Liquid continued velocity profiles

Many of the considerations discussed in the previous section can be applied to vertical bubble flow. Two cases are of interest first, one in which there is no downward or upward liquid flow (Ql = 0), for example, on a distillation tray. With a shallow liquid layer there will be no liquid velocity profile, and hence bubble-rise velocities for uniformly sized bubbles will be the same at each cross-section. It can be shown (see Nicklin, N2) that if the rising velocity of a swarm of bubbles not continuously generated in a still liquid is Vo, then the rising velocity relative to the tube of a swarm of continuously generated bubbles will be, for... 

Second, when a liquid is flowing in a long tube, a liquid velocity profile will develop, and bubbles near the center of the tube will rise more rapidly than those near the wall. In slug flow, all slugs rise with nearly the maximum velocity, but in dispersed bubble flow, the variations in speed tend to cancel and give a component to the bubble velocity due to the average liquid velocity. As before, from continuity, this mean liquid velocity at any cross section must be (Qo + Qt)A, so that the bubble rise velocity is... 

In all the above derivations in this section, the influence of viscosity is neglected so that analytical solutions for velocity and pressure profiles can be obtained. When the viscosity of fluid is taken into account, it is difficult to obtain any analytical solution. Kuts and Dolgushev [35] solved numerically the flow field in the impingement of two axial round jets of a viscous impressible liquid ejected at the same velocity from conduits with the same diameter and located very close to each other. The mathematical formulation incorporated the complete Navier-Stokes equations transformed into stream and velocity functions in cylindrical coordinates r and z, with the assumption that the velocity profiles at the entrance and the exit of the conduit were parabolic. The continuity equation is given by Eq. (1.22) and the equations for motion in dimensionless form are ... 

Equations (3),(4) are supplemented by the conjugate boundary conditions (velocity and shear stress continuity) at the interface. The numerical solution of the hydrodynamic equations yields the liquid film thickness as well as velocity profiles in each phase. They are used for the description of mass and heat transfer. 

There is a great deal of theoretical and experimental information from micrometeorological research on the transfer of momentum, heat, and mass at solid and liquid surfaces and across their associated air boundary layers (hence the term boundary layer models for relationships arising from this approach). Based on the analogy between transfer of momentum and mass, it has been shown that k is proportional to the friction velocity in air (u ) and that k is also proportional to Sc. Apart from an assumption that the surface was smooth and rigid, it was also necessary to assume continuity of stress across the interface in order to convert the velocity profile in air to the equivalent profile in the water (Deacon, 1977). The relationship developed by Deacon is as follows ... 

Answer Use the postulated form of the one-dimensional velocity profile developed in part (a) and neglect the entire left side of the equation of motion for creeping flow conditions at low rotational speeds of the solid sphere. The fact that does not depend on cp, via symmetry, is consistent with the equation of continuity for an incompressible fluid. The r and 9 components of the equation of motion for incompressible Newtonian fluids reveal that dynamic pressure is independent of r and 9, respectively, when centrifugal forces are negligible. Symmetry implies that does not depend on cp, and steady state suggests no time dependence. Hence, dynamic pressure is constant, similar to a hydrostatic situation. Fluid flow is induced by rotation of the solid and the fact that viscous shear is transmitted across the solid-liquid interface. As expected, the -component of the force balance yields useful information to calculate v. The only terms that survive in the (/ -component of the equation of motion are... 

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. 

Model III relies on constant eddy viscosity (which is a function of gas velocity and column diameter) but predicts the power law form of the gas holdup profile. It accomplishes that by using both gas and liquid continuity equations, and be establishing that the cross-sectional integral of the slip velocity is a constant multiple of the superficial gas velocity. The agreement of model III with the velocity data of Hills is excellent as is its prediction of Hills s measured voidage profiles. 

Figure 2. Particle size effect on solid profiles, isoparaffin, iron oxide, N2. (a) Batch system, zero liquid velocity. (b) Continuous system, positive liquid velocity.

The experimental data also showed that when the superficial liquid velocity is zero or very small, the presence of partition plates gives the stepwise profile of solid particles. This avoids uniform distribution and gives large solid holdup. Continuous operation in this range, therefore, results in a higher mean residence time of the solid particles than that in the column without partition plates. 

In the absence of any obstacles in the channel, the simple fluid develops a parabolic profile of speed of flow the fluid flows the fastest in the center of the channel, and rests at the walls of the capillary. When a droplet or bubble is introduced into this flow, the velocity field is modified. Because the droplet (bubble) separates the continuous liquid that is in front of it, from the liquid behind it, the parabolic profile can no longer be sustained, and close to the caps of the droplet addition recirculation of the continuous fluid is created. In addition, there is circulation of the liquid inside the droplet, and there is some flow along the droplet (or bubble) (see Fig. 2). All these effects increase the viscous dissipation in the carrier (continuous) fluid and the liquid inside a droplet. As a result, it demands a higher pressure drop along the capillary to maintain the same average speed of flow as without the bubble or droplet inside of it. Equivalently, for a constant pressure drop along the capillary the speed of flow decreases after the addition of the droplet (or bubble). This can be described by an increased resistance to flow in that capillary, or by an additional charge of resistance carried by the droplet (or bubble). 

Equations 5.7 and 5.10 are valid only under the assumption of continuity of the velocity and shear stress profiles at the gas-liquid interface. If discontinuity in velocity and shear stress at gas-liquid interface is assumed, Equations 5.7 and 5.10 become... 

---