Slip velocity, definition

Using the definition of slip velocity given by equation (1), it is seen that the Field-Davidson model requires P to be identically zero, while the Joshi-Sharma modification suggests the unrealistic result that P is negative if the slip velocity definition is obeyed. 

What this shows is that, from the definition of off-bottom motion to complete uniformity, the effect of mixer power is much less than from going to on-bottom motion to off-bottom suspension. The initial increase in power causes more and more solids to be in active communication with the liquid and has a much greater mass-transfer rate than that occurring above the power level for off-bottom suspension, in which slip velocity between the particles of fluid is the major contributor (Fig. 18-23). 

Substituting these expressions into the definition of the slip velocity (and dividing by the entering velocity, V, to make the results dimensionless) gives... 

We can then solve for the slip velocity, using the definition of viscosity as p = —pu X, where X is... 

In (3.427) the dispersed phase velocity occurs as an undetermined variable. The phasic velocities are related to the mixture velocity through the mixture velocity definition (3.421). The dispersed phase velocity is computed from the continuous phase velocity Vc and a relative (slip) velocity v fc, in accordance with the definitions ... 

The just-suspended state is defined as the condition where no particle remains on the bottom of the vessel (or upper surface of the liquid) for longer than 1 to 2 s. At just-suspended conditions, all solids are in motion, but their concentration in the vessel is not uniform. There is no solid buildup in comers or behind baffles. This condition is ideal for many mass- and heat-transfer operations, including chemical reactions and dissolution of solids. At jnst-snspended conditions, the slip velocity is high, and this leads to good mass/heat-transfer rates. The precise definition of the just-suspended condition coupled with the ability to observe movement using glass or transparent tank bottoms has enabled consistent data to be collected. These data have helped with the development of reliable, semi-empirical models for predicting the just-suspended speed. Complete suspension refers to nearly complete nniformity. Power requirement for the just-suspended condition is mnch lower than for complete snspension. 

In the foregoing, the reservations against the terminal velocity-slip velocity approach of Harriott (1962a) have been elaborated. This was, however, an important stage of theorizing the role of turbulence in mass transfer processes. It was a definite improvement on the dynamic similarity or power input-based empirical correlations. As pointed out by Nienow (1975), the estimation of the characteristic slip velocity involves a complicated process. Further, this approach cannot be used when the solid-fluid density difference approaches zero. 

To estimate (ys), one may note that the velocity profile in the liquid is influenced by the soUd zones only in a region of their size, a, in all directions (see Fig. 1) this behavior actually reflects the Laplacian character of the Stokes equation obeyed by the fluid velocity. One, therefore, expects (ys) U/a, where U is the slip velocity of the fluid on the shear-free zones, so that we eventually obtain Ff = A(f>sr ilJ/a. Now if one recalls the definition of the effective slip length, as given by the Navier BC, Ff also reads Ff = ArnV/bee, with F [/ the averaged slip velocity over the superhydrophobic surface. Combining the two independent estimates, one deduces ... 

Equation (8.1) represents the solids velocity at choking and includes the assumption that the slip velocity UsUp is equal to Uj (see Section 8.1.4 below for definition of slip velocity). Equations (8.1) and (8.2) must be solved simultaneously by trial and error to give ecu and LZch-... 

We have to be careful in the definition of gas and particle velocities and in the relative velocity between them, the slip velocity. The terms are often used loosely in the literature and are defined below. 

Here, Pc is the mixture density of the dense phase. U up i is defined by J Uf-U/), where Uf and U are mean velocities of the dilute and dense phases, respectively. This definition of mesoscale slip velocity differs a little bit from that in the cluster-based EMMS model, because the continuous phase transforms from the dilute phase to the dense phase. And their quantitative difference is l-f)PgUgc/Pc, which is normally negligible for gas-solid systems. Similarly, the closure of Fdi switches to the determination of bubble diameter. And it is well documented in literature ever since the classic work of Davidson and Harrison (1963). Compared to cluster diameter, bubble diameter arouses less disputes and hence is easier to characterize. The visual bubbles are normally irregular and in constantly dynamic transformation, which may deviate much from spherical assumption. Thus, the diameter of bubble here can also be viewed as drag-equivalent definition. 

MjUi -D = -UitijNj + qiNj, or with (4.14) and the definition of the slip velocity. 

Upon evaluating this at L/2, using the definition of slip velocity (4.27), and Equation 4.29, the relationship between the boundary velocity, fluctuation velocity, and stress ratio may be written as... 

The definition of friction factor using mean fluid properties has been most widely used because it reduces to the correct single-phase value for both pure liquid and pure gas flow. This technique is very similar to the so-called homogeneous model, because it has a clear physical significance only if the gas and liquid have equal velocities, i.e., without slip. Variations of this approach have also been used, particularly the plotting of a ratio of a two-phase friction factor to a single-phase factor against other variables. This approach is then very similar to the Lockhart-Martinelli method, since it can be seen that (G4)... 

One consequence of the inability of the continuum description to resolve the region nearest the boundary is that the continuum variables extrapolated toward the boundary from the two sides may experience jumps or discontinuities. This is definitely the case at a fluid interface, as we shall see. Even at a stationary, solid boundary, the fluid velocity u may appear to slip when the fluid is a high-molecular-weight material or a particulate suspension.3... 

Clearly, the solution breaks down in the limit r —> 0. In fact, according to (7-80), an infinite force is necessary to maintain the plane = 0 in motion at a finite velocity U, and this prediction is clearly unrealistic. Presumably, one of the assumptions of the theory breaks down, although a definitive resolution of the difficulty does not exist at the present time. The most plausible explanation is that the no-slip boundary condition is inadequate in regions of extremely high shear stress. However, as discussed in Chapter 2, this issue is still subject to debate. 

The relationship between holdup and slip (relative) velocity between the two phases is, by definition,... 

Extension to 2D and 3D Systems In the majority of microfluidic cases where 1/k is much smaller than the channel height, the Helmholtz-Smoluchowski equation provides a reasonable estimate of the flow velocity at the edge of the double layer field. As such when modeling two- and three-dimensional flow systems, it is common to apply this equation as a slip boundary condition on the bulk flow field. Since beyond the double layer by definition... 

For the pendant drop in figure I, the equations above are subjected to the no-slip boundary conditions at solid surfaces and the kinematic condition on the free surfaces. The kinematic condition implies that there is no liquid crossing the boundary into the gas phase, or in other words forms a definite boundary between the phases. For creeping flows into an atmosphere of gas with minimal velocities there will be no interfacial shear stress tangential to the surface and the normal stress inside the fluid is balanced by the surface tension as described by the famous Young-Laplace equation. 

---