I n d . Eng. Chem. Res. 1995,34, 1371-1382
1371
Boundary Layer Solution for a Bubble Rising through a Liquid Containing Surface-Active Contaminants Graham F. Andrews and Shu-Lun S. Wong* Biotechnologies Department, Idaho National Engineering Laboratory, Lmkheed Idaho Technologies Company, P.O. Box 1625, Idaho Falls, Idaho 83415-2203
Millimeter-sized bubbles rise through most liquids with Reynolds numbers of several hundred.
A boundary layer solution is given for flow over the upper surface of spherical or spherical-cap bubbles when surface-active solutes adsorb on the bubble interface and inhibit its motion. Diffusion of adsorbed surfactant along the interface is shown to be negligible. The surfactant accumulates not as a cap around the bottom of the bubble, as in the creeping-flow case, but in a band around the flow separation point from where it desorbs back into solution. The bubble interface is stationary at, or slightly above (if diffusion along the interface is considered), the flow separation point. As much surfactant desorbs from as adsorbs to the interface in the boundary-layer region, and since this equality must hold for the whole bubble, it is also true for the wake region. The flow separation angle moves from 130”to 80”with increasing surfactant concentration. This may, however, also reflect a change in bubble shape since the accumulation of surfactant reduces the interfacial tension and could be producing the sharp corner characteristic of the flow separation point in spherical-cap bubbles.
Introduction Predicting the hydrodynamics around a bubble rising through a liquid containing surface-active solutes is a problem with a long history (Clift et al., 1978; Levich, 1962). It is an important problem because the hydrodynamics influence not only the bubble rise velocity but also the gadiquid mass transfer rate, vital in fennentation and gadliquid reaction processes, and the particle capture rate by the bubbles in flotation processes (Reay and Ratcliff, 1973). It is also a very difficult problem involving not only the laws of fluid mechanics but also the rate of mass transfer of the surfactant between the bubble interface and the bulk liquid (Borwanker and Wasan, 1988) and the laws of surface thermodynamics that dictate how the adsorbed surfactant changes the interfacial tension. The resulting gradients of interfacial tension around the bubble have a strong influence on the mobility of the interface and thus on the hydrodynamics. A complete mathematical model requires that the Navier-Stokes and surfactant diffusion equations be solved simultaneously, with boundary conditions arising from the force and surfactant mass balances on the bubble interface. This is difficult even if the bubble is assumed t o remain spherical. Accounting for the way in which adsorbed surfactant may change its shape to ellipsoidal or cap-shaped adds new levels of complexity. Most theoretical work on this problem has focused, for reasons of mathematical simplicity, on spherical bubbles in the creeping-flow hydrodynamic regime. Harper (1972) gives a rigorous discussion and a complete solution, and several approximate solutions are also available. Most of these approximate models avoid the surfactant mass transfer and thermodynamic parts of the problem by assuming a form for the variation of the interfacial tensions around the bubble (Schecter and Farley, 1963). In the stagnant-cap model (Sadhal and Johnson, 1983) the surfactant adsorbed on the bubble interface is assumed to form an immobile cap around the rear stagnation point. In the absence of surfactant the cap does not exist and the bubble behaves hydrodynamically as a fluid sphere. With increasing surfactant concentration in the liquid the cap grows up around
the bubble until it covers the entire surface, at which point the bubble behaves as a solid sphere. The creeping-flow regime is satisfactory for the study of drops and for small bubbles in a very viscous liquids, but is totally unrealistic for most practical situations. For a bubble rising in water R e = 1 corresponds to a diameter of a few microns. Bubbles in real processes are commonly a few millimeters to a centimeter in diameter ( R e = 200-1000), an awkward regime in which the bubbles vary in shape from spherical to spherical cap. Few solutions are available for this range of Reynolds numbers, and those that do exist are essentially interpolations between the known solutions for completely fluid spheres (no surfactant) and solid spheres (assumed by extension of the creeping-flowcase to be the situation a t high surfactant concentrations). Ruckenstein (1964) based his interpolation on integral boundary-layer hydrodynamics but did not consider the surfactant mass transfer and thermodynamic sections of the problem explicitly. Lochiel’s (1965)approach was more rigorous in these aspects of the problem, but he used a perturbation of the fluid-sphere hydrodynamic solution that may not be valid very far toward the solidsphere limit. This paper presents a solution to the problem for millimeter-sized bubbles based on ‘boundary-layer and wake” hydrodynamics. The first part gives a semiquantitative discussion based solely on the force and surfactant mass balance equations at the bubble interface. It shows that the “stagnant-cap” model is a consequence of creeping flow hydrodynamics and that, at higher Reynolds numbers, it must be replaced by a “stagnantring” model in which the adsorbed surfactant accumulates not around the rear stagnation but around the flow separation point. It also shows that this model is consistent with the appearance of a “corner” a t the flow separation point as the bubble changes from spherical to cap-shaped. This discussion also provides an essential boundary condition for the second part of the paper in which integral boundary layer theory is applied to the hydrodynamics and surfactant mass transfer sections of the problem. The solution applies to the upper, ‘boundary-layer” region of spherical and
0888-588519512634-1371$09.00/00 1995 American Chemical Society
1372 Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995
spherical-cap bubbles. In the absence of a proper description of the wake behavior (Fan and Tsuchiya, 1990) it cannot predict the bubble rise velocity. However, after further development, it should give reasonable estimates for gasAiquid mass transfer, very little of which happens in the wake region, and excellent predictions for the capture of particles by the bubble, all of which happens on the upper surface (Reay and Ratcliff, 1973).
Variation of Adsorbed Surfactant and Interface Velocity Whatever formulation is used for the hydrodynamic and surfactant transport parts of the problem, the equations describing the mass balance for adsorbed surfactant and the tangential and radial force balances on the interface remain the same. They therefore provide the starting point for consideration of the problem. The Interface Force Balance. If the shear arising from the viscosity of the gas can be ignored (reasonable for large bubbles), then the shear stress exerted on the interface by the liquid must be balanced by the surface force arising from the interfacial tension gradient. The resulting equation for a spherical interface is
Since adsorbed surfactant (r)reduces the interfacial tension (a),the term in brackets is always positive. Consider first the creeping-flow hydrodynamic solution. In this case the velocity gradient a t the interface V ’ 2 0 everywhere, so eq 1gives drldd L 0 everywhere. Since the adsorbed surfactant concentration can only increase around the bubble, the surfactant must be accumulating in, and desorbing from, the region around the rear stagnation point. Now consider the very different ‘(boundary-layer and wake” type of hydrodynamics that are more appropriate for millimeter-sized bubbles in most liquids. In the upper, boundary-layer region the situation is similar to that described above. V ’ is positive and the resulting shear stress pushes the interface down around the bubble. r is increasing (eq 1) and the interfacial tension, a, decreasing around the interface. However, in the wake region V ‘ is negative, which tends to push the interface, and its adsorbed surfactant, back up around the bubble. Equation 1shows that r is decreasing and a increasing around the bubble in this region. The adsorbed surfactant is accumulating not around the rear stagnation point, but in a ring around the flow separation point, where V ‘ = 0 (by definition), and r is a maximum (by eq 1). Symmetry still requires dl7dO = 0 at the rear stagnation point, but this now corresponds not to a maximum adsorbed concentration, as in the creeping-flow case, but a minimum. Shear from the recirculating wake is continuously creating fresh interface at the rear stagnation point. The adsorbed surfactant concentration on this interface may be large, particularly when the surfactant concentration in the wake exceeds its critical micelle concentration (Stebe and Maldarelli, 19941,but it must be less than the adsorbed concentration a t the flow separation point. The distribution of adsorbed surfactant around a millimeter-sized bubble in lowviscosity liquids is very different from that predicted by creeping-flow models.
One consequence of this distribution concerns the effect of surfactants on bubble shape. Analyses of bubble shape in the absence of surfactant show a smooth transition from spherical to ellipsoidal t o cap-shaped with increasing bubble size and decreasing interfacial tension (Ryskin and Leal, 1984), with flow separation occurring where the interface has a large curvature. The radial force balance relates the local curvature to the ratio Apla, where Ap is the pressure difference across the interface. According to the above discussion, adding surfactant not only reduces the equilibrium interfacial tension, it also creates a variation of interfacial tension around the bubble with a minimum at the flow separation point. Surfactants should therefore accentuate the transition from spherical to cap-shaped (compared with a bubble of equal volume in a liquid of the same equilibrium interfacial tension) by increasing the curvature at the flow separation point. Indeed a cap shape could be defined as one with a sharp corner at the flow separation point. The Interface Surfactant Mass Balance. Consider the upper sector of a spherical or spherical-cap bubble shown in Figure 1. The stream tube is the one that passes with a distance 6 , (the mass transfer boundary layer thickness) at an angle 8 from the front stagnation point. Mass conservation requires that the outflow rate of adsorbed surfactant by a combination of interfacial motion and surface diffusion equals the rate of transfer of surfactant to the interface and the rate at which surfactant is being lost from the liquid.
Note that there is no accumulation term. All of the models considered here describe the steady-state situation in which adsorption of surfactant to one region of the interface is balanced by desorption from other regions. The distance that a real bubble must rise before it accumulates enough adsorbed surfactant for this steady state t o be established is an open question. Simplified models can be characterized by the terms that they leave out of this equation. Lochiel (1965) ignored liquidinterface mass transfer (the third term) on the grounds that adsorbed surfactant can be transported around the interface by interfacial motion much faster than it can be transferred to the interface from the bulk liquid. On the other hand, both Harper (1972) and the present work conclude that diffusion of adsorbed surfactant along the interface (the second term) can safely be ignored. Ignoring both of these terms leaves uT = 0, implying that the interface can be divided into immobile regions where adsorbed surfactant accumulates (u = 0, r > 0) and other regions that are free of surfactant but where the interface is mobile (r = 0, u * 0). Sadhal and Johnson (1983) used this equation to justify the ((stagnation-cap”model, in which adsorbed surfactant forms an immobile cap over the rear of a drop. However the discussion in the previous section shows this “cap” is a consequence of creeping flow type hydrodynamics, and should not be applied to millimetersized bubbles. In the range of Re of interest here, the adsorbed surfactant accumulates not around the rear stagnation point but around the flow separation point. A “stagnation rinf around the flow separation is more appropriate than a “stagnation cap”, although there may
Ind. Eng. Chem. Res., Vol. 34,No. 4,1995 1373 Velocity = U Surfactant Concentration = S
S t r e a m Tube
-P
Interface Velocity = u Adsorbed Surfactant Conc. =
l
r
Velocity = V(y)
o
I:
h
0
O
Surfactant Canc. = C(y)
O
0
10
Flow in S t r e a m Tube =
Sb. s,".
2 r r ( R + y ) sin@ V(y) dy
0
Outflow o f S u r f a c t a n t in B o u n d a r y L a y e r
=
2r(R+y)
s i n 0 V(y) C(y) dy
Figure 1. Control volume for surfactant mass balance.
be cases where the surfactant concentration in the wake is so high that the two are indistinguishable in practice. Boundary Conditions. In the limiting cases of the solid and fluid spheres, boundary-layer solutions can be found for the hydrodynamics over the front part of the sphere (Kolansky et al., 1977; Moore, 1963). These solutions are useful because they give considerable insight into the problem without involving the complexities of the wake region. The question is whether a similar solution (including all the terms in eq 2) can be obtained for the flow around the upper surface of a spherical or spherical-cap bubble rising through a solution containing surfactants. The difficulty is essentially one of boundary conditions. The solid-sphere and fluid-sphere cases are initial value problems, all of the boundary conditions being provided by the symmetry requirement at the front stagnation point. The same cannot be true of the bubble problem. Surfactant is adsorbing from the liquid to the bubble interface around the front and rear stagnation points and is being transported from both directions to the region around the flow separation point, from where it desorbs back into the bulk liquid. The hydrodynamics are so entwined with the transport of surfactant that the two parts of the problem must be solved simultaneously. Also the desorption rate around the flow separation point influences, via the interface force balance, the mobility of the interface and thus the
hydrodynamics throughout the boundary layer. This is inherently a boundary-value problem. At least one boundary condition must be specified a t the flow separation point, and the "stagnant-ring" picture suggests a form for this condition. The interface is being pushed down the bubble (u > 0) by the shear stress on the interface in the boundary-layer region and pushed up the bubble by the recirculating flow in the wake (u < 0). This suggests that at the separation point
e = e,
u=v'=o
(3)
A more rigorous boundary condition at the point where u = 0 is derived later. Within the accuracy of the present analysis it was found to give results indistinguishable from eq 3 for all realistic parameter values. An interesting consequence of this boundary condition is that it makes the left-hand side of eq 2 zero at the flow separation point (dr/d€J= 0 where V ' = 0 from eq 1). What this means is that there is no net transfer of surfactant t o the interface in the boundary-layer region; the adsorption of surfactant around the front stagnation point is exactly balanced by desorption from the region further down the interface but above the flow separation point. The same balance between surfactant adsorption and desorption must also hold for the wake region. Note that caution is needed when comparing theories
1374 Ind. Eng. Chem. Res., Vol. 34,No. 4, 1995
such as that presented here with experimental data. Surfactants have a direct effect on observable quantities (mass transfer coefficients, bubble shapes, etc.) by creating tangential gradients of interfacial tension around the bubble. However they also have an indirect effect due to the changes they cause in the bubble rise velocity. The direct effect of the addition of surfactant on bubble shape suggested above may not in fact be observable because the presence of surfactant also slows the bubble down. The analysis of the boundary-layer region presented here gives no information about the rise velocity, since the form drag arising from the pressure in the wake cannot be predicted. Thus comparison with experimental data on bubble shape or gas/ liquid mass transfer coefficients must await either an extension of the analysis to the wake region, or a data set in which bubble rise velocity is measured simultaneously with the other variable. Work is proceeding in our laboratory on measurements of the capture of fine particles by bubbles, a phenomenon that happens only in the boundary-layer region (Reay and Ratcliff, 1973).
The Model The Effect of Adsorbed Surfactant on Interfacial Tension. The relationship between the interfacial tension, u, and the amount of surfactant adsorbed on the bubble interface, I?, is an essential component of the analysis. Like many other components (hydrodynamics, mass transfer, etc.) it is a complex subject in itself, and must be rather oversimplified in order to obtain a solution to the overall problem. Despite its k n o w n limitations, the Gibbs equation is the common starting point for such analyses (Lochiel, 1965; Harper, 1972; Levich, 1962).
(4) This must be combined with an adsorption isotherm that describes the equilibrium between the adsorbed surfactant concentration, r, and the equilibrium concentration in the liquid s*.Tajima et al. (1971) have shown that this equilibrium can be described by the Langmuir isotherm.
r,s* r = s, + s* Combining eqs 4 and 5
centration at which the model remains valid is that producing r of order 0.9r, near the separation point, which is found to happen when S is of order 5S,. A simplified asymptotic solution for S > S,, based on a two-phase adsorption isotherm (similar to the “insoluble surfactant’’assumption) has been published previously (Andrews, et al., 1988). The use of this two-phase isotherm simplified not only eq 6 but also the boundary condition, eq 3 being replaced by the requirement that I‘ = r, at the flow separation point. However, the Langmuir form of the isotherm is a much better approximation to reality. Eliminating r between eqs 5 and 7 gives a relationship between the surface tension of an interface and the concentration of surfactant in the liquid at equilibrium with the surfactant adsorbed on the interface. Since adsorptive equilibrium implies S* = S, this can be written
o* - oo= -RzT, I n ( l +
s/s,)
(8)
This is the Szyszkowski equation, and it has two uses. Compilations of measured parameter values (Reid, et al., 1977)allow estimates of S , for different compounds. Also, for solutions in which the surfactant is not well characterized (e.g., fermentation broths) it allows our theoretical results to be expressed in terms of the equilibrium surface tension, u*, an easily measurable parameter, rather than S. Surfactant Mass Transfer. The layer of adsorbed surfactant on the bubble interface is assumed to be very thin, from which it follows that the radial velocity at its outer edge can be ignored (Andrews et al., 1988). Adsorptive equilibrium is assumed, so the concentration of surfactant in the liquid at the outer edge of the layer is S*,related to r by eq 5 , and also a function of 8. The transfer (adsorption and desorption) of surfactant between the bulk liquid and this layer is itself a complex process, particularly for ionic surfactants whose adsorption produces electrical double layer effects near the interface (Bonvanker and Wasan, 1988). In order to manage the complexity of the overall problem, it is treated here as a simple mass transfer process that can be characterized by a concentration boundary layer of thickness 6,. In order to evaluate the third and fourth terms in eq 2, the concentration profile in the boundary layer is approximated by the fourth-order polynomial that satisfies the k n o w n limits C = S* at y = 0 and C = S, dCldy = d2C/dy2= 0 at y = 6,:
This can be substituted directly into eq 1and is all that is actually needed for the analysis. Further discussion is given only to establish the limitations and utility of the model. Integrating eq 6 gives the surface pressure as oo - o =
-Rm, In(1 - rm,)
(7)
The adsorbed surfactant concentration, r, is a maximum at the flow separation point. At high concentrations of dissolved surfactant, S, this value approaches the saturation value r, and eq 7 would predict that CJ becomes negative, a ridiculous result arising from the breakdown of the assumptions underlying the Gibbs equation. The practical upper limit of surfactant con-
Applying the concentration boundary layer equation at gives
y =0
Here the second equality follows from differentiating eq 9. It is important to realize the difference between
Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1376 these equations and conventional integral mass transfer boundary-layer theory. In the conventional theory that would apply, for example, to gasfliquid mass transfer froni a bubble, the equilibrium dissolved concentration, S*, is a constant. Hence the “mass transfer shape factor” M = 0 (eq 10) and the concentration profile is greatly simplified. In the case of surfactant mass transfer to and from the interface, S* is the aqueous phase concentration in equilibrium with the adsorbed surfactant concentration (eq 5) which varies around the bubble. Hence dS*/d8 is not zero and M cannot be ignored. Another difference from the conventional theory is that, in evaluating the last term in eq 2, no presumption is made that 6, is much smaller than the hydrodynamic boundary layer thickness, 6. Despite the high Schmidt number for surfactant, this assumption may not be valid for mobile interfaces where 6 is small. Hydrodynamics. It is possible, in principle, to solve all the above equations simultaneously with the NavierStokes equations for flow around a spherical surface. This would be extremely difficult, particularly since it is a boundary-value problem. The simpler, integral boundary-layer approach to the hydrodynamics was therefore adopted. There axe, however, two important differences from the conventional applications of this theory. First the Reynolds number regime of 100-1000 is the lowest for which boundary-layer theory can be expected t o give reasonable results. Second, we are dealing with a boundary layer on a surface that is not only moving, but whose velocity varies around the bubble. These difficulties required modifications to the conventional theory that are described in this section. Note that one advantage of integral boundary-layer theory is that it can readily be extended to nonspherical axisymmetric surfaces. This allows future extension of the model, incorporating the radial force balance on the interface, t o predict the shape of rising bubbles. The form of the integral force-momentum balance is unaffected by the motion of the interface. For a spherical surface it can be written
The velocity profile in the boundary layer is given by the conventional Pohlhausen fourth-order polynomial, modified to allow for motion of the interface
Here Z3 = l - u N f is a dimensionless interface rigidity parameter. The profile shape is described by the parameter N rather than the conventional shape factor A(= NIZ3) because A would be infinite for a non-trivial velocity profile on a fluid sphere with u = Vf. N is found by applying the boundary layer equation at y = 0 (the outer edge of the interfacial layer) again assuming that the radial velocity is zero:
For a solid sphere (u = 0) this reduces to a simple algebraic equation for N . With the mobile interface it becomes a differential equation for u. To allow for the low Re regime and the consequent thick boundary layers, terms of order 6IR are retained in the model. This was done in the last term of eq 2, and similar terms appear in the exact definitions of the displacement and momentum thickness for an axisymmetric boundary layer:
6, = 6, =
f(1 - E)(1 + $) dY
f;( (); 1-
);
1+
(14)
dY
In conventional boundary-layer theory, the velocity at the outer edge of the boundary layer, Vf, is set equal to the inviscid flow solution at the interface (=1.5U sin 8 for a sphere). However, measurements and numerical simulations of the flow around solid spheres at Reynolds number in the hundreds show that this is a considerable overestimate, and that the pressure in the boundary layer is correspondingly higher than that given by the conventional theory (Seeley, et al., 1975; Kolansky, et al., 1977). The thicker the boundary layer the greater the discrepancy, suggesting a correction factor of the form
Vf = 1.5Ux sin 8
(15)
x = 1 - a(6lR) The best value for the semiempirical constant, a, was found as follows. It could be argued that Vf should be set equal to the inviscid flow solution not at y = 0 but a t y = 6 (see Figure 2a; drawn for the sake of clarity for 6IR = 0.3, the thickest boundary layer for which reasonable results can be expected). The value a = 0.8 then gives a reasonable linear approximation t o the inviscid flow solution over the expected range of boundary-layer thicknesses 0 < 6IR < 0.3. Computer runs showed the solution to be insensitive to a in the range 0.5 < a < 0.8,and a value a = 0.7 was adopted based on comparisons of the model with known solutions for solid spheres (Wong, 1991). It was found that this correction term also improves the model’s ability to simulate the shape of the velocity profile on a moving interface. The actual velocity profile must go through a maximum (Seeley et al., 1975) if the profile near the surface, where V increases withy, is to match the free-stream profile where V decreases with y (Figure 2b). As the interface becomes more mobile, the position of this maximum moves down toward the interface until, for the case shown in Figure 2c, it occurs at y = 0. This is the boundary layer on a fluid sphere described by Moore (1963) with zero shear stress on the interface (V’ = 0), the desired limit of our model at zero surfactant concentration. Equation 12 will predict a velocity profile with a maximum if A = NIZa > 12. The difficulty with conventional boundary-layer theory is that this maximum is necessarily larger than the inviscid flow solution at y = 0 (=1.5U sin 8). This is both physically impossible and sufficiently mathematically disruptive to stop execution of the computer program whenever A = 12. Reducing Vf by the correc-
1376 Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 (a) Solid Sphere
(u=O : V'>O)
(b) Mobile Interface (O
