Trajectory tear-off

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. 

In both cases, the trajectory tear-off point of sharp reversible distillation in the intermediate extractive section should lie at side 1-3 and the trajectory of intermediate section is a line, which is a geometric locus of points where the straight lines passing through a given point of pseudoproduct are tangent to residue curves. This trajectory reaches side 1-3 at the tear-off point and vertex 2 is the node... 

I. Define the main notions (1) region of reversible distillation of top Reg g bottom Reg -e, and intermediate sections Reg g (2) regions of trajectory tear-off Reg and Reg (3) region of possible product points of sharp reversible distillation Reg, Reg, and Reg and (4) node of trajectory bundle of reversible distillation Nrev-... 

To overcome these defects, it was necessary to apply the conception of sharp separation and to develop the theory of distillation trajectory tear-off from the boundary elements of concentration simplex at sharp separation (Petlyuk, Vinogradova, Serafimov, 1984 Petl50ik, 1998) and also to develop the geometric theory of section trajectories joining in feed cross-section in the mode of minimum reflux that does not contain simplifications and embraces mixtures with any number of components and any splits (Petlyuk Danilov, 1998 Petlyuk Danilov, 1999b Petlyuk Danilov, 2001a Petlyuk Danilov, 2001b). 

In conclusion, at the limit (boundary) value of reflux number the product point Xd approaches side 1 -2 (sharp separation, the second class of fractioning Fig. 5.2b). At the same time, the saddle stationary point S (trajectory tear-off point x from side 1-2) appears at side 1-2. Therefore, at boundary reflux number in... 

At D = Dpr and at i = R in both sections, there are two zones of constant concentrations - in the feed point Xf and in the trajectory tear-off points of sections x from the boundary elements of concentration simplex. For a three-component mixture there is a transition from the first class of fractioning right away into the third class, omitting the second class. At further increase of reflux number, the product compositions do not change any more. 

AtD < Dpr and R = in the top section, there are two zones of constant concentrations in feed point xp and in trajectory tear-off point from the boundary element of concentration simplex and in the bottom section there is one zone in feed point xp. At D > Dpr and R = on the contrary, in the bottom section there are two zones of constant concentration and in the top the section there is one zone. In both cases there is a transition from the first class of fractioning to the second one (i.e., in one of the sections, zone of constant concentrations in feed cross-section disappears, and in the other section, the zone is preserved, but the composition in it starts to change with the change of R). 

At further increase of R at direct separation, top product point xd begins to move along side 1-2 to vertex 1 till component 1 will be completely in top product. After that, further movement of product points xd and xb is stopped (i.e., the third class of fractioning ensues). At indirect separation, bottom product point Xb moves to vertex 3 till component 3 will be entirely in bottom product. At the second class of fractioning, trajectory tear-off point x of one of the sections is not changed and, for mixtures with constant relative volatilities, part of trajectory of this section x s S Ai+ is also not changed (Stichlmair et al., 1993). 

At nonsharp separation, the stationary points of section working regions, except the stable node N+, are located outside the concentration simplex (the direction of trajectory from the product is accepted). At sharp separation, other stationary points - trajectory tear-off points x from the boundary elements of concentration simplex - are added to the stable node. These are the saddle points S and, besides that, if the product point coincides with the vertex corresponding to the lightest or to the heaviest component, then this point becomes an unstable node N. ... 

Trajectory Tear-Off Theory and Necessary Conditions of Mixture Separability... 

If the problem is stated in this way, it is necessary to determine what product compositions xd and xb of sharp separation are feasible at distillation of nonideal zeotropic and azeotropic mixtures. The theory of distillation trajectory tear-off from the boundary elements of concentration simplex answers this question. 

Conditions of Distillation Trajectory Tear-Off at Sharp Splits... 

Let s examine two constituent parts of section distillation trajectory at the example of sharp preferable split of three-component ideal mixture (Fig. 5.6a) the part located in the boundary element (the side of concentration triangle), and the part located inside concentration simplex (triangle). There is a trajectory tear-off point from the boundary element x between these two parts. 

Trajectory tear-off points x are a special kind of stationary points that can be called pseudostationary ones because, in the vicinity of these points, the concentrations of components, absent in the boundary element, along the distillation trajectory in the direction to the product decrease monotonously at infinite number of separation stages > xf, where is a separation stage closer to the... 

Figure 5.7. Variations in component concentration ratios at neighboring trays about stationary point (x ) for any components i and about top section trajectory tear-off (pseudostation-ary) point (xj.) for absent in the boundary element components j. Kf is the phase equilibrium coefficient of absent component) in the pseudo-stationary point.

We see that these conditions differ from those in other stationary points (Eq. [5.6]). The difference in change of concentrations ratio of the components at two neighboring plates in the stationary points and of the components absent in the boundary element in trajectory tear-off points x is shown for the top section in Fig. 5.7. 

Inequalities (Eqs. [5.9] and [5.10]) for the components absent in the product and in the boundary element are valid inside concentration simplex not only in the vicinity of trajectory tear-off points x from the boundary elements, but also in other trajectory points that are not stationary. 

Equations (5.9), (5.10), (5.15), and (5.16) are necessary and sufficient conditions of trajectory tear-off from the boundary element of concentration simplex. Equations (5.9) and (5.10) can be called operating ones because they depend on separation mode, and Eqs. (5.15) and (5.16) can be called structural ones because they depend only on the structure of the field of phase equilibrium coefficients. 

Trajectory Tear-Off Regions and Sharp Distillation Regions... 

In trajectory tear-off points of the top section x( phase equilibrium coefficients of the components present in the product Kj should be greater than those of the components absent in the product Kj, and vice versa in the bottom section. Therefore, tear-off of trajectories from the boundary elements of concentration simplex is feasible only if in the vicinity of this boundary elements there are component-... 

We call the region where trajectory tear-off is feasible trajectory tear-off region ... 

Figure 5.10. Component-order regions Reg, trajectory tear-off regions Reg and Re , and sharp split regions Reg and Reg (hatched for bottom section and shaded for top section, hatched and shaded for two section) for the acetone(l)-chloroform(2)-methanol(3) mixture for splits (a)l,3 1,2 (Reg / and Reg j ) and (b) 2,3 1,2 (Reg ...

If the split without distributed components is set, then the mentioned trajectory tear-off regions of sections should be boundary elements of one sharp split region... 

Product points can belong only to those boundary elements of concentration simplex that contain trajectory tear-off regions. Along with that, product points should be located at these boundary elements within the limits of some region,... 

The notions of trajectory tear-off regions Reg or Reg possible product... 

To understand the structure of section trajectory bundles for multicomponent mixtures and their evolution with the increase of reflux number, let s examine first three-component mixtures, basing on the regularities of distillation trajectory tear-off at finite reflux and the regularities of location of reversible distillation trajectories. 

At L/V = K2, there is second bifurcation, trajectory tear-off from vertex 1 along sides 1-2 and 1-3 becomes feasible (i.e., inside concentration triangle vertex turns into unstable node N ) (Fig. 5.11c), distillation process for the product point under consideration becomes feasible, trajectory bundle with the saddle point S at side 1-2 and, with the stable node at side 1-3 in the vicinity of vertex 1, appears S ... 

L/V < (L/y) g point xd at side 1-2 cannot be the distillation top product point. Along with that, at side 1 -2 there is a stable node (Fig. 5.14a). At (L/ E) = (L/ y), the bifurcation goes on. Distillation trajectory tear-off from side 1-2 inside concentration triangle and the distillation process becomes feasible. The stable node at side 1-2 turns into a saddle in point x[. The trajectory bundle appears with saddle point S at side 1-2 and with the stable node A+ inside concentration triangle at reversible distillation trajectory in the vicinity of point x[. At further increase... 

The most important pecuUarities of location of reversible distillation trajectories of azeotropic mixtures, influencing the evolution of distillation trajectories at the change of the parameter L/V, are limitedness of trajectory tear-off segment... 

It is expedient to discuss the influence of these pecuharities on the evolution of distillation trajectories at the concrete example of azeotropic mixture, such as acetone(l)-benzene(2)-chloroform(3). At side 2-3, there is reversible distillation trajectory tear-off segment of the bottom section from vertex 2 to 0 13-point... 

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