Knudsen number disperse phase

The phase-density ratio is defined by i = Pp/pg and, since cpi 1, the buoyancy term is negligible. The two new dimensionless numbers generated in this process are the phase-velocity ratio 03 = t/p/t/, and the disperse-phase Knudsen number KUp = uItJL. In addition, the dimensionless form ofEqs. (1.10),... 

Following the convention used in gas dynamics, we will define the velocity ratio and disperse-phase Knudsen number differently according to whether the disperse phase is subsonic or supersonic ... 

Disperse-phase Knudsen number Kjip = sJndpl 2apgQL) for Map < 1 Krip = Jndp 2apgQL)y[. ip for Map > 1... 

The reader may be surprised not to And a Reynolds number defined speciflcally for the disperse phase. This is because the disperse-phase viscosity is well defined only for Knp 1 (i.e. the collision-dominated or hydrodynamic regime). In this limit, Vp oc oc Knp/Map so that the disperse-phase Reynolds number would be proportional to Map/Krip when Map < 1. However, in many gas-particle flows the disperse-phase Knudsen number will not be small, even for ap 0.1, because the granular temperature (and hence the collision frequency) will be strongly reduced by drag and inelastic collisions. In comparison, molecular gases at standard temperature and pressure have KUp 1 even though the volume fraction occupied by the molecules is on the order of 0.001. This fact can be... 

The disperse-phase Knudsen number can also be infinity for dense, elastic systems due to initial conditions. For example, if one releases a dense assembly of particles that are initially at rest in a vacuum, the particles will accelerate due to gravity such that they all have the same velocity. For this case, the granular temperature is null... 

Figure 1.3. Ranges of validity of hydrodynamic models for the velocity kinetic equation in terms of the disperse-phase Knudsen number. NS, Navier-Stokes. Adapted from Bird (1994) by permission of Oxford University Press.

Knudsen number for continuous phase (relative to particle diameter) Knudsen number for continuous phase (relative to particle radius) Knudsen number for disperse phase... 

Both diffusion coefficients and mass transfer are important, but they depend on the different solids, drainage (flow), and particles porosity. The effective diffusion involves Knudsen and convective diffusion, which depends on the phase of fluid (gas or liquid) and pore size (large or small). These coefficients are characterized by Peclet number (Pe), which depends on the axial or radial dispersion and diffusivity. Depending on the velocity profile, these coefficients can vary radially or axially. The diffusion and dispersion coefficients can also vary due to its dependence on the radial position. If the coefficients vary along the reactor, as in heterogeneous reactors, for example, the velocity is not constant. Thus, the axial dispersion occurs. 

---