Bubble Formation at an Orifice in Surfactant Solutions under Constant

Ind. Eng. Chem. Res. 2000, 39, 1473-1479

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Bubble Formation at an Orifice in Surfactant Solutions under Constant-Flow Conditions Shu-Hao Hsu, Wei-Hua Lee, Yu-Min Yang,* Chien-Hsiang Chang, and Jer-Ru Maa Department of Chemical Engineering, National Cheng Kung University, Tainan, Taiwan 70101

Bubble formation in surfactant solutions is frequently encountered in systems of practical importance but has seldom been investigated systematically. In this work a simple modified two-stage spherical model, by which the dynamic surface tension at the bubble surface plays a role, was developed for bubble formation under constant-flow conditions. Equilibrium and dynamic surface tension data and the values of bubble volume at detachment were obtained experimentally for aqueous solutions of sodium dodecyl sulfate and Triton X-100. The predicted results are shown to agree well with experimental ones, especially for concentrations less than the critical micelle concentration. This implies that the surface relaxation phenomenon is important for bubble formation in surfactant solutions. For accurate prediction of bubble formation in surfactant solutions, dynamic surface tension rather than equilibrium data should be taken into consideration. Introduction Processes with gas-liquid contact are numerous industrially. The accurate prediction of the bubble volume is very important for the precise design of masstransfer equipment such as a bubble column. Many investigations of the problem of the bubble formation have been undertaken in the past. Reviews of this subject are available.1-3 Several models of bubble formation from a submerged single orifice under constant-flow conditions, in which the gas flow rate through an orifice is constant throughout the bubble formation, have ever been proposed.4-9 Davidson and Schuler4,5 presented one-stage spherical bubble formation models for viscous and inviscid liquids, respectively, by neglecting the surface tension force. Two-stage spherical bubble formation models with different forms of the drag coefficient in Newton’s second law of motion were thereafter proposed. As the drag coefficient, Stokes’ equation for a solid sphere was used by Ramakrishnan et al.,6 a modified Stokes’ equation by Ruff,7 and the Hadamard-Rybczynsk equation for a fluid sphere by Takahashi and Miyahara.8 On the other hand, a nonspherical bubble formation model was proposed by Terasaka and Tsuge9 for predicting both bubble volumes and shapes without detachment conditions being determined experimentally as in previous two-stage models. It is noteworthy, however, that most models available in the literature as previously mentioned dealt tacitly with bubble formation in pure liquids. Traces of surface-active contaminants are known to be able to depress the surface tension of liquids drastically. The measures required to purify such systems and the precautions needed to ensure no further contamination are so stringent that one must accept the presence of surface-active contaminants in most systems of practical importance. Therefore, an important question among others which pertain to the bubble formation arises as, do the models developed for pure liquids still * To whom correspondence should be addressed. Tel.: 8866-2757575, ext. 62633. Fax: 886-6-2344496. E-mail: ymyang@ mail.ncku.edu.tw.

hold for the case of surfactant solutions even if the change of equilibrium surface tension in the models is taken into consideration? Nevertheless, this problem has seldom been systematically studied. In this work, a simple modified two-stage spherical model, by which the dynamic surface tension at the bubble surface plays a role, was developed for bubble formation under constant-flow conditions. Equilibrium and dynamic surface tension data and values of bubble volume at detachment were obtained experimentally for surfactant solutions. The predicted results were then compared with experimental ones. To the authors’ knowledge, this is the first attempt to account for surfactant effects in the bubble formation. Theoretical Considerations The two-stage spherical bubble formation model of Ruff7 is described in brief, and a modified model, which takes the variation of surface tension during bubble formation into consideration, is proposed. Model by Ruff. The bubble formation is assumed to take place in two stages. In the early stage of bubble growth, the bubble remains attached to the orifice up to the point when the resultant of the lifting forces just exceeds the resultant of the restraining forces (expansion stage). A further increase in the bubble volume causes an imbalance of forces, and the bubble lifts while still being connected to the orifice through a neck (detachment stage). The neck contracts, and the bubble finally detaches itself from the orifice. The condition for detachment is when the distance between the bubble center and the orifice plate is equal to the bubble radius at the end of the first stage such that the subsequent expanding bubble does not coalesce with it. After the evaluation of various constants, the first stage equation for this situation is

Ve5/3 ) 0.05773

( )

Qg2 µl + 2.418 Q V 1/3 + g Fl g g e

( )

()

µl πDσ 2/3 V + 0.2041 Fl g e Fl

10.1021/ie990578l CCC: $19.00 © 2000 American Chemical Society Published on Web 03/23/2000

1/2

Qg3/2Ve1/6 (1) g

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Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000

The time of detachment td during the second stage is given as

td )

( )[ ( Qg1/5 3/5

g

1+4

)] 3/4

µl 1/5

Flg Qg

3/5

(2)

The final volume Vb can be evaluated by finding individually the value of the force-balance bubble volume Ve (by trial and error) and the volume entering the bubble during detachment stage Qgtd and then adding the two. Thus

Vb ) Ve + Qgtd

(3)

Modified Ruff Model. When the surface of a surfactant solution is freshly formed, the amount of surfactant molecules per unit area is less than that at equilibrium and the surface tension is correspondingly higher. The value of the dynamic surface tension and the rate at which it approaches equilibrium are determined by relaxation processes taking place in or near the surface. Reviews of this subject are available.10-13 During equilibration the only quantity measured is the surface tension as a function of time for the given constant solution concentration far from the interface, i.e.,

σ ) σ(t)

(4)

The coupling of the dynamic surface tension into the Ruff model for bubble formation under constant-flow conditions is possible by considering the variation of surface tension with time as given by eq 4. The solution procedure is shown in Figure 1. However, one should note that the Harkins-Brown correction factor was not considered in eq 1 and neglect of this correction factor may cause an error. Because we intend to develop an approach to improve the applicability of the Ruff model, eq 1 originally used in the Ruff model was retained in this study for comparison purposes.

Figure 1. Estimation procedure for the bubble volume at detachment in the modified Ruff model.

Experimental Section Pure water was obtained by treating tap water with double reverse osmosis units. The water-soluble surfactants used are sodium dodecyl sulfate (SDS, C12H25SO4Na) supplied by Sigma Chemical Co. and polyoxyethylated tert-octylphenol (Triton X-100, CH3C(CH3)2CH2C(CH3)2C6H4(OCH2CH2)nOH, n ) 9-10) supplied by Aldrich Chemical Co. They were used as received without further purification. Experiments were conducted in a room in which the temperature was controlled within 20 ( 2 °C. Measurement of the Bubble Volume. Figure 2 shows the apparatus for the measurement of volumes of a bubble growing at the orifice. The constant-flow orifices were made of stainless steel capillary tubes of 0.915 and 0.554 mm i.d. The 1 m long capillary tube was inserted through a silicone rubber stopper fitted to the lower end of an 8 × 8 cm, 50 cm tall square bubble column made of Plexiglass. The nitrogen gas was first saturated with water vapor by passing through a flask containing water. The flow rates of the nitrogen stream were adjusted by the needle valves and measured by a mass flowmeter. The bubbling apparatus was illuminated from behind using a lamp, and a piece of tracing paper attached to the outside of the rear tank wall served as a diffuser. A

Figure 2. Schematic diagram of the experimental apparatus for bubble formation.

DALSA CAD1 high-speed camera (Coredo Inc., Canada) was used to visualize the bubbling phenomena and bubble ebullition cycle. Using a maximum pixel clock of 16 MHz, a maximum frame rate of 2926 frames/s in a resolution of 64 × 64 is possible. The video from the camera is acquired into a single-frame buffer of an image grabber board F64. The F64 is capable of acquir-

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1475

Figure 3. Bubbling cycle in pure water. The frame numbers are shown correspondingly. Di ) 0.915 mm; bubble frequency ) 13 s-1; film speed ) 2926 frames s-1.

Figure 5. Equilibrium surface tensions of SDS and Triton X-100 solutions.

Figure 4. Bubble formation under constant-flow conditions.

ing consecutive frames into different locations in VRAM memory. This is useful for performing high frame rate acquisition and performing analysis on the images after they are in memory. The relevant parameters of bubble dynamics were measure by examining the photographs frame by frame using a software Optimas version 5.22. The capillary tube, which was also photographed, was employed as a reference length for the determination of the actual dimensions. A stroboscope was also used to measure the bubble frequency for the purpose of independent determination of the bubble volume at detachment. Measurement of Equilibrium and Dynamic Surface Tensions. Equilibrium surface tension data of the aqueous surfactant solutions were determined with a surface tension meter (CBVP-A3 Type, Kyowa Interface Science Co. Ltd., Japan) by the Wilhelmy plate method. Many different techniques exist to measure the dynamic surface tension of liquid interfaces, covering the time interval from milliseconds to hours and days. A bubble pressure tensiometer BP (Kruss, Germany) based on the principle of maximum bubble pressure was used in this work. At minimum bubble radius the gas pressure necessary to create the bubble is maximum and can be related to surface tension using the YoungLaplace equation.14 The bubble pressure tensiometer measures the dynamic surface tension as a function of the bubble frequency, which varies between 0.5 and 10 Hz in steps of 0.1 Hz. It should be noted that the

Figure 6. Dynamic surface tensions of SDS solutions. (a) Effect of bubble frequency on the surface tension. (b) Relaxation of surface tension with time.

behavior of a nitrogen gas bubble in measuring the dynamic surface tension by the method of maximum bubble pressure may not be exactly the same as that of a growing bubble under constant-flow conditions. However, a close similarity between them is to be expected.

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Figure 7. Relaxation of surface tension with time for Triton X-100 solutions.

In fact, this is the reason the authors chose it from the available methods for measuring the dynamic surface tension. Furthermore, what we can obtain from the bubble pressure tensiometer is the surface tension measured at the maximum bubble pressure corresponding with a bubbling frequency or bubbling time. Thus, for a bubbling frequency or bubbling time, the measured surface tension was somewhat higher than the surface tension prevailing at the point of bubble separation, with the extent depending on the tension relaxation behavior of a solution and the bubble frequency. Results and Discussion Constant-Flow Condition. For bubble formation under a constant-flow condition, the gas flow rate through an orifice is constant throughout the whole period. The bubbling cycle in pure water is shown in Figure 3 for example. When the photographs were examined, the volumes of growing bubbles were measured and are shown in Figure 4 for a few cycles. The orifice used under the experimental conditions in Figure 4, that is, Di ) 0.915 mm and orifice length L ) 1 m, satisfies the constant-flow condition proposed by Takahashi and Miyahara,8 L/Di4 ) 1.43 × 1012 > 1 × 1012 m-3. Although a small “waiting period” existed for each cycle, the bubbles were found to be formed regularly and the bubble volumes increased linearly. The constantflow conditions were therefore reasonably supposed to be satisfied by the experimental conditions. It is noteworthy that the results of the bubble volume at detachment and the bubble period as shown in Figure 4 agree well with those determined independently by the measurements of the gas flow rate and bubble frequency (deviation within (5%). Equilibrium and Dynamic Surface Tensions of Surfactant Solutions. Because the concentration is usually very low, the addition of surfactants to water causes no significant change in physical properties except that the surface tension is depressed considerably. Figure 5 shows the equilibrium surface tensions for aqueous solutions of SDS and Triton X-100. This figure shows how SDS and Triton X-100 lower the surface tension of water quite appreciably even at low concentrations. The discontinuity in the σ-composition

Figure 8. Comparison of bubble volumes obtained experimentally and theoretically by the Ruff model.7

curve is identified with the critical micelle concentration (cmc), at which micellization begins. Because the micelles themselves are not surface-active, the surface tension remains approximately constant beyond the cmc. This degree of sharpness, however, is observed only for highly purified compounds. If impurities are present, the curve will display a slight dip at this point. The maximum bubble pressure method of this work measures the dynamic surface tension for bubble frequency between 0.5 and 10 s-1. Figure 6a shows the frequency effect on the dynamic surface tension of the SDS solution. This effect is more evident if the experimental data in Figure 6a are expressed as a plot of surface tension versus average bubble lifetime. As shown in Figure 6b, relaxation in surface tension due to adsorption of surfactant molecules at the gas-liquid interface is clearly observed. The surface tension of a surfactant solution will relax from that of water and finally reaches the equilibrium value. It is desirable that a proper relationship between the surface tension and time be developed. If eq 4 is expressed in the form

σ ) σw -

atc 1 + btc

(5)

the constants a, b, and c may be determined by using dynamic surface tension data through a regression analysis. This approach is also applicable to aqueous solutions of Triton X-100 as shown in Figure 7. It is obvious that the dynamic surface tension depends on the species and concentration of the surfactant. In the range of concentrations studied, however, the depression of surface tension can hardly approach the equilibrium one even at the available longest time t ) 2 s. For shorter times the surface tension at the interface is surely far from equilibrium. From Figures 6 and 7, it is clear that the empirical equation (5) with appropriate parameters is most applicable for the dynamic surface tensions of SDS and Triton X-100 solutions in the time scales of 0.1-2 s. In the following section, eq 5 will be applied to predict the surface tensions in the time scales of 0.04-0.4 s.

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1477

Figure 9. Comparison of bubble volumes obtained experimentally and theoretically by the Ruff model7 with those computed by the modified Ruff model.




However, the potential error caused by the application of eq 5 to the time scale of less than 0.1 s would not be significant, especially for solutions with slower surface tension relaxation behavior, because eq 5 reduces to a surface tension value of pure water as t f 0, as expected. Estimation of the Bubble Volume. The applicability of the Ruff7 model is examined by pure water data of this work and highly viscous liquid data of Terasaka and Tsuge.9 As shown in Figure 8 while the bubble volume in pure water is well estimated by this model, those in glycerol and 88 wt % glycerol solution are underestimated by it with average errors of 16% and 10%, respectively. The agreement between experimental data and predictions is satisfactory in view of the wide range of viscosity considered. As is also shown in Figure 8, the reproducibility of pure water data is quite good.

Figure 9 shows the relation between the bubble volume at detachment and the gas flow rate for a 0.4 mM SDS solution. The experimental results obviously deviate from those predicted by considering the equilibrium surface tension (σ ) 57 mN/m). On the other hand, the modified Ruff model gives a much better fit than the original model. This implies that the surface relaxation phenomenon plays an important role in the bubbling process. For an accurate prediction of the bubble formation in a surfactant solution, the dynamic surface tension rather than equilibrium data should be taken into consideration. Similar results were obtained for a 2 mM SDS solution, as shown in Figure 10. Figure 11 shows the results for 7 and 10 mM SDS solutions. As revealed by Figure 5 these concentrations are well beyond the cmc of SDS. For the bubble formation in a micellar solution, dissociation of micelles

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Conclusions


Experiments of the bubble formation under constantflow conditions in aqueous solutions of SDS and Triton X-100 were conducted, and the bubble volume at detachment can be determined. Equilibrium and dynamic surface tensions of the surfactant solutions were also measured. A simple modified two-stage spherical model was proposed by considering the variation of the surface tension with time during bubble formation. Two conclusions can be drawn as follows. 1. The two-stage spherical model of Ruff,7 which tacitly deals with bubble formation in pure liquids, is found to underestimate the bubble volumes at detachment in aqueous solutions of SDS and Triton X-100. 2. The modified Ruff model, by which the dynamic surface tension at the bubble surface plays a role, predicts the results with satisfactory agreement. This implies that the surface relaxation phenomenon is important for the bubble formation in surfactant solutions. For accurate predictions of the bubble formation in surfactant solutions, dynamic surface tension rather than equilibrium data should be taken into consideration. Acknowledgment This study was partially supported by the National Science Council of the Republic of China through Grant NSC 88-2214-E-006-023. Y.-M.Y. thanks Mr. Hsin-Ku Huang for his assistance in preparing the manuscript. Nomenclature a, b, c ) constants in eq 5 D ) bubble diameter, m Di ) nozzle inner diameter, m g ) gravitational acceleration, m/s2 L ) nozzle length, m Qg ) gas flow rate, m3/s t ) time, s td ) time during the detachment stage, s Vb ) bubble volume, m3 Ve ) bubble volume at the end of the expansion stage, m3 Greek Letters


µl ) liquid viscosity, mPa s Fl ) liquid density, kg/m3 σ ) surface tension, mN/m σw ) surface tension of water, mN/m

Literature Cited is involved, and much more complicated surface relaxation phenomena may occur. The data in Figure 11 show that the modified Ruff model overestimates 13% and 12% for 7 and 10 mM SDS solutions, respectively. The original Ruff model, on the other hand, underestimates 19% and 16% respectively for 7 and 10 mM SDS solutions. Meanwhile, the original Ruff model is incapable of distinguishing 7 mM from 10 mM because their equilibrium surface tensions are exactly the same. Figures 12-14 show the relation between the bubble volume at detachment and the gas flow rate for Triton X-100 solutions at concentrations of 0.007, 0.2, and 1 mM, correspondingly. The same conclusions can be drawn from these results. It should be noted that this is also true for the situation of a capillary orifice with Di ) 0.554 mm.

(1) Kumar, R.; Kuloor, N. R. The Formation of Bubbles and Drops. In Advances in Chemical Engineering; Drew, T. B., Hoopes, J. W., Jr., Eds.; Academic Press: New York, 1970; Vol. 8. (2) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles; Academic Press: New York, 1978. (3) Tsuge, H. Hydrodynamics of Bubble Formation from Submerged Orifices. In Encyclopedia of Fluid Mechanics; Cheremisinoff, N. P., Ed.; Gulf Publishing Co.: New York, 1986; Vol. 3. (4) Davidson, J. F.; Schuler, B. O. G. Bubble Formation at an Orifice in a Viscous Liquid. Trans. Inst. Chem. Eng. 1960, 38, 144. (5) Davidson, J. F.; Schuler, B. O. G. Bubble Formation at an Orifice in an Inviscid Liquid. Trans. Inst. Chem. Eng. 1960, 38, 335. (6) Ramakrishnan, S.; Kumar, R.; Kullor, N. R. Studies in Bubble FormationsI. Bubble Formation under Constant Flow Conditions. Chem. Eng. Sci. 1969, 24, 731. (7) Ruff, K. Formation of Gas Bubbles at Nozzles with Constant Throughput. Chem. Ing. Tech. 1972, 44, 1360.

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1479 (8) Takahashi, T.; Miyahara, T. Bubble Volume Formed at Submerged Nozzles: Constant Gas Flow Condition. Kagaku Kogaku Ronbunshu 1976, 2, 138. (9) Terasaka, K.; Tsuge, H. Bubble Formation under Constant Flow Conditions. Chem. Eng. Sci. 1993, 48, 3417. (10) Defay, R.; Petre, G. Dynamic Surface Tension. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1971; Vol. 3.

Interfacial Tensions of Surfactant and Polymer Solutions. Adv. Colloid Interface Sci. 1994, 49, 249. (13) Chang, C. H.; Franses, E. I. Adsorption Dynamics of Surfactants at the Air/Water Interfaces: A Critical Review of Mathematical Models, Data, and Mechanisms. Colloids Surf. A 1995, 100, 1. (14) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; Wiley: New York, 1990.

(11) Van den Tempel, M.; Lucassen-Reynders, E. H. Relaxation Processes at Fluid Interfaces. Adv. Colloid Interface Sci. 1983, 18, 281.

Received for review August 3, 1999 Revised manuscript received January 31, 2000 Accepted February 1, 2000

(12) Miller, R.; Joos, P.; Fainerman, V. B. Dynamic Surface and

IE990578L

Bubble Formation at an Orifice in Surfactant Solutions under Constant

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