The Slip Velocity Approach

According to Harriott, Equations 6.14 and 6.15 should yield the minimum value of the mass transfer coefficient, (value of atu= Vp). Harriott s (1962a) smdy can be broadly divided into two parts (i) the effect of physical properties, in particular the liquid viscosity, density, and solute diffusivity (components of Schmidt number) along with the particle diameter, was derived from the variation of as calculated from Equation 6.14 or 6.15 for the respective parameters, (ii) The effect of power input (obtained from the impeller diameter, type, and speed of agitation) was derived 

For very large particles such that settling occurs in Newton s law region, is 

On the other hand, for very small particles (Rep lor 0), mass transfer can be depicted by diffusion in an infinite quiescent medium. For such a situation, the limit of Shp=2 implicit in Ranz and Marshall s correlation suggests that with decreasing particle diameter, must increase in a hnearly reciprocal manner. 

FIGURE 6.2 Shows solid-liquid mass transfer coefficients for particles settling in quiescent water. (Reproduced with permission from Harriott (1962a). American Institute of Chemical Engineers. 1962.) 

The dependence of on the density difference between the solid and liquid 

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In the presence of a surfactant (Terpineol), Rice et al. (2) obtained a very good fit between theory and experiment using ( )( ) = (l- ) which is the Wallis model with n = 2. In the work cited, average bubble size was around 1 mm diameter. This particular structure shows, according to equation (3), that the slip velocity approaches terminal rise velocity as voidage becomes small, as one expects. 

Cartesian coordinates. Furthermore, the following analyses are restricted to microchannels of a rectangular cross section, 2w x 2 h (width by height), which is close to the real shape of microchannels made using the microfabrication technologies. With these considerations, the model, based on the slip velocity approach, describing the velocity field of such flow can be further simplified to... 

Stokes second problem and is specifically referred to as the slip velocity approach in electroosmotic flows. More general discussions of the applicability of such slip velocity approach in electrokinetic flows were provided elsewhere [6]. Using the slip velocity approach, the steady velocity field of a fully developed flow driven by an applied electric field, E, and a pressure gradient, dp/dz, is governed by the Stokes equation, expressed as... 

Equation 14 shows that for steady, fully developed electroosmotic flow in an open-end rectangular microchannel, the slip velocity approach leads to a pluglike velocity profile, given by the Smoluchowski equation ... 

As flooding is approached, the slip velocity continues to decrease until at the flood point is zero and the following relationship apphes ... 

The hydrodynamics were modelled simply, using an average hold-up and slip velocity approach... 

To include the slip in this model, we use the slip length. In this approach, if slip occurs, the slip velocity v, is proportional to the velocity gradient at the wall... 

As outlined above, steady-state theories for the liquid-solid mass transfer are largely classified into two categories i.e., those based on Kolmogoroff s theory and those based on the terminal velocity-slip velocity approach. 

The model based on terminal and slip velocity approach is rather tenuous. It breaks down as the density difference between liquid and solid approaches zero. Under highly turbulent conditions, an accurate estimation of slip velocity is rather difficult, and there is disagreement on whether or not the relative velocity between the solid and liquid alone is enough to obtain an accurate estimate of the mass-transfer coefficient. 

Figure 6. Slip length b deduced from the data of Fig. 4 and 5 as a function of the slip velocity Vg. The threshold for the onset of strong slip appears as a kink in the b (Vg) curve, at the critical velocity V. Above V, b increases with Vg. following a power law with an exponent 0.8 0.04. At very high shear rates, b deviates from the power law and the Vg dependence tends to saturate, indicating that a new regime of linear strong slip is approached.

The results set out in 3. show that a fiiction law must make allowance for the remarks made in 3.2 in order to represent fiiction with macroscopic shp in the case of polymer melts. An initial approach was made by Chernyak and Leonov in 1986 [18], and then Leonov in 1990 [20]. They proposed relations for modelling the bellshaped curve with its maximum and minimi minima. It appeared worthwhile to adapt these relations to take into accoimt the existence of a positive stress at rest and the decrease in stress at the wall when the slip velocity increases, for low shp regimes. With given temperature and pressure, these relations are written as follows ... 

The other approach is the terminal velocity-slip velocity theory . It is postulated that the slip velocity will be of the order of the particle terminal velocity uj when the agitator speed is Nj. Subsituting, in equation (17.13c), a mass transfer coefncient k can be calculated and if the postulate is correct kn determined experimentally should be constant and of the order of ky. As already indicated, is found to be constant and in addition kjs /kj is found to... 

The curves show that the velocity at the membrane surface (A = 1) is 0 when is 0 as expected. As the slip velocity increases with increasing (f), the wall shear decreases, and the velocity profiles become flatter, approaching those for plug flow [12-14]. The effect of slip coefficient on axial pressure gradient (P) is as follows an increase in (j) results in a decrease in wall shear stress so that P also decreases. In addition, the transition from laminar to turbulent flow at a porous surface occurs at a Re of less than 2000, which is also the case with membrane systems. 

Velocity Distribution of the Steady Electroosmotic Flow in Open- and Closed-End Microchannels Slip Velocity Approach... 

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. 

The particle-fluid slip velocity approach based on Tchen s equation is a refinement of the terminal settling velocity-slip velocity approach (e.g., Lee 1981, 1984). However, most investigators made simplifying assumptions that may not be tenable. The assumptions were inevitable since the original equation is far too complex. Even with these simplifications, determination of characteristic particle-fluid slip velocity (representative velocity) responsible for mass transfer still required numerical solution of the ranaining equations. Further, it required information on turbulence characteristics over a wide range of conditions and is therefore tedious. 

Some phenomenological approaches for handling the slip boundary condition have been proposed, where the slip velocity is takes as an empirical function of the wall shear stress Rosenbaum and Hatzikiriakos (1997) introduced a power-law relation for slip velocity ... 

When the size of a particle approaches the same order of magnitude as the mean free path of the gas molecules, the setthng velocity is greater than predicted by Stokes law because of molecular shp. The slip-flow correc tion is appreciable for particles smaller than 1 [Lm and is allowed for by the Cunningham correc tion for Stokes law (Lapple, op. cit. Licht, op. cit.). The Cunningham correction is apphed in calculations of the aerodynamic diameters of particles that are in the appropriate size range. 

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