Collision Mechanics of Solids

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 

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In a collision between two spheres of different temperatures, heat conduction occurs at the interface. The contact area is usually negligibly small compared to the cross-sectional area of the spheres. Since the duration of the impact is also very short, the temperature change of the colliding particles is confined to a small region around the contact area. Therefore, the heat conduction between the two particles can be treated as that between two semiinfinite media. It is also assumed that there is no thermal resistance between the contact surfaces. Hence, the temperature and heat flux distributions are continuous across the contact area. The surfaces outside the contact area are assumed to be flat and insulated. For general information on collision mechanisms of solids, readers may refer to Chapter 2. 

The book is arranged in two parts Part I deals with basic relationships and phenomena, including particle size and properties, collision mechanics of solids, momentum transfer and charge transfer, heat and mass transfer, basic equations, and intrinsic phenomena in gas-solid flows. Part II discusses the characteristics of selected gas-solid flow systems such as gas-solid separators, hopper and standpipe flows, dense-phase fluidized beds, circulating fluidized beds, pneumatic conveying systems, and heat and mass transfer in fluidization systems. 

According to the analysis in the previous sections, the primary particle size in flame reactors is determined by the relative rates of particle collision and coalescence. For highly refractory materials, the characterislic coalescence time (12.6) depends on the solid-state diffusion coefficient, which is a very sensitive function of the temperature. The mechanisms of solid-.staie diffusion depend in a complex way on the structure of the solid. For example, a perfect cubic crystal of the substance AB consists of alternating ions A and B. Normally there are many defects in the lattice structure even in a chemically pure single crystal defect types are shown schematically in Fig. 12.8. The mechanism of diffusion in cry.stalline solids depends on the nature of the lattice defects. Three mechanisms predominate in ionic... 

The model as formulated in this section cannot be used to predict a priori the solids entrainment rate into the jet because of the two empirical constants in Eq. (61). Lefroy and Davidson (1969) have developed a theoretical model based on a particle collision mechanism for entrainment of solid particles into a jet. The resulting equation for particle entrainment velocity is... 

In conduction, heat is conducted by the transfer of energy of motion between adjacent molecules in a liquid, gas, or solid. In a gas, atoms transfer energy to one another through molecular collisions. In metallic solids, the process of energy transfer via free electrons is also important. In convection, heat is transferred by bulk transport and mixing of macroscopic fluid elements. Recall that there can be forced convection, where the fluid is forced to flow via mechanical means, or natural (free) convection, where density differences cause fluid elements to flow. Since convection is found only in fluids, we will deal with it on only a limited basis. Radiation differs from conduction and convection in that no medium is needed for its propagation. As a result, the form of Eq. (4.1) is inappropriate for describing radiative heat transfer. Radiation is... 

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