Particle trajectory simulations

It can be seen that in many of these particle-level phenomena, sources will be a function of continuous phase variables. Such a situation results in strong coupling between particle trajectory simulations and simulation of the continuous phase flow field. Detailed modeling of each of these can be accomplished following conventional practices and will not be discussed here. More information can be found in textbooks on chemical reaction engineering and heat and mass transport processes (Levenspiel, 1972 Westerterp et al., 1984 Kuo, 1986 Kunii and Levenspiel, 1991). 

If we wish to know the number of (VpV)-collisions that actually take place in this small time interval, we need to know exactly where each particle is located and then follow the motion of all the particles from time tto time t+ bt. In fact, this is what is done in computer simulated molecular dynamics. We wish to avoid this exact specification of the particle trajectories, and instead carry out a plausible argument for the computation of r To do this, Boltzmann made the following assumption, called the Stosszahlansatz, which we encountered already in the calculation of the mean free path ... 

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

Simulations. In addition to analytical approaches to describe ion—soHd interactions two different types of computer simulations are used Monte Cado (MC) and molecular dynamics (MD). The Monte Cado method rehes on a binary coUision model and molecular dynamics solves the many-body problem of Newtonian mechanics for many interacting particles. As the name Monte Cado suggests, the results require averaging over many simulated particle trajectories. A review of the computer simulation of ion—soUd interactions has been provided (43). 

Despite the little experimental data, there are two models available in the literature. Adams etal. (1992) considered dense phase conveying. They tried to predict the amount of attrition as a function of conveying distance by coupling a Monte Carlo simulation of the pneumatic conveying process with data from single-particle abrasion tests. Salman et al. (1992) focused on dilute phase conveying. They coupled a theoretical model that predicts the particle trajectory with single particle impact tests (cf. Mills, 1992). 

The integration of forces between all molecules over several thousand time-steps produces particle trajectories from which time-averaged macroscopic properties can be computed. In MD the simulation is limited by the computer storage capacity and speed, so that short-lived phenomena (100-1000 ps) are generally calculated. 

Evident progress in studies of liquids has been achieved up to now with the use of computer simulations and of the models based on analytical theory. These methods provide different information and are mutually complementary. The first method employs rather rigorous potential functions and yields usually a chaotic picture of the multiple-particle trajectories but has not been able to give, as far as we know, a satisfactory description of the wideband spectra. The analytical theory is based on a phenomenological consideration (which possibly gives more regular trajectories of the particles than arise in reality ) in terms of a potential well. It can be tractable only if the profile of such a well is rather... 

In actual applications, the gas flow in a gravity settler is often nonuniform and turbulent the particles are polydispersed and the flow is beyond the Stokes regime. In this case, the particle settling behavior and hence the collection efficiency can be described by using the basic equations introduced in Chapter 5, which need to be solved numerically. One common approach is to use the Eulerian method to represent the gas flow and the Lagrangian method to characterize the particle trajectories. The random variations in the gas velocity due to turbulent fluctuations and the initial entering locations and sizes of the particles can be accounted for by using the Monte Carlo simulation. Examples of this approach were provided by Theodore and Buonicore (1976). 

Figure 20 Simulations data adapted from Sitz and Mullins [71]. Plot of the mean lifetime of methon particle on the Ir(l 11) and Ir(l 1 0)—(1 x 2) surfaces as a function of 1 /Ts as calculated from trajectory simulations.

The particle trajectories can be simulated using a random force in the generalized Langevin equation that is constant during a small time step ts with values given by a Gaussian distribution. The memory function for this form of random force is (12)... 

The Monte Carlo simulation of Brownian coagulation involves the evaluation of the ensemble average of the coagulation rate over a large number of particle pairs, through the generation of particle trajectories. The inter-particle forces due to the van der Waals attraction and Born repulsion are accounted for in the description of the relative motion [40] two Particles. The relative Brownian motion of two particles is described by the... 

First, and most important, nonlinear dynamics provides an intellectual framework to pursue the consequences of nonlinear behavior of transport systems, which is simply not possible in an intellectual environment that is based upon a linear mentality, characterized by well-behaved, regular solutions of idealized problems. One example that illustrates the point is the phenomenon of hydrodynamic dispersion in creeping flows of nondilute suspensions. It is well known that Stokes flows are exactly reversible in the sense that the particle trajectories are precisely retraced when the direction of the mean flow is reversed. Nevertheless, the lack of reversibility that characterizes hydrodynamic dispersion in such suspensions has been recently measured experimentally [17] and simulated numerically [18], Although this was initially attributed to the influence of nonhydrodynamic interactions among the particles [17], the numerical simulation [18] specifically excludes such effects. A more general view is that the dispersion observed is a consequence of (1) deterministic chaos that causes infinitesimal uncertainties in particle position (due to arbitrarily weak disturbances of any kind—... 

Figure 2.5 shows a Molecular Dynamics simulation, which is the counterpart of the previous example. Particle trajectories are shown for fluid particles, interacting via an law (as in the Lennard-Jones interaction), with a structureless "hard" wall. A striking feature is that the diffusion between the... 

Two of the most common classes of particle-dynamic simulations are termed hard-particle and soft-particle methods. Hard-particle methods calculate particle trajectories in response to instantaneous, binary collisions between particles and allow particles to travel ballistically between collisions. This class of... 

Once the instantaneous velocity is obtained, particle trajectories can be simulated. To introduce two-way coupling, it is necessary to calculate the source terms in the balance equations of mass, momentum and energy for the continuous phase. With such source terms, the continuous phase flow field needs to be solved again, which is later used to calculate new trajectories. The number of iterations between turbulence and particulate modules to obtain convergence is typically three. However, in strongly coupled flows, convergence may be difficult to reach (Kohnen et ai, 1994). 

To calculate dispersed phase particle trajectories it will be necessary to solve a set of coupled ordinary differential equations (Eqs. 4.1 and 4.9). Any standard initial value ODE solvers can be used for this purpose. These methods are not discussed here. Necessary details may be found in texts such as Numerical Recipes (Press et al., 1992). When calculating the trajectories of dispersed phase particles, any other auxiliary equations to account for heat transfer or chemical reactions can also be solved following similar procedures. Care must be taken to ensure that the time steps used for integration are sufficiently small and the trajectory integration is adequately time accurate. It is often necessary to use different time steps to simulate transients in the continuous flow field and trajectories of dispersed phase particles. 

With a Eulerian-Lagrangian approach, processes occurring at the particle surface can be modeled when simulating particle trajectories (for example, the process of dissolution or evaporation can be simulated). However, as the volume fraction of dispersed phase increases, the Eulerian-Lagrangian approach becomes increasingly computation intensive. A Eulerian-Eulerian approach more efficiently simulate such dispersed multiphase flows. 

It must be noted here that most industrial fluidized bed reactors operate in a turbulent flow regime. Trajectory simulations of individual particles in a turbulent field may become quite complicated and time consuming. Details of models used to account for the influence of turbulence on particle trajectories are discussed in Chapter 4. These complications and constraints on available computational resources may restrict the number of particles considered in DPM simulations. Eulerian-Eulerian approaches based on the kinetic theory of granular flows may be more suitable to model such cases. Application of this approach to simulations of fluidized beds is discussed below. 

The claim by G A that only one of these traditions developed techniques to imitate real-world conditions is quite misleading. Both traditions used the cloud chamber to manufacture an artificial environment that approximates known phenomena. For the Cavendish physicists, the cloud chamber became one of the defining instruments of particle physics, precisely because the laboratory phenomena were modeled on the movement of the charged particles. The knotty clouds blended into the tracks of alpha particles and the threadlike" clouds simulate beta-particle trajectories (Galison Assmus, 1989, p. 268). Of course, G A are correct that these physicists aspired to dissect nature into its fundamental components, reflecting the long tradition of the corpuscular conception of matter. 

For details on the derivation of this formula the reader is referred to section 4.5.1 of Frenkel and Smit (1996). The physical message to be taken away from this discussion is that by carefully observing the statistics of particle trajectories it is possible to reconcile the relevant microscopic motions seen in a molecular dynamics simulation with the macroscopic reflection of these motions, namely, the existence of a diffusion constant. On the other hand, we have not fully owned up to the rarity of diffusion events as measured on the time scale of atomic vibrations. This topic will be taken up again in chap. 12. 

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