Low Slip Velocity

Finally, consider the case when the magnitude of the slip velocity between the particles and the gas is close to umf/e everywhere in the fluidized bed. With the vertical pressure drop equal to the particle weight, the following holds for any value of the particle Reynolds number, 

when u0/umf and Fr are identical for two beds and the slip velocity is close to umf/e, the dimensionless drag coefficient is also identical for two beds. 

For all three limiting cases identified above, similitude can be obtained by maintaining constant values for the dimensionless parameters, 

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Fig. 23. Evolution of the extrapolation length at low slip velocity, fr0, as a function of the surface density of grafted chains for the experiments reported in Fig. 22. The fact that b0 appears independent of o when y increases linearly with a indicates that in this range of surface densities, the surface layer has saturated the number of melt chains it can capture...

Figure 4. Evolution of the slip velocity. Vs, as a function of the top plate velocity, Vt, for a PDMS melt of molecular weigt 9.6 lO in contact with a silica surface pretreated by grafting an almost dense monolayer of OTS. The two dotted lines are respectively Vj =V, and Vs= VtA/d. Clearly, slip is always present, as indicated by the fact that the measured Vj is always above the average velocity one would have inside a layer of thickness A, in the case of a linear velocity gradient and no slip (lower dotted line). At very high shear rates V, becomes comparable to Vt and the flow is almost a plug flow. The experimental relative uncertainty on Vj is larger for the low slip velocities (close to 20%) than for the larger ones (10%).

These AC drive systems require the inverters to operate with either low-slip induction motors or reluctance-type synchronous-induction motors.. Such systems are u.sed where DC commutator motors are not acceptable. Examples of such applications are motor operations in hazardous atmospheres and high motor velocities. 

The above results show close agreement between the experimental and theoretical friction factor (solid line) in the limiting case of the continuum flow regime. The Knudsen number was varied to determine the influence of rarefaction on the friction factor with ks/H and Ma kept low. The data shows that for Kn < 0.01, the measured friction factor is accurately predicted by the incompressible value. As Kn increased above 0.01, the friction factor was seen to decrease (up to a 50% X as Kn approached 0.15). The experimental friction factor showed agreement within 5% with the first-order slip velocity model. 

Liquid metals represent a special case in the estimation of two phase void fractions. Because of the great differences in vapor and liquid density, very low qualities correspond to high void fractions, and for the same reason very large slip velocity ratios occur. In a recent paper. Smith et al. (S13) review previous measurements of void fractions in... 

The experimental data obtained at low surface densities, for end grafted chains, are in very good agreement with these theoretical predictions, not only for the overall evolution of the slip velocity vs the shear rate or of the slip length vs the slip velocity as shown in Fig. 19, but also for the molecular weight dependence of the critical velocity V which do follow exactly the laws implied by... 

Chen8,9 studied the gas holdup of a 7-cm i.d. 244-cm long column randomly packed with open-end screen cylinders of various sizes (1.27 cm x 1.27 cm and 1.9 cm x 1.9 cm) and screen meshes (8-14 mesh). The results with an air-water system were obtained in the bubble-flow regime. The screen cylinders were found to reduce the gas holdup. The results showed that for t/0g < 4 cm s, the gas holdup was a linear function of gas velocity, a result similar to the one obtained in an unpacked bubble-column bul not in a column packed with Raschig rings or other conventional packings. He also showed that for low gas velocity, l/0G < 3.64 cm s 1 the parameter (hG - 1ig)//ig was a unique linear function of liquid velocity (independent of gas velocity). Here, /iG is the gas holdup at zero liquid velocity. He also obtained a relationship between the gas holdup and the slip velocity between gas and liquid. All the data were graphically illustrated, however, no analytical correlation was presented. 

Terminal velocity-slip velocity theory In this theory, the steady slip velocity between solid and liquid is used in the correlation for the Sherwood number. For low particle Reynolds number, Friedlander33 gave... 

For low flow regimes, the absorbed chains at the wall are entangled with the bulk polymer. Slippage is then very small and wall stress increases with slip velocity. 

It should be underlined that it was possible to obtain slip at low stress values, by considering the flow of a polydimethylsiloxane (PDMS) through a silica die with walls grafted by a fluorinated monolayer [11]. It can also be found in the experimental study published in 1993 by Migler et al. [12], which moreover validates the model prediction in term of extrapolation length, for sufficiently high slip velocities. 

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 following method is recommended for calculating slip velocity in static extractors at low dispersed-phase holdup ... 

Figures 17b and 17c show the response in the lateral and normal directions to a lateral constant velocity drive for the stick slip regime that occurs at low driving velocities. This behavior is similar for the presently discussed model. The separation between the plates, which is initially Zq at equilibrium, starts growing before slippage occurs and stabilizes at a larger interplate distance as long as the motion continues. Since the static friction is determined by the amplitude of the potential corrugation exp(l — Z/A), it is obvious that the dilatancy leads to a decrease of the static friction compared to the case of a constant distance between plates.

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