Particle interaction forces

The fluid-particle interaction force, omitting the virtual mass term and combining the pressure terms in the equation of motion becomes... 

The volumetric fluid-particle interaction force F0 in Eq. (28) is calculated from the forces acting on the individual particles in a cell ... 

Note that depending on the manner in which the drag force and the buoyancy force are accounted for in the decomposition of the total fluid particle interactive force, different forms of the particle motion equation may result (Jackson, 2000). In Eq. (36), the total fluid-particle interaction force is considered to be decomposed into two parts a drag force (fd) and a fluid stress gradient force (see Eq. (2.29) in Jackson, 2000)). The drag force can be related to that expressed by the Wen-Yu equation, /wen Yu> by... 

Equations (5.139) to (5.142) are the basic equations for a gas-solid flow. More detailed information on both the fluid-particle interacting force Fa and the total stresses T and Tp must be specified before these equations can be solved. One approach to formulate the fluid-particle interacting force FA is to decompose the total stress into a component E representing the macroscopic variations in the fluid stress tensor on a scale that is large compared to the particle spacing, and a component e representing the effect of detailed variations of the point stress tensor as the fluid flows around the particle [Anderson and... 

In a packed bed, the fluid-particle interaction force is insufficient to support the weight of the particles. Hence, the fluid that percolates through the particles loses energy due to frictional dissipation. This results in a loss of pressure that is greater than can be accounted for by... 

Because the surface-particle interaction forces are important only near to the surface of the collector, it is natural to divide the domain into two regions. The inner region, called the interaction-force boundary layer, has a thickness determined by the range over which interaction forces are important (less than about 10-5 cm). Because of the proximity of the surface, convection is negligible. Outside the... 

In this paper it is shown that the rate of deposition of Brownian particles on the collector can be calculated by solving the convective diffusion equation subject to a virtual first order chemical reaction as a boundary condition at the surface. The boundary condition concentrates the surface-particle interaction forces. When the interaction potential between the particle and the collector experiences a sufficiently high maximum (see f ig. 2) the apparent rate constant of the boundary condition has the Arrhenius form. Equations for the apparent activation energy and the apparent frequency factor are established for this case as functions of Hamaker s constant, dielectric constant, ionic strength, surface potentials and particle radius. The rate... 

Fig. 6. Effect of the solvent quality on the PDMS suriace layer thickness and particle-particle interaction force a) PDMS surface layer swelling due to good solvency of the liquid medium b) compression of PDMS layer due to low solvency of the liquid medium.

This term is often referred to as a particle-particle interaction force and has the effect of keeping the particles apart above a maximum possible particle packing. The particle-particle interaction coefEcient G ag) is named the modulus of elasticity. A survey of different particle-particle interaction force models are given by Massoudiet et al [98]. 

Enwald and Almstedt [40] adopted a relation for the particle-particle interaction force proposed by Bouillard et al [16] ... 

The change of momentum for a particle in the disperse phase is typically due to body forces and fluid-particle interaction forces. Among body forces, gravity is probably the most important. However, because body forces act on each phase individually, they do not result in momentum transfer between phases. In contrast, fluid-particle forces result in momentum transfer between the continuous phase and the disperse phase. The most important of these are the buoyancy and drag forces, which, for reasons that will become clearer below, must be defined in a consistent manner. However, as detailed in the work of Maxey Riley (1983), additional forces affect the motion of a particle in the disperse phase, such as the added-mass or virtual-mass force (Auton et al., 1988), the Saffman lift force (Saffman, 1965), the Basset history term, and the Brownian and thermophoretic forces. All these forces will be discussed in the following sections, and the equations for their quantification will be presented and discussed. 

Generally, there are six important types of particle-particle interaction forces that can exist in a dispersion as summarized in Table 7.1. 

Zypman FR (2006) Exact expressions for colloidal plane-particle interactions forces and energies with applications to atomic force microscopy. J Phys Condens Matter 18 2795-2803... 

Wherever particles are involved, the problem of fouling, or deposition on surfaces arises. In the case of heat exchangers for combustion gases, thermal transfer efficiency may be drastically reduced by the deposition of the relatively highly insulating soot and flyash particles. In other contexts sulfuric acid and other corrosive vapor droplets diffuse or impact upon conduit surfaces thereby shortening their useful lifetime [1.22,25]. In all of these cases, numerous questions of kinetic theory arise including all the phoretic forces. In addition, particle interaction forces are ultimately responsible for delivery to the surface in question. 

In biophysics, the questions of cellular and membrane interactions are important. Here, particle interaction forces are very similar, and much the same type of treatment is of relevance in both cases [1.45-47]. 

During the last decade or so, new knowledge has emerged from research both within and outside aerosol investigations that refines the conventional picture s domains of applicability and extends the understanding of aerosol phenomena. These considerations have important implications for very practical questions some of which were discussed in Chap.l. Likewise, model calculations on transport to transition and free-molecular regime particles clarify the importance of the details of particle interaction forces and particle transport and may have implications for coagulation, condensation, and sorption processes of aerosols. 

These recent advances in the understanding of particle interaction forces are due to improved definition of the pertinent variables. Therefore, this review will begin with an inventory and discussion of the variables affecting aerosol particle interaction forces. 

When the forces between a particle and another condensed species are treated, thermodynamics and statistical mechanics of the aerosol (particles plus gas) enter through temperature dependence of the interaction forces. However, actual aerosol particle interaction forces may be altered in a fundamental way if one or both of the particles or surfaces absorb molecules from the suspending gas. ASH et al. [5.4] considered nonionic systems in which the relative velocity of the particle and surface or other particle is "sufficiently small, in relation to the rates of absorption and desorption, that absorption equilibrium is maintained as the particles move together, collide and then either adhere or separate." They, therefore, assume constant temperature for the entire aerosol system implying at least several nonabsorbing gas molecular collisions with the sorbent species between each sorbate interaction that is to say, the sorbate must be a minority (< 10 percent) species in the gas. By use of conventional equilibrium thermodynamics they derive the expression for the excess force (beyond van der Waals and electrostatic) between two bodies due to sorption as... 

The interactions of uncharged species have been touched upon above. Since the van der Waals forces dominate these interactions, they will be discussed at length in the final section of this chapter. It suffices here to say that these forces arise from the frequency-dependent electric and magnetic susceptibilities of the interacting species, and it is precisely these susceptibilities which are responsible for the spectral properties of the molecules comprising the particle. Thus, molecular (or chemical) specificity of particle interaction forces is of central importance. From another standpoint, this can be understood by considering an individual molecule as the limiting case of a particle. Then for the intermolecular van der Waals force to be consistent as two such particles (molecules) approach to the point of orbital overlap, their interaction must reduce to the relevant chemical interaction force which is fundamentally dependent upon chemical specificity. 

The simplest van der Waals forces involving free charge carriers occur where only a single substance has the free charges and no other substance in the system being considered can make an electrostatic contribution. For example, an electrolyte sphere coated with a nonpolar hydrocarbon near a pure water aerosol particle is such a system. In this case, the two-particle interaction force can be computed by use of local dielectric permeabilities whenever the charge carrier s plasma frequency is less than the lowest absorption frequency of the system [e.g., in (5.38)]. 

For the understanding and description of aerosol particle interaction processes, the two most important questions are what the two-particle interaction forces are and what the particle-surface interaction forces are. Since a substantial fraction of condensational aerosol particles from all sources nucleate to form spheres prior to subsequent coagulation and deposition, the description of the interaction of spheres is of central interest. 

Thus, in order to render the stability theory completely determinate, we need to specify in an unequivocal form both the conservation equations governing macroscopic suspension flow and all the rheological equations of state. This is easily seen to be possible for coarse dispersions of small particles. For such dispersions, normal stresses in the dispersed phase may be approximately described in terms of the particulate pressure as explained in Section 4, and this pressure can be evaluated for uniform dispersion states with the help of Sections 7 and 8. As a result, particulate pressure appears to be a single-valued function of mean variables characterizing the uniform dispersion state under study and of the physical properties of its phases. This single-valued function involves neither unknown quantities nor arbitrary parameters. On the other hand, if the particle Reynolds number is small, all interphase interaction force constituents also can be expressed in an explicit consummate form with help from the theory in reference [24]. This expression for the fluid-particle interaction force recently has been employed as well in stability studies for flows of collisionless finely dispersed suspensions [15,60]. 

Colloids are primarily characterized by their dimensions. Several properties—for example, the interfacial area per unit mass of dispersed material and hence the capacity as adsorbent, particle-particle interaction forces, and rheological behavior—are... 

It is noted that the buoyancy force is also included in Eq. (53) as one of the fluid-particle interaction forces. Due to the small particle size, the Saffman and Magnus forces are ignored in Eq. (53). 

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