Particles interaction between fluids

A simple model for interactions between particles in an associating bulk fluid consists of a particle-particle potential and the interactions between sites belonging to different molecules. Supposing that each molecule has M sites, the potential of interaction between molecules 1 and 2 is [14]... 

Now, we would like to investigate adsorption of another fluid of species / in the pore filled by the matrix. The fluid/ outside the pore has the chemical potential at equilibrium the adsorbed fluid / reaches the density distribution pf z). The pair distribution of / particles is characterized by the inhomogeneous correlation function /z (l,2). The matrix and fluid species are denoted by 0 and 1. We assume the simplest form of the interactions between particles and between particles and pore walls, choosing both species as hard spheres of unit diameter... 

Detailed consideration of the interaction between particles and fluids is given in Volume 2 to which reference should be made. Briefly, however, if a particle is introduced into a fluid stream flowing vertically upwards it will be transported by the fluid provided that the fluid velocity exceeds the terminal falling velocity m0 of the particle the relative or slip velocity will be approximately o- As the concentration of particles increases this slip velocity will become progressively less and, for a slug of fairly close packed particles, will approximate to the minimum fluidising velocity of the particles. (See Volume 2, Chapter 6.)... 

As the concentration is increased above q> 0.01, hydrodynamic interactions between particles become important. In a flowing suspension, particles move at the velocity of the streamline corresponding to the particle centre. Hence particles will come close to particles on nearby streamlines and the disturbance of the fluid around one particle interacts with that around passing particles. The details of the interactions were analysed by Batchelor17 and we may write the viscosity in shear flow as... 

These generally behave as non-Newtonian fluids. The Einstein theory has been extended to this case by considering the hydrodynamic interaction between particles. Formulae of the following type are proposed ... 

In gas-solid flows, flow patterns of both phases depend not only on the initial conditions and physical boundaries of the system but also on the mechanisms of momentum transfer or the interacting forces between the phases. The forces controlling the motions of particles may be classified into three groups (1) forces through the interface between fluid and particles, (2) forces due to the interactions between particles, and (3) forces imposed by external fields. Although interparticle forces and field forces do not directly change the course of the fluid motion, they may indirectly influence the motion via particle-fluid interactions. 

Compared to small molecules the description of convective diffusion of particles of finite size in a fluid near a solid boundary has to account for both the interaction forces between particles and collector (such as van der Waals and double-layer forces) and for the hydrodynamic interactions between particles and fluid. The effect of the London-van der Waals forces and doublelayer attractive forces is important if the range over which they act is comparable to the thickness over which the convective diffusion affects the transport of the particles. If, however, because of the competition between the double-layer repulsive forces and London attractive forces, a potential barrier is generated, then the effect of the interaction forces is important even when they act over distances much shorter than the thickness of the diffusion boundary layer. For... 

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 interaction between particles and fluids is important in many industrial processes. The most relevant size distribution in such cases is, therefore, based on the sedimentation behavior of particles in fluids and the applicable size is the settling or Stokes diameter. 

The effect of the particle movement on the fluid phase, and interactions between particles are neglected. This is justified because the number of particles in the freeboard is low. Particles are allowed to bounce off walls, where a coefficient of restitution of 0.8 is applied, which is based on the elasticity of sand and metal collisions but can be varied considerably (from 0.5 to 0.9) without affecting the entrainment results noticeably. 

An optimal choice of weights can be found by measuring the local dif-fusivity of a random walk along the reaction coordinates and applying the feedback method to shift weight towards the bottlenecks in the simulation. This generalized ensemble optimization approach has recently been illustrated for the simulation of dense Lennard-Jones fluids close to the vapor-liquid equilibrium [21]. The interaction between particles in the fluid is described by a... 

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