Ind. Eng. Chem. Fundam. 1985, 2 4 , 44-49
44
c = critical i, j = component i, j m = mixture (pseudocritical) value
R = reduced value s = solvent 1,2 = (reference) 1,2 Superscripts 0 = infinite dilution value
*
= tracer value
Literature Cited Caldwell, C. S.; Babb, A. L. J . Phys. Chem. 1956, 60, 51. Darken, L. S. Trans. Am. Inst. Minlng Metall. Eng. 1948, 175, 184. Dullien, F. A. L. AIChE J . 1972, 18, 62. Ely, J. F.; Hanley, H. J. M. Ind. Eng. Chem. Fundam. 1981, 2 0 , 323 Ertl, H. H. Ph.D. Thesis, University of Waterloo, London, Ont.. 1973. Ertl, H. H.; Dullien, F. A. L. AIChE J . 1073, 19. 1215. Fair, J. R.; Lerner, B. J. AIChE J . 1976, 2 , 13. Haluska, J. L.; Colver, C. P. Ind. Eng. Chem. Fundam. 1971, 10, 610. Hanley, H. J. M. Cryogenics 1976, 16, 643. Hildebrand, J. Science 1971, 174. 490. Johnson, P. A.; Babb, A. L. J . W y s . Chem. 1956, 6 0 , 14. Kruger, G. J.; Weiss, R. Z . Naturforsh. 1070, 2 5 a , 777. Leffler, J.; Cullinan, H. T. Ind. Eng. Chem. Fundam. 1966, 5 , 189. Leland, T. W.: Chappelear, P. S. Ind. Eng. Chem. 1966, 60(7), 15. Loflin, T.; McLaughin E. J . Phys. Chem. 1989, 73, 186.
Mathur, G. P.; Thodos, G. AIChEJ. 1065. 11. 613. Mo, K. C.; Gubbins, K. E. Mol. Phys. 1976, 31, 825. Murad, S.; Gubbins, K. E. Chem. Eng. Sci. 1977, 3 2 , 499. Murad, S. Chem. Eng. Sci. 1961, 3 6 , 1867. Pitzer, K. S.; Llppman, D. A.; Curl, R. F.; Huggins, C. M.; Peterson, D. E. J . Am. Chem. SOC. 1955, 7 7 , 3433. Reid, R . C.;Prausnitz, J. M.; Sherwood, T. K. "The Properties of Gases and Liquids", 3rd ed.; Wiley: New York, 1977. Sanni. S. A.; Fell, C. J. D.; Hutchison, H. P. J . Chem. Eng. Data 1971, 16, 424. Sanni, S. A.: Hutchinson, H. P. J . Chem. Eng. Data 1973, 18, 317. Slattery, J. C.; Bird, R. B. AIChE J . 1958, 4 , 137. Teja, A. S.;Rice, P. Ind. Eng. Chem. Fundam. Wela, 2 0 , 7 7 . Teja, A. S.; Rice, P. Chem. Eng. Sci. 196lb, 36, 417. Tyn, M. T.; Calus, W. F. J . Chem. Eng. Data 1975, 2 0 , 106. Umesi, N. 0.: Danner. R. P. Ind. Eng. Chem. Process Des. Dev. 1981, 2 0 , 662. Vadovic, C. J.; Colver, C. P. AIChE J . 1973, 19, 546. Vlgnes, A. Ind. Eng. Chem. Fundam. 1966, 5 , 189. Wilke, C . R. Chem. Eng. Prog. 1940, 45(3), 218. Wilke, C. R.; Chang, P. AIChE J . 1955, 1 , 264. Witherspoon, P. A.; Bonoli, L. Ind. Eng. Chem. Fundam. 1969, 8 , 589.
Received for review April 26, 1983 Revised manuscript received January 20, 1984 Accepted July 2, 1984
Part of this work was carried out under Grant No. CPE 8104201 from the National Science Foundation
Theory and Experiment for Interbubble Gas Diffusion in Foam Hslng C. C h e w t and Robert Lemlich" Department of Chemical & Nuclear Engineering, Universify of Cincinnati, Cincinnati, Ohio 4522 1
Results of Lemlich's (1978) theoretical approach for predicting the changes in bubble size distribution that occur over time due to interbubble gas diffusion in an otherwise stable enclosed foam are mutually compared for three generalized dimensionless initial distributions. They are also compared against corresponding results obtained from an extension of de Vries' (1957) theory. Also, the effect of a distribution with a temporarily unchanging number of bubbles is examined, and the influence of lamellar mobility on Permeability is discussed. The results of the theoretical approach are supported by experiment.
Introduction
Under some conditions, a confined foam may be quite stable against the rupture of lamellae. Nevertheless, gas will diffuse from small bubbles to large bubbles, thus enlarging the latter while shrinking the former, even to the point of disappearance. De Vries (1957,1972) proposed a simply theory for predicting the rate of this disappearance. His theory assumes that diffusion occurs only from small bubbles to other bubbles that are, by comparison, always infinitely larger. Also, it does not deal with the growth of these larger bubbles. By expanding on a view first proposed by Clark and Blackman (1948), Lemlich (1978) developed a more detailed theory which predicts the rate of change in size for all the bubbles. As did de Vries, he took the pressure within the bubble as the driving force for diffusion. However, Lemlich's theory considers diffusion as first occurring between each bubble and the liquid. The liquid is assumed to have in solution a spatidy invariant effective average dissolved gas concentration that varies with time. [Neither theory incorporates second-order effects of a Brookhaven National Laboratory, Upton, NY 11973. 0196-4313/85/1024-0044$01.50/0
stabilizing nature that may be significant in an extremely fine foam that consists primarily of micron size bubbles (Ward et al., 1982)l. Further results of Lemlich's theory were subsequently displayed by Ranadive and Lemlich (1979), and the computational procedure was improved by Cheng and Lemlich (1980). The purpose of the present study is threefold to explore further the effect of initial bubble size distribution, to compare results of one theory against the other, and to compare theory against experiment so as to examine the magnitude and behavior of the effective permeability. In this way, further insight can be gained into the manner in which the bubble size distribution changes with time as foam ages. This, in turn, bears on such matters as surface area, adsorbed holdup, and liquid content in foam. There is also some analogy between the aging of foam and the aging of certain other aggregates (Vengrenovitch, 1982; Rivier, 1983), including emulsions, precipitates, metal grains, and tissue cells. Theory Initial B u b b l e S i z e D i s t r i b u t i o n s . Three different
forms of initial continuous bubble size distribution will be considered. They are the empirical distribution proposed 0 1985
American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 24,No. 1, 1985 45 1.0
by de Vries (1957,1972) for foam produced by a high-speed mixer, the theoretical Maxwell-Boltzmann-like distribution attributed to Bayens by Gal-Or and Hoelscher (1966), and the simple Boltzmann-like distribution recently proposed by Lemlich (1982) which exhibits the unusual property of stereological invariance. Cast in generalized dimensionless form convenient for the present analysis, the distributions are given by eq 1-3 respectively. 2.082R de Vries: 4(R,O) = (1 0.347R2)*
I
+
Bayens:
32 4(R,O) = -R2 exp(-4R2/r) l?
Lemlich:
?r
4(R,O) = 5 R exp(-7rR2/4)
(2) (3)
R is the bubble radius, r, divided by the initial arithmetic number-mean radius, r1,JO), and 4(R,O) is the dimensionless initial number-frequency distribution function of bubble radii. By virtue of their dimensionless form, eq 1-3 are free of parametric constants. De Vries’ Approach with Present Extension. De Vries combined the classical law of Laplace and Young with a form of Fick’s law of diffusion to yield eq 4 for the lifetime, 7 , of a small spherical bubble
T Figure 1. Fractional number of bubbles remaining as a function of dimensionless time starting with the initial size distributions of (1) de Vries (1957,1972), (2)Bayens (Gal-Or and Hoelscher, 1966), and (3)Lemlich (1982). The broken curves are based on an extension of de Vries’ (1957, 1972) theoretical approach. The solid curves are based on Lemlich‘s (1978) theoretical approach. W
D -2
r
.I
7=-
(4)
2yJRT
where P,!yJRT is taken as constant. P, is the prevailing surrounding absolute pressure (such as atmospheric), y is the surface tension, J is the effective permeability to gas transfer as defined by Lemlich (1978) which is twice that of de Vries, R is the gas constant, and T is the absolute temperature. Since all bubbles of a size equal to or smaller than r will disappear within time 7,eq 4 was then combined with eq 1 (originally in dimensional form) to yield an expression for the fractional number of bubbles surviving at time 7. In present dimensionless form his result is given by eq 5. (5)
T is the dimensionless time which is defined as 2JyRT~/P,rq,~(0), and N(T)/N(O)is the number of bubbles surviving a t T divided by the initial number of bubbles. In order to examine the effects of other initial distributions, the present authors now apply de Vries’s approach to the generalized initial distributions represented by eq 2 and 3, in turn. The final results are eq 6 and eq 7, respectively.
(7) Equations 5-7 are shown as broken curves in Figure 1. Lemlichs Approach and Present Results. In contrast, Lemlich’s theory yields eq 8 for every bubble, whether it is shrinking or growing. dr - K d7 - (r:,
:>
K is defined as 2JyRT/P,, and r2,1is the number-mean
R Figure 2. Distributions of dimensionlessradii of bubbles computed from theory at various successive dimensionlesstimes, based on the number of bubbles present at zero time, starting initially with the distribution of Lemlich (1982).
bubble radius by second and first moments. Thus r shrinks when r < r2,1and r grows when r > r2,1. For the purpose at hand, eq 8 is recast in dimensionless form as 1 -a= - - 1 (9) dT R2,l R where Rz,l is defined as r2,1/rl,o(0).Equation 9 is solved numerically by finite differences via digital computation, which implicitly takes K as constant. Of course, R is necessarily constrained to positive values. As an illustration, Figures 2 and 3 show results at successive dimensionless times for the most recently proposed of the three initial distributions which is eq 3. While @(R,T)is the dimensionless number-frequencydistribution function of R at time T based on the number of bubbles present initially, +(R,T) is the corresponding dimensionless number-frequency distribution function based on the number of bubbles present ”currently”,that is, at time T. Note that 4(R,T) and +(R,T) simply equal their corresponding dimensional distribution functions multiplied by rl,o(0). Thus the areas under the successive curves of Figure 2 plainly illustrate the decline in the number of bubbles, while the profiles and peaks in Figure 3 clearly show the progressive shift in the distribution toward larger radii. Returning to Figure 1, the solid curves show the prediction of Lemlich’s theory starting with each of the three initial distributions, eq 1-3. The less rapid decline at the
46
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985 m
foam, this seeming relative insensitivity to variation in continuous initial distribution is worthy of note and of further investigation. A recent paper by Monsalve and Schechter (1984) appears to have embarked in this direction. In the meantime, an average curve (drawn dashed) for S(T)/S(O) is added to Figure 4. It is well fitted empirically by
R Figure 3. Distributions of dimensionlessradii of bubbles computed from theory at various successive dimensionless times, based on the number of bubbles present at each such time, starting initially with the distribution of Lemlich (1982).
in
E
0
1
2
3
4
5
6
T Figure 4. Progressive increase in key dimensionless mean bubble radii (left ordinate) and progressive decrease in relative total surface area (right ordinate), all as computed from theory, starting with the initial size distribution of (1) de Vries (1957, 1972), (2) Bayens (Gal-Or and Hoelscher, 1966), and (3) Lemlich (1982).
higher values of T compared to that of de Vries reflects the allowance in Lemlich's theory for the temporary growth of intermediately sized bubbles before their shrinkage and ultimate disappearance. Continuing with the said theory, Figure 4 depicts the progressive changes in two key statistical parameters. Rl,o is the dimensionless arithmetic number-mean radius, defined as the arithmetic number-mean radius at time T divided by rl,o(0). R3,2is the dimensionless Sauter mean which is defined at the number-mean radius by the third and second moments at time T divided by rl,o(0). Omitted for the sake of clarity is R2,1. It is correspondinglydefined and falls approximately midway between Rl,o and R3,2. Similarly omitted, and also correspondingly defied, is R3,1. It falls roughly midway between Rzqland R3,2. With all three initial distributions, Figure 4 shows that the various dimensionless all increase with T. In addition, these curves show t i e relative effects on RjSkof starting with different distributions. For a unit volume, S(T)/S(O) = R3,2(0)/&2(r), where S(T) is the total bubble surface area at time T and S ( 0 ) is the initial total bubble surface area. Figure 4 also shows the progressive decrease in S(T)/S(O). The effect of the choice of initial bubble size distribution on this fractional decrease as a function of T is evidently much more modest than on the other distribution parameters. Since total bubble surface is perhaps the most important property of
According to eq 10, the dimensionless half-life of fresh foam surface is very roughly 6, barring lamellar rupture of course. Discontinuous Distributions. All of the foregoing distributions are unimodal. They are also continuous down to a bubble size of zero. Results from an initial distribution that does not extend to zero bubble size can be quite different-even qualitatively contrary. For example, generally dRl,o/dT> 0 for the former (as may be seen from Figure 4), while dRl,o/dT C 0 for the latter. This latter behavior can be most easily appreciated by considering a simple system consisting of just two spherical bubbles of nearly equal size. With conservation of gaseous volume, the combined surface area will decrease with time as gas diffuses from the smaller bubble into the larger. However, the arithmetic number-mean radius will decrease until the smaller bubble completely disappears. At that point, it will (abruptly) increase. The decrease in Rl,owith T for a distribution that does not extend to zero size is derived in a more general fashion via population balance in the Appendix. Some other generalizations are also confirmed in the Appendix. Comparison with Experiment Previous Work. De Vries (1957, 1972) reported reasonable agreement between photographic experimentswith foams generated by a high speed mixer and the prediction of N(T)/N(O)from his dimensional form of eq 5 when the permeability was assumed to be governed by Fickian diffusion through a liquid film. However, the limitations of his theory preclude meaningful comparisons for bubble surface or size distribution. In a recent study, Nishioka et al. (1983) generated, in turn, a very fast draining foam of continuously varying bubble size distribution with an initial arithmetic number-mean radius of 154 pm and a very slow draining foam of continuously varying bubble size distribution with an initial arithmetic number-mean radius of 6 pm (Nishioka, 1982). The progressive change in bubble surface area was obtained indirectly by measuring the pressure change in the head space over the foam in the chamber (Nishioka and Ross, 1981) and applying the Ross-Tate law which is an equation of state for foam (Ross, 1969). The experimental results were then compared against theoretical results computed from eq 8 with the initial distribution taken to be that of de Vries, that is, eq 1 in specific dimensional form. The said theoretical results are thus the particular dimensional equivalent of the lowest curve for S(T)/S(O) in Figure 4. For the fast draining foam, the comparison yielded a high effective J. This can be attributed partly to preferential accumulation of liquid in the Plateau borders and unreplaced gravitational drainage, which together thin the lamellae, and partly to the very high lamellar mobility which induces considerable micromixing in all the interstitial liquid. Furthermore, rapidly thinning lamellae are prone to rupture. Any rupture that might have occurred would have necessarily increased the actual rate of
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985
47
Table I. Experimental Results for the Aging of Foams with Initial Polymodal Bubble Size Distributions bimodal trimodal tetramodal time, s
rl,o, cm
r3,1,
0 1200 2400
0.0859 0.0857 0.0854
0.0894 0.0901 0.0910
cm
r3,2,
cm
0.0906 0.0914 0.0923
cm
r 3 , 1 ~cm
r3,2,
cm
rl,otcm
r3,1,
0.0921 0.0917 0.0911
0.1006 0.1015 0.1039
0.1039 0.1045 0.1088
0.0920 0.0882 0.0877
0.0942 0.0950 0.0956
rl,O,
surface reduction. The theory does not incorporate rupture. For the slow draining foam, the comparison yielded a low effective J. This can be attributed to the greater foam stability, the minimal lamellar mobility, and the presumably more spherical bubbles. Furthermore, the presence of any significant surface [skin] resistance would also decrease J. Finally, the second-order stabilizing effects mentioned in the introduction may have exerted some appreciable influence a t early times when the average bubble size was smallest. Present Work. As pointed out in an earlier paper concerned with errors in measurement (Cheng and Lemlich, 1983),the bubble size distribution at a containing wall is not representative of the distribution in the bulk foam. Furthermore, the presence of such a wall will itself influence the change in bubble sue distribution with time. (The same can also be said of a free surface). Nevertheless, in an effort to obtain at least a rough but direct comparison between theory and experiment which would yield more insight into J , the authors photographed the successive changes in bubble size as seen through the transparent wall of a vertical Pyrex pipe of 5.1 cm i.d. that contained foam generated by sparging bubbles of approximately predetermined size through a pool which consisted of the nonionic surfactant Triton X-100 in an aqueous solution of 0.01 M KC1 and 10 vol % USP glycerine. Measurements of foam density and electrical conductivity were also taken. Through the proper choice and use of bubblers, initial distributions were obtained that consisted mostly of just two, three, or four different bubble sizes, as desired. The apparatus is described in the aforementionedpaper (Cheng and Lemlich, 1983). Further details are on file (Cheng, 1980). The bubbles in the field of view were taken to be a microcosm. Therefore, no correction for sampling error was required. However, the microcosm was not truly isolated. A stabilizing liquid feed to the top of the column of foam drained down through the foam under observation. This necessarily affected the results to some extent but ensured that there was no rupture of lamellae. Since these approximately discrete, initial, bubble size distributions were very unlike the continuous distributions of eq 1-3, the theoretical results of Figures 1-4 could not be applied to them. Accordingly, the theory represented by eq 8 was applied directly. This was done, via finite differences and digital computation, for several constant values of K. The curves of Figure 5 show the theoretical results for various key mean radii that stem from the initial bimodal distribution. Figures 6 7 show comparable results for the trimodal and tetramodal distributions. In all three figures, the decrease in the arithmetic number-mean radius is evident, as are the increases in r3,1 and r3,2. From its definition, rz,l = r:sl/r3,2.(By way of auxiliary comparison, if the bimodal distribution were perfectly discrete with the r3,1, and same initial key mean radii, the changes in r3,2 over the given time span would be about half those indicated in Figure 5 ) . The corresponding experimental results are listed in Table I and shown by the points in Figures 5-7. Com-
I 0095
cm
r3,2,
cm
0.0964 0.0975 0.0986 I
K. cm21sec
-
1.6 X 1 0' -----3.4 X l o - ' '3. I
0 090
5.1
E
: I
I
i -
0.085
0.080
-
0
1000
2000
3000
r, sec
Figure 5. Comparison of theory with experiment in terms of key mean bubble radii for a bimodal size distribution.
0.085
0
1000
2000 K,
3000
sec
Figure 6. Comparison of theory with experiment in terms of key mean bubble radii for a trimodal size distribution. K, cm8/rec -1.6X10'
---4.4
xi07
5.2 x i 0 7
0.090
0.085
1
'
I I
I
1000
2000
3000
r, sec
Figure 7. Comparison of theory with experiment in terms of key mean bubble radii for a tetramodal size distribution.
parison against theory indicates an effective K of the order of lo-' to lo4 cm2/s for the experimental systems. Substituting into the definition of K yields an effective J of the order of to 10-l' s/cm.
48
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985
Table 11. Experimental Bubble Counts for the Aging of a Foam with an Initial, Fairly Continuous, Bubble Size Distribution time, s radius X 39.4, cm 0 2400 3600 4 1 1 0-0.085 3 6 0.085-0.125 2 4 10 5 0.125-0.165 13 7 0.165-0.205 I 15 22 30 0.205-0.285 6 15 0.285-0.325 18 2 8 16 0.325-0.355 12 4 0.355-0.435 45 10 6 0.4354.545 68 10 6 0.545-0.655 48 9 100 12 0.655-0.885 11 12 40 0.885-1.005 42 14 12 1.005-1.2 15 11 38 18 1.215-1.435 14 6 15 1.435-1.665 18 22 18 1.665-2.075 27 0 36 2.075-3.005 10 0 0 3.005-3.455
0.002
i 0 N ”
0.001
L
0.000
0
1200
2400
3600
7,s Figure 8. Experimental results for the progressive shrinkage of small bubbles, each surrounded by much larger bubbles.
For additional comparison, Table I1 lists initial and subsequent bubble-size counts, as photographed through the glass wall, for a foam of fairly continuous bubble size distribution generated by a sintered glass bubbler. For surface survival, the experimental data yield S(T)/S(O) = r3,2(0)/r3,2(T) = 0.57 a t 2400 s and 0.49 a t 3600 s. Substituting in eq 10 and combining with the definition of T yields an effective J of about 1 X lo-” s/cm a t each time. This falls within the above range for J . Finally, photographs were taken of some small bubbles each of which was surrounded by much larger bubbles (Cheng, 1980). According to de Vries’ theory (eq 4) and according to Lemlich’s theory [eq 8 with r2,1 m], these should exhibit a linear variation of r2 vs. time with a slope of -K and -2K respectively. Figure 8 shows the results. The slopes are approximately equal to -4 X lo-’ cm2/s, s/cm according which yields a value of J of about 4 X to Lemlich’s theory. This value of J also agrees well with to lo-” s/cm obtained above. the range of If surface resistance to diffusion is negligible, and if the remaining resistance can be viewed as that of a stagnant film equal in thickness to the volumetric average thickness, then Lemlich suggested eq 11 for the independent evaluation of J for spherical bubbles. 30(1 - 33) J= H33r3.2
-
D is the diffusivity of the gas through the liquid, H is Henry’s constant, Zl is the volumetric fraction of foam that is liquid, and r3,2is the Sauter mean bubble radius. However, eq 11underestimates J , partly because much of the liquid in the foam accumulates in the Plateau
borders, thus thinning the lamellae. Accordingly, the right-hand side of eq 11was divided by the fraction of the interstitial liquid that resides in the lamellae (Datye and Lemlich, 1983) as determined in the manner of Agnihotri and Lemlich (1981) from measurements of Zl and electrical conductivity. Also, a minor adjustment was incorporated by changing the spherical coefficient of 3 to the regular dodecahedral coefficient of 3.3 to account for bubbles being out of round. The then resulting J was of the order of s/cm. This independent J of s/cm is generally less than the J of to s/cm obtained above from Lemlich’s theory and the successive photographs of the aging foam. However, agreement is within an order of magnitude. The difference is attributed, at least in part, to the movement of interstitial liquid within the foam, which causes micromixing and thus increases the effective J over what it would be were the liquid completely stagnant. However, the assumptions of theory and the proximity of the retaining wall are not to be overlooked as possible contribution factors. Conclusions According to Lemlich’s theory for interbubble gas diffusion in otherwise stable foam, variations among reasonable continuous initial bubble-size distributions produce relatively little effect on the fractional decrease in total bubble surface area as a function of dimensionless time. If the parameter K does not vary with time, the said fractional decrease is given approximately by eq 10. Increases in key mean radii are shown in Figure 4. In a foam where the total number of bubbles does not change with time, the results may be qualitatively contrary to those for a foam in which they do change. Worth noting among such results is a decrease in the arithmetic number-mean radius of the bubbles with time before any bubbles shrink to zero size. Drainage, in all its aspects, greatly influences the effective permeability, J. However, with proper J , Lemlichs theoretical approach is supported by experiment. Appendix For a closed system with an unchanging total number of bubbles, J/(R,T) = @(R,T). In present dimensionless form, the population balance (Randolph and Larson, 1971) for such a system yields
where U = dR/dT and $ = $(R,T) for the sake of brevity. Combining eq 9 with eq 12 gives
Integrating with respect to R via the moment transformation method
Carrying through the integration by parts, and noting that $ = 0 for R = 0 and for R = m, yields
where
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985 49
For n = 0, eq 15 gives dpo/dT = 0. This accords with the stated condition of no change in the total number of bubbles. For n = 1, eq 15 becomes
sented by the subscripts when such are present, cm
R = dimensionless equivalent of r defined correspondingly R = specific gas constant, cm2/szK S = total surface area of bubbles per unit volume, cm-I T = absolute temperature, K U = rate of change of bubble radius, cm/s Greek Letters
surface tension, g/s2 4 = dimensionless number-frequency distribution function of bubble radii based on the number of bubbles present initially p = moment of dimensionless bubble radius 7 = time, s T = dimensionless time, K7/rq,0(0) = dimensionless number-frequency distribution function of bubble radii based on the number of bubbles present at current time
Now, Hulburt and Katz (1964) suggest that eq 18 is often an adequate approximation for P - ~ .
y =
Also, by virtue of normalization, po = 1. Substituting in eq 17 yields
+
Subscripts
since 2p: - I.L~> 0 by eq 18 and 1.112 - p2 equals the negative of the variance. The result dpl/dT < 0 implies dRl,o/dT < 0 which was presented earlier in the present paper for particular discrete distributions. Thus,dRlp/dT < 0 is now shown to be a more general result, provided of course there is no change in the total number of bubbles with time. For n = 2, eq 15 together with the aforesaid relationship for the variance give dP2 -- 2(P12 - P2) < _
(20)
dT P2 since = 1. Equation 20 confirms the decrease of bubble surface area with time. Finally, for n = 3, eq 15yields dp3/dT = 0. This accords with the condition of conservation of total gaseous volume. Nomenclature D = diffusivity of gas through liquid, cm2/s D = volume of liquid in foam divided by total volume of foam H = Henry's constant, cm2/s2 J = effective permeability to transfer of gas between bubble and midplane of lamella, s/cm K = parameter 2JyRT/P,, cm2/s n = exponent N = total number of bubbles Pa = absolute pressure of surroundings, g/cm s2 r = radius of bubble when subscripts are absent, or mean radius of bubble taken by moments of the orders repre-
j , k , n, -1, 0, 1, 2, 3 =order of moment
Literature Cited Agnihotri, A. K.; Lemiich, R. J . Colloid Interface Sci. 1981,8 4 , 42. Cheng, H. C. Ph.D. Dissertation, University of Cincinnati, Cincinnati, OH,
1980. Cheng, H. C.; Lemlich, R. Ind. Eng. Chem. Fundam. 1980, 19, 133. Cheng, H. C.; Lemiich, R. Ind. Eng. Chem. Fundam. 1983,22, 105. Clark, N. 0.; Blackman, M. Trans. Faraday SOC. 1948,44, 1. Datye, A. K.; Lemllch, R. Int. J . MuMphase Flow 1983,9 , 627. de Vries, A. J. "Foam Stability"; Rubber-Stichting: Delft, 1957. de Vrles, A. J. I n "Adsorptive Bubble Separation Techniques", Lemlich, R., Ed.; Academic Press: New York. 1972;pp 7-31. Gal-Or, B., Hoelscher, H. E. A I C M J . 1986, 12, 499. Hulburt. H. M.; Katz, S. Chem. Eng. Sci. 1964, 19, 555. Lemlich, R. Ind. Eng. Chem. Fundam. 1978, 17, 89. Lemiich, R. Chem. Eng. Common. 1882, 76, 153. Monsake, A.; Schechter, R. S. J . Colloid Interface Sci. 1984, 9 7 , 327. Nishioka, 0.; Ross, S. J . Colloid Interface Sci. 1981. 8 1 , 1. Nishioka, 0. M. Private communication, Owens-Corning Fiberglass Corp., Granville, OH 43023,1982. Nishdca, G. M.;Ross, S.; Whitworth, M. J . Collokd Interface Sci. 1983,9 5 ,
435. Ranadive, A. Y.; Lemiich, R. J . Colloid Interface Sci. 1979, 70, 392. Randolph, A. D.;Larson, M. A. "Theory of Particulate Processes"; Academic Press: New York, 1971. Ross, S . Ind. Eng. Chem. 1969,61(10), 48. Rivier, N. Phil. Mag. B 1983,47(5), L45. Vengrenovitch, R. D. Acts. Metsll. 1982,30, 1079. Ward, C. A.; Tikuisls, P.; Venter, R. D. J . Appl. fhys. 1982, 53, 6076.
Received for review May 23, 1983 Revised manuscript received April 18, 1984 Accepted April 23, 1984
This material is based upon work supported by the National Science Foundation under Grant No. ENG77-09887.
