Section trajectory bundles intermediate

For the intermediate spfits, it is possible to take into consideration in the best way the regularities of location of section trajectory bundles, using two- or three-stage algorithm of search for (L/y) with gradual precise of the value of this parameter. 

The example of tangential pinch for four-component mixture is quasisharp separation of azeotropic mixture acetone (l)-benzene (2)-chloroform (3)-toluol (4) of composition Xf (0,350 0,250 0,150 0,250) at intermediate split 1,3(2) 2,4(3) (admixture components heavy and light key are in brackets correspondingly) at the following composition the products xd (0,699 0,001 0,300,0) and xb (0 0,500 10 0,500). The same top product composition, as in the previous example (Fig. 5.18b) of separation of three-component mixture in the top section, is accepted for convenience of analysis. In this case, the boundary elements of top section trajectory bundle, located in face 1-2-3, completely coincides with top section trajectory bundle at separation of previously mentioned three-component mixture. 

In Chapter 5, we saw that the distillation process in a column section is feasible only if there are reversible distillation trajectories inside concentration simplex and/or at several of its boundary elements, because only in this case a section trajectory bundle with stationary points lying at these trajectories of reversible distillation arises in concentration simplex. This condition of feasibility of the process in the section has general nature and refers not only to the top and the bottom, but also to intermediate sections. Therefore, pseudoproduct points can... 

This develops the general algorithm of calculation of minimum reflux mode for the columns with two feed inputs at distillation of nonideal zeotropic and azeotropic mixtures with any number of components. The same way as for the columns with one feed, the coordinates of stationary points of three-section trajectory bundles are defined at the beginning at different values of the parameter (L/V)r. Besides that, for the intermediate section proper values of the system of distillation differential equations are determined for both stationary points from the values of phase equihbrium coefficients. From these proper values, one finds which of the stationary points is the saddle one Sm, and states the direction of proper vectors for the saddle point. The directions of the proper vectors obtain linear equations describing linearized boundary elements of the working trajectory bundle of the intermediate section. We note that, for sharp separation in the top and bottom sections, there is no necessity to determine the proper vectors of stationary points in order to obtain linear equations describing boundary elements of their trajectory bundles, because to obtain these linear equations it is sufficient to have... 

We now examine the structure of intermediate section trajectory bundle for four-component mixtures at the example of ideal mixture (ATi > K2 > > K4). 

The dimensionality of intermediate section trajectory bundle is equal to n -m + 1, where n is total number of components, and m is summary number of components of top product and entrainer. Pseudoproduct point is located at the continuation of the boundary element, formed by all the components of the top product and entrainer. Reversible distillation trajectories and the stationary points are located at the mentioned pseudoproduct boundary element and at all boundary elements whose dimensionality is bigger by one (at m = n — 1, they are located inside concentration simplex). 

The general approach to design calculation of extractive distillation columns is similar to the approach applied for two-section columns. We use our notions about the structure of intermediate section trajectory bundles (see Sections 6.4 6.6), about possible compositions at the trays adjacent to the feed cross-section from above and below (see Section 7.2), and about possible directions of calculation... 

Let s examine now the structure of trajectory bundles of sharp reversible distillation for the intermediate (extractive) section of the column with two feedings at separation of different types of azeotropic mixtures, the way we did it for the top and the bottom sections (Fig. 4.21). While composing these diagrams, we used, just as we did before, the data on the phase equilibrium coefficients of present and absent components at the sides of the concentration triangle and the general regularities of the location of the trajectory bundles of sharp reversible distillation. 

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-... 

In contrast to the product point, the pseudoproduct point can be located not only inside or at the boundary of the concentration simplex, but also outside it. The latter case refers to colunms of extractive distillation with two feeds, which leads to new regularities of location of trajectory bundles and their stationary points, that differ from regularities of location of top and bottom section trajectories. Therefore, we pay a lot of attention in this chapter to trajectory bundles of intermediate sections in extractive distillation colunms. 

As in that case, to develop a general algorithm of calculation of minimum reflux mode for columns with several feed inputs, we need to understand the location of reversible distillation trajectories of intermediate sections and the structure of trajectory bundles for these sections. 

We examine the structure of trajectory bundles of intermediate sections (i.e., location and character of the stationary points of these bundles). 

Figure 6.5 shows stationary points Sm and iV+ of trajectory bundles of intermediate section Reg j and separatrixes of saddle stationary point Sm obtained by means of calculation for ideal mixture pentane(l)-hexane(2)-heptane(3) at the composition of pseudoproduct x), = -1.0 x)) 2 = 1-5 3 = 0.5 and at the value... 

In contrast to nonsharp separation in the top and bottom sections, the intermediate section has at reversible distillation trajectory not just one node stationary point, but there are saddle point Sm and node point Nm- Separatrixes of the saddle points Sm divide concentration triangle into four regions Reg t trajectory bundles of intermediate section, one of which is the working one Reg jj,t-... 

Because in the mode of minimum reflux the intermediate section should be infinite, its trajectory should pass though one of its stationary points Sm or A+. Therefore, the following cases are feasible in minimum reflux mode (1) point A+ coincides with the composition at the tray above or below the cross-section of control feed (2) composition point at the trays of the intermediate section in the cross-section of control feed lies on the separatrix line, surface, or hypersurface of point Sm (i.e., in separatrix min-reflux region of intermediate section Reg , filled of trajectory bundle Sm — A+). In both cases, composition point at the tray of the top or bottom section, adjacent to the control feed, should lie in the separatrix min-reflux region of this section Re (5 - A+). 

Theoretical analysis of the separation of azeotropic mixtures with the help of extractive distillation was carried out in the works (Levy Doherty, 1986 Knight Doherty, 1989 Knapp Doherty, 1990 Knapp Doherty, 1992 Wahnschafft Westerberg, 1993 Knapp Doherty, 1994 Wahnschafft, Kohler, Westerberg, 1994 Bauer Stichlmair, 1995 Rooks, Malone, Doherty, 1996 Stichlmair Fair, 1998 Doherty Malone, 2001). Characteristic peculiarities of the process of extractive distillation of binary azeotropic mixtures were investigated in these works. More general conception of the processes of extractive and autoextractive distillation on the basis of the theory of intermediate section trajectory tear-off from boundary elements of concentration simplex was introduced in the works (Petlyuk, 1984 Petlyuk Danilov, 1999). Trajectory bundles of intermediate section for multicomponent mixtures were examined in the latter work. 

As for the top and bottom sections, working trajectory bundle of the intermediate section should be located in a sharp distillation region where the order of the components is the same as at the trajectory tear-off region i.e. segment... 

The analysis carried out above (see Figs. 6.8 and 6.9) allows for a general conclusion about dimensionality, structure, and evolution of trajectory bundles (regions) of intermediate section at sharp extractive distillation for any multicomponent mixtures. 

For four-component mixtures at nim = 3 and at two components in the bottom product (Fig. 6.9b), the conditions of joining in the case of bottom control feed are defined by the dimensionality of trajectory bundles N - Sm(d = 1) and (d = 1) and are similar to those of joining of sections trajectories of two-section column in the mode of minimum reflux at intermediate split (see Section 5.6). Point X/-1 should lie on the separatrix min-reflux region Re (N - Sm) and point Xf should lie on the separatrix min-reflux region Re ( - N ). 

Trajectory bundles of bottom and intermediate sections in the mode of minimum reflux should join with each other in the concentration space of dimensionality (n - 1). Therefore, joining is feasible at some value of the parameter (L/T) if the summary dimensionality of these bundles is equal to (n - 2). 

We have a considerable limitation of sharp extractive distillation process in the column with two feeds the process is feasible if the top product components number is equal to one or two. This Umitation arises because, in the boundary element formed by the components of the top product and the entrainer, there is only one point, namely, point iV+, that belongs to the trajectory bundle of the intermediate section. If Eq. (6.11) is valid, then the joining of the trajectories of the intermediate and top sections takes place as at direct split in two-section columns in the mode of minimum reflux. If Eq. (6.12) is valid then joining goes on as at split with one distributed component. 

We express the value of the parameter (E/D) through limit conditions when the point of tear-off of reversible distillation trajectory coincides with the saddle point of trajectory bundle of the intermediate section (e.g., = Sm at Fig. 

Before examining minimum reflux mode for complexes with branching of flows, we discuss complex columns with side withdrawals of flows. Side products of such columns cannot be pure components at finite reflux, but the number of components in each side product can differ from the number of components in the other side products, in the initial mixture, and in the top and bottom products. In such complex columns in each section, the number of components at the exit from the section is smaller, than at the entrance. The simplest example of separation is 1 1, 2 3 (Fig. 6.14). In this case, side product 1,2 is withdrawn above feed. Such splits are sharp. We confine oneself to examining of complex columns with sharp splits. The pseudoproduct of each intermediate section of the column with side withdrawals of products is the sum of all the products above (below) the section under consideration, if this section itself is located above (below) feed. For such splits, all the pseudoproduct points of the intermediate sections are located at the boundary elements of concentration simplex. Therefore, the structure of trajectory bundles for the intermediate sections does not differ from the structure of trajectory bundles for the top or bottom sections at sharp separation. 

Figure 6.14 shows trajectories of the intermediate section for separation 1 1, 2 3 at different modes. Pseudoproduct points ( > — Dj+D) is located at side 1-2, and joining of the intermediate and bottom sections in the mode of minimum reflux goes on in the same way as for the simple column at indirect split. Trajectory of the intermediate section r tears off from side 1-2 in point Sn, and point of side product xd can coincide with point Sn (Fig. 6.14a) or lie at segment 1 - Sri (Fig. 6.14b). The first of these two modes is optimal because the best separation between top and side products (the mode of the best separation) is achieved at this mode. Zones of constant concentrations in the top and intermediate sections arise in point Sri = AC2- Therefore, in the mode of minimum reflux in the intermediate section, there are two zones of constant concentrations. At the reflux bigger than minimum, point 5 1 moves to vertex 2 and at i = oo this point reaches it (i.e., at i = 00, pure component 2 can be obtained in the infinite column as a side product). Therefore, for the colunuis with side withdrawals of the products, the mode of the best separation under minimum reflux corresponds to joining of sections in points 5 1 and of the trajectory bundle of the intermediate section (at sharp separation) or in its vicinity (at quasisharp separation). The trajectory of the column with a side product at minimum reflux at best separation may be described as follows ...

The important peculiarity of the sequence at Fig. 6.16b consists in the fact that column section between the cross-section of feed input and the top of the column by its nature is not the top but the intermediate section of the column with two feeds (see Section 6.3.). This leads to important peculiarities of the trajectory bundle (Reg int) of this section. 

In the stationary points of the trajectory bundle of the intermediate section, the liquid-vapor tie-lines should be directed to pseudoproduct point of this section that is, in the given case, the point of water phase from decanter xli = Such quasistationary point is point qSm, where the calculated trajectory of the intermediate section changes its direction sharply, and the compositions at neighboring trays are very close to each other (quasizones of constant concentrations), and stationary point Af+ that coincides with the point of ternary heteroazeotrope and stationary point N that coincides with the point of binary azeotrope benzene(l)-isopropanol(2). Point Sm is located at reversible distillation trajectory of the intermediate section joining mentioned points N and Its location at this trajectory... 

For intermediate sections of columns with side products, with side sections, and of Petlyuk columns location of the stationary points of separatrix trajectory bundles (regions Reg jjfj) is the same as for simple columns, product compositions of which coincide with pseudoproduct compositions of these intermediate sections (possible product regions Rego and Reg of simple columns and possible pseudoproduct regions Regn and Reg of intermediate sections coincide). This extends the use of methods of minimum reflux mode calculation worked out for the simple columns to the previously mentioned complex columns and complexes. 

The location of intermediate sections trajectories of columns with two feeds, including those at extractive, heteroazeotropic, and heteroextractive distillation, has fundamental distinctions from that of section trajectories of the simple columns. At sharp extractive or heteroextractive distillation, pseudoproduct point x), of the intermediate section should be located at the continuation of the boundary element, to which components of top product and of entrainer belong. If this condition is valid, the whole trajectory bundle of the intermediate section including trajectory tear-off point x[ from the mentioned boundary element is located in the region Reg where the top product components are more volatile and the entrainer components are less volatile than the rest of components. The trajectory tear-off point of the intermediate section is the stable node x[ = A+). The conditions of intermediate section trajectory tear-off in different points of trajectory tear-off region Reg allow to determine limit modes of extractive distillation for each mixture - the mode of minimum flow rate of the entrainer min, and for the... 

Therefore, the condition of joining of the top section separatrix min-reflux bundle Re with the trajectory of the intermediate section is -1 =... 

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