8024
Ind. Eng. Chem. Res. 2009, 48, 8024–8029
The Behavior of an Oscillating Particle Attached to a Gas-Liquid Surface Paul Stevenson,† Seher Ata, and Geoffrey M. Evans Centre for AdVanced Particle Processing, UniVersity of Newcastle, Callaghan, NSW 2308, Australia
A force balance is presented for a particle attached to the surface of a larger bubble that oscillates such as it would immediately after a coalescence event. Similar previous analyses have neglected the Basset history force, the added mass of the particles, and the drag force imparted by the fluid upon the particles. Methods of crudely estimating the upper-bound of the Basset history force and the drag are proposed, and it is shown that these, along with the d’Alembert force due to particle acceleration, govern the value of the compensating capillary force and therefore the position of the three-phase contact line. The weight, buoyancy, and hydrostatic force due to meniscus depression are insignificant. The position of the three-phase constant line demonstrates an approximately first order lag to a stimulus. Because of this, the true position of the three-phase contact line can be estimated, and this must require that the three-phase contact angle demonstrates oscillatory behavior. It is possible that this phase-lag may underpin the process of particle detachment. 1. Introduction When bubbles in a flotation froth coalesce, some of the particles previously attached to the bubble surfaces detach. In their model of the entire froth phase of flotation, Neethling and Cilliers1 assumed that all particles that had been attached to a ruptured film detach, whereas those residing upon intact films stay attached, although this was not experimentally tested. Moreover, it was assumed that once the particles had left the bubble they would remain unattached. Ata2 conducted an experimental study upon the behavior of hydrophobic particles attached to bubbles as they coalesced. Bubbles of air of diameter of approximately 2.1 mm were formed at the tip of each of two capillary tubes submersed in water. A film formed between the two bubbles which eventually ruptured causing the two bubbles to coalesce to form one larger bubble. One or both of the bubbles were covered with soda lime glass beads of arithmetic mean diameter of 66 µm and a 90th percentile diameter of 92 µm. The coalesced bubble oscillated in various modes, and some particles were observed to detach and be thrown into the liquid from the bubble. When only one bubble was covered in particles, the spheres were seen to rapidly redistribute evenly across the surface. In this paper we will present a force balance for particles attached to the surface of a coalescing bubble as a precursor to understanding detachment due to coalescence of bubbles in the froth phase in flotation. In addition, we will provide a simple explanation for the rapid redistribution of particles after the coalescence event. It should be noted that attached particles also experience oscillation due to Reynolds stresses imparted by turbulent eddies in flotation, and the comments made herein are generally applicable to this situation as well. Princen3 was possibly the first worker to attempt to quantify the forces on an attached particle. Kondrat’ev and Izotov4 wrote a force balance upon a particle attached to a bubble that oscillates due to exposure to turbulent eddies. They allowed for particle weight, buoyancy, and the capillary force and used these to develop an equation of motion for the particle. Nguyen5 proposed a similar analysis, but added the hydrostatic force due to meniscus depression. We assert that there are three important components that have not been included in Nguyen’s analysis: † To whom correspondence should be addressed. E-mail:
[email protected]. Tel: +61 (0) 2 4921 6192.
1. The Basset history force 2. The “added mass” of the particle 3. The drag imparted on the particle by the liquid We will further assert that the hydrostatic force due to meniscus depression, that was suggested by Nguyen5 is vanishingly small (for Ata’s experiments) compared to the force components that he neglected. In fact, Ralston et al.6 recognized that velocity dependent drag forces were required in the force balance. However, since the three-phase contact angle has a finite response time to an external stimulus, they held that contact angle dynamics had to be linked to an analysis that took into account the drag force, and this problem was considered intractable. In this paper we suggest methods of estimating, however crudely, the forces imparted upon an oscillating particle attached to bubble, and suggest a method of simulating the position of the three-phase contact line, assuming that the expansion of the contact line is instantaneous. The system that we study is highly stylized since an oscillation is imposed upon the particle, and the time varying forces imparted are explored; in flotation the particle oscillation is due to bubble coalescence and time varying Reynolds stresses caused by eddies. It is shown that the Basset history force is significant under certain conditions. Moreover, by recognizing that the position of the three-phase contact line responds to stimuli approximately as a first order lag, we can suggest a way of estimating the actual amplitude of oscillation of the position of the meniscus. The corollary is that the three-phase contact angle must vary over the cycle of oscillation. Finally, we offer an explanation for the rapid redistribution of particles over the bubble surface that was observed by Ata.2 2. Definition of the Theoretical Problem Consider a spherical particle of radius rp that is attached onto the underside of a bubble that has such a large radius that its undeformed surface appears planar to the particle (see Figure 1). The contact angle is θ and the angle between the half angle subtended by the particle surface at the center of the particle is R. The depth of meniscus depression is denoted by H. Initially the particle assumes its equilibrium position, but it is then excited vertically in simple harmonic motion of amplitude so that the peak deflection from the equilibrium position is a,
10.1021/ie900085d CCC: $40.75 2009 American Chemical Society Published on Web 04/29/2009
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
8025
where the minus sign is due to the force acting downward. Hydrostatic Force Due to Meniscus Depression. The curvature of the meniscus means that the three-phase contact line lies below the plane of the undeformed surface. As a consequence the disk that shares a perimeter with the three phase contact line experiences a pressure, relative to the undeformed plane, of FfgH, where H is the depth of the meniscus depression. This can be calculated numerically, but Nguyen and Schulze7 give a useful approximation which will be adopted in the present analysis: H ) 2L sin
( θ -2 R )(1 + r sinL R )
-1/2
(7)
p
Thus, the hydrostatic force due to meniscus depression is Figure 1. Schematic of a spherical particle attached to the underside of a large bubble, after Nguyen and Schulze.7
and the angular velocity of the excitation is ω. Thus the deflection from the equilibrium position is x ) a sin ωt (1) where the dimension x is measured positive upward), the velocity of the particle is x˙ ) aω cos ωt and the acceleration of the particle is
(3) x¨ ) -aω sin ωt By considering a force balance on the particle, we will calculate how R varies with time during the first cycle of simple harmonic oscillation of the particle. It is assumed that deformations of the meniscus are instantaneous and that there is a unique three phase contact angle, θ; in fact, there exists hysteresis between advancing and receding contact angles, so the instantaneous contact angle will, in fact, vary between these two values. It will be demonstrated later that the lag between the position of the meniscus and the stimuli thereof also changes the contact angle. The assumption of instantaneous response will be relaxed later in this paper. 3. Forces Capillary Force. The tension of the gas liquid surface, σ, exerts a force at the three-phase contact line. Because the system exhibits rotation symmetry about a vertical axis passing through the center of the particle, this force exhibits only a vertical component. The length of the three phase contact line is 2πrp sin R and the tension acts at an angle of θ - R to the horizontal plane. Thus the vertical component of the capillary force (measured positive upward) is (4)
Buoyancy. The buoyancy is given by the product of the density of the liquid, acceleration due to gravity, and the volume of liquid displaced by the submerged section of the particle, that is, πrp3 (2 + 3 cos R - cos3 R) 3 Particle Weight. The particle weight is F b ) Ff g
4πrp3 Fw ) -Fp g 3
(5)
-1/2
(8)
p
where L is the capillary length given by L)
σ Ff g
(9)
Therefore, by writing the force balance, F c + Fb + F w + Fh ) 0
(2)
2
Fc ) 2πσrp sin R sin(θ - R)
( θ -2 R )(1 + r sinL R )
Fh ) 2rp sin RFf gL sin
(10)
the value of Re may be calculated, given relevant system properties, for (undisturbed) static equilibrium. However, because our stylized system experiences simple harmonic motion, we must also consider the drag force imparted by the fluid upon the particle, the Basset history force, and a d’Alembert force due to particle acceleration so that one can rewrite a quasistatic force balance. Drag Force. Even though the particle may be oscillating at high frequency, we assume that the particle is sufficiently small for creeping flow assumptions to be made when estimating the drag force imparted by the fluid upon the particle. To exactly calculate this drag, one would require knowledge of the flowfield around the particle that takes into account the presence of the gas-liquid surface, but such a calculation is beyond the scope of the current work. Instead, we obtain an estimate for the drag force by integrating the shear and normal stresses predicted by Stokes8 analysis for the drag on an isolated sphere over only the wetted surface of the particle at low Reynolds number (thus we assume creeping flow). Stokes gave the deviation pressure acting on a surface of a sphere moving at velocity x˙ in a quiescent liquid as p′ ) -
3µx˙ cos β 2rp
(11)
where β is the angle subtended at the particle surface by the arc connecting a particular position on the surface and the stagnation point of the flow. By integrating the streamwise components of the pressure, the drag force due to the deviation pressure for the attached particle is calculated as -
∫
π-R
0
p'2πrp2 sin β cos β dβ ) πµx˙rp(1 + cos3 R) (12)
(6)
where the negative sign recognizes that the drag force is in the opposite direction to the velocity of the particle. Applying the same procedure for the longitudinal shear stress, τ:
8026
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
Fa ) -x¨
πrp3 [8Fp + Ff (2 + 3cos R - cos3 R)] 6
(19)
4. Simulation of Position of the Three-Phase Contact Line By including the drag force, Basset history force, and the d’Alembert force, one can write the quasi-static force balance: Fc + Fb + Fw + Fh + Fbas + Fa ) 0
Figure 2. Values of 1F2 with arguments 1, {1.25, 1.75}, -0.25(ωt)2 for the two cycles of oscillation as calculated by Mathematica; χ ) ωt/2π.
τ)-
3µx˙ sin β 2rp
(13)
so the drag due to the shear stress is -
∫
π-R
0
( R2 )
τ2πrp2 sin2 β dβ ) 4πµx˙rp(2 - cos R) cos4
(14) The combined estimated drag force is
( R2 ))
(
FD ) -πµx˙rp 1 + cos3 R + 4(2 - cos R) cos4
(15) which, when R ) 0, reduces to FD ) -6πµx˙rp
(16)
which is Stokes’ solution, as expected. It should be noted that Stokes’ solution for the drag on a solid sphere is only valid for cases where the Reynold’s number is much lower than unity. It will be shown below that this is not the case for the experiments of Ata.2 Basset History Force. Account must be taken for the Basset9 history force which, for a sphere in a quiescent liquid (i.e., the convective derivative of the liquid is zero), is generally given by Fbas ) -6rp2√πFf µ
∫
t
x¨
-∞
√t - T
dT
(17)
where T is the dummy variable of integration. Making the gross assumptions that the particle is located entirely within the liquid and that the presence of the bubble surface has no influence upon the flowfield around the particle, by substituting for x¨ (eq 3), we calculate the Basset history force to be Fbas ) 14.2arp2ω3(Ff µ)1/2t3/2 1F2
(18)
where 1F2 is the hypergeometric function with arguments 1, {1.25, 1.75}, -0.25(ωt)2; this function is plotted over the first two cycles of oscillation in Figure 2; χ, the label of the abscissa, denotes the number of cycles of oscillation. d’Alembert’s Force. The absolute d’Alembert (inertial) force, Fa, is given by the effective mass (i.e., the particle mass plus added mass8) multiplied by the acceleration of the particle, and acts in the opposite direction to the acceleration. Since the added mass is given by one-half of the displaced mass of liquid, the d’Alembert force is estimated to be
(20)
Thus, by assuming that the three-phase contact line responds to stimuli instantaneously one can calculate the value of R and thus the position of the three-phase contact line. It is pertinent to estimate the forces that were exerted in the experiment of Ata’s2 in which one bubble covered in particles coalesced with a particle-free bubble. The bubbles exhibited a complex damped oscillatory behavior, but the dominant mode of oscillation occurred with a period of approximately 12 ms (i.e., ω ) 524 s-1). The peak amplitude of the oscillations was approximately half of the bubble radius (i.e., 503 µm). The viscosity and density of the water were 1 cP and 1000 kg · m-3 respectively; the forces upon a particle of arithmetic mean radius (i.e., 33 µm) are calculated herein. The density of the particles are 2500 kg · m-3. The contact angle was measured at between 27 and 30°; a value of 28.5° is assumed herein. The surface tension of pure water at STP, 0.072 N · m-1 is adopted. We will calculate the forces on a particle attached to the underside of a bubble although the force balance proposed herein has general applicability if appropriately modified. Figure 3 shows a plot over the first two cycles of oscillation of the drag, d’Alembert, and Basset history forces. The maximum absolute hydrostatic force due to meniscus depression over the cycle, Fh, is 1.8 × 10-5 µN, the maximum absolute buoyancy is 1.5 × 10-3 µN, and the particle weight is 3.7 × 10-3 µN. These are small compared to the capillary, Basset history, drag, and d’Alembert forces as shown in Figure 3. The corresponding angles, R, which give the position of the meniscus over the first two cycles, are shown in Figure 4; the peak-topeak amplitude is approximately 5°. In particular it should be noted that the Basset history force is significant. This force is often insignificant in fluid-particle systems. For example, in an analysis of particle motion on submerged oscillating plates10 the history force is insignificant compared to the drag and d’Alembert forces due to the relatively low angular frequency of the system. However, in this case the angular frequency is relatively high which means that the Basset history force must be considered. The value of the Basset history force over the first five cycles of oscillation is given in Figure 5. The 90th percentile particle radius in Ata’s experiments was 46 µm; the significant forces upon these larger particles under the same oscillation conditions are plotted in Figure 6. In addition, the corresponding values of R are plotted on Figure 4. The deviation of R is seen to be slightly greater than for the smaller particle, and the phase of oscillation of R is different. It is possible to estimate the validity of the use of Stokes’8 solution in the calculation of the drag. The maximum velocity of the particle throughout the oscillation cycle is given by aω. Thus, the maximum Reynolds number, based upon the liquid kinematic viscosity and a particle radius of 33 µm is approximately 8.7 which is beyond the range of applicability of Stokes’ solution. In fact, the correlation of Schiller and Naumann11 indicates that the drag coefficient should be approximately 1.6 times lower than that calculated using Stokes’
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
8027
Figure 3. Plots of Basset history force, d’Alembert’s force, drag force, and capillary force over two cycles of oscillation for particles of radius 33 µm.
will proceed by finding the response of the three-phase contact line to a Heaviside step function, and, by differentiating, obtain the impulse response. Making the assumption that the response is linear, we will assess the frequency response of the motion of the three-phase contact line to determine whether the effect is significant. Stechemesser and Nguyen12 presented experimental data and a model for the step response of the position of a three-phase contact line. The experimental data indicates that the three-phase contact line responds approximately as a first-order lag with time constant of between 2 and 10 ms, that is, the impulse response is g(t) ) exp(-t/tc) Figure 4. Position of the three phase contact line over two cycles of oscillation for particles of radius 33 µm (bold line) and radius 46 µm (fainter line).
where the time constant tc is between 2 and 10 ms. Their experimental results suggest that the response is approximately linear. By taking the Laplace transform of eq 21 we obtain the transfer function G(s) )
Figure 5. Basset history force over five cycles of oscillation for particles of radius 33 µm; χ ) ωt/2π.
solution. However, given the uncertainty in the estimate of the drag force, the Stokes’ solution is retained in the interest of simplicity. 5. Accounting for Finite Response Time of the Three-Phase Contact Line The position of the three-phase contact line does not respond to stimulus instantaneously, and, as mentioned above, Ralston et al.6 asserted that this represents a confounding issue to the modeling of the behavior of particles attached to an interface. We propose a method herein to approximately estimate the effects of the finite kinetics of the three-phase contact line. We
(21)
1 1 + t cs
(22)
For illustrative purposes we assume that tc takes the intermediate value of 5 ms and we plot the frequency response of eq 10 in the Bode plot of Figure 7. The corner frequency, ωc, is 200 s-1 which is somewhat lower than the frequency of 524 s-1 observed by Ata2 for the oscillation of bubbles after the coalescence event. Thus we see that the phase lag at 524 s-1 is -69° and more importantly, the gain is -4.5 dB. The implications of this are as follows: 1. The actual response of the three-phase contact line is 69° out of phase with the response if the contact line kinetics were instantaneous. 2. The amplitude of the response is 36% of the amplitude response if the contact line kinetics were instantaneous. Thus, the peak-to-peak amplitude of the variation in R across a cycle of oscillation is estimated to be 36% of 5° which is 1°48′. In addition, because the response is 69° out of phase with the stimulus, it is clear that the three-phase contact angle, θ, must vary throughout the cycle of oscillation in order to maintain quasi-static equilibrium given by eq 20. The selection of tc of 5 ms is, admittedly, arbitrary. Had a time constant of 2 ms been assumed the gain at 524 s-1 would have been -1.6 dB (i.e., 69%) and a phase lag of -46°, whereas a time constant of 10 ms would have yielded a -7.3 dB gain
8028
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
Figure 6. Plots of Basset history force, d’Alembert’s force, drag force, and capillary force over two cycles of oscillation for particles of radius 46 µm.
Figure 7. Bode plots for a first-order lag with a time constant of 5 ms. At the frequency of 524 s-1 the gain is -4.5 dB and the phase lag is -69°.
(i.e., 19%) and a phase lag of -79°. Further work is required to enable estimates of the time constant as a function of system properties. 6. Further Comments about Experimental Observations Redistribution of Particles after the Coalescence Event. Ata2 noticed that, when a bubble covered in particles coalesced with a demineralized bubble, the particles rapidly redistributed evenly over the surface of the bubble within several tens of milliseconds of the coalescence event. It is clear that significant forces are exerted upon the particles in the azimuthal direction. We speculate that these are due to spatial gradients in surface energy due to variations of particle concentration on the bubble surface; parts of the coalesced bubble that are particle free have a higher surface energy than those that are significantly mineralized. Thus, the spatial gradient of surface energy imparts a force upon the particles that act to rapidly redistribute them, in a similar fashion to the redistribution of surfactants at a gas-liquid surface due to Marangoni stresses.
Damped Harmonic Motion of the Bubbles. In the analysis, the amplitude and period of the first oscillation observed by Ata was assumed for all subsequent cycles. In fact the oscillations exhibit damped harmonic motion so the period of oscillation stays approximately constant, whereas the amplitude becomes progressively lower. The oscillation of the particlecovered bubbles attenuates more rapidly than the demineralized bubbles and the amplitude of oscillation lowers. We suppose that the reasons for these observations are as follow: 1. The surface energy of particle-covered bubbles is lower than that of demineralized ones so that, when surface area is lost due to two bubbles coalescing to become one, there is less energy to dissipate by the bubbles oscillating in the liquid, so the amplitude of oscillation is lower. 2. The surfaces of particle-covered bubbles are rigid and therefore maintain a no-slip boundary condition. Consequently, the surface of the bubble is able to support greater shear stress than a clear liquid-gas surface and therefore more able to dissipate energy, thereby increasing the rate of attenuation. Particle Detachment after the Coalescence Event. Ata observed that a number of particles were thrown off the coalescing bubbles. In the above analysis, particle detachment would occur if R approached zero, which it does not. Thus, we have not explained why detachment of some particles was observed, although the phase-lag of the position of the meniscus may underlie the physics of detachment. We also note that, as observed by Stover et al.,13 bubbles coalesce in a complex manner and exhibit multiple modes of oscillation which complicates the mechanical analysis. In addition, it is highly likely that the attached particles interact with each other in a manner not considered herein. Finally, we note that we have only obtained crude estimates of the drag and Basset history forces and it is not claimed we have done this with any significant accuracy. 7. Conclusions 1. A force balance has been presented for a particle attached to the meniscus of an oscillating liquid-gas surface. Crude estimates of the Basset history force and the drag force imparted upon the particle by the fluid are made. In addition, we include the added mass in the inertial d’Alembert force. 2. It is demonstrated that, for the conditions observed by Ata,2 that the Basset history force, drag force, d’Alembert force, and capillary force are large compared to the particle weight,
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
buoyancy, and hydrostatic force due to meniscus depression. The latter force is vanishingly small. 3. By solving for a quasi-static force balance and assuming that the meniscus responds instantaneously to stimuli, the position of the meniscus throughout the cycle of oscillation can be calculated. 4. When the particle size is greater, the oscillation of the meniscus on the particle surface increases, and the phase changes. 5. By noting that the response of the meniscus is not instantaneous and, in fact, approximately exhibits the characteristics of a first order lag, the actual movement of the meniscus upon the surface of the particle throughout the cycle of oscillation can be estimated. 6. The oscillation of the meniscus is attenuated due to the first order lag and is out of phase with the stimulus. The corollary of this observation is that the three-phase contact angle must necessarily change throughout the cycle of oscillation in order to maintain quasi-static equilibrium. We speculate that the phase lag of the meniscus position may underlie the physics of particle detachment. Nomenclature a ) amplitude of oscillation [m] Fa ) d’Alembert force [N] Fb ) buoyancy [N] Fbas ) Basset history force [N] Fc ) capillary force [N] FD ) drag force [N] Fh ) hydrostatic force due to meniscus depression [N] FW ) particle weight [N] g ) acceleration due to gravity [m · s-2] H ) meniscus depression [m] L ) capillary length defined in eq 9 [m] p′ ) deviation pressure [Pa] rp ) radius of a spherical particle [m] t ) time since oscillation commenced [s] tc ) time constant of a first-order lag [s] T ) dummy variable of integration in eq 17 [s] x ) vertical displacement of the particle (+ve upward) [m] Greek R ) half angle subtended by the three-phase contact line at particle center [ ]
8029
Re ) value of the above half angle at static equilibrium [ ] β ) longitudinal angle of a position on the particle surface [ ] θ ) three-phase contact angle [ ] µ ) liquid dynamic viscosity [Pa · s] Ff ) liquid density [kg · m-3] Fp ) particle density [kg · m-3] σ ) surface tension [N · m-1] τ ) shear stress [N · m-2] χ ) number of oscillation cycles [ ] ω ) angular frequency [s-1] ωc ) corner frequency [s-1]
Literature Cited (1) Neethling, S. J.; Cilliers, J. J. Modelling Flotation Froths. Int. J. Miner. Process. 2003, 72, 267–287. (2) Ata, S. Coalescence of Bubbles Covered by Particles. Langmuir 2008, 24, 6085–6091. (3) Princen, A. Surfactant and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1969; Vol. 2, Chapter 2. (4) Kondrat’ev, S. A.; Izotov, A. S. Influence of Bubble Oscillations on the Strength of Particle Adhesion, with an Accounting for the Physical and Chemical Conditions of Flotation. J. Min. Sci. 1988, 34, 459–465. (5) Nguyen, A. V. New Method and Equations for Determining Attachment Tenacity and Particle Size Limit in Flotation. Int. J. Miner. Process. 2003, 68, 167–182. (6) Ralston, J.; Fornasiero, D.; Hayes, R. Bubble-Particle Attachment and Detachment in Flotation. Int. J. Miner. Process. 1999, 56, 133–164. (7) Nguyen, A. V.; Schulze, H. J. Colloidal Science of Flotation; Marcel Dekker: New York, 2004. (8) Stokes, G. G. On the Effect of the Internal Friction of Fluids on the Motion of Pendulums, Camb. Philos. Trans. 1851, 9, 8–106. (9) Basset, A. B. A Treatise on Hydrodynamics; Dover: Mineola, NY, 1888; Vol. 2. (10) Stevenson, P.; Thorpe, R. B. Incipient Motion of a Bed of Particles Resting on a Submerged Oscillating Plate. Chem. Eng. Sci. 2004, 59, 1295– 1300. ¨ ber die Grundlegenden Berechnungen (11) Schiller, L.; Naumann, A. U bei der Schwerkraftaufbereitung. Zeit. Ver. Deutsch. Ing. 1935, 77, 318– 325. (12) Stechemesser, H.; Nguyen, A. V. Time of Gas-Solid-Liquid ThreePhase Contact Expansion in Flotation. Int. J. Miner. Process. 1999, 56, 117–132. (13) Stover, R. L.; Tobias, C. W.; Denn, M. M. Bubble Coalescence Dynamics. AIChE J. 1997, 43, 2385–2392.
ReceiVed for reView January 18, 2009 ReVised manuscript receiVed April 10, 2009 Accepted April 14, 2009 IE900085D
