Brazilian Journal of Chemical Engineering - Fluid Dynamics of Bubbles in Liquid
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Brazilian Journal of Chemical EngineeringPrint version ISSN 0104-6632
Braz. J. Chem. Eng. vol.16 n.4 São Paulo Dec. 1999
http://dx.doi.org/10.1590/S0104-66321999000400003
Fluid dynamics of bubbles in liquid
C.M. SCHEID1, F.P. PUGET
2 , M.R.T. HALASZ
2 and G.
MASSARANI2*
1DTQ/IT/UFRRJ,
2PEQ/COPPE/UFRJ, Caixa Postal 68502, CEP
21945-970,
Rio de Janeiro - RJ, Brazil, phone: +55 (21) 590-2241, fax:
+55 (21) 590-7135,
E-mail: [email protected]
(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.
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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).
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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), Dp 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 mw = 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)
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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).
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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 Dt 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 QG 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.
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LiquidrL
(103 kg/m
3)
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.
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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, eG, the parameter which
characterizes bubble concentration, may be measured by the reduction of the mixture,
volume when air feed is interrupted.
(6)
where VT 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
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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 < eL < 0.95 (Figure 5).
Table 2: Evaluation of Richardson and Zaki correlation (1954) for air bubbling in
liquid
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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
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A cross section area of the bubble column [L2]
Dt tube diameter [L]
Eo Eotvos number [-]
M Morton number [-]
QG volumetric flow of gas [L
3/q ]
Re Reynolds number [-]
Rp volumetric radius of bubble [L]
U¥ terminal rise velocity [L/T]
VL volume of stagnant liquid [L
3]
VT total volume of expanded system [L
3]
e volumetric fraction of liquid
f sphericity [-]
m l dynamic viscosity of liquid phase [M/Lq]
m w dynamic viscosity of water [M/Lq]
rl liquid density [M/L
3]
s surface tension [ML/q2]
ACKNOWLEDGEMENTS
The authors are grateful for the support received from CNEN and CNPq.
REFERENCES
Almeida, O.P., Estudo do Efeito de Fronteiras Rígidas Sobre a Velocidade Terminal de
Partículas Isométricas, M.Sc. Thesis, PEQ/COPPE/UFRJ (1995) [ Links ]
Astarita, G. and Apuzzo, G., Motion of Gas Bubbles in Non-Newtonian Liquids, A.I.Ch.E.
Journal, 11, No. 5, p. 815 (1965) [ Links ]
Clift, R., Grace, J.R., and Weber, M.E., Bubbles, Drops, and Particles, Academic Press, New
York (1978) [ Links ]
Davidson, J.F. and Schüler, B.O., Bubbles Formation at an Orifice in a Viscous Liquid, Trans.
Instn. Chem. Engrs., 38, p. 144 (1960) [ Links ]
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Gaudin, A.M., Flotation, McGraw-Hill, New York, 2nd
Edition (1957) [ Links ]
Govier, G.W. and Aziz, K., The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold,
New York (1972) [ Links ]
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) [ Links ]
Karamanev, D.G., Rise of Gas Bubbles in Quiescent Liquids, A.I.Ch.E. Journal, 40, No. 8, p.
1418 (1994) [ Links ]
Massarani, G., Fluidodinâmica em Sistemas Particulados, Editora UFRJ, Rio de Janeiro
(1997) [ Links ]
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) [ Links ]
Richardson, J.F. and Zaki, W.N., Sedimentation and Fluidization: Part I, Trans. Instn. Chem.
Engrs, vol. 32, p. 35 (1954) [ Links ]
* To whom correspondence should be adressed
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