230
Langmuir 1986,2, 230-238
Hydrodynamics, Enrichment, and Collapse in Foams Ganesan Narsimhan and Eli Ruckenstein" Department of Chemical Engineering, State University of New York a t Buffalo, Buffalo, New York 14260 Received September 11, 1985. I n Final Form: December 30, 1985 The semibatch foam fractionation of a nonionic surfactant is examined on the basis of a model which accounts for (i) the gravity drainage from the plateau borders, (ii) the thinning of the liquid lamellae (films) caused by the capillary pressure, the plateau border suction, and the disjoining pressure, and (iii) the van der Waals mediated rupture of the thin films. The enrichment factor and the liquid holdup profile have been calculated for different values of the inlet bubble size, superficial gas velocity, inlet concentration of the surfactant, viscosity, and surface viscosity. The results indicate the existence of an optimum inlet bubble size and an optimum inlet concentration of the surfactant for maximum enrichment. An increase in the superficial gas velocity, viscosity, or surface viscosity decreases the enrichment factor. The residence time which is necessary for the collapse of the foam bed as a result of the rupture of the thin films (caused by van der Waals interaction and thermal perturbations) decreases with the increase in the superficial gas velocity and surface viscosity and increases when the viscosity becomes larger. The calculated values of this critical residence time for the collapse of the foam bed provide an estimate for the maximum operable residence time.
Introduction In recent years, there has been a revival of interest in the use of foam fractionation as a viable separation technique. It has found application in the removal of impurities from waste water, elimination of radioactive contaminants from dilute effluents, selective separation of certain complex organic molecules such as proteins, etc. Foam fractionation is based on the selective adsorption of one or several surface-active solutes a t the gas-liquid interface, which is generated by a rising ensemble of bubbles through the solution. This ensemble of bubbles forms a foam bed (atop the liquid pool) which carries preferentially the surface-active solutes. The prediction of the separation efficiency of the surface-active solutes in a foam fractionator is rendered difficult because of the simultaneous occurrence of various phenomena such as (i) gravity drainage of the liquid from the plateau borders, (ii) drainage of the liquid from the lamellae caused by the capillary pressure, plateau border suction, and van der Waals disjoining pressure, (iii) coalescence of the bubbles as a result of the rupture of the lamellae, and (iv) gas diffusion from the smaller to the larger bubbles. Previous attempts to model the foam beds have assumed either the liquid drainage or the gas diffusion as the predominant phenomenon. Earlier inve~tigatorsl-~ calculated the foam density by considering constant bubble size and by accounting only for the gravity drainage of the liquid from the plateau border; they either assumed that the walls of the plateau borders are rigid (i.e., of infinite surface viscosity) or included a parameter to account for the surface mobility. Leonard and L e m l i ~ hHaas , ~ and J o h n ~ o nand ,~ Desai and Kumar6 investigated the effect of surface viscosity on the rate of drainage from the plateau borders. Hartland and Barber7 accounted for both the gravity drainage from the plateau borders and the thinning of the lamellae, by assuming rigid plateau border walls. This (1) G. D., Miles, Schedlovsky; Ross, J. J. Phys. Chem. 1945, 49, 93. (2) Jacobi, W. H.; Woodcock, K. E.; Grove, C. S. Ind. Eng. Chem. 1956, 48,9046. (3) Steiner, L.; Hunkeler, R.; Hartland, S. Trans. Inst. Chem. Eng. 1977, 55, 153. (4) Leonard, R. A.; Lemlich, R. AIChE J. 1965, 11, 18. (5) Haas, P. A.; Johnson, H. F. Ind. Eng. Chem. Fund. 1967,6, 225. (6) Desai, D.; Kumar, R. Chem. Eng. Sci. 1982, 37, 1361. (7) Hartland, S.; Barber, A. D. Trans. Inst. Chem. Eng. 1974,52,43.
0743-7463 186 l2402-0230$01.50/0
assumption was relaxed by Desai and Kumar,*~~ who also included the effect of surface viscosity in the calculation of the liquid holdup profile and exit foam densities. They, however, neglected the effect of the capillary pressure and of the van der Waals interactions on the thinning of the liquid lamellae. In contrast to the previous authors, Lemlich'O and Monsalve and Schechter,'l ignoring the effect of gravity drainage, have examined the change in the bubble size distribution in a foam bed caused by the interbubble gas diffusion. Several investigators have studied the instability and the eventual rupture of thin liquid films. When the thickness of the film is reduced due to thinning to values smaller than about 1000 A, van der Waals attractive interactions and double-layer repulsive interactions strongly influence the process of thinning. Scheludko12and Vrij,13 employing energy considerations, derived expressions for the critical thickness of rupture on the basis of a mechanism of film rupture involving surface thermal perturbations and the van der Waals disjoining pressure. The instability of thin films has been investigated extensivel ~ . ' ~ - However, '~ since these studies disregard the effect of thinning on film rupture, they predict much smaller critical thicknesses than those observed experimentally. Ivanov and Dimitrovlg and othersZ0J1have improved the previous calculations by including the effect of thinning on the critical thickness of rupture. In spite of such extensive studies on the stability of isolated thin free films, the previous authors have not attempted to couple the (8) Desai, D.; Kumar, R. Chem. Eng. Sci. 1983, 38, 1525. (9) Desai, D.; Kumar, R. Chem. Eng. Sci. 1984,39, 1559. (10) Lemlich, R. Ind. Eng. Chem. Fund. 1978, 17, 89. (11) Monsalve, A.; Schechter, R. J. Colloid Interface Sci. 1984,97,327. (12) Scheludko, Proc. K.Ned. Akad. Wet., Ser. B: Phys. Sci. 1965, 65, 76. (13) Vrij, A. Discuss. Faraday SOC.1966, 42, 23. (14) De Vries Red. Trau. Chim. Pays-Bas 1958, 77, 441. (15) Vrij, A.; Overbeek, J. J. Am. Chem. SOC.1968, 90, 3074. (16) Ruckenstein, E.; Jain, R. K. J. Chem. SOC.,Faraday Trans. 2 1974, 70, 132. (17) Jain, R. K.; Ruckenstein, E. J. Colloid Interface Sci. 1976,54, 108. (18) Lucassen, J.; Van den Temple, M.; Vrij, A.; Hesselink, F. T., h o c . K. Ned. Akad. Wet., Ser. B: Phys. Sci. 1970, 73, 109. (19) Ivanov, I. B.; Dimitrov, D. S.Colloid Polymn. Sci. 1974,252, 982. (20) Gurman, R. J.; Homsey, G. M. Chem. Eng. Commun. 1975,2,27. (21) Radoev, B. P.;Scheludko, A. D.;Manev, E. D. J. Colloid Interface Sci. 1983, 95, 254.
0 1986 American
Chemical Sociptv
Langmuir, Vol. 2, No. 2, 1986 231
Hydrodynamics, Enrichment, and Collapse in Foams
z+
u BREAKER
FILMS
J.
TOP PRODUCT
PLATEAU BORDERS
Z
Figure 2. Schematic diagram of a differential volume element of a foam bed.
I GAS
LIQUID POOL
t
I
Figure 1. Schematic diagram of a semibatch foam fractionator. hydrodynamics of the foam bed with the instability of the thin films to predict the conditions for foam collapse. In the present paper, a model for the hydrodynamics of a foam bed is suggested. The model accounts for (i) the gravity drainage from the plateau border, (ii) the thinning of the liquid lamellae (films) caused by the capillary pressure, the plateau border suction, and the disjoining pressure, and (iii) the van der Waals mediated rupture of the thin films. In contrast to the previous treatments, the present one accounts also for (1)the effects of capillary pressure and disjoining pressure on the thinning of the films and (2) the van der Waals mediated rupture of the film which results in foam collapse. The enrichment factor for a nonionic surfactant and the liquid holdup profile have been calculated by using the above model, for different operating conditions and physical properties of the system, in order to determine the optimum operating conditions. The model has also been used to predict the critical residence time for the collapse of the foam bed. The model is presented in the next section. Results of the calculations for a semibatch foam fractionator and the conclusions are presented in the subsequent two sections.
Model for a Semibatch Foam Fractionator A semibatch foam fractionator is presented schematically in Figure 1. An inert gas is sparged into a liquid pool in order to produce a foam which moves through the bed carrying with it some of the pool liquid. The foam bed usually consists of an ensemble of bubbles of different sizes. For the sake of simplicity, it will be assumed that the bubbles have the same size. More complicated calculations to be published later, which have accounted for the size distribution of bubbles, indicated that this assumption can be used in many cases. The liquid that is carried by the moving foam is distributed between the thin lamellae (films) separating the bubbles and the plateau borders. As the foam moves through the bed, the liquid drains from the lamellae to the neighboring plateau borders under the action of the capillary pressure, plateau border suction, and disjoining pressure, and the liquid in the plateau borders drains under the action of gravity.22 As a result, the liquid holdup decreases with height. In addition, as a result of drainage, the thickness of the lamellae can become of the order of a few hundred angstrom, thickness at which, because of the van der Waals interactions, the films become unstable to perturbations (of thermal or mechanical origin). This hastens the further thinning of the lamellae leading to their eventual breakup and coalescence of the neighboring bubbles and thus to foam collapse. The liquid pool is considered to contain (22) Bikerman, J. J. "Foams"; Springer-Verlag: New York, 1973.
FILM
INTERSECTION OF HORIZONTAL PLANE
PLATEAU BORDER
FILM [INTERSECTION OF FACES OF TWO ADJACENT POLYYHEORAI
Figure 3. Regular dodecahedral structure of a bubble and cross-sectional view of a plateau border. two components, of which one, being surface active, is preferentially adsorbed a t the gas-liquid interface. The present calculations consider only nonionic surfactants. The foam at the top of the bed is sent to a foam breaker to obtain the top product which is enriched in the surface-active component. The following simplifying assumptions are employed in the calculations: (i) The gas phase consists of dodecahedral bubbles of the same size surrounded by thin films and plateau borders. (ii) The foam bed moves in plug flow. (iii) The total volume of the liquid pool is so large compared to the flow rate of the top product that the surfactant concentration in the liquid pool can be assumed constant. (iv) The plateau borders are randomly oriented. (v) The surface concentration of the surfactant is the equilibrium concentration. (vi) The thin lamellae (films) separating the bubbles are subjected to thermal perturbations only. Consider a volume element of foam of unit cross section between z and z + dz, as shown schematically in Figure 2. The liquid is distributed between the films and plateau borders. The liquid in the films drains into the plateau borders under the action of the capillary pressure, plateau border suction, and disjoining pressure due to the van der Waals interactions. The foam as a whole is moving upward with the superficial velocity G. The liquid in the plateau borders is draining, however, downward under the action of gravity. An idealized regular dodecahedral shaped gas bubble is shown in Figure 3. The faces of adjacent bubbles border the liquid lamellae (films), which intersect in plateau borders. A schematic cross-sectional view of a plateau border is also shown in Figure 3. For the dodecahedralshaped bubbles, the coordination number is 12. Therefore, the number of films nF per bubble is equal to 6. Since a plateau border is formed by the intersection of three liquid films, the number of plateau borders per bubble nPis equal to 10. As can be seen from Figure 3, the number of plateau borders per bubble on a horizontal plane ng is equal to 2 ( = n , / 5 ) . If the film thickness is denoted by xF, the bubblefilm contact area by AF, the number of bubbles per unit volume by N , and the number of bubbles that flows in unit time per unit area of the foam bed by 7, the volume of liquid in the films per bubble is nFAFxF. A material balance for the liquid in films over a volume element between z and z + dz yields
232 Langmuir, Vol. 2, No. 2, 1986
Narsimhan and Ruckenstein
In the above equation, the first term represents the change in the volume due to convection, whereas the second term is the change in the volume due to the drainage of the liquid from the films. In order to calculate the rate of gravity drainage of the liquid from the plateau borders, the total cross section of the plateau borders at a given height of the column must be calculated. The total number of bubbles M per unit area of the cross section of the bed is given by
M = 2NR
(2)
where R is the radius of a bubble. Therefore, the total number of plateau borders intercepted by a unit area of bed cross section is 2NRn’,
(3)
The plateau border can, however, have any orientation. If one assumes that they are oriented at random, then the probability for their orientation to be between 0 and 0 + dO is 2 sin 0 cos 0 d0. Therefore, the volumetric flow rate in a plateau border oriented between 0 and 0 + d0 is equal to 2a,u sin 0 cos26 de, where a p is the cross-sectional area of a plateau border and u is the velocity of drainage under the action of gravity. Therefore, the average volumetric flow rate per plateau border = 2aPuJ3” cos2 0 sin 0 d0 = 2/3apu.The rate of gravity drainage of liquid with respect to a moving foam is therefore 4/3Nn’,a,uR. The material balance for the liquid in plateau borders can then be written as d 4 d - - [qa,n,l] + - - [Nn’,a,uR] + dz 3 dz
the material balance equations for films and plateau borders to obtain the liquid holdup profile. They, however, neglected the effects of the capillary pressure and of the van der Waals disjoining pressure on the rate of thinning of the films. Since the capillary pressure is inversely proportional to the bubble size, it cannot be neglected compared to the plateau border suction for sufficiently small bubble sizes. Moreover, because of the increase in the radius of curvature of the plateau borders caused by gravity drainage, the capillary pressure may become comparable to the plateau border suction in the final section of the foam bed even for large bubble sizes. In addition, van der Waals attractive interactions strongly influence the rate of thinning of the liquid films when the film thickness is of the order of lo3 A or smaller and, as discussed later, also accelerate the eventual rupture of the films. It is therefore necessary to include the effects of capillary pressure and van der Waals interactions for the accurate prediction of the liquid holdup and enrichment factor. The average velocity of gravity drainage u through the plateau border, obtained by numerically solving the Navier-Stokes equations for the appropriate boundary conditions, is given by6 where c,, the velocity coefficient, is defined as the ratio of the average velocity to the average velocity through the plateau border for an infinite value of the surface viscosity. The velocity coefficient c, is a function of the inverse of the dimensionless surface viscosity y defined as y = 0.4387p~,‘~~/p,
= bi,
+ bil(y - Ti) + biz(Y - Ti)’ + b i B ( ~- yiI3
where yi 5 y 5 7i+l, i = 1, 2, ...5 . Five different sets of coefficients bij 0 = 0-3) have been used for five different intervals of y. The values of the coefficients j = 0-3) and the constants yi can be found elsewhere.
{ti,,
Adsorption Isotherm of Nonionic Surfactants at the Air-Water Interface The equation of state for the alkanols (C3-C1,,) at the liquid-bubble interface has the formz3
P, = -RgTrm[ In (1 -
where c and r are the bulk concentration and the surface excess of the surfactant, respectively. In the above equation, the first and the third term represent the convective change in the amount of surfactant in the plateau borders and the films, respectively. The second term represents the change in the amount of surfactant due to gravity drainage. The last term is the change in the amount of surfactant a t the gas-liquid interface due to convection. Combining eq 5 with eq 1and 4, it can easily be shown that the change in the concentration with the axial distance is related to the change in the gas-liquid interfacial area with the axial distance. Since the present model assumes bubbles of constant size (no coalescence), there is no change in the gas-liquid interfacial area before the collapse of the foam bed and hence no change in the bulk concentration with the axial distance. Equations 1 and 4 are similar to those of Desai and K ~ m a rwho , ~ solved
(7)
The numerical results could be represented as the following functional dependence of c, on y cv
where 1 is the length of a plateau border. For a dodecahedral arrangement 1 = 6R, 6 being a constant (=0.816).s In the above equation, the first term represents the change in the volume due to convection, whereas the second and third terms represent the change in the volume due to gravity drainage and the drainage from the films to the plateau borders, respectively. Similarly, a material balance for the surfactant can be written as
(6)
u = c,a,pg/[20(3~/2)]
$)+$(k ) ]
(8)
where r is the surface excess, r, is the surface excess at saturation, R, is the gas constant, T i s the absolute temperature, Hsis the partial molar free energy of surface mixing at infinite dilution, and the surface pressure P, is defined as u = 6, - P, (9) u being the surface tension and usbeing the surface tension of pure water. The resulting equilibrium adsorption isotherm has the formz3
(23) Lucassen-Reynders, E. H. “Progress in Surface and Membrane Science”; Edited by Cadenhead, D. A., Danielli, J. F., Eds.; Academic Press: New York, 1976; Vol. 10, p 253.
Hydrodynamics, Enrichment, and Collapse in Foams
Langmuir, Vol. 2, No. 2, 1986 233
where K , = (P,/y)p,+, (y being the mole fraction of the surfactant in the bulk) is a constant. Velocity of Thinning of a Foam Film A foam film thins because of the capillary pressure and plateau border suction. In addition, when the film thickness is of the order of lo3 angstrom or smaller, van der Waals interactions influence the process of thinning through the disjoining pressure. Thinning of a circular plane parallel film is adequately described by the Reynolds equation if the film surface is rigid. However, for a mobile surface, one has to account for the surface viscosity as well as the surface and bulk diffusion of surfactants in order to predict the rate of thinning. The analysis of Ivanov et al.19 for the thinning velocity accounting for the above factors is employed in what follows. The velocity of thinning, which is determined by the pressure drop A p between the center of the film and the plateau border and the surface mobility a t the gas-liquid interface, is given by
by these external disturbances gives rise to two opposite forces: (1)A restoring force caused by the local capillary pressure which tends to smoothen the film surface and (2) a force due to the increase in the absolute value of the negative disjoining pressure (caused by van der Waals interactions) which tends to deform the film further. If the film is thick, the former effect dominates and the film tends to return to its plane parallel shape. However, for a disturbance of a given wavenumber, there exists a transition thickness below which the perturbations grow leading to the eventual rupture of the film. From the moment of the onset of the instability (thickness equal to the transition thickness xt) the amplitude of the perturbation grows whereas the mean thickness of the film decreases because of drainage. This continues until the amplitude becomes equal to half of the mean thickness (critical thickness) a t which point the film ruptures instantaneously. This critical thickness is determined by the dominant wavenumber for which the growth rate of the perturbation is maximum. Ivanov et al.,I9 assuming only thermal perturbations, have derived an expression for the critical thickness a t which the film ruptures instantaneously. The equations for the dominant wavenumber l,, the transition thickness xt, and the critical thickness xCrit are given by ~ 1 , ' - 2 ( d ~ / d ~ ) , ==, 0~
(16)
Here, RF is the radius of the planar face of the film, the eigenvalues A, are solutions of the equations Jo(Xn)
kn =
=0
(12)
in/RF
(13)
and Jo, J1, and J2are Bessel functions. The surface mobility is accounted for in eq 11 via the surface viscosity ps and the parameter a a=-
w+w, [ 3Dp
1+
]
~~,(ar/ac), DXF
(14)
which compares the relative magnitudes of the bulk and the surface diffusion with that of the Marangoni effect. In eq 14 D and D, are the bulk and surface diffusion coefficients of the surfactant, respectively, and (du/dc), and (dr/dc), refer to the gradients of the surface tension and the surface excess, respectively, at the bulk concentration. The pressure drop A p is given by A p = 20/R
+ u/R, + A / ~ ~ T X F ~
and
where
(15)
where A is the Hamaker constant and the radius of curvature R, of the plateau border can be calculated, on the basis of geometric considerations, from the equation+ R, = -1.732~ +~[(1.732xF)' - 0.644(0.433xF2- a,)]'/2/0.322 (1 5 4 In eq 15, the first term on the right-hand side represents the capillary pressure acting on the surface of the film, the second term represents the plateau border suction, and the last term is the disjoining pressure due to van der Waals attraction. Critical Thickness for Film Rupture A foam film is subjected to external disturbances which lead to the corrugation of the film. The fluctuations of the film thickness can be represented as a superposition of an infinite number of waves of different frequencies and amplitudes. The displacement of the film surface caused
and x (the disjoining pressure) is related to the film thickness x via the expression K
= -A/6rx3
(21)
Let us note that A,, k,, and a in the above equations are given by eq 12-14, respectively. The above equations are to be solved in conjunction with those describing the hydrodynamics of the foam bed in order to evaluate the critical thickness for film rupture. The material balance equations (1)and (4)are solved to determine the film thickness xF and the plateau border area a, as a function of the axial distance z. The radius of curvature of the plateau border R, is determined from eq 15a. The pressure drop A p for film drainage is then calculated by using eq 15. Equations 16-18 are then solved simultaneously to evaluate xcrit,xt, and I,. Because of drainage, the film thickness x F continuously decreases with the axial distance. A t a certain axial distance (or, equivalently, residence time), the film thins to the critical
234 Langmuir, Vol. 2, No. 2, 1986
Narsimhan and Ruckenstein
thickness xcdt resulting in the rupture of the films and the collapse of the foam bed. Dimensionless quantities are defined as N N* = N / N o , N* = N / N o = (1 - €)No a*p = ap/apo, R* = R / R o
Boundary Conditions In order to solve the above equations, one should specify the values of the inlet parameters Ro,to,$FO, xF0, co, G , nF, np, and AFO. The structure of the foam determines the values of nF,np,and AFO. For a dodecahedral structure,8 nF = 6 ,
AFO = 1.152R02
FO
where the subscript 0 refers to the quantity at the inlet. The material balance equations can then be written as --d
dZ
_-
--
g)
= 0 (24)
[N*a*@*u*C*]-
where
and AI'*
(28)
and the planar area of the film AFO is given by
XF
u* = u / u , = cya*p, x* = -
AFOXFO N*A*,( aponpGR0
n p = 10
= ( 2 o / R + u / R , - n)/(ao/Ro)
The liquid holdup t at any height is given by the sum of the liquid in the plateau borders and the films. Therefore, E = to(1 - dFo)N*ap*R* + nFNoAFoXF$J*A*FX* ( 2 6 ) where to and dFOare the inlet liquid holdup and the fraction of liquid distributed in films a t the inlet, respectively, and AFO is the planar area of the film a t the inlet. The enrichment factor e , which is the ratio of the concentration of the surfactant in the top product to the inlet concentration co,depends on the total amount of surfactant in the bulk as well as a t the gas-liquid interface at the outlet of the foam bed. Therefore, e = c * f ( l+ (KN&FonF/€)N*AF*)
(27)
where K = F/c is the equilibrium constant for the adsorption of the surfactant and C*f = cf/co, cf being the bulk concentration of the surfactant in the liquid at the outlet of the foam bed. In the above equation, the second term is the enhancement in the surfactant concentration of the top product due to the surfactant adsorbed at the gasliquid interface. Since in the present treatment the bulk concentration of the surfactant does not change with the axial position (no coalescence), C*f in eq 27 is equal to unity. Equations 23-27 are to be solved with the appropriate boundary conditions in order to obtain the profile of the liquid holdup and the enrichment factor of the foam bed.
The inlet bubble size Ro and the superficial gas velocity G depend on the sparger and the gas flow rate, respectively. Of course, the inlet concentration co is known. The inlet liquid holdup to and the fraction of the liquid distributed in the films $FO a t the pool liquid-foam interface depend upon the entrainment of the liquid in the plateau borders and films at that point. It is difficult to evaluate the above quantities from hydrodynamics. However, for a closepacked structure of the foam bed, one can assume that eo = 0.26. From an overall material balance for the liquid at steady state, it can be seen that the difference between the rate of uptake of liquid and the rate of drainage of liquid at the pool liquid-foam interface should be equal to the flow rate of the top product. Since the flow rate of the top product is very much smaller than the rate of uptake as well as the rate of drainage at the pool liquidfoam interface, one can approximate the rate of uptake of liquid at the pool liquid-foam interface as equal to the rate of gravity drainage. At the pool liquid-foam interface, the rate of uptake of liquid per unit cross section of the bed is G t o / ( l - to), while the rate of drainage through the plateau border per unit cross section of the bed is (4/ 3)Non',ap&ouo. Therefore,
The above mass balance equation determines the inlet distribution of the liquid between the films and the plateau borders. If one assumes that the inlet thickness of the film is the same for all the bubbles, then
and Rpo = - 1 . 7 3 2 ~+~ [(1.732xFO)' ~ - 0.644(O.433xFo2 - apo)]l/z (33) 0.322 Equations 29-32 are to be solved simultaneously in order to determine $FO, xFO,and apowith the constraint that Rpo L 0.
Holdup Profile and Performance of a Semibatch Foam Fractionator The material balance equations (23)-(25) have been numerically solved subject to the initial conditions (29)-(32)in order to determine the film thickness and the plateau border area as a function of the axial distance. The liquid holdup profile and the enrichment factor are calculated by using eq 26 and 27, respectively. All the cal-
Langmuir, Vol. 2, No. 2, 1986 235
Hydrodynamics, Enrichment, and Collapse in Foams ~~~
0 15-
N
w
5
L
VISCOSITY (poise) 10-2
CURVE 1 2
0
010-
0.5
0 v) v)
W 1
z
-
1
005-
z
w
z0 0 003
I
1
I
0.07 0 09 LIQUID HOLDUP E
005
0 11
013
Figure 6. Liquid holdup profiles for two different viscosities at 1 atm and 293 K for kLs = lo4 sP, G = 0.1 cm/s, 8 = 30 s, Ro = 0.1 cm, and co = I
I
I 1 1 1 1 1 1
I
I
I 1 1 1 1 1 1
10-7
-8
I
I
(g mol)/cm3.
I l l l l
1
10-6
5
C (gm moles/cc)
Figure 4. Adsorption isotherm of nonanol-water system a t an air-liquid interface a t 293 K.
E
301 20
0
io-*
LIQUID H O L D U P E
,//
,,,,/
, , , ,,,,
10-7
Figure 7. Liquid holdup profiles for two different surface viscosities a t 1atm and 293 K for p = lo-' P, G = 0.1 cm/s, 8 = 30 (g mol)/cm3. s, Ro = 0.1 cm, and co =
10-6
C(gm moles/cc)
0 15-
Figure 5. Plot of surface pressure vs. bulk concentration for nonanol-water system a t an air-liquid interface a t 293 K.
2 N
Table I. Values of System Parameters Used in the Calculations 1 R, cm 0.05-2.5 P, atm 293 8, s 30 T,K W5 c, (g mol)/cm3 5 x 1O-*-IO4 D, cmz/s D,, cm2/s A , ergs 4.38 x 10-13 rm,(g mol)/cm2 6.127 X 10-2-1 P, p 2.7 X lo6 Psr sp 10-4-10-1 Kp, dyn/cm 1.8 0.05-5 HB/R,T G, cmjs
culations were performed for the nonanol-water system whose adsorption isotherm and surface pressure are plotted in Figures 4 and 5, respectively. The inlet bubble size, the superficial gas velocity, the inlet concentration, the viscosity, and the surface viscosity are the parameters whose effect on the enrichment factor have been investigated. The range of variation of these variables is given in Table I. Typical calculated liquid holdup profiles for two different viscosities are presented in Figure 6. It can be seen that the liquid holdup decreases rapidly near the foampool liquid interface and less rapidly in the rest of the bed. As expected, the holdup is higher a t higher viscosities. Liquid holdup profiles are presented in Figure 7 for two different surface viscosities. Again, as expected, the liquid holdup increases when the surface viscosity increases. The effect of the superficial gas velocity on the liquid holdup
G CURVE 1
0 W
cm/sec 0 05 0 io
Narsimhan and Ruckenstein
236 Langmuir, Vol. 2, No. 2, 1986
.i;
1
BUBBLE SIZE
8i
2
(cm) 0.1 0.5
3
io
CURVE 1
7
i
'
SURFACE VISCOSITY (sp)
CURVE
1
2 3
5 ~ i O - ~ 10-3
I
1I
I
I
I
0 10-2
1
io-
'
1
VISCOSITY (poise)
Figure 11. Effect of viscosity on the enrichment factor a t 1 atm and 293 K for G = 0.1 cm/s, Ro = 0.1 cm, 6 = 30 s, and co = loW7 ( g mol)/cm3.
\
G = 0.05c d s e c a
CURVE
'[
1 2 3
5
1
I
I
2 BUBBLE SIZE (cm) 1
I
J
3
0
0.05 0. i
0.5 1 .o
4 5
I
ob
VISCOSITY (poise) 0.01
t
10-4
10-3
SURFACE
10-2
io-f
VISCOSITY (sp)
Figure 10. Effect of the inlet bubble size on the enrichment fador a t 1 atm and 293 K for p = sP,6 = 30 s, and co P, ps = = io-' (g mol)/cm3.
Figure 12. Effect of surface viscosity on the enrichment factor at 1 atm and 293 K for G = 0.1 cm/s, Ro = 0.1 cm, 0 = 30 s, and co = (g mol)/cm3.
to obtain large enrichment factors. A typical dependence of the enrichment factor on the superficial gas velocity is plotted in Figure 9 for different bubble sizes. The enrichment factor decreases exponentially when the superficial gas velocity increases because of the increased liquid holdup. Plots of the enrichment factor vs. bubble size for different superficial gas velocities are presented in Figure 10. This figure indicates the existence of an optimum bubble size for sufficiently low superficial gas velocities at which the enrichment factor is maximum. However, the enrichment factor is fairly insensitive to the bubble size a t larger values of G (>0.5cm/s). The rate of drainage of the liquid is influenced by the initial distribution of the liquid between the films and the plateau borders. Since the time scale of drainage from the films is much smaller than the time scale of gravity drainage from the plateau borders, the higher the fraction of the liquid distributed
in the films, the larger is the overall drainage rate. As the bubble size increases, a larger proportion of the liquid is distributed in the film, thus resulting in a faster drainage rate. On the other hand, an increase in the bubble size results in a decrease in the interfacial area per unit volume. Because of the above two opposing effects, there exists an optimum bubble size. The dependence of the enrichment factor on viscosity and surface viscosity are shown in Figures 11 and 12, respectively. The increase in either the viscosity or the surface viscosity decreases the liquid drainage rate, resulting in a higher liquid holdup and therefore a lower enrichment factor. At high values of the surface viscosity the interfacial mobility of the film becomes negligible and as a result the enrichment factor becomes insensitive to the surface viscosity, as can be seen from Figure 12. At very low surface viscosities (viscosity), the enrichment factor becomes insensitive to the viscosity
Langmuir, Vol. 2, No. 2, 1986 237
Hydrodynamics, Enrichment, and Collapse in Foams
6t a 5
3
c
0' 10-8
0.i 0.25 1.0
i 2
' '
i ' ' ' ' ' '
10-7
'
I
400 0
t l l l l J
Co(gm moles /cc)
Figure 13. Effect of the inlet concentration on the enrichment factor at 1 atm and 293 K for p = lo-* P, pels = lo-* sP,0 = 30 s, and Ro = 0.1 cm.
Figure 14. Effect of the superficial gas velocity on the critical residence time for foam collapse at 1atm and 293 K for p = 10" P, plS= lo4 sP,Ro = 0.1 cm, and co = (g mol)/cm3.
(surface viscosity) for small values of the latter quantity, because the velocity in the film is uniform in such cases. The effect of the inlet concentration of nonanoi on the enrichment factor is examined in Figure 13. An increase in the inlet concentration results in the increase of the surface concentration of the surfactant and hence in a decrease in the interfacial tension. The former effect tends to increase the enrichment factor, whereas the latter effect decreases the liquid drainage rate and therefore decreases the enrichment factor. Consequently, there exists an optimum inlet concentration at which the enrichment factor is maximum. Foam Collapse As the foam bed moves through the column, the film thickness decreases because of drainage. At a critical residence time Bo the film thickness becomes equal to the critical thickness for rupture and therefore the film ruptures resulting in the collapse of the foam bed. The film thickness is calculated as a function of the axial distance (or, equivalently, the residence time) by solving equations 23-27 describing the hydrodynamics of the foam bed. Assuming that the foam bed is subjected only to thermal perturbations, the critical thickness for film rupture xcrit, the transition thickness xt, and the dominant wavenumber for film rupture I , are calculated by solving eq 16-18 in conjunction with the hydrodynamic equations (23)-(27). The residence time (or, equivalently, the axial distance) a t which the film thickness becomes equal to the critical thickness for film rupture yields the critical residence time 0, for the collapse of the foam bed. The time Bc is plotted vs. the superficial gas velocity, viscosity, and surface viscosity in Figures 14, 15, and 16, respectively. As the superficial gas velocity increases, the initial film thickness decreases. As a result, at higher superficial gas velocities less time is required for the film to drain to the critical thickness for rupture. Consequently, the critical residence time for the collapse of the foam bed Bc decreases when the superficial gas velocity increases. At high superficial gas velocities, the initial film thickness becomes insensitive to the superficial gas velocity. Consequently, the critical residence time reaches an asymptotic value. As expected, Bc increases with the increase in viscosity, because of the
1.0
0.5
SUPERFICIAL GAS VELOCITY (cm/sec)
10-6
t
i Oio-2 L E ; - - - L L L L L u I J
I
lo-'
p (poise)
Figure 15. Effect of viscosity on the critical residence time for foam collapse at 1atm and 293 K for pCs= lo4 sP,G = 0.1 cm/s, Ro = 0.1 cm, and co = lo-' (g mol)/cm3.
550k
400 10-4
10-~
10-2
lo-'
SURFACE VISCOSITY (sp)
Figure 16. Effect of surface viscosity on the critical residence time for foam collapse at 1atm and 293 K for p = lo-* P,G = 0.1 cm/s, Ro = 0.1 cm, and co = lo-' (g mol)/cm3. decreased rate of film drainage. However, an increase in the surface viscosity decreases the critical residence time (Figure 16). This happens because such an increase results in smaller initial film thicknesses which more than compensates for the decreased rates of drainage. The residence time Bc is more sensitive to the viscosity than to the surface viscosity. One may note that these calculations of the
238 Langmuir, Vol. 2, No. 2, 1986
Narsimhan and Ruckenstein
critical residence time 8, disregard any external mechanical disturbances that may arise in the foam bed from the flow field. In addition, they disregard nonlinear effects which are important when the growing thermal perturbations become large. Therefore, the reported values of 8, are to be considered as an upper bound for the successful operation of the foam column. Desai and K ~ m a have r ~ ~recently reported critical liquid holdups for the collapse of a recirculating semibatch foam bed. Since the foam bed was operated a t fairly high superficial gas velocities with a recirculating flow, considerable mechanical disturbances were generated by the flow field. Therefore, their experimental results could not be compared with the present calculations which involve only thermal perturbations. It is, however, likely that the calculated values of BC provide a fairly good estimate for the maximum operable residence time at low superficial gas velocities, because of the absence of large mechanical interfacial disturbances under such conditions. Conclusions A model for the hydrodynamics of a semibatch foam fractionator is presented. The model accounts for (i) the gravity drainage from the plateau border, (ii) the thinning of the liquid lamellae (films) due to the capillary pressure, plateau border suction, and van der Waals disjoining pressure, and (iii) the rupture of the thin films. The model employs the simplifying assumption that the gas phase consists of dodecahedral bubbles of the same size. The liquid holdup profile and the enrichment factor for the nonionic surfactant nonanol was calculated for different values of the inlet bubble size, superficial gas velocity, inlet concentration of the surfactant, viscosity, and surface viscosity. The enrichment factor is found to decrease with the increase of superficial gas velocity, viscosity, and surface viscosity. Calculations indicate the existence of an optimum inlet bubble size and an optimum inlet concentration for the maximum enrichment factor. Calculations of the critical residence time for foam collapse have been carried out for different values of the superficial gas velocity, viscosity, and surface viscosity under the assumption that the gas-liquid interface is subjected to thermal perturbations only. The critical residence time for foam collapse is found to decrease with the increase in the superficial gas velocity and surface viscosity and to increase with the increase in viscosity. The calculated values of the critical residence time provide an estimate for the maximum operable residence times at low superficial gas velocities. Notation
2 AF
c c"
D DS e g
cross-sectional area of plateau border, cm2 Hamaker constant, ergs area of film, cm2 bulk concentration of surfactant, (g mol)/cm3 velocity coefficient for gravity drainage of plateau border diffusion coefficient, cmz/s surface diffusion coefficient, cmz/s enrichment factor acceleration due to gravity, cm/s2
(24) Desai, D.; Kumar,
R. Chem. Eng. Sci.
1985, 40, 1305.
superficial gas velocity, cm/s partial molar free energy of surface mixing at infinite dilution, ergs wavenumber of the perturbation at the gas liquid interface adsorption equilibrium constant for the surfactant, cm capillary constant for the surfactant, dyn/cm length of plateau border, cm = 0.816R dominant wavenumber for rupture of films height of foam bed, cm number of bubbles per unit area of the cross section of the foam bed, cm-2 number of films per bubble number of plateau borders per bubble number of plateau borders per bubble on a horizontal plane = n,/5 number of bubbles per unit volume of the foam bed, cm-3 number of bubbles per unit volume of the gas phase, cm-3 pressure, atm surface pressure, dyn/cm quantity defined by eq 19 radius of bubble, cm radius of the film, cm gas constant, cm3 atm g mol-' K-' radius of the curvature of the plateau border, cm time, s temperature, K velocity of gravity drainage, cm/s velocity of gravity drainage for infinite surface viscosity, cm/s velocity of film drainage, cm/sec critical thickness for film rupture, cm film thickness, cm transition thickness of the film, Le., the thickness corresponding to neutral stability, cm axial distance, cm dimensionless axial distance
Greek Symbols quantity defined by eq 14 quantity defined by eq 20 P inverse of dimensionless surface viscosity defined Y by eq 7 surface excess of the surfactant, (g mol)/cm2 surface excess of the surfactant at saturation, (g mol)/ cm2 constant = 0.816 density of the liquid, g/cm3 number of bubbles per unit area of the foam bed per unit time, cm-2 s-l viscosity, P surface viscosity, sP surface tension, dyn/cm surface tension of pure water, dyn/cm eigenvalue defined by eq 12 amplitude of the perturbation, cm initial amplitude of the perturbation, cm liquid holdup disjoining pressure fraction of liquid in films residence time, s critical residence time for the collapse of foam bed, s Subscript 0 inlet f outlet a
