W. M. JACOBP, K. E. WOODCOCK, and C. S. GROVE, JR. Chemical Engineering Department, Syracuse University, Syracuse 10, N. Y.
Theoretical Investigation of Foam Drainage A theoretically derived equation Correlates foam drainage with liquid viscosity Helps predict drainage behavior of fire-fighting foams
I N GENERAL, the stability of aqueous foam is determined by two different phenomena: the rate a t which liquid drains from a foam, and the rate at which the body of the foam breaks down. If the foam is unstable, much of the liquid draining from the foam is liquid released by the rupture of bubbles; this is the case in the very unstable foams formed by ethyl alcohol and water solutions, for example. In the more stable foams, there is no perceptible breakdown of the bubble until after the great majority of the liquid has drained from the foam. In this condition, the drainage from the foam is almost completely unaffected by the bubble breakdown. Behavior of this type is characteristically exhibited by foams generated from solutions of many of the common detergents and surfactants, and by fire-fighting foams. In fire-fighting foams, where the high heat capacity and low viscosity associated with set foams are important, the rate of liquid drainage from the foam becomes a paramount measure of stability. These studies were originally undertaken in an attempt to predict and correlate Present address, Atomic Power Division, Westinghouse Electric Corp. Pittsburgh, Pa. 1
2046
the drainage behavior of proposed firefighting foams. Numerous authors have considered the phenomena of foam drainage, and have presented equations, both theoretical and empirical, with which to represent these phenomena. Recently Bikerman (3) has given an excellent critical review of this literature. Several authors (7, 5, 7, 9) have used the equation V = 1 - e-ct
(1)
VO
to correlate experimental drainage data with such diverse foaming solutions as beer, wine, various concentrations of saponin, egg albumin, peptone, and gelatin. Radow (72) has found it unsuitable for correlating drainage data of foams from commercial fire-fighting liquids. Miles, Shedlovsky, and Ross ( 7 7 ) have treated foam as a series of capillary tubes. The drainage rate from these tubes is supposedly regulated by viscosity (following Poiseuille's law), and by the difference between the liquid height in these tubes and the equilibrium capillary height. On the basis of this model, they derived the following drainage equation (V,
INDUSTRIAL AND ENGINEERING CHEMISTRY
-
V)
- alog(Vo -
V)
4b
= ct
(2)
which has been successfully used to represent the experimental data presented by its authors. Gibbs discussed the drainage of single films (8) and compared this phenomenon with drainage of liquid from between two parallel, immovable plates. In this configuration, the average velocity of the draining liquid is u = px 62/12 i.r
(3)
Ross (73) has treated foam drainage using a model similar to Gibbs. He also considered foam drainage as the floiv of liquid from between parallel plates, but allowed the separation of these plates to vary in proportion to the liquid content of the foam at any given time. O n this basis, the following equation was obtained: l / ( V o - V)' - 1/Vo2
kt
(4)
As far as is known, there is no record in the literature of the successful application of this equation. It has been found to be inconsistent with the experimental data of this study.
Development of Theory In the drainage of liquid from a foam the liquid might be more properly thought of as draining through the
AQUEOUS FOAMS foam, rather than from it. Liquid draining from the upper layers of a foam column drains through every bubble wall beneath it. It is logical to expect, then, that after a short drainage period the liquid concentration in the upper portion of a foam column would be much less than the liquid concentration in the lowest few layers of bubbles. In the model assumed here, foam drainage is considered as liquid flowing from between plates; however, the distance between the plates is allowed to vary with height as well as time. In this manner, allowance is made for greater liquid concentrations that exist in the lower portions of foam columns. Let 1 represent the liquid content at some point in the column of foam, expressed as the volume fraction of liquid. Z denotes the distance from this point to the top of the column of foam. The assumption is made that the liquid flow rate through the foam column at height Z is determined by an average bubble wall thickness, 6, and further, that this average bubble wall thickness is a linear function of the liquid content. This latter assumption may be expressed mathematically by 6 = kl
(5)
where k is a constant for any given bubble diameter. In an actual foam system it is reasonable to assume that there is some average bubble diameter such that the corresponding k value will accurately represent the system. The weight rate of liquid flow between two vertical plates is given by
where w is in grams per second, and S is length of plates in centimeters. By combining Equations 5 and 6, and considering the mean bubble wall thickness, 6, as equivalent to the plate separation, xo, the flow rate of liquid through some point in the foam may be expressed as a function of the liquid content. Volume units are used for the drainage rate, and the equation is put on a basis of unit cross-section area of foam by defining S' as the liquid film length per square centi,meter.
the lower boundary by R
balance across the slice may be written as
If Equation 7 is differentiated and substituted into Equation 8, final differentia1 equation representing liquid drainage within foam column is obtained: a1 at
p8k3l2S' a1
40
az
(9)
This may now be integrated, using as boundary conditions the initial values - .
dl of 1 and at Some arbitrary point, bZ
z = z,.
A particular solution of the equation may be easily found by the method of separation of variables, with the result
The time required for any given volof liquid to drain from a foam varies directly with the viscosity of the liquid. This is typical of viscous flow theories of drainage, and has received limited experimental verification (6, 70). In their present form, the application of these equations to the correlation of exDerimental data is cumbersome. However, by considering the height of the foam column constant during drainage process (usually a close approximation), it is possible to lump the height terms with other constants of the equations and produce simplification.
,-
1 =
-+_____ z fa
(10)
dpg:! +b
where a and b are constants. By solving for them in terms of the stated initial conditions, at Z , and to, one obtains
42
- zo
I =
tip-
+ 10/2 (&) (I1)
1 +
2
(&)
10
There are several points of interest concerning this equation. If the initial gradient of liquid concentration with height approaches zero, the equation predicts that liquid concentration at that point will remain constant in time. Physically, this condition may not be stable, and any random variation in liquid distribution may cause a shift to the normal situation with a positive liquid gradient. It is apparent, however, that the initial variation in liquid content along the height of the foam column will play an important part in determining its drainage rate. A more useful form of Equation 11 may be obtained by substituting it into Equation 7 to obtain drainage rate. If the initial conditions applied correspond to those at the base of the foam column, the equation then represents the drainage from the over-all column
(7)
R = Equation 7 represents liquid drainage through foam as a point condition dependent upon liquid concentration at that point. I n order to obtain a time dependence of this drainage rate, it is necessary to incorporate Equation 3 into a liquid balance over a differential vertical slice of foam dZ. If the flow across the upper boundary of this slice is represented by R, and the flow across
1
+ 5, the liquid Z
1
(12) The drainage rate may now be integrated with respect to time to obtain the drainage volume from the foam system.
v = 113 [.-
' 0
+ 102'
(
x
R =
13
(Yt
+ 1)"Z
(14) 1
213 Y
(15)
In these forms, with only two empirical constants, the equations may be readily applied to the correlation of drainage data. Constant 0 corresponds to the initial drainage rate, while the time dependence of the drainage rate is a function of y. Experimental Apparatus
The foam meter used in these studies is a typical pneumatic device, similar to several of those previously described (6, 7). Essentially, it consists of a glass tube 7 2 inches long and 20 mm. in inside diameter, with a sintered glass disk of 40- to GO-micron porosity affixed to the lower end. The initial 50 cc. of tube volume are calibrated in 1-cc. intervals, and the remainder of the tube in the 25-cc. intervals. The total tube volume is 750 cc. The tube is surrounded by a water jacket, and water is circulated from a constant temperature bath to maintain the tube at a constant temperature of 25" C. A drawing of the apparatus is shown in Figure 1. I n the operation of this apparatus, measured volumes of air were passed through the porous glass dispersion disk. Air was obtained from the laboratory compressed air system, passed through a felt filter, reduced in pressure through regulating values to a constant pressure of 15 pounds per square inch gage, and throttled through a needle valve into the foam tube. The rate of air flow was measured by means of a small rotameter located iust before the needle valve. so that t h i metering pressure was held constant a t all times. VOL. 48, NO. 11
0
NOVEMBER 1956
2047
Figure 1.
Tests were conducted by introducing a known volume of foam solution into the base of the tube. Air flow was then started and maintained at a constant rate until all the liquid had been converted into foam. At this point, the air flow was cut off, and a stop watch M-as started to record the times taken by varying volumes of liquid to drain from the foam mass. In general, the expansion (ratio of foam volume to liquid volume) of the foam produced could be controlled by using differing air rates. If the foam was being produced by a large air rate, much of the liquid was carried up in suspension between the bubble walls, and a relatively low expansion resulted. At lower air rates, excess liquid drained from the bubbles as they were formed, and higher expansion foams resulted. By simply varying air rates in this manner: a very wide range of expansions could be produced. The drainage data
2048
Diagram of apparatus
reported here, however, were all obtained from foams generated with a constant air rate of 90 cc. per minute. Surfactant Solution Studied
The previous theoretical discussion has assumed a model of foam drainage based on viscous liquid flow; it does not depend in any way upon the chemical structure of the foaming agent. Thus, while the chemical and physical properties of the foam agent may affect the bubble diameter. the bubble wall thickness, and fluid viscosity. they should not affect the mechanism by which the foam body drains. I n an effort to check this assumption, tests have been conducted on several foam agents of widely differing chemical properties. Readily available commercial preparations have been used in most cases. Because the purpose of the study \vas to examine the basic
INDUSTRIAL AND ENGINEERING CHEMISTRY
phenomenon of drainage, rather than to determine the drainage properties of particular chemical compositions, no attempt was made to obtain or prepare chemically pure surfactants. Tests were conducted using nonionic surfactants (Triton X-loo), anionic surfactants (fatty alcohol sulfates), and amphoteric substances (Mearlfoam, Miranol), where the charge of the surface active ion depends on the p H of the environment. Triton X-100 is a condensate of ethylene oxide and octyl phenol, and is supplied by the Rohm & Haas Co. Mearlfoam is a foam concentrate used for making mechanically generated fire-fighting foam solution. It is composed of protein hydrolyzates, with iron salts added as a foam stabilizer, and is typical of a number of commercially available foam liquids of fire fighting purposes. Mearlfoam is supplied by the Mearl Manufacturing Co.
AQUEOUS F O A M S 50
The Miranol compounds are organic detergents supplied by the Miranol Chemical Co., Irvington, N. J. The manufacturer gives their structure as
\
.
R
I
C-N-CH2CH20
/
Na
/\
OH CHzCOONa
where R is a straight-chain alkyl group. R is a 10-carbon chain in Miranol SM, a 12-carbon chain in Miranol HM, and a 14-carbon chajn in Miranol MM. Fatty alcohol sulfates tested were research samples supplied by Du Pont.
p
Correlation of Experimental Results
I n order to correlate experimental results, the theoretical drainage rate equation has been used. Taking Equation 14, and raising both sides to the -2/3 power, one obtains R-wa =
6-313
(Yt
+ 1)
CONSTANTS L I S T E D IN TABLE
I
c
Figure 2.
Correlation of drainage data
(16)
The application of this equation to experimental data involves graphical differentiation of drainage curves, and is subject to some inaccuracies. However, by plotting R-2/3 as a function of time on ordinary graph paper, trial values of constants p and y may be obtained. Usually only slight adjustments are necessary to obtain the optimum fit to the drainage curve. Figures 2 through 5 show typical drainage data obtained from the foam meter, using a wide range of differing surfactant solutions. In all these experimental runs, 50 cc. of solution were used, and a standard air rate of 90 cc. per minute was used for generation of the foam. The times recorded are based on zero time at the cessation of air input upon complete foaming. Each run has been correlated by means of Equation 15, and the resulting constants are given in Table I ; the correlations obtained may be observed in Figures 2 to 5. The solid lines in these figures are plots of Equation 15, using the correlating constants of Table I, while the experimental points have been taken at random from the drainage data. In every case, an excellent fit has been obtained up to the time when at least 60y0 of the liquid has drained from the foam body. Beyond this point, most of the experimental drainage points begin to show deviation from the theoretically fitted curve. There are at least two possible causes for this: changes in the bubble sizes and the structures of the foam mass, or the thinning of bubble walls to such an extent that simple viscous forces are no longer the controlling factor in the drainage rate. Bubble breakdown might be expected
50
40 ,X
ci
u
0-
w
30.
1 a
’ x , x
LL
n w 20
a Y
J
f
J
$
lo
i
X- 2 % MEARLFOAM
f
- CORRELATING LINE, OBTAINED BY FITTING EQUATION 15 TO EXPERIMENTAL DATA. CONSTANTS LISTED IN TABLE
I.
0 0
4
8
12
16
20
VOL. 48,
24
NO. 11
28
32
NOVEMBER 1956
36
2049
50
Figure 5. Correlation of drainage data
40
0
0
d 30 w z a
E D
2
20
3
-I
0
> IO
TO E X P E R I M E N T A L DATA. CONSTANTS L I S T E D IN TABLE 0
2
0
4
6
a
12
IO
14
I.
16
18
TIME, MINUTES
to increase the rate of drainage, rather than to retard it, as observed, so it may be assumed that the influence of bubble breakdown is not appreciable in these cases. The diverse surfactant solutions tested all resulted in drainage curves of the same form. I t may be inferred from this that the basic mechanism of drainage was similar in all these cases, and did not vary with solution composition. Although the forms of the drainage curves were similar, wide variations were observed in drainage rates. This variation is most readily illustrated by the correlation constant, p, shown in Table I. /3 (the initial drainage rate) ranges from a minimum value of 6.55
W
0 020
I3
w
Other Forms of Drainage Curves While the theoretical equations appear to be reasonably successful in correlating
Table I. Solution
Sodium dodecyl sulfate, 2 wt. Yo Sodium tetradecyl sulfate, 2 wt. % Sodium hexadecyl sulfate, 2 wt. % Miranol SM (CIO), 2 vol. % Miranol HM (cia), 2 vol. % Miranol M M (cir), 2 vol. 70 Triton X-100, 2 vol. % Merlfoam, 2 vol. %
X
- 2 Yo SAPONIN SOLUTION
-
OF A R B U Z O V AND GREBENSHCHIKOV (Reference 2 )
-
DATA 0.016
z_ a
-
K
-
n
z 0
0.012
!-
0
Q
K LL
p
0.008
Q
-
K
w
c3
a 5
3.004
Q LT
Y
-
n 0
2050
INDUSTRIAL AND ENGINEERING CHEMISTRY
Correlating Constants Expansion
0
5.0
31.5
1,OO
5.0
30.0
0.95
9.1
23.0
0.83
5.3
27.0
0.79
6.1
24.0
0.70
8.4
16.0
0.45
10,5
12.5 6.55
0.29 0.20
9.6
y
I
I
-
z
5
for a Mearlfoam solution, to a maximum of 31.5 for a sodium dodecyl sulfate solution. These variations in initial drainage rate appear to be partially dependent upon the expansion of the foam. Examination of the table reveals that, in general, foams with the highest expansions exhibit the lowest drainage rates; the converse is also true. This is in qualitative agreement with the theory, which predicted that drainage rate should increase strongly with increasing liquid content.
X
-7
Figure 6. Drainage rate of saponin solution, obtained different type of foam meter
AQUEOUS FOAMS 50
I
I
I
I
2 % SAPONIN SOLUTION DATA OF ARBUZOV AND GREBENSHCHIKOV (Reference 2 )
045. 40 .
Figure 7. Correlation of 2% saponin data (Equation 16)
std -i
e 35
a LT
y
30
LT
25 -
a za n
e
u
20
e
15 -
e
10
0
the drainage data of this study, drainage curves with inflection points have been reported by some previous observers, and it is obvious that these cannot be correlated by the equations developed. Drainage curves of this type typically show a rapid increase in the drainage rate from the initial value, a maximum drainage rate being attained after a relatively short time. After maximum is reached, drainage rate steadily declines toward zero with increasing time. It is suggested that this initial increase in drainage rate results from foam columns where the water distribution is uniform, or even topheavy; as soon as the drainage process starts, this liquid may tend to redistribute, so that the liquid concentration is greatest a t the base of the column. As this redistribution occurs, the bubble walls at the base of the column become enlarged, and an increase in drainage rate is observed. Once an “equilibrium” distribution of liquid is attained drainage rate should steadily decrease according to theory. Bikerman ( 4 ) has reproduced drainage curves obtained by Arbuzov and Grebenshchikov (2) for saponin solutions; these are excellent examples of the inflected curves just described. I n their experiments, foam was formed by sucking a saponin solution through a porous glass disk at the top of the drainage tube. Thus the most freshly generated foam was always a t the top of the foam column, and a t the conclusion of the foaming operation, the liquid concentration in the upper portions of the column would be expected to be at least as great as that a t the bottom. This is in direct contrast to the operation
IO
20
30
40
of the more common pneumatic foam meters, where foam is formed at base of tube, and foam a t top of the foam column has always partially drained a t conclusion of foaming process. The Arbuzov and Grebenshchikov drainage data for a foam of 2% saponin solution have been graphically differentiated to obtain the drainage rates, and these are shown in Figure 6. R-113 has then been plotted with respect to time in Figure 7 . Following the initial increase in drainage rate, this curve very rapidly approaches the straight line predicted by Equation 16. Thus at least latter stages of drainage process are in agreement with theory, and support is lent to the idea that maxima in foam drainage curves are result of redistribution of liquid within column. Acknowledgment
The authors wish to acknowledge with thanks the help of the Mearl Manufacturing Corp., Roselle Park, N. J., and of the Engineer Research and Development Laboratories, DA-44-009 Eng. 1422, Project No. 8-76-04-009, in making this investigation possible by sponsorship of the experimental studies. Nomenclature
g
k a
= gravitational acceleration cm./sec. = proportionality constant, cm. = volume liquid fraction i n foam
R = drainage rate, cc./sec./sq. cm. S = length, cm. S’ = liquid film length per unit area, cm.-1 t = time, seconds u = velocity, cm. per second
50
60
70
80
90
V = drained liquid volume, cc. per sq. cm. Vo = original liquid content of foam, cc. per sq. cm. w = weight rate of drainage, grams per second x o = plate separation, cm. 2 = distance from top of foam column, cm. 6 = average bubble wall thickness, cm. p = density, grams per cc. p = viscosity, poises p, y, a, 6, c = constants literature Cited
(1) Amerine, M. A., Martini, L. P., De Mattei, W., IND. ENG. CHEM.34, 152 11942). Arbuzov, K. N., Grebenshchikov, B. N., J . Phys. Chem. (U.S.S.R.) 10, 32 (1937). (3) Bikerman, J. J., “Foams. Theory and Industrial Application,” pp. 98113, Reinhold, New York, 1953. (4) Bikerman. J. J.. Trans. Faraday SOC. 34, 634 (1938). (5) Blom, J., J. Inst. Brewing 43, 251 (1937 ). \ - - -
I -
(6) Brady, A. P., Ross, S., J. Am. Chem. Soc. 66, 1348 (1944). (7) Clark, G. L., Ross, S., IND.END. CHEM.32, 1594 (1940). (8) Gibbs, J. W., “Collected Works,” vol. I, pp. 300-14, Yale University Press. New Haven. 1948. Gray, P. P., Stone; I., Wallenstein Labs. Communs. 3, 159 (1940). McBain, J. W., Ross, S., Brady, A. P., Natl. Advisory Comm. Aeronaut., Tech. Note 1840 (1949). Miles, G. D., Shedlovsky, L., Ross, J., J . Phys. Chem. 49, 93 (1945). Radow, N., unpublished M.Ch.E. thesis, Syracuse University, 1953. Ross, S., J . Phys. Chem. 47,266 (1943). RECEIVED for review November 25, 1955 ACCEPTED September 25, 1956
(END OF SYMPOSIUM) VOL. 48, NO. 11
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