Pseudoproduct points

For the intermediate section, point x plays the same role as the product point XDi for the top section (i.e., at reversible distillation in each cross-section of the intermediate section, the continuation of the liquid-vapor tie-line goes through point x j j). Let s call point the pseudoproduct point. It is seen from Eq. (4.22) that, in contrast to the product point, the pseudoproduct point can lie without the concentration triangle (i.e., the values x can be negative or greater than 1). 

Let s accept that the top product point coincides with vertex 1 (xo = 1). Then it follows from Eq. (4.22) that the pseudoproduct point should he on the straight line passing though side 1 (top product)-3 (entrainer). 

Nrev of the trajectory bundle at different pseudoproduct points. The location of point and of the whole trajectory of extractive reversible distillation depends on that of the pseudoproduct point x (i.e., on the ratio E/F between the flow rates of the entrainer and the main feeding). Changing the parameter E/F, we get the trajectory bundle of extractive reversible distillation that, for an ideal mixture, fills up the whole concentration triangle. 

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. 

Pseudoproduct points of intermediate sections of these complexes are always located inside or at the boundary of the concentration simplex. Therefore, the... 

Locations of reversible distillation trajectories depends on position of pseudoproduct point (i.e., on compositions and on flow rates of feeds and of separation products, as is seen from Eq. [6.3]). Difference from the top and bottom sections appears, when the pseudoproduct point of the intermediate section is located outside the concentration simplex (i.e., if concentrations of some components x j)i obtained from Eq. [6.3], are smaller than zero or bigger than one), which in particular takes place, if concentration of admixture components in separation products are small components (i.e., at sharp separation in the whole column). The location of reversible distillation trajectories of the intermediate sections at x j i < 0 or x, > 1 differs in principle from location of ones for top and bottom sections, as is seen from Fig. 6.3 for ideal three-component mixture (Ki > K2 > K3) and from Fig. 6.4 for ideal four-component mixture (Ki > K2 > K3 > K4). 

As far as pseudoproduct point x d and liquid-vapor tie-line in all points of reversible distillation trajectory should lie at one straight line, pseudoproduct point x at Fig. 6.3, can lie behind side 2-3 or side 1-2 and at Fig. 6.4, they can lie behind face 1-2-3 or face 2-3-4. 

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... 

K2) for two-feed column, pseudoproduct point, Sm and A, stationary points of intermediate section regions Reg j. 

This allows us to actively influence the location of the pseudoproduct point of the intermediate section in order to maintain sharp separation (i.e., separation at which the intermediate section trajectory ends at some boundary element of the concentration simplex). This is feasible in the case when inside concentration simplex there is one trajectory of reversible distillation for pseudoproduct point x ) that ends at mentioned boundary element, and there is the second trajectory inside this boundary element. To maintain these conditions, pseudoproduct point x j) of the intermediate section should be located at the continuation of the mentioned boundary element, because only in this case can liquid-vapor tie-hues in points of reversible distillation trajectory located in this boundary element he at the lines passing through the pseudoproduct point x jy. We discuss these conditions in Chapter 4. It was shown that in reversible distillation trajectory tear-off point x[ev e from the boundary element the component absent in it should be intermediate at the value of the phase equUibrium coefficient between the components of the top product and of the entrainer rev,D > Kevj > Kev.s)- This condition is the structural condition of reversible distillation trajectory tear-off for the intermediate section. Mode condition of tear-off as for other kinds of sections consists of the fact that in tear-off point the value of the parameter (LfV) should be equal to the value of phase equilibrium coefficient of the component absent at the boundary element in tear-off point of reversible distillation trajectory ((L/V)m =... 

Intermediate section trajectory tear-off point x should lie on the reversible distillation trajectory in the boundary element Reg g (x e Reg g ) farther from pseudoproduct point x/, than all tear-off points x. g from it of reversible distillation trajectories into adjacent boundary elements. 

Figure 6.7 shows for comparison the trajectories of quasisharp (a) and sharp (b) reversible distillation in the intermediate section for ideal mixture (Ki > K2> K3). This figure shows that, at movement of pseudoproduct point from the vicinity of continuation of side 1-3 to this continuation itself, there is transformation of reversible distillation trajectory it disintegrates into two parts - into one that lies inside the triangle and into the part that lies at side 1-3. We note that similar transformation also takes place at passage from quasisharp distillation to sharp one for the top and bottom sections. The stationary point Sm at this transformation stays at the internal part of reversible distillation trajectory. Point N+ passes to side 1-3. Point N passes from vertex 3 to its vicinity at side 1-3. 

In accordance with the structural conditions of trajectory tear-off, three following variants of extractive distillation are feasible for such a mixture (1) the top product is component 1, the entrainer is component 4, and the pseudoproduct point x j) lies in the continuation of edge 1-4 (Fig. 6.9a) (2) the top product is mixture 1,2, the entrainer is component 4, and the pseudoproduct point hes in continuation of face 1-2-4 (Fig. 6.9b) and (3) the top product is component 1, the entrainer is mixture 3,4, and the pseudoproduct point x lies in the continuation of face 1-3-4 (Fig. 6.9c). 2,3... 

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). 

Figure 6.11. About calculation minimum entrainer flow rate E/D)ram- Ki, j and E/D as functions x j (a,b,c, respectively) for extractive distillation of the acetone(l)-water(2)-methanol(3) azeotropic mixture. x[ j = x g concentration of component 1 in tear-off point of intermediate section reversible distillation trajectory on side 1-2 Ki, phase equilibrium coefficient of component i in point j, x), j and E/D, concentration of component 1 in pseudoproduct point and ratio of entrainer and overhead flow rates, respectively, if tear-off point of intermediate section trajectory xj j on side 1-2 coincide with point...

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 ...

Figure 6.15. Column with two strippings as three two-section columns (a), section trajectories of the first two-section column (b), and section trajectories of the second two-section column (c). Dotted line, an imaginary part of section trajectories between pseudoproduct points and tear-off points. Attraction regions...

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... 

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... 

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