Particles Stokes

Particles of constant size Gas film diffusion controls, Eq. 11 Chemical reaction controls, Eq. 23 Ash layer diffusion controls, Eq. 18 Shrinking particles Stokes regime, Eq. 30 Large, turbulent regime, Eq. 31 Reaction controls, Eq. 23... 

For a solid particle, Stokes Law gives the terminal velocity (i ,g) as... 

Stokes diameter Diameter of a sphere of the same density as the particle in question having the same settling velocity as that particle. Stokes diameter and aerodynamic diameter differ only in that Stokes diameter includes the particle density whereas the aerodynamic diameter does not. 

Stokes considered the resistance experienced by a sphere moving uniformly through an incompressible viscous fluid. Viscous flow implies a low Reynolds number Re. He assumed an infinite medium, rigid particle, no slipping at the surface of the particle. Stokes pointed out that his solution was erroneous in the case of a cylinder. 

Finally, a steric repulsive force = E i must be included to keep particles from overlapping. This steric force would not be required if the hydrodynamic forces were treated more realistically, because as particles approach each other closely, strong lubrication forces are produced by the solvent that must be squeezed from between the particles (Bonnecaze and Brady 1992a). However, these lubrication forces are omitted from one-particle Stokes hydrodynamics, and a steric force must be introduced. 

Biological particles. Stokes law applies to spherical particles, which are large in comparison with the molecules that comprise the liquid medium, and are present at a concentration low enough to avoid modification of the liquid viscosity. Most biological particles are not spherical, and Strokes law must be modified to take this into account. One approach to this problem is to consider that the biological particles shapes could be approximated by ellipsoids of revolution , or spheroids with one major and one minor axis. Calculations show that the frictional force over these ellipsoids is greater than that expected for spherical particles of the same volume.3... 

In summary, the Boussinesq-Basset, Brownian, and thermophoretic forces are rarely used in disperse multiphase flow simulations for different reasons. The Boussinesq-Basset force is neglected because it is needed only for rapidly accelerating particles and because its form makes its simulation difficult to implement. The Brownian and thermophoretic forces are important for very small particles, which usually implies that the particle Stokes number is near zero. For such particles, it is not necessary to solve transport equations for the disperse-phase momentum density. Instead, the Brownian and thermophoretic forces generate real-space diffusion terms in the particle-concentration transport equation (which is coupled to the fluid-phase momentum equation). 

If the NDF is a function of the particle velocity then the solution of the GPBE provides the modeler with the essential information for calculating the real-space advection term. This approach is used whenever the particle Stokes number is not small, and will result in the development of a particle-velocity distribution. More details on this topic can be found in Chapter 8. An alternative approach consists of integrating the NDF with respect to the particle velocity. Let us consider, for example, a generic NDF n(t,x, p, p), which is a function of the time t, space x, particle velocity Vp, and internal coordinates p. By integrating out the particle velocity the following NDF is obtained ... 

When the particle Stokes number is not small, the truncated expansion for the particle velocity mean particle velocity must be calculated from the disperse-phase momentum equation described in Section 4.3.7. Let us for the time being consider a very dilute population of identical particles. The mean velocity of these particles can be found by solving Eq. (4.91). For small particle... 

In summary, the Eulerian two-fluid model is represented by Eqs. (5.112) and (5.113) in addition to a constitutive model for the fluid stress tensor Tf. As already mentioned, Eq. (5.112) was derived under the assumption that the particle-velocity distribution is very narrow (i.e. small particle Stokes number), and the particles must have the same internal coordinates. If these simplifications do not hold, for example under dense conditions when particle-particle collisions become important, then particle-velocity fluctuations must be taken into account, as discussed at the end of Chapter 4. 

There are several ways to calculate the PDI. If there are no density inhomogeneities and size is the only variable, then the polydispersity is completely defined by the particle-size distribution. If diis is known, then the particle Stokes numbers for the smaller and largest particles can be calculated in terms of dio% and dgo%. These... 

The pseudo-homogeneous or dusty-gas model very small particle Stokes number and limited polydispersity (momentum-balance equation only for the continuous phase if the system is dilute or for the mixture of continuous and disperse phases if the system is dense). 

The Eulerian two-fluid model with particle-phase velocity that is based on the mean particle size small particle Stokes number and limited polydispersity (both in dilute and in dense systems). 

In terms of the fluid mass seen by the particle f, conservation of mass at the mesoscale leads to the following mesoscale models in the limit of zero particle Stokes number (i.e. u = U = Uf) ... 

In particular, the left-hand side can be interpreted as the particle velocity conditioned on the particle size, and then the right-hand side is a local (in real space) second-order approximation of the conditional velocity. However, in order to accurately reproduce the moment fluxes, the approximation in Eq. (8.107) must be close to the true velocity. Or, to put it a different way, the conditional variance of the particle velocity must be small so that the conditional velocity distribution function is tightly centered at u(t,x,f). This will be true, for example, when the particle Stokes number is below the critical value at which PTC begin to occur (i.e. particles with very small inertia). 

In gas—particle flows, h depends on the particle Stokes number as... 

Schmidt number Sherwood number Stokes number particle Stokes number... 

Nasr-El-Din and Shook (58) studied solids distribution in a vertical pipe downstream of a 90° elbow. They tested sand-water slurries of various solid concentrations and particle sizes. The slurry flows were turbulent, and the particle Stokes number (inertia parameter) based on the pipe diameter and bulk velocity varied from 0.5 to 3. The solids distribution downstream of the elbow was found to be a function of the radius of curvature of the elbow, solids concentration, and particle size. 

The frictional coefficient / contains information about the size and shape of the particle. For spherical particles. Stokes s law, given in Equation 34-10, holds for laminar-flow conditions... 

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