Rising bubbles potential flow

Single gas bubbles in an inviscid liquid have hemispherical leading surfaces and somewhat flattened wakes. Their rise velocity is governed by Bernoulli s theory for potential flow of fluid around the nose of the bubble. This was first solved by G. I. Taylor to give a rise velocity Ug of ... 

As Wallis (1969) points out, the upper limit of region 4 is with very large bubbles when their rise is dominated by inertial forces. Under these conditions, the terminal rise velocity is readily calculated from potential flow theory and is given by... 

Consider a bubble rising in a fluidized bed. It is assumed that the bubble is solids-free, is spherical, and has a constant internal pressure. Moreover, the emulsion phase is assumed to be a pseudocontinuum, incompressible, and inviscid single fluid with an apparent density of pp(l — amf) + pamf. It should be noted that the assumption of incompressibility of the mixture is not strictly valid as voidage in the vicinity of the bubble is higher than that in the emulsion phase [Jackson, 1963 Yates et al., 1994]. With these assumptions, the velocity and pressure distributions of the fluid in a uniform potential flow field around a bubble, as portrayed by Fig. 9.10, can be given as [Davidson and Harrison, 1963]... 

A bubble-bubble interaction model based on potential flow over bluff bodies was developed and incorporated in a code called, BuDY (for Bubble DYnamics). The model is based on an assumption that the instantaneous velocity of an individual bubble in a fluidized bed can be obtained by adding to its rise velocity in isolation, the velocity which the emulsion phase would have had at the nose of the bubble, if the bubble was absent. The details of model development, model equations and solution procedures are described in Ranade (1997a). Appropriate representation of bubble formation, coalescence and exit of bubbles from the dense bed were included in the model. With the knowledge of initial bubble positions and bubble size, subsequent bubble positions can be tracked to predict instantaneous velocities and bubble positions within the dense bed. 

Suppose that a is sufficiently small, i.e., We is sufficiently large, that surface tension plays no role in determining the bubble shape, except possibly locally in the vicinity of the rim where the spherical upper surface and the flat lower surface meet. Further, suppose that the Reynolds number is sufficiently large that the motion of the liquid can be approximated to a first approximation, by means of the potential-flow theory. Denote the radius of curvature at the nose of the bubble as R(dX 6 = 0). Show that a self-consistency condition for existence of a spherical shape with radius R in the vicinity of the stagnation point, 0 = 0, is that the velocity of rise of the bubble is... 

The hydrodynamic field used does not comprise the interesting case of large rising bubbles under the potential flow. Ej and Ej are sensitive to the hydrodynamic field and the bubble surface state, which was not yet been taken into account. 

Potential Flow Solutions for Gas Bubbles Which Rise through Incompressible Fluids That Are Stagnant Far from the Submerged Objects. A nondeformable bubble of radius R rises through an ideal fluid such that... 

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. 

At infinite Reynolds number, potential flow theory may be used to investigate bubble motion. As a consequence of D Alembert s paradox (see, e.g., Ref. 10), potential flow theory predicts no drag on a steadily rising bubble. However, it provides a result for the force on an accelerating sphere ... 

When combined with boundary layer analysis, potential flow theory provides estimates for the velocity of a rising bubble. The lowest-order result for the terminal velocity of a freely rising bubble is... 

Now visualize the action of a submerged jet of air in liquid. At very low air flow velocities, the bubbles are large. They rise as independent bubbles at the orifice. When the velocity of air through the orifice increases, the air projects as a cone into the liquid and small bubbles shear off. The maximum velocity is reached when the ratio of absolute hydrostatic pressure outside the orifice divided by the absolute air pressure in the orifice is 0.528. This determines sonic velocity. Four regions of the air cone or jet are conceptually drawn in Fig. 11. Region I is called the potential core of air with a uniform velocity. The outermost annular cone, Region II, is an intermittency zone in... 

The characteristic TMP history for a submerged membrane with cross flow (bubbling) is depicted in Figure 10.2a. Due to deposition TMP rises slowly, and the rate of rise is less with more imposed shear or lower solids. In an idealized situation TMP would remain unchanged at subcritical flux conditions. However, for various reasons (see Section 10.5.1) some degree of TMP rise tends to occur at aU fluxes so the interest is in the acceptable rate of rise and the sustainable flux. Figure 10.2a also includes the potential for a sudden TMP jump that can be observed in prolonged operation at constant flux (discussed further in Section 10.5.1). 

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