Phase trajectory analysis

The complex flow within an impactor can be studied by using the concept of phase trajectory analysis where the paths of particles with different initial locations and velocities are determined. By analyzing these paths, conclusions can be drawn about a particle s fate as it travels through an impactor. Because in this analysis ideal streamline flow conditions are assumed (which actually may not be the case), phase trajectory analysis helps show how predictions from ideal assumptions may be modified by real-world conditions. A fairly simple case is chosen to illustrate the method. 

Montgomery J A Jr, Chandler D and Berne B J 1979 Trajectory analysis of a kinetic theory for isomerization dynamics in condensed phases J. Chem. Phys. 70 4056... 

Simple analysis of the motion of mapping point through the phase trajectory (Fig. 2.13) indicates that X represents the stationary concentration of chemisorbed radicals corresponding to a given external conditions. 

The problem is reduced to finding the phase trajectories of the equation system (104) at the (g, 0)-plane at different y values (dimensionless reaction rate) and values of p (relationship of the rates of relaxation g and heat removal at T = Tq). Dependence of the solution on x and in the physically justified ranges of their variation (tj > I at q qi ij< 1) turns out to be relatively weak. The authors of ref 234 applied the well-known method of analysis of specific trajectories changing at the bifurcational values of parameters [237], In the general case, the system of equations (104) has four singular points. The inflammation condition has the form... 

The solutions of x = Ax can be visualized as trajectories moving on the (x,y) plane, in this context called thep/iaseplane. Our first example presents the phase plane analysis of a familiar system. 

The analysis of linearized sytem thus allows, when conditions (1)—(3) are met, us to find the shape of phase trajectories in the vicinity of stationary (singular) points. A further, more thorough examination must answer the question what happens to trajectories escaping from the neighbourhood of an unstable stationary point (unstable node, saddle, unstable focus). In a case of non-linear systems such trajectories do not have to escape to infinity. The behaviour of trajectories nearby an unstable stationary point will be examined in further subchapters using the catastrophe theory methods. 

Phase plane analysis indicates that the two limit cycles possess different sensitivities toward perturbations. It is much easier to pass from the small limit cycle to the large one than to achieve the reverse transition. This differential sensitivity results from the relative sizes of the attraction basins of the two cycles. Moreover, to pass from the large cycle to the small one, the quantity of substrate must be sufficient to cross the border defined by the unstable trajectory, but not so large so as to avoid bringing the system across the basin of the small cycle, into the other side of the attraction basin of the large cycle in such a case, the perturbation would only cause a phase shift of the large oscillations. In... 

D. Chandler,. Chem. Phys., 68, 2959 (1978). Statistical Mechanics of Isomerization Dynamics in Liquids and the Transition State Approximation. J. A. Montgomery, jr., D. Chandler, and B. J. Berne, /. Chem. Phys., 70,4056 (1979). Trajectory Analysis of a Kinetic Theory for Isomerization Dynamics in Condensed Phases. 

For the consecutive reaction A B two types of trajectories are studied (i) phase trajectories in the complete variable (concentration) space and (ii) temporal trajectories of each variable separately. Phase trajectories neither intersect nor merge, but temporal trajectories may or may not intersect. At a point of intersection, the concentrations of two or three reaction components are equal. Yablonsky et al. (2010) performed an analysis of the theoretically possible types of temporal concentration intersections and the conditions of their occurrence. 

For a qualitative analysis of projections of the chaotic phase trajectory onto the plane it seems suffisant ( ) to consider a dynamical system truncated at the three harmonics. This system can be easily written in an explicit form. 

Ovod, V. L, Hydrodynamic Focusing of Particle Trajectories in Light-Scattering Counters and Phase-Doppler Analysis, Parr. Part Syst Charact, 1995, 12,207-211. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Chymotrypsin, 170,171, 172, 173 Classical partition functions, 42,44,77 Classical trajectories, 78, 81 Cobalt, as cofactor for carboxypeptidase A, 204-205. See also Enzyme cofactors Condensed-phase reactions, 42-46, 215 Configuration interaction treatment, 14,30 Conformational analysis, 111-117,209 Conjugated gradient methods, 115-116. See also Energy minimization methods Consistent force field approach, 113 Coulomb integrals, 16, 27 Coulomb interactions, in macromolecules, 109, 123-126... 

This analysis is limited, since it is based on a steady-state criterion. The linearisation approach, outlined above, also fails in that its analysis is restricted to variations, which are very close to the steady state. While this provides excellent information on the dynamic stability, it cannot predict the actual trajectory of the reaction, once this departs from the near steady state. A full dynamic analysis is, therefore, best considered in terms of the full dynamic model equations and this is easily effected, using digital simulation. The above case of the single CSTR, with a single exothermic reaction, is covered by the simulation examples THERMPLOT and THERM. Other simulation examples, covering aspects of stirred-tank reactor stability are COOL, OSCIL, REFRIG1 and REFRIG2. Phase-plane plots are very useful for the analysis of such systems. 

In compositional analysis of very small precipitates, or in interface segregation studies, using a probe-hole type atom-probe, one is always faced with the fact that the probe-hole may cover both the matrix and the precipitate phases, or the interface as well as the matrix. Thus any abrupt compositional changes will be smeared out by the size of the probe-hole and also by the effect of ion trajectories. A similar uncertainty seems to exist in the compositional analysis of nitride platelets formed in nitrided Fe-3 at.% Mo alloy, aged between 450 and 600°C, where Wagner ... 

To all intents and purposes, the Hopf prediction is the exact result. It is not difficult to construct an intersection between the maximum and minimum which is a stable stationary state. For instance, with y = 0.02, k = 0.12, and /i = 0.2 we have 0X = 1.042, 9C = 2400, and 0SS = i/k = 5/3. The corresponding phase plane nullclines are shown in Fig. 5.10, together with a trajectory spiralling in to the stationary-state intersection. The trace of the Jacobian matrix is negative for this solution (tr(J) = —4.1 x 10 2) indicating its local stability. This is not, however, a particularly fair test of the relaxation analysis because the parameters /i and k are not especially small. In the vicinity of the origin (where is small) both approaches converge. 

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