Mode of minimum reflux

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. 

In the mentioned works, it is suggested that tray by tray method should be used only for the part of the column located between zones of constant concentrations. The special equations, taking into account phase equilibrium between the meeting vapor and liquid flows, are applied to such zones. Approximations to the mode of minimum reflux are estimated by means of gradual increase of theoretical plates number in that part of the column for which tray by tray method is used. 

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. 

The appearance of the tangential pinch in the mode of minimum reflux was investigated in the works (Levy Doherty, 1986 Fidkowski, Malone, Doherty, 1991). 

The approach to calculation of the minimum reflux mode, based on eigenvalue theory, was introduced in the work (Poellmann, Glanz, Blass, 1994). In contrast to the above-mentioned works of Doherty and his collaborators this method calculates the mode of minimum reflux not only for direct and indirect, but also for intermediate split of four-component mixtures. 

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). 

The main achievement of Underwood consists in the proof of equality of parameters (j) and ir in Eqs. (5.1) and (5.2) in the mode of minimum reflux. Sum up these equations is following main equation ... 

At a greater number of components, the trajectory bundle fills up some distillation simplex Reg p. In two-section columns, each section has its distillation simplex Regf or Reg, and the availabihty of the common roots of the equations of Underwood for two sections means that these simplexes in the mode of minimum reflux adjoin to each other by their vertexes, edges, faces, or hyperfaces. 

At some value of parameter (L/ trajectory bundles of sections Reg and Regi adjoin each other by their boundary elements - separatrix min-reflux regions Reg , s (5 => A+, shortly 5 - A+) and Re = (S N, shortly - N+), mostly remote from product points xd and xb, if one uses for determination of (L/ y) " the model in Fig. 5.29b, or if validity of condition (Eq. [5.18]) is achieved between some points of these boundary elements, if one uses the model in Fig. 5.29a. At this value of the parameter (L/F)J ", the distillation process becomes feasible in infinite column at set product compositions. Such distillation mode is called the mode of minimum reflux. It follows from the analysis of bundle dimensionality 5 - 1V+ and that, at separation without distributed... 

Taking into consideration the symmetry of these sphts, we confine the discussion to the direct split. In the mode of minimum reflux, point x/ should coincide with the stable node A +, and point x/ i should belong to rectifying minimum reflux bundle 5 - Nf (Fig. 5.30). Along with that, Eq. (5.18) should be valid. The search for the value (L/ F) is carried out in the following way at different values (L/V)r, he coordinates of point X/ = Nf are determined by means of the method tray by tray for bottom section and then the coordinates of point x/ i are determined by Eq. (5.18). At(L/F), < (L/F) , points X/-1 and xd are located on different sides from the plane or hyperplane - Nf and, at (L/ V)r > (L/ y)f", these points are located on one side. That finds approximate values (L/ F) (not taking into consideration curvature of bundle 5 - Nf) and approximate coordinates of points X/ i and x/. To determine exact values (L/ F) and coordinates X/-1 and Xf, one varies the values of (LjVjr in the vicinity of found approximate value (L/V)f. Then one realizes trial calculations of top section trajectories by means of the method tray by tray from feed cross-section to the product. If at... 

The analysis of dimensionaUty of sections trajectory separatrix bundles shows that for splits with one distributed component trajectory of only one section in the mode of minimum reflux goes through corresponding stationary point or (there is one exception to this rule, it is discussed below). The dimensionality of bundle 5 - A4+ is equal to A - 2, that of bundle — iV+ is equal to n — A — 1. The total dimensionality is equal to n - 3. Therefore, points x/ i and Xf cannot belong simultaneously to minimum reflux bundles at any value of LlV)r. If only one of the composition points at the plate above or below the feed cross-section belongs to bundle 5 - A + and the second point belongs to bundle 5 - 5 - A+, then the total dimensionality of these bundles will become equal n - 2 therefore, such location becomes feasible at unique value oi(LjV)r (i.e., in the mode of minimum reflux). 

At quasisharp separation with one distributed component in the mode of minimum reflux zone of constant concentrations is available only in one of the sections (in that, trajectory of which goes through point 5 ). 

In the mode of minimum reflux, i min at sharp distillation without distributed components traj ectory of the top (bottom) section goes from the product point xd xb) to the trajectory tear-off point Sj Sj) into the boundary element, containing one additional component referring to product components, that is the closest one by phase equilibrium coefficient, then it goes from point S (Sj) to the point of trajectory tear-off S (S ) inside concentration simplex, then it goes from point to point X/ I (xf) in the feed cross-section of the column. Along with that, material balance should be valid in the feed cross-section. 

In the mode of minimum reflux adiabatic sections trajectories intersect reversible distillation trajectories in points Therefore, the separation process between product point and point can be carried out in principle, maintaining phase equilibrium between meeting flows of vapor and liquid in the cross-section at the height of the colunm by means of differential input or output of heat. We call such a separation process, with the same product compositions as at adiabatic distillation, a partially reversible one. A completely reversible process is feasible only for the preferable split that is rarely used in practice. Nonadiabatic distillation used in industry is a process intermediate between adiabatic and partially reversible distillation. Summary input and output of heat at nonadiabatic and adiabatic distillation are the same, and the energetic gain at nonadiabatic distillation is obtained at the transfer of a part of input or output heat to more moderate temperature level, which uses cheaper heat carriers and/or coolants. 

The trajectory of the intermediate section in a three-section column connects the trajectories of the top and bottom sections, and should join them in the cross-sections of the top and the bottom feeds (i.e., at trays above and below these cross-sections, the material balance should be valid). In the mode of minimum reflux, only two of three sections adjacent to one of the feeds Xpi or Xp2, called control ... 

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+). 

We now examine the conditions of joining of sections trajectories at a set flow rate of entrainer (i.e., at set value of the parameter E/D) for a three-component mixture in the mode of minimum reflux. Each of two feeds can be the control one, and the intermediate section trajectory in the mode of minimum reflux in both cases should pass through the saddle point Sm because this trajectory passes through the node point not only in the mode of minimum reflux, but also at reflux bigger than minimum (point arises at the boundary element of the concentration simplex because the extractive distillation under consideration is sharp). 

Therefore, point x/ i (composition at the tray of the intermediate section that is higher than the feed cross-section) in the mode of minimum reflux should lie on the separatrix N - Sm (Fig. 6.10). If it turns out that oint x/ i is outside the working... 

Therefore, the joining of trajectories of the bottom and the intermediate sections in the mode of minimum reflux for the case when the bottom feed is the control one is similar to that of section trajectories of two-section column at direct split (see Section 5.6). In this mode, zones of constant concentration arise in the bottom and in the intermediate sections. The column trajectory may be put in brief as follows ... 

If the top feed is the control one, two variants are feasible in the mode of minimum reflux. For the first of them, point Xe-i (composition at the tray higher than input of entrainer) should coincide with point iV+ and point Xg (composition at the tray of the intermediate section lower than input of entrainer) as at any mode should coincide with point iV+. For this variant, a zone of constant concentration arises in the top section in the cross-section of the input of the entrainer, and a pseudozone of constant concentrations, caused by the sharpness of separation but not by the value of parameter (LlV)m, arises in the intermediate section. 

We now examine the four-component mixture for = 2 (Fig. 6.9a). The conditions of joining of section trajectories at a set flow rate of the entrainer in the mode of minimum reflux in the cases of top or bottom control feed do not differ from the conditions for three-component mixtures discussed above. In the case of bottom... 

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 ). 

We now examine the general case of separation of a multicomponent mixture by means of sharp extractive distillation in a column with two feeds at a set flow rate of entrainer in the mode of minimum reflux. The conditions of sections joining are similar to the conditions of sections joining of the two-section column and depend on the number of components in the product or in the pseudoproduct of each section (trir, trim, and tits) (i.e., on the dimensionality of the working and separatrix bundles of the sections). 

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. 

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 ...

While solving this task, we act the way we did when we deternained points x/ i and X/ in the mode of minimum reflux (see Section 5.6). 

The above-described algorithm can be somewhat modified for more precise determination of the value Rmin and the coordinates of the segments [x/ ijim and [x/]iin at nonsharp separation. The modified algorithm should take into consideration that at nonsharp separation in the mode of minimum reflux each product contains only key non-product components as impurity ones (i.e., for split l,2...k k- -l,k- -2...n, the top product contains impurity component A -I-1,... 

We note, however, that the main algorithm described above can be used in the majority of cases. It will be used in the examples given below. The modified algorithm is necessary only for the modes close to the mode of minimum reflux. 

The described algorithm for nonsharp separation and for the modes that are close to the mode of minimum reflux can be made more rigorous in the same way as it was described above for the intermediate splits without distributed components. The content of impurity component k - -1 in the top product and the content of impurity component k - 1 in the bottom product should be considered while determining points 5, 5. .. 1V+ and Sf... N+ and by the modified algorithm. 

Because minimum values of these parameters were determined before at the stage of calculation of the mode of minimum reflux (see Section 6.8), design their values should be chosen reasoning from economic considerations taking into account energy and capital expenditures. This choice is similar to that of optimal reflux excess coefficient for two-section columns. Along with that, the equaUty of vapor flow rates in the second and third columns in the cross-section of output of side product is taken into consideration. 

Ratios of the flow rates of liquid and vapor in two-section colunms are determined by reasoning from the calculated mode of minimum reflux and specified set of values of reflux excess coefficients, taking into consideration the output of heat by pumparounds decreasing the vapor flow rate passing from one two-section column into another. The amount of heat withdrawn by pumparounds is determined through application of the pinch method (Liebmann et al., 1998). 

Calculation of the mode of minimum reflux in the first two-section column. 

Thermodynamic losses caused by mixing of flows of different composition in the feed cross-section of the colunm (A2). These losses always arise at separation of multicomponent mixture at any split without distributed components. The losses are absent only at the preferable split when the compositions of the liquid and vapor parts of feeding coincide (in the mode of minimum reflux) or are close (at the reflux bigger than minimum) to the composition of the liquid flow from the top section of the column and to the composition of vapor flow from the bottom section of the column, respectively. 

Thermodynamic losses caused by input into the column of unequiUbrium flows of reflux from condenser and of vapor from reboiler (A3). To exclude A3, it is necessary to replace condenser and reboiler by the input of liquid and vapor from the other columns (i.e., to turn from the flowsheet in Fig. 6.12d to the flowsheets in Fig. 6.12c,e,f). At such passage, parts of section trajectories Xd S and xb Sj, at which nonequilibrium of liquid and vapor flows being mixed at the trays is especially big, are excluded. It is very clearly seen in Fig. 8.5 (Petlyuk Platonov, 1965), which shows working and equilibrium lines for each of three components at the preferable split and mode of minimum reflux (ais = 5 0-23 = 2 Xfi = 0.1 Xfi = 0.6 xpi = 0.3 xbi = 0.0001 xz)3 = 0.0004 Lmin/F = 0,25). As is evident from the figure, the nonequilibrium at the end parts of the column xd S and xb Sj, if working with a condenser and a reboiler (the shaded regions correspond to them), exceeds many times the nonequilibrium in the middle part of the column at parts Sj x/ i and Xf. 

After identification of several preferable sequences, choosing among the optimum sequences, taking into consideration possible thermodinamic improvements and thermal integration of columns, arises. This task is similar to the synthesis of separation flowsheets of zeotropic mixtures (see Section 8.3), and it should be solved by the same methods (i.e., by means of comparative estimation of expenditures on separation). The methods of design calculation, described in Chapters 5 7 for the modes of minimum reflux and reflux bigger than minimum, have to be used for this purpose. In contrast to zeotropic mixtures, the set of alternative preferable sequences for azeotropic mixtures that sharply decreases the volume of necessary calculation is much smaller. 

To choose the entrainer among a number of alternative entrainers, it is necessary to carry out comparative estimation of expenditures on separation. For preliminary estimation at extractive distillation, the value of minimum flow rate of entrainer can be used and, at the sequence in Fig. 8.22b, the length of possible product composition segment at the side of the concentration triangle can be used. For more precise estimation, it is necessary to calculate the summary vapor flow in the columns in the mode of minimum reflux at several values of excess factor at the flow rate of entrainer. 

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