Fluid dynamics of bubbles in liquid

Fluid dynamics of bubbles in liquid

C.M. SCHEID¹, F.P. PUGET² , M.R.T. HALASZ² and G. MASSARANI2^*

¹DTQ/IT/UFRRJ, ²PEQ/COPPE/UFRJ, Caixa Postal 68502, CEP 21945-970,

Rio de Janeiro - RJ, Brazil, phone: [phone], fax: [phone],

E-mail: [e-mail]

(Received: July 23, 1999; Accepted: August 26, 1999)

Abstract - Results gathered from the literature on the dynamics of bubbles in liquid are correlated by means of a formulation traditionally employed to describe the dynamics of isometric solid particles. It is assumed that the shape of the bubble depends, by means of the Eotvos number, on its diameter and on the gas-liquid surface tension. The analysis reported herein includes the dynamics of the isolated bubble along with wall and concentration effects. However, the effects of gas circulation in the bubble, which result in terminal velocities higher than those of a rigid sphere, are not being considered. A limited number of experimental points are obtained employing a modified version of the Mariotte flask which permits the precise measure of bubble volume. A classic bubble column is also employed in order to measure gas holdup in the continuous phase. Experiments were carried out employing air, with distilled water, potable water, water with variable amounts of surfactant and glycerin as the liquid phase.

Keywords: bubbles, fluid dynamics, terminal velocity, concentration effects.

INTRODUCTION

Some characteristic aspects of the dynamics of air bubbles in water, extracted from the work of Gaudin (1957) about flotation, are presented in Figure 1. The figure shows that the bubble behaves like a rigid sphere when its diameter is smaller than 0.5 mm and like a rigid spherical cap (with a sphericity of approximately 0.57) when its diameter is greater than 15 mm. Between these limits, the greater the diameter of the bubble the greater the internal circulation of gas, which affects the bubble-water boundary conditions and induces a decrease in the drag force on the bubble. As a consequence, the terminal velocity of the bubble is high than that of the equivalent rigid sphere. Beyond Dp=1.5mm, internal circulation decreases, the bubble is deformed and the drag force increases until the bubble behaves like a rigid spherical cap (Astarita and Apuzzo, 1965; Clift et al., 1978).

Figure 1:
Terminal rise velocity of air bubble in water: (A) The particle is a rigid sphere, (B) The bubble in distilled water, (C) The bubble in aqueous solution of terpeniol (3.7 10^-3 kg/m³), (D) The bubble in aqueous solution of terpeniol (2.2 10^-2 kg/m³), (E) The bubble is a spherical cap.

Figure 1 also shows that the presence of surfactant in water accelerates deformation and reduction of internal circulation. Qualitatively, the water is said to be "contaminated" when the bubble shows the same behavior as that of a rigid particle which is deformed as its diameter increases, changing from a sphere to a spherical cap (Grace et al., 1976).

It is well known that water presents peculiar physicochemical properties. However, bubbles are expected to present to a certain extent the effects of gas circulation under different conditions, as in diluted solutions (Astarita and Apuzzo, 1965).

This contribution focuses on the study of the dynamics of bubbles in liquid for systems in which bubbles may be treated as solid particles. Regarding bubbles in extended liquid media, unification by sphericity of the correlation developed by Grace et al. (1976) for fluid systems and the correlation for solid particles proposed by Massarani (1997) was studied. Following this same strategy, wall and concentration effects in solid particle dynamics (Almeida, 1995; Richardson and Zaki, 1954) were evaluated based on experimental evidence presented by Govier and Aziz (1972).

The formulation proposed in this survey to estimate the terminal velocity of bubbles in liquid is evaluated using experimental data previously reported by Peebles and Garber (1953) along with experimental results obtained from a modified Mariotte flask and a classic porous plate bubble column.

FORMULATION

Grace et al. (1976) established the following empirical correlation which permits estimating the terminal velocity of an isolated bubble in liquid (U¥ ) using the Reynolds number:

(1)

where

where

The correlation is valid for Eotvos number Eo<60 and Morton number M<10^-3. In Equation (1), D_p is the equivalent sphere diameter, r_L and m_L are density and viscosity of the liquid (continuous phase), s is the gas-liquid surface tension and m_w = 0.9 10^-3 Pa.s is constant.

If the bubble is considered to be an isometric solid particle, its terminal velocity may be calculated with the following empirical correlation (Massarani, 1997):

(2)

where

where f is the sphericity, a shape factor of the bubble defined as

The equivalence of empirical correlations (1) and (2) may be established by sphericity, assuming that the shape of the bubble depends only on the Eotvos number (Grace et al., 1976; Karamanev, 1994). The procedure involves the computation of the sphericity value for which both correlations lead to the same terminal velocity U¥ . With the aid of the Statistica software,

(3)

where

As both correlations discussed are empirical, the second correlation (Equations 2 and 3, Figure 2) will be evaluated opportunely by direct comparison with experimental data. Figure 2 refers to the air-water system at 298 K and includes experimental data for contaminated water reported by Clift et al. (1978).

Figure 2:
Air-water system at 298 K: estimation of terminal rise velocity from
correlations valid for isometric solid particles (Equations 2 and 3).

The terminal rise velocity of bubble U is affected by the confinement of the liquid. In this contribution the correlation proposed experimentally by Almeida (1995) for the displacement of solid particles along the main axe of a tube with diameter D_t will be evaluated.

(4)

where

,

and

The rise velocity of the bubble in the bubbling process is also substantially affected by the presence of adjacent bubbles. In this contribution the classic correlation of Richardson and Zaki (1954), valid for rounded solid particles with a narrow granulometric distribution, will be evaluated. This correlation, when restricted to the stagnant continuous phase, reduces to the following (Massarani, 1997):

(5)

where

Here Q_G is the gas flow, A the cross sectional area of the bubble column and e_L the liquid volumetric fraction.

MATERIALS AND METHODS

The formulation proposed in this work is evaluated employing experimental data obtained by Peebles and Garber (1953) for some aqueous and nonaqueous systems, along with experimental results obtained at LSP/COPPE for several systems, described in Table 1. In all experiments air bubbles were employed.

Table 1: Physicochemical properties of the air-liquid systems studied at LSP/COPPE.

Liquid	rL (103 kg/m3)	mL (10-3 Pa.s)	s (10-3 N/m)	Analysis
Distilled Water	1.00	0.947	71.0	Bubbling
Potable water	1.00	0.947	63.9	Wall effect
Water + lauryl (2ppm)	1.00	0.947	62.6	Wall effect Bubbling
Water + lauryl (5ppm)	1.00	0.947	60.8	Wall effect Bubbling
Water + lauryl (15ppm)	1.00	0.947	46.0	Wall effect Bubbling
Glycerin	1.25	370	58.8	Wall effect

The experiments were carried out in a set of Mariotte flasks with inner diameters of 11, 20, 40.3 and 50.8 mm and a height of 1.2 m and in a classic bubble column with a porous plate with an inner diameter of 70 mm and a height of 1m. The equipment employed is sketched in Figure 3, which shows that the bubbles originated in a capillary tube in the Mariotte flasks. It can also be noticed that both systems present a visualization section with parallel plates for photographic evaluation of the shape and dimension of the bubbles.

Figure 3:
Mariotte flask and bubble column with porous plate

Mariotte flasks permit the precise determination of the volume of the bubble from liquid displacement. Additionally the terminal velocity of the bubble can be calculated by displacement and time measures during its rising trajectory. As shown in the remarkable work of Davidson and Schüler (1960), bubble volume is not very sensitive to capillary tube diameter if the experiments are carried out with constant gas flow. After some preliminary experiments, a capillary tube with an inner diameter of 1 mm was adopted, resulting in bubbles with a diameter of approximately 4 mm in a liquid with surface/interfacial tension of s > 60 10^-3 N/m.

The volumetric fraction occupied by the gas phase during bubbling, e_G, the parameter which characterizes bubble concentration, may be measured by the reduction of the mixture, volume when air feed is interrupted.

(6)

where V_T is the total volume of the expanded system and V_L is the volume of stagnant liquid.

In the experiments carried out at LSP/COPPE, characterized by Eo<3, the bubble photographs show round and asymmetric flat shapes, suggesting ellipses with an eccentricity of approximately 0.5. The geometric shape of the bubble remains an open question. However, considering the bubble to be either an oblate or a prolate spheroid its sphericity is close to 1, a value which characterizes the spherical shape. This evidence confirms the estimate in Equation 3.

RESULTS AND CONCLUSIONS

The formulation used to describe the dynamics of isometric solid particles (Massarani, 1997), Equation 2, is extended in order to cope with the dynamics of bubbles in liquid. In this context, Equation 3 permits estimating the sphericity of the bubble from the Grace et al. empirical correlation (1976), Equation 1. Figure 4 permits the evaluation of the proposed formulation using experimental data obtained at LSP/COPPE (15 points) and those reported by Peebles and Garber (1953) (24 points), for aqueous and nonaqueous systems, in accordance with the following conditions:

, ,

, ,

Figure 4:
Comparison between experimental and estimated values (Equations 2 and 3) for terminal rise velocity of the air bubble in liquid.

It has been observed that the proposed correlation permits terminal rise velocity estimation with an average error of 10%.

Concerning the Richardson and Zaki correlation (1954), Equation 5, the experiments on air bubbling in stagnant liquid (Table 1) resulted in the terminal rise velocity values and the corresponding bubble diameters shown in Table 2, in the range of liquid volumetric fraction 0.82 < e_L < 0.95 (Figure 5).

Table 2: Evaluation of Richardson and Zaki correlation (1954) for air bubbling in liquid

Air-liquid surface tension, s (10-3 N/m)	Bubble terminal rise velocity, U¥ (10-2 m/s)	Diameter of the spherical bubble, Dp (10-3 m)
71.0	33.1	3.6
62.6	32.3	3.4
60.8	30.4	3.0
45.4	28.3	2.6

Figure 5:
Bubbling of air in stagnant liquid, s = 60.8 10^-3 N/m: determination of terminal rise velocity of isolated bubble using Equation 5

Photographic visualization shows that the bubbles are almost spherical and confirms the order of magnitude of the diameters calculated (shown in Table 2).

NOMENCLATURE

A cross section area of the bubble column [L²]

D_t tube diameter [L]

Eo Eotvos number [-]

M Morton number [-]

Q_G volumetric flow of gas [L³/q ]

Re Reynolds number [-]

R_p volumetric radius of bubble [L]

U¥ terminal rise velocity [L/T]

V_L volume of stagnant liquid [L³]

V_T total volume of expanded system [L³]

e volumetric fraction of liquid

f sphericity [-]

m _l dynamic viscosity of liquid phase [M/Lq]

m _w dynamic viscosity of water [M/Lq]

r_l liquid density [M/L³]

s surface tension [ML/q²]

ACKNOWLEDGEMENTS

The authors are grateful for the support received from CNEN and CNPq.

• Almeida, O.P., Estudo do Efeito de Fronteiras Rígidas Sobre a Velocidade Terminal de Partículas Isométricas, M.Sc. Thesis, PEQ/COPPE/UFRJ (1995)
• Astarita, G. and Apuzzo, G., Motion of Gas Bubbles in Non-Newtonian Liquids, A.I.Ch.E. Journal, 11, No. 5, p. 815 (1965)
• Clift, R., Grace, J.R., and Weber, M.E., Bubbles, Drops, and Particles, Academic Press, New York (1978)
• Davidson, J.F. and Schüler, B.O., Bubbles Formation at an Orifice in a Viscous Liquid, Trans. Instn. Chem. Engrs., 38, p. 144 (1960)
• Gaudin, A.M., Flotation, McGraw-Hill, New York, 2^nd Edition (1957)
• Govier, G.W. and Aziz, K., The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York (1972)
• Grace, J.R., Wairegi, T. and Nguyen, T.H., Shapes and Velocities of Single Drops and Bubbles Moving Freely Through Immiscible Liquids, Trans. Instn. Chem. Engrs., 54, p. 167 (1976)
• Karamanev, D.G., Rise of Gas Bubbles in Quiescent Liquids, A.I.Ch.E. Journal, 40, No. 8, p. 1418 (1994)
• Massarani, G., Fluidodinâmica em Sistemas Particulados, Editora UFRJ, Rio de Janeiro (1997)
• Peebles, F.N. and Garber, H.J., Studies on the Motion of Gas Bubbles in Liquids, Chem. Engr. Progress, 49, No. 2, p. 88 (1953)
• Richardson, J.F. and Zaki, W.N., Sedimentation and Fluidization: Part I, Trans. Instn. Chem. Engrs, vol. 32, p. 35 (1954)

*

To whom correspondence should be adressed

Publication Dates

• Publication in this collection
  01 Mar 2000
• Date of issue
  Dec 1999

History

• Accepted
  26 Aug 1999
• Received
  23 July 1999