Limiting trajectory

When following the mean temperature evolution along the limiting trajectory, it was observed that a flame is definitely stable if the temperature attains... 

Figure 12.8 Calculated trajectories of fluid particles in the combustor with flame holder (solid lines) and the curves of constant dimensionless residence time t/tr (dashed curves). The residence time tr is defined as the time taken for the fluid particle to reach the turning point at the limiting trajectory (marked by the arrow). Conditions are similar to Fig. 12.66... 

LaFrance and Grasso [29] report an application of MD methods (again, termed trajectory analysis ) to the dissolved air flotation of nitrocellulose particles of 2.3 fim diameter. This work also neglected Brownian motion considerations, but included electrostatic, van der Waals, the Lewis acid-base interaction forces, and hydrodynamic forces. Lafrance and Grasso found limiting trajectories by successions of forward integrations, and from this calculated the capture efficiency per air bubble as a function of solution chemistry. 

The negative effect of the centrifugal force can be summarised by the negative effect of SRHI, which is an essential deviation from Sutherland s formula. A common action of these factors appear if the limit trajectory ends not at the equator but at 0 = 0,. Results of such common action are shown in Fig. 10.15 for a fixed bubble radius a = 0.04cm and for a number of critical film thicknesses H,. = h,. / a,. 

The mass flow rate to the sphere is found by calculating the rate at which mass impinges on the sphere between the stream surface IT = 0 and the stream surface defined by the loci of limiting trajectories, illustrated in Fig. 8.3.2 for the cylindrical case. The loci of limiting trajectories are determined by setting r = a + ap and 6 = ttII, whence... 

The distance E yi l e limiting trajectory from the axis 6 = 0 as r—as shown in Fig. 8.3.2, follows directly from the definition of the stream function and the fact that lp = E yiaU. Note that the corresponding limiting trajectory distance for the spherical collector is ph. ... 

The limiting trajectory dp = tt) is seen to lie in the range of large h/Up for large values of the adhesion group N j-... 

Integrating this equation along the limiting trajectory and taking the cube root of the solution give... 

One important objective of trajectory analysis is to determine the limiting trajectory and thus the critical radius of particle capture. As shown in Fig. 3.7-2, the limiting trajectory is defined as the exact path that divides the particle trajectories into those leading to capture by the collector and those passing by the collector. The distance between the limiting trajectory and the axis is then defined as the critical radius (R ), which can be used to determine the removal efficiency of magnetic filtration (Watson, 1973) ... 

Fig. 3.7-2 Limiting trajectory for smaii paramagnetic particies captured by a wire a is the radius of the coiiector.

The perspective exploited by transition path sampling, namely, a statistical description of pathways with endpoints located in certain phase-space regions, was hrst introduced by Pratt [27], who described stochastic pathways as chains of states, linked by appropriate transition probabilities. Others have explored similar ideas and have constructed ensembles of pathways using ad hoc probability functionals [28-35]. Pathways found by these methods are reactive, but they are not consistent with the true dynamics of the system, so that their utility for studying transition dynamics is limited. Trajectories in the transition path ensemble from Eq. (1.2), on the other hand, are true dynamical trajectories, free of any bias by unphysical forces or constraints. Indeed, transition path sampling selects reactive trajectories from the set of all trajectories produced by the system s intrinsic dynamics, rather than generating them according to an artificial bias. This important feature of the method allows the calculation of dynamical properties such as rate constants. 

Shown in Fig. 10.6, is the distance Ecyia of limiting trajectory from the axis 0 = 0 at r 00, so that = aEcyiU. Note that for a sphere the distance of the limiting trajectory from the axis 0 = 0 at r oo is equal to a. 

The basic difference between particle capture in the case of non-zero molecular and hydrodynamic forces (Fig. 10.9) from particle capture in the case when these forces are ignored (Fig. 10.7) is in the limiting trajectory dividing the family of trajectories into those that lead the particles to collide with the cylinder, and those that lead the particles away from the cylinder. In the first case the limiting trajectory ends at the back critical point of the cylinder Op = tt, and in the second case -at the point of contact between particle Up and the cylinder surface at Op = njl. 

In the case Nad 1, it is possible to assume that the limiting trajectory passes through the region h Up. When these inequalities are satisfied, the equation of inertialess particle motion normally to the cylinder surface, with due consideration for the expressions (10.86), (10.93), (10.94), (10.96), and (10.99), will become... 

Thus, Eqs. (10.103) and (10.104) describe trajectories of particles Up flowing around the cylinder surface, including the limiting trajectory, at the conditions 1 and h/ttp 1. 

Let us show how one can And the limiting trajectory and collision frequency of particles with the cylinder without solving these equations. The limiting trajectory (see Fig. 10.9) ends at the back point Op =n, y = y. To determine y, one should proceed as follows. When the particle is moving along the critical trajectory near the back stagnation point, the velocity is Ur = = 0 and since the particle mo-... 

The solution to this equation can be easily obtained. In particular, along the limiting trajectory,... 

If inertia of particles is neglected, the limiting trajectory passes at a distance Up from the body. In Ref [58] this effect is called the effect of hooking. It has been shown in the same work that, for a potential flow around the sphere we have... 

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. 

Taking various values Ao and Oq, it is possible to determine trajectories of motion of drop S2 relative to Si. During the motion, the drop S2 either collides with Si or passes by. Trajectories corresponding to these cases, form two families trajectories terminating at r = 1 2 + l i from the center of sphere Si (collision), and trajectories passing by Si and extended to infinity. These two families are separated by the limiting trajectory. 

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