted as a function of time for two limiting cases. I n the first a sufficiently high value of depth, h, was assigned (108 cm) so that gravitational settling was unimportant. In the second case, the depth (1.0 cm) was so shallow that gravitational settling became important. The calculations in Figure 1 are for a particle distribution having a n initial mean radius of 0.329 p and an initial concentration of 108 particles/cm3. It is apparent that for large values of time the curve for h = 108 shows that linear relation as predicted by Smoluchowski. On the other hand, for heights of 1.0 em, where gravitational settling is important, Smoluchow-
l------
No I O 6 P A R T I C L E S I C C
01 10.’
8 10-2
REDUCED
10-1
NUMBER
IO
DENSITY ( N / N o )
Figure 3. Mean aerosol size as a function of time and initial size
ski’s relation is not obeyed. Although the derivation of Smoluchomki’s theory indicates that his coagulation rate is in error (Levich, 1962) by a factor of [ l (r/l/?rDt)], which is significant for small values of time, this correction is insufficient to explain the deviation of the calculated results from Smoluchowski’s theory. Thus far experiments have not tested the apparent deviation from Smoluchowski’s theory for small times. For initial particle concentrations of 106 i t would be necessary to have several measurements \Tithin the first few seconds to test the theory. I n summary, a straight line relation between the reciprocal of number density and time is predicted for the condition where experiments have been made. According to theory, the smaller particles groiv more rapidly than the larger particles, owing to thermal coagulation. The coagulation constant for 0.1-p particles is twice as large as that for 0.33-p particles. As the size becomes smaller the discrepancy becomes greater. This trend is valid regardless of whether the Fuchs correction factor is included (Dames, 1966). I n Figures 2 and 3 the reduced mean radius ratio (the mean radius divided by the initial mean radius) is presented as a function of the reduced particle number (the ratio of the total number of particles to the initial number of particles). For these plots the initial numbers of particles were 106 and 108, and the initial value of was 1.5. These figures confirm that the smaller particle sizes (i.0 = 0.01 p) grow faster than particles of larger size. Also, Figure 2 indicates that the mean radius of the largest particles (70 = 0.329 p ) begins to level off
+
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
487
-7
I
1.01-
029
1.06
-
1.04-
e . _ 1.5 097
-
...-bo.*
-
0
o
r..0329
B 1.025
MICRON
r..o.io
N. = I O~PARTICLESKL. r. .O 3 2 9 M I C R O N
1.000
092( 10
I
Id
I 02
-
0
Id
'I 0
3 - P O I N T S O L U T I O N METHOD 7-POINT
0 12-POINT 0 IO-POINT
0.98
T I M E (SECONDS1
Figure 4. The standard deviation parameter as a function of time and number density
I
0.96
0
102
I'0
I o4
. . 4
1 shift the time scale. However, the scale was not shifted two cycles as one might expect. Also, except for the smallest size where noncontinuum effects are important, the smaller the concentrations, the less the magnitude of variation in p was marked. Judging from Figure 3 we can conclude that the larger the size of the particle, the greater the magnitude of 1.5 the deviation in fl and the more rapidly it occurs. A smaller concentration density does not appear to alter this conclusion. I n Figure 5 the variation of the standard deviation variable is presented as a function of the reduced number density. These data are essentially the same as those presented in Figure 4. Only the abscissa has been changed. The graph presents results for initial particle sizes of 0.01, 0.1, and 0.329 0921 . . p for two different values of the initial particle concentration. IO* 10-2 16' The results indicate that the shapes of the curve are similar REDUCED NUMBER DENSITY (N/Na) in both cases and are identical for very small particles. The Figure 5. The standard deviation parameter as a function primary conclusion that can be drawn from this curve is that, of reduced number density and number density for large particles where gravitational settling may be important, the amplitude and period of curve are dependent on particle size. For very small particles the shape of the curve, and starts to decrease when the particle is 6.1 times its initial as plotted in Figure 5, is independent of the initial concentrasize. This is due to the effect of gravitational settling on the tion. I n Figure 4 the same data give two different curves for particle size distribution. For a smaller initial concentration 0.01-p particles. On this basis, the data for p should be pre( N o = 106) Figure 3 shows that the reduced radius reaches a sented on a reduced concentration plot. maximum of 2.6. This difference is due to the fact t h a t gravitaThe effect of using a 16-poiiit solution method rather than tional settling is more important for smaller concentrations, other points, such as 3-, 7-, and 12-point solutions, in the since the thermal coagulation rate is less. least-squares approximation is illustrated. Figure 4 shows the relation between the standard deviation The ordinate and abscissa in Figure 6 are the standard parameter p (the standard deviation in the log-narmal disdeviation parameter and time in seconds. The 16-point tribution is actually In p) as a function of time. The initial method will integrate exactly a 31-degree polynomial, whereas value of p was 1.5 and the values of the ordinates are the the 3-point method will exactly integrate a 5-degree polyvalues p/1.5. Plots were made for three different initial parnomial. This figure gives a measure of the calculational acticle sizes and two different compositions ( N o= lo6and 106). curacy of the method. There appears to be no difference in the For particles having initial sizes of 0.329 p , the value of p shapes of the curves. As the points in the approximation increases with time until i t reaches a maximum value and method are increased, the p curve moves down toward the then decreases. In other words, initially, thermal coagulation smaller standard deviation parameter. I n other words, results in a broader distribution, but as gravitational settling the curve for the 3-point solution is located above that of occurs, the distribution narrows. The behavior for very small the 7-point solution. B u t the 16-point curve lies between particles, 0.01 p, is not as simple. Initially, the parameter p the curves of 7 and 12 points but closer to the 12-point curve. decreases, which indicates that the distribution is becoming Therefore, we can conclude that a 12-point solution is ademore narrow; then i t begins to increase, resulting in a broader quate. distribution. Our time studies were not carried to a sufficient I n Figure 7 the reduced standard deviation parameter fl is length to indicate whether one would eventually have a presented as a function of reduced particle number density. decrease. Horyever, i t is difficult to see any reason why one The plot shows the effect of including the Fuchs correction would not expect behavior similar to that shown by the larger for deviations from continuum mechanics. By comparing the particles. Except for the smallest particle size distribution of curves for the largest particle size, 70 = 0.329 p, with and 0.01 p, the major effect of varying the concentration was to I
, . 1 . ( , 1
488
I
I
I
I
I
I
.
I
I
3
I
l
l
t
l
l
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
l
3
I
.
.
,
.
over existing models. Finally, we have developed a general calculation procedure that can be extended to include the effect of electrical charges on coagulation. Nomenclature r.
A
Y
0 9 2 L , . . . . , ’
310*
16’
.
A1 A ( T ’ ,r”)
= constant for Cunningham correction
B(r‘, r”)
=
B, C(r’)
=
c, D *
. . * , , . 1
I62
REDUCED
Figure 7. The effect
NUMBER
”
~
”
”
lo-’
’
’
“‘‘‘*I
DENSITY(N/Nd
of including Fuchs correction
f
G(r’, r”) 9
h k
KO without Fuchs correction, it is seen that t h e two curves almost overlap, which means that the correction has little effect on coagulation constant for this particle size. For the initial particle size 0.1 F , the curve with the Fuchs correction maintains the same shape as that without the correction, but the magnitude of the reduced standard deviation parameter was slightly higher, which means the broadness of particle size distribution was wider due to correction factor. I n the curve for a 0.01-r particle the parameter had a completely different shape. The standard deviation parameter on the curve with Fuchs correction decreases shortly a t first and then starts to increase until i t reaches p/1.5 = 1.0, whereas the parameter of the curve without correction decreases to a minimum value at N / N o = 0.04. It is interesting that the particle size distribution with the Fuchs correction is broader than t h a t without the correction for small particle size. We can conclude from Figure 7 t h a t the Fuchs correction has no significant effect for the large particle sizes, but the correction strongly effects the size distribution of small particles. The effect of including the Fuchs correction is t h a t the particle size distribution is broader. Conclusion
A method for calculating the time-dependent distribution of a n aerosol undergoing coagulation has been developed. The method is approximate, but differs from other specific distribution models in that three parameters [ N ( t ) ,a ( t ) ,and / 3 ( t ) ] instead of one are time dependent. The equations and derivation of the method were discussed. -4number of numerical calculations were made for diffzrent initial concentrations and particle sizes. The calculations allowed a n evaluation of the effect of gravitational settling and the inclusion of the Fuchs correction for deviation from continuum mechanics. Of considerable interest was the effect on the standard deviation parameter of changes in the initial average radius. It was concluded that the inclusion of the Fuchs correction factor has a very significant effect for very small particles. The parameter p underwent significant changes; calculations based on models which do not account for variation in the standard ?Leviation of particle size are in error. On the other hand, starting with different values of p and the same initial mean radius, the values of p tend to converge to one value. This is, of course, in accord with such models. We believe that the incorporation of the correction for the variation of the standard deviation results in a significant improvement
L X(t)
N O
=
= = = = = = = = = = =
term accounts for the increase of particle of mass by thermal coagulation of two smaller particles with radii of r’ and r” term accounts for the increase of particle of mass by gravitational coagulation constant for Cunningham correction Cunningham correction undetermined parameter diffusivity abbreviation of function, f{r, a ( t ) , p ( t ) } the Fuchs correction for noncontinuum effects acceleration of gravity height of container Boltsmann constant coagulation coefficient nonlinear operator total number of particles a t time t initial total number of particles the number of particles having a volume between v and v dv a t time t constant constant residual function radius of coarse particle mean radius of particles radius of the particle in the upper limit radius of the particle in the lower limit particle radii
n(v, t )
= =
P
=
R(Cj,@ j )
R
= = =
r
=
Tu
r, r’
r” T
= = = = ( ~ 33 11s = temperature, O
t
=
Q
rL
tk
V V+
TV(k) 2
+
K
time, see = abscissas for Gauss-Hermite integration = volume = mean volume = weight for Gauss-Hermite integration =
Qv/t
GREEKLETTERS = mean value of the natural logarithm of the radius of particles = standard deviation in the log-normal distribup(t) tion e(r’, r”) = collection efficiency for two particles of sizes r’,
a(t)
D Dl
x
PP
f
r” = viscosity of gas medium
reduced volume, v/v+ mean free path = density of aerosol particles = density of surrounding gas = trial solution to nonlinear equatioii = =
Literature Cited
Ames, W. F., “Xonlinear Partial Differential Equations in Engineering,” Academic Press, New York, N. Y., 1965. Castleman, A. W., Jr., Horn, F. H., Lindauer, G. C., Proceedings of the International Congress on the Diffusion of Fission Prod. ucts, Saclay, France, November 4-6, 1969. Davies, C. N., Aerosol Sci. 35 (1966). Derjaguin, B., Vlasenko, G., Dokl. Akad. S a u k SSSR 6 3 , 155 (1948). Findersen, W., Meteorol. 2. 56, 356 (1939). Finlayson, B. A,, Scriven, L. E., Chem. Eng. Sci. 20, 395 (1965). Friedlander, S. K., J . Meteorol. 17, 479 (1960). Friedlander, S. K., J . Meteorol. 18, 753 (1961): Friedlander, S. K., Wang, C. S., J . Collozd Scz. 22, 126 (1966). Friedman, S., Chem. Eng. Progr. 48, 118 (1952). Hidy, G. M., J . Colloid Sci. 20, 123 (1965). Keller, K., Schikarski, W., “Zur Theorie des Verhalters nuklearer Aerosol in geschlossenen System,” presented at Aerosol Technology at Frankfurt, Germany, 1970. Langmuir, I., Blodgett, K., J . Meteorol. 5 , 175 (1948). Levich, V. G., “Physicochemical Hydrodynamics,” PrenticeHall, Englewood Cliffs, N. J., p 209, 1962. Ind. Eng. Chern. Fundarn., Vol. 1 1 , No. 4, 1972
489
Levin, L., Dokl. Akad. Nauk SSSR 94, 1045 (1954). Lindauer, G. C., Castleman, A. W., Jr., Nucl. Sci. Eng. 42, 58 (1970). Lindauer, G. C., Castleman, A. W., Jr., Nucl. Sci. Eng. 43, 212 (1971a). Lindauer, G. C., Castleman, A. W., Jr., J . Aerosol Sci. 2, 85 (1971b). Martynov, G. A., Bakanov, S. P., in “Research in Surface Forces,” Derjaguin, B., Ed., translated by the Consultants Bureau. New York. 1963. Muller, H., Kollozd-Rezh. 26, 257 (1928a). Muller, H., Kolloid-Bezh. 27, 223 (192813). Ranz, W., Wong, J., Arch. Ind. Hyg. Occup. Med. 5, 464 (1952). Ritz, W. F., Rezne Anaew Math. 135. 1 (1908). Rundell, D:, Gentry, >., unpublished data. Schuman, T. E. W., Quart. J . Roy. Meteorol. SOC.66, 195 (1940). Smoluchowski, M.von, Phys. 2.17, 557, 585 (1916). ’
Smoluchowski, M. von, 2. Phys. Chem. 92, 129 (1918). Swift, D. L., Friedlander, S. K., J . Colloid Sci. 19, 621 (1964). Todes, 0. bl.,in “Collection: Problems in Kinetics and Catalysis VII,” Izd-vo ANSSSR, 1949. Walton, W., Woolcock, A., “Aerodynamic Capture of Particles,” p 129, Pergamon Press, Elmsford, N.Y., 1960. Zebel, G., Kolloid 2. 156, 2 (1987). Zebel, G., Kolloid 2. 157, 37 (1958). RECEIVED for review July 12, 1971 ACCEPTED August 3, 1972 Computer time was supported through the facilities of the Computer Science Center of the University of Maryland under NASA Grant 398. C. A. Wishes to acknowledge support by the M n t a Martin Foundation. J. G. wishes to acknowledge support by the National Science Foundation through Grant GK-5449.
Surface Waves and Surfactant Effects in Horizontal Stratified Gas-liquid Flow Tupil V. Narasimhan and E. James Davis* Department of Chemical Engineering, Clarkson College of Technology, Potsdam, N . Y . 13676
The effects of gas and liquid flow rates and surfactant concentrations on the flow regimes in horizontal gasliquid flow are presented. A theoretical analysis of two-dimensional waves is shown to b e in good agreement with measured wavelengths and wave frequencies.
S i n c e Jeffreys’ pioneering work (1924, 1925a,b) on the formation of water waves by wind there has been a steadily increasing interest in the phenomenon, partly stimulated by the applications of gas-liquid flow in the chemical process industry. A better understanding of the fluid mechanics of two-phase flow is necessary for the prediction of heat and mass transfer rates associated with cooler-condensers, thin film distillation, and other two-phase flow systems. I n their study of heat transfer to wavy films in gas-liquid concurrent flow, Frisk and Davis (1972) briefly reviewed the literature on heat and mass transfer associated with two-phase flow. Surface waves have been found to enhance interfacial mass transfer by more than 200% in some cases, b u t there are wide variations reported for heat and mass transfer enhancement. The variations are most likely due to the variety and complexity of fluid dynamic characteristics. It is the purpose of this paper to examine some of the wave characteristics t h a t can affect heat and mass transport across wavy interfaces. There is little need to completely review the extensive literature on the interaction of wind and waves, but it is useful to identify three types of studies related to the wave phenomena of interest here. The three groups may be classified as: (1) stability analyses; (2) fundamental experimental studies; and (3) momentum integral analyses. I n the first group are the linear stability analyses of Lock (1954), Feldman (1957), Phillips (1957), Miles (1957, 1959a,b, 1960, 1962, 1967), and Benjamin (1959). More recently Craik (1966) examined the hydrodynamic stability of thin liquid films experimentally and theoretically, and Craik 490
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(1968) and Smith Bnd Craik (1971) analyzed and experimentally studied the effects of surface-active agents on the stability of horizontal liquid films. I n the latter two papers the authors considered the role of the surface-active agent to be t h a t of a viscoelastic membrane and the emphasis is on the surface elasticity. Cohen and Hanratty (1965) also analyzed the instability arising from irreversible transfer of energy from the air flow to small surface disturbances. The object of the various linear stability analyses that have been performed has been to determine criteria for the onset of wave motion and to determine the normal and tangential stresses over the waves. The theoretical difficulties associated with predicting a priori t h e Reynolds stresses due to gas phase turbulence have precluded extension of the theoretical analyses to rigorously account for gas phase turbulence, which is usually encountered in applications. Jeffreys, Phillips, bliles, and Benjamin considered turbulence inasmuch as it determines the mean shear flow. The second group of studies germane to the present work are the experimental investigations which have been performed to measure the onset of instability, to elucidate the flow regimes encountered in gas-liquid flow, and to provide more information about the turbulence characteristics of the gas phase and the structure of the gas-liquid interface. Hanratty and his coworkers have reported several experimental investigations of gas-liquid flow in a horizontal wind tunnel of large aspect ratio (width t o height’) (Hanratty and Engen, 1957; Hanratty and Hershman, 1961; Lilleleht and Hanratty, 1961a,b; Cohen and Hanratty, 1968; Woodmansee and Hanratty, 1969). Hanratty and Engen (1957) observed
