Feed cross-section

A mass balance over the feed cross section (Fig. 2-46) gives... 

This intersects with the 45 diagonal at point D (y-x = xp), with the abscissa at point E ( = 0, X = Xp/f) and with the enrichment and the stripping line at point S (see Fig. 2-44). (The coordinates of S, Xg and yg describe the composition of reflux and vapor above the feed cross section.)... 

The caloric factor / is obtained from a heat balance over the feed cross section. Since for the preheating of a subcooled feed to boiling conditions the condensation of vapor AG, is necessary. 

In Fig. 2.6a, Ri =0,Ri> R2 > Ri. With the increase of R, while maintaining D/F ratio, points and x,b become remote from point xf, maintaining the constant concentration area of both sections in the feed cross-section. Such a mode is called the first class of fractionation. Its specific feature is that the feed composition and the compositions in the areas of constant concentrations of both sections, adjoining the feed tray, coincide. 

With further increase of R, we immediately pass to the third class of fractionation. For binary mixtures, the second class of fractionation is unavailable. The third class of fractionation is characterized by the fact that, in the case of R increase, the compositions of the separation products are not changed and the areas of constant concentrations in feed cross-section disappear (Fig. 2.6b). In the case of R changing, the compositions on the trays will change as well (in Fig. 2.6b, R(i = 00, R(i > R5 > R4 > R3). 

Let s consider the change of compositions of three-component ideal mixture products in the concentration triangle (Fig. 2.7) under the same conditions as before for the binary one. With the increase of R in the first fractionation class, points xd and xb are moving in opposite directions and transferred along the straight line passing through the vapor-liquid feed tie-line xp yp (Rg- 2.7a). The zones of constant concentrations of the column are in the feed cross-section (Fig. 2.8a). 

Under the conditions of the second fractionation class, the compositions of products change, but the composition on the feed cross-section differs from the composition of the feed. 

The second stage, including the selection of the best reflux numbers and the quantity of column section trays, will be the important one. The geometric distillation theory makes it possible to determine the feasible compositions that are to be in trays above and below the feed cross-section, then make the design calculations of the trajectory of sections and determine the best ratio of section tray numbers. The new algorithms allow for an increase in the design quality and apart from that, they make it possible to lower the separation costs and to practically exclude the human participation in the process of calculation. 

The main peculiarity of thermodynamically reversible distillation process consists of the fact that flows of two different phases (vapor and liquid) found in any cross-section are in equilibrium, and flows found in the feed cross-section are of the same composition as feed flows. 

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 4.20. 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.

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. 

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. 

However, this method is not effective for the mixtures with component numbers greater than four and, besides that, does not take into consideration the leap of concentration in feed cross-section. 

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. 

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). 

AtD < Dpr and R = in the top section, there are two zones of constant concentrations in feed point xp and in trajectory tear-off point from the boundary element of concentration simplex and in the bottom section there is one zone in feed point xp. At D > Dpr and R = on the contrary, in the bottom section there are two zones of constant concentration and in the top the section there is one zone. In both cases there is a transition from the first class of fractioning to the second one (i.e., in one of the sections, zone of constant concentrations in feed cross-section disappears, and in the other section, the zone is preserved, but the composition in it starts to change with the change of R). 

So the distillation process in two-section column may be feasible, it is necessary that sections trajectories are joined with each other (i.e., that there is material balance between sections flows at the plates above and below feed cross-section). 

The mixture of two flows of liquid goes into the plate, located below feed cross-section the hquid part of feeding and of liquid, following down from top section bottom plate (Fig. 5.29a). Therefore, between liquid leaving top section and liquid going into bottom section, there is a leap of concentrations in accordance with the equation of material balance in feed cross-section ... 

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). 

The following cases of location of composition points at plates above and below feed cross-section x/ i and xf. (1) point Xf-i lies in rectifying minimum reflux bundle 5 - A+, and point Xf lies inside the working trajectory bundle of the bottom section (at nonsharp separation) or m separatrix bundle... 

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...

The process of separation becomes possible only if trajectory bundles of the sections become sufficiently big for the fact that the conditions of sections joining could be valid (i.e., that the material balance would be valid in feed cross-section of the column between the certain trajectory points of the sections). 

Usage of nonadiabatic columns (i.e., column with intermediate at height inputs or outputs of heat), broadens conditions of separability of mixtures, having two reversible distillation traj ectory tear-off points. If, for example, heat is brought and drawn off in feed cross-section (Fig. 5.36), as it was offered in the work (Poellman Blass, 1994), then it is possible in one of the sections of the column, where there is limitation at the value of the parameter L/V or V/L, to keep the corresponding allowed value of this parameter, and joining of trajectories of the sections can be maintained at the expense of increase of vapor and liquid flows in the second section (Petlyuk Danilov, 1998). In a more general case, when there are limitations of the values of the parameters L/V and V/L in both sections, it is possible to use intermediate inputs and outputs of heat in the middle cross-sections of both sections. 

Figure 5.36. The column with the heat output (Qinter) in the feed cross-section.

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. 

Minimum values of the parameters L/V)r and (y/L)s in the feed cross-section of the column and compositions at trays above and below this cross-section x/ i and Xf at adiabatic and nonadiabatic distillation remain the same. The stationary points Sr and Ss also coincide, but at parts of reversible distillation trajectories between column ends and stationary points Sr and Ss the additional stationary points and Nf"-, corresponding to the points of intermediate inputs and output of heat (Fig. 6.2a), appear. 

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... 

We now discuss the general algorithm of calculation of minimum reflux mode for the column with several side withdrawals located above and below feed cross-section at sharp separation in each section and at the best separation between products. 

Minimum reflux mode is determined by the conditions of joining of trajectories of two sections adjacent to the feed cross-section. Therefore, the interconnected parameters (L/V) " and (V/L) " are determined initially for these two sections. Compositions in points x and x g are calculated preliminarily for these sections at set requirements to compositions of all the products at the conditions of sharp or quasisharp separation in each section. Minimum reflux mode is calculated in the same way as for the simple column that separates initial raw materials into products of compositions x jy and x g. Liquid and vapor flow rates for the other sections are calculated at the obtained values of (L/V) " and (L/L)f" with the help of material balance equations (strictly speaking, with the help of equations of material and thermal balance). 

Figure 6.15 shows the simple example of separation of a four-component mixture into four pure components in a column with side strippings. As for the columns with side withdrawals of the products, the calculation of minimum reflux mode should be started with determination of the conditions of joining of trajectories of two sections adjacent to feed cross-section. For section ri, the pseudoproduct equals the sum of top and two side products. The minimum reflux mode for the first two-section column is calculated the same way it is done for the corresponding simple column with split 1,2,3 4 (indirect split). In a more general case, when the bottom product contains more than one product component, the intermediate split will be in this column. 

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