affect the deformation. The analysis includes the material response to explicit strain in liquids, strain from the preferred configuration which the liquid particles seek upon removal of forces. When the Deborah number is small an exponential relaxation of the initial velocities is predicted. Quantitative experimental verification of the above responses was obtained for a Newtonian corn syrup solution and for a viscoelastic solution of Separan AP30. The experimental technique provides a scheme for measuring elastic moduli of liquids; Le., for the Separan solution tested, E was 4.0 dyn/cm2. As a result of this analysis the terms “solid” and “liquid” have little meaning in discussing viscoelastic materials unless the time scale of the process is also included.
t
T X
x
V
ND,
N R ~ Pn
’I
E
Acknowledgment
V,
The assistance of Dr. N. M. Howe in computing the position coordinates shown and in providing many useful suggestions in the analysis is gratefully acknowledged. Assistance in obtaining the experimental data was given by C. Lawrence, Jr.
X
P
E
E(1)
F(1 )
= deformation gradient matrix = relative deformation gradient matrix
=
r
xc 1 )
V
= infinitessimal strain tensor = relative strain tensor
E F
f,
4r,t) Q
Nomenclature
elastic modulus
G(t) = shear relaxation modulus J&), Jl(p) = zero- and first-order ordinary Bessel functions of the argument p, respectively n = numerical index of terms, n = 1, . . ., a P = arbitrary constant indicating hydrostatic pressure r , 8, = cylindrical coordinate system R = vessel radius
dummy time variable time stress tensor, contravariant components Ti’ spatial coordinate system, vector = spatial time derivative of the coordinate position x with components vi, Le., velocity. 1 indicates the spatial acceleration of the coordinate position x = velocity, vector. The components of v are vk (contravariant) and its spatial gradient is denoted in component form by u,jk = Deborah number, dimensionless, expressing the ratio of a characteristic process time to a characteristic material time. = Reynolds number = eigenvalues = viscosity coefficient = preferred coordinate system, vector = components of E = density = function which maps the material coordinates into soatial coordinates. the motion of the body,’vector = function which maps the spatial coordinates into the material coordinates, the motion of the body, vector = angular velocity, function of r and t = angular velocity, constant = gradient operator = unit matrix = transpose of A = total material time derivative of A = = = =
S
1 AT
-
A
literature Cited
Coleman, B. D., Noll, W., Rev. Mod. Phys. 33, No. 2, 239 (1961). Metzner, A. B., et al., Chem. Eng. Progr. 6 2 , No. 12, 81 (1966); A.I.Ch.E. J. 1 2 , 863 (1966). See also Metzner, A. B., Trans. Soc. Rheol. 1 2 , No. 1 (1968). RECEIVED for review June 11, 1968 RESUBMITTED June 2, 1970 ACCEPTEDFebruary 25, 1971
Continuous Foam Drainage and Overflow Fang-Shung Shih’ a n d Robert Lemlich2 Department of Chemical and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 45221
Stationary, overflowing, and refluxing columns of liquid foam are examined with systems of various surface viscosity. Bubble sizes are measured photographically, and foam density is measured via electrical conductivity. Liquid content, internal coalescence, and overflow rate are determined a t steady state. The experimental results are generally in accord with the theory for foam drainage and overflow presented earlier b y the authors’ group.
O v e r the years, a number of relationships have been proposed for predicting the rate of foam drainage and/or overflow (Brady and Ross, 1944; Haas, 1965; Haas and Johnson, 1967; Jacobi, et al., 1956; Miles, et al., 1945; Morgan, 1969; Ross, 1943; Rubin, et al., 1967). However, these relationships 1 2
Present address, Dow Chemical Co., Midland, Mich. To whom correspondence should be sent.
254 Ind.
Eng. Chem. Fundam., Vol. 10, No. 2, 1971
are based on highly simplified models for the drainage channels (such as vertical cylinders or parallel planes) and generally involve empirical constants peculiar to the particular system. Accordingly, several years ago, Leonard and Lemlich (1965a) proposed a more fundamental theory for foam drainage - and overflow. This theory is based on interstitial Newtonian flow through randomly oriented capillaries (Plateau borders) bounded by convexly curved gas-liquid surfaces
(walls) of finite surface viscosity. Later, Fanlo and Lemlich (1965) devised a small modification regarding the prediction of foam overflow rate. Equation 1 shows the resulting theoretical relationship for predicting the steady rate of overflow of a fairly dry foam (meaning a foam of low liquid content) from a vertical column. (The foam is generated by sparging a pool of surfactant solution a t the bottom.)
The theoretical function, + (Fanlo and Lemlich, 1965), which relates the two dimensionless groups, is given over the range of interest by the solid curve of Figure 1. For foams of greater liquid content, the proper theoretical relationship is more complicated (Leonard and Lemlich, 1965a), but it can be satisfactorily approximated by simply finding Q from Equation 1 and then multiplying it by the approximate (but theoretical) wetness correction factor of (1 3Q/G) to give a corrected Q. If the bubble sizes are not uniform, d for use in Equation 1 is found from Equation 2.
+
If there is appreciable coalescence within the rising foam, d is evaluated at the top of the vertical column. For heavy coalescence, or for marked deviation from plug flow for the bulk foam, the theory breaks down. The theory also predicts the bulk density (liquid content) of a stable stationary foam with steady liquid feed a t the top and steady drainage through to the bottom. To facilitate the calculations, a n information flow diagram has been published (Shih and Lemlich, 1967). Further discussion can be found in Lemlich (1968a, 1971). The theory has been tested against experiment by Fanlo and Lemlich (1965), Hoffer and Rubin (1969), Leonard and Lemlich (1965b), Lemlich (1968~)using the data of Rubin,
7
2.4 2.2
Figure 1. Comparison of theory for the rate of steady foam overflow against present experimental results and other work Symbol
w
A 0
\ &
Surfactant Surface viscosity, dyn sec/cm Saponin 4.2 10-4 Albumin 2.6 10-4 Triton X-1 00 1.0 10-4 From Figure 4 of Rubin, et al. (1 9 6 7 ) From Figure 5 of Rubin, et al. ( 1 9 6 7 )
x x x
et al. (1967), and Shih and Lemlich (1967) using the data of other investigations. Agreement between theory and experiment has generally been good, especially in view of the wide range of variables involved. However, there is one important variable which has not been varied widely. It is the surface viscosity. The experimental studies used only surfactants which are known to be of low surface viscosity. Films formed with such surfactants are two-dimensionally mobile (Mysels, et al., 1959). Accordingly, the present experimental study was undertaken to see how well the theory applies with a surfactant such as saponin or albumin which in water reportedly gives solutions of high surface viscosity (Joly, 1964). The aforementioned theory is based on a model of polyhedral bubbles. Such bubbles are approximated in a foam of low liquid content and fairly uniform bubble size. However, if the bubbles vary too much in size and/or the foam is too wet, the bubbles will be more spherical than polyhedral. Accordingly, a second objective of the present study is t o see how well the theory applies to rounder bubbles of nonuniform size. Experimental Details
Figure 2 shows the overall experimental setup. The foam fractionation column was constructed from two 1.5-ft long Pyrex pipes and a n inverted U-bend flanged together atop a 12-1. round-bottomed flask. The column was of 2-in. internal diameter. Eight pairs of platinum electrodes were inserted through the walls of the column in order to measure the electrical conductivity, and thus the bulk foam density (liquid content), in situ. The electrodes were equispaced vertically, and protruded only about 2 mm into the rising foam. Thus these electrodes constituted less of a local disturbance than the earlier complete immersion of two-faced electrodes in the foam (Fanlo and Lemlich, 1965). The electrodes were platinized (Willard, et al., 1964). The electrodes were calibrated against each other and against foams of known density. The conductivity was measured a t 1000 Hz with a Model RC 16B2 bridge manufactured by Industrial Instruments, Inc. Sufficient conductivity was obtained by adding 0.04 L!4 KCl to the feed solution for each run. Control runs showed that a t this concentration the presence of KC1 had no other significant effect. Nitrogen from a pressure cylinder passed in turn through control valves, a rotameter calibrated separately by water displacement, a pair of packed humidifying columns to eliminate any spurious evaporative effects, a psychrometer, and a bubbler, enteiing the liquid pool below the foam column. For most of the runs, the bubbler was an extracoarse sintered glass sparger (200-1 openings). A single glass capillary or a multi-orificed metal spinneret were used for the remainder. The charge in the flask consisted of 0.04 M KCl in distilled water, plus either bovine serum albumin (Armour), saponin (Eastman), or Triton X-100 (Rohm and Haas) as the surfactant. The average molecular weights were taken as 70,000 (Altman, 1961), 432, and 633, respectively. [Caution: saponin is toxic. Ingestion, inhalation, or contact with skin or clothing should be avoided.] The interface between pool and foam was always in the lower portion of the column proper-never in the flask, lest the wider cross section of the flask, with its lower upward foam velocity, induce premature drainage in the foam. The foam rose up through the column and exited through Ind. Eng. Chem. Fundom., Vol. 10,
NO. 2, 1971 255
Figure 2.
Overall experimental setup (not to scale)
A.
Nitrogen supply 8. Control valves C. Flow meter D. Mercury manometer E. Trap F. Packed humidifying columns G. Psychrometer H. Humidifled nitrogen I. Bubbler (sparger) J. Liquid pool K. Foam 1. Foam breaker
the overhead U-bend which was equipped with a variablespeed impeller-type foam breaker and a Nylon mesh sack. Collapsed foam from the sack was returned t o the pool a t the bottom in order to assure steady-state operation and/or pumped to the top as external reflux. Its flow rate was measured by timing the volume collected upon momentary interruption of flow. Bubble sizes were measured photographically in the foam with a n Exakta camera fitted with a 2-in. extension tube, and Kodak Plux-X roll film (ASA 125) a t f/22 and */lw sec. Illumination was provided by a 500-W G.E. photoflood reflector lamp above the camera a t a n incident angle of 45’ and approximately 15 cm from the object. Care was exercised to minimize the period of illumination so as not to heat the foam unduly. As a result of room air conditioning, the operating temperature was generally maintained a t 25 & 1 O C . Steady state was reached within 1 hr. A Beckman DU spectrophotometer as used to check the invariance of the albumin concentration in the pool. Liquid viscosities were measured with a n Ostwald viscosimeter. They proved to be very nearly equal to the viscosity of pure water. A total of 104 runs of three types were conducted, namely, overflowing foam without external reflux, overflowing foam with external reflux, and stationary foam fed steadily a t the top with liquid through the reflux inlet. The gas rate ranged from 3.6 to 20.3 cc/sec for the first and second types of operation, and of course was zero for the third. Further details of experimentation and results have been placed on file (Shih, 1968). 256 Ind. Eng. Chem. Fundom., Vol. 10, No. 2, 1971
M. N.
0. P. Q.
R. S. T. U. V.
W. X. Y.
ReAux Pump Return Foamate sampler Pool sampler Electrodes Distributer switch Conductivity bridge Mobile photographic unit Spectrophotometer Variable speed motor Flow meter (buret) Foam column
Results and Discussion
Controls and Ranges. The linear rate of foam ascent, taken as G/A(1 - D) from the gas rotameter readings, agreed within =t4% with the rate found by directly timing the ascent as viewed through the glass wall of the column. This indicates the foam rose up the column in essentially plug flow. For the runs with foam overflow, the average bubble diameter ranged from 0.077 to 0.30 cm, the rate of total foam overflow on a collapsed (gas-free) basis ranged from 0.015 to 1.03 cc/sec, and the external reflux ratio ranged from zero -11 for to infinity. The pool concentration was 5.23 X Triton X-100, and ranged from 1.85 X lop6 to 11.4 X M for albumin and from 3.13 X to 5.68 X M for saponin. For the runs with a stationary foam, the average bubble diameter ranged from 0.08 to 0.13 cm, the liquid feed rate ranged from 0.118 to 2.17 cc/sec, and the resulting volumetric liquid fraction in the foam ranged from 0.032 to 0.135. The pool concentration was 5.23 x JP for Triton X-100, ’If for saponin. ?ci for albumin, and 8.68 X 11.4 X Figure 3 shows the ranges of pool concentration relative t o the independent critical micelle concentrations as the latter may be inferred from the breaks in the three surface tension curves. These breaks indicate that the critical micelle concentration is roughly 2 x for Triton X-100, 1 X 10” JP for albumin, and 2 x 10-4 *If for saponin. The surface tension measurements, which were conducted with a du Nuoy ring tensiometer, also show that the presence of the 0.04 M KC1 was without significant effect on the surface tension within the surfactant concentration ranges of interest.
FREQUENCY
0o
0.25
0.20
5
\
DISTRIBUTION FUNCTION,
5
IO
I
I
cm-'
\ \ \
\
\Z
0.15
-
e
'%
POOL CONCENTRATIOI 1.85 I Id'M
e
5E y -9
-0
-7
-6
-4
-5
-3
-2
IO~~~(CONCENTRATIO g-mole/ N, I1
Figure 3. Surface tension. pool concentration 0 , albumin without KCI A, saponin without KCI
0.10
m m 3 m
0.05
Arrows indicate ranges of
W, albumin with KCI saponin with KCI The curve for Triton X-1 00 i s from an earlier study (Leonard and Lemlich, 1965bl
0
e,
Some auxiliary shaking experiments demonstrated that the presence of the KC1 was also without apparent effect on foamability or foam stability. Finally, in some extra control runs, the presence of the KC1 was without significant effect on the rate of foam overflow from the column. This accords with a previous investigation with KaCl (Fanlo and Lemlich, 1965). The average diameter of the bubbles decreased considerably with pool concentration, and increased slightly with gas flow rate. The solid curves of Figure 4 illustrate this for albumin. Individual bubble sizes varied appreciably. The broken curve of Figure 4 shows typical results. The fairly wide distribution of bubble sizes made for rather rounded bubbles, since small bubbles can readily fit among large bubbles with a minimum of contact and hence a minimum of distortion from sphericity. This is especially so when the liquid content in the foam is high. The photographs confirmed it. However, the degree of roundness was not so great as it appeared t o the naked eye, or even in the photographs a t first glance. Careful inspection through the large bubbles faintly revealed blunted polyhedra on the other side. I n other words, the shape as seen a t the column wall can be somewhat misleading. If the apparent bubble shape can be misleading, so indeed can the apparent distribution of bubble sizes. On statistical (sampling) grounds alone, the average bubble diameter within the foam should be about 25% smaller than that observed a t the surface for a frequency distribution such as that in Figure 4 because a surface sample discriminates against the smaller bubbles (de Vries, 1957). Of course this still leaves some question regarding distortion and segregation a t the column wall. So in the absence of further information, the average bubble size was determined directly from the photographs, with no correction other than that for optical magnification. Although the pool concentration was varied between groups of runs, it remained constant during the course of a run (by virtue of recycle). This was checked with the spectrophotometer which showed that the concentration of albumin in the pool did not change significantly from the original charge during the course of a run. Foam Density and Coalescence. The aforementioned theory predicts t h a t the liquid content of a stable (noncoalescing) foam should be uniform along the vertical
I
I
I
I
I
5
IO
15
20
25
GAS
FLOWRATE, cclsec
Figure 4. Diameter of bubbles with the sintered glass sparger and albumin solution The three solid curves show the average diameter as a function of gas Row rote and pool concentration; the broken curve shows the approximote frequency distribution with a pool of 3.6 X 10" M concentration and gas Raw rates of 7 to 15 cc/sec
portion of a steadily overflowing foam column (or a steadily fed stationary foam) of uniform horizontal cross section. Present conductivity measurements indicate that this is indeed the case. The solid curves of Figure 5 show Jf albumin in the pool. typical results with 11.4 X Except a t low gas flow rate which promotes coalescence (Lemlich, 1968a, 1968b), the volumetric fraction of liquid in the foam was essentially uniform up the column until the U-bend was approached. Corresponding results were obtained when reflux was employed. However, when the comparatively low concentration of 1.85 X JI albumin was employed, coalescence in the column became significant and the liquid fraction in the foam decreased up the column even a t higher gas flow rates. This is illustrated by the broken curves of Figure 5. The coalescence also evidenced itself in the photographs b y a n increase in average bubble diameter up the column. I n the present experimental work, severe coalescence appeared t o stem more from film rupture (Kitchener, 1964) than from gas diffusion between bubbles (de Vries, 1957, 1971). The foam 0.207
I
I
I
I
I
HEIGHT ABOVE T H E POOL-FOAM INTERFACE, cm
Figure 5. Volumetric fraction of liquid in the foam within the column, without reflux, for various gas flow rates and for two different concentrations of albumin in the pool Solid curves are for 1 1.4 X 10- M and broken curves are for 1.85 X 1 0C6 M . The sharp drop-offs a t the right are due to the U-bend Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
257
experimental results (after making the appropriate inverse application of the wetness factor, 1 3&/G). In making the test, ps for the solution of Triton X-100 was taken as 1.0 x dyn sec/cm. This value was obtained in earlier work (Leonard and Lemlich, 196513) by the independent method of timing the rate of rise of the black spots that form in a thin draining film (Mysels, et al., 1959). Cursory examination of such a draining film of Triton X-100 in the present investigation yielded no disagreement with that value. (The films were studied in well-drained foam.) However, draining films of albumin or saponin revealed no black spots a t any concentration, either with or without KCI. The faint interference patterns observed by daylight appeared to move slowly a t first and then became quite rigid (implying a high p s ) . Accordingly, p s values for the albumin and the saponin systems were obtained by combining the experimental results with the theory, ,and calculating p s . This yielded an average ps of 2.6 X dyn sec/cm for the albumin, and 4.2 x dyn sec/cm for the saponin. (Similar dyn sec/cm for Triton X-100). calculation yields 0.84 X These values of p s for albumin and saponin, while higher than ps for Triton X-100, are still smaller than what might otherwise have been expected. The reason for this is not clear a t present. It may stem from some idealization in the theoretical model, or from the time-dependent or other nonNewtonian behavior which is known often to accompany high surface viscosity. The transit time through a capillary (and hence perhaps the lifetime of part of its wall) is typically less than 1 sec, while surface viscometry involves surfaces that are much older. High surface viscosities are known frequently to increase considerably with time. Good agreement in p s is obtained for Triton X-100 which shows highly mobile draining films and low surface viscosity. I n any case, Figure 1 shows generally good agreement between present theory and experiment, subject to the foregoing remarks concerning p s . The prediction of Haas and Johnson (1967), recast by Lemlich (1968~)into dimensionless form for foams of low liquid content, is also shown for comparison. It is a horizontal line because it does not allow for differing surface viscosities. For further comparison, Figure 1 also shows as shaded regions the clouds of data from the independent experimental work of Rubin, et al. (1967), as analyzed by Lemlich (1968~). This work involved the anionic surfactant, Aresket 300, in water. Figure 6 shows a combined test for overflow rate with the present experimental data, plus data which were previously analyzed by Shih and Lemlich (1967) for Aresket-300 (Brunner and Lemlich, 1963), sodium dodecylbenzene sulfonate (Banfield, et al., 1965), and Triton X-100 (Fanlo and Lemlich, 1965; Leonard and Lemlich, 196513). [The aforementioned data of Rubin, et al. (1967) could not be included in this figure because the values of the individual parameters were not published. ] Figure 6 compares the theoretical overflow rate (per unit column cross section) determined from Equation 1 and etc., against the experimentally measured rate. The diagonal is the line of theoretical perfection. Good agreement between theory and experiment is evident over the wide range of variables involved. The average deviation of theory from experiment is +9'% for the present experimental results and i15% for all the results combined. within an The original theory also permits estimation of overflowing column with or without reflux. Since the computations are a bit involved, a pair of information flow dia-
+
-4
-5
-3
-2
-I
lo9,0(EXPERIMENTAL Q/A,
0
crn/sec)
Figure 6. Test of the theoretical prediction of Fanlo and Lemlich ( 1 9 6 5 ) for the rate of steady foam overflow Symbol
A 0
v
+
-4
Surfactont Saponin Albumin Triton X-1 00 Aresket 300 N a DBS
Surface viscosity, dyn rec/cm 4.2 10-4
x x 10-4 1 . 0 x 10-4 1 . 8 x 10-4 0.82 x 10-4 2.6
I
I
-3
-2
log,,(EXPERIMENTAL
1
-I
0
W/A, crn/sec)
Figure 7. Test of the theory of Leonard and Lemlich ( 1 9 6 5 a ) for the rate of steady drainage through a stationary foam. Symbol
A
+0
Surfactant Saponin Albumin Triton X-1 00 Triton X-1 00 (earlier work)
Surface viscosity, dyn reclcm 4.2 10-4 2.6 10-4
1.0 1.0
x x x x
10-4 10-4
holdup time for the overflow runs was usually only on the order of several minutes, and auxiliary photographs of steadily fed stationary foam taken a t 5-min intervals showed little evidence of bubble shrinkage and growth which are characteristic of interbubble gas diffusion. Overflow Operation. Probably the most important use of the aforementioned theory is t o predict the rate of foam overflow. Accordingly, this aspect of the theory was thoroughly tested. Figure 1 shows the test with the present 258 Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
grams were developed, along with computer programs. The same values of p 8 as for overflow rate (Figures 1 and 6) were employed. The resulting theoretically estimated 3 agreed with experimentally measured 3 within a n average deviation of =t38yOwhen reflux was employed. I n the absence of reflux, theoretical D was always lower than experimental 3 ; theoretically, D should be about twice &/(& G), but the experiments showed it to be about thrice (Shih, 1968). Stationary Operation. Of more fundamental interest is the drainage through a stationary column of stable foam. T h e original theory of Leonard and Lemlich (1965a) relates D to the rate of steady drainage, W / A . This rate can be conveniently found from and the other variables by means of the information flow diagram (Shih and Lemlich, 1967) mentioned earlier. Function 4 is not involved. Figure 7 shows a comparison of theoretically predicted W / A against experimentally measured W / A . The same values of p s as for Figures 1 and 6 were employed. As before, the diagonal is the line of theoretical perfection. Fairly good agreement between theory and experiment is evident, the average deviation being &24% for the present experimental results and +26% for all results combined. This agreement is gratifying in view of the wide range of variables involved and the important fact that the same ps values were employed as were employed for overflow.
+
Conclusions
The effective surface viscosity for the Triton X-100 agrees well with that found independently from the movement of black spots in mobile film. However, the effective surface viscosity for the albumin and the saponin are smaller than what one might expect for a rigid film. With due regard for surface viscosity, the theory appears t o be quite reliable in predicting the rate of foam overflow, and the rate of steady drainage through a stationary foam. Acknowledgment
This work was partly supported by F.W.P.C.A. Research Grant WP-00161 from the U. S. Department of the Interior. Nomenclature
A
=
horizontal cross-sectional area of the foam column, cm2
9 = volume fraction of liquid in the foam
bubble diameter, cm individual bubble diameter, cm volumetric flow rate of the gas, cm3/sec g = acceleration of gravity, cm/sec2 ni = number of bubbles of diameter di
d di G
= = =
& = volumetric flow rate of liquid in the entire overflowing foam, cc/sec
W = volumetric rate of liquid draining through the foam, cc/sec liquid viscosity, dyn sec/cm2 = surface viscosity, dyn sec/cm 4 = function P = liquid density, g/c8m3 z = summation c c =
Literature Cited
Altman, P. L., “Blood and Other Body Fluids,” Dittman, D. S., Ed., Biological Handbooks, Federation of American Societies for Experimental Biology, Washington, D. C., 1961. Banfield, D. L., Newson, I. H., Alder, P. J., A.Z.Ch.E.-Z.Chem.E. Symp. Ser. (London), 1, 3 (1965). Brady, A. P., Ross, S., J . Amer. Chem. SOC.66, 1348 (1944). 2,297 Brunner, C. A., Lemlich, R., IND.ENQ.CHEM.,FUNDAM. (1963). de Vries, A. J., “Foam Stability,” Rubber-Stichting, Delft, 1957, pp 12, 24. de Vries, A. J., “Adsorptive Bubble Separation Techniques,” Lemlich. R.. Ed., Academic Press, New York, N. Y., 1971. Chapter 2. Fanlo, S., Lemlich, R., A.Z.Ch.E.-Z. Chem. E. Symp. Ser. (London), 9, 75, 85 (1965). Haas, P. A., U. S. Atomic Energy Comm. ORNL-3527 (1965). 6,225 Haas, P. A., Johnson, H. F., IND.ENQ.CHEM.,FUNDAM. I
,
,.
(1967)
Hoffer. M. S.. Rubin., E.,, IND.ENQ. CHEM..FUNDAM. 8. 483 (1969). Jacobi, W. M., Woodcock, K. E., Grove, C. S., Jr., Znd. Eng. Chem. 48, 2046 (1956). Joly, M., “Recent Progress in Surface Science,” Danielli, J. F., Pankhurst, K. G. A., Riddiford, A. C., Ed., Academic Press, New York, N . Y., 1964, Chapter 1, pp 33, 45. Kitchener, J. A., “Recent Progress in Surface Science,” Danielli, J. F.. Pankhurst. K. G. A.. Riddiford. A. C.. Ed..’ Academic Press; New York,’ N. Y., 1964, Chapter’2, p 68: Lemlich, R., ‘[Progress in Separation and Purification,” Perry, E. S., Ed., Interscience, New York, N. Y., 1968a, pp 1-56. Lemlich, R., Ind. Eng. Chem. 60, 16 (1968b). Lemlich, R., Chem. Eng. Sci. 23, 932 (1968~). Lemlich, R., “Adsorptive Bubble Separation Techniques,’’ Lemlich, R., Ed., Academic Press, New York, N. Y., 1971, ChaDter 3. Leonaid, R. A., Lemlich, R., A.Z.Ch.E. J . 11, 18 (1965a). Leonard, R. A,, Lemlich, R., A.Z.Ch.E. J . 11, 25 (1965b). Miles, G. D., Shedlovsky, L., Ross, J., J . Phys. Chem. 49, 93
- -- .
1194.5) , , I
Morgan, C. R.. Ph.D. Dissertation. Universitv of Iowa. Iowa Ciiy, Iowa, 1969. Mysels, K. J., Shinoda, K., Frankel, S., “Soap Films, Studies of their Thinning and a Bibliography,” Pergamon Press, New York, N. Y., 1959, pp 10, 85. Ross, S., J . Phys. Chem. 47, 266 (1943). Rubin, E., La Mantia, C. R., Gaden, E. L., Jr., Chem. Eng. Sci. 22, 1117 (1967). Shih, F. S Ph.D. Dissertation, University of Cincinnati, Cincinnati, ahio, 1968. Shih, F. S., Lemlich, R., A.Z.Ch.E. J . 13, 751 (1967). Willard, H. H., Merritt, L. L., Dean, J. A., “Instrumental Methods of Analysis,” 4th ed, Van Nostrand, New York, N. Y., 1964. RECEIVED for review November 17, 1969 ACCEPTED January 25, 1971
Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
259
