Hydrodynamics fluid-particle

Broadly speaking, for G/S systems, three modes of particle-fluid contacting may be recognized to take place simultaneously as shown in Fig. 43 bubbles containing sparsely disseminated particles, emulsion of densely suspended particles, and defluidized (transient as well as persistent) particles not fully suspended hydrodynamically by the flowing gas. For all intents and purposes, it is desirable to suppress bubbles and to prevent defluidization. 

The motion of a particle in the flow field can be described in the Lagrangian coordinate with the origin placed at the center of the moving particle. There are two modes of particle motion, translation and rotation. Interparticle collisions result in both the translational and the rotational movement, while the fluid hydrodynamic forces cause particle translation. Assuming that the force acting on a particle can be determined exclusively from its interaction with the surrounding liquid and gas, the motion of a single particle without collision with another particle can be described by Newton s second law as... 

Moreover, the influence of the motions of the particles on each other (i.e., when the motion of a particle affects those of the others because of communication of stress through the suspending fluid) can also influence the measured diffusion coefficients. Such effects are called hydrodynamic interactions and must be accounted for in dispersions deviating from the dilute limit. Corrections need to be applied to the above expressions for D and Dm when particles interact hydrodynamically. These are beyond the scope of this book, but are discussed in Pecora (1985), Schmitz (1990), and Brown (1993). 

The EMMS model was first proposed for the hydrodynamics of concurrent-up particle-fluid two-phase flow. Though it is based on a rather simplified physical picture of the complex system (Li, 1987 Li and Kwauk, 1994), it harnesses the most intrinsic complexity in the system, the meso-scale heterogeneity, and this is why it allows better predictions to the critical phenomena in the system which is obscured in other seemingly more comprehensive models. 

G.K. Batchelor, Brownian diffusion of particles with hydrodynamic interaction, J. Fluid Mech. 74 (1976) 1-29. 

An appropriate understanding of particle-fluid two-phase flow rests on an adequate analysis of the local hydrodynamics corresponding to the scale of bubbles or clusters, as well as the overall hydrodynamics corresponding to the scale of the retaining vessel. Local hydrodynamics, designated as phases, is rooted in the intrinsic characteristics of particle-fluid systems,... 

This section will first deal with the phases in particle-fluid two-phase flow by developing a mathematical model to quantify local hydrodynamic states. This analysis will reveal the insufficiency of the conditions for the conservation of mass and momentum alone in determining the hydrodynamic states of heterogeneous particle-fluid systems, and calls for a methodology different from what is used in analyzing dilute uniform flow. For this purpose the concept of multi-scale interaction between particles and fluid and the principle of energy minimization are proposed. 

The six equations in F,(XXi = 1, 2,..., 6) just described in terms of force balance and continuity, are, however, not sufficient to determine the hydrodynamic state of a heterogeneous particle-fluid system, because with these, multiple solutions would result among which only one is valid to represent the stable state. An additional condition is needed to define this solution. 

If we consider the set of six equations on mass and momentum conservation on the one hand, and the characteristic energy extrema for stability of the three broad regimes of operation on the other, mathematical modeling for local hydrodynamics of particle-fluid two-phase flow beyond minimum fluidization needs therefore to satisfy the following constraints ... 

While local hydrodynamics of particle-fluid two-phase flow—phases, regimes and patterns—is concerned with its intrinsic stability, of direct engineering significance is its overall hydrodynamics, which deals with its space-dependent characteristics subject to the boundary conditions set by the retaining vessel. For the axisymmetric equipment generally employed in engineering, overall hydrodynamics is resolved into the axial and the radial... 

The theoretical description of translational diffusion in a lipid bilayer depends on the size of the diffusing particle. Theoretical descriptions based on fluid hydrodynamic theory (51, 52) have been shown to be applicable to particles whose radius in the plane of the bilayer is significantly larger than the radius of the lipid molecules that constitute the bilayer, in which case the diffusion coefficient may be given by ... 

The main assumptions made in arriving at eq 10 were (1) no interaction between the globules and (2) no slippage at the particle-fluid interface. Most emulsions of practical interest exceed the concentration for which eq 10 is valid. With increasing concentration, the hydrodynamic interaction between the globules increases, and eventually mechanical interference occurs between the particles as packed-bed concentrations are approached. To take into account the increased hydrodynamic interaction, many investigators (30, 38-43) have expanded eq 10 into the polynomial form ... 

For the mechanical behavior of two particle-fluid systems to be simitar, it is necessary to have geometric, hydrodynamic, and particle trajectory similarity. Hydrodynamic similarity is achieved by fixing the Reynolds number for the flow around the collector. By (4.26), similarity of the particle trajectories depends on the Stokes number. Trajectory similarity also requires that the particle come within one radius of the surface at the same relative location. This means that the interception parameter, R = dp/L, must also be preserved. 

The dynamic particle-fluid interaction is characterized by the drag force, which is defined as the projection of the resultant of all hydrodynamic forces on the flow direction ... 

The heat transfer characteristics in multiphase systems depend strongly on the hydrodynamics, which vary significantly with particle properties. The particle size, size distribution, and shape affect the particle and fluid flow behavior through particle-fluid and particle-particle interactions. A discussion of the hydrodynamic characteristics of packed and fluidized beds follows. 

On the other hand, k is the mass transfer coefficient for diffusion and should be capable of being estimated from one of the many mass transfer relationships available. Thus the limiting case of diffusion control can be determined theoretically or experimentally. The main difficulty in the theoretical approach is in deciding which of the many mass transfer equations to use and what is the particle-fluid slip (or relative) velocity for the complex hydrodynamics in the stirred tank. 

Reciprocal screening length Larger-to-smaller particle size ratio Characteristic wavelength Stokes law correction factor Electrochemical potential of an ion Reference electrochemical potential of an ion Chemical potential of a particle Reference chemical potential of a particle Fluid kinematic viscosity Hydrodynamic pressure tensor Pressure between plates (disjoining pressure)... 

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