Trajectory variational equations approach

In his early survey of computer experiments in materials science , Beeler (1970), in the book chapter already cited, divides such experiments into four categories. One is the Monte Carlo approach. The second is the dynamic approach (today usually named molecular dynamics), in which a finite system of N particles (usually atoms) is treated by setting up 3A equations of motion which are coupled through an assumed two-body potential, and the set of 3A differential equations is then solved numerically on a computer to give the space trajectories and velocities of all particles as function of successive time steps. The third is what Beeler called the variational approach, used to establish equilibrium configurations of atoms in (for instance) a crystal dislocation and also to establish what happens to the atoms when the defect moves each atom is moved in turn, one at a time, in a self-consistent iterative process, until the total energy of the system is minimised. The fourth category of computer experiment is what Beeler called a pattern development... 

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, REFRIG and STABIL. 

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 solutions of these equations (the trajectories) will for long times (i.e., after transient effects associated with switching on the external parameters have decayed) approach so-called limit sets, which may be classified into fixed points (stationary states), limit cycles (periodic oscillations), mixedmode oscillations, quasiperiodic oscillations, and chaotic behavior. Transitions between these states may occur upon variation of the external parameters pk and are called bifurcations. Experimental evidence for these effects with the system CO + 02/Pt(110) will be briefly presented without going further into details of the underlying general theory (see 16, 17). 

This equation tells us that both short and long time measurements are important to acquire a full lineshape of D(t) where continuous variation of t is ideal. If our goal is a full-recording of D(t), the best approach is to calculate MSD from the statistical accumulation ofX data, real-time movements (trajectory) of molecules. Therefore, fast detection is better because the MSD in a long period can be obtained from the summation of Ax in any range. However, when Axt becomes short with fast observation, finer spatial resolution is also required. This is the reason why the dilemma between time resolution and spatial resolution still remains a problem in handling the real-time movements of substances in any biological system. 

The second broad area in Section 6.2 is concerned with particles. For the separation of particles from a fluid or fractionation of particles, one can adopt an Eulerian appmach to determine the particle concentration variation as observed by an observer located at a fixed coordinate (jqy,z). In such an approach, the fluid velocity is also what is determined by an observer at (jqy,z) as a function of time. However, an alternative approach, the Lagrangian approach, is frequently preferred and will be adopted often. The Lagrangian description of particle motion is obtained by an observer who rides on the particle. The geometrical coordinates (jqy,z) of the particle/observer change with time as the particle changes its location in the device in response to fluid motion and other forces, external and/or diffusive, acting on the particle. In such an approach, the coordinates (x,y,z) of a particle are dependent variables whose values as a function of time in the separation device are of interest. These equations, called trajectory equations, are also provided in Section 6.2. 

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