Comparison of Theoretical and Experimental Characteristics of

Ind. Eng. Chem. Res. 2002, 41, 6235-6245

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Comparison of Theoretical and Experimental Characteristics of Oscillating Bubbles F. J. Montes,† M. A. Gala´ n,† and R. L. Cerro* Chemical and Materials Engineering Department, University of Alabama in Huntsville, Huntsville, Alabama 35899

Mass transfer in typical areation devices such as sieve plate reactors and bubble columns takes place predominantly within the oscillating bubble regime. Mass-transfer rates from oscillating bubbles can be several times larger than mass-transfer rates from rigid bubbles. The hydrodynamics of oscillating bubbles has been described by Tsamopoulos and Brown (J. Fluid Mech. 1983, 127, 519-537) and later by Feng (SIAM J. Appl. Math. 1992, 52 (1), 1-14), using the domain perturbation technique first developed by Joseph (Arch. Ration. Mech. Anal. 1973, 51, 295-303). A high-speed video imaging system was used to determine the shapes of oscillating bubbles for a wide range of fluids and operating conditions. Air bubbles introduced through an orifice were recorded while rising inside a liquid. When the liquid was water, a hydrogen bubble technique was used to determine pathlines around the rising bubble. The experimental data on shapes and pathlines were compared with bubble shapes and pathlines created using theoretical models. 1. Introduction Many biological processes use common aeration devices such as stirred-tank reactors, bubble columns, and sieve plate reactor towers to provide the necessary oxygen. Bioprocesses involving cell cultures can be sensitive to high hydrodynamic stresses,1,2 leading to an important decrease in cellular viability. Bubble columns and sieve plate towers are the most adequate aeration devices for these types of cultures. Frequently, sieve plate towers perform better than bubble columns because of their lower power consumption per unit volume and larger gas holdup, leading to superior gasliquid contact. Most of the bubbles generated in a sieve plate are oscillating bubbles; that is, the bubble shape changes with time as the bubble travels along the column. Oscillating bubbles result in much larger mass-transfer rates than constant-shape bubbles.3 Thus, characterization of oscillating bubbles is of practical importance in the determination of gas-liquid mass-transfer rates. Oscillations of drops and bubbles have been the subject of extensive research and modeling.4-8 Oscillations are described9 as the result of wake shedding, with the onset of oscillations coinciding with the onset of vortex shedding from the wake.10 Oscillating behavior has been also reported as being a consequence of acoustic stresses, interactions with other bubbles, etc.8 From a hydrodynamic point of view, oscillating regimes of gas bubbles have been experimentally characterized on the basis of the Weber and Morton numbers11 * To whom correspondence should be sent. Phone: (256) 824-7313. Fax: (256) 824-6839. E-mail: [email protected]. † Departamento de Ingenierı´a Quı´mica y Textil, Universidad de Salamanca, 37008 Salamanca, Spain. E-mail addresses: [email protected], [email protected].

NWe )

DbUb2F σ

NMo )

gµ4 Fσ3

Because terminal velocity is also a function of bubble diameter, once the gas-liquid system under study is defined, the presence of oscillating bubbles depends on the average bubble diameter. Thus, the choice of the size of the orifice where the bubble is generated is critical in the determination of bubble regimes. Mean bubble diameters of 3 mm or less usually lead to rigid bubbles. On the other end, bubble diameters above 2 cm are characteristic of spherical-cap bubbles. The computation of theoretical velocity profiles surrounding an oscillating bubble is the first and crucial step in the development of accurate mass-transfer correlations. Once the expression for the convective term of the mass conservation equation is known, the continuity equation can be solve to yield, with the appropriate boundary conditions, a correlation for the masstransfer coefficient. In a previous paper,11 the convective term was calculated using the model proposed by Tsamopoulus and Brown7 for nonlinear oscillating bubbles. The goal of the present paper is to comparesin terms of bubble shapes, oscillation modes, and fluid fieldss the results of a one-parameter expansion model for the velocity potential proposed by Tsamopoulos and Brown7 with the two-parameter expansion developed by Feng.8 Theoretical results can be checked against our experimental results obtained using high-speed video combined with specific visualization devices to determine the main oscillation modes involved in the movement of an oscillating bubble rising through a stagnant liquid. 2. Experiments A sketch of the experimental setup is shown in Figure 1. A rectangular cell (10 × 10 × 25 cm) with flat glass

10.1021/ie011009s CCC: $22.00 © 2002 American Chemical Society Published on Web 03/30/2002

6236

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002

Figure 1. Experimental setup used for the observations of bubble shape and velocity profiles.

walls had a small, hollow metal cylinder centered at the bottom. The cylinder had a stainless steel cover plate with a small (typically 1 mm in diameter) orifice connected to an air supply. The flat glass walls allowed an undistorted view of the bubbles from the side using a high-speed video-imaging system. A gas chamber was placed between the metering valve of the gas pipe and the recipient’s cylinder to act as a buffer for fluctuations in the gas flow rate. A pressure transducer with a digital readout monitored the pressure inside the gas chamber. A rotameter connected to an inverted funnel measured the gas flow rate. A Kodak Ektapro high-speed video system was used to record up to 1000 frames per second. The frames captured by the video system were recorded with a Panasonic AG-7300 video cassette recorder and then captured by a TARGA-1000 board. The liquids used in the experiments were water and silicon oil (dimethyl polysiloxane) with viscosities of 5, 20, and 100 cSt. Breathing-quality air from a cylinder was used to generate the bubbles at three flow rates (40, 100, and 200 mL/min). When the experiments were done with water, hydrogen bubbles were used to visualize the flow patterns around the bubbles. A discontinuous DC power supply connected to a thin tungsten wire and to a carbon electrode provided the electrical current for an electrolytic reaction where small hydrogen bubbles were generated on the surface of the tungsten wire. The rising terminal velocity of these tiny (about 10 µm in diameter) hydrogen bubbles can be neglected against

the terminal velocity of the air bubble. Thus, we can assume that the movement of the hydrogen bubbles is due only to the movement of the air bubble. The velocity patterns of the hydrogen bubbles are representative of the velocity patterns of the fluid. Figure 2a and b shows consecutive frames containing the shapes of oscillating air bubbles rising in water and 5-cSt silicon oil, respectively . Figure 2a demonstrates the visualization of flow patterns around the oscillating bubbles using the hydrogen bubble technique. Periodic vortex formation and shedding behind the bubble coincides with the development of oscillations. Bubbles of constant shape were generated for different air flow rates with silicon oil of viscosities 20 and 100 cSt. The results of those experiments are shown in Figure 3a and b. The different hydrodynamic behaviors of the air bubbles shown in Figures 2 (oscillating) and 3 (rigid) are mapped in Figure 4 as a function of the Morton and Weber numbers. These dimensionless numbers include the main parameters involved in bubble formation, oscillation, and rising. The Morton number relates physical properties only and includes density, viscosity, surface tension, and gravity. The Weber number is the ratio of inertial to capillary forces. For small Morton and Weber numbers, capillary forces dominate over viscous and gravitational forces, and bubbles behave as rigid spheroids. For small Morton numbers but large Weber numbers, inertial forces overcome capillarity, and the

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002 6237

Figure 2. Consecutive frames of oscillating air bubbles rising in (a) water and (b) 5-cSt silicon oil. Figure 2a shows the flow patterns around the oscillating bubbles using the hydrogen bubble technique.

bubbles show sustained oscillations. For large Morton numbers, the bubbles behave as rigid ellipsoids. 3. Theory The main complexity of the problem of an oscillating bubble rising in a stagnant fluid is the fact that bubble boundaries, that is, the surfaces between the gas and liquid phases, are time-dependent. A powerful approach to this type of problem is the method of domain perturbations first developed by Joseph.12,13 Tsamopoulos and Brown,7 and later Feng,8 applied the method of domain perturbation to analyze oscillating bubbles. The method of domain perturbations maps the variable domain into a fixed domain substituting the radial coordinate (r) of the moving domain by a new coordinate, η. A dimensionless shape function, F(θ,φ,t) defines the

radial coordinate of the bubble in a spherical coordinate system, as shown in Figure 5. θ and φ are the meridional and azimuthal angles, respectively, and t is the time. R is the radius of the unperturbed spherical bubble. The functional relationships between moving and fixed domain variables and the domain boundaries for the liquid phase are

r ) ηF(θ,φ,t) F(θ,φ,t) e r e∞

(1)

1 eη e ∞ Under these definitions, the bubble will appear to be a rigid bubble in the domain (r, θ, φ). The following step is to formulate a perturbation expansion of the shape

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Figure 3. Bubbles of constant shape generated for different air flow rates using (a) 20- and (b) 100-cSt silicon oil.

Figure 4. Map of different types of rising bubbles as a function of Morton and Weber numbers. The experimental air bubbles obtained in this work are also shown.

function and the velocity field using a small perturbation parameter. The solution of the different order problems contains the rigid bubble and the potential flow around it as the zeroth-order solution. Bubble deformation is produced by two different factors: oscillation and rising. Tsamopoulos and Brown7 represented bubble oscillations as a series expansion of a small parameter. Feng8 discussed the inadequacy of using a one-parameter series expansion when two different physical processes induce bubble deformation. Thus, dynamical pressure acting on the bubble cap shape as a result of the rising phenomenon must be taken into account separately from oscillation. If only one parameter is used, an artificial constraint is introduced into the problem, forcing the existence of an ordering relation between two independent physical effects (oscillation and rising). This analysis is the basic

Figure 5. Spherical coordinate system for the perturbed and unperturbed domains.

difference between Feng’s8 formulation and the previous work of Tsamopoulus and Brown.7 The equations resulting from Tsamopoulus and Brown’s7 one-parameter expansion and Feng’s8 twoparameter expansion are reproduced here using a single notation. For the reader unfamiliar with these works, this presentation will serve as an introduction to this type of analysis. For the reader familiar with the work of these authors, it will provide a consistent framework for comparison with experimental data.


Assume that the center of our coordinate system is attached to the bubble. An inviscid fluid surrounds the bubble moving upward in irrotational, incompressible motion. The velocity of the fluid far from the bubble is constant and equal to U∞. The main dimensionless variables and their dimensional counterparts are

( ) ( ) ( )

r ) r′/R ω ) ω′ U∞ ) U′∞

1/2

FR σ

1/2

σ FR3

NWe 2

)

( )

Φ ) Φ′ 1/2

F σR

1/2

(2)

R σ

p ) p′

where Φ is the velocity potential; ω is the oscillation frequency; p is the pressure; and F and σ are the density and surface tension of the liquid, respectively. The dimensional terminal velocity can be computed, following Mendelson,14 as a function of bubble diameter. To find a solution for the velocity potential Φ, the shape function F, and the oscillation frequency ω, one must solve Laplace’s equation subject to the balance of capillary and dynamic forces; to a kinematic boundary condition; and, because this is an exterior problem, to a far-field condition. Tsamopoulus and Brown7 considered a constant-volume, two-dimensional axisymmetric bubble. Feng8 used a polytropic pressure-volume relationship and allowed for the possibility of 2D and 3D bubbles. One-Parameter Asymptotic Expansion (Tsamopoulos and Brown7). The one-parameter expansion leads to the following expressions for the velocity potential and the axisymmetric shape function

Φ(r,θ,t;) )

F(θ,t;) )

∞

k

∞

k

Φ[k](η,θ,t) ∑ k! k)0

∑

(3) F(k)(θ,t)

k)0k!

where is the oscillation amplitude and t is the time. The Φ[k] terms are the kth-order total derivatives of Φ with respect to evaluated at ) 0. These terms can be expressed as the sum of a contribution calculated on the spherical domain (where Φ(k) represents the kthorder partial derivatives of Φ with respect to evaluated at ) 0) and some other terms that account for the deformation of the domain and include the corresponding expansion terms of the shape function. The first two terms of the expansion for Φ are

Φ

|

|

|

Φ[1] )

∂Φ ∂Φ(0) ∂Φ(0) + F(1) ) Φ(1) + F(1) ∂ )0 ∂η )0 ∂η )0

[2]

(2)

)Φ

(2)

+F

|

∂Φ(0) ∂η

(4)

+ )0

F(1)

2

|

∂2Φ(0) ∂η2

)0

+ 2F(1)

∂Φ(1) ∂η

The F(k) terms are expressed as functions of Legendre polynomials. An infinite number of Legendre polynomials are solutions of Laplace’s equation in spherical coordinates. Thus, for each order of the expansion, an infinite number of solutions exist for the F(k) functions, one for each mode of oscillation (n). The first correction

term for the shape function is

Fn(1)(θ,tn) ) Pn[cos(θ)] cos(tn)

(5)

The dimensionless time, tn, is a function of the oscillation frequency, ωn, characteristic of each oscillation mode

tn )

( ) σ FR3

(1/2)

ωnt′

(6)

The inclusion of a second-order correction term yields no appreciable difference in the resulting shape function.7 Then, as F(0) ) 1 is the static bubble solution, the shape function for a first-order approximation is

Fn(θ,tn) ) 1 + Pn[cos(θ)] cos(tn)

(7)

The first-order correction to the oscillation frequency is identical to the Rayleigh-Lamb solution 4

F(θ,t;) ) 1 +

∑ CnPn[cos(θ)] cos(tn)

(8)

n)2

In the next section, we will show that experimental bubble shapes are better represented by a linear combination of single-mode shapes resulting from eq 7 4

F(θ,tn;) ) 1 +

∑ CnPn[cos(θ)] cos(tn)

(9)

n)2

Mode n ) 1 represents a rigid bubble rising through a stagnant liquid. Thus, if the coordinate system moves with the bubble, this mode can be obviated. On the other hand, modes for n > 4 are not considered because, as the oscillation frequency increases, the amplitude decreases rapidly with time6 and the oscillation is promptly damped. The coefficients Cn modulate the contribution of each mode to the final shape. In fact, these coefficients modify the generic amplitude , adapting its value to the different modes of oscillation. Prosperetti6 pointed out that this correction properly connects the oscillation amplitude with the mode of oscillation. The dimensionless coefficients, 0 e Cn e 1, are obtained experimentally by comparing theoretical and experimental bubble shapes. Solving Laplace’s equation and transforming the perturbed-domain variable η back to the radial coordinate r (eq 1) allows for the computation of the radial and angular velocities

Uosc(r,θ,tn) ) ∇φ(r,θ,tn)

(∂U ∂r ) 1 ∂U (r,θ,t ) ) ( ) r ∂θ

Uosc r (r,θ,tn) ) Uosc θ

(10)

n

In our experiments, the bubble moves in a coordinate system fixed with respect to the laboratory coordinates. Thus, the velocity of the rising bubble is added to the radial and angular components of the velocity vector to result in a velocity field and pathlines to with the ones shown in Figure 2a. In potential flow theory, it is common to describe the flow around an immersed

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sphere by a uniform stream plus a point doublet.15 The velocity components calculated from this potential and mapped using eq 1 to the surroundings of the bubble with shape given by eq 9 are

[ ( )] [ ( )]

Uris r (r,θ,tn) ) -cos(θ) 1 Uris θ (r,θ,tn)

F3(θ,tn) r3

F3(θ,tn) 1 ) sin(θ) 2 + 2 r3

(11)

Finally, the global velocity field is obtained from the summation of the components given by eqs 10 (oscillation) and 11 (rising). Two-Parameter Asymptotic Expansion (Feng8). The two-parameter asymptotic expansion leads to expressions for the velocity potential and for the 3D shape function of the type

1j2k Φ[j,k](η,θ,φ,t) Φ(r,θ,φ,t;1,2) ) j)0 k)0 j!k! ∞

1j2k F(θ,φ,t;1,2) ) F(j,k)(θ,φ,t) j)0 k)0 j!k! ∞

∞

∞

(12)

where 1 and 2 are the dimensionless velocity given by eq 2 and the oscillation amplitude, respectively. The Φ[j,k] terms are the (j + k)th-order total derivatives of Φ, j times with respect to 1 evaluated at 1 ) 0 and k times with respect to 2 evaluated at 2 ) 0. The Φ[j,k] terms can be expressed as the sum of a contribution calculated on the spherical domain (Φ(j,k)) and other terms accounting for the deformation of the domain, which include the corresponding expansion terms of the shape function

∂Φ(j-1,k) + ∂η ∂Φ(j-2,k) 1 j(j - 1)ηF(2,0) + ... 2 ∂η

∂Φ(j,k-1) 1 ∂Φ(j,k-2) + k(k - 1)ηF(0,2) + ... ∂η 2 ∂η (13)

... + kjη2F(0,1)F(1,0)

∂Φ(j-1,k-1) + ... ∂η2

The partial derivatives Φ(j,k) satisfy Laplace’s equation, and the solution for these derivatives includes spherical harmonics (Ylm) based on Legendre polynomials (Plm)

Φ(j,k)(η,θ,φ,tj,k) ) ∞

l

∑ ∑ l)0 m)-l Ylm(θ,φ) )

[

η(l+1)

Ylm(θ,φ) - δ1jδ0kη cos(θ)

]

(2l + 1) (l - |m|)! 4π (l + |m|)!

1/2

Plm[cos(θ)]eimφ (14)

where δ1j and δ0k are Kronecker deltas, ζ(j,k) lm is a time-

1/2

t′

(15)

l

R(j,k) ∑ ∑ lm (tj,k) Ylm(θ,φ) l)0 m)-l

(16)

The shape coefficients R(j,k) lm are obtained from the normal stress balance across the bubble interface. Solutions for Different Expansion Orders. Cal(j,k) culation of the coefficients R(j,k) lm and ζlm for the increasing approximation orders leads to increasingly accurate expressions for the bubble shape and velocity potential. For comparison, we consider the first two terms for each of the expansion parameters and the first cross term. The solution of order (0,0) is the static, spherical bubble. The solution of order (1,0) is a correction for the velocity potential describing potential flow around a sphere immersed in an infinite fluid. The potential flow theory description is a uniform stream plus a point doublet. The solution of order (2,0), the second correction to the bubble shape, is influenced by the effect of dynamical pressure from the uniform external flow field. From experimental observations, we know that, at this order, the main term affecting bubble shape corresponds to l ) 2 and m ) 0 and is timeindependent. This description is equivalent to the classical linear solution of bubble oscillations about a spherical shape developed by Lamb16 and also obtained by Tsamopoulos and Brown7 2 1/2 ω(0,1) lm ) [(l - 1)(l + 2)]

R(0,1) lm ) cos(ωlmt(0,0))

(17)

∂R(0,1) l lm l + 1 ∂t(0,0)

(

ζ(0,1) lm ) -

ζ(j,k) lm (tj,k)

[ ] σ (FR3)

The partial derivatives of the shape function with respect to 1 and 2 (F(j,k)) are also expressed in terms of the spherical harmonics

F(j,k)(θ,φ,tj,k) )

∑∑

... + kηF(0,1)

tj,k ) 1j2k

∞

∑∑

Φ[j,k] ) Φ(j,k) + jηF(1,0)

dependent coefficient obtained from the kinematic boundary condition at each order of the expansion, and Plm represents the associated Legendre polynomials. The subscript l denotes the main oscillation modes, and m indicates the asymmetric and axisymmetric parts of those modes. To take into account the relation between the oscillation frequency and the amplitude, it is convenient to define the dimensionless time tj,k as a function of the two characteristic expansion parameters. This definition depends also on the order of the expansion (j,k)

)

Solutions of order (0,1) are identical to the results of Tsamopoulos and Brown7 for the nonlinear oscillations of an inviscid bubble. Finally, in solutions of order (1,1), the two-expansion parameters interact, determining the bubble shape and velocity potential. Using the kinematic boundary condition and the normal stress balance


Figure 6. Experimental and theoretical shapes of bubbles oscillating in (a) water and (b) 5-cSt silicon oil with oscillation amplitude equal to 0.2 and linear coefficients C2 ) 0.2, C3 ) 0.5, and C4 ) 0.3. Theoretical bubbles were calculated using the single-parameter expansion.

boundary condition, the following shape coefficients result (0,1) (0,1) ∂Rl-1,m ∂Rl+1,m (1,1) (1,1) Rlm(1,1) ) A-1 (l,m) + A+1 (l,m) ∂t(0,0) ∂t(0,0)

(1,1) A-1 (l,m) ) -

(1,1) (l,m) ) A+1

[

2

2

]

3 (2l - 1) (l - m ) 2 l(3l - 2) (4l2 - 1)

[

ζ(1,1) lm

[

(1,1) B-1 (l,m) ) -

1/2

l g1 2

2

]

(2l + 1) ((l + 1) - m ) 3 2 (l + 2)(3l - 2) (4(l + 1)2 - 1)

and the velocity potential coefficients are

(1,1) (l,m) ) B+1

1/2

(18)

]

(1,1) (0,1) ∂Rl,m -1 ∂Rl,m (1,1) (0,1) ) + + B-1 (l,m)Rl-1,m + ∂t(1,0) (l + 1) ∂t(0,0)

[ [

(1,1) (0,1) (l,m)Rl+1,m B+1

]

2 2 3 (l - m ) 2 (4l2 - 1)

1/2

(19)

]

((l + 1)2 - m2) 3l 2(l + 1) (4(l + 1)2 - 1)

1/2

4. Comparison of Theory with Experimental Data Single-Parameter Expansion. Typical experimental shapes of bubbles rising and oscillating in water and

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Figure 7. Representation of a bubble rising in water and its related pathlines from a static point of view using the single-parameter model. The dimensional time step between the frames is 2 ms.

5-cSt silicon oil are shown in Figure 6a and b, respectively. Also shown above each one of the experimental shapes are theoretical bubble shapes computed using eq 9. For the two fluids used in our experiments, the oscillation amplitude was chosen to be equal to 0.2, and the linear combination parameters were C2 ) 0.2, C3 ) 0.5, and C4 ) 0.3. These are the experimental

values that most closely represent the observed bubble shapes. A single-parameter model represents more closely the initial shape of a bubble, when the bubble has not yet reached terminal velocity. Once a bubble approaches terminal velocity, its shape appears more like an oblate spheroid, with its minor axis along the direction of the uniform velocity field. This is an obvious


Figure 8. Axisymmetric theoretical and experimental shapes of bubbles rising and oscillating in (a) water and (b) 5-cSt silicon oil with oscillation amplitude equal to 0.1. Theoretical bubbles were calculated using the two-parameter model.

consequence of the single-parameter model, as the uniform flow field is not considered to have any influence on the bubble shape. The rise of a bubble with respect to the laboratory coordinate system is sketched in Figure 7. Also shown are some of the pathlines generated from the combination of eqs 10 and 11. These sketches must be compared with the experimental images shown in Figure 2a. The theoretical model reproduces very well the typical recirculation generated at the bottom of any rising bubble with a Reynolds number above 200.9 Two-Parameter Perturbation Solution. Figure 8a and b shows axisymmetric theoretical (top) and experimental (bottom) shapes of rising, oscillating bubbles in water and in 5-cSt silicon oil, respectively. The oscillation amplitude 2 was set equal to 0.1. The influence of the dynamic pressure on the bubble shape is proportional to the Weber number and, consequently, to the bubble terminal velocity as correlated by Mendelson.14 Breathing mode l ) 0, i.e., oscillations that generate changes in bubble volume, is not included. There is no

experimental evidence of sizable changes in bubble volume. Mode l ) 1, representing a rigid bubble rising in an infinite liquid, is also not included. Oscillation modes 2-4 are included because they describe most of the detected experimental phenomena. Higher modes have no appreciable influence on bubble shape, because of the rapid decrease in oscillation amplitude. For m ) 0, the bubbles are axisymmetric, as is every case shown in Figure 8. The two-parameter model does not represent accurately the initial shapes of the bubbles because 2 is considered to be constant with time. In practice, bubbles reach terminal velocity after a brief period of time following the detachment of the bubble from the orifice. After a brief period of time, experimental and theoretical shapes match closely, basically reproducing the shape of an oblate spheroid. Figure 9 is a sketch of a single bubble rising in water in the laboratory coordinate system. Six points representing fluid particles are tracked during the sequence of bubble movement. The pathlines generated from the velocity field obtained by a combination of eqs 13 and

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Figure 9. Representation of a bubble rising in water and its related pathlines from a static point of view using the two-parameter model. The dimensional time step between the frames is 2 ms.

14 must be compared with the experimental pathlines shown in Figure 2a. The two-parameter model provides much closer agreement with experiments than the oneparameter expansion when bubble shapes are compared, and it captures the essence of liquid recirculation at the bottom of the rising bubble.

5. Conclusions Pathlines are slightly different when the one-parameter and two-parameter models are compared, but the main structure of the flow is described by the two models in essentially the same way. If one pays attention to details, the contribution of the two-parameter


model is a more realistic description of bubble shapes after a short period of time following their detachment from the orifice, and consequently, it provides a more accurate representation of the velocity field surrounding the bubble. Liquid recirculation behind the bubbles is described by both models, although there seems to be closer agreement between theory and experimental data when the two-parameter model is used. In short, both models capture the essence of the physical phenomena, but the two-parameter model has more flexibility in accounting for nonlinear interactions between the bubble shapes and velocity profiles. However, the main use of these models is in the analysis of heat- and mass-transfer rates, where oscillations account for the large departure between measured and predicted rates when rigid-bubble models are used. A mass-transfer model of oscillating bubbles was first introduced by Montes et al.11 using a penetration theory approach. A more precise, detailed analysis will require substitution of the oscillating bubble velocity profiles into the mass-transfer equations or simultaneous resolution of momentum- and mass-transfer balances. Because the description is essentially correct, although not as accurate, the simpler one-parameter model would be our choice for such an undertaking. Notation A+1, A-1 ) coefficients depending on oscillation mode B+1, B-1 ) coefficients depending on oscillation mode Cn ) weight factors F ) shape function g ) acceleration of gravity constant, m/s2 n ) oscillation mode NMo ) Morton number (gµ4/Fσ3) NWe ) Weber number (DU2F/_σ) p ) pressure, Pa Plm ) associated Legendre polynomials Pn ) Legendre polynomials R ) bubble radius, m r ) radial coordinate, m t ) time, s tn, tj,k ) dimensionless time U ) liquid velocity, m/s U∞ ) bubble terminal velocity, m/s Ylm ) spherical harmonics Greek Letters Rlm ) shape coefficients δ ) liquid density, kg/m3 δ1j, δ0k ) Kronecker deltas ζ(j,k) lm ) time-dependent coefficient for velocity potential

1 ) expansion parameter 2, ) oscillation amplitude φ ) azimutal coordinate Φ ) velocity potential, m2/s η ) perturbed-domain radial coordinate µ ) liquid viscosity, Pa s θ ) meridional coordinate σ ) air-liquid surface tension, N/m ω ) oscillation frequency, 1/s

Literature Cited (1) Davis, P. F.; Remuzzi, A.; Gordon, E. J.; Dewey, C. F.; Gimbrone, M. A., Jr. Turbulent Fluid Shear Stresses Induces Vascular Endothelial Cell Turnover in Vitro. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 2114. (2) Hua, J.; Erickson, L. E.; Yiin, T. Y.; Glasgow, L. A. A Review of the Effects of Shear and Interfacial Phenomena on Cell Viability. CRC Crit. Rev. Biotechnol. 1993, 13, 305. (3) Calderbank, P. H. Mixing: Theory and Practice; Academic Press: New York, 1967; Vol. 2, Chapter 6). (4) Schroeder, R. R.; Kintner, R. C. Oscillations of Drops Falling in a Liquid Field. AIChE J. 1965, 11 (1), 5-8. (5) Miller, C. A.; Scriven, L. E. The Oscillations of a Fluid Droplet Immersed in Another Fluid. J. Fluid Mech. 1968, 32 (3), 417-435. (6) Prosperetti, A. Free Oscillations of Drops and Bubbles: The Initial Value Problem. J. Fluid Mech. 1980, 100 (2), 333-347. (7) Tsamopoulos, J. A.; Brown, R. A. Nonlinear Oscillations of Inviscid Drops and Bubbles. J. Fluid Mech. 1983, 127, 519-537. (8) Feng, J. Q. The Oscillations of a Bubble Moving in a Inviscid Fluid. SIAM J. Appl. Math. 1992, 52 (1), 1-14. (9) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops and Particles; Academic Press: New York, 1978; Chapter 7. (10) Edge, R. M.; Grant, C. D. The Terminal Velocity and Frequency of Oscillation of Drops in Pure Systems. Chem. Eng. Sci. 1971, 26, 1001-1012. (11) Montes, F. J.; Gala´n, M. A.; Cerro, R. L. Mass Transfer from Oscillating Bubbles in Bioreactors. Chem. Eng. Sci. 1999, 54, 3127-3136. (12) Joseph, D. D. Parameter and Domain Dependence of Eigenvalues of Elliptic Partial Differential Equations. Arch. Ration. Mech. Anal. 1967, 24, 325-351. (13) Joseph, D. D. Domain Perturbations: The Higher Order Theory of Infinitesimal Water Waves. Arch. Ration. Mech. Anal. 1973, 51, 295-303. (14) Mendelson, H. The prediction of bubble terminal velocities from wave theory. AIChE J. 1967, 13, 250-254. (15) White, F. M. Fluid Mechanics, 3rd ed.; McGraw-Hill: New York, 1994. (16) Lamb, H. Hydrodynamics; Cambridge University Press: Cambridge, U.K., 1932.

Received for review December 13, 2001 Revised manuscript received February 8, 2002 Accepted February 12, 2002 IE011009S

Comparison of Theoretical and Experimental Characteristics of

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