Velocity shear-induced

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

In the stirred tank, the final mean size of particles was reduced by the increase of stirring rate, being consistent with increased fluid shear induced particle disruption relative to aggregation. Use of three different gas velocities in the bubble column, however, results in no significant difference in agglomerate size but since the size is relatively small, it may simply reflect an asymptotic value. 

The velocity, viscosity, density, and channel-height values are all similar to UF, but the diffusivity of large particles (MF) is orders-of-magnitude lower than the diffusivity of macromolecules (UF). It is thus quite surprising to find the fluxes of cross-flow MF processes to be similar to, and often higher than, UF fluxes. Two primary theories for the enhanced diffusion of particles in a shear field, the inertial-lift theory and the shear-induced theory, are explained by Davis [in Ho and Sirkar (eds.), op. cit., pp. 480-505], and Belfort, Davis, and Zydney [/. Membrane. Sci., 96, 1-58 (1994)]. While not clear-cut, shear-induced diffusion is quite large compared to Brownian diffusion except for those cases with very small particles or very low cross-flow velocity. The enhancement of mass transfer in turbulent-flow microfiltration, a major effect, remains completely empirical. 

In order to obtain solutions with the desired flow properties, shear-induced degradation should be avoided. From mechanical degradation experiments it has been shown that chain scission occurs when all coupling points are loose and the discrete chains are subjected to the velocity field. Simple considerations lead to the assumption that this is obtained when y) is equal to T sp(c-[r ]) (Fig. 18). The critical shear rate can then easily be evaluated [22]. 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

Leighton, D. Acrivos, A. Viscous resuspension. Chem. Eng. Sci. 1986, 41 (6), 1377-1384. Leighton, D. Acrivos, A. Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. Fluid Mech. 1987, 177, 109-131. Leighton, D. Acrivos, A. The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 1987, 181, 415-439. Altobelli, S.A. Givler, R.C. Fukushima, E. Velocity and concentration measurements of... 

The DMF module (PaU Corp., New York) which consists of several disks mounted on the same shaft [2, 135]. The reported studies of the apphcation of the rotating disk dynamic membrane indicate that high shear-induced filtration is much less sensitive to the solids concentration. Advantage of the rotational system is that it permits operation at both very low transmembrane pressure-drop, and low upstream mass velocity, without loss in depolarization efficiency. It is thus possible with this system to achieve cleaner separation of solute components, than is achievable with conventional systems. Equipment for this process is, unfortunately, significandy more costly, and maintenance costs much higher also, than those for conventional membrane systems. 

Here the radii of the two colliding particles are termed af and ay the mean velocity gradient inducing contacts by fluid shear is given as G. Other parameters are as defined in Table 1. 

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