## Distillation Trajectories and Minimum Reflux Mode in Two Feed Columns with Nonsharp Separation in Intermediate Sections

Columns with several inputs of feed are used in a number of different cases: (1) when flows with the same set of components but different compositions come to the unit; (2) when the raw materials are gradually warmed and are put into the column in several flows different in temperature, or when they are gradually evaporated or condensed, and after separation into liquid and vapor phases they are put into the column in several flows, different in temperature, composition, and phase state (units of petroleum refining, units of productions of ethylene and propylene); (3) when an absorbent is used for separation (units having absorbers or fractioning absorbers); and (4) when an entrainer that is put into the column of extractive distillation in a separate flow is used for separation.

Gradual heating and evaporation is used in the case of separation of mixtures with a wide interval of boiling, when heat is put in at a lower, and cold is put in at a higher temperature, compared with their input in the reboiler and condenser. This allows for a decrease of total energy consumption in separation.

Absorption is used in the case of extraction of liquid components from the gas phase, when the usage of distillation is unprofitable because of the necessity for too low temperatures in condensers.

Extractive distillation is used to increase the relative volatility of components being separated of nonideal mixtures and to separate azeotropic mixtures that cannot be separated by means of simple distillation.

Columns with several inputs of feed have one or several intermediate sections, located between these inputs of feed. To calculate minimum reflux mode of such columns at separation of mixtures with constant relative volatilities and molar flows, the Underwood method (Barnes, Hanson, & King, 1972; Nikolaides & Malone, 1987) was used.

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

It follows from general thermodynamic considerations that at one and the same product compositions the column with several feed flows of different composition should require less energy for separation than the column with one feed flow formed by mixing all the feed flows. It follows from the fact that summary entropy of all feed flows should be smaller than that of the mixed flow because the mixing of flows of different composition increases the entropy and the separation of flows decreases it. Therefore, the minimum reflux number for the column with several feed inputs should be smaller than that for the column with one mixed feed flow (i.e., it is unprofitable to mix flows before their separation).

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.

As in that case, to develop a general algorithm of calculation of minimum reflux mode for columns with several feed inputs, we need to understand the location of reversible distillation trajectories of intermediate sections and the structure of trajectory bundles for these sections.

6.3.1. Location of Reversible Distillation Trajectories of Intermediate Sections

Locations of reversible distillation trajectories depends on position of pseudo-product 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'Di 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'Di < 0 or x'Di > 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 (K > 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'D 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.

Figure 6.3. Reversible distillation trajectories of ideal ternary mixtures (K1 > K2 > K3) for intermediate section of two-feed column: (a) x'D 1 < 0; (b) x'D 3 < 0. Solid lines with arrows, tie-lines liquid-vapor; x'D 1 and x'D 3, concentrations of components 1 and 3, respectively, in pseudoproduct point.

Figure 6.3. Reversible distillation trajectories of ideal ternary mixtures (K1 > K2 > K3) for intermediate section of two-feed column: (a) x'D 1 < 0; (b) x'D 3 < 0. Solid lines with arrows, tie-lines liquid-vapor; x'D 1 and x'D 3, concentrations of components 1 and 3, respectively, in pseudoproduct point.

Reversible distillation trajectories at Fig. 6.3 should connect vertexes 1 and 2 or 2 and 3, and at Fig. 6.4, they should connect vertexes 2 and 4 or 1 and 3.

In this section, we examine only the nonsharp distillation in an intermediate section, when all flows of the feed contain all the components (in the sections to follow, we examine sharp extractive distillation in intermediate section, when entrainer and main feed have different sets of components).

At nonsharp distillation in the intermediate section, as in top and bottom sections, there is only one reversible distillation trajectory, but in the intermediate section it has two node points Nrev in vertexes of concentration simplex, and in the top and bottom sections it has one node point in one vertex.

Figure 6.4. Reversible distillation trajectories of ideal four-component mixtures (K1 > K2 > K3 > K4) for intermediate section of two-feed column: (a) x'D 1 < 0; (b) x'D 4 < 0. Solid lines with arrows, tie lines liquid-vapor.

Figure 6.4. Reversible distillation trajectories of ideal four-component mixtures (K1 > K2 > K3 > K4) for intermediate section of two-feed column: (a) x'D 1 < 0; (b) x'D 4 < 0. Solid lines with arrows, tie lines liquid-vapor.

6.3.2. The Structure of Trajectory Bundles of Intermediate Sections

We examine the structure of trajectory bundles of intermediate sections (i.e., location and character of the stationary points of these bundles).

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 x'D can be located only the way it is shown in Figs. 6.3 and 6.4, (it is result of direction of

tie-lines liquid-vapor) and the points of top xD or bottom product xB, along with that, in accordance with Eq. (6.3) can be located only in the vicinity of sides 1-2 or 2-3 in Fig. 6.3 or facets 1-2-3 or 2-3-4 in Fig. 6.4. Hence, it follows that feasible splits for columns with one or two feeds are the same (i.e., if the flows of several feeds are mixed before separation, we can only get the same products as in a column with several feeds, but the energy consumption for separation will be bigger).

Figure 6.5 shows stationary points Sm and N+ of trajectory bundles of intermediate section Regiint and separatrixes of saddle stationary point Sm obtained by means of calculation for ideal mixture pentane(1)-hexane(2)-heptane(3) at the composition of pseudoproduct x'D 1 = -1.0; x'D 2 = 1.5; x'D 3 = 0.5 and at the value of L/V = 1.2.

In contrast to nonsharp separation in the top and bottom sections, the intermediate section has at reversible distillation trajectory not just one node stationary point, but there are saddle point Sm and node point Nm. Separatrixes of the saddle points Sm divide concentration triangle into four regions Reg«t filled trajectory bundles of intermediate section, one of which is the working one Reg* int.

6.3.3. Control Feed at Minimum Reflux Mode

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 xF1 or xF2, called "control"

one, should be infinite. One of the sections (the top or bottom one) remains finite in this mode. At the increase of reflux number (i.e., of the parameter [L/V]r) the stationary points of three-section trajectory bundles move along reversible distillation trajectories for set points of products xD and xB and of pseudoproduct x'D. The general regularities of this movement for top and bottom sections were examined in detail in Chapter 5. The stationary points of intermediate section move along the unique trajectory of reversible distillation. At some value of (L/V)r = (L/V)\, there is joining of intermediate section trajectory with the trajectory of the first of the rest of the sections. At some greater value of (L/V)r = (L/V)2, there is joining of the intermediate section trajectory with the remaining section at infinite number of trays in these sections (Nm = to and Nr = to or Ns = to). The found values of (L/V)2 are minimum for the separation in a three-section column. The feed located between the intermediate and the last section, with which there was a joining, is the control one. Along with that, the first section, with which there was a joining, is finite and its joining at (L/V)2 with the intermediate section goes on as at a reflux bigger than minimum (regularities of joining at a reflux bigger than minimum are examined in Chapter 7).

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 N+. Therefore, the following cases are feasible in minimum reflux mode: (1) point N+ 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 - N+). 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 Regmp1'* (S2 - N+).

6.3.4. 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 equilibrium 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

Figure 6.6. Section trajectories of ideal ternary mixtures (Ki > K2 > K3) for two-feed column with direct split 1: 2,3.

Figure 6.6. Section trajectories of ideal ternary mixtures (Ki > K2 > K3) for two-feed column with direct split 1: 2,3.

coordinates of the stationary points that are located inside and at the boundary elements of the concentration simplex. Such necessity exists for the intermediate section because, at nonsharp separation, part of the stationary points located outside the concentration simplex and their coordinates cannot be determined.

The rest of the algorithm is similar to that for columns with one feed. Conditions of joining of trajectories in cross-sections of both feeds are checked at various values of the parameter (L/V)r and the value (L/V)2, at which there is a joining in the cross-section of control feed, is found. Both feasible cases of joining described above (see Subsection 6.3.3) are checked. The first case corresponds to direct or indirect split in the column with one feed, and the second case corresponds to intermediate split.

It was shown in the work (Glanz & Stichlmair, 1997) that in some cases expenses on separation are smaller if feed with a higher bubble temperature is brought into the higher feed cross-section of the column. Figure 6.6 features such a case. In this figure part of the trajectory of the intermediate section is directed into the side, opposite the part of the top section trajectory (control feed is bottom, the trajectory passes through point Sm). At this part of the trajectory of the intermediate section, there is an increase of temperature at the trays of the column in the upward direction, which is indicative of the process inverse to distillation (see Chapter 2). The trajectory in Fig. 6.6 may be briefly described as follows:

RegB Regf Reg|

Reg1

min,R

Regmpnnt n Regs Reg;

Therefore, the conceptual design calculation of columns with several feeds includes the determination of the best succession of bringing in these feeds along

the height of the column, which requires the calculation of the minimum reflux mode at different successions. For the column with two feeds, one has to begin the calculations with the regular succession (that means bringing in the feed with lower bubble temperature into the higher cross-section of the column). If it turns out that the energy consumption at separation is smaller than in the column with one mixed feed, then one can leave the inverse succession unexamined. Otherwise, one has to carry out the calculations for the inverse succession.

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