Particle trajectories

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 ... 

For many applications, it may be reasonable to assume that the system behaves classically, that is, the trajectories are real particle trajectories. It is then not necessary to use a quantum distribution, and the appropriate ensemble of classical thermodynamics can be taken. A typical approach is to use a rnicrocanonical ensemble to distribute energy into the internal modes of the system. The normal-mode sampling algorithm [142-144], for example, assigns a desired energy to each normal mode, as a harmonic amplitude... 

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]). 

Here r(t) is the stress at a fluid particle given by an integral of deformation history along the fluid particle trajectory between a deformed configuration at time f and the current reference time t. 

After identification of the elements that contain feet of particle trajectories the old time step values of F at the feet are found by interpolating (or extrapolating for boundary nodes) its old time step nodal values. In the example shown in Figure 3.6 the old time value of Fat the foot of the trajectory passing through A is found by interpolating its old nodal values within element (e). 

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). 

Figures 18-36, 18-37, and 18-38 show some approaches. Figure 18-36 shows velocity vectors for an A310 impeller. Figure 18-37 shows contours of kinetic energy of turbulence. Figure 18-38 uses a particle trajectory approach with neutral buoyancy particles.

Fig. 3.48. Schematic diagram of particle trajectories undergoing scattering at the surface and channeling within the crystal. The depth scale is compressed relative to the width of the channel, to display the trajectories [3.120].

For a given particle of size d, from the point M where the equilibrium line meets the line of zero vertical velocity (see Fig. 13.4), the critical path of the particle may be defined. All particles of this size between points D and G are entrained in the downward stream and are collected. The remaining particles of this size join in the upward-moving stream of fluid and penetrate the cyclone. The point D may be obtained by tracking back the particle trajectory from the point M using the equation of the particle trajectory, which is given by... 

By integrating Eq. (13.35) step by step in time, the particle trajectory of the particle may be obtained. In the integration, the interaction between the particle and the wall may be approximated as being fully elastic however, when the particle hits the sidewall of the cyclone, the particle may be treated as being collected and the computation for the particle may terminated in order to save the computational time that may be required to track the particle to the bottom of the cyclone. If the particle trajectories for a range of particle diameters at different rates of fluid flow through the cyclone are determined, then the particle efficiency curve and the cut-off particle diameter of the cyclone may be obtained. 

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. 

FIG. 14 Brownian dynamics. The arrows indicate that the particle trajectories show diffusive behavior. 

Detailed balance is a chemical application of the more general principle of microscopic reversibility, which has its basis in the mathematical conclusion that the equations of motion are symmetric under time reversal. Thus, any particle trajectory in the time period t = 0 to / = ti undergoes a reversal in the time period t = —ti to t = 0, and the particle retraces its trajectoiy. In the field of chemical kinetics, this principle is sometimes stated in these equivalent forms ... 

In the purely non-adiabatic limit the phase (5.52) coincides with that calculated in [203] and for very long flights (rt b,v" v) or high energies (.E e) it reduces to what can be obtained from the approximation of rectilinear trajectories. However, there is no need for these simplifications. The SCS method enables us to account for the adiabaticity of collisions and consider the curvature of the particle trajectories. The only demerit is that this curvature is not subjected to anisotropic interaction and is not affected by transitions in the rotational spectrum of the molecule. 

From the image sequences, information on the velocities of nano-particles can be extracted. The statistical effect of Brownian motion on the flowing speed of the mixed liquid is found small enough to be ignored as shown in Fig. 37 where most of the particles trajectories in the liquid are straight lines and parallel with the wall basically. Therefore, Brownian diffusive motion is ignorable. 

In general, all electrostatic separator systems contain at least four components (i) a chargingdischarging mechanism (ii) an external electric field (iii) a nonelectrical particle trajectory device and (iv) feed and product collection systems. Depending primarily on the charging mechanism involved, the electrostatic separator systems are classified into three categories (i) free fall separators (ii) high tension separators and (iii) conduction separators. 

Let us consider the SHV mode of the TCP flow as the base state of a dynamical system described by three velocity components Vr, V0 and Vz relative to the fixed cylindrical co-ordinate system depicted in Figure 4.4.7(b). This dynamical system is described mathematically by the equations of motion of the particle trajectories in the 3D (r, 0, z) coordinate system ... 

Regular and high-speed movies were taken of the tracer particle movement around the jets at different velocities and different solid loadings. The tracer particles used are red plastic pellets of similar size and density to the bed material. The movies were then analyzed frame by frame using a motion analyzer to record the particle trajectories and the particle velocities. 

Typical particle trajectories observed in the movies are shown in Fig. 47 for a jet velocity of 62.5 m/s and a solid loading of 1.52. The time elapsed between dots shown in Fig. 47 was typically 5 movie frames, while the movie speed was 24 frames/s. The colored tracer particles were followed in the vicinity of the jet until they disappeared into the jet, as... 

The solids entrainment rate into a jet in a fluidized bed can be calculated from Eqs. (61) and (23) if the empirical constants C1 and C2 and the jet half-angle 6 are known. The jet half-angle 6 can be taken to be 10° as suggested by Anagbo (1980), a value very close to 7.5° obtained from solid particle trajectories reported here. The real jet half-angle will be larger than 7.5° because of the truncation of the jet by the front plate of the semicircular bed. 

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). 

Houghton, G., Particle Trajectories and Terminal Velocities in Vertically Oscillating Fluids, Can. J. Chem. Eng., 44 90-95 (1966)... 

At any given instant the equation S(x, t) = const, defines a surface in Euclidean space. As t varies the surface traces out a volume. At each point of the moving surface the gradient, VS is orthogonal to the surface. In the case of an external scalar potential the particle trajectories associated with S are given by the solutions mx = VS. It follows that the mechanical paths of a moving point are perpendicular to the surface S = c for all x and t. A family of trajectories is therefore obtained by constructing the normals to a set of... 

---