Slip velocity of particles

When a 1, the slip velocity of particles relative to the gas is negligible in comparison with the particle velocity, so that 9f/ fe is nearly equal to <7s/cse- With this equality Eq. (6-8) is modified to the form ... 

The particle Reynolds number based on maximum terminal velocity in air (maximum slip velocity of particles suspended in air) can be estimated by... 

Solid-Liquid Mass Transfer There is potentially a major effect of both shear rate and circulation time in these processes. The sohds can either be fragile or rugged. We are looking at the slip velocity of the particle and also whether we can break up agglomerates of particles which may enhance the mass transfer. When the particles become small enough, they tend to follow the flow pattern, so the slip velocity necessary to affect the mass transfer becomes less and less available. 

From this, the velocities of particles flowing near the wall can be characterized. However, the absorption parameter a must be determined empirically. Sokhan et al. [48, 63] used this model in nonequilibrium molecular dynamics simulations to describe boundary conditions for fluid flow in carbon nanopores and nanotubes under Poiseuille flow. The authors found slip length of 3nm for the nanopores [48] and 4-8 nm for the nanotubes [63]. However, in the first case, a single factor [4] was used to model fluid-solid interactions, whereas in the second, a many-body potential was used, which, while it may be more accurate, is significantly more computationally intensive. 

For particles ranging in size from 30 nm to 2 pm, the particle dissolution is influenced mainly by Brownian motion and experimentally Dr values were found to vary from 1.0 to 2.0. For bigger particles (non-Brownian) between 10 pm and 5 mm, dissolution is mainly controlled by the relative slip velocities between particles and surrounding fluid. The bigger the particles, the higher is the relative slip velocity resulting in faster dissolution rate. Therefore values of Dr for these particles (2.0 < Dr < 3.0) are higher than those affected by Brownian motion. For both the Brownian and non-Brownian particles, the dissolution is by erosion of the external surface. 

The zeta-potential of the particle refers to the potential at the outer limit of the boundary layer, often called the shear plane or slipping plane. It may be determined from measurement of the velocity of particles in an electric field E. Since the precise location of the shear plane is difficult to define, the zeta-potential is an ambiguous measure of the potential at the surface of the particle (Figure 3.5). 

Equations (167) and (168) express the fact that the force and torque on each particle are linear vector functionals of the slip velocities of all the particles. It should be clearly noted that these operators are calculable solely from a knowledge of the intrinsic translational and rotational solutions of Stokes equations resulting from motion of the yth particle when all the other particles, as well as the fluid at inflnity, are at rest. In the case where u = 0, Eqs. (167) and (168) reduce to (148) and (149), respectively. As yet, there exists no system for which these operators are known. 

Another quantity of importance is the axial slip velocity of the particle,... 

The kinematics and dynamics boundary conditions at the interfaces close the hydrodynamic problem (l)-(2). On the solid-liquid boundary the non-slip boundary conditions are applied -the liquid velocity close to the particle boundary is equal to the velocity of particle motion. In the case of pure liquid phases the non-slip boundary condition is replaced by the dynamic boundary condition. The tangential hydrodynamic forces of the contiguous bulk phases, nx(P+Pb) n, are equal from both sides of the interface, where n is the unit normal of the mathematical dividing surface. The capillary pressure compensates the difference between the... 

The sedimentation velocity of particles under gravity can be measured and compared with the predicted values based on no-slip and slip flow boundary conditions. The ratio of the sedimentation velocity as a function of slip length can be derived as... 

Concerning the liguid/solid mass transfer coefficient ks, two different theories have been used to correlate data from so-lid/liquid suspensions one uses the terminal slip velocity between particles and the liquid. The other approach is based on Kolmogoroff s theory of isotropic turbulence and uses energy dissipation rate C= Go8 correlation. Sanger [30] recommend the equation... 

Harriott s data on in stirred tanks were found to be 1.5-8 times higher as compared to those estimated from the steady-state correlations of Ranz and Marshall (1952b) and Friedlander (1957) using the particle Reynolds number based on in Equations 6.14 and 6.15, respectively. In view of the complex procedure for calculating the true slip velocity of a particle suspended in stirred tank from Equation 6.16, Harriott (1962a) suggested a modified, simple procedure to estimate (i) compute... 

Once the slip velocity is known, the drift flux can be obtained by using Eq. (128), which is termed the hydrodynamic curve. On the other hand, for the given operating conditions with specified superficial velocities of particles and fluid, the drift flux can also be directly obtained through the operating line given by... 

By substituting the relative velocity, or slip velocity, between particles and fluid for the velocity of classical fluidization, Kwauk (1963,1992) extended... 

An average upward particle velocity in the core, V p, is defined and can be obtained, making the usual assumption that the slip velocity of the particles in the core is equal to their terminal velocity. 

The great challenge now is how to translate the fundamental expression of the second term of Eq. (5) into an equation— in terms of known variables (particle size, slip velocity, physical properties) and/or the particle Reynolds number (in terms of slip velocity and particle size)— that can be used with confidence in the stability analysis or computational simulation of interest. In computational simulations, spatial and temporal resolution may create additional issues as these scales may interfere with the time and length scales of fluid flow and particle behavior. 

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