Minimum reflux mode

When R = i min (minimum reflux mode), the number of stages is infinite (in the feed point, the step between stages becomes equal to zero - this is an area of constant concentrations or pinch). 

Generally speaking, for the first and second fractionation classes under the minimum reflux mode, the points of compositions in the zones of constant concentrations (i.e., stationary points of the trajectory bundles) should be arranged at the trajectories of reversible distillation built for the product points. It follows from the conditions of the material balance and the phase equilibrium in the zones of constant concentrations. Figure 2.11b illustrates the partially reversible process (it is reversible in the colunm parts that are from the constant concentration zones for the minimum reflux mode up to the column ends). 

The synthesis of optimum sequences for the multicomponent azeotropic mixture is the issue of the distillation theory. Geometric theory of distillation overcomes the principal part of this problem - the determination of possible splits for each potential distillation column that may be included into the synthesized sequence. The best feasible sequences selection is carried out on the basis of the criteria of a minimum number of columns, as well as minimum liquid and vapor flows, under the minimum reflux mode. 

The analysis of the minimum reflux mode is used at the stage of sequence selection, as well as at the stage of determination of optimum reflux ratios and the quantity of column trays. The geometric theory of distillation makes it possible to develop the general methods of calculation of minimum and more reflux mode. 

How many trays should be there in the column under minimum reflux mode ... 

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). 

Unfortunately, the method of Underwood cannot be applied to nonideal mixtures and even to ideal ones, relative volatilities of the components that depend on the temperature. Therefore, tray by tray method was used for the calculation of minimum reflux mode for such ideal mixtures (Shiras, Hanson, Gibson, 1950 Erbar Maddox, 1962 McDonough Holland, 1962 Holland, 1963 Lee, 1974 Chien, 1978 Tavana Hanson, 1979) and others. 

However, numerous questions remained unsolved in these works (1) the methods of prediction of possible product compositions for a given feed composition were absent, which does not allow to calculate minimum reflux mode (2) the methods of calculation were good only for two special splits direct and indirect ones, but these methods were not good for the intermediate splits (3) the peculiarities arising in the case of availability of a-lines, surfaces, and hypersurfaces that are characteristic of nonideal and azeotropic mixtures were not taken into consideration and (4) the sudden change of concentrations in the feed cross-section was not taken into consideration. 

The approximate calculation method of minimum reflux mode (Koehler, Aguirre, Blass, 1991) - the method of the smallest angle, which holds good for mixtures with any component numbers and for any sphts, including frequently found at azeotropic mixtures separation cases of tangential pinch, is based on the calculation of reversible distillation trajectories for the given product compositions. 

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. 

The approximate method of calculation of the minimum reflux mode for three-component mixtures at the absence of tangential pinch was suggested in the work (Stichlmair, Offers, Potthofk 1993). 

The previously enumerated methods of calculation of the minimum reflux mode for nonideal zeotropic and azeotropic mixtures have considerable defects (1) they presuppose preliminary setting of possible separation product compositions, which is a comphcated independent task for azeotropic mixtures (2) they embrace only three- and four-component mixtures or only special splits and (3) they do not take into consideration the leap of concentrations in feed cross-section. 

In practice, the enumerated calculation methods are hardly used when designing distillation units because of these defects. Calculation of the minimum reflux mode is not conducted at aU, and the working reflux number and number of plates in the sections are chosen, as a rule, arbitrarily, based on the designer s intuition and experience, which can lead to considerable overstating of separation costs. 

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... 

The conducted analysis of product points evolution, depending on R for ideal mixtures, determines a number of the important qualitative regularities of the minimum reflux mode the existence of three classes of fractioning, the availability of one or two zones of constant concentrations in each section of the column, feasibility of various splits by means of a corresponding choice of two parameters of the mode -otR and D/F. 

These qualitative regularities have a general nature and apply not only to ideal mixtures, but also to nonideal ones. Only the possibility of analytic solution for the minimum reflux mode (Underwood equation system) and Unearity of separatrixes and of dividing surfaces and hypersurfaces of section regions Reg specific for ideal mixtures. The latter circumstance was also extended to nonideal mixtures in a number of approximate methods (Levy et al., 1985 Julka Doherty, 1990 Stichlmair et al., 1993). 

For nonideal zeotropic and azeotropic mixtures, the solution of the task of minimum reflux mode calculation in such a statement run across the insurmountable calculating difficulties in the majority of cases. 

The development of distillation traj ectory bundles theory at finite reflux showed that the task of minimum reflux mode calculation for nonideal zeotropic and azeotropic mixtures can be solved in another statement at set composition xf and thermal state q of feeding, it is necessary to determine minimum reflux number i min for the set product compositions xd and xb of sharp separation and set permissible concentrations of admixtures in the products. 

The algorithm of calculation of minimum reflux mode at tangential pinch has some peculiarities. At tangential pinch in top section (L/y) = =... 

Figure 5.35 is carried out according to the results of calculation of (L/y) " for equimolar mixture pentane(l)-hexane(2)-heptane(3)-octane(4) were made at separation of it with distributed component at spht 1,2 23,4 at different distribution coefficients of component 2 between products. This figure shows the location of rectifying plane S - S - and of bottom section trajectory in minimum reflux mode at several characteristic values of distribution coefficient of component 2 (1) at joining at the type of direct spht (1 2,3,4) (Fig. 5.35b X02 = 0.1, x/ = Aj+, zone of constant concentrations is located in feed cross-section in bottom...

As far as the second assumption is concerned, as was mentioned in Section 5.5, it does not influence the compositions in the stationary points. Therefore, it does not influence the first two stages of the described algorithms of calculation of minimumreflux mode. This assumption could have some influence only at the third stage of the algorithms, when curvature of separatrix trajectory bundles should be taken into consideration. Therefore, the assumption about equilibrium plates at calculation of minimum reflux mode is even more justified than at calculation of finite columns. 

Therefore, the stated algorithm of calculation of minimum reflux mode, based on the geometry of the trajectory bundles in concentration space, are potentially as one likes precise and most general, because they embrace any spUts on mixtures with any components number and any degree of nonideahty. 

Determine other possible sharp splits for this mixtures, calculate the minimum reflux mode for each. 

Therefore, the conceptual calculation of infinite column with intermediate input and/or output of heat consists in two stages (1) calculation of minimum reflux mode for adiabatic column, and (2) determination of opt 7, opt opt 7, and opt < 5, ( pinch method ). 

Distillation Trajectories and Minimum Reflux Mode in Iwo-Feed Columns with Nonsharp Separation in Intermediate Sections... 

For nonideal three-component mixtures, the methods of calculation of minimum reflux mode was developed in the works (Glanz Stichhnair, 1997 Levy Doherty, 1986). The simplifled method that was offered before for the columns with one feed (Stichlmair, Offers, Potthof, 1993) was developed in the work (Glanz Stichlmair, 1997). 

In Chapter 5, to develop a general algorithm of calculation of minimum reflux mode for columns with one feed, we had to understand the location of reversible distillation trajectories and the structure of top and bottom section trajectory bundles. 

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+). 

General Algorithm of Calculation of Minimum Reflux Mode... 

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... 

For the second variant of minimum reflux mode with top control feed — (LfV)m = Kj that is, not a pseudozone arises in the intermediate section, but a true zone of constant concentrations, caused by the value of the parameter (L/F)m- In the top section, for this variant, no zone of constant concentrations... 

The general algorithm of calculation of the minimum reflux mode for columns of extractive distillation with two feeds requires the check-up of the conditions of trajectories joining for the cases of bottom and top control feed and requires the determination of the values of (E/E) bigger of these two... 

In particular, for the most widespread spht with one-component entrainer and one-component top product nim = 2, wzr = 1), the joining of intermediate section trajectories with the trajectories of the top and the bottom sections goes on the way it is at direct split in two-section colunms. This uses the simplest modification of the algorithm of calculation of the minimum reflux mode. 

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. 

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