Section Trajectories

Figure 2.7. A location of product points and trajectories under minimum reflux for given three-component feed xp (a) first class of fractionation, (b) second class of fractionation, (c) third class of fractionation. Ri < R2 < R3 < R4 < Rs < Re < = 00 sphts xo(i) xb(i) at Ri, xo(2) xb(2) aiR2,XD(i)-XB(3) atiis = 7, x i(4) xb(4) at R4, xd(5) xb(S) at R = at R(, and R-j = 00, x and xl — tear-off points of rectifying and stripping section trajectories.

Figure 2.14a shows a flowsheet of the column of extractive distillation and, in Fig. 2.14b, an example of acetone(l)-water(entrainer)(2)-methanol(3) mixture with section trajectories is shown. This mixture, which is impossible to separate sharply into acetone (xd) and methanol-water mixture (xb) in the single-feed column, may be separated into these products in the column with an extractive section located between two feed inlets. 

Figure 4.1. Location of reversible section trajectories and Kquid-vapor tie-lines in arbitrary tray cross-section illustrating that extended tie-lines pass through product points Xb and yo- x and y, composition in arbitrary tray cross-section (httle circles).

Figure 4.2. Location of reversible section trajectories of an ideal four-component mixture at sharp split and liquid-vapor tie-line of the feed point (xp - yp).

It is clear from Fig. 4.1 that the location of a reversible distillation section trajectory is determined only by the location of its product point (i.e., it does not depend on any parameter). One component is absent at sharp reversible distillation [i.e., the product point of the section is located at some (n - 1) component edge, face, or hyperface of the concentration simplex]. That is why, in all the points of a section trajectory at sharp reversible distillation, liquid-vapor tie-lines should be directed to this edge, face, or hyperface or from it [i.e., one and the same component should... 

Figure 4.7. Sharp reversible section trajectories of ace-tone(l)-benzene(2)-chloroform(3) mixture (a) rectifying section, (b) bottom section. Double line, possible overhead product Reg gy thick solid line, possible bottom product Reg gy 5 dotted lines with arrows, reversible section trajectories dotted lines without arrows, lines of stationarity.

Figure 4.9. Reversible section trajectories of acetone (l)-benzene(2)-chlorofonn(3) mixtnre forgiven product points (a) rectifying section, and (b) bottom section Xij(i), xd(2). xoQ), xb, product points x. , tear-off points. 

If the feed point lies on the a-line, a-surface, or a-hypersurface, then the liquid-vapor tie-Une of feeding is directed to some (n — 2)-component boundary element or from some (n — 2)-component boundary element. It along with that the liquid-vapor tie-Une is directed to the possible product composition region at this boundary element or from this region, then the product of reversible distillation section can contain n — 2 components. For example, if the feed point in Fig. 4.10c lies on the 23-Une within the true bundle of bottom section trajectories then the liquid-vapor tie-Une of feeding is directed to vertex 1 [i.e., the component 1 = Reg can be a product of the section (the product contains n — 2 components)]. 

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). 

Besides the location of reversible distillation trajectories in the concentration simplex, the character of the liquid and vapor flow rates changing is of great importance. In accordance with the formulas [Eqs. (4.11) and (4.13)], the ratio of liquid and vapor flow rates in each cross-section in the top section should be equal to the phase equilibrium coefficient of the heaviest component and in the bottom section to that of the lightest component. For ideal mixtures, these phase equilibrium coefficients should change monotonously along the sections trajectories, which leads to maximum liquid and vapor flow rates in the feed cross-section (see Figs. 4.3 and 4.6). 

Figure 4.11. Sharp reversible section diagrams of some structures of three-component mixtures. 1,3.4a,..., classification according to Gurikov (1958). Double line, possible composition of overhead product Reg thick solid line, possible composition of bottom product Reg dotted lines with arrows, reversible section trajectories 123,132,312. regions of component order Reg dotty lines, separatrixes thin hues, a-lines.

Therefore, an infinitesimal amount of heat should be drawn off in each cross-section of the top section and should be brought in in each cross-section of the bottom section. For azeotropic mixtures, the phase equilibrium coefficients field is of complicated character, which leads to nonmonotony of the Liquid and vapor flow rates changing along the sections trajectories (i.e., to the necessity of input or output of heat in various cross-sections of the section). Such character of the flow rates changing at reversible distiUation influences on the conditions of minimum reflux mode in adiabatic columns, which results in a number of cases in the phenomenon of tangential pinch (see Chapter 5). 

Figure 4.12. Iteration of component concentration at calculation reversible section trajectories, xf and xf...

Figure 4.19. Reversible intermediate section trajectories of sharp auto-extractive distillation ofideal ternary mixture ( fi > K2 > K3) (a) D < E, and (b) D > E. Component 1, overhead prodnct component 3, entrainer x and y, composition in arbitrary cross-section x j), composition of psendoprodnct.

Figure 4.20. Reversible section trajectories for acetone(l)-water(2)-methanol(3) extractive distillation. Short segments with arrows, liquid-vapor tie-lines in arbitrary cross-sections of stripping and intermediate sections little circles, composition in main and entrainer feed cross-section.

Let K2 > Ki > in the feed point. In which regions of components order Reg can be located (1) top reversible section trajectory, (2) bottom reversible section trajectory, (3) intermediate reversible section trajectory ... 

Numerous works (Levy, Van Dongen, Doherty, 1985 Levy Doherty, 1986 Julka Doherty, 1990) in which distillation trajectory bundles of three-and four-component mixtures for two sections of distillation column were used at hxed product compositions and at different values of reflux (vapor) number, are of great importance. They defined the conditions of two section trajectories joining in the feed cross-sections of the column in the mode of minimum reflux, and they developed the methods of this mode calculation for some splits. 

Calculation investigations (Petlyuk, 1978 Petlyuk Vinogradova, 1980 Shafir et al., 1984) determined the conditions under which saddle and saddle-node stationary points of sections trajectory bundles at finite reflux arise inside the concentration simplex, but not only at its boundary elements, promoted the development of this trajectory bundles theory. 

As far as stationary points of section trajectory bundles should be located at the trajectories of reversible distillation, the systematic analysis of these trajectories locations was of great importance (see Chapter 4). 

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). 

A number of regularities of the minimum reflux mode are common for the ideal, nonideal, and even azeotropic mixtures. Among these regularities is the following each section trajectory at minimum reflux and at sharp separation is partially... 

In a more general case, when there are several distributed components, it is necessary to obtain from Eq. (5.3) the common roots for two sections. After the substitution of each of these roots into Eq. (5.1) or (5.2), we obtain the system of hnear equations relatively to di and or bi and the solution of which determines separation product compositions and internal vapor and liquid flows in the column sections. In addition, one can find the compositions of equilibrium phases in the cross-sections of constant concentration zones (i.e., stationary points of sections trajectories bundles). 

Figine 5.6. Preferrable split of a ideal mixture at minimum reflux (a) the section trajectories, and (b) the pinches in column (shaded). Boundaries for material balances discussed in the text are indicated by dotted-dashed Hues. 

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.

Structure and Evolution of Section Trajectory Bundles for Three-Component Mixtures... 

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. 

Let s note that the top product point cannot be located at sides 2-3 and 1-3 because Eq. (5.15) is not vahd for these sides (i.e., Xo i [2-3] and xd [1-3]). Let s also note that ihc separatrix sharp split region of section trajectories bundle... 

Evolution of Section Trajectory Bundies for Three-Component Mixture 139... 

General regularities of the evolutions of sections trajectory bundles, discussed in the previous section for three-component mixtures, are valid also for the mixtures with bugger number of components. Figure 5.23 shows evolution of top section trajectory bundle at separation of four-component ideal mixture, when the product is pure component (i.e., at direct split) Ki > K2 > >... 

Stationary points 5, 5, and iV+ (5 = A ) move along the edges of concentration tetrahedron. The section trajectory bundle may be presented in the following brief form (the bundle s direction is indicated by the double arrow, its stationary points around it) ... 

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