Stochastic Trajectory Models

In trajectory models, the particle turbulent diffusion can be considered by calculating the instantaneous motion of particles in the turbulent flow field. In order to simulate the stochastic characteristics of the instantaneous gas velocity in a turbulent flow, it is required to generate random numbers in the calculation process. 

Gaussian probability density distribution. Thus, when a particle passes through a turbulent eddy, we have 

by coupling Eqs. (5.247), (5.248), and (5.249) with the governing equations for the gas phase, the instantaneous particle velocities can be obtained. Moreover, the stochastic trajectories of the particles can also be obtained by 

In the numerical integration of a particle trajectory, the selection of the integral time step is important. A typical way of choosing the integral time step is based on the interacting duration between the turbulent eddy and the particle. This interacting duration Tj may be determined by 

In computation using the stochastic trajectory model, the Monte Carlo approach is commonly employed. It is necessary to calculate several thousands, or even tens of thousands, of trajectories to simulate the particle flow field. The central issue in developing the stochastic trajectory model is how to model the instantaneous turbulent gas flow field. The method 

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Two basic trajectory models, i.e., the deterministic trajectory model [Crowe etal., 1977] and the stochastic trajectory model [Crowe, 1991], are introduced in this section. The deterministic approach, which neglects the turbulent fluctuation of particles, specifically, the turbulent diffusion of the mass, momentum, and energy of particles, is considered the most... 

As pointed out earlier in Chapter 3 and will be covered in more detail later on in this chapter, the flow in a rotary kiln is typically gas-solid turbulent flow with chemical reactions, mainly combustion. The building blocks behind the user-defined functions (UDF) in commercial CFD codes applied to rotary kiln combustion modeling consist of "renormalization group" (RNG) k-s turbulent model for gas phase and, in the case of pulverized combustion particles, the statistical (stochastic) trajectory model for homogeneous volatile and heterogeneous solid-phase char combustion. The underlying equations are discussed in the next section. 

Wang, L.-P. and Stock, D.E. (1992). Stochastic Trajectory Models for Djibulent Diflusion Monte-Carlo Process versus Markov Chains. Atmos. Environ., Vol. 26, pp. 1599-1607. Wang, L.-P. and Stock, D.E. (1993). Dispersion of Heavy Particles by Tuibulent Motion. J. Atmos. Sd., Vol. 50, pp. 1897-1913. 

To give an impression of the virtues and shortcomings of the QCL approach and to study the performance of the method when applied to nonadiabatic dynamics, in the following we briefly introduce the QCL working equation in the adiabatic representation, describe a recently proposed stochastic trajectory implementation of the resulting QCL equation [42], and apply this numerical scheme to Model 1 and Model IVa. 

Fig. 2.16. The random trajectory in the stochastic Lotka model, equation (2.2.76). Parameters are fco//3 = fijk = 10, the initial values Na = Nb = 10. When the trajectory touches the Na axis, the predators B are dying out and the population of the prey animals A infinitely...

In this section we analyze experimental data and make comparisons with theory. Data were obtained for 100 CdSe-ZnS nanocrystals at room temperature.1 We first performed data analysis (similar to standard approach) based on the distribution of on and off times and found that a+= 0.735 0.167 and v = 0.770 0.106,2 for the total duration time T = T = 3600 s (bin size 10 ms, threshold was taken as 0.16 max I(t) for each trajectory). Within error of measurement, a+ a k 0.75. The value of a 0.75 implies that the simple diffusion model with a = 0.5 is not valid in this case. An important issue is whether the exponents vary from one NC to another. In Fig. 13 (top) we show the distribution of a obtained from data analysis of power spectra. The power spectmm method [26] yields a single exponent apSd for each stochastic trajectory (which is in our case a+ a apSd). Figure 13 illustrates that the spread of a in the interval 0 < a < 1 is not large. Numerical simulation of 100 trajectories switching between 1 and 0, with /+ (x) = / (x) and a = 0.8, and with the same number of bins as the experimental trajectories, was performed and the... 

It can easily be observed that the considered case (4.149) corresponds to the situation for T = 0 only one marked particle evolves through a stochastic trajectory (a type 1 displacement vith Vj speed). This example corresponds to a Dirac type input and the model output response or the sum Pi(Ze,t) + P2(0, t), represents the distribution function of the residence time during the trajectory (see also application 4.3.1). 

Simpler BGK kinetic theory models have, however, been applied to the study of isomerization dynamics. The solutions to the kinetic equation have been carried out either by expansions in eigenfunctions of the BGK collision operator (these are similar in spirit to the discussion in Section IX.B) or by stochastic simulation of the kinetic equation. The stochastic trajectory simulation of the BGK kinetic equation involves the calculation of the trajectories of an ensemble of particles as in the Brownian dynamics method described earlier. 

As mentioned in the introduction, a dynamic MC simulation constmcts a stochastic trajectory Ai— X2—> —>X,—-through the (configurational) phase space, X, of the chosen model. The precise meaning of X depends on the character of the model for example, for an off-lattice bead-spring model, X just means the set of coordinates of all monomers of all N polymer chains f, ... 

This discussion suggests that even the reference trajectories used by symplectic integrators such as Verlet may not be sufficiently accurate in this more rigorous sense. They are quite reasonable, however, if one requires, for example, that trajectories capture the spectral densities associated with the fastest motions in accord to the governing model [13, 15]. Furthermore, other approaches, including nonsymplectic integrators and trajectories based on stochastic differential equations, can also be suitable in this case when carefully formulated. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

Particle trajectories can be calculated by utilizing the modern CFD (computational fluid dynamics) methods. In these calculations, the flow field is determined with numerical means, and particle motion is modeled by combining a deterministic component with a stochastic component caused by the air turbulence. This technique is probably an effective means for solving particle collection in complicated cleaning systems. Computers and computational techniques are being developed at a fast pace, and one can expect that practical computer programs for solving particle collection in electrostatic precipitators will become available in the future. 

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