Friction coefficient solid sphere

A drop rises or falls through a liquid as the result of buoyancy or gravity forces opposed by frictional resistance to motion. The latter is characterized by a drag coefficient, the knowledge of which makes it possible to calculate drop velocity and hence the residence time of the drop in the column. Although the variation of drag coefficients with Reynolds number is well known for solid spheres, this is not the case for liquid drops, as shown by the bottom line of Fig. 2. 

Fig. 3 Hydrodynamic interactions including lubrication, with particles of radius a = 2.5b. The solid symbols are the LB friction coefficients, and, for the relative motion of two spheres along the line of centers left) and perpendicular to the line of centers (right). Results are compared with essentially exact results from a multipole code [21] in the same geometry (solid lines)...

Fig. 7 Decay of translational (U) and rotational (12) velocity correlations of a suspended sphere. The time-dependent velocities of the sphere are shown as solid symbols the relaxation of the corresponding velocity autocorrelation functions are shown as open symbols (with statistical error bars). A sufficiently large fluid volume was used so that the periodic boundary conditions had no effect on the numerical results for times up to r = 1,000 in lattice units (h = b = 1). The solid lines are theoretical results, obtained by an inverse Laplace transform of the frequency-dependent friction coefficients [175] of a sphere of appropriate size (a = 2.6) and mass (pj/p = 12) the kinematic viscosity of the pure fluid = 1/6...

Here, p is the density of the fluid, V is the relative velocity between the fluid and the solid body, and A is the cross sectional area of the body normal to the velocity vector V, e.g., nd1/4 for a sphere. Note that the definition of the drag coefficient from Eq. (11-1) is analogous to that of the friction factor for flow in a conduit, i.e.,... 

In a hard sphere approach, particles are assumed to interact through instantaneous binary collisions. This means particle interaction times are much smaller than the free flight time and therefore, hard particle simulations are event (collision) driven. For a comprehensive introduction to this type of simulation, the reader is referred to Allen and Tildesley (1990). Hoomans (2000) used this approach to simulate gas-solid flows in dense as well as fast-fluidized beds. There are three key parameters in such hard sphere models, namely coefficient of restitution, coefficient of dynamic friction and coefficient of tangential restitution. Coefficient of restitution is discussed later in this chapter. Detailed discussion of these three model parameters can be found in Hoomans (2000). 

DPMs can also be used to understand the influence of particle properties on fluidization behavior. It has been demonstrated that ideal particles with restitution coefficient of unity and zero coefficient of friction, lead to entirely different fluidization behavior than that observed with non-ideal particles. Simulation results of gas-solid flow in a riser reactor reported by Hoomans (2000) for ideal and nonideal particles are shown in Fig. 12.8. The well-known core-annulus flow structure can be observed only in the simulation with non-ideal particles. These comments are also applicable to simulations of bubbling beds. With ideal collision parameters, bubbling was not observed, contrary to the experimental evidence. Simulations with soft-sphere models with ideal particles also indicate that no bubbling is observed for fluidization of ideal particles (Hoomans, 2000). Apart from the particle characteristics, particle size distribution may also affect simulation results. For example, results of bubble formation simulations of Hoomans (2000) indicate that accounting... 

In this expression Y = Yi + Y2 12 where Yj and Y2 are the surface tensions of the sphere and flat, respectively, and Yj2 is the surface tension of the interface between them. The JKR expression recognizes the fact that even in the absence of a normal force (N = 0) siuface tension will cause some elastic deformation of the surfaces producing a finite contact area. This fact alone renders the concept of a coefficient of friction meaningless since it implies that there is some finite friction force between solids even under zero normal force. 

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