Particle trajectory crossing

The occurrence of particle trajectory crossing (PTC) is associated with the free-transport term in the collisionless KE (and, by extension, the GPBE), and leads to a multi-velocity state that is difficult to capture with Eulerian solvers (Sachdev et at, 2007 Saurel et al, 1994). In a ID velocity phase space, the simplest KE for the NDF n t, x, v) is 

We immediately observe that the free-transport term in the KE leads to a closure problem in the moment-transport equation if we truncate the moment set at m2N-i, then, in order to solve Eq. (8.5), we must provide a moment closure for m2N- Eor the example given in Eq. (8.4), it is straightforward to verify that the analytical expression for the moments satisfies Eq. (8.5). In fact, for this example, we can observe that a two-point quadrature with Wi(r,x) = 6(x - t + 1), W2(f, x) = 6 x + t + 1), f t,x) = 1, and f2(.t,x) = -1 exactly reproduces the moment mk(t,x) for arbitrary k. Thus, starting from the moment set (mo,mi,m2,ms), the two-point quadrature is the optimal closure for m4. The moment-transport equations needed for a two-point quadrature are 

In summary, although the weakly hyperbolic nature of Eq. (8.7) has been shown rigorously only for ID phase space (i.e. one velocity component in the KE), experience strongly suggests that the full 3D system is also weakly hyperbolic. This observation implies that the numerical schemes used to solve the moment-transport equations closed with QBMM must be able to handle local delta shocks in the moments. Qne such class of numerical schemes consists of the kinetics-based finite-volume solvers presented in Section 8.2. As a final note, we should mention that the work of Chalons et al (2012) using extended Gaussian quadrature (see Section 3.3.2) and kinetics-based finite-volume solvers to close Eq. (8.7) suggests that the system with 2A + 1 moments is fully hyperbolic and thus does not exhibit 

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For gas-particle flows, fhe mosf obvious manifestation of non-equilibrium behavior is particle trajectory crossing (PTC) at KUp = oo (i.e. no collisions). PTC occurs in the very-dilute-flow regime ( p c 1) and is most easily identified when fhe granular femper-afure is null (Map = c ). An example of PTC is shown in Figure 1.4. The panel on fhe... 

Stokes numbers, particle trajectory crossings (PTC) are not important, in which case it is possible to neglect velocity fluctuations (i.e. upUp] 1). In this limit, the disperse-phase momenrnm balance reduces to... 

In the remainder of this section, we introduce the principal modeling issues related to spatial transport using moment methods. First, we discuss the realizability of the NDF and of moment sets (which are related to the numerical errors discussed above). Second, we introduce the phenomenon of particle trajectory crossing (PTC) that occurs with the inhomogeneous KE (and is exactly captured by the NDF), and describe how it leads to a closure problem in the moment-transport equations. Next, we look at issues related to coupling between spatial and phase-space transport in the GPBE (i.e. due to correlations between velocity and internal coordinates such as particle volume). Finally, we introduce KBFVM for solving the moment-transport equations in connection with QBMM, and briefly discuss how they can be used to ensure realizability as well as to capture PTC and to treat coupled moments. 

Figure 8.1. An example of particle trajectory crossing in a 2D domain. The fully developed particle number density moo is shown together with the mean particle velocity vector (mio/moo, moi/moo). This example was computed using QBMM with a KBFVM (Yuan Fox, 2011) and realizable second-order fluxes (Vikas et al, 2011a). Note that the mean particle velocity is not defined in regions where the density is exactly zero. Owing to a smali amount of numerical diffusion, the density is very small, but nonzero, outside of the two jets.

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. 

Figure 8.2 Particle trajectory crossing several orbits in a centrifuge...

Fig. 7.7 Sample particle trajectories in a twin vortex flow. The agitator location is set by the amplitude, which is 0.5 (i.e., it is at midpoint between the center and the perimeter) and marked by the crosses. The dimensionless period for each vortex is 0.5. The mixing protocol is to activate one agitator for a period of time and then switch to the other agitator. [Reprinted by permission from H. Aref, Stirring Chaotic Advection, J. Fluid Meek, 143, 1-21 (1984).)...

Figure 2. The effects of sa7npling under isokinetic (A), low-flow (B), and high-flow (C) conditions. The cross-hatched regions represent the sampling orifice large and small symbols represent large and small aerosol particles, respectively solid lines are air streamlines ar dashed lines are heavy particle trajectories.

The crossing trajectory effect refers to the impact of the continuous change of the fluid eddy-particle interactions as the heavy particle trajectory might go through numerous eddies reflecting different flow properties. Hence it follows that the velocity history of heavy particles may differ from that of a marked fluid particle. Similar closure models for the drift velocity and the velocity co-variances have been derived from kinetic theory by Koch and co-workers [38, 39] and Reeks [62, 63]. 

This is far less simple than it sounds after a particle has crossed the barrier, say from left to right, its fate is not yet determined. It is only after subsequent relaxation leads it toward the bottom of the right well that its identity as a product is established. If this happens before the particle is reflected back to the left, that crossing is reactive. Assumption (2) in fact states that all equilibrium trajectories crossing the barrier are reactive, that is, they go from reactants to products without being reflected. For this to be a good approximation to reality two conditions should be satisfied ... 

Figure 4.15 Schematic diagram of particle trajectories downstream from the exit of the converging nozzle, (a) Large particles follow their initial motion and their paths cross, producing a divergent beam, (b) Intermediate size particles bend tow ard the axis under the influence of the gas, producing a focused beam, (c) Very small particles follow the gas motion, producing a divergent beam.

Here, Ls is the width of the image of the spatial filter in the receiving optics which depends on the slit width and the magnification of the optics, D is the diameter of the considered particle size class and r(Dj) is the particle size-dependent radius of the measurement volume (Figure 7-32). For any other trajectory of the particle through the measurement volume the effective cross-section perpendicular to the particle trajectory is obtained with the particle trajectory angle Ok. 

It is conventional to define a dimensionless collection efficiency by comparing the actual diffusional mass flow rate to the mass flow rate of particles to the collector for straight particle trajectories, that is, the mass swept out by the projected cross-sectional area of the collector. From this definition the spherical collector efficiency is... 

Having found the law describing the flow around the body, we can obtain from (10.119) the family of particle trajectories, and then determine the limiting trajectory and cross section of particles collisions with the body. 

Therefore, two particles cannot be present at the same place and conversely we exclude from description such phenomena as a tearing or a penetrating of the bodies, destruction and origin of new particles and trajectories, crossing trajectories, etc. A typical quantity ir we are interested in (which may be scalar, vector, or tensor) is a field, i.e.,... 

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