494
Langmuir 1986,2, 494-508
estimates the role of surfactant concentration driven Marangoni motion and the surface viscosity in prolonging the time of rupture, (d) overestimates the stabilization due to the presence of a lighter fluid on top of a denser fluid, (e) overestimates the destabilization due to the RayleighTaylor instability (heavier fluid on top), and (f) overestimates the critical and dominant wavelengths of the disturbances. It is found that for almost all situations of practical interest, viz., 171 < 1, factors a and b are dominant over factors c and e and, consequently, the time of rupture is always found to be less than that given by the linear theory. For example, a wetting film with an initial amplitude of disturbance equaling 0.4 of the thickness and q = -0.5 ruptures in about one-tenth of the time predicted by the linear theory (Figure 4). The result that the surfactants are able to prolong the time of rupture of a wetting film by an order of magnitude appears to be of particular interest in controlling the rate of flotation and the rate of cooling of a hot metal that is cooled by quenching it in a liquid that boils due to the release of heat. Conclusions a and b were also arrived at by Williams and Davis16 through direct numerical simulations for a single film devoid of solutes. Quantitatively, the analytical expression from the present theory yields results for a pure wetting film that are within 15% of the numerical simulations even for amplitudes as large as 0.6 of the film thickness.
The methodology is applied also to thin free films and thus finds application in predicting the lifetime of foams. Again, it is deduced that the nonlinearities here prefer shorter wavelengths (as compared to the linear theory) both for the finite-amplitude neutrally stable and dominant waves. Just as in the case of a wetting film, the nonlinear theory predicts a greater destabilization due to the van der Waals interactions as compared to the linear theory. In contrast to a wetting film, however, the nonlinearities act to diminish the stabilizing influence of the Marangoni motion and the surface viscosity. The linear theory thus overestimates the stabilizing effects of surfactants on the time of rupture by as much as a factor of about 4. The overall effect of nonlinearities is thus always to reduce the time of rupture as compared to the one given by the linear theory. For a free film with surfactants, the time of rupture may be as much as about 1 order of magnitude smaller than that given by the linear theory, depending on the amplitude of the perturbation. Finally, the paramount strength of the method is in the ease of derivations and computations that it offers. These are only slightly more involved than the linear stability analysis and yet give explicit analytical expressions for all of the quantities of interest. It is in view of this that it appears attractive to exploit the present methodology to other thin-film situations.
Effect of Bubble Size Distribution on the 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 January 13, 1986. I n Final Form: April 7, 1986 The performance and stability of a semibatch foam fractionator for the separation of a nonionic surfactant are investigated on the basis of a model that accounts for (i) the size distribution of bubbles, (ii) the gravity drainage from the plateau borders, (iii) the thinning of the liquid lamellae (films) caused by the capillary pressure, plateau border suction and disjoining pressure, (iv) the coalescence of bubbles due to the rupture of the thin films, and (v) the interbubble gas diffusion. The enrichment factor and the profiles of the liquid holdup, bubble size distribution, number of bubbles per unit volume, and bulk concentration of the surfactant have been calculated for different values of the inlet mean bubble size, coefficient of variation of the inlet bubble size distribution, superficial gas velocity, viscosity, and inlet concentration of the surfactant. When the inlet bubble size distribution is narrow, the enrichment factor is found to be almost the same as that for equal size bubbles of mean size. As the inlet bubble size distribution becomes broader, the enrichment factor decreases because of the coalescence of the small bubbles, particularly at lower superficial gas velocities, lower viscosities, smaller mean bubble sizes, and lower inlet concentrations. For very broad inlet bubble size distributions, a rapid coalescence of bubbles is found to occur, resulting in the collapse of the foam bed. The calculations provide an estimate of the maximum coefficient of variation of the inlet bubble size distribution below which the foam bed is stable, at low superficial gas velocities, and an upper bound for this coefficient at relatively high superficial gas velocities. The results indicate that the simplifying assumption of equal size bubbles becomes more valid and thus can be employed for the prediction of the performance and stability of the foam beds at high superficial gas velocities, high viscosities, larger inlet mean bubble sizes, and higher inlet concentrations. Introduction Foam fractionation is a viable separation technique for the concentration and the separation of surface-active components from dilute solutions. It has found application in waste water treatment, removal of radioactive contaminants from dilute effluents, selective separation of proteins, etc. It is based on the principle of selective adsorption of one or more surface active solutes at the gas0743-7463/86/2402-0494$01.50/0
liquid interface. The gas-liquid interface is generated by bubbling an inert gas through the solution thus forming an ensemble of gas bubbles (foam) which preferentially carries with it the surface-active solutes. The breakup of foam (either by mechanical or chemical means) results in the enrichment of the surface-active solute because of the recovery of the solute adsorbed at the gas-liquid interface. The prediction of the enrichment of the surface-active 0 1986 American Chemical Society
Langmuir, Vol. 2, No. 4, 1986 495
Enrichment and Collapse in Foams solute in a foam fractionator should account for various phenomena such as (i) gravity drainage of the liquid from the plateau borders, (ii) drainage of liquid from the films as a result of the capillary pressure, plateau border suction, and van der Waals disjoining pressure, (iii) coalescence of the bubbles as a result of the rupture of thin films, and (iv) interbubble gas diffusion. Earlier inve~tigatorsl-~ calculated the foam density by assumingequal size bubbles and by accounting only for the gravity drainage of the liquid from the plateau borders. These investigations either assumed that the plateau border walls are rigid (infinite surface viscosity) or accounted for their surface mobility through an adjustable parameter. Several investigator~~"have studied the effect of surface viscosity on the rate of gravity drainage from the plateau borders. Hartland and Barber,' assuming rigid plateau border walls, included the effects of liquid drainage both from the plateau borders as well as from the films in their calculation of exit foam densities. Desai and K u m d S relaxed the above assumption by including the effect of surface viscosity in their calculation of the liquid holdup profile. They, however, neglected the effect of the capillary pressure and of the van der Waals interactions on the drainage of liquid from the films. The authorslo have included the above effects in their calculation of the enrichment factors and the liquid holdup profiles for various operating conditions and physical properties of the system. The effect of interbubble gas diffusion on the evolution of the bubble size distribution has been investigated by Lemlich" and by Monsalve and Schechter12 by ignoring the effect of liquid drainage both from the plateau borders as well as from the films. The instability and the eventual rupture of isolated thin films due to van der Waals interactions have been studied exten~ively.'~-'~ Most of these studies predict much smaller critical thickness for film rupture than those observed experimentally because they ignore the effect of thinning on film rupture. Ivanov and Dimitrov20 and others2l* have accounted for the effect of thinning in their calculation of the critical thickness of rupture of isolated thin films. The authorslo have coupled the hydrodynamics of the foam bed with the instability of thin films under the assumption of equal size bubbles in order to predict the critical residence times for the collapse of the foam bed. (1) Miles, G. D.; Schedloveky;Rose, J. J. Phys. Chem. 1945, 49, 93. (2) Jambi, W. H.; Woodcock, K.E.; Grove, C. 5. I d . 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. AZChE J . 1965,11, 18. (5) Hans, 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. (8) Desai, D.; Kumar,R. Chem. Eng. Sci. 1983, 38, 1525. (9) Desai, D.; Kumar,R. Chem. Eng. Sci. 1984, 39, 1559. (10) Narsimhan, G.; Ruckenstein, E. Langmuir 1986,2, 230. (11) Lemlich, R. Ind. Eng. Chem. Fund. 1978,17, 89. (12) Monsalve, A.; Schechter, R. J. Colloid Interface Sci. 1984,97,327. (13) Scheludko, Proc. K. Ned. Akad. Wet., Ser. B: Phys., Sci. 1965, 65, 76. (14) Vrij, A. Diecuse. Faraday SOC.1966, 42, 23. (15) De Vries, Red. Trau. Chim. Pays-Bas 1958, 77,441. (16) Vrij, A.; Overbeek, J. J. Am. Chem. SOC.1968, 90,3074. (17) Ruckenstein, E.; Jain, R. K. J. Chem. SOC.,Faraday Trans. 2 1974, 70, 132. (18) Jain, R. K.; Ruckenstein, E. J. Colloid. Interface Sci. 1976,54, 108. (19) Lucaesen, J.; Van den Temple, M.; Vrij, A,; Heseelink, F. T. Roc. K . Ned. Akad. Wet., Ser. B: Phys. Sci. 1970, 73, 109. (20) Ivanov, I. B.; Dimitrov, D. S. Colloid Polym. Sei. 1974,252,982. (21) Gumerman, R. J.; Homsy, G. M. Chem. Eng. Commun. 1975,2, 27. (22) Radoev, B. P.; Scheludko, A. D.; Manev, E. D. J. Colloid. Interface Sci. 1983, 95, 254.
LIQUID POOL
GAS
Figure 1. Schematic diagram of a semibatch foam fractionator.
The present paper investigates the effect of bubble size distribution both on the enrichment factor as well as on the stability of the foam bed by accounting for (i) gravity drainage from the plateau borders, (ii) drainage of the liquid from the films as a result of the capillary pressure, plateau border suction, and van der Waals disjoining pressure, (iii) coalescence of the bubbles as a result of the rupture of thin films, and (iv) interbubble gas diffusion. The model proposed in the present investigation is more complete than the previous model employed by the aut h o d o in that it also accounts for (1) the bubble size distribution, (2) the evolution of bubble size distribution due to coalescence, and (3) interbubble gas diffusion. The enrichment factor and the profiles of liquid holdup, bubble size distribution, number of bubbles per unit volume of the foam bed, and the bulk concentration of the surfactant have been calculated by using the above model for a nonionic surfactant, for different inlet bubble size distribution, operating conditions, and physical properties of the system. The calculations seem to elucidate the effect of the inlet bubble size distribution, superficialgas velocity, and physical properties of the system on the stability 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 containing a surface-active component. A foam is thus produced that moves up the bed entraining some of the liquid from the pool. The foam bed consists of an ensemble of bubbles of different sizes. The size distribution of the bubbles at the pool liquid-foam interface depends upon the nonuniformity in the openings of the sparger through which the inert gas is introduced. The surfaceactive component is preferentially adsorbed at the bubble-liquid interface. Only nonionic surfactants are considered in the present calculations. The liquid that is entrained 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 in the films drains into the plateau borders under the action of the capillary pressure, plateau border suction, and disjoining pressure. On the other hand, the liquid in the plateau borders drains under the action of gravity. As a result, the liquid holdup decreases with height. In addition, the inert gas diffuses through the thin lamellae from the smaller to the larger bubbles because of the difference in their capillary pressures. Consequently, the larger
496 Langmuir, Vol. 2, No. 4, 1986
-Liquid 1 &
Gas Phase
"V 8 interbubble
0 diffusion
Gas Uptake
Narsimhan and Ruckenstein
Films
~ : z z = ~ ~ : : = z z
coalescence
I Plateau I Borders I
A I Liquid Uptake
Gravity Drainage
Figure 2. Schematic diagram of a differential volume element of a foam bed.
bubbles grow at the expense of the smaller ones. Moreover, the rate of drainage of the films is not uniform because bubbles of different sizes exert different capillary pressures on the intervening films. As a result, there is a distribution of film thicknesses at any cross section of the foam bed. As soon as the film thickness, because of drainage, becomes of the order of a few hundred angstroms, the film becomes unstable to perturbations (of thermal or mechanical origin). This instability, induced by van der Waals interactions, accelerates the further thinning of the lamellae leading to their eventual breakup and subsequent coalescence of the neighboring bubbles. This, in turn, leads to the decrease in the gas-liquid interfacial area and the transfer of the adsorbed surfactant from the disappearing gas-liquid interface to the liquid. As a result of the above phenomena, the bubble size distribution, the gas-liquid interfacial area, the liquid holdup, and the concentration of the surfactant are functions of the axial position in the foam bed. The foam at the top of the bed is sent to a foam breaker in order to obtain the top product which is enriched in the surface-active component. The foam bed is assumed to move in plug flow. If the total volume of the pool liquid is large compared to the flow rate of the top product, one can assume that the concentration of the surface-active component in the pool liquid is constant. The foam bed consists of randomly packed gas bubbles of different sizes. If the bubble size distribution is not very broad, it is reasonable to assume that the bubbles are arranged on the average in a dodecahedral configuration (as they are when they are of equal size), so that the average coordination number is 12 (Figure 2). Furthermore, the plateau borders are expected to be oriented randomly because of the random packing of the gas bubbles. At any axial position of the foam bed, the plateau borders will have more or less the same cross-sectional area but will have a distribution of lengths because of the size distribution of the bubbles. It is reasonable to assume that the distribution of plateau border lengths is the same as the bubble size distribution. For surfactants the time scale required to achieve adsorption equilibrium at the gas-liquid interface is very small, much smaller than the residence time of the moving foam. It is, therefore, reasonable to assume that the surface concentration of these surfactants is the equilibrium Concentration. For complex macromolecules such as proteins, the time required to attain adsorption equilibrium is not small compared to the residence time of the moving foam. In such cases, the rates of adsorption and desorption should be explicitly accounted for, in order to determine their surface concentrations. In the present calculations, it is assumed that the surface concentration of the surfactant nonanol is the equilibrium concentration. In addition, the instability of the thin lamellae (films) is
assumed to be caused only by the growth of thermal perturbations (Le., the interfacial mechanical perturbations are negligible). In order to obtain the variation of the bubble size distribution, gas-liquid interfacial area, liquid holdup, and surfactant concentration with the axial position, it is necessary to write a population balance equation for the bubble size distribution as well as conservative equations for the liquid holdup in the films and in the plateau borders and for the surfactant. These equations should, of course, account for the structure of the foam bed. The description of the structure of the foam bed and the development of the balance equations are discussed in the following sections. Structure of the Foam Bed Since every two bubbles are separated by a liquid lamella, the number of films, CF,per unit volume of foam bed is given by CF = NnF (1) where N is the number of bubbles per unit volume of the foam bed and nF is the number of films per bubble (equal to 6 in the dodecahedron geometry). The size distribution of the bubbles at any axial position z is described by the number density Nf(R,z),where f(R,z)dR is the fraction of the bubbles whose radii are between R and R + dR. Therefore, the probability that a film is in contact with bubbles whose radius is between R1 and R1 + dR1is equal to NnFf(R1,z)dR,/C,. Therefore, the number of films per unit volume of the foam between bubbles of radius, R1,R1 + dR1 and those of radius R2,R2 + dR2 denoted by Cp(R1,R2)dR1 dR2 is equal to the product of the above probability and the number of films in contact with bubbles of size R2,R2 + dR2,i.e.,
The area of the liquid film separating the bubbles of radii R, and R2 will be a function of both R1 and R2. If the bubbles are of the same radius, the area of the film AUF(R),for a dodecahedral arrangement, is given bys A,F(R) = 1.152R2 (3) If the area of the film AF(Rl,R2) separating the bubbles of radii R1and Rz is equally influenced by the two bubbles, one can expect AF(R1,R2) to be the geometric mean of AuF(&)and Aup(R2),i.e., AF(R182) = [ A u F ( R I ) A u F ( R ~=) I1.152RiR2 ~'~ (4) Therefore, the average area of the film surroundings a bubble of radius R1 is 1.15m1RB being the average radius of the bubbles. In other words, the random packing of the gas bubbles behaves as if every bubble is in direct contact with bubbles of average size. From eq 2 and 4, the interfacial area AT per unit volume of the foam bed is given by
= 1 . 1 5 2 N n ~ S , ~ S , ~ R 1 R 2 ( R , , z ) f ( R dR1 2 , Z )d R 2 =
1.152 NnFfj2
(5) Population Balance Equation for Bubble Size Distribution Let us now consider a volume element of a foam bed of unit cross-sectional area between z and z + dz,as shown
Langmuir, Vol. 2, No. 4,1986 497
Enrichment and Collapse in Foams INTERSECTION OF HORIZONTAL PLANE PLATEAU BOROER
PLATEAU BORDER (INTEF6ECTIMI OF ME FILMS)
PFILY,, ppB
JRP
PGAS BUBBLE FILM [INTERSECTION OF FACES OF TWO ADJACENT POCWEDRA)
Figure 3. Regular dodecahedral structure of a bubble and cross-sectionalview of a plateau border.
schematically in Figure 3. The foam bed consists of bubbles of different sizes in a dodecahedral arrangement with intervening lamellae (films) of different planar interfacial areas. The size distribution of the bubbles changes due to (i) the coalescence caused by the rupture of the films and (ii) the change in size of the bubble because of gas diffusion from the smaller to the larger ones. Let us denote the number of bubbles that flows per unit area of the foam bed and unit time by q , the fraction of bubbles whose volume is between u and u du by f,(u,z) du, the rate of coalescence between bubbles of volume u1 and u2 per unit volume of the bed by @(u1,u2),and the rate of growth of a bubble of volume u by ri. A balance of the number of bubbles whose volume is between u and u + du in the volume element between z and z dz yields the following population balance equation:
+
+
In the above equation, the first term represents the change in the number of bubbles whose volume is between u and u + du, per unit volume of the foam bed, due to convection, the second term represents their change due to growth, and the third and the last terms represent the production and the loss of bubbles of volume u, u + du, respectively, due to coalescence. The density functions fu(u,z) and f(R,z) in terms of bubble volume and bubble radius, respectively, are related by the following equation: f(R,z) = 4?rR2f,[(4/3)?rR3,z1
(7)
The population balance equation can be written in terms of the density function f(R,z) by taking a number balance over the volume element between z and z + dz for bubbles whose radius is between R and R + dR as
nFNf(R,z)JmB(R,R ?f(R ',z) dR' (6a) where @(R1,R2) is the rate of coalescence between bubbles of radii R1 and R2per unit volume of the bed and R is the rate of growth of a bubble of radius R. The terms in the above equation have the same physical significance as the corresponding terms in eq 6. Integrating eq 6a with respect to the bubble radius, one obtains a balance for the total number of bubbles of all sizes as
second term is the change in the number of bubbles due to the disappearance of small bubbles caused by shrinkage, and the last term is the decrease in the total number of bubbles due to coalescence. Mass Balance Equation for Films The liquid is distributed between the films and the plateau borders. The liquid in the films drains into the plateau borders under the action of the capillary pressure, plateau border suction, and van der Waals disjoining pressure. Since bubbles of different sizes exert different capillary pressures on the bordering films, the rate of drainage and hence the thickness of the films will depend upon the sizes of the bubbles in contact with the films. Denoting by xF(R1,R2,z) the thickness of the film in contact with bubbles of radii R1 and R2 at the axial distance z , a material balance for the liquid in such films over a volume element of the foam bed between z and z dz (Figure 3) yields d - - [ ~ ~ F A F ( R ~ , R ~ ) X F ( R ~-, R ~ , ~ ) ~ dz N ~ F A F ( R ~V(R132,z) ,R~) = 0 (9)
+
where V(R1,R2,z)is the velocity of drainage of the film. In the above equation, the first term represents the change in the volume of the liquid in the film due to convection, whereas the second term is the change in the volume due to drainage. Mass Balance Equation for Plateau Borders The foam as a whole is moving upward with a superficial gas velocity G. The liquid in the plateau border is draining, however, downward under the action of gravity. A plateau border is formed by the intersection of three liquid films. Since the films separating the bubbles of different sizes have different planar interfacial areas, the plateau borders at a given cross section will have a distribution of lengths. As pointed out earlier, it is reasonable to assume that the plateau borders at any axial position have the same cross-sectional area ap and that the distribution of their lengths is the same as the bubble size distribution. As can be seen from Figure 2, for a regular dodecahedral structure, the number np of plateau borders per bubble is equal to 10 and the number n g of plateau borders per bubble on a horizontal cross section is equal to 2 (=np/5). In order to calculate the rate of drainage (due to gravity) of the liquid from the plateau borders, the total cross section of the plateau borders at a given axial position of the column must be calculated. With the bubbles approximated as spherical, the number of spheres with a radius between R and R dR that are intercepted by a horizontal plane at a distance betweeen X and X + dX from their centers is given by 2dXNf(R,z) dR. If r is the radius of the circle intercepted by the horizontal plane through a bubble of radius R, then the number of circles per unit area with radii between r and r + dr intercepted on bubbles with radii between R and R + dR is
+
F dR dr = 2Nf(R,z) dR I
I dR =
(ax)R
Therefore, the number of circles n(r) dr with radii between r and r + dr is given by In eq 8, the first term refers to the change in the total number of bubbles per unit volume due to convection, the
n(r) dr =
1
"2Nf(R,z) dR r dr (R2 - p2)1/2
Narsimhan and Ruckenstein
498 Langmuir, Vol. 2, No. 4, 1986
For the total number of bubbles M that are intercepted by the horizontal plane per unit area one thus obtains
Consequently, the total number of plateau borders intercepted by a unit area of bed cross section is 2I@n',,. The plateau borders can, however, have any orientation. If one assumes that they are randomly oriented, then the probability for their orientation to be between 6' and B d6' is 2 sin B cos B de. Therefore, the volumetric flow rate in a plateau border oriented between B and B d6' is equal to 2apu sin B cos2 B de, where u is the velocity of drainage under the action of gravity. As a result the average volumetric flow rate per plateau border is 2 a , ~ J $cos2 ~ B sin B dB = 2/3apuwhich, combined with eq 10, yields for the rate of gravity drainage of the liquid 4 / f l n ' , a p u R . Because of coalescence and disappearance of small bubbles due to shrinkage, both the total number of bubbles per unit volume and the corresponding number of plateau borders decrease. The liquid in the annihilated plateau borders is redistributed among the remaining plateau borders. The total number of plateau borders annihilated due to coalescence and shrinkage in a volume element between z and z + dz is given by
Mass Balance Equation for the Surfactant Similarly, a material balance for the surfactant can be written as d 4 d - -(qapnpic) + - -(Nn',,a@uc) dz 3 dz
dR2] -
+
+
(N/2)nF~pSomSomB(R1,R2)f(R1,Z)f(R2,Z)a 1
(15) where c and I? are the bulk concentration and the surface excess of the surfactant, respectively. In eq 15, the first and the third term represent the convective change in the amount of surfactant in the plateau borders and 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 at the gas-liquid interface due to the bubble coalescence and growth. Combining eq 15 with eq 9 and 14, one obtains
[
iNn\a@u
- Tapnpi-
-
Nnpf(R,z)IR=o (11) Since the bubbles in the foam bed are assumed to be randomly arranged, the length distribution of the annihilated plateau borders can be expected to be the same as that of all the plateau borders. Therefore, the volume of liquid in the annihilated plateau borders that is redistributed in the volume element between z and z + Az is given by
w
(~/2)nFn,a,~So"So"~(~1,~z,f(~l,z)f(~2,z)
Therefore, the change in the bulk concentration of the
- ~ n p a p i k f ( ~ , z (12) ) ~ R ~ osurfactant ~ is related to the amount of surfactant redistributed from the destroyed gas-liquid interface (second where 1 is the average length of the plateau borders. For term in eq 16) as well as to the amount redistributed from a dodecahedral arrangement, the average length is given by8
i = sR
(13) 6 being a constant (=0.816). In the present model, it is assumed that the volume of liquid in the plateau borders annihilated by coalescence and shrinkage is redistributed among the remaining plateau borders proportional to their lengths. The material balance for the liquid in the plateau borders over a volume element between z and z + dz can now be written as d 4 d - -(Ta,n,i) + - -(Nn;aPRu) dz 3 dz
+
In the above equation, the first term represents the change in the volume of liquid in the plateau borders 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. The last two terms represent the volume of the liquid redistributed because of coalescence and shrinkage, respectively.
the annihilated plateau borders (third term in eq 16) because of coalescence and shrinkage. Equations 6,8,9, 14, and 16 are to be solved with the appropriate boundary conditions, to be discussed later, in order to determine the size distribution of the bubbles f(R,z),the film thickness XF(R1$2,%),the plateau border area ap,the number of bubbles per unit volume N , the bulk concentration of the surfactant c, the gas-liquid interfacial area, and the total amount of surfactant as a function of the axial position. In order to accomplish this, the velocity due to gravity drainage u , the velocity of drainage of the the rate of growth of bubble size due to films V(R1,R2,z), interbubble gas diffusion R, and the coalescence frequency B(R1,R2)should be known. The evaluation of the above quantities is discussed next.
Velocity of Gravity Drainage through Plateau Border The average velocity of gravity drainage u through the plateau border, obtained by the solution of the NavierStokes equation with the appropriate boundary conditions, is given by6 (17) where cv, the velocity coefficient, is defined as the ratio
Langmuir, Vol. 2, No. 4, 1986 499
Enrichment and Collapse in Foams of the average velocity to the average velocity through the plateau border for a large (infinite) value of the surface viscosity. The velocity coefficient c, is a function of the inverse of the dimensionless surface viscosity y defined as
area a p , on the basis of geometric considerations, as8 R, = -1.732f.F [(1.732fF)2- 0 . 6 4 4 ( 0 . 4 3 3 f .-~ ~ 1 , ) ] ’ / ~ / 0 . 3 2 2 (24)
+
where where p and pa are the viscosity and the surface viscosity, respectively. The numerical results of the functional dependence of c, on y are expressed as a spline function CV
= bio
+ bil(Y - T i ) + b i 2 ( ~-
+ b i 3 ( ~- y J 3
(19)
where yi d y d ~ i + i~ =, 1, 2, ...5. Five different sets of spline coefficients bij 0’ = 0,1,2,3)have been given for five different intervals of y. The values of the coefficients (bij, j = 0,1,2,3)and the constants ( y i ,i = 1,2,...5) can be found elsewhere?
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 becomes of the order of lo3 A or smaller, van der Waals interactions influence the process of thinning through the disjoining pressure. Let us consider the film separating two bubbles of radii R1 and R2 and denote by pot the pressure at which the inert gas is introduced into the liquid. The pressures inside the two bubbles will be greater than p o by their capillary pressures, Le., the pressures inside the bubbles of radii R1 and R2 will be p o + 2a/R1and p o + 2a/R2,respectively, a being the surface tension. Because of the difference in the capillary pressures, the film between the two bubbles of radii RIand R2 will not be planar but will be concave toward the smaller of the two bubbles. Let R, be the radius of curvature of the film. The pressure p inside the film (without the disjoining pressure which is introduced later) will then be given by
Therefore, the radius of curvature R, of the film is given by _1 -- - 1- - 1 R2 d R1 R , 2R2 2R1’
-
When R1 = R2, R, 0 3 ; i.e., the film is planar. The pressure drop ( A p ) , due to the capillary pressure can now be written as = -2a- - =2a- + -2a= - +2a- a a (22) R2 R, R1 R , R1 R2 Since the radius of curvature R , of the film is much larger than the fiim thickness, the velocity of drainage can be calculated as for a planar film. The total pressure drop Ap responsible for the thinning of the film is given by
where R , is the radius of curvature of the plateau border and A is the Hamaker constant. In the above equation, the first two terms represent the contributions from the capillary pressure, the third and the fourth term representing the contributions from the plateau border suction and the disjoining pressure, respectively. The radius of curvature of the plateau border R , can be expressed in terms of the mean film thickness f F and plateau border
f F
=
S,mS,mAF(R1.R3f(R1P)f(R*’t) M1 a
2
(25) Reynolds’ equation adequately describes the thinning of a circular plane parallel film if the film surface is rigid. In order to predict the velocity of thinning for a mobile interface, however, one has to account for the surface viscosity as well as for the surface and bulk diffusion of surfactants. The analysis of lvanov and Dimitrovmfor the velocity of thinning accounting for the above factors is employed in the present calculations. The velocity of thinning V(R1,R2,z)of the film separating bubbles of radii Rl and R2 in terms of the pressure drop Ap (as given by eq 23) and the interfacial mobility is given by
Lt.
( 6 +~ P5J2n2aXF(R1,R21Z)J2(x,)
I-’
(26)
+ 6~ + P 5 ~ , 2 ~ X ~ ( ~ i ~ ~ y z ) ] x , 3 J i ( x , )
n-1 [ ~ C L
where the eigenvalues An are the solutions of the equation Jo(X,) = 0 (27) kn
= h/RF(R1$2)
(28)
and Jo,J1,and J2 are Bessel functions. In eq 26 the surface mobility is accounted for by the parameter a defined as
where D and DS are the bulk and surface diffusion coefficients of the surfactant, respectively, and (da/ac),and (dI’/dc)o refer to the gradients of the surface tension and the surface excess, respectively, at the bulk concentration. The parameter a compares the relative magnitudes of the bulk and surface diffusion with that of the Marangoni effect.
Rate of Growth of Bubbles due to Interbubble Gas Diffusion As pointed out earlier, the inert gas diffuses through the thin lamellae from the smaller to the larger bubbles because of the difference in their capillary pressures. As a result, the larger bubbles grow at the expense of the smaller bubbles. Let us consider the diffusion of the inert gas between bubbles of radii R1and R2 through the intervening The number of moles liquid film of thickness xF(R1,Rp). of inert gas transferred per unit time J I 2is given by
where D,and H are the diffusion coefficient of the inert gas and the Henry’s law constant, respectively. The diffusion of the inert gas will be from the bubble of radius R1to the bubble radius R2, or vice versa, depending on
500 Langmuir, Vol. 2, No. 4, 1986 whether R, is less or greater than R2. Since the bubbles are assumed to be arranged randomly with an average coordination number, the number of moles Jl of inert gas transferred per unit time from the bubble of radius R, is
The number of moles of inert gas N , in a bubble of radius R1 is given by
Narsimhan and Ruckenstein perturbations are small under such conditions. The analysis of Ivanov and DimitroP for the evaluation of the critical thickness for film rupture due to thermal perturbations is employed in the present calculations. The equations for the dominant wavenumber 1,(R1,R2),the transition thickness xt(Rl,R2)(at which the perturbations begin to grow), and the critical thickness xCIit(R1,R2) (at which the film ruptures instantaneously) for a film bordering bubbles of radii R1 and R2 are given by
R, being the gas constant. A mass balance of the inert gas in a bubble of radius R1 therefore yields
1.e.;
and
It is clear that the rate of growth is negative for bubbles of small sizes and positive for large bubbles. Critical Thickness for Film R u p t u r e Thin liquid films (lamellae) separating bubbles of different sizes are continually subjected to thermal perturbations. In addition, mechanical perturbations are also generated by the flow field, particularly at high superficial gas velocities. These perturbations can be represented as a superposition of an infinite number of waves of different frequencies and amplitudes. The corrugation of the film surface caused by these external perturbations gives rise to two opposing forces: (1) a force due to the increase in the magnitude of the negative disjoining pressure (caused by van der Waals interactions) which tends to deform the film further and (2) a restoring force caused by the local capillary pressure which tends to smoothen the film surface. For a thick film, the restoring force due to the capillary pressure predominates so that the perturbations die out resulting in the return of the film to its plane parallel shape. However, for a disturbance of a given wavenumber, there exists a transition thickness below which the amplitude of the perturbations grows because of the predominant effect of the disjoining pressure. In addition, the mean thickness of the f i decreases because of film drainage. This continues until the amplitude of the growing perturbations equals half the mean film thickness (critical thickness) at which point the film ruptures instantaneously. This critical thickness is governed by the dominant wavenumber for which the growth rate of the perturbations is maximum. Of course, the critical thickness depends on the radius of the film, the physical properties of the system such as surface tension, surface tension gradient, density, viscosity, and the surface viscosity and the rate of f i i drainage. As pointed out earlier, the radius and the velocity of drainage of the film depend on the sizes of the bubbles it borders. Therefore, the critical thickness for rupture of a film will be also dependent upon these sizes. In the present calculations, it is assumed that the films are subjected only to thermal perturbations. This assumption is reasonable especially for low superficial gas velocities because the mechanical
where (37)
z +( 6 +~+/ J~&&, ~, %Q Xx )) JA~~(J&~)( X , )(38) m
( $ =n=1(6p
6pc~
(the disjoining pressure), in terms of the film and thickness x , is r = -- A (39) 6rx3 In the above equations, Ap is given by eq 23 and A,, k,, and a are given by eq 27-29, 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 rupture of the T i s separating bubbles of different sizes. The population balance equation, eq 6, the balance equation for the total number of bubbles, eq 8, and the material balance equations, eq 9, 14, and 15, are to be solved to determine the film thickness xF(Rl,R2)between bubbles of different sizes and the plateau border area up as a function of the axial distance z. The radius of curvature of the plateau border R, is evaluated from eq 24. The pressure drop Ap for the drainage of a film between bubbles of any sizes is then calculated by using eq 23. Equations 34-36 are then solved xt(Rl,R2),and 1,simultaneously to evaluate xcIit(R1,R2), (Rl,Rz)for all sets of values of R, and R2. Since the films thin continuously because of drainage, the thickness of the films between bubbles of certain sizes may become equal to the critical thickness at a certain axial distance resulting in their rupture and the subsequent coalescence of the bubbles. Coalescence Frequency Let us consider a film separating bubbles of radii R1 and R2. A t a certain axial distance, or, equivalently, a certain
Enrichment and Collapse in Foams
Langmuir, Vol. 2, No. 4, 1986 501
Dimensionless Balance Equations
time tc(R1,R2)spent by the moving foam in the foam bed, the film thickness XF(R&,Z) becomes equal to the critical thickness xcrit(R1,R2).As a result, the film ruptures instaneously leading to the coalescence of the bubbles of radii R1and RP. The probability P that the bubbles of radii R1 and R2 coalesce during the residence time interval t and t dt can be written as
+
bubbles of radii R1 and R2 coalesce during t, t + d t
]
= P(R182) dt
where P(R1,R2)is the coalescence frequency.
1
coalescence between more than two = O(dt) p [ bubbles to occur during t, t + d t i.e., at any instant of time, multiple coalescence events in the assemblage of bubbles are negligible. The average number of pairs of bubbles in one dodecahedral assemblage of bubbles is nF. During an infinitesimal time interval between t and t + dt, a binary coalescence event can occur between any one of the nF pairs of bubbles in a dodecahedral assemblage, provided the thickness of the film separating the bubbles becomes equal to the critical thickness. Therefore, the expression for the coalescence frequency P(R1,R2)can be written as P(R1P2) = (l/nF)a[t - tc(R1Pz)l (40) where 6 is the Dirac delta function, t is the time spent by the moving foam to reach the axial position z, and tc(Rl,R$ is the value of t at which the thickness of the film between the bubbles of radii R1 and R2 equals the critical thickness xcrit(R1,R2).The coalescence frequency is a Dirac delta function because the rupture of the film of critical thickness is instantaneous. Coalescence events will change the bubble size distribution and also will result in the rearrangement of the assemblage of bubbles. During an infinitesimal time interval, only one binary coalescence event per a dodecahedral assemblage is likely to occur. Consequently, the number of binary coalescence events during an infinitesimal time interval is small and the resulting rearrangement of the assemblage is also small. It is, therefore, reasonable to assume that the rearrangement of the bubbles (due to coalescence) will very likely preserve the thickness of the film.
-
d -[N*a*$*] dZ
Adsorption Isotherm of Nonionic Surfactants at the Air-Water Interface The surface pressure Pa for alkanols (C3-Clo) at the bubble-liquid interface has the form23
where I' is the surface excess, rm is the surface excess a t saturation, R, is the gas constant, Tis the absolute temperature, Ha is the partial molar free energy of surface mixing at infinite dilution, and the surface pressure P, is the difference between the surface tensions a, of the pure water and u of the surfactant solution u = us - P, (42) The adsorption isotherm has the formz3
where Kp = ( P , / Y ) ~(y, being ~ the mole fraction of the surfactant in the bulk) is a constant. (23) Lucassen-Reynders,E. H. h o g . Surf. Sci. 1976, 10, 253.
and
--
4 us d + 156G -[N*a*$*u*] + dz
502 Langmuir, Vol. 2, No. 4, 1986
Narsimhan and Ruckenstein
where
and AP* = [ ( a / R i )+ ( a / & )
Boundary Condition In order to solve eq 45-51, the values of the inlet parameters Rot K , €0, 4 ~ 0 cg, , G,nF, np,and AuF(RO)must be known. The structure of the foam determines the values of nF,np, and AUF(RO).For a dodecahedral structure8 nF = 6
+ ( a / R p )+
A/6ax3~(Ri,R2,z)I / (ao/Ro) The liquid holdup t at any height is the sum of the liquid in the plateau borders and the films. Therefore, € = €o(l - c$Fo)N*a*&?*+ nFNduF(Rg)xFfl*
1=A*F(R*,R*z)x 0
1
*F
x
f*(R*1,R*2,z)f*(R*l,z)f*(R*z,Z) dR*l d R * 2 (50) where to and @FO are the inlet liquid holdup and the fraction of the liquid distributed in the films at the inlet respectively. The enrichment factor e , which is defined as the ratio of the concentration of the surfactant in the top product to the inlet concentration cot depends both on the total amount of surfactant in the bulk and at the gas-liquid interface at the outlet of the foam bed. Therefore,
f*(R*z,l)dR*l d R * 2
I
(51)
where K = r / c and c * ~= cf/co, cf being the bulk concentration of the surfactant 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 gas-liquid interface. Equations 45-51 must be solved with the appropriate boundary conditions and inlet bubble size distribution in order to obtain the enrichment factor and the profiles of liquid holdup, bubble size distribution, number of bubbles per unit volume, gas-liquid interfacial area, and bulk concentration of the surfactant. Inlet Bubble Size Distribution The bubble size distribution at the pool liquid-foam interface is determined by the nonuniformities in the openings of the sparger through which the inert gas is introduced. The inlet bubble size distribution can be satisfactorily described by a r distribution as
np = 10 and AUF(RO) = 1.152R:
The inlet mean bubble radius Ro, the coefficient of variation of the inlet bubble size distribution K , and the superficial gas velocity G depend upon the sparger and the inlet gas flow rate. The inlet liquid holdup eo and the fraction of the liquid distributed in the films 4F0are governed by the entrainment of the liquid in the plateau borders and the films at the pool liquid-foam interface. It is difficult to evaluate the above quantities from hydrodynamics. However, for a closed packed structure of the foam bed, one can assume that to = 0.26. An overall steady-state material balance for the liquid indicates 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. The flow rate of the top product is much smaller than the rate of uptake as well as the rate of drainage at the pool liquid-foam interface. Therefore, the rate of uptake of liquid at the pool liquid-foam interface can be approximated as equal to the rate of gravity drainage. At the pool liquidfoam interface, the rate of uptake of liquid per unit cross section of the bed is Gto/(l - eo), while the rate of drainage through the plateau border per unit cross section of the bed is 4 / 1 J V o n p a ~ o uTherefore, o. 4 GtO -Nonpadouo = 15 (1 - €0)
The inlet distribution of the liquid between the films and the plateau borders is governed by the above mass balance equation. If the inlet film thickness is assumed to be the same for all bubbles, then xFO
=
€ o c $ ~ o / n ~ N d u ~SomSmA*~(R*i,R*2)f*o(R*l)f*o(R*z) ( R o )0 dR*l . m * 2 (57)
No = (1 - eo)/y3~Ro3&mR*3f*0(R*) dR* apo
where a and b are the parameters of the r distribution. The inlet mean bubble size Ro and the coefficient of variation of the inlet distribution K (the ratio of the standard deviation and the mean) are given by Ro = ab K
= l/a112
(53)
The dimensionless inlet bubble size distribution fCo(R*) can then be expressed in terms of the coefficient of variation K as
(55)
= (1 - 4FO)~O/NOnp~R0
(58) (59)
and R, = - 1.732X~g+ [ ( 1 . 7 3 2 ~-~0.644(O.433xFo2 ~)~ - a,)]1/2/0.322 (60) where A*F(R*I,R*2) = R*1R*2and f*o(R*)is given by eq 54. Equations 56-60 are to be solved simultaneously in order to determine xm, and a@ with the constraint that Rpo 2 0. Effect of Inlet Bubble Size Distribution on the Performance of a Semibatch Foam Fractionator The population balance equation, eq 45, balance equation for the number of bubbles, eq 46, and material balance equations, eq 47-49, have been solved numerically subject
Langmuir, Vol. 2, No. 4, 1986 503
Enrichment and Collapse in Foams
Table 1. Values of System Parameters Used in the Calculations quantity value 1 293 10-5 10-5 10-6 10-2-0.5
lo4 0.1-1.0 0.1,0.5 30 (5 x 10-8)-104 4.38 x 10-13 6.127 X 2.7 x 106 1.8 1.296 X lo6 L 2 1[ 8
' ' '
" " 10-7 "
" ' 1 1 '
Inlet M e a n
7
' ' ' 10-6 ' " l l l L l 10-5 C (gm moles/cc) Figure 4. Adsorption isotherm of nonanol-water system at an air-liquid interface at 293 K. sb-8
Curve
I
0.4 6 w
5
L
e V 9
4
c
c
E
r
.-u
k W
3 2
'4
00
0
10-8
,d, ,,,,
, , ,,
10-7
,,,A 10-6
C(gm mdeslcc)
Figure 5. Plot of surface pressure vs. bulk concentration for nonanol-water system at an air-liquid interface at 293 K.
to the inlet bubble size distribution eq 54 and the boundary conditions eq 56-60. Thus, one can determine the bubble size distribution f*(R*,Z),the number of bubbles per unit volume N*, the thickness of films X*F(R*~,B*~,Z), the plateau border area u * ~and , the bulk concentration of the surfactant c* as a function of the axial distance. The dominant wavenumber Zn(R1,B2),the transition thickness xt(Rl,R2), and the critical thickness xcrit(R1,R2)for the rupture of the films separating the bubbles of various sizes have been obtained by solving eq 34-36 in conjunction with the material balance equations. The coalescence frequency for bubbles of various sizes was then evaluated from eq 40. The liquid holdup profile and the enrichment factor were calculated from eq 50 and 51, respectively. All the calculations were performed for the nonanol-water mixture. The adsorption isotherm and surface pressure of nonanol a t the air-water interface are given in Figyes 4 and 5, respectively. The inlet mean bubble radius Ro,the coefficient of variation K of the inlet bubble size distribution, the superficial gas velocity G, the inlet surfactant concentration co, and the viscosity p are the parameters whose effect on the enrichment factor have been investigated. The range of variation of these variables and the values of the other physical parameters are given in Table
0 05
0 40
0 45
Narsimhan and Ruckenstein 8
Superficial Gas Velocity
Curve
Curve
Coefficient of
Variation
R
kdSeCl
7-
0.i 0.25 0.50
i
2
i
6-
40
al
8
5-
c 0
e c
2 4-
5
3
8t
4
.- 37 W
' 1
2-
00
-0
005
0 io
Coefficient of Variation
Superficial Gas Velocity (cmlsec)
0 i5
K
Figure 7. Effect of the coefficient of variation of the inlet bubble size distribution on the enrichment factor at 1 atm and 293 K for p = 10-* P, p8 = lo4 sP,co = lo-' (ng mol)/cmg,8 = 30 s, and Ro = 0.5 cm.
smaller mean inlet bubble sizes, as can be seen from Figure 6. For very broad inlet bubble size distributions (large values of the coefficient of variation), the foam bed becomes unstable because of rapid coalescence followed by collapse. The coefficient of variation at which the foam bed becomes unstable is indicated by arrows in Figure 6. For smaller inlet mean bubble sizes, the foam bed becomes unstable at a smaller coefficient of variation (Figure 6) because of the higher rate of film drainage and subsequent coalescence. It is to be noted that 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 (which is governed by the capillary pressure, plateau border suction, and disjoining pressure) is much smaller than the time scale of gravity drainage from the plateau borders, the overall rate of drainage is higher when a larger fraction of liquid is distributed in the films. As the inlet mean bubble size increases, a larger proportion of the liquid is distributed initially in the films and, as a result, the drainage rate becomes higher. For an inlet mean bubble size of 0.5 cm, the decrease in the liquid holdup (because of the above mentioned increased drainage rate) more than offsets the decrease in the gas-liquid interfacial area per unit volume and, therefore, the enrichment factor is higher than that for an inlet mean bubble size of 0.1 cm. The effect of the coefficient of variation of the inlet bubble size distribution on the enrichment factor for different superficial gas velocities is shown in Figure 7. The higher the superficial gas velocity, the faster is the uptake of liquid. Consequently, at higher superficial gas velocities, the plateau border area and the liquid holdup are larger. The increase in the liquid holdup at higher superficial gas velocities results in a smaller enrichment factor, as can be seen from Figure 7. Broader inlet bubble size distributions (higher coefficients of variation) decrease the enrichment factor because of the coalescence of the smaller bubbles, as already noted before. The coefficient of variation has a stronger effect on the enrichment factor at smaller superficial gas velocities. The effed of superficial gas velocity on the enrichment factor for different Coefficients of variation of the inlet distribution is shown in Figure 8. For narrow inlet bubble size distributions,the enrichment factor decreases with an
io
05
Figure 8. Effect of the superficial gas velocity on the enrichment factor at 1 atm and 293 K for p = lo-* P, pa = lo4 sP,co = lo-' (g mol)/cm*,8 = 30 8, and go= 0.5 cm. Viscosity (poise)
Curve
7-
I 2
1
10-2
2 x io-2
W
2i-
0;
I
I
I
0 05
0 io
0 i5
Coefficient of Variation
K
Figure 9. Effect of the coefficient of variation of the inlet bubble size distribution on the enrichment factor at 1 atm and 293 K for p = lo4 sP,co = lo-' (g mol)/cm*, G = 0.1 cm/s, 8 = 30 s, and = 0.5 cm.
ko
increase in the superficial velocity because of the increased liquid holdup. However, for a broad inlet distribution (coefficient of variation of 0.10),the decrease in the enrichment factor for higher superficial velocities is due both to larger liquid holdups and to the coalescence of the smaller bubbles. Since the effect of coalescence is more pronounced at smaller superficial gas velocities, the enrichment factor exhibits a minimum. The effect of the coefficient of variation of the inlet distribution on the enrichment factor for different values of viscosity is shown in Figure 9. Broader inlet bubble size distributions, because of coalescence, decrease the enrichment factor. The coefficient of variation has a stronger effect on the enrichment factor at lower viscosities. This happens because the faster rate of film drainage leads to a larger number of coalescence events. For liquid visP, the foam bed becomes cosities of and 2 X unstable (because of coalescence and collapse) for coefficients of variation greater than 0.1, as indicated by the arrows in Figure 9. The enrichment factor is plotted vs. viscosity for different coefficients of variation in Figure 10. As the vis-
Langmuir, Vol. 2, No. 4, 1986 505
Enrichment and Collapse in Foams
'1 6
Curve
Coefficient of Variation K
i 2
0 0.05
A
Curve
Coefficient of Variation I
i 2
8-
o.oror
3
1
0.05 0.0707
al
0.40
4
'"I
I L
Q
6-
E
4-
9 c .-0
E
W
2-
OL 40-8
I
40-7
I
IO-'
Inlet Concentration C&m mole/cc)
01
' I
I
io-'
io-2
Figure 12. We&of inlet concentration on the enrichment factor at 1atm and 293 K for p = P, pa = lo4 sP,G = 0.1 cm/s, 0 = 30 s, and Ro = 0.5 cm. 4
Viscosity (poise)
Figure 10. Effect of viscosity on the enrichment factor at 1atm = lo-' sP,co = lo-' (B mol)/cm3,G = 0.1 cm/s, and 293 K for ~.r, 0 = 30 8 , and Ro = 0.5 cm. 8
Curve
Inlet Concentration
(gmmole/cc) r.5 x {o-~ 5 x 40-8 40-7 2 x 40-7 5x 40-7
4 4
7r
40-6
'I01 0
I
I
0.05
O'iO
Coefficient of Variation
C IC
Figure 11. Wect of the coefficientof variation of the inlet bubble size distribution on the enrichment factor at 1 atm and 293 K forp=10-zP,p,=104sP,G=0.1cm/s,0=30s,and~o=0.5 cm.
cosity becomes higher, the liquid holdup increases because of a lower rate of liquid drainage. For narrow inlet bubble size distributions, the decrease in the enrichment factor at higher viscosities is mainly due to the greater liquid holdup. However, for broader inlet distributions, the decrease in the enrichment factor is due both to the greater liquid holdup and to the smaller gas-liquid interfacial area (caused by coalescence). The effect of coalescence being greater at lower viscosities (because of increased rate of film drainage), the plot of the enrichment factor against viscosity exhibits a minimum when the inlet distribution is sufficiently broad. However, when the inlet distribution is very broad, the foam bed becomes unstable as a result of coalescence and collapses. Figure 11shows the enrichment factor as a function of the coefficient of variation of the inlet distribution, for different values of the inlet concentrationof the surfactant. Higher inlet concentrations of the surfactant decrease the
0
R" Figure 13. Inlet and exit bubble size distributionsof a semibatch foam fractionator at 1 atm and 293 K for c~ = 5 X P,pa = lo4 sP,co = lo-' (g mol)/cm3, G = 0.1 cm/s, 0 = 30 8, Ro = 0.5 cm, and K = 0.05. Dimensionless Bubble Radius
interfacial tension and, consequently, the rate of film drainage. As a result, less coalescence occurs at higher inlet concentration and the effect of the coefficient of variation on the enrichment factor is less pronounced. The enrichment factor is plotted in Figure 12 against the inlet concentration of nonanol for different coefficients of variation of the inlet distribution. A higher inlet concentration results in a greater surface concentration of surfactant and therefore in a smaller interfacial tension. The former effect increasesthe enrichment factor, whereas the latter decreases it as a result of lower liquid drainage rate. Consequently, there exists an optimum inlet concentration at which the enrichment factor is maximum. Typical inlet and exit bubble size distributions for stable foam beds are plotted in Figure 13. It can be seen that the outlet bubble size distribution shifts toward larger
506 Langmuir, Vol. 2, No. 4, 1986 Curve
Narsimhan and Ruckenstein 2 .o
5
b c f f i c m t of
Curve
VDlollon r
I 5-
2 3
0 05 0 4504 0 2236
2 3 4
*a
Axial Distance (cm)
i.i78~40-~ 2.366xiO-3 i.937~40-~
i.5
1
,*
*
.-c
c
v)
p:
4.0
v) 0) v)
Figure 14. Profiles of the enrichment factor and the liquid holdup in a semibatch foam fractionator at 1 atm and 293 K for p = 5 x sP,co = lo-' (g mol)/cm3,G = 0.1 cm/s, 6 = P, p = 30 s, and io = 0.5 cm.
bubble sizes, because of the coalescence of the smaller bubbles. Typical profile of the enrichment factor and liquid holdup for three different values of the coefficient of variation of the inlet distribution are shown in Figure 14. For the values of 0.05 and 0.1581 of the coefficient of variations the foam bed is stable. The liquid holdup decreases more rapidly near the pool liquid-foam interface for the broader of the two inlet distributions. This is due to the higher rate of film drainage, which occurs because of a larger number of small bubbles. The resulting coalescence of small bubbles leads to the decrease in the gas-liquid interfacial area. Therefore, the subsequent rate of film drainage becomes slower for the broader inlet distribution and the decrease in the liquid holdup is less rapid. Consequently, the enrichment factor is lower for the broader inlet distribution. For K = 0.2236, the liquid holdup decreases dramatically near the pool liquid-foam interface because of the very high rate of film drainage caused by the large number of small bubbles. The coalescence of these bubbles leads to a large decrease in the number of bubble per unit volume and in the gas-liquid interfacial area. The resulting dramatic increase in liquid holdup and the associated dramatic decrease in the enrichment factor lead eventually to the collapse of the foam bed. The evolution of the bubble size distribution before the eventual collapse is shown in Figure 15. Due to coalescence, the size distribution shifts to the larger bubble sizes. The size distribution evolves into a bimodal distribution function and the fraction of larger bubbles keeps increasing leading eventually to the collapse of the bed. The variation of the number of bubbles per unit volume, the mean bubble size, and the gas-liquid interfacial area per unit volume with the axial distance is shown in Figure 16. Because of bubble coalescence,the number of bubbles per unit volume decreases continuously and the mean bubble size increases continuously leading eventually to the collapse of the foam bed. The interfacial area per unit volume increases near the pool liquid-foam interface because of the dramatic decrease in the liquid holdup. However, because of the opposite effect of coalescence,the interfacial area per unit volume reaches a maximum value at a certain axial position and decreases further continuously. The rate of interbubble gas diffusion is extremely small, except when the film thickness is of the order of a few hundred angstroms. As a result, the effect of interbubble gas diffusion on the evolution of the bubble size distribution and on the stability of the foam bed is expected to be significant only for very large residence times, especially in the top section of the bed.
.-s E E
v)
0.5
0 0
2 .o
i.0
? 3
Dimensionless Bubble Radius R* Figure 15. Evolution of bubble size distribution with the axial distance in a semibatch foam fractionator at 1 atm and 293 K for p =-5 X lo-' P, pLs= lo4 sP,co = lo-' (g mol)/cm3,G = 0.1 cm/s, Fl, = 0.5 cm, and K = 0.2236.
/ I
4.4k
-6
-5
-4
-3
"F
2
0.2
0-40
i
0 0.03 6
i2
48
24
30
Axial Distance z x io4(cm)
Figure 16. Variation of the number of bubbles per unit volume, the mean bubble size, and the interfacial area per unit volume with the axial distance in a semibatch foam fractionator at 1atm and 293 K for I-= 5 X P, p a = lo4 sP,co = lo-' (g mol)/cm3, G = 0.1 cm/s, Ro = 0.5 cm, and K = 0.2236.
It is to be noted that in the present calculations, the films are assumed to be subjected only to thermal perturbations. As pointed out earlier, the above assumption is reasonable at low superficial gas velocities because of the absence of large mechanical interfacial disturbances under such conditions. Therefore, the present calculations are likely to provide the maximum value of the coefficient of variation of the inlet bubble size distribution for which the operation of the foam bed is stable, when the superficial gas velocity is low. The calculations also provide an upper bound for this coefficient for operating conditions under which mechanical disturbances are not negligible.
Enrichment and Collapse in Foams
Conclusions A model for the hydrodynamics of a semibatch foam fractionator is developed. The model accounts for (i) the size distribution of bubbles, (ii) the gravity drainage from the plateau border, (iii) the thinning of the liquid lamellae (films) due to the capillary pressure, plateau border suction, and van der Waals disjoining pressure, (iv) the coalescence of bubbles due to the rupture of thin films, and (v) interbubble gas diffusion. The enrichment factor and the profiles of the liquid holdup, bubble size distribution, number of bubbles per unit volume, and bulk concentration of the surfactant are calculated for the nonionic surfactant nonanol for different values of the inlet mean bubble size, coefficient of variation of the inlet bubble size distribution, superficial gas velocity, viscosity, and inlet concentration of the surfactant. When the inlet bubble size distribution is narrow the enrichment factor is almost the same as that for equal size bubbles of mean size. As the inlet bubble size distribution becomes broader, the enrichment factor is found to decrease because of the coalescence of the small bubbles. For very broad inlet bubble size distributions, a rapid coalescence of bubbles is found to occur, resulting in a dramatic increase in the liquid holdup. This is accompanied by a dramatic decrease in the enrichment factor leading eventually to the collapse of the foam bed. The bubble size distribution in such cases, because of coalescence, becomes bimodal with the fraction of large bubbles progressively increasing before the eventual collapse of the foam bed. The effect of the coefficient of variation of the inlet bubble size distribution on the enrichment factor is found to be more pronounced at lower superficial gas velocities, lower viscosities, smaller mean bubble sizes, and lower inlet concentrations. The effect of interbubble gas diffusion on the evolution of the bubble size distribution and on the stability of the foam bed is significant only at large residence times. The calculations provide an estimate of the maximum coefficient of variation of the inlet bubble size distribution below which the operation of the foam bed is stable, at low superficial gas velocities. They also provide an upper bound for this coefficient at relatively high superficial gas velocities. The calculations do indicate that the simplifying assumption of equal size bubbles can be employed for the prediction of the enrichment factor and of the stability of the foam bed when the inlet bubble size distribution is narrow, especially at high superficial gas velocities, high viscosities, larger inlet mean bubble sizes, and high inlet concentrations.
2
AF(RI,Rz) AuF(R) AT C
CF C"
Cp(R18z)
D D, Ds e
Notation cross-sectional area of plateau border, cm2 Hamaker constant, erg area of the film between bubbles of radii R1 and R2, cm2 area of the film for equal size bubbles of radius R, cm2 gas-liquid interfacial area per unit volume of the foam bed, cm-' bulk concentration of surfactant, (g mol)/cm3 number of films per unit volume of the foam bed, cm-3 velocity coefficient for gravity drainage of plateau border number density for the number of films between bubbles of radii R1 and R2 per unit volume of the foam bed, cmm3 diffusion coefficient of the surfactant, cm2/s diffusion coefficient of the inert gas, cm2/s surface diffusion coefficient of the surfactmt, cmz/s enrichment factor
Langmuir, Vol. 2, No. 4, 1986 507 density for the bubble size distribution in terms of bubble radius R, cm-2 density for the bubble size distribution in terms of bubble volume u acceleration due to gravity, cm/s2 superficial gas velocity, cm/s Henry's law constant partial molar free energy of surface mixing at infinite dilution, erg number of moles of inert gas transferred per unit time between bubbles of radii R1 and R2, (g mol)/s average numer of moles of inert gas transferred per unit time from a bubble of radius R1, (g mol)/s wavenumber of the perturbation at the gasliquid interface adsorption equilibrium constant for the surfactant, cm capillary constant for the surfactant, dyn/cm mean plateau border length, cm dominant wavenumber for the rupture of films between bubbles of radii R1 and R2 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 p / 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 37 bubble radius, cm mean bubble radius, cm radius of curvature of film, as defined by eq 21, cm radius of the film between bubbles of radii R1 and R2, cm gas constant, cm3 atm (g mol)-' K-' radius of curvature of the plateau border, cm rate of growth of bubble radius, cm s-l time, s time at which the thickness of the film between bubbles of radii R1 and Rz becomes equal to the critical thickness, s temperature, K velocity of gravity drainage, cm s-l velocity of gravity drainage for infinite surface viscosity, cm s-l Velocity of drainage of the film between bubbles of radii R1 and R2 at the axial position z, cm s-l bubble volume, cm3 mean bubble volume, cm3 rate of growth of bubble volume, cm3 s-l critical thickness for the rupture of the film between bubbles of radii R1 and R2, cm transition thickness of the film between bubbles of radii R1 and R2, cm thickness of the film between bubbles of radii Rl and R2, cm mean film thickness, cm axial distance. cm Greek Symbols a quantity defined by eq 29 coalescence frequency for binary coalescence P(Ri8d of bubbles of radii R1.and R2, cm-3 s-l inverse of dimensionless surface viscosity, Y defined by eq 18 r surface excess of the surfactant, g mol cm-2
Langmuir 1986,2,508-513
508
r,
surface excess of the surfactant at saturation, gm moles cm-2 density of the liquid, g cm-3 constant = 0.816 coefficient of variation of the inlet bubble size distribution number of bubbles per unit area of the foam bed and unit time, cm-2 s-l viscosity, P surface viscosity, SP surface tension, dyn/ cm
P
6 K
9 P Pa U
=a All
t
7r
4F 0
surface tension of pure water, dyn/cm eigenvalues defined by eq 27 liquid holdup disjoining pressure quantity defined by eq 38 fraction of liquid in films residence time, s
Subscripts 0 f
inlet outlet
Wettability of Polyacetylene: Surface Energetics and Determination of Material Properties Anthony Guiseppi-Elie*t and Gary E. Wnek Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Sheldon P. Wesson Owens-Corning Fiberglas, Technical Center, Granville, Ohio 43023 Received January 6,1986. I n Final Form: April 29, 1986 The critical surface tension for wetting, yo of predominantly cis-polyacetylene has been found from contact angle measurements to be 40.1 mN m-l. This is the largest yc found to date for a purely hydrocarbon polymeric solid. When polyacetylene is iodine-doped to (CHh.,),, ye increases modestly to 44.2 mN m-l. Changes in the limiting slope in the vicinity 71- yc of the Zisman plot suggest that the doped polymer is better wetted by polar liquids. Using ye found for cis-(CH), and empirical correlations established with / ~ . is in good agreement with the the solubility parameter, 6, produced a value of 6 = 9.94 (cal ~ m - ~ ) lThis value of 9.7 (calcm4)1/2calculated from group contributions. From establishedempirical correlationsbetween yc and the Lorentz-Lorentz function, the dielectric constant of cis-(CH), was found to be 3.6. This value is in excellent agreement with previously reported direct measurements. The dispersion component of the substrate surface energy,,:y has been found, by using the method of Fowkes, to be 58 mN m-l for cis-(CH), and 90 mN m-l for (CH10.20)x.
Introduction The surface chemistry of polyacetylene is implicated in many of ita suggested te~hnologies.~ Battery applications, liquid junction solar cells, and fuel cells are all influenced significantly by the chemistry at the material's surface. In addition, polyacetylene is also of fundamental interest, since a polymeric system of conjugated double bonds has only rather recently been available as film for the study of surface energetics. Energetics of solid surfaces can best be approached from a study of the interaction of probe phases (liquid or gaseous) with the unkown solid. Substrate wettability can be evaluated from the contact angle made between an appropriate probe liquid and the substrate. This approach allows a comparison of the wettability of various substrates with respect to the particular probe liquid and has the advantage of requiring a minimum of instrumentation and comparative experimental facility. A second more useful approach is that credited to ZisThe Zisman method allows the evaluation of a semiempirical property of the substrate, the critical surface tension for wetting, yc. The critical surface tension for wetting is determined from a plot of the cosine of the observed contact angle made between a series of probe Present address: Molecular Electronics Corporation, Torrance,
CA 90503-2417.
liquids of known surface tensions at the unknown solid surface vs. the surface tension of the probe liquids. The critical surface tension for wetting is defined as that value of surface tension below which all liquids make zero contact angle with the substrate and is determined graphically by an extrapolation of nonzero contact angle data to cos Bobd equal unity. These generalized approaches allow a qualitative ranking of substrates as more or less wettable but do not provide fundamental information on the energetics of the solid surface. A more fundamental approach involves the application of the modified form3of the long-standingYoung equation4 YS
-re =~
s l +~ l COS v Be
(1)
and the Girifalco and Good6geometric mean rule approach to interfacial tensions.
(1) Guiseppi-Elie,A.; Wnek, G. E. J . Polymn. Sci., Polym. Chem. Ed. 1985,23, 2601. ( 2 ) Zisman, W . A. Ind. Eng. Chem. 1963,55 (lo), 19. (3) Barton, A. F. M. Handbook of Solubility Parameters and Other Cohesion Parameters; CRC Press: Boca Raton, FL, 1983; p 426. (4) Young, T . Philos. Trans. R. SOC.London 1805, 95, 15. (5) Girifalco, L. A.; Good, R. J. J . Phys. Chem. 1957, 61, 904. (6) Good, R.J. J. Colloid Interface Sci. 1975, 52, 308. (7) Kinlock, A. J. J. Mater. Sci. 1980, 15, 2141.
0 1986 American Chemical Society
