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Ind. Eng. Chem. Res. 1996,34, 366-372
A Numerical Investigation of Bubbles Rising at Intermediate Reynolds and Large Weber Numbers Alex C. Hoffmann* and Herman A. van den Bogaard Department of Chemical Engineering, University of Groningen, Groningen, The Netherlands
The steady state rise of bubbles at intermediate Reynolds numbers and large Weber numbers has been investigated using finite element simulations. The velocity and stress fields are discussed. Near-constant normal stress profiles over a region around the bubble apex could be generated by optimizing the wake angle of spherical cap bubbles. These wake angles agreed with relations in the literature. The flow in the region around the bubble apex was near-inviscid; further away deviation from inviscid flow was severe. A toroidal vortex was seen in the wake region, well approximated by Harper and Moore’s modification to Hill’s spherical vortex, confirming Bhaga and Weber’s experimental findings. Nevertheless, a constant normal stress over the entire front of the rising spherical cap bubble could not be generated. This was, however, possible by optimizing an oblate elliptical cap shape. Data for this shape are given for a Reynolds number of 20.
Introduction The rise of bubbles in liquids is a problem of continued interest to the process industry. In many applications detailed knowledge of the rates of mass or heat transfer between a disperse gas phase and a continuous liquid phase is essential for design calculations. Also the modeling and scale-up of processes involving fluidized beds depends on knowledge of the solid and gas flow in and around fluidization bubbles. In liquids very small bubbles take on a spherical shape due t o the effect of surface tension. For larger bubbles forces arising from gravity and from the flow of liquid relative to the rising bubble become significant and cause deviation from the spherical. Three dimensionless numbers, the Weber number, the Eotvos number, and the Morton number, are used to describe the relative significance of these forces:
where UBis the bubble rise velocity, el and pi are the density and viscosity of the liquid, respectively, g is the acceleration due to gravity, de is the diameter of a sphere with the same volume as the bubble (volume equivalent diameter), and u is the surface tension of the liquid. The first two numbers are bubble characteristics and measures of the flow force and the gravitational force, respectively,relative to forces arising from surface tension. The Morton number is a characteristic only of the liquid and also indicates the relative significance of the surface tension. If W e and Eo are large enough ( W e > 20 according to Haberman and Morton as quoted by Collins (1967) and Eo > 40 according to Clift et al. (1978)),then the effect of forces arising from surface tension on the bubble shape is negligible. It turns out that the bubble then will take on a spherical or elliptical cap shape. The flow pattern of the fluid relative to a rising bubble of a given shape is, in an infinite medium, dependent on the bubble Reynolds number: E @ldeUdp1 (2) If ReB is in the range of about 1.2 to about 100 and We and Eo are large enough (this is true for bubbles in
* Author to whom correspondence should be addressed.
many viscous liquids such as, for instance, heavier oil fractions, molten metals, or polymers) the spherical or elliptical cap bubble will have associated with it a closed laminar wake of recirculating material which follows the bubble as it rises (Bhaga and Weber, 1981; Clift et al., 1978). There are also indications that fluidization bubbles can be viewed as belonging to this category; this will be the subject of further work. Davies and Taylor (1950) considered the flow of the liquid relative to a rising spherical cap bubble at its apex; they assumed inviscid flow of the liquid in this region. This led them, when they applied the condition that the pressure should be constant in the neighborhood of the bubble apex, to a relation for the bubble rise velocity:
UB = 2/!J@! (3) where R is the radius of curvature of the bubble front. They showed this relation t o be consistent with experimental bubble rise velocities. Collins (1966) calculated a second approximation bringing the pressure distribution in the neighborhood of the apex of the bubble nearer to the ideal of a constant value. Grace and Harrison (1967) calculated rise velocities corresponding to a variety of shapes of the bubble front in two and three dimensions. Other workers considered the flow not only in the neighborhood of the apex of the bubble but also behind it. Rippin and Davidson (1967) considered a situation of free streamline flow around the front of the bubble continuing past the bubble, creating a stagnant wakelike region behind the bubble. On the experimental side Clift et al. (1978) collected a large number of experimentally determined values of the so-called wake angle (8, in Figure 1)and they give the empirical relation 8, = 50
+ 190 e~p(-0.62Re$.~) (Eo > 40, Re, > 1.2) (4)
which fits the data very well. Grace et al. (1976) and Bhaga and Grace (1981)report experimental studies of the shape and terminal velocity of rising spherical cap bubbles, as well as visualization of the flow pattern around them.
0888-588519512634-0366$09.00/00 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 367
E
Du'
(7)
where
Fr
= U:/gL
and Re
gU,J/p
(8)
Thus for dynamic similarity between geometrically similar configurations, Reynolds number similarity and Froude number similarity is required. A simple dimensional analysis of geometrically similar bubbles rising at their terminal velocity reveals, however, that for this case Figure 1. Schematic of a spherical cap bubble with the wake
Fr = flRe), for geometrically similar bubbles rising at their terminal velocity (9)
There have been a number of numerical studies of rising bubbles or drops, most of them concentrating on spherical or near-spherical shapes; some are mentioned in Hartholt et al. (1994). Ryskin and Leal (1984)studied with the use of the finite difference method bubbles deviating somewhat from the spherical, the most deformed being of a kidney shape. They showed that a closed recirculatory wake region formed behind some of the bubbles. More references will be mentioned in the Results and Discussion section.
(Wegener and Parlange (1973); note that eq 9 agrees with the analysis of Grace (19731, since his group: E o O . ~ ~ 0.25, / M where M is the Morton number, can, if Qg
