Collisional force

The effect of the collisional force due to the impact of particles should be included when accounting for the motion of a particle except in a very dilute gas-solid flow situation. Basic mechanisms of collision between two particles or between a particle and a solid wall are discussed in Chapter 2. The collisional force between a particle and a group of neighboring particles in a shear suspension is discussed in 5.3.4.3. In a very dense system where particle collisions dominate the flow behavior, collisional forces can be described by using kinetic theory, as detailed in 5.5. The key equations derived in other chapters pertaining to the collisional forces can be summarized in the following. 

For elastic spheres, the maximum collisional force in a collinear impact between two particles, Fc (= fm in Eq. (2.132)), is given by 

The averaged collisional stress between a particle and a group of neighboring particles of the same diameter in a shear flow can be expressed by —(/ipAp), where /Lip is given by 

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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. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

Consider a collision between two frictionless elastic spheres so that only normal force and normal velocities are involved i. e., no tangential forces or tangential velocities need to be accounted for in this case. A general case is shown in Fig. 2.15, where two spheres with different sizes, velocities, and material properties collide with each other. Only the collisional force is considered. 

Brittle erosion is the loss of material from a solid surface due to fatigue cracking and brittle cracking caused by the normal collisional force Fn. Materials with very limited capacity for elastic and plastic deformation, such as ceramics and glass, respond to particle impacts by fracturing. The yield stress for brittle failure Fb for normal impacts is about... 

As mentioned, the erosion of a solid surface depends on the collisional force, angle of incidence, and material properties of both surface and particles. Although abrasive erosion rates cannot be precisely predicted at this stage, some quantitative account of erosion modes which relates various impact parameters and properties is useful. In the following, a simple model for the ductile and brittle modes of erosion by dust or granular materials suspended in a gas medium moving at a moderate speed is discussed in light of the Hertzian contact theory [Soo, 1977]. 

The forces due to the liquid bridge comprise attractive capillary force and repulsive viscous force. The algorithms for the calculation of these forces were implemented in the DEM model along with the collisional forces that arise during interparticle contact. 

An expression for this random collisional force can be derived by taking into account that 1) it must be linear in w since the fluctuations are assumed to be sufficiently weak, and 2) it may contain only those components directed along unit vectors marking preferable directions, at a given physical point of the suspension under study. There are usually two such directions, and these directions are determined by acceleration g of external body forces and by mean fluid slip velocity u). When these directions are essentially different, the corresponding general expression for the collisional force is presented in reference [23]. If vectors g and (u) are collinear, as is specific to fluidized beds and to other vertical flows of suspensions, this expression takes the form ... 

To make Equation 7.1 fully determinate, we must 1) specify functions F.(( )) involved in the definition of the interphase interaction force in Equation 3.2 2) introduce a statistical model for concentrational fluctuations which would allow us to get the spectral density of these fluctuations and 3) determine the unknown coefficients in the collisional force expression according to Equation 3.1. 

To characterize the gas flow in the fluidized bed, important forces are viscous and inertia as well as par-ticle-to-gas forces. For the particles, the important forces include gravity, particle inertia, gas interaction with the particles such as drag, collisional forces between particles and between particles and wall, and particle surface forces such as electrostatic and adhesion forces. For larger particles, in Geldart groups B or D, the particle surface forces can be neglected. The question is complicated for smaller particles because the surface forces are difficult to quantify. We must... 

Colloid The dispersion of small particles in a second material. In a fluid the particle suspension is controlled by collisional forces rather than gravity. 

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