Analysis Using Phase Trajectories

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. 

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. 

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. 

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 ... 

In contrast to the subsystem representation, the adiabatic basis depends on the environmental coordinates. As such, one obtains a physically intuitive description in terms of classical trajectories along Born-Oppenheimer surfaces. A variety of systems have been studied using QCL dynamics in this basis. These include the reaction rate and the kinetic isotope effect of proton transfer in a polar condensed phase solvent and a cluster [29-33], vibrational energy relaxation of a hydrogen bonded complex in a polar liquid [34], photodissociation of F2 [35], dynamical analysis of vibrational frequency shifts in a Xe fluid [36], and the spin-boson model [37,38], which is of particular importance as exact quantum results are available for comparison. 

Nonlinear analysis requires the use of new techniques such as embedding of data, calculating correlation dimensions, Lyapunov exponents, eigenvalues of singular-valued matrices, and drawing trajectories in phase space. There are many excellent reviews and books that introduce the subject matter of nonlinear dynamics and chaos [515,596-599]. 

Using the expression (5.22) together with Tables 5.5 and 5.6 on the base of the general principles reported in Sect. 5.2 one can carry out an exhaustive classification of the four-component systems as it has been already done for terpolymerization in Sect. 5.3. However, when the forth monomer is added, the number of the system types increases from 7 (see Fig. 6) to 41 (see Fig. 9) and that is a reason why the results of the complete theoretical analysis cannot be represented in the framework of this review. Without appealing to the classification and using only the algorithm described in Sect. 5.2 one may present a phase portrait of any concrete four-component system and hence predict the qualitative character of its dynamic behavior before the computer calculations of trajectories x(p) are performed. 

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