Particle, trajectory equation

We may now use this result in the particle trajectory equation resulting from combining equations (7.3.19a) and (7.3.21a) ... 

The particle trajectory equation (see equation (6.2.45) in such a device is as follows ... 

Dividing equation (7.3.258b) by equation (7.3.259b), we get the particle trajectory equation ... 

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... 

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 ... 

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 ... 

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... 

The dependence of the electron ion recombination rate constant on the mean free path for electron scattering has also been analyzed on the basis of the Fokker Planck equation [40] and in terms of the fractal theory [24,25,41]. In the fractal approach, it was postulated that even when the fractal dimension of particle trajectories is not equal to 2, the motion of particles is still described by difihsion but with a distance-dependent effective diffusion coefficient. However, when the fractal dimension of trajectories is not equal to 2, the motion of particles is not described by orthodox diffusion. For the... 

The AGDISP model is based on actually tracking the motion of discrete particles. The dynamic equations governing the particle trajectory are developed and integrated. The equations include the influence of the aircraft dispersal system configuration, aircraft wake turbulence, atmospheric turbulence, gravity, and evaporation. 

Example 3.3 Consider a rotating gas flow in a cylindrical chamber with a small particle injected into the flow. Assume that the gas rotates as a rigid body with a constant angular velocity co and the only driving force is the Stokes drag [Kriebel, 1961]. Initially, the relative particle velocity is normal to the flow. Develop the equations for the particle trajectory in this rotating flow and discuss the effect of particle sizes on the trajectory. 

Note that /ep in Eq. (5.238) is replaced with /Ep for Eq. (5.240), where /Ep is the heat generated by thermal radiation per unit volume and Qap is the heat transferred through the interface between gas and particles. Thus, once the gas velocity field is solved, the particle velocity, particle trajectory, particle concentration, and particle temperature can all be obtained directly by integrating Eqs. (5.235), (5.237), (5.231), and (5.240), respectively. Since the equations for the gas phase are coupled with those for the solid phase, final solutions of the governing equations may have to be obtained through iterations between those for the gas and solid phases. 

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). 

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)... 

Applicable to investigation of equilibrium properties Random alterations of dispersion states, which are accepted or rejected based on a probability function that depends on the free energy change Applicable to systems with simple geometry Generates particle trajectories based on fundamental equations of motion (e.g. Newton s equation)... 

In terms of the quantum-potential formulation particle trajectories can be associated with the quantum HJ equation (6) in exactly the same way as in the classical case [34, 35]. As before, particle trajectories associated with the phase S may be obtained by constructing the normals to S, each one distinguished by its initial coordinates. By this procedure Bohm managed to revive the pilot-wave model of De Broglie. It means that a point particle of mass m on a trajectory x = x(t), is now associated with the physical... 

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