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Effect of bubble contamination on rise velocity and mass transfer
Chemical Engineering Science 60 (2005) 1 – 9 www.elsevier.com/locate/ces
Effect of bubble contamination on rise velocity and mass transfer S.S. Alves∗ , S.P. Orvalho, J.M.T. Vasconcelos Department of Chemical Engineering, Instituto Superior Técnico, Centro de Eng. Biológica e Química, 1049-001 Lisboa, Portugal Received 28 January 2004; received in revised form 5 July 2004; accepted 6 July 2004 Available online 15 September 2004
Abstract An apparatus where individual bubbles are kept stationary in a downward liquid flow was adapted to simultaneously (i) follow mass transfer to/from a single bubble as it inevitably gets contaminated; (ii) follow its shape; and (iii) periodically measure its terminal velocity. This apparatus allows bubbles to be monitored for much longer periods of time than does the monitoring of rising bubbles. Thus, the effect of trace contaminants on bubbles of low solubility gases, like air, may be studied. Experiments were done with air bubbles of 1–5 mm initial equivalent diameter in a water stream. The partial pressure of air in the liquid could be manipulated, allowing bubbles to be either dissolving or kept at an approximately constant diameter. Both drag coefficient and gas–liquid mass transfer results were interpreted in terms of bubble contamination kinetics using a simplified stagnant cap model. Drag coefficient was calculated from stagnant cap size using an adaptation of Sadhal and Johnson’s model (J. Fluid Mech. 126 (1983) 237). Gas–liquid mass transfer modelling assumed two mass transfer coefficients, one for the clean front of the bubble, the other for the stagnant cap. Adjusted values of these coefficients are consistent with theoretical predictions from Higbie’s and Frössling’s equations, respectively. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Bubble; Drag coefficient; Hydrodynamics; Mass transfer; Stagnant cap; Surfactant
1. Introduction Gas–liquid systems, with the gas as the disperse phase, are as important as they are complex. An important contribution to the complexity of bubble interaction with the liquid environment is the effect of trace amounts of surfactant. Both bubble rise velocity and gas–liquid mass transfer are known to be slowed down by surface-active contaminants. While the two phenomena are certainly related, both being due to the accumulation of the surfactant at the bubble surface, they tend to be separately studied. The rise velocity of air bubbles in water strongly depends on water purity, especially in the spherical and ellipsoidal range (0.5–17 mm equivalent diameters). Values up to 0.36 m s−1 terminal velocities have been found in so-called
∗ Corresponding author. Tel.: +351-21-8417188; 8499242. E-mail address: [email protected] (S.S. Alves).
fax:
+351-21-
0009-2509/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.07.053
‘hyper-clean’ water for air bubbles of 1.6–1.8 mm diameter (Duineveld, 1995), while less than one-half of that is usually measured in tap water. The greatest cause for the spreading in terminal velocity data (see Clift et al., 1978) is attributed to surface contamination. Clean bubble interfaces are freely mobile. However, the presence of even small amounts of impurities leads to the adsorption and collection of surfactants at the bubble surface which hinders its mobility. Frumkin and Levich (1947) proposed this to be the result of the uneven distribution of surfactant due to surface advection by the main flow from the bubble front end to the rear stagnation point. The concentration gradient results in a tangential gradient of surface tension that causes a tangential stress, also known as Marangoni stress, to oppose the flow shear stress. Surface motion is thus retarded and the drag coefficient is increased towards that of a rigid sphere. This is the so-called stagnant cap hypothesis, which has been successful in explaining bubble rise velocity data, both from experiments where an average velocity is calculated by measuring the time taken by a rising bubble to travel a
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fixed distance and from more detailed studies involving the instantaneous monitoring of the bubble velocity (Takemura and Yabe, 1999; Zhang and Finch, 2001; Zhang et al., 2001), thus providing the bubble time-history. Most mathematical models proposed to simulate the increase in drag coefficient as a function of surface contamination assumes the stagnant cap hypothesis. The model of Sadhal and Johnson (1983) was developed for creeping flow past a spherical bubble with a stagnant cap created by surfactants of low solubility. With these assumptions a simple relationship between drag force and the stagnant cap angle could be analytically deduced. Bischof et al. (1991) used this relationship together with a simple model for bubble contamination to simulate the transient behaviour of a bubble rising in quiescent water. Relaxing of Sadhal and Johnson’s assumptions involves numerical methods. Cuenot et al. (1997) abandoned creeping flow and investigated the effect of contaminant bulk mass transfer, adsorption and desorption at Re = 100, but kept bubble sphericity in their model. McLaughlin (1996) considered the effect of an insoluble surfactant, but this time on the flow around a deforming bubble. Both the simulations of Cuenot et al. and those of McLaughlin produce relationships between cap angle and drag coefficient similar to that of the simpler model of Sadhal and Johnson. Gas–liquid mass transfer between a bubble and surrounding liquid also depends upon overall surface mobility, which is affected by contamination by surfactants. Upper values of mass transfer coefficient, kL , occur for a bubble with a totally mobile surface, for which its value may be predicted using Higbie’s equation (Bird et al., 1960): u 1/2 mobile kL = 1.13 (1) D , d where d is the bubble diameter, u is the bubble–liquid relative velocity (slip velocity) and D is the diffusivity. Lower values of mass transfer coefficient occur for a bubble with a totally rigid surface, for which its value may be predicted using an equation proposed by Frössling (1938) from the laminar boundary layer theory: u 2/3 −1/6 rigid kL = c , (2) D d where c ≈ 0.6 and is the kinematic viscosity of the liquid. The stagnant cap has also been used to interpret and theorize about gas–liquid mass transfer to bubbles (Ramirez and Davis, 1999; Takemura and Yabe, 1999), the value of the mass transfer coefficient lying within the limits imposed by Eqs. (1) and (2). Experimental data are usually obtained from bubbles in free rise through a column of liquid (Baird and Davidson, 1962; Leonard and Houghton, 1963, Bischof et al., 1991), using either photography/video (Deindoerfer and Humphrey, 1961; Takemura and Yabe, 1999), or the constant volume technique (Koide et al., 1974), or LDA (Brankovic et al., 1984). This limits the phenomenon characteristic time to a few seconds. To be able to follow bubble
dissolution for much longer periods of time, allowing the effect of trace surfactants on dissolution of slowly dissolving gases, Vasconcelos et al. (2002) adapted the kind of apparatus used by Schulze and Schlünder (1985) and Moo-Young et al. (1971) where individual bubbles are kept stationary in a downward liquid flow. The liquid flow is in a closed circuit, allowing continuous cleaning of the liquid. While both terminal velocity and liquid film coefficient are known to depend on surface contamination, few experimental studies exist on the simultaneous decrease in mass transfer and terminal velocity as a bubble gets contaminated. Raymond and Zieminski (1971) showed that bubble mobility, hydrodynamic characteristics and mass transfer are functionally related. A clear relationship between the drag coefficient and the mass transfer coefficient was later confirmed by Koide et al. (1974) and Takemura and Yabe (1999). In these cases, free rise of bubbles of a highly soluble gas, CO2 , was monitored, thus limiting experiments to a maximum period of a few seconds. In this work, an apparatus where individual bubbles are kept stationary in a downward liquid flow was adapted to simultaneously (i) follow mass transfer to/from a single bubble as it inevitably gets contaminated; (ii) follow its shape as it increases its sphericity; and (iii) periodically measure its terminal velocity along the process. A simple model of contamination kinetics based on the stagnant cap hypothesis is used to simultaneously interpret the terminal velocity and the mass transfer data.
2. Model 2.1. Surfactant accumulation The model follows the stagnantly cap model assumption that adsorbed surfactant molecules are dragged towards the rear of the bubble by adjacent liquid (Griffith, 1962). If surface convection is fast compared to both bulk diffusion and both adsorption and desorption, the adsorbed surfactant is collected in a stagnant cap region, leaving the frontal region virtually uncontaminated and thus freely mobile (Cuenot et al., 1997). It is assumed that diffusion controls the surfactant transport from the bulk, as if it were insoluble (Saddhal and Johnson, 1983; Ponoth and McLaughlin, 2000). The rate of diffusion of surfactant from the bulk to the surface is described in terms of a mass transfer coefficient, kS . The accumulation rate of surfactant on the stagnant cap also depends on the surfactant concentration in the bulk, C∞ , and on the available clean surface area Ao (see Fig. 1). Since the latter is the total surface area, A, minus the cap surface area, Acap , the following equation may be written: dnS = kS (A − Acap )C∞ , dt
(3)
S.S. Alves et al. / Chemical Engineering Science 60 (2005) 1 – 9
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2.2. Drag coefficient and terminal velocity Sadhal and Johnson’s model (1983) is used. These authors related the drag coefficient for a partly contaminated bubble with the drag coefficients for contaminated and clean bubbles, and with the stagnant cap angle, : CD () =
CD,rigid − CD,mobile 2 × 2 + sin() − 2 sin(2) − + CD,mobile .
Fig. 1. Stagnant cap model.
where nS is the number of surfactant moles in the stagnant cap. If aS is the area occupied by a mole of surfactant, then Eq. (3) becomes (Acap = nS · aS ) dAcap = kS aS (A − Acap )C∞ . dt
(4)
Since we are considering mass transfer to a mobile surface, the mass transfer coefficient kS must obey Higbie’s relationship with some characteristic length, assumed to be the bubble equivalent diameter de : −1/2
kS = k de
,
(5)
k
where is a constant for a given gas diffusivity and given terminal gas–liquid slip velocity, and de may be calculated, for an ellipsoid, from the major axis, a, and the minor axis, b: de = (a 2 b)1/3 .
(6)
1 sin(3) 3
(10)
CD,rigid and CD,mobile are the drag coefficients for bubbles with a rigid and mobile surface, respectively, and may be calculated from correlations of bubble terminal velocity correlations in Clift et al. (1978). This model was derived for spherical bubbles. To use it for non-spherical bubbles, angle is reinterpreted as the cap angle for the spherical bubble with the same equivalent diameter. The time-dependent cap angle (t) for the equivalent spherical bubble is geometrically related with the clean surface area Ao at time t, thus with actual A(t) and Acap (t) (Vasconcelos, 2002): 2 = − cos−1 1 − 2 (A − Acap ) . (11) de 2.3. Bubble dissolution Neglecting differences in diffusivity between oxygen and nitrogen, and gas-side mass transfer resistance, bubble dissolution may be described by dn rigid = −[kLmobile (A − Acap ) + kL Acap ](C ∗ − C), dt
(12)
where n is the number of gas moles, C ∗ is the saturation concentration and C is the bulk concentration of dissolved rigid gas. kLmobile and kL are the mass transfer coefficients for bubbles with mobile and rigid surfaces, respectively, which are expected to be given by Eqs. (1) and (2), respectively.
Hence, dAcap −1/2 (A − Acap ), = k aS C∞ d e dt
3. Experimental (7)
where factor k , molar area aS and bulk concentration C∞ may be lumped into a single parameter k which may be assumed constant for a given contaminant at a given bulk concentration: k = k aS C∞ .
(8)
The resulting equation for the growth of the contaminated surface with time is the following: dAcap −1/2 (A − Acap ). = k de dt
(9)
In this work, modification of a previous set-up (Vasconcelos et al., 2002) has been made, so that a bubble is captured against a descending water stream in a vertical tube 1.60 m long with internal diameter 22 mm (see Fig. 2). At its bottom end, the tube expands to 31 mm internal diameter in a conical section 0.14 m high. The liquid flow rate is adjusted to maintain the bubble in the conical section at a specified level for observation. The observation section is involved in a squared section water jacket for elimination of optical distortion while measuring the bubble by video technique. Water is pumped in closed circuit from an open-air reservoir after adjustment of the dissolved air concentration C to the desired level. This is carried out through
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Fig. 3. Initial and final bubble’s terminal velocity in distilled, clean distilled and clean millipore water. (dark symbols—initial; open symbols—final).
position. Alternatively, slight bubble dissolution or bubble growing may be obtained for comparison purposes.
4. Results and discussion 4.1. Initial and final terminal velocities
Fig. 2. Experimental set-up: (1) observation section; (2) vertical tube; (3) rotameters; (4) vacuum vessel; (5) pressure controller and water jet pump; (6) screw pump; (7) heat exchanger; (8) 1 m particle retention filter; (9) activated carbon filter (optional); (10) overhead tank; (11) DO-meters; (12) galvanic probes; (13) video camera; (14) lamp and diffuser; (15) PC and monitor; (16) syringe.
the working pressure in a vacuum vessel (item 4 in Fig. 2), together with the use of a fine bubble air disperser and a D.O.-meter mounted in the reservoir (item 10 in Fig. 2). The liquid passes through a 1 m particle retention filter and, optionally, a cleaning filter of activated carbon for organic materials adsorption (items 8 and 9 in Fig. 2). Distilled water and millipore water were used as working liquids, in conjunction with the activated carbon filter. Contamination stems therefore from trace molecules/particles collected by the water at the open tank surface or at equipment walls between filter and test tube (item 1 in Fig. 2). The experimental set-up allowed a bubble to be visually followed for an indefinite period time as it gathers contaminant. Periodic measurement of terminal velocity as a function of bubble age was easily achieved just by stopping the pump and allowing the bubble to ascend along the tube. Restarting the pump brings the bubble back to the observation section. The time taken for the bubble to travel between two marks 1.190 m apart from each other was shown to be within ±0.1 s repetitivity. Bubble volume could be kept constant by adjusting the reservoir level relative to bubble
Fig. 3 presents the initial and the final terminal velocities measured in clean water (distilled water and millipore water). The initial terminal velocities of bubbles of 1.3–4.6 mm equivalent diameter agree with values predicted for pure water by the uppermost curve of Fig. 7.3 in Clift et al. (1978) (curve A in Fig. 3). Clift et al. (1978) reviews extensive experimental data from several authors on bubble terminal velocity. For smaller bubbles, experimental terminal velocities are also in good agreement with the left-hand side of the pure water curve in Fig. 7.3 (curve B in Fig. 3), although more scattered. Maximum terminal velocities of 0.35 m s−1 were found for 1.3–1.4 mm initial bubble diameter. Final terminal velocities agree with the contaminated water curve of Fig. 7.3 in Clift et al. (1978) (curve C in Fig. 3). Initial and final terminal velocities shown in Fig. 3 were obtained in two types of experiments: dissolving bubble experiments and (approximately) constant diameter experiments. Typical terminal velocity vs. time trajectories of typical bubbles for these two kinds of experiments are presented in Fig. 4. 4.2. Constant diameter experiments: simulation of drag coefficient and terminal velocity A first set of experiments were carried out, in which water saturation and hydrostatic pressure at the bubble level were controlled to minimize change in the bubble diameter. The bubble was kept stationary in the downward flow of water and had its terminal velocity periodically measured.
S.S. Alves et al. / Chemical Engineering Science 60 (2005) 1 – 9
Fig. 4. Terminal velocity evolution in distilled water.
Fig. 5 presents data obtained from a typical (approximately) constant diameter experiment. Scatter in diameter for the ∼ 4 mm bubble is due to bubble oscillation. As
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expected, terminal velocity decreases as the bubble gets contaminated. Time for contamination is of the order of 1000 s. Contamination is also detectable through the change in bubble aspect ratio a/b. Constant diameter experiments provide less information than dissolving bubble experiments, as they do not supply mass transfer data. Still, enough information is provided to determine the single unknown parameter, k , necessary to simulate contaminant accumulation in bubbles. Using Eqs. (9)–(11), a single value of k = 5.4 × 10−5 m1/2 s−1 was adjusted to obtain the best fit of drag coefficient for a set of 14 experiments with continuously cleansed distilled water. This value is characteristic of the medium and system. The drag coefficient (at steady-state) was calculated using the bubble projected area: 4 de d e 2 . (13) CD = g 2 3 u a Fig. 6 superimposes on experimental data simulations of the change in the drag coefficient and terminal velocity with time for a typical experiment. Also shown are the calculated values of u and CD for totally mobile and totally rigid bubbles. They were determined using curves in Fig. 3 (from Clift et al., 1978) for instantaneous measured values of de and a/b. Agreement between experiment and simulation is on the whole quite reasonable, as shown by the parity diagram, Fig. 7. Fig. 8 shows experimental and simulated terminal velocities and normalized drag coefficients for two different bubble diameters. Drag coefficients were normalized using ∗ CD () =
mobile CD − C D rigid
CD
mobile − CD
.
(14)
Fig. 5. Bubble equivalent diameter, de (), terminal velocity, u (), and aspect ratio, a/b (x), as a function of bubble age, for (i) a ∼ 1.4 mm diameter bubble and (ii) a ∼ 4 mm bubble (oscillating).
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∗, Fig. 8. Bubble terminal velocity, u, and normalized drag coefficient, CD as a function of bubble age. Lines correspond to simulation: —–, 4.3 mm, - - - - -, 1.9 mm. ∗ (), Fig. 6. Bubble terminal velocity, u (), and drag coefficient, CD as a function of bubble age. ——–, simulation; — — —, values for contaminated bubble; - - - - - - - - -, values for clean bubbles.
The simulation presented in Fig. 8 shows that, as expected, smaller bubbles contaminate quicker than larger bubbles. Despite some dispersion of the data, experimental data agree well with simulation. 4.3. Dissolving bubbles: simultaneous simulation of hydrodynamics and mass transfer
Fig. 7. Drag coefficient parity diagram. - - - - -, lines corresponding to 30% error. Different symbols correspond to different bubbles studied.
CD,mobile and CD,rigid , as CD , were obtained using the bubble projected area, i.e., Eq. (13). Scatter in points for the ∼ 4 mm bubble is attributed due to bubble oscillation.
In a second set of experiments, the partial pressure of the air dissolved in the liquid was decreased, allowing the bubble to dissolve and mass transfer data to be obtained, in addition to hydrodynamic data. Fig. 9 presents data from a dissolving bubble. Simulations for CD and for u are superimposed, using the value of parameter k (Eq. (8)) obtained in the constant diameter experiments, to validate the combination of the simple model of contaminant accumulation with Sadhal and Johnson’s model for drag coefficient. Reasonable agreement is obtained. Also superimposed in Fig. 9 are experimental and simulated values of bubble diameter vs. time. A simulation was carried out using Eqs. (1)–(9) and (12), with two adjusted parameters: (i) k(g) = kL,mobile de , assumed constant (see Eq. (1)) which allows kL,mobile to be calculated as a function of diameter; 1/2
S.S. Alves et al. / Chemical Engineering Science 60 (2005) 1 – 9
Fig. 9. Bubble equivalent diameter, de (, experimental; —, simulation), terminal velocity, u (, experimental; , simulation), and drag coefficient, CD (, experimental; , simulation), as a function of bubble age.
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Fig. 10. Mass transfer coefficient kL vs. de for clean and contaminated bubbles.
(ii) kL,rigid for the contaminated (rigid) part of the bubble. kL,rigid was considered not to vary with diameter. The process was repeated for 13 bubbles of different di and k ameters, with optimization of k(g) L,rigid for each bubble. The agreement obtained between experiment and simulation for these experiments is similar to that shown in Fig. 9. The resulting optimized values of kL,mobile for the initial (clean) bubbles and kL,rigid for the final (rigid) bubbles are shown in Fig. 10, as a function of bubble diameter, superimposed on curves from Higbie and Frössling equations (1) and (2). It is apparent that optimized values of kL,mobile are close to the Higbie curve, while optimized values of kL,rigid are close to the Frössling line (Fig. 10). The simultaneous effect of contamination on hydrodynamics and mass transfer is clearly illustrated in Fig. 11 where values of drag coefficient and mass transfer coefficient, both normalized to quantify the degree of contamination, are plotted against each other. The normalized drag coefficient, calculated using Eq. (14) takes the value zero when the bubble is clean (fully mobile) and the value one when it is rigid due to contamination. The normalized mass transfer coefficient, kL∗ , calculated using the equation below, takes the value one for mobile bubbles and zero for rigid bubbles: kL∗ =
kL − kL,rigid . kL,mobile − kL,rigid
(15)
In Fig. 11 theoretical relationships resulting from the model proposed in this paper and that proposed by Takemura
∗ as a function of normalFig. 11. Normalized mass transfer coefficient kL ∗ . ——-, simulation; - - - - -, correlation from ized drag coefficient, CD Takemura and Yabe (1999).
and Yabe (1999), which can be approximated by a simple relationship, are superimposed on the experimental values ∗ 1/2 ) . kL∗ = (1 − CD
(16)
Agreement between the results of the two models is striking. Experimental results from this work, despite considerable dispersion, are consistent with theory, as were results from Takemura and Yabe (1999) despite the fact that these were obtained for different bubbles under very different conditions, namely, spherical CO2 bubbles of less than 1 mm in diameter, which were absorbed in times of the order of seconds, at the most.
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5. Conclusions
Greek letters
It may be concluded that: (1) An apparatus where individual bubbles are kept stationary for long periods of time in a downward liquid flow may be used to (i) follow mass transfer to/from a single bubble of a low solubility gas, as it inevitably gets contaminated; (ii) follow its shape; and (iii) periodically measure its terminal velocity. (2) Both drag coefficient and gas–liquid mass transfer results may be interpreted in terms of bubble contamination kinetics using a simplified stagnant cap model. Drag coefficient may be reasonably estimated using Sadhal and Johnson’s (1983) relationship, adapted for non-spherical bubbles in terms of an equivalent diameter. (3) Adjusted gas–liquid mass transfer coefficients for the mobile and rigid parts of bubble surface are consistent with theoretical predictions from Higbie’s and Froessling’s equations, respectively. ∗ can be theoretically related to (4) Normalized kL∗ and CD each other. Experimental results are consistent with theory. A simple relationship obtained by Takemura and Yabe (1999), under very different conditions from those in this work, is applicable.
u
Notation a aS A A Acap b C C∗ CD ∗ CD C∞ d de D g k k kL kL∗ kS n nS t
major axis of ellipsoid, m area per surfactant mole on the stagnant cap, m2 mol−1 total bubble area, m2 bubble area that is clean, m2 area of the stagnant cap surface, m2 minor axis of ellipsoid, m bulk concentration of dissolved gas, mol/m3 saturation concentration of dissolved gas, mol/m3 drag coefficient, dimensionless normalized drag coefficient, dimensionless bulk concentration of contaminant, mol/m3 bubble diameter, m bubble equivalent diameter, m gas diffusivity, m2 /s acceleration of gravity, m/s2 constant in Eq. (5) constant in Eq. (8) liquid-side mass transfer coefficient, m s−1 normalized liquid side mass transfer coefficient, m s−1 surfactant mass transfer coefficient, m s−1 number of gas moles the number of surfactant moles in the stagnant cap time, s
slip velocity, ms−1 stagnant cap angle kinematic viscosity, m s−1
Superscripts and subscripts mobile rigid
mobile interface rigid interface
Acknowledgements The first author gratefully acknowledges financial support by FCT (Grant No. SFRH/BD/3229/2000). This work was supported by FCT—Fundação para a Ciência e a Tecnologia and Program FEDER (Project POCTI/EQU/47689/2002).
References Baird, M.H.I., Davidson, J.F., 1962. Gas absorption by large rising bubbles. Chemical Engineering Science 17, 87–93. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport Phenomena. Wiley, New York, p. 541. Bischof, F., Jahn, M., Sommerfeld, M., 1991. Calculation of transient bubble rise in contaminated water. In: Proceedings of the Second International Conference on Multiphase Flow ’95-Kyoto, PD2-17. Brankovic, A., Curie, I.G., Martin, W.W., 1984. Laser-Doppler measurements of bubble dynamics. Physics of Fluids 27, 348. Clift, R., Grace, J.R., Weber, M.E., 1978. Bubbles, Drops, and Particles, Academic Press, London. Cuenot, B., Magnaudet, J., Spennato, B., 1997. The effects of slightly soluble surfactants on the flow around a spherical bubble. Journal of Fluid Mechanics 339, 25. Deindoerfer, F.H., Humphrey, A.E., 1961. Mass transfer from individual gas bubbles. Industrial and Engineering Chemistry 53, 755–759. Duineveld, P.C., 1995. The rise velocity and shape of bubbles in pure water at high Reynolds number. Journal of Fluid Mechanics 292, 325–332. Frössling, N., 1938. Über die verdünstung fallenden tropfen, (Evaporation of falling drops). Gerlands Beitage zur Geophysik 52, 170–216 (cited in Griffith, 1962). Frumkin, A., Levich, V.G., 1947. On surfactants and interfacial motion. Zhurnal Fizicheskoi Khimii 21, 1183–1204 (cited in Griffith, 1962) (in Russian). Griffith, R.M., 1962. The effect of surfactants on the terminal velocity of drops and bubbles. Chemical Engineering Science 17, 1057–1070. Koide, K., Orito, Y., Hara, Y., 1974. Mass transfer from single bubbles in Newtonian liquids. Chemical Engineering Science 29, 417–425. Leonard, J.H., Houghton, G., 1963. Mass transfer and velocity of rise phenomena for single bubbles. Chemical Engineering Science 18, 133–142. McLaughlin, J.B., 1996. Numerical simulation of bubble motion in water. Journal of Colloid and Interface Science 184, 614–625. Moo-Young, M., Fulford, G., Cheyne, I., 1971. Industrial and Engineering Chemistry Fundamentals 10, 157. Ponoth, S.S., McLaughlin, J.B., 2000. Numerical simulation of mass transfer for bubbles in water. Chemical Engineering Science 55, 1237–1255.
S.S. Alves et al. / Chemical Engineering Science 60 (2005) 1 – 9 Ramirez, J.A., Davis, R.H., 1999. Mass transfer to a surfactant-covered bubble or drop. American Institute of Chemical Engineers Journal 45 (6), 1355–1358. Raymond, D.R., Zieminski, S.A., 1971. Mass transfer and drag coefficients of bubbles rising in dilute aqueous solutions. American Institute of Chemical Engineers Journal 17, 57–65. Sadhal, S.S., Johnson, R.E., 1983. Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film—exact solution. Journal of Fluid Mechanics 126, 237. Schulze, G., Schlünder, E.U., 1985. Physical absorption of single gas bubbles in degassed and preloaded water. Chemical Engineering and Processing 19, 27–37.
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Takemura, F., Yabe, A., 1999. Rising speed and dissolution rate of a carbon dioxide bubble in slightly contaminated water. Journal of Fluid Mechanics 378, 319–334. Vasconcelos, J.M.T., Orvalho, S.P., Alves, S.S., 2002. Gas–liquid mass transfer to single bubbles: effect of surface contamination. American Institute of Chemical Engineers Journal 48 (6), 1145–1154. Zhang, Y., Finch, J.A., 2001. A note on single bubble motion in surfactant solutions. Journal of Fluid Mechanics 429, 63–66. Zhang, Y., McLaughlin, J.B., Finch, J.A., 2001. Bubble velocity profile and model of surfactant mass transfer to bubble surface. Chemical Engineering Science 56, 6605–6616.
Effect of bubble contamination on rise velocity and mass transfer S.S. Alves∗ , S.P. Orvalho, J.M.T. Vasconcelos Department of Chemical Engineering, Instituto Superior Técnico, Centro de Eng. Biológica e Química, 1049-001 Lisboa, Portugal Received 28 January 2004; received in revised form 5 July 2004; accepted 6 July 2004 Available online 15 September 2004
Abstract An apparatus where individual bubbles are kept stationary in a downward liquid flow was adapted to simultaneously (i) follow mass transfer to/from a single bubble as it inevitably gets contaminated; (ii) follow its shape; and (iii) periodically measure its terminal velocity. This apparatus allows bubbles to be monitored for much longer periods of time than does the monitoring of rising bubbles. Thus, the effect of trace contaminants on bubbles of low solubility gases, like air, may be studied. Experiments were done with air bubbles of 1–5 mm initial equivalent diameter in a water stream. The partial pressure of air in the liquid could be manipulated, allowing bubbles to be either dissolving or kept at an approximately constant diameter. Both drag coefficient and gas–liquid mass transfer results were interpreted in terms of bubble contamination kinetics using a simplified stagnant cap model. Drag coefficient was calculated from stagnant cap size using an adaptation of Sadhal and Johnson’s model (J. Fluid Mech. 126 (1983) 237). Gas–liquid mass transfer modelling assumed two mass transfer coefficients, one for the clean front of the bubble, the other for the stagnant cap. Adjusted values of these coefficients are consistent with theoretical predictions from Higbie’s and Frössling’s equations, respectively. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Bubble; Drag coefficient; Hydrodynamics; Mass transfer; Stagnant cap; Surfactant
1. Introduction Gas–liquid systems, with the gas as the disperse phase, are as important as they are complex. An important contribution to the complexity of bubble interaction with the liquid environment is the effect of trace amounts of surfactant. Both bubble rise velocity and gas–liquid mass transfer are known to be slowed down by surface-active contaminants. While the two phenomena are certainly related, both being due to the accumulation of the surfactant at the bubble surface, they tend to be separately studied. The rise velocity of air bubbles in water strongly depends on water purity, especially in the spherical and ellipsoidal range (0.5–17 mm equivalent diameters). Values up to 0.36 m s−1 terminal velocities have been found in so-called
∗ Corresponding author. Tel.: +351-21-8417188; 8499242. E-mail address: [email protected] (S.S. Alves).
fax:
+351-21-
0009-2509/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.07.053
‘hyper-clean’ water for air bubbles of 1.6–1.8 mm diameter (Duineveld, 1995), while less than one-half of that is usually measured in tap water. The greatest cause for the spreading in terminal velocity data (see Clift et al., 1978) is attributed to surface contamination. Clean bubble interfaces are freely mobile. However, the presence of even small amounts of impurities leads to the adsorption and collection of surfactants at the bubble surface which hinders its mobility. Frumkin and Levich (1947) proposed this to be the result of the uneven distribution of surfactant due to surface advection by the main flow from the bubble front end to the rear stagnation point. The concentration gradient results in a tangential gradient of surface tension that causes a tangential stress, also known as Marangoni stress, to oppose the flow shear stress. Surface motion is thus retarded and the drag coefficient is increased towards that of a rigid sphere. This is the so-called stagnant cap hypothesis, which has been successful in explaining bubble rise velocity data, both from experiments where an average velocity is calculated by measuring the time taken by a rising bubble to travel a
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fixed distance and from more detailed studies involving the instantaneous monitoring of the bubble velocity (Takemura and Yabe, 1999; Zhang and Finch, 2001; Zhang et al., 2001), thus providing the bubble time-history. Most mathematical models proposed to simulate the increase in drag coefficient as a function of surface contamination assumes the stagnant cap hypothesis. The model of Sadhal and Johnson (1983) was developed for creeping flow past a spherical bubble with a stagnant cap created by surfactants of low solubility. With these assumptions a simple relationship between drag force and the stagnant cap angle could be analytically deduced. Bischof et al. (1991) used this relationship together with a simple model for bubble contamination to simulate the transient behaviour of a bubble rising in quiescent water. Relaxing of Sadhal and Johnson’s assumptions involves numerical methods. Cuenot et al. (1997) abandoned creeping flow and investigated the effect of contaminant bulk mass transfer, adsorption and desorption at Re = 100, but kept bubble sphericity in their model. McLaughlin (1996) considered the effect of an insoluble surfactant, but this time on the flow around a deforming bubble. Both the simulations of Cuenot et al. and those of McLaughlin produce relationships between cap angle and drag coefficient similar to that of the simpler model of Sadhal and Johnson. Gas–liquid mass transfer between a bubble and surrounding liquid also depends upon overall surface mobility, which is affected by contamination by surfactants. Upper values of mass transfer coefficient, kL , occur for a bubble with a totally mobile surface, for which its value may be predicted using Higbie’s equation (Bird et al., 1960): u 1/2 mobile kL = 1.13 (1) D , d where d is the bubble diameter, u is the bubble–liquid relative velocity (slip velocity) and D is the diffusivity. Lower values of mass transfer coefficient occur for a bubble with a totally rigid surface, for which its value may be predicted using an equation proposed by Frössling (1938) from the laminar boundary layer theory: u 2/3 −1/6 rigid kL = c , (2) D d where c ≈ 0.6 and is the kinematic viscosity of the liquid. The stagnant cap has also been used to interpret and theorize about gas–liquid mass transfer to bubbles (Ramirez and Davis, 1999; Takemura and Yabe, 1999), the value of the mass transfer coefficient lying within the limits imposed by Eqs. (1) and (2). Experimental data are usually obtained from bubbles in free rise through a column of liquid (Baird and Davidson, 1962; Leonard and Houghton, 1963, Bischof et al., 1991), using either photography/video (Deindoerfer and Humphrey, 1961; Takemura and Yabe, 1999), or the constant volume technique (Koide et al., 1974), or LDA (Brankovic et al., 1984). This limits the phenomenon characteristic time to a few seconds. To be able to follow bubble
dissolution for much longer periods of time, allowing the effect of trace surfactants on dissolution of slowly dissolving gases, Vasconcelos et al. (2002) adapted the kind of apparatus used by Schulze and Schlünder (1985) and Moo-Young et al. (1971) where individual bubbles are kept stationary in a downward liquid flow. The liquid flow is in a closed circuit, allowing continuous cleaning of the liquid. While both terminal velocity and liquid film coefficient are known to depend on surface contamination, few experimental studies exist on the simultaneous decrease in mass transfer and terminal velocity as a bubble gets contaminated. Raymond and Zieminski (1971) showed that bubble mobility, hydrodynamic characteristics and mass transfer are functionally related. A clear relationship between the drag coefficient and the mass transfer coefficient was later confirmed by Koide et al. (1974) and Takemura and Yabe (1999). In these cases, free rise of bubbles of a highly soluble gas, CO2 , was monitored, thus limiting experiments to a maximum period of a few seconds. In this work, an apparatus where individual bubbles are kept stationary in a downward liquid flow was adapted to simultaneously (i) follow mass transfer to/from a single bubble as it inevitably gets contaminated; (ii) follow its shape as it increases its sphericity; and (iii) periodically measure its terminal velocity along the process. A simple model of contamination kinetics based on the stagnant cap hypothesis is used to simultaneously interpret the terminal velocity and the mass transfer data.
2. Model 2.1. Surfactant accumulation The model follows the stagnantly cap model assumption that adsorbed surfactant molecules are dragged towards the rear of the bubble by adjacent liquid (Griffith, 1962). If surface convection is fast compared to both bulk diffusion and both adsorption and desorption, the adsorbed surfactant is collected in a stagnant cap region, leaving the frontal region virtually uncontaminated and thus freely mobile (Cuenot et al., 1997). It is assumed that diffusion controls the surfactant transport from the bulk, as if it were insoluble (Saddhal and Johnson, 1983; Ponoth and McLaughlin, 2000). The rate of diffusion of surfactant from the bulk to the surface is described in terms of a mass transfer coefficient, kS . The accumulation rate of surfactant on the stagnant cap also depends on the surfactant concentration in the bulk, C∞ , and on the available clean surface area Ao (see Fig. 1). Since the latter is the total surface area, A, minus the cap surface area, Acap , the following equation may be written: dnS = kS (A − Acap )C∞ , dt
(3)
S.S. Alves et al. / Chemical Engineering Science 60 (2005) 1 – 9
3
2.2. Drag coefficient and terminal velocity Sadhal and Johnson’s model (1983) is used. These authors related the drag coefficient for a partly contaminated bubble with the drag coefficients for contaminated and clean bubbles, and with the stagnant cap angle, : CD () =
CD,rigid − CD,mobile 2 × 2 + sin() − 2 sin(2) − + CD,mobile .
Fig. 1. Stagnant cap model.
where nS is the number of surfactant moles in the stagnant cap. If aS is the area occupied by a mole of surfactant, then Eq. (3) becomes (Acap = nS · aS ) dAcap = kS aS (A − Acap )C∞ . dt
(4)
Since we are considering mass transfer to a mobile surface, the mass transfer coefficient kS must obey Higbie’s relationship with some characteristic length, assumed to be the bubble equivalent diameter de : −1/2
kS = k de
,
(5)
k
where is a constant for a given gas diffusivity and given terminal gas–liquid slip velocity, and de may be calculated, for an ellipsoid, from the major axis, a, and the minor axis, b: de = (a 2 b)1/3 .
(6)
1 sin(3) 3
(10)
CD,rigid and CD,mobile are the drag coefficients for bubbles with a rigid and mobile surface, respectively, and may be calculated from correlations of bubble terminal velocity correlations in Clift et al. (1978). This model was derived for spherical bubbles. To use it for non-spherical bubbles, angle is reinterpreted as the cap angle for the spherical bubble with the same equivalent diameter. The time-dependent cap angle (t) for the equivalent spherical bubble is geometrically related with the clean surface area Ao at time t, thus with actual A(t) and Acap (t) (Vasconcelos, 2002): 2 = − cos−1 1 − 2 (A − Acap ) . (11) de 2.3. Bubble dissolution Neglecting differences in diffusivity between oxygen and nitrogen, and gas-side mass transfer resistance, bubble dissolution may be described by dn rigid = −[kLmobile (A − Acap ) + kL Acap ](C ∗ − C), dt
(12)
where n is the number of gas moles, C ∗ is the saturation concentration and C is the bulk concentration of dissolved rigid gas. kLmobile and kL are the mass transfer coefficients for bubbles with mobile and rigid surfaces, respectively, which are expected to be given by Eqs. (1) and (2), respectively.
Hence, dAcap −1/2 (A − Acap ), = k aS C∞ d e dt
3. Experimental (7)
where factor k , molar area aS and bulk concentration C∞ may be lumped into a single parameter k which may be assumed constant for a given contaminant at a given bulk concentration: k = k aS C∞ .
(8)
The resulting equation for the growth of the contaminated surface with time is the following: dAcap −1/2 (A − Acap ). = k de dt
(9)
In this work, modification of a previous set-up (Vasconcelos et al., 2002) has been made, so that a bubble is captured against a descending water stream in a vertical tube 1.60 m long with internal diameter 22 mm (see Fig. 2). At its bottom end, the tube expands to 31 mm internal diameter in a conical section 0.14 m high. The liquid flow rate is adjusted to maintain the bubble in the conical section at a specified level for observation. The observation section is involved in a squared section water jacket for elimination of optical distortion while measuring the bubble by video technique. Water is pumped in closed circuit from an open-air reservoir after adjustment of the dissolved air concentration C to the desired level. This is carried out through
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Fig. 3. Initial and final bubble’s terminal velocity in distilled, clean distilled and clean millipore water. (dark symbols—initial; open symbols—final).
position. Alternatively, slight bubble dissolution or bubble growing may be obtained for comparison purposes.
4. Results and discussion 4.1. Initial and final terminal velocities
Fig. 2. Experimental set-up: (1) observation section; (2) vertical tube; (3) rotameters; (4) vacuum vessel; (5) pressure controller and water jet pump; (6) screw pump; (7) heat exchanger; (8) 1 m particle retention filter; (9) activated carbon filter (optional); (10) overhead tank; (11) DO-meters; (12) galvanic probes; (13) video camera; (14) lamp and diffuser; (15) PC and monitor; (16) syringe.
the working pressure in a vacuum vessel (item 4 in Fig. 2), together with the use of a fine bubble air disperser and a D.O.-meter mounted in the reservoir (item 10 in Fig. 2). The liquid passes through a 1 m particle retention filter and, optionally, a cleaning filter of activated carbon for organic materials adsorption (items 8 and 9 in Fig. 2). Distilled water and millipore water were used as working liquids, in conjunction with the activated carbon filter. Contamination stems therefore from trace molecules/particles collected by the water at the open tank surface or at equipment walls between filter and test tube (item 1 in Fig. 2). The experimental set-up allowed a bubble to be visually followed for an indefinite period time as it gathers contaminant. Periodic measurement of terminal velocity as a function of bubble age was easily achieved just by stopping the pump and allowing the bubble to ascend along the tube. Restarting the pump brings the bubble back to the observation section. The time taken for the bubble to travel between two marks 1.190 m apart from each other was shown to be within ±0.1 s repetitivity. Bubble volume could be kept constant by adjusting the reservoir level relative to bubble
Fig. 3 presents the initial and the final terminal velocities measured in clean water (distilled water and millipore water). The initial terminal velocities of bubbles of 1.3–4.6 mm equivalent diameter agree with values predicted for pure water by the uppermost curve of Fig. 7.3 in Clift et al. (1978) (curve A in Fig. 3). Clift et al. (1978) reviews extensive experimental data from several authors on bubble terminal velocity. For smaller bubbles, experimental terminal velocities are also in good agreement with the left-hand side of the pure water curve in Fig. 7.3 (curve B in Fig. 3), although more scattered. Maximum terminal velocities of 0.35 m s−1 were found for 1.3–1.4 mm initial bubble diameter. Final terminal velocities agree with the contaminated water curve of Fig. 7.3 in Clift et al. (1978) (curve C in Fig. 3). Initial and final terminal velocities shown in Fig. 3 were obtained in two types of experiments: dissolving bubble experiments and (approximately) constant diameter experiments. Typical terminal velocity vs. time trajectories of typical bubbles for these two kinds of experiments are presented in Fig. 4. 4.2. Constant diameter experiments: simulation of drag coefficient and terminal velocity A first set of experiments were carried out, in which water saturation and hydrostatic pressure at the bubble level were controlled to minimize change in the bubble diameter. The bubble was kept stationary in the downward flow of water and had its terminal velocity periodically measured.
S.S. Alves et al. / Chemical Engineering Science 60 (2005) 1 – 9
Fig. 4. Terminal velocity evolution in distilled water.
Fig. 5 presents data obtained from a typical (approximately) constant diameter experiment. Scatter in diameter for the ∼ 4 mm bubble is due to bubble oscillation. As
5
expected, terminal velocity decreases as the bubble gets contaminated. Time for contamination is of the order of 1000 s. Contamination is also detectable through the change in bubble aspect ratio a/b. Constant diameter experiments provide less information than dissolving bubble experiments, as they do not supply mass transfer data. Still, enough information is provided to determine the single unknown parameter, k , necessary to simulate contaminant accumulation in bubbles. Using Eqs. (9)–(11), a single value of k = 5.4 × 10−5 m1/2 s−1 was adjusted to obtain the best fit of drag coefficient for a set of 14 experiments with continuously cleansed distilled water. This value is characteristic of the medium and system. The drag coefficient (at steady-state) was calculated using the bubble projected area: 4 de d e 2 . (13) CD = g 2 3 u a Fig. 6 superimposes on experimental data simulations of the change in the drag coefficient and terminal velocity with time for a typical experiment. Also shown are the calculated values of u and CD for totally mobile and totally rigid bubbles. They were determined using curves in Fig. 3 (from Clift et al., 1978) for instantaneous measured values of de and a/b. Agreement between experiment and simulation is on the whole quite reasonable, as shown by the parity diagram, Fig. 7. Fig. 8 shows experimental and simulated terminal velocities and normalized drag coefficients for two different bubble diameters. Drag coefficients were normalized using ∗ CD () =
mobile CD − C D rigid
CD
mobile − CD
.
(14)
Fig. 5. Bubble equivalent diameter, de (), terminal velocity, u (), and aspect ratio, a/b (x), as a function of bubble age, for (i) a ∼ 1.4 mm diameter bubble and (ii) a ∼ 4 mm bubble (oscillating).
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∗, Fig. 8. Bubble terminal velocity, u, and normalized drag coefficient, CD as a function of bubble age. Lines correspond to simulation: —–, 4.3 mm, - - - - -, 1.9 mm. ∗ (), Fig. 6. Bubble terminal velocity, u (), and drag coefficient, CD as a function of bubble age. ——–, simulation; — — —, values for contaminated bubble; - - - - - - - - -, values for clean bubbles.
The simulation presented in Fig. 8 shows that, as expected, smaller bubbles contaminate quicker than larger bubbles. Despite some dispersion of the data, experimental data agree well with simulation. 4.3. Dissolving bubbles: simultaneous simulation of hydrodynamics and mass transfer
Fig. 7. Drag coefficient parity diagram. - - - - -, lines corresponding to 30% error. Different symbols correspond to different bubbles studied.
CD,mobile and CD,rigid , as CD , were obtained using the bubble projected area, i.e., Eq. (13). Scatter in points for the ∼ 4 mm bubble is attributed due to bubble oscillation.
In a second set of experiments, the partial pressure of the air dissolved in the liquid was decreased, allowing the bubble to dissolve and mass transfer data to be obtained, in addition to hydrodynamic data. Fig. 9 presents data from a dissolving bubble. Simulations for CD and for u are superimposed, using the value of parameter k (Eq. (8)) obtained in the constant diameter experiments, to validate the combination of the simple model of contaminant accumulation with Sadhal and Johnson’s model for drag coefficient. Reasonable agreement is obtained. Also superimposed in Fig. 9 are experimental and simulated values of bubble diameter vs. time. A simulation was carried out using Eqs. (1)–(9) and (12), with two adjusted parameters: (i) k(g) = kL,mobile de , assumed constant (see Eq. (1)) which allows kL,mobile to be calculated as a function of diameter; 1/2
S.S. Alves et al. / Chemical Engineering Science 60 (2005) 1 – 9
Fig. 9. Bubble equivalent diameter, de (, experimental; —, simulation), terminal velocity, u (, experimental; , simulation), and drag coefficient, CD (, experimental; , simulation), as a function of bubble age.
7
Fig. 10. Mass transfer coefficient kL vs. de for clean and contaminated bubbles.
(ii) kL,rigid for the contaminated (rigid) part of the bubble. kL,rigid was considered not to vary with diameter. The process was repeated for 13 bubbles of different di and k ameters, with optimization of k(g) L,rigid for each bubble. The agreement obtained between experiment and simulation for these experiments is similar to that shown in Fig. 9. The resulting optimized values of kL,mobile for the initial (clean) bubbles and kL,rigid for the final (rigid) bubbles are shown in Fig. 10, as a function of bubble diameter, superimposed on curves from Higbie and Frössling equations (1) and (2). It is apparent that optimized values of kL,mobile are close to the Higbie curve, while optimized values of kL,rigid are close to the Frössling line (Fig. 10). The simultaneous effect of contamination on hydrodynamics and mass transfer is clearly illustrated in Fig. 11 where values of drag coefficient and mass transfer coefficient, both normalized to quantify the degree of contamination, are plotted against each other. The normalized drag coefficient, calculated using Eq. (14) takes the value zero when the bubble is clean (fully mobile) and the value one when it is rigid due to contamination. The normalized mass transfer coefficient, kL∗ , calculated using the equation below, takes the value one for mobile bubbles and zero for rigid bubbles: kL∗ =
kL − kL,rigid . kL,mobile − kL,rigid
(15)
In Fig. 11 theoretical relationships resulting from the model proposed in this paper and that proposed by Takemura
∗ as a function of normalFig. 11. Normalized mass transfer coefficient kL ∗ . ——-, simulation; - - - - -, correlation from ized drag coefficient, CD Takemura and Yabe (1999).
and Yabe (1999), which can be approximated by a simple relationship, are superimposed on the experimental values ∗ 1/2 ) . kL∗ = (1 − CD
(16)
Agreement between the results of the two models is striking. Experimental results from this work, despite considerable dispersion, are consistent with theory, as were results from Takemura and Yabe (1999) despite the fact that these were obtained for different bubbles under very different conditions, namely, spherical CO2 bubbles of less than 1 mm in diameter, which were absorbed in times of the order of seconds, at the most.
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5. Conclusions
Greek letters
It may be concluded that: (1) An apparatus where individual bubbles are kept stationary for long periods of time in a downward liquid flow may be used to (i) follow mass transfer to/from a single bubble of a low solubility gas, as it inevitably gets contaminated; (ii) follow its shape; and (iii) periodically measure its terminal velocity. (2) Both drag coefficient and gas–liquid mass transfer results may be interpreted in terms of bubble contamination kinetics using a simplified stagnant cap model. Drag coefficient may be reasonably estimated using Sadhal and Johnson’s (1983) relationship, adapted for non-spherical bubbles in terms of an equivalent diameter. (3) Adjusted gas–liquid mass transfer coefficients for the mobile and rigid parts of bubble surface are consistent with theoretical predictions from Higbie’s and Froessling’s equations, respectively. ∗ can be theoretically related to (4) Normalized kL∗ and CD each other. Experimental results are consistent with theory. A simple relationship obtained by Takemura and Yabe (1999), under very different conditions from those in this work, is applicable.
u
Notation a aS A A Acap b C C∗ CD ∗ CD C∞ d de D g k k kL kL∗ kS n nS t
major axis of ellipsoid, m area per surfactant mole on the stagnant cap, m2 mol−1 total bubble area, m2 bubble area that is clean, m2 area of the stagnant cap surface, m2 minor axis of ellipsoid, m bulk concentration of dissolved gas, mol/m3 saturation concentration of dissolved gas, mol/m3 drag coefficient, dimensionless normalized drag coefficient, dimensionless bulk concentration of contaminant, mol/m3 bubble diameter, m bubble equivalent diameter, m gas diffusivity, m2 /s acceleration of gravity, m/s2 constant in Eq. (5) constant in Eq. (8) liquid-side mass transfer coefficient, m s−1 normalized liquid side mass transfer coefficient, m s−1 surfactant mass transfer coefficient, m s−1 number of gas moles the number of surfactant moles in the stagnant cap time, s
slip velocity, ms−1 stagnant cap angle kinematic viscosity, m s−1
Superscripts and subscripts mobile rigid
mobile interface rigid interface
Acknowledgements The first author gratefully acknowledges financial support by FCT (Grant No. SFRH/BD/3229/2000). This work was supported by FCT—Fundação para a Ciência e a Tecnologia and Program FEDER (Project POCTI/EQU/47689/2002).
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