Applications of Kinetic Theory

Abstract

The overall objective of this investigatfm is to develop experimentdly verified models for circulating fluidized bed (CFB) combustors. This report presents the author's derivation of analytical solutions useful in understanding the operation of a CFB. The report is in a form of a chapter that reviews the kinetic theov applications. DISCLAIMERThis report was prepared as an account of work sponsored by at1 agency of the (Jnitcd States Government. Ncither the United Statcs Government nor any itgency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refcrence herein to any specific cominercial product, process, or service by t r d e name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its cndorsetnent, recornmendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. APPLICATIONS OF KINETIC THEORY GRANULAR SHEAR FLOWThe simplest solution to the Nsvier-Stokes' equation for a fluid is that of shear or Couettc flow (Scldichting, 1966). It is assunied that the flow is steady, incompressible and fully developed. The latter condition insures that the velocity is nonzero in one direction, say in the x direction only. Then the continuity eqiiation with coIistant porosity shows that the gradient of velocity is zero, I-Icncc all accelerations vanish and the momentum balance for granular flow is simplywhere P is given by Equation (9-250). Furthermore, anadlogous to the fluid pressure one assumes the solids pressure in the x direction to be constant. The momentum balance then assumes the simple formIn the simple shear flow one assumes a no slip boundary condition at one w d l .The second wail is assumed to move with velocity U. With the fluid adhering to the wall the boundary conditions are The velocity profile is then us = Uy/h 2 (9 -' 275) (9 -275) ..llthotigh Equation (9-274) docs not permit a direct dctcrmination of viscosity as is done in the Poiseuille viscorrieter in which the pressure in Equation (9-273) varies with lcngth of flow, it turns out that thc granular tcnperature biLltlncc, Equation 8, ( 9 2 1 1 ) permits one to estimate the viscosity, as developed below, 19~1. D ISWith the same assumptions made for the simple shear flow discussed in the previous paragraph, the granular temperature Equation (9-21 1) with 110 production due to fluid turbulence or molecular fluid collisions is as follows.(9 -276)(9 -2 7 6~)In obtaining the dissipation by the wall in Equation ( 2 -h q it was assumed that the granular heat flux in Equation (9-211) is a constant and that the granular heat flow wits developed, as is done in the thermal forced convection problems.For an elastic...

Study on the Numerical Model of Dense Solid Suspension Driven by a Coaxial Mixer

“…In addition, other researchers − also emphasized the importance of solid stress on simulation results. In order to describe instantaneous collisions between particles, Gidaspow proposed the kinetic theory of granular flow (KTGF), in which the particle flow was analogized to thermal movement of gas molecules. Barrue et al combined the EE method with the KTGF model, and the approach successfully predicted the solid distributions in dense solid–liquid stirred tanks.…”

Circulating fluidized bed hydrodynamics experiments for the multiphase fluid dynamics research consortium (MFDRC).

“…The interparticle stress employs a close-packed particle volume fraction of 0.68, a particle-to-wall momentum-loss factor of 0.1, a specular-incident-angle coefficient of 1, and diffuse coefficient of 0 (Williams et al, 2002;Williams et al, 2003). A Gidaspow-type particle-fluid drag law is used, which includes particle-Reynolds-number and particle-volumefraction effects (Gidaspow, 1994;Williams et al, 2002. The gravitational acceleration is downward with a value of 9.8 m/s 2 .…”

“…The particle momentum equation is given by (6.3.5) where D p characterizes the fluid-particle drag and τ p is the interparticle stress. Several models for fluid-particle drag are considered (Snider et al, 1998;Snider, 2001;Williams et al, 2002), including the Stokes relation, the Gidaspow model (Gidaspow, 1994), and the Ergun relation. The latter two models include the effects of finite Reynolds number and of the close-packed particle volume fraction θ cp .…”

Gas Bubble Diameter and Rise Velocities in Tapered Fluidized Beds: Experiment and Computational Fluid Dynamics Simulation

“…The shear viscosity for the solid phase consists of kinetic viscosity resulting from the translational movement of solid particles, collisional viscosity arising from the collision of solid particles, and frictional viscosity. The shear viscosity for solid can be given by μ normals = μ normals , kin + μ normals , col + μ normals , fr in which the kinetic term can be evaluated by μ normals , kin = 10 ρ normals d normals normalΘ s π 96 ε normals ( 1 + e ss ) g 0 , ss true[ 1 + 4 5 g 0 , ss ε normals ( 1 + e ss ) true] 2 the collisional term by μ normals , col = 4 5 ε normals ρ normals d normals g …”

“…The shear viscosity for solid can be given by μ normals = μ normals , kin + μ normals , col + μ normals , fr in which the kinetic term can be evaluated by μ normals , kin = 10 ρ normals d normals normalΘ s π 96 ε normals ( 1 + e ss ) g 0 , ss true[ 1 + 4 5 g 0 , ss ε normals ( 1 + e ss ) true] 2 the collisional term by μ normals , col = 4 5 ε normals ρ normals d normals g 0 , ss ( 1 + e…”