Mixing advection

Mixed advection is usually written in the form (Reeks, 1983, 1991, 1992 Zaichik, 1997 Zaichik eta/., 2008)... 

For example, in a PBE the velocity internal coordinate is absent, so only advection, diffusion, and the source term will appear. Likewise, for a KE only the free-transport, mixed advection, and source terms will appear. 

The mixed advection term in Eq. (B.4) involves both a spatial and a phase-space gradient of the NDF, and thus it is more complicated than the pure advection term described in Section B.3 of this appendix. Nevertheless, by working with the transport equation for the moments, we can show that it can be treated like an advection term with special properties. 

Similarly to what we did in the case of free transport, with mixed advection we define the half-moment sets needed to compute the weights and abscissas used in Eqs. (B.50) and (B.51) as... 

Notice that for mixed advection both the extended quadrature and the regular quadrature use N nodes. This is possible because with mixed advection the highest-order moment appearing in the spatial fluxes has a lower order due to the division by V (i.e. instead of multiplication by Vi for free transport). However, this observation only holds when A does not depend on v. The mixed advection formulas in the y and z directions are analogous to those derived above for the x direction. Likewise, the comments in Section B.4 of this appendix concerning the use of the CQMOM for free transport apply also for mixed advection. 

A key technical point for mixed advection is the dependence of A on v, in particular when any one of the components of v tends towards zero. For example, /l(v) in Eq. (B.52) has three possible limiting behaviors ... 

In proposed applications of mixed advection to gas-particle flows (Pandya Mashayek, 2003b Reeks, 1977, 1983, 1993 Zaichik, 1997, 1999 Zaichik et al, 2008), A is taken to be independent of v, and hence it behaves as an effective pressure. Therefore, in order for the time step computed from Eq. (B.52) to be nonzero, it will be necessary for A to be zero at spatial locations where either or is zero. Since these abscissas are computed from the half-moment sets and respectively, they can be zero only if the kernel density functions used in tlie EQMOM have zero width. (See Section 3.3.2... 

Note that, since L has units (m/sf, the nonnegative function h ) would be dimensionless. With this model for A the realizability condition in Fq. (B.52) would always yield a nonzero upper bound on At when h ) is finite. physically, E is null in the limit of pure particle trajectory crossing where the true NDF is a sum of Dirac delta functions. On the other hand, when E reaches its maximum value, the NDF is Gaussian. Thus, since mixed advection is associated with random particle motion, the model in Fq. (B.56) also makes physical sense. Nonetheless, the potential for singular behavior in the update formula makes the treatment of mixed advection problematic. 

In summary, in this section we have shown that the realizable scheme for mixed advection is very similar to those introduced for advection and free transport. However, the numerical properties of the scheme are very dependent on the functional form used for A. In order for the standard models for mixed advection to be well conditioned, the functional form for A must compensate for velocity abscissas located too close to the origin. With multivariate FQMOM, the model given in Fq. (B.56) can be used for this purpose. The potential for singular behavior with mixed advection makes it nevertheless problematic, and alternative schemes may be required to properly handle such behavior. 

Moore WS (2000) Determining coastal mixing rates using radium isotopes. Cont Shelf Res 20 1995-2007 Moore WS, Shaw X J (1998) Chemical signals from submarine fluid advection onto the continental shelf J GeophysRes 103 21543-21552... 

The RTD quantifies the number of fluid particles which spend different durations in a reactor and is dependent upon the distribution of axial velocities and the reactor length [3]. The impact of advection field structures such as vortices on the molecular transit time in a reactor are manifest in the RTD [6, 33], MRM measurement of the propagator of the motion provides the velocity probability distribution over the experimental observation time A. The residence time is a primary means of characterizing the mixing in reactor flow systems and is provided directly by the propagator if the velocity distribution is invariant with respect to the observation time. In this case an exact relationship between the propagator and the RTD, N(t), exists... 

Reactors which generate vortex flows (VFs) are common in both planktonic cellular and biofilm reactor applications due to the mixing provided by the VF. The generation of Taylor vortices in Couette cells has been studied by MRM to characterize the dynamics of hydrodynamic instabilities [56], The presence of the coherent flow structures renders the mass transfer coefficient approaches of limited utility, as in the biofilm capillary reactor, due to the inability to incorporate microscale details of the advection field into the mass transfer coefficient model. 

Advection is important in fragmentation processes, and an initially homogeneous system may evolve spatial variations due to spatially dependent fragmentation rates. For example, Fig. 36 shows the spatial distribution of eroded clusters in the journal bearing flow operating under good mixing... 

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