Trajectory bundles

Caustics The above formulae can only be valid as long as Eq. (9) describes a unique map in position space. Indeed, the underlying Hamilton-Jacobi theory is only valid for the time interval [0,T] if at all instances t [0, T] the map (QOi4o) —> Q t, qo,qo) is one-to-one, [6, 19, 1], i.e., as long as trajectories with different initial data do not cross each other in position space (cf. Fig. 1). Consequently, the detection of any caustics in a numerical simulation is only possible if we propagate a trajectory bundle with different initial values. Thus, in pure QCMD, Eq. (11), caustics cannot be detected. 

A particularly convenient notation for trajectory bundle system can be introduced by using the classical Liouville equation which describes an ensemble of Hamiltonian trajectories by a phase space density / = f q, q, t). In textbooks of classical mechanics, e.g. [12], it is shown that Liouville s equation... 

In a way, the limit set is thus the entire funnel between the two extreme cases qlc, and g o, Fig. 5. This effect is called Takens-chaos, [21, 5, 7]. As a consequence of this theorem each momentum uncertainty effects a kind of disintegration" process at the crossing. Thus, one can reasonably expect to reproduce the true excitation process by using QCMD trajectory bundles for sampling the funnel. To realize this idea, we have to study the full quantum solution and compare it to suitable QCMD trajectory bundles. 

A root velocity can be defined as the rate of advance of a given value of an H function root. For any given composition, roots with lower index numbers have lower velocities. An arbitrary initial noncoherent boundary involving variations of all roots thus is resolved, upon undisturbed development, into separate variations of the roots. This is shown by schematic trajectories of root values in a distance-time diagram in Figure 6. After resolution, each trajectory bundle involves variation of... 

Figure 7. TDSCF trajectory bundle result for HeI2Ne. The upper curves show the force on the He on two typical trajectories, while the lower curves show the IS I2 trajectory displacements. Short time behavior is shown, before the inphase collision occurs.

Trajectory Bundles Under Infinite Reflux Distillation Diagrams... 

Figure 2.4. Trajectory bundles under infinite reflux for (a) three-component ideal and (b) azeotropic mixtures. xd(x) xb(V),xd(2) - a b(2), possible splits solid lines, trajectories dotty line, separatrix under infinite reflux.

In Fig. 2.4b, another example of the trajectory bundles is shown (let s call the picture of trajectory bundles a distillation diagram), but already for a three-component azeotropic mixture acetone(l)-benzene(2)-chloroform(3). 

In this case, we have two trajectory bundles, differing by their unstable nodes and separated from each other with a specific trajectory, which begins not at the unstable node, but in a saddle (azeotrope 13 of maximum temperature) and is called the separatrix. 

Figure 2.5. Trajectory bundles under finite reflux of ace-tone(l)-benzene(2)-chloroform(3) azeotropic mixture for (a) rectifying and (b) stripping section. Solid lines with arrows, trajectories solid Une, a-hne dotty hne, separatrix under infinite reflux big circles, stationary points nnder infinite reflux tittle circles, stationary points nnder finite reflnx.

The given trajectory belongs to some trajectory bundle bounded by its fixed points (points A(r, Sr and N+ of Fig. 2.5a and points Ss and N+ of Fig. 2.5b), the separatrixes of the saddle points S and the sides of the concentration triangle. 

Knowledge about the regularities of the trajectory bundles arrangement under the finite reflux provides an opportunity to develop the reliable and fast-acting algorithm to fulfill design calculations of distillation to determine the required number of trays for each section. 

Generally speaking, for the first and second fractionation classes under the minimum reflux mode, the points of compositions in the zones of constant concentrations (i.e., stationary points of the trajectory bundles) should be arranged at the trajectories of reversible distillation built for the product points. It follows from the conditions of the material balance and the phase equilibrium in the zones of constant concentrations. Figure 2.11b illustrates the partially reversible process (it is reversible in the colunm parts that are from the constant concentration zones for the minimum reflux mode up to the column ends). 

In the case of the reflux ratio alteration and conservation of the product composition, the stationary points of trajectory bundle are traveling along the reversible distillation trajectories built for a given product, so the trajectories may be called lines of stationarity. Thus, the analysis of the reversible distillation trajectory arrangement in the concentration simplex is decisive in general geometric theory of distillation. 

What is the stationary point of the distillation trajectory bundle ... 

What is the arrangement of the distillation trajectory bundles under infinite and finite reflux modes dependent on ... 

At R = oo and N = oo, distillation trajectories bundles fill up distillation regions Reg°° in concentration simplex limited by node and saddle stationary points (points of components and azeotropes) and by boundary elements of various dimensionality, part of which are located at boundary elements of concentration simplex and part of which are located inside it. 

The analysis of the thermodynamically reversible process of distillation for multicomponent azeotropic mixtures was made considerably later. Restrictions at sharp reversible distillation were revealed (Petlyuk, 1978), and trajectory bundles at sharp and nonsharp reversible distillation of three-component azeotropic mixtures were investigated (Petlyuk, Serafimov, Avet yan, Vinogradova, 1981a, 1981b). 

As far as stationary points of trajectory bundles of distillation at finite reflux lay on trajectories of reversible distillation, these trajectories were also called the lines of stationarity (pinch lines, lines of fixed points) (Serafimov, Timofeev, Balashov, 1973a, 1973b). These lines were used to deal with important applied tasks connected with ordinary and extractive distillation under the condition of finite... 

Significance of reversible distillation theory consists in its application for analysis of evolution of trajectory bundles of real adiabatic distillation at any splits. Numerous practical applications of this theory concern creation of optimum separation flowsheets determination of optimum separation modes, which are close to the mode of minimum reflux and thermodynamic improvement of distillation processes by means of optimum intermediate input and output of heat. 

Such a set of trajectories for which not more than one trajectory passes through each nonsingular point is convenient for understanding as a distillation trajectory bundle. This notion is different for various modes of distillation because the number of parameters influencing the location of the trajectories is different. 

Therefore, one or several trajectory bundles, filUng up that region inside the concentration simplex, where one and the same component is the heaviest (for the top section) or the Ughtest (for the bottom section), will correspond to all the product points located at one and the same (n — 1) component edge, face, or hyperface of concentration simplex. We call this region the region of reversible... 

Let s illustrate the location of trajectory bundles of reversible distillation by the example of three-component acetone(l)-benzene(2)-chloroform(3) mixture with one binary saddle azeotrope with a maximum boiling temperature (Fig. 4.7)... 

The region of reversible distillation can contain one or several reversible distillation trajectory bundles (lines of stationarity). Some of these bundles can be true some of them can be fictitious. Fictitious bundles always have two node points and true ones have one node point or no node point. 

Such an approach on the basis of product points will be necessary at the analysis of the location of adiabatic sections trajectories bundles (at finite reflux) which products consist less (n - 1) components (see Chapter 5). 

Locations of trajectories bundles Regr. of node points of these bundles Nrev, and of possible product segments Reg Wd Regf can be shown in diagrams of three-component azeotropic mixtures sharp reversible distillation for various types of such mixtures (Fig. 4.11). 

The location of trajectory bundles and possible product composition segments at reversible distillation of three-component mixtures determines the location of trajectory bundles, and of possible product composition regions of multicomponent mixtures and the locations of trajectory bundles of real adiabatic columns. 

Trajectories Bundles of Reversible Distillation for Multicomponent Mixtures 93... 

It was shown (Petlyuk, 1986) that the diagram of reversible distillation of any three-component mixture can be forecasted by scanning only the sides of the concentration triangle, defining at each point the values of phase equilibrium coefficients of all the components and using Eqs. (4.19) and (4.20). The latter way defines trajectory tear-off segments Reg or Reg(. y and possible product segments Reg or Reg The node points iV v of the trajectory bundles are determined hypothetically on the basis of the data on the location of azeotropes points and a-points. 

Let s examine the analysis of structure of reversible distillation trajectory bundles at the concrete example of four-component mixture acetone(l)-benzene(2)-chloroform(3)-toluene(4). At the beginning, the segments of the components order Regff at the edges of the concentration tetrahedron are defined by means of scanning and calculation of the values Ki (Fig. 4.13a). The corresponding regions of components order Reg in the tetrahedron are shown in Fig. 4.13b and in its faces - in Fig. 4.14. The whole face 1-2-3, where the component 4 that is absent... 

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