Particle force equations

The various electrostatic forces acting between particles, particles and surfaces, and liquid interfaces in the presence of electric fields having been the subject of numerous theoretical and experimental investigations. While the fundamental force mechanisms between materials have been identified (Lapple 1970 Krupp 1967 Adamson 1976), there remains practical limitations to their application because of the uncertainty of detailed descriptions at contact points such as the number and size of asperities, close contact separation distance and contact area, presence of films, and gas breakdown from electric fields. Complications arise from the presence of other permanent forces such as van der Walls and contact electronic forces or if there is a distribution of particle sizes. Dielectrophoretic effects resulting from field gradients and dielectric present yet another electrostatic force factor (Jones, 1995). 

In contrast to contacting particles, an isolated particle of charge Q that interacts with a local field Eis simply evaluated from the relationship f =Q E. Such electrostatic forces for isolated particle can be simply evaluated by levitation, i.e., utilizing gravitational forces and steady-state experiments (Colver, 1976 Jones, 1986, Tombs and Jones, 1990, 1993). Charged droplets have been levitated for the studies of surface tension and viscosity (Rhim and 

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The particle motion along curvilinear pathways and the subsequent deposition rate on nearby bodies are calculated from dimensionless particle force equations. A key parameter that derives from these equations is the Stokes number,... 

From the continuity and momentum equations for the fluid and solid phases along with the boundary conditions, the following groups of independent dimensionless parameters are found to control the hydrodynamics, noting our assumption that the particle-particle forces are only dependent on hydrodynamic parameters,... 

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... 

Suppose we have two particles with masses mi and m2, and position vectors R1 and R2 in the laboratory frame, respectively, that interact through the isotropic intermolecular potentential V(R), with R = R2--RiI, with no other forces acting on them. For each particle, an equation of motion may be written down,... 

In the above calculation the system has been treated as though the nucleus were stationary and the electron moved in a circular orbit about the nucleus. The correct application of Newton s laws of motion to the problem of two particles with inverse-square force of attraction leads to the result that both particles move about their center of mass. The center of mass is the point on the line between the centers of the two particles such that the two radii are inversely proportional to the masses of the two particles. The equations for the Bohr orbits with consideration of motion of the nucleus are the same as those given above, except that the mass of the electron, m, is to be replaced by the reduced mass of the two particles, /, defined by the expression 1/m = 1/m + 1/M, where M is the mass of the nucleus. 

The force acting on a particle within a centrifugal field is defined by Newton s fundamental force equation FM = mu. Acceleration acting upon the particle, directed loward the center of rotation is a = nr2. Therefore, the centrifugal force ading on the particle is F — mru 2. or expressed as multiples of gravity. 

The simplest theory of impact, known as stereomechanics, deals with the impact between rigid bodies using the impulse-momentum law. This approach yields a quick estimation of the velocity after collision and the corresponding kinetic energy loss. However, it does not yield transient stresses, collisional forces, impact duration, or collisional deformation of the colliding objects. Because of its simplicity, the stereomechanical impact theory has been extensively used in the treatment of collisional contributions in the particle momentum equations and in the particle velocity boundary conditions in connection with the computation of gas-solid flows. 

Equation (4.2) is a particle-settling equation that is derived from two equations of force. One force equation is the buoyancy force FB, and the other is the drag force FD, produced by the particle movement in the surrounding fluid. As the particle or water droplet velocity increases... 

The external resistance force of a particle in equation (7.1) is split into two terms, the first of which is equal to ((v,J — v3i r ) - the resistance in a corresponding monomer liquid, and the second one, T) , is connected with the neighbouring macromolecules and satisfies the equation, which can be written in the simplest covariant form (see Section 8.4 and Appendix D). 

The breakage theory of spheres is a reasonable approximation of what may occur in the size reduction of particles, as most size-reduction processes involve roughly spherical particles. An equation for the force required to crush a single particle that is spherical near the contact regions is given by the equation of Hertz (Timoschenko and Goodier, Theory of... 

We have performed numerical experiments using a three dimensional relativistic kinetic electromagnetic particle-in-cell code The code works from first principles by solving the Lorentz force equation for the particles and the Maxwell s equations for the electromagnetic fields. 

Each particle in a bed of porous particles is surroimded by a laminar sublayer (Figure 5.4), through which mass transfer takes place only by molecular diffusion. On one side, this layer is exposed to the flowing mobile phase and is entirely accessible. On the other side, it wraps the particle wall and is accessible from the particle inside only at the pore openings. The thickness of this layer, hence the mass transfer coefficient, is determined by hydrodynamic conditions and depends on the flow velocity. The mass transfer rates can be correlated in terms of the effective mass transfer coefficient, fcy, defined according to a linear driving force equation ... 

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. 

The force that is exerted on a charged particle carrying a charge q, possessing an instantaneous velocity v and submitted to the action of a magnetic induction 6, is given by the Lorentz force equation ... 

For solids with continuous pores, a surface tension driven flow (capillary flow) may occur as a result of capillary forces caused by the interfacial tension between the water and the solid particles. In the simplest model, a modified form of the Poiseuille flow can be used in conjunction with the capillary forces equation to estimate the rate of drying. Geankoplis (1993) has shown that such a model predicts the drying rate in the falling rate period to be proportional to the free moisture content in the solid. At low solid moisture contents, however, the diffusion model may be more appropriate. 

In the general case, the direction of movement of the particle relative to the fluid may not be parallel with the direction of the external and buoyant forces, and the drag force then makes an angle with the other two. In this situation, which is called two-dimensional motion, the drag must be resolved into components, which complicates the treatment of particle mechanics. Equations are available for two-dimensional motion, but only the one-dimensional case, where the lines of action of ail forces acting on the particle are collinear, will be considered in this book. 

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