Bubble in a Translational Flow

In the case of a spherical bubble in a translational flow at small Reynolds numbers, the solution of Oseen s equation (2.3.1) results is a two-term asymptotic expansion for the drag coefficient [476]  

The drag coefficient monotonically decreases as the Reynolds number increases. For high Reynolds numbers, one can use the approximation of ideal fluid for solving the problem on the flow past a bubble. In this case, the leading term of the asymptotic expansion of the drag coefficient has the form [291] 

In [283], a number of experimental data are presented on the rise of air bubbles in clean mixtures of distilled water and pure, reagent grade, glycerin covering a wide range of the Morton numbers. 

The motion of air bubbles in free-surface turbulent shear flows is considered in [83] in detail. 

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Spherical bubble as Re —> 0, 0 < Pe < oo. The problem of mass transfer to a spherical bubble in a translational flow as Re 0 was studied numerically in [321], The results for the mean Sherwood number can be approximated well by the expression... 

Spherical bubble at any Peclet numbers for Re > 35. For a spherical bubble in a translational flow at moderate and high Reynolds numbers and high Peclet numbers, the mean Sherwood number can be calculated by the formula [94]... 

In this section, some interpolation formulas are presented (see [367, 368]) for the calculation of the mean Sherwood number for spherical particles, drops, and bubbles of radius a in a translational flow with velocity U at various Peclet numbers Pe = aU /D and Reynolds numbers Re = aU-Jv. We denote the mean Sherwood number by Shb for a gas bubble and by Shp for a solid sphere. 

Formula (5.6.4) is valid for an arbitrary laminar flow without closed streamlines for particles and drops of an arbitrary shape. The quantity Sh(l,Pe) corresponds to the asymptotic solution of the linear problem (5.6.1) at Pe > 1. For spherical particles, drops, and bubbles in a translational or linear straining shear flow, the values of Sh(l, Pe) are shown in the fourth column in Table 4.7. 

In chemical technology one often meets the problem of a steady-state motion of a spherical particle, drop, or bubble with velocity U in a stagnant fluid. Since the Stokes equations are linear, the solution of this problem can be obtained from formulas (2.2.12) and (2.2.13) by adding the terms Vr = -U cos6 and V = U[ sin 6, which describe a translational flow with velocity U, in the direction opposite to the incoming flow. Although the dynamic characteristics of flow remain the same, the streamline pattern looks different in the reference frame fixed to the stagnant fluid. In particular, the streamlines inside the sphere are not closed. 

In the case of mass exchange between a bubble and a translational Stokes flow of a quasi-Newtonian power-law fluid (n is close to unity), one can use the following simple approximate formula for calculating the mean Sherwood number at high Peclet numbers ... 

Of course the feature that differs in this case is the form of the velocity field u. For simple translation of a bubble through a quiescent fluid (that is, the uniform-flow problem) at zero Reynolds number, this is solely a consequence of the change from no-slip conditions for a solid body to the condition of vanishing tangential stress at the surface of a clean bubble (recall that the shape remains spherical for Re

[Pg.669]

In the remainder of this section, we evaluate the coefficient c in (9 274) for the specific case of a translating gas bubble. In this case, the full creeping-flow solution for the velocity field is... 

Formula (4.4.21) is quite general and holds for solid particles, drops, and bubbles of arbitrary shape in a uniform translational flow at any Re as Pe —F 0. 

To construct approximate formulas for the Sherwood number in the case of a translational shear flow past particles and bubbles in the entire range of Peclet numbers, one can use formulas (4.7.9) and (4.7.10) where Shpoo and Shboo must be replaced by the right-hand sides of Eqs. (4.9.3), (4.9.4) and (4.9.5), (4.9.6), respectively, with 0 = 0. 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

Lamella Division. Lamella or bubble division proceeds by subdividing foam bubbles or lamellae. Thus, mobile foam bubbles must preexist. Division is illustrated in Figure 6. A translating foam bubble encounters a point where flow branches in two directions (Figure 6a). The interface stretches around the branch point and enters both flow paths. The initial bubble divides into two separate bubbles (Figure 6b) that continue to move downstream. 

One should realize that these calculations are based on an expression for Vr which corresponds to potential flow past a stationary nonde-formable bubble, as seen by an observer in a stationary reference frame. However, this analysis rigorously requires the radial velocity profile for potential flow in the Uquid phase as a nondeformable bubble rises through an incompressible liquid that is stationary far from the bubble. When submerged objects are in motion, it is important to use liquid-phase velocity components that are referenced to the motion of the interface for boundary layer mass transfer analysis. This is accomplished best by solving the flow problem in a body-fixed reference frame which translates and, if necessary, rotates with the bubble such that the center of the bubble and the origin of the coordinate system are coincident. Now the problem is equivalent to one where an ideal fluid impinges on a stationary nondeformable gas bubble of radius R. As illustrated above, results for the latter problem have been employed to estimate the maximum error associated with the neglect of curvature in the radial term of the equation of continuity. 

Interfacial Electrokinetic Flow, Fig. 6 Schematic illustration of the front cap of a bubble translating in a long capillary tube (After Chang [16])... 

An initial and important result is that the accumulation of surfactant molecules at the rear stagnation point of a translating droplet imposes a surface stress which opposes the outer flow (see Fig. 2). A direct implication is that for a gas bubble rising in a quiescent liquid, the surface is immobilized and the rise velocity is more similar to the rise of a rigid disc than to an inviscid bubble. 

As mentioned previously, another interesting phenomenon that can be observed around an oscillating bubble is microstreaming, which typically is a vortical flow structure in the vicinity of the bubble (see Fig. 6). Microstreaming is a second-order flow field originated from the phase difference between the radial and translational motion of the bubble. 

Results here imply that there may be an optimal LAD screen weave, at least in terms of maximizing bubble point, because the 450 mesh outperforms the other two mesh screens. This implies that this 450 x 2750 screen is capable of delivering a much higher flow rate over the 325 x 2300 screen since bubble point pressure translates into total allowable flow rate from a LAD to a transfer line. Using a 450 x 2750 does in fact increase margin over the low baseline value for the 325 x 2300 screen. 

Division of gas flow. There has been considerable controversy regarding the amount of gas which flows through each phase in a bubbling fluidized bed. The amount of gas carried by translation of bubble voids can be written as... 

Yet, Eq. (14) does not describe the real situation. It must also be taken into account that gas concentration differs in the solution and inside the bubble and that, consequently, bubble growth is affected by the diffusion flow that changes the quantity of gas in the bubble. The value of a in Eq. (14) is not a constant, but a complex function of time, pressure and bubble surface area. To account for diffusion, it is necessary to translate Fick s diffusion law into spherical coordinates, assign, in an analytical way, the type of function — gradient of gas concentration near the bubble surface, and solve these equations together with Eq. (14). 

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