Analysis of Fine Bubble Attachment onto a Solid Surface within the Framework of Classical DLVO Theory

Journal of Colloid and Interface Science 219, 69 – 80 (1999) Article ID jcis.1999.6421, available online at http://www.idealibrary.com on

Analysis of Fine Bubble Attachment onto a Solid Surface within the Framework of Classical DLVO Theory Chun Yang,* Tadeusz Dabros,† Dongqing Li,* Jan Czarnecki,‡ and Jacob H. Masliyah§ ,1 *Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta T6G 2G8, Canada; †CANMET, Western Research Center, P.O. Bag 1280, Devon, Alberta T0C 1E0, Canada; ‡Syncrude Canada Ltd., Edmonton Research Center, Edmonton, Alberta T6N 1H4, Canada; and §Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 2G6, Canada Received February 19, 1999; accepted July 12, 1999

Most of existing theoretical analyses of air flotation processes focus on modeling micron-size particles captured by a single, large gas bubble (being of the order of millimeters in diameter) (1– 4). Usually, the collection efficiency is introduced to quantify the flotation rate. Such a collection efficiency, however, depends on complex processes (or steps) of the particle– bubble collision, interaction, attachment, and detachment, and hence it is characterized by too many interplay factors. To simplify the theoretical treatment, it is acceptable, to some extent, to treat each step separately. This means the particle– bubble collision, attachment, and detachment can be evaluated independently (1– 4). Each of these steps has been extensively studied in the literature. Excellent monographs were presented by Derjaguin et al. (5) and Schulze (6). However, in accordance with the theoretical analyses, the corresponding experimental results, mostly based on the column flotation technique, always exhibit the net effect of these steps. In practice, it is difficult to split the bubble–solid attachment process into the individual steps. This, to some extent, makes measurement results difficult to directly compare with theoretical predictions. In order to gain some insight into basic mechanisms involved in the process of bubble–solid surface interaction and attachment, it is appropriate to analyze a related but simpler and better-defined system. The impinging jet technique provides such an example. It was pioneered by Dabros and van de Ven (7) and has been used extensively to study particle deposition (7–13). Such a technique allows us to directly visualize the deposition process in the vicinity of the stagnation point where hydrodynamic conditions are well defined and controlled. Moreover, the corresponding theoretical model characterizing mass flux to the collector is greatly simplified in the region of the stagnation point so that the measured results can be directly compared with theoretical predictions. Importantly, the bubble–solid surface attachment can be modeled directly based on the continuity equation without artificially splitting it into particular steps. It seems that Harwot and van de Ven (14) were the first to

Fine bubble attachment onto a solid surface in an impinging jet flow was analyzed within the framework of DLVO theory. The effects of hydrodynamic convection, van der Waals (VDW) interaction, electrostatic double-layer (EDL) interaction, and gravitational force on bubble attachment rate (in terms of the Sherwood number) were examined in detail. The analyses showed that due to large Peclet number and gravity number for gas bubbles the behavior of the bubble attachment is significantly different from that of colloidal particle deposition in some aspects. Specifically, it was demonstrated that within a certain range of physicochemical conditions, gas bubbles can attach onto a solid surface despite the existence of repulsive VDW interaction force and the fact that the surfaces of both the bubble and the solid collector carry the same sign of electrostatic potentials. This is attributed to the role played by the short-range attractive asymmetric EDL interaction and the strong hydrodynamic and gravity forces, without any need for the so-called hydrophobic interaction force. In addition, it was also shown that the models derived for the impinging jet system can be used to evaluate transport of fine gas bubbles onto a large particle surface, suggesting that the information extracted from the impinging jet geometry can be applied to the analysis of flotation processes. © 1999 Academic Press Key Words: bubble transport; bubble–solid attachment; impinging jet; DLVO theory; asymmetric double-layer interaction.

INTRODUCTION

Study of gas bubble attachment onto solid surfaces (referred to as collectors) is of significant importance to fundamental interfacial science and practical processes. Interest in this subject stems from its relevance to air flotation used in many industrial and environmental separation processes such as mineral extraction and waste water treatment. A fundamental understanding of the underlying mechanisms governing bubblecollector interaction and attachment is essential to the process control and optimization of the aforementioned technological processes. 1

To whom all correspondence should be addressed.

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0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.

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use the impinging jet technique to study deposition phenomena related to flotation processes. They investigated the deposition of carboxylated latex particles on an air/water interface in the presence of calcium oleate to model deinking flotation. The surface of carboxylated latex particles resembles that of ink particles, whereas the air/water interface is representative of a bubble in water. Recently, Dabros et al. (15) successfully extended the impinging jet technique to study fine bubble (the average size usually ranging from 20 to 40 mm in diameter) attachment onto a solid surface. Their experiments have amply demonstrated that fine bubble attachment is critically influenced by the physicochemical and hydrodynamic parameters. From a physics point of view, the process of gas bubble attachment onto a solid surface in the impinging jet geometry can be analogous to bubble–particle interaction and attachment occurring in flotation processes. The flow pattern created around the stagnation point is very similar to those existing at the foremost part of a spherical collector (a solid particle) exposed to an induced uniform flow or setting due to gravity. In addition, the quantitative nature of the short-range interfacial forces obtained from the impinging jet experiments is pretty much the same as that which happened in the bubble– particle interaction and attachment. In contrast to well-documented studies of particle deposition onto different collectors (16, 17), such as spherical collectors, cylindrical collectors, and rotating disks, it seems that there is relatively little information available on bubble attachment in the literature. In particular, there is no systematic theoretical analysis of the influence of physicochemical and hydrodynamic parameters on this subject. Due to the large gravity number and Peclet number for gas bubbles, the bubble attachment is expected to be substantially different from colloidal particle deposition. Furthermore, despite the increasingly important applications of microbubble flotation (18), quantitatively little is known about the characteristics of fine bubble attachment, where the bubble diameter is in the range of micrometers. Therefore, in this work, an attempt is made to analyze fine bubble attachment onto a solid surface in the impinging jet geometry under varying physicochemical and hydrodynamic conditions. A previously developed mass transfer model for rigid spherical particles is extended to fine bubbles. The conditions for validity of such an extension will be discussed. In particular, our model of describing surface forces is restricted within the framework of DLVO theory, which means that only van der Waals (VDW) and electrostatic double-layer (EDL) interaction forces need to be taken into account. Here we ignore the hydrophobic interaction force between bubbles and solids that may play some roles in air flotation processes. We will demonstrate that in certain circumstances where no hydrophobic interaction forces are present, bubbles can still attach onto a solid surface even though there exists a repulsive VDW interaction force and both bubble and solid surfaces carry the same sign of electrostatic potentials. In

FIG. 1. A simple illustration of bubble attachment onto a collector in the stagnation flow. Due to radial symmetry, the fluid flow and bubble transfer can be specified by cylindrical coordinates (r, z). All basic geometric parameters are given in this figure.

fact, Dabros et al. (15) performed bubble attachment experiments and they observed bubble attachment occurring even to a hydrophilic surface within a certain range of physiochemical parameters. In this study, we attribute this fact to the role played by the so-called asymmetric electrical double-layer interaction. The asymmetric electrical double-layer interaction refers to the existence of a short-range (approximately the same order of magnitude as the Debye length) attractive interaction between two charged interfaces that have different surface potentials but are of the same sign. A physical interpretation of the presence of the short-range attraction was furnished by Overbeek (19). In addition, we are going to illustrate that the models derived for the impinging jet system can be used to estimate the transport of fine gas bubbles onto a large particle surface. TRANSPORT EQUATIONS IN THE IMPINGING JET

A schematic representation of an impinging jet geometry is illustrated in Fig. 1. Due to axisymmetry, the system can be described by incorporating cylindrical coordinates (r, z) shown in Fig. 1. Assuming zero radial components of colloidal forces acting on the bubble and neglecting radial diffusion of bubbles, the governing convective diffusion equation for the steady-state transport of spherical bubbles from stagnationpoint flow toward a solid collector takes the dimensionless form (20) ~1 1 H!Pe f 3 ~H!n# 5

d dH

H F f 1 ~H!

dn# 1 1 ~1 1 H! 2 Pe f 2 ~H!n# 2 F# z ~H!n# dH 2

GJ

,

[1]

71

FINE BUBBLE ATTACHMENT ONTO A SOLID SURFACE

where H 5 ( z 2 a p)/a p is the dimensionless gap width between a bubble and the collector (a p is the radius of the bubble), f 1 (H), f 2 (H), and f 3 (H) are the universal hydrodynamic correction functions to account for hydrodynamic interactions between the bubble and the collector surface (21, 22), n# 5 n/n ` is the dimensionless bubble concentration (n ` is the bubble bulk number concentration), Pe 5 2 a a p3 /D ` is the Peclet number, a represents the intensity of the stagnationpoint flow (7, 20). D ` is the bubble diffusion coefficient in the bulk phase, which can be estimated from the Stokes–Einstein equation D` 5

kT , 6 pm a p

[2]

where k is the Boltzmann constant, T is the absolute temperature, and m is the dynamic fluid viscosity. F# z 5 F z a p/kT in equation [1] denotes the dimensionless normal component of the forces acting on a bubble. In this study, the bubble– collector interaction is treated essentially in terms of DLVO theory. Therefore, the forces are assumed to consist of gravitational force F# GR and DLVO forces: i.e., London–van der Waals force F# VDW and electric double-layer force F# EDL. F# z 5 F# GR 1 F# VDW 1 F# EDL

[3]

The dimensionless gravitational force F# GR is expressed by F GR 5 2Gr 5

2 D r ga 3p , 9 mD`

[4]

where Dr 5 r p 2 r l is the difference between the bubble, r p, and the liquid, r l, density, respectively. In Eq. [4], the negative sign denotes that in the impinging jet setup, the gravitational force acts in opposition to bubble flowing jet as shown in Fig. 1. The dimensionless London–van der Waals dispersion force between a spherical bubble and a flat plate is approximated as (23) F VDW 5 2Ad

l# ~ l# 1 22.232H! , H 2 ~ l# 1 11.116H! 2

[5]

where Ad, referred to as the adhesion number, is defined as Ad 5

A 123 6kT

and A 123 is the Hamaker constant for interaction between bubble phase 1 and collector phase 3 in liquid medium 2, l# 5

l /a p is a dimensionless parameter accounting for VDW retardation effects, l is London retardation wavelength (usually having the order of 10 27 m). To quantitatively describe the electrostatic double-layer force between a bubble and a collector, the approximate HHF (following the names of Hogg, Healy, and Fuerstenau) expression (24), assuming constant potential for EDL interactions, is chosen, F# EDL 5 Dlt

F

G

exp~2t H! exp~22 t H! , 2 Da 1 1 exp~2t H! 1 2 exp~22 t H!

[6]

where Dl is the double layer parameter defined as Dl 5

e 0e ra pzczp , kT

t 5 k a p is the ratio of the particle radius to the Debye length k 21 or is called the ionic strength and Da is the double-layer asymmetry parameter given by Da 5

~ z c 2 z p! 2 . 2 z czp

The double-layer asymmetry parameter characterizes the electrostatic interaction due to the difference in z-potential values between two surfaces. In the derivation of Eq. [6] (24), linear approximation was used to solve the Poisson–Boltzmann equation, and the Derjaguin model was implemented to calculate interaction forces so that the use of Eq. [6] is restricted to low values of the zeta potentials and a large value of t. Numerical solutions of the nonlinear Poisson–Boltzmann equation show that this HHF approximation is quite accurate even when z p and z c are as high as 60 mV and t is as low as 5. A detailed discussion about the validity of the HHF expression was given by Overbeek (19), Chan and White (25), and Baouch et al. (26). The reason we are incorporating the constant potential approximation, that has a sound thermodynamics basis, in our calculations is solely due to its simplicity (17). From a physical point of view, the constant potential interaction is expected to be relevant to many real situations with large exchange current. It is worth noting that in the literature no general agreement exists on a quantitative description of the electrostatic doublelayer interaction in dynamic attachment processes. Specifically, as yet there is not enough information regarding whether the interacting surfaces maintain a constant potential or a constant charge density during the attachment process. Extensive discussion on this issue can be found elsewhere (16, 27). As suggested by Adamczyk and Warszynski (28), an alternative to the HHF expression is to use either the linear superposition method or the complete numerical solution of the nonlinear Poisson–Boltzmann equation.

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In order to solve the bubble transport Eq. [1], one should also specify the boundary conditions at the collector and in the bulk liquid. The appropriate boundary conditions are given as n# 5 1 n# 5 0

as H 3 `

d at H o 5 d# 5 ap

[7a]

(ii) The hydrodynamic boundary layer should be thicker than the bubble diffusion boundary layer. It can be shown (see Appendix) that for the axisymmetric stagnation-point flow, the laminar hydrodynamic boundary thickness d n is approximately given by

d n 5 4.8R Re 20.5, [7b]

The second condition, refered to as “perfect sink” approximation, is widely used to model particle deposition (16). It states that all particles are irreversibly captured by the collector surface once they have reached a certain separation distance d# . Further discussions concerning the validity of “perfect sink” and other types of boundary conditions can be found in Ref. (16). It should be pointed out that analogous to particle deposition, in this work bubble attachment is considered to occur when a bubble can arrive at such an interception distance d# . In order words, this means that we are dealing with bubble transport under the influences of buoyancy and hydrodynamic forces, as well as colloidal surface forces, to a separation distance larger than d# , beyond which the process involved many complex phenomena (some of them are even difficult to define). In principle, all of the theories mentioned above were originally developed for describing deposition of colloidal particles onto a solid surface (7, 20). To extend these theories to fine bubbles, the following additional conditions need to be satisfied: (i) The bubble has to retain a spherical shape. The bubble shape may be deformed due to either the fluid inertia or the viscous force. When the viscous force is dominant, the deformation of the bubble shape can be best characterized by the capillary number, Ca 5 m U p/g, measuring the ratio of the viscous force to the interfacial tension force (here U p is the relative bubble–fluid velocity, and g is the interfacial tension which is taken as 72.7 mJ/m 2 for completely mobile air–water interface at 20°C). In the impinging jet system, the appropriate relative bubble–fluid velocity is of the order of a a p2. Taking the upper bound of the laminar flow intensity a to be of the order of 1.0 3 10 7 s 21 m 21 (9), one can readily calculate the capillary number to be of the order of Ca 5 0.01 for a bubble of radius a p 5 50 mm. On the other hand, as the inertia effect is a major concern, the bubble deformation is controlled by the Weber number, Wb 5 r U p2a p/g, representing the ratio of the inertia force to the interfacial tension force. In a similar manner, the Weber number can be estimated to be of the order Wb 5 1.0 3 10 24. Therefore, one can conclude that both viscous and inertia forces are small compared to interfacial forces, and therefore the bubbles should remain nearly spherical and undeformed in the present study conditions.

[8]

where R is the radius of the capillary and Re is the Reynolds number defined as Re 5 r V ` R/ m (here V ` is the mean fluid velocity in the capillary). Depending on the Reynolds number, the hydrodynamic boundary-layer thickness usually is about a few hundred micrometers. For example, when the Reynolds number varies from 100 to 1000, the corresponding hydrodynamic boundary thickness falls in the range 640 to 200 mm. On the other hand, it is difficult to quantitatively estimate the thickness of the bubble diffusion boundary layer, since the particle concentration profile is quite complex in the vicinity of the wall due to hydrodynamic and colloidal interactions (20). Simple geometrical interpretation of the diffusion boundary layer does not hold anymore. Nevertheless, as suggested in Ref. (17), for micron-sized particles, the diffusion boundarylayer thickness is of the order of 1 to 10 mm for moderate flow rates, which is very thin compared to the hydrodynamic boundary-layer thickness. Specifically, for the case considered in this work, because of the large value of the Schmidt number for gas bubbles, the bubble diffusion boundary layer should be even thinner. Thus, the condition that the hydrodynamic boundary layer should be thicker than the bubble diffusion boundary layer is fulfilled. (iii) The Stokes law can be applied to the bubble motion. Actually, there are twofold implications which can be drawn from this condition: First, the Reynolds number based on the gas bubble radius should be small, i.e., Re p 5 r U pa p/ m , 1. Following the same approach used to estimate the capillary number and the Weber number, the typical bubble Reynolds number can be calculated as of the order of Re p 5 0.1 (here the bubble radius is chosen as 50 mm). Second, the nonslip boundary condition must be satisfied at the bubble surface. The condition opens the old “Pandora Box” regarding bubble motion in liquids, namely, whether the Rybczynski–Hadamard formula or the Stokes drag formula is confirmed by experiments (29). This subject has attracted the attention of scientists for decades (30). An excellent introduction was recently published (31). Nevertheless, based on all evidence provided in the references (29 –32), there is no doubt that small bubbles below 100 mm in diameter created in the impinging jet system can certainly be treated as rigid spheres, i.e., objects with nonslip boundary conditions. (iv) The separation distance d# chosen should be physically meaningful, and should be shorter than the critical rupture thickness of a thin liquid film as well. This condition is associated with the “perfect sink” boundary condition, Eq.

73

FINE BUBBLE ATTACHMENT ONTO A SOLID SURFACE

[7b]. Physically, the boundary condition means that once a bubble has arrived at such a separation distance, the bubble will not be free to return to the solution anymore. In other words, the bubble is irreversibly captured by the collector surface and permanent attachment is achieved. Therefore, to make the “perfect sink” boundary condition valid, the separation distance H o 5 d# shown in Eq. [7b] usually should be chosen around the primary minimum energy where attractive interactions are present. Computation tests were performed to estimate the influence of the separation distance d# on the attachment results. It was found that the results are insensitive to the choice of d# when d# is less than 0.001. Therefore, in the numerical calculations, d# was chosen as 0.0005. For a bubble radius of 10 mm, this d# value corresponds to 5 nm, which falls into the proximity of the primary energy minimum well in most practical situations. On the other hand, the bubble attachment process is related to drainage of the thin liquid film between the bubble and the collector surface. If the thin liquid film is unstable due to attractive interaction forces or both buoyancy and hydrodynamic forces exerted on the bubble by the fluid flow, it will keep thinning until the film thickness reaches the so-called critical rupture thickness. Once the film breaks, a three-phase contact line is created. When radial hydrodynamic forces are not very strong, tangential forces caused by contact angle hysteresis, surface roughness, and interfacial tension between gas bubble and liquid will render bubbles immobile. In the literature, the value of the critical rupture thickness is found in the range 30 –200 nm depending on the surface characteristics of the solid surface and the aqueous solution. Undoubtedly such reported critical rupture thickness is much larger than the chosen separation distance d# in our calculations. RESULTS AND DISCUSSION

One is unlikely to find analytical solutions to the bubble transport Eq. [1] due to its mathematical complexity. In this regard, the numerical approach has to be employed. The detailed procedure for numerically solving the equation was described in Ref. (33). In addition to complementing numerical solutions, the limiting analytical solution of the bubble transport equation can provide valuable insight into the bubble attachment process and can be used as a criterion for examining the numerical solution as well. Since both the Peclet number Pe and gravity number Gr are very large for gas bubbles, the bubble attachment rate (flux) is basically determined by the bubble trajectories, which are deterministic. In fact, Spielman and Fitzpatrick (34) developed a model to analyze the trajectories of non-Brownian particles near a rotating disc. They gave the following approximate formula by assuming large Peclet number, i.e., Pe @ 1, and neglecting the diffusion term ( f 1 dn# /dH ' 0) in the mass transfer equation [1].

Sh 5

HEF

1 Pe exp 2 2

2

`

d#

1 11H

GJ

~1 1 H! f 3 ~H! dH ~1 1 H! 2 f 1 ~H! f 2 ~H! 2 2f 1 ~H!F# z ~H!/Pe

[9]

If one is to further assume that there are no hydrodynamic and colloidal surface interactions, namely, f 1 (H) 5 f 2 (H) 5 f 3 (H) 5 1, F# VDW 5 F# EDL 5 0, and F# z 5 2Gr, one can show that Eq. [9] can be simplified to Sh 5

1 ~1 1 d# ! 2 1 2 Gr/Pe . Pe 2 ~1 1 d# ! 2

[10]

Noting that d# is very small, i.e., d# ! 1, then Eq. [10] becomes Sh 5

1 Pe 1 Gr, 2

[11]

which indicates that if the Peclet number is large and the colloidal interactions are weak or negligible, the bubble mass transfer is completely controlled by the hydrodynamic convection and gravity forces. Numerical calculations for the convective diffusion mass transfer Eq. [1] with boundary conditions [7a, 7b] were performed to examine the effects of physicochemical parameters on bubble attachment. As a unique feature of gas bubble attachment, the adhesion number Ad was assumed to be negative, reflecting the fact that the van der Waals (VDW) interaction is repulsive. In the literature, it is well established that repulsive VDW interaction exists between gas bubbles and solid surfaces through aqueous medium (35, 36). The mechanisms of negative Hamaker constant and the conditions for repulsive VDW interaction were discussed extensively by Visser (37) and van Oss (38). In addition, when EDL interaction parameters were formulated, specific attention was given to the so-called asymmetric EDL interaction, described by Da in Eq. [6]. Such asymmetric EDL interaction was mostly ignored in previous theoretical treatments of particle deposition studies. Physically, the asymmetric EDL interaction implies the existence of a short-range attractive interaction between two charged interfaces with different surface potentials even though they carry the same sign of electrostatic potential. This short-range attractive interaction can be predicted from the HHF formula (24) and the more “rigorous” complete solution to the Poisson–Boltzmann equation as well (25, 39). Recently, based on a more sophisticated model, the attractive interaction between similarly charged colloidal particles was theoretically examined by Chu and Wasan (40). Experimentally, Dabros et al. (15) observed that the bubbles attach with a small but fine

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YANG ET AL.

to 20.1). This suggests that the magnitude of the adhesion number does not significantly affect the bubble attachment rate. This is desirable as an accurate determination of the adhesion number (or the Hamaker constant) is always difficult in the modeling bubble attachment. Effect of Hydrodynamic Condition

FIG. 2. Calculated Sherwood number Sh versus Peclet number Pe for different adhesion numbers Ad with gravity number Gr 5 10 5. The limiting solution of geometric interception is given by the dotted line.

rate even onto a hydrophilic solid surface. Clearly, in their experiment no hydrophobic interaction was present between the gas bubbles and the hydrophilic solid surface. If the VDW interaction is repulsive, one may conclude that the electrostatic double-layer interaction should be attractive. Therefore, the scenario in which gas bubbles attach onto a hydrophilic solid surface may be explained using the asymmetric EDL attractive interaction argument.

Figure 3 illustrates the dependence of the normalized bubble flux j o/n ` on the bubble radius a p for different hydrodynamic conditions (characterized by the Reynolds number Re). Again the repulsive VDW interaction was considered by choosing the negative value for the adhesion number (Ad 5 20.1) or the Hamaker constant ( A 5 22.5 3 10 221 J) in the calculation. By definition, both the Peclet number Pe and the gravity number Gr, representing hydrodynamic convection and buoyancy forces, respectively, are proportional to a p4. Therefore, as bubble size becomes larger, higher bubble attachment flux was observed, suggesting significant positive contributions of hydrodynamic convection and buoyancy forces to the bubble attachment in impinging jet flow. It also can be seen that for low Reynolds number Re, due to the presence of an energy barrier (defined as the maximum of the bubble– collector interaction energy), there exists a minimum critical bubble size for attachment. Above this size, the hydrodynamic convection and buoyancy forces are strong enough to such a degree that bubbles can overcome the energy barrier and achieve attachment. However, such a minimum critical bubble size does not exist for high Reynolds number, for instance Re 5 1000,

Effect of van der Waals Interaction In Fig. 2, the calculated Sherwood number Sh is plotted as a function of the Peclet number Pe for various adhesion numbers Ad. Here the dotted line arising from Eq. [11], the limiting solution of geometric interception, was drawn as a reference. In the calculation, the EDL interaction parameters and the gravity number were chosen as Dl 5 1000, Da 5 2, t 5 1000, and Gr 5 10 5. These typical values correspond to the bubble (being 10 mm in radius) and the collector surface with a surface potential of about 10 mV and 60 mV in a 1:1 ion-type electrolyte having ionic concentration of 10 23 M. It should be noted that, for a small Peclet number Pe (Pe , 10 2), the Sherwood number Sh approaches the gravity number Gr, as indicated by Eq. [11]. This reveals that for Pe , 10 2, bubble transport is dominated by gravity buoyancy. Moreover, it can be observed that the Sherwood number Sh for different adhesion numbers Ad falls in a narrow range (the maximum change of Sh is only 5%) irrespective of the two orders of magnitude difference in the adhesion number (Ad changes from 20.001

FIG. 3. The normalized particle deposition flux #j o 5 j o/n ` (cm/s) versus bubble radius a p for different Reynolds numbers with the EDL interaction. All the curves are plotted for the repulsive VDW interaction with Ad 5 20.1.

FINE BUBBLE ATTACHMENT ONTO A SOLID SURFACE

75

indicating that under such circumstances the attachment process is overwhelmingly dominated by hydrodynamic convection (at least for bubble sizes larger than 5 mm). By closer examination of Fig. 3, it is also interesting to note that the dependence of j o/n ` on Re for bubbles is totally different from that of particles reported in literature (41). For instance, the value of j o/n ` corresponding to Re 5 10 is even higher than that of Re 5 100. Such unusual phenomena are attributed to the coupling effects of high buoyancy force and the radial component of the stagnation flow. A detailed discussion of this was recently given by Yang et al. (20). Effect of Gravitational Force Figure 4a presents the dependence of Sherwood number as a function of the Peclet number for various gravity numbers at fixed EDL parameters, Dl, Da, and t. A strong impact of Gr number on flux is observed. However, the influence of Gr on Sh in the presence of energy barriers is significantly different from that in the absence of energy barriers reported by Prieve and Ruckenstein (42) and Yang et al. (20). For small values of Gr (e.g., Gr 5 10 4), attachment rate is reduced until hydrodynamic convection (characterized by Pe) becomes strong enough to push bubbles to overcome the energy barrier. While in the case of large values of Gr (e.g., Gr 5 10 6), it is shown that the Sherwood number Sh is almost independent of the Peclet number Pe, indicating that the bubble attachment process under such conditions is completely controlled by gravity. The results shown in Fig. 4a can be better illustrated by presenting the corresponding bubble– collector interaction force profile, as seen in Fig. 4b. It should be mentioned that, in addition to VDW and EDL force interactions, the interaction force constructed here includes the gravitational force interaction. This is because, based on the bubble transport Eq. [1], the gravitational force interaction plays exactly the same role as both VDW and EDL force interactions do. In fact, Fig. 4b, clearly shows that Gr has tremendous effects on the shape of the interaction force profile and hence the bubble attachment rates. For instance, under strong gravitational force (e.g., Gr 5 10 6), the interaction force is always negative (attractive) and no energy barrier is present so that significant bubble attachment can be observed. Effect of Electrostatic Double-Layer Interaction The effects of EDL interaction on bubble attachment are plotted in Figs. 5a and 6a. Their corresponding interaction force profiles are shown in Figs. 5b and 6b, respectively. Basically, the impact of Dl and t on bubble attachment follows the same trend as that reported for particle deposition (20, 43, 44). In Fig. 5a, the influence of EDL parameter Dl on Sh is presented as a function of Pe for Gr 5 10 5. It should be noted that a larger Dl usually yields a lower attachment rate. This is

FIG. 4. (a) Calculated Sherwood number Sh versus Peclet number Pe for different gravity numbers Gr under combined effects of the repulsive VDW interaction (Ad 5 20.1) and the EDL interaction (Dl 5 1000, t 5 1000, and Da 5 2). (b)Dimensionless bubble– collector interaction force profiles corresponding to Fig. 4a. The dotted line shows the interaction force without the asymmetric EDL interaction (Da 5 0).

due to the presence of a higher energy barrier (shown in Fig. 5b), revealing that in such circumstances, the resistance arising from the energy barrier dominates the bubble attachment process. For instance, the energy barrier for the curve of Dl 5 3000 is much higher than that for the curve of Dl 5 1000.

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EDL parameter Dl (shown in Fig. 5a), suggesting that the influence of EDL interaction is significant to bubble attachment. For a fixed particle size, a larger value of t usually implies a thinner EDL thickness and hence a weaker EDL interaction. Therefore, a higher attachment rate can be achieved.

FIG. 5. (a) Calculated Sherwood number Sh versus Peclet number Pe for different EDL parameters Dl under combined effects of the repulsive VDW interaction (Ad 5 20.1) and the EDL interaction (t 5 1000 and Da 5 2). (b) Dimensionless bubble– collector interaction force profiles corresponding to Fig. 5a. The dotted line shows the interaction force without the asymmetric EDL interaction (Da 5 0).

Accordingly, there is very low bubble attachment for Dl 5 3000 unless strong hydrodynamic convection takes place. Figure 6a displays the influence of EDL ionic strength t on the Sherwood number Sh plotted against the Peclet number Pe. It is interesting to note that in Fig. 6a dependence of the shape of Sh on the EDL ionic strength t is very similar to that on the

FIG. 6. (a) Calculated Sherwood number Sh versus Peclet number Pe for different EDL ionic strength t under combined effects of the repulsive VDW interaction (Ad 5 20.1) and the EDL interaction (Dl 5 1500 and Da 5 2). (b) Dimensionless bubble– collector interaction force profiles corresponding to Fig. 6a. The dotted line shows the interaction force without the asymmetric EDL interaction (Da 5 0).

77

FINE BUBBLE ATTACHMENT ONTO A SOLID SURFACE

In addition, it should be pointed out that in Figs. 4b, 5b, and 6b, we also plotted the interaction force profile without consideration of the asymmetric EDL interaction, i.e., Da 5 0, which is shown as the dotted line. It clearly shows that in the absence of the asymmetric EDL interaction, the interaction force profile forms so high an energy barrier that it is difficult for bubbles to overcome it to complete attachment. Nevertheless, evidence strongly demonstrates that with repulsive VDW interaction, it is the asymmetric EDL interaction that yields short-range attraction and makes it possible for bubbles to achieve attachment. This suggests that, despite the repulsive van der Waals interaction and the fact that both the bubble and the collector carry the same sign of electrostatic potential, there exists a realistic window within which bubble attachment can occur without need of the so-called hydrophobic forces interaction. Application of the Bubble Transport Equation to Fine Gas Bubble Attachment onto a Solid Particle So far we have performed a generalized analysis of the effects of van der Waals interaction, hydrodynamic condition, gravity force, and electrostatic double-layer interaction on bubble attachment. In the following we will show that these results are very meaningful for practical applications. Consider in an aqueous solution a falling particle with radius of R o 5 1 mm and fine gas bubbles rising from below. From a hydrodynamics point of view, the flow pattern (created by the falling particle) near the foremost part of the sphere is equivalent to that existing in the impinging jet geometry. Therefore, the bubble transport equation [1] can be used here to estimate whether the fine bubble would attach onto the falling particle or not. When the particle reaches the terminal velocity U o, the gravity force is precisely balanced by the viscous drag on the particle. Then the friction factor f is expressed as (45) f5

8 gR o r s 2 r l , 3 U 2o rl

[12]

where r s is the density of the particle and is chosen as r s 5 2200 Kg/m 3 . For the considered particle size, the drag does not follow the Stokes law. Alternatively, as suggested by Bird et al. (45), the friction factor f is approximated by f>

12.2 . Re 3/5 s

[13]

Combining Eqs. [12] and [13], one can estimate the particle terminal velocity as U o ' 0.27 m/s, leading to the particle Reynolds number of Re s 5 270. In such a case, an analytical solution to the disturbance flow induced by the falling particle

does not exist, since neither the creeping flow conditions nor the potential flow conditions hold. However, as mentioned above, we are only interested in the region near the stagnation point where the flow pattern of the uniform flow is very similar to that created by the impinging jet flow. Following Levich (29), at a small distance z from the surface of the particle, the stream function c is simplified as

c>

3 U z 2 sin 2 u . 4 o

[14]

The velocity components are defined as un 5 2 ut 5

1 ­c R 2 sin u ­ u

­c 1 , R sin u ­R

[15a]

[15b]

where R 5 R o 1 z. Noting that z ! R o, we may further assume that R ' R o, sin u ' r/R o, and cos u ' 1 (here we are using the cylindrical coordinates defined earlier). Substituting Eq. [14] into equations [15a, 15b] yields the velocity distributions in the stagnation region, un 5 2 ut 5

z2 3 Uo 2 2 Ro

3 zr Uo 2 . 2 Ro

[16a]

[16b]

By comparing the velocity distributions given by Eqs. [16a, 16b] with those defined for the impinging jet flow (7, 20), we are able to obtain the intensity of the stagnation point flow as

a5

3 Uo , 2 R 2o

[17]

and for the given conditions a is about 4.0 3 10 5 1/ms. Once we have the stagnation flow intensity a, we can now calculate both the bubble Peclet number Pe and the gravity number Gr defined earlier. Two bubble sizes (a p 5 10 mm and a p 5 20 mm) are chosen in the calculations. For the sake of consistency, we consider the same parameters for van der Waals and electrostatic double-layer interactions as used before, and they are Ad 5 20.1, Dl 5 1000, Da 5 2, and t 5 1000. Putting these parameters back into the bubble transport Eq. [1], we can approximately estimate the mass transfer rate of fine bubble attachment onto the falling particle. The results are listed in the Table 1. From the values given in the table, one can observe that the bubble of a p 5 10 mm will attach onto the

78

YANG ET AL.

TABLE 1 Results of Mass Transfer Rate of Fine Gas Bubble Attachment onto a Large Solid Particle by Using the Bubble Transport Equation [1] Ro (mm)

ap (mm)

Ad

D1

t

Da

Pe

Gr

Sh

1.0 1.0 1.0 1.0

10 10 20 20

20.1 20.1 20.1 20.1

1000 1000 1000 1000

1000 1000 1000 1000

0 2 0 2

3.7 3 10 4 3.7 3 10 4 6.0 3 10 5 6.0 3 10 5

1.0 3 10 5 1.0 3 10 5 1.6 3 10 6 1.6 3 10 6

0 3.5 3 10 4 5.7 3 10 5 5.9 3 10 5

particle if the asymmetric double-layer interaction is considered (Da 5 2). Otherwise, the bubble attachment will not happen; i.e., Sh ' 0 when Da 5 0. This indicates that there is a significant impact of short-range attraction due to the asymmetric double-layer interaction on bubble attachment. It also should be noted that as the bubble radius increases to 20 mm, both Pe and Gr are increased by one order of magnitude. The driving forces generated by the large Pe and Gr are sufficiently strong to overcome the energy barrel formed from repulsive van der Waals and electrostatic double-layer interactions. Accordingly, our model predicts that bubbles of a p 5 20 mm can always attach onto the particle surface. These results clearly demonstrate that under certain circumstances it is not necessary to include additional attraction forces, e.g., hydrophobic interaction force, to explain bubble–solid attachment.

APPENDIX

In the following, we will derive Eq. [8], which is used to approximately estimate the hydrodynamic boundary layer thickness in the impinging jet flow. To begin, we assume that the bubble concentration is very dilute so that the Navier– Stokes equations can be applied to describe the undisturbed fluid flow field. According to Schlichting (46), the governing equations and boundary conditions, based on boundary-layer approximation, are expressed as continuity equation 1 ­ ­vz ~rvr ! 1 5 0, r ­r ­z momentum balance in the boundary layer leading to

CONCLUSION

1. The particle transport equation developed for the impinging jet stagnation flow has been extended to analyze fine bubble attachment onto a solid surface within the framework of the classical DLVO theory. 2. The results showed that due to the large values of Pe and Gr for gas bubbles, the behavior of bubble attachment in many aspects is significantly different from that of colloidal particle deposition. 3. More importantly, we have demonstrated that without hydrophobic interaction, bubble attachment is feasible even though there exists a repulsive VDW interaction force and both bubble and solid surfaces carry the same sign of electrostatic potential. The results presented in this study are useful for interpretation of the experimentally observed phenomenon that fine bubbles can attach onto a hydrophilic solid surface. 4. As an illustration, we considered a practical case of fine bubble transport onto a falling particle. It was found that due to similarity in hydrodynamic conditions, the models derived for the impinging jet flow could be used to calculate the mass transfer rate of fine bubble attachment onto a falling particle. This suggests that the information derived from the impinging jet geometry can be used to understand flotation processes.

[A-1]

vr

­vr ­vr ­ 2 vr 1 ­P 1 vz 52 1n , ­r ­z r ­r ­z2

[A-2]

and boundary conditions z50

vr 5 0

z 3 `

vz 5 0 vr 5 U

[A-3a] [A-3b]

Assume that the velocity variation of the potential flow outside the boundary-layer region is U 5 br

[A-4]

where b is a constant representing the flow intensity. According to the Bernoulli equation, the hydrodynamic pressure is expressed by 2

1 ­p dU 5U 5 b 2 r 5 b U. r ­r dr

[A-5]

By introducing new parameters, h 5 =b / n z and f( h ) 5

FINE BUBBLE ATTACHMENT ONTO A SOLID SURFACE

=1/ bn c /r 2 (c being the stream function of the viscous boundary-layer flow) and doing a similarity transformation, the boundary-layer equation [A-2] can be simplified to an ordinary differential equation

S D

d 3f d 2f df 1 2f 3 2 2 dh dh dh

2

1 1 5 0,

[A-6]

which is subject to the following boundary conditions:

h50

df 50 dh

f50

h 3 `

df 5 1. dh

[A-7b]

Î

n . b

[A-8]

ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the Andrew Stewart Memorial Graduate Scholarship of University of Alberta, to Mr. C. Yang, and the Natural Sciences and Engineering Research Council of Canada and Syncrude Canada Ltd., to Professor J. Masliyah.

REFERENCES 1. 2. 3. 4. 5.

To determine the flow intensity b, one should recall that the potential flow pattern outside the viscous boundary-layer region is given by Eq. [A-4], from which the stream function w of the potential flow can be determined,

w 5 b zr 2 .

[A-9]

6.

7. 8. 9.

Applying Eq. [A-9] to the exit of the capillary in the impinging jet system, one can write

10.

at r 1 5 0 and z 1 5 2 R

w1 5 0

[A-10a]

11. 12.

at r 2 5 R and z 2 5 2 R

w 2 5 2 b R 3.

[A-10b]

13.

Noting that by definition the stream function can be related to the volumetric flow rate, for the impinging jet system, it can been shown that

14. 15.

2 p ~ w 2 2 w 1 ! 5 Q n 5 pn R Re.

16.

[A-11]

Substituting Eqs. [A-10a] and [A-10b] into Eq. [A-11] yields

n Re b5 . 4R 2

[A-12]

17. 18. 19. 20. 21.

Finally, the hydrodynamic boundary-layer thickness can be obtained by putting Eq. [A-12] into Eq. [A-8],

[A-13]

It is interesting to note that for a fixed Re, the hydrodynamic boundary-layer thickness in the impinging jet geometry is constant. It also should be pointed out that the hydrodynamic boundary-layer thickness estimated from Eq. [A-13] is based on an assumption of hyperbolic flow distribution (46) outside the boundary layer (i.e., Eq. [A-4]), which may not be always true in the practical impinging jet system.

[A-7a]

One may obtain the solution to Eq. [A-6] in the form of a power series, from which the boundary-layer thickness is given by

d n 5 2.4

d v 5 4.8R Re 20.5.

79

22.

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