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Ind. Eng. Chem. Res. 2007, 46, 9232-9237
GENERAL RESEARCH Heat, Mass, and Momentum Transfer to a Rising Ellipsoidal Bubble Abdullah Abbas Kendoush Department of Mechanical and Aerospace Engineering, UniVersity of Florida, P.O. Box 116300, GainesVille, Florida 32611-6300
Semianalytical solutions were obtained for the heat, mass, and momentum transfer to an oblate ellipsoidal bubble. The flow field around the bubble was assumed irrotational and inviscid. The solutions were validated by their asymptotic approach to the limit of the single spherical bubble, as well as by their comparisons with data obtained by other authors. 1. Introduction Interest in the motion of thermal or isothermal bubbles in a fluid medium has existed for many decades and has resulted in hundreds of experimental and theoretical investigations for the laws that govern this phenomenon, which, from a practical point of view, is of considerable importance in various industrial processes such as absorption of gases in liquid columns, fermentation, agitation, stirring, sewage purification, directcontact heat exchangers, desalination, power generation, and so on. Oblate-ellipsoidal bubbles (OEBs) are the intermediate state between spherical and spherical-cap bubbles. Bubbles originate as small spheres and, when the dynamic pressure is strong enough, in comparison to the surface tension, they deform to oblate ellipsoids and eventually to spherical-cap bubbles. Rising bubbles whose surfaces are free of contamination by surfactants can be treated analytically by making the shear stress at the bubble surface equal to zero, because the viscosity of the gas or vapor inside the bubble is much smaller than the viscosity of the surrounding liquid. A potential flow field incorporated into the viscous dissipation integral can provide a solution of the momentum transfer over the bubble surface.1 This method is based on the balance at steady state between the rate of working on the fluid due to bubble motion and the rate of dissipation of mechanical energy in the fluid that is due to thermal energy. The analytical difficulties introduced by the oblate ellipsoidal shape of the bubble under consideration rendered the amount of published analytical work in this field rather limited. An outstanding contribution was presented by Tomiyama et al.,2 who measured and modeled the rise velocities of ellipsoidal bubbles, dimpled hemispherical-cap bubbles, and distorted spheroid bubble. Dandy and Leal3 numerically solved the Navier-Stokes equations around an OEB of fixed shape in the Reynolds number range of Re e 200. Their computations showed that, beyond a critical aspect ratio x ) a/b of the bubble, a standing eddy could be observed within a finite Re range. The eddy is a consequence of an accumulation of vorticity near the bubble surface. * To whom correspondence should be addressed. E-mail:
[email protected].
Fan and Tsuchiya4 recommended the following correlation for the aspect ratio:
{
1 Ta < 1 3 x ) {0.81 + 0.206 tanh[2(0.8 - log10Ta)]} 1 eTa e39.8 0.24 Ta > 39.8
}
where
()
Ta ) g1/4
FL σ
3/4
dU
Ta is called the Tadaki number, and d is defined as the bubble formation diameter. The rise of oblate ellipsoidal bubbles in highly purified water has been studied experimentally by Duineveld5 and Ybert and Di Meglio.6 Both groups found good agreement between the experimental values of the rise velocity and those obtained by equating the buoyancy force acting on the bubble with the drag force corresponding to Moore’s7 predictions. Moore’s7 analytical solution of the drag forces on the surface of the oblate ellipsoidal bubble was done using boundary layer analysis, based on expansions in powers of Re-1/2. Many authors (e.g., Meiron,8 Miksis et al.,9 and Benjamin10) agreed that the Weber number (We) approaches a limiting value of ∼1.8 asymptotically at high values of oblatness. The Weber number represents the ratio of the dynamic pressure that is causing bubble distortion to the surface tension pressure that is available to resist it. The reported results of heat and mass transfer to ellipsoidal bubbles are rather scarce. The two papers by Lochiel and Calderbank11 and Feng and Michaelides12 examine the subject. The first paper solved the energy equation using boundary-layer approximations, and their solution is valid for Peclet numbers of Pe > 102. The second paper solved the unsteady conduction equation, and their solution is valid for Pe , 1. The present work focuses on the analytical solution of heat mass and momentum transfer to the OEB at high Re values. 2. Method of Analysis Consider a freely rising oblate ellipsoidal bubble in an infinite fluid under the influence of gravity. The flow velocity around
10.1021/ie070687x CCC: $37.00 © 2007 American Chemical Society Published on Web 11/17/2007
Ind. Eng. Chem. Res., Vol. 46, No. 26, 2007 9233 Table 1. Numerical Values of the Integrals Given in Eqs 9 and 14 x
Wea
H(x)
G(x)
1.00 1.25 1.50 1.75 1.84 1.95
0.000 0.873 1.155 1.333 1.384 1.438
4.000 3.735 3.639 3.633 3.646 3.671
4.000 3.968 3.868 3.855 3.855 3.856
a The Weber number (We) values were obtained from eq A4 in the Appendix.
The shear stress on the bubble surface is calculated from the following relation:
[ ( )
[τrθ]r)R(θ) ) µ r Figure 1. Geometry of the oblate ellipsoidal bubble.
[ ( )] r 1 re + re 2 r
[ ( )] [ ( )]
Uθ ) -U sin θ 1 + 0.5
∫0
π
2reZ7(θ)
E ) -3πreU2µ
∫0π K(θ) dθ ) ∫0π
Z(θ)5
e
) H(x)
E U
(10)
The drag coefficient is defined as
The radius of the ellipsoidal bubble was given by Meiron8 as follows: 2
(9)
This integral was solved numerically, and the values are given in Table 1. The drag force (D) is given by dividing eq 9 by U, as follows:
(3)
(5)
[ (643 )We f(θ)] ) r Z(θ)
∫0π K(θ) dθ
(1 + 2Z(θ)3)sin3 θ dθ
D 0.5FU2πre2
CD )
(6)
Here, the Weber number is given by We ) U(Fre/σ)1/2, where σ is the coefficient of surface tension. The function f(θ) is equal to 1 + 3 cos(2θ).
(11)
Substituting eq 10 into eq 11 gives
CD )
(4)
dA ) 2πR(θ)2 sin θ dθ
(8)
Substituting eqs 3, 5, and 8 into eq 4 gives
(2)
where the dA term represents an elementary area on the bubble surface, which is given as
R(θ) ) re 1 -
-3U2µ sin3 θ(1 + 2Z3(θ))
D)
3
(τrθUθ)r)R(θ) dA
(7)
(1)
This velocity field has been utilized previously by the present author14 to obtain the virtual mass coefficient of the OEB. 2.1. Drag Force Derivation. The viscous dissipation provides a method for the determination of the drag force acting on the bubble surface:15
E)
[τrθ]r)R(θ) )
3
re r
r)R(θ)
where
2
Here, re is the equivalent radius of the oblate ellipsoidal bubble (re ) [3V/(4π)]1/3, where V is the volume of the bubble), and r and θ are polar coordinates whose origin moves with the bubble, as shown in Figure 1. The velocity components of the flow are obtained from Ur ) -∂φ/∂r and Uθ ) -∂φ/(r ∂θ), which give the following relations:
re Ur ) U cos θ 1 r
]
Substituting eqs 2, 3, and 6 into eq 7 yields
the bubble is U, the semi-major axis of the bubble is a, and the semi-minor axis is b, as shown in Figure 1. Consider the boundary separation at the bubble surface to be negligible. The domain of the solution is for values of x e1.65 and for Re values up to 103. These limits ensure that no standing eddy occurs in the wake region of the OEB, according to the numerical results of Blanco and Magnaudet.13 The velocity potential of an OEB was given by Meiron8 as follows:
φ ) -Ure cos θ
∂Ur ∂ Uθ + ∂r r r ∂θ
12H(x) Re
(12)
where Re ) 2reFU/µ. Equation 12 reduces to the spherical bubble’s equation (that is, CD ) 48/Re) upon letting x ) 1 and We f 0. Figure 2 shows the variation of H(x) with x. It should be noted that the terminal velocity of the OEB can be obtained directly from equating the drag force D to the buoyancy force, B ) 4/3πre3(F - Fg)g; hence, the terminal velocity of the OEB becomes U ) 4re2(F - Fg)g/9µH(x). 2.2. Heat or Mass Transfer. Clift et al.16 proposed an equation for the mass transfer as a result of resolving the diffusion for the spherical bubble. The equation was slightly modified to be applied to the ellipsoidal bubble as follows:
Sh ) 0.798
[PeU ∫ [U ] π
0
θ r)R(θ)
sin2 θ dθ
1/2
]
(13)
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Ind. Eng. Chem. Res., Vol. 46, No. 26, 2007
Figure 2. Variation of (0) H(x) and (O) G(x) with x.
where Sh is the Sherwood number. Substituting eq 3 into eq 13 yields
Sh ) 0.564(PeG(x))1/2
(14)
where
G(x) )
∫0π
[1 + 2Z3(θ)] sin3 θ dθ Z3(θ)
Here, Pe ) ScRe ) 2reU/D (where Sc is the Schmidt number). The integral in eq 14 was solved numerically, and the values are given in Table 1. Equation 14 reduces to the single spherical bubble equation (that is, Sh ) 1.128xPe of Boussinesq and Higbie17) upon letting x f 1 (see Table 1). Figure 2 shows the variation of G(x) with x. The equivalence of eq 14 in heat transfer may be obtained, simply by replacing the Sherwood number (Sh) by the Nusselt number (Nu). 3. Discussion and Validation A comparison between the present solution (eq 12), the boundary-layer solution of Moore,7 and the numerical solution of Blanco and Magnaudet,13 where the agreement is satisfactory, is shown in Figure 3. Gupalo et al.18 (cited in the work of Polyanin et al.19) produced a solution for the mass transfer from an oblate ellipsoidal bubble using a potential flow model. Their solution does not converge to the single spherical bubble solution when the oblateness is removed. A comparison of the present solution (eq 14) with that of Gupalo et al.18 is shown in Figure 4. Gupalo et al.’s solution was not in total agreement with the present solution. Miller20 used the following equation for the mass transfer to an ellipsoidal bubble:
Sh ) 2 + Re1/2Sc1/3
(15)
Here, Sh ) kL2re/D. Miller20 neglected the factor of 2 in this equation and calculated the following ratio of the mass transfer coefficient from this equation to that calculated from Boussinesq’s equation:
Figure 3. Drag coefficient (CD), as a function of the Reynolds number (Re): (A) x ) 1.50, (B) x ) 1.75, and (C) x ) 1.95. The solid line represents the present solution, the dashed line represents Moore’s analytical solution,7 and the data points (noted as squares) represent Blanco and Magnaudet’s13 numerical solution.
k*L )
kL kL,calculated
) 0.97Sc-1/6
(16)
When eq 15 was substituted into the present eq 14, the following ratio was obtained:
k*L ) 0.904Sc-1/6
(17)
The result from this equation is close to that of Miller20 by 6.8%.
Ind. Eng. Chem. Res., Vol. 46, No. 26, 2007 9235
Figure 4. Comparison between (s) the present solution (eq 14) and (- ‚ - ‚ -) the solution of Gupalo et al.18
Figure 5. Comparison between (s) the present solution (eq 14) and (- - -) the solution of Montes et al.21
Figure 5 shows a comparison between the present solution (eq 14) and the following theoretical equation of Montes et al.21
Sh ) 1.128Pe1/2(1.1 + 0.027We1/2)
(18)
Figure 6. Comparison of the present solution (eq 21, represented by the solid line) with the experimental data of various authors for gas bubbles in water (reference sources and corresponding symbols are given in the table legend below the figure).
The comparison was based on the following fixed parameters: x ) 1.75 and We ) 1.333. Comparison is made with the experimental data collected by Clift et al. in their Figure 7.15.16 They used a mass-transfer factor of (kA/AexD) for bubbles in water. Equation 14 was modified to obtain this factor as follows:
kL(2re)
xD
) 0.564[G(x)U(2re)]1/2
(19)
The term A/Ae is equal to the ratio of the surface area of the ellipsoidal bubble to the surface area of the volume-equivalent sphere; that is,
A b (x + 2) ) Ae 3re2 2
2
(20)
Multiplying both sides of eq 19 by eq 20 and rearranging yields
0.752(a(x)U)1/2b2(x2 + 2) kA ) (de)2.5 AexD
(21)
When this equation is plotted against the experimental data, as in Figure 6, the values estimated using eq 21 straddles across the experimental data. Ellipsoidal bubbles with a size of de > 0.5 cm experiences oscillation in their rise due to the change in the ratio of inertial forces to capillary forces. Bubble oscillation is beyond the scope of this paper. Lochiel and Calderbank11 derived a solution for the mass transfer from an OEB, based on the inviscid flow equations of
Figure 7. Comparison between (s) the present solution (eq 22), (- - -) the Lochiel and Calderbank11 solution, and (- - -) the experimental data of Skelland and Cornish.34
Zahm.33 Figure 7 shows a comparison between the following present solution (eq 22), the experimental data of Skelland and Cornish,34 and the solution of Lochiel and Calderbank,11 where the agreement is acceptable.
(Sh)OEB (Sh)sphere
) 0.5(G(x))1/2
(22)
In a previous report, the author35 derived the following equation for the heat or mass transfer from the spherical-cap bubble:
Sh ) 2.113(Pe)1/2
(23)
For an OEB of oblateness of x ) 1.95, Table 1 gives G(x) )
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Ind. Eng. Chem. Res., Vol. 46, No. 26, 2007
3.856; the present eq 14 for heat and mass transfer to an OEB then becomes the following relationship:
Sh ) 1.107(Pe)1/2
(24)
The heat or mass transfer from a spherical bubble is given as Sh ) 1.128(Pe)1/2; therefore, the heat or mass transfer rate from the spherical-cap bubble is the highest, whereas that from the OEB is the lowest for the same Pe ranges. Zhao et al.36 and Haut and Cartageb37 produced solutions for the mass-transfer rate from OEB; however, their solutions were not in a form that was suitable for comparison with the present solution. 4. Conclusions
Greek Symbols φ ) velocity potential (m2/s) µ ) absolute viscosity (Ns/m2) θ ) polar coordinate F ) density of fluid (kg/m3) σ ) surface tension (N/m) τ ) shear stress (N/m2) Subscripts e ) equivalent r ) radial θ ) angular g ) vapor or gas Acknowledgment
Equations were derived for the drag coefficient, the terminal velocity, and the convective heat and mass transfer to a single oblate ellipsoidal bubble. All the previously mentioned derived equations were functions of the Weber number (We). The derived solutions were validated by comparison with the experimental and theoretical works of other authors. Appendix: The Relationship between x and the Weber Number (We) The aspect ratio of the bubble (as shown in Figure 1) is given as
x)
a b
(A1)
Using eq 6, we have
[
3 (We)2 46
]
(A2)
(π2) ) a ) r [1 + 323 (We) ]
(A3)
R(0) ) b ) re 1 and
R
2
e
Substituting eqs A2 and A3 into eq A1 yields
x)
32 + 3We2 32 - 6We2
Nomenclature Parameters a ) major axis of the OEB (m) b ) minor axis of the OEB (m) CD ) drag coefficient D ) mass diffusion coefficient (m2/s) d ) bubble deformation diameter (m) E ) viscous dissipation integral (Nm/s) g ) acceleration due to gravity (m/s2) r ) polar coordinates (m) Pe ) Peclet number Re ) Reynolds number Sc ) Schmidt number Sh ) Sherwood number U ) velocity of the OEB (m/s) We ) Weber number x ) aspect ratio
(A4)
I wish to thank the Department of Mechanical and Aerospace Engineering, Thermal Science and Fluid Dynamics Group and the Institute of International Education: Scholar Rescue Fund at the University of Florida for their continued support of my work. Literature Cited (1) Kendoush, A. A. Hydrodynamic model for bubbles in a swarm. Chem. Eng. Sci. 2001, 56, 235. (2) Tomiyama, A.; Celata, G. P.; Hosokawa, S.; Yoshida, S. Terminal velocity of single bubbles in surface tension force dominant regime. Int. J. Multiphase Flow 2002, 28, 1497. (3) Dandy, D. S.; Leal, L. G. Boundary layer separation from a smooth slip surface. Phys. Fluids 1986, 29, 1360. (4) Fan, L.-S.; Tsuchiya, K. Bubble Wake Dynamics in Liquids and Liquid-solid Suspension; Butterworth-Heinemann: Boston, 1990. (5) Duineveld, P. C. The rise velocity and shape of bubbles in pure water at high Reynolds number. J. Fluid Mech. 1995, 292, 325. (6) Ybert, C.; Di Meglio, J. M. Ascending air bubbles in protein solutions. Eur. Phys. J. 1998, B4, 313. (7) Moore, D. W. The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 1965, 23, 749. (8) Meiron, D. I. On the stability of gas bubbles rising in an inviscid fluid. J. Fluid Mech. 1989, 198, 101. (9) Miksis, M.; Vanden-Broeck, J.-M.; Keller, J. B. Axisymmetric bubble or drop in a uniform flow. J. Fluid Mech. 1981, 108, 89. (10) Benjamin, T. B. Hamiltonian theory of bubbles in an infinite liquid. J. Fluid Mech. 1987, 181, 239. (11) Lochiel, A. C.; Calderbank, P. H. Mass transfer in the continuous phase around axisymmetric bodies of revolution. Chem. Eng. Sci. 1964, 19, 471. (12) Feng, Z.-G.; Michaelides, E. E. Unsteady heat and mass transfer from a spheroid. AIChE J. 1997, 43 (3), 609. (13) Blanco, A.; Magnaudet, J. The structure of the axisymmetric highReynolds number flow around an ellipsoidal bubble of fixed shape. Phys. Fluids 1995, 7 (6), 1265. (14) Kendoush, A. A. The virtual mass of an oblate ellipsoidal bubble. Phys. Lett. A 2007, 366, 253. (15) Kendoush, A. A. Calculation of flow resistance from a spherical particle. Chem. Eng. Process 2000, 39, 81. (16) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, drops and particles; Academic Press: New York, 1978. (17) Higbie, R. The rate of adsorption of pure gas into a still liquid during short period of exposure. Trans. Am. Inst. Chem. Eng. 1935, 31, 365. (18) Gupalo, Yu.; Ryazantsev, Yu. S.; Sergeev, Yu. A. Diffusion flux to a distorted gas bubble at large Reynolds number. Fluid Dynam. 1976, 11, 548. (19) Polyanin, A. D.; Kutepov, A. M.; Vyazmin, A. V.; Kazenin, D. A. Hydrodynamics, Mass and Heat Transfer in Chemical Engineering; Taylor and Francis: London, 2002. (20) Miller, D.N. Scale-up of agitated vessels gas-liquid mass transfer. AIChE J. 1974, 20, 445. (21) Montes, F. J.; Galan, M. A.; Cerro, R. L. Mass transfer from oscillating bubble in bioreactors. Chem. Eng. Sci. 1999, 54, 3127. (22) Jackson, R. Chem. Eng. 1964, 178, 107.
Ind. Eng. Chem. Res., Vol. 46, No. 26, 2007 9237 (23) Weber, M. E. McGill University, Montreal, Canada. Unpublished calculations, 1975. (24) Chuang, S. C.; Goldschmidt, V. W. Bubble formation due to a submerged capillary tube in quiescent and coflowing streams. J. Basic Eng. 1970, 92, 705. (25) Davidson, J. F.; Harrison, D. Fluidised Particles, Cambridge University Press: London and New York, 1963. (26) Bellman, R.; Pennington, R. H. Effect of surface tension and viscosity on Taylor instability. Q. Appl. Math. 1954, 12, 151. (27) Lane, W. R. Shatter of drops in streams of air. Ind. Eng. Chem. 1951, 43, 1312. (28) Zheleznyak, A. S. J. Appl. Chem. USSR 1967, 40, 834. (29) Rajan, S. M.; Heideger, W. J. Drop formation mass transfer. AIChE J. 1971, 17, 202. (30) Van Krevelen, D. W.; Hoftijzer, P. J. Studies of gas bubble formation. Chem. Eng. Prog. 1950, 46, 29. (31) Lanauze, R. D.; Harris, I. J. Gas bubble formation at elevated system pressures. Trans. Inst. Chem. Eng. 1974, 52, 337. (32) Gillespie, T.; Rideal, E. On the adhesion of drops and particles on impact at solid surfaces. II. J. Colloid Sci. 1955, 10, 281.
(33) Zahm, A.F. Natl. AdVis. Commun. Aeronaut., Washington, 1926, Report 253. (34) Skelland, A. H. P.; Cornish, A. R. H. Mass transfer from spheroids to an air stream. AIChE J. 1963, 9, 73. (35) Kendoush, A. A. Theory of convective heat and mass transfer to spherical-cap bubbles. AIChE J. 1994, 40, 1440. (36) Zhao, B.; Wang, J.; Yang, W.; Jin, Y. Gas-liquid mass transfer in slurry bubble systems. I. Mathematical modeling based on a single bubble mechanism. Chem. Eng. J. 2003, 96, 23. (37) Haut, B.; Cartageb, T. Mathematical modeling of gas-liquid mass transfer rate in bubble columns operated in the heterogeneous regime. Chem. Eng. Sci. 2005, 60, 5967.
ReceiVed for reView May 15, 2007 ReVised manuscript receiVed September 14, 2007 Accepted September 20, 2007 IE070687X
