We studied the effects of design parameters in the two-column configuration. It is always possible to produce high purity products using this two-column flowsheet (shown in Fig. 6.8). Now we consider a more simple process configuration that has only a single column, which is shown in Figure 6.14. If we consume most of the heavy reactant (HK, A) toward the bottoms of the column, the heavier product (IK, C) can be obtained from the column base, and the lighter product (LK, B) can be withdrawn from the overhead. Then, the question becomes, can the same purity level be obtained for both products using a single reactive distillation column? Feasibility analysis is useful in exploring this question.

Feasibility analysis helps us to determine whether a single reactive distillation process will produce on-spec products. It is carried out at the very beginning of the design and can provide guidance for conceptual design. One useful approach is the difference point method7 in which tray-by-tray calculations are carried out to determine the feasibility.

Figure 6.14 One-column flowsheet for decomposition reaction A , B + C (HK , LK + IK), with aB/aC/aA = 4/2/1.

7J. W. Lee, S. Hauan, K. M. Lien, and A. W. Westerberg, Difference points in extractive and reactive cascades. II. Generating design alternatives by the lever rule for reactive systems, Chem. Eng. Sci. 55, 3161-3174 (2000).

Figure 6.14 One-column flowsheet for decomposition reaction A , B + C (HK , LK + IK), with aB/aC/aA = 4/2/1.

An advantage of this method is that the interaction between reaction and separation can be visualized. The entire procedure can be carried out using relatively simple calculations. Moreover, the theory is based on basic material balances.

A brief introduction is given here, and readers can refer to the original literature for greater detail. Let us use the case A , B + C (HK , LK + IK) to illustrate the analysis. In the ternary composition space shown in Figure 6.15, H denotes the heavy key component, I is the intermediate key component, and L stands for the light key. We count the number of trays from bottoms up, that is, the reboiler is tray 0, followed by tray 1, and so on (Fig. 6.15a). Consider the material balance around the envelope shown in Figure 6.15a, where there are n trays with reaction in this section and all flowrates are molar flows. There are two streams leaving the system (vapor flowrate Vn and bottoms flowrate B) and one liquid flow entering the system (Ln+i). Here, jn is the total extent of reaction on all of the trays from tray 1 to n, including the column base.

The variables yn, xB, and xn+i are corresponding composition vectors of Vn, B, and Ln+i, respectively; y is the vector chemical stoichiometric, that is, (B, C, A) = (L, I, H) = (1, 1, — 1); and yT is the sum of chemical stoichiometric, that is, yT = 1. Thus, the overall molar balance and component material balances can be expressed as

Because uT = 1, it will be dropped for subsequent development. Moving negative terms in Eqs. (6.6) and (6.7) to the right-hand side (RHS), we have

Dividing the left-hand side (LHS) of Eq. (6.9) with the LHS of Eq. (6.8) and the RHS of Eq. (6.9) with the RHS of Eq. (6.8), 8„+1 can be defined.

Rearranging the first two terms in Eq. (6.10), we have dn+1 = VT^ + Vn+B XB (6:11)

Similarly, manipulating the first and third terms in Eq. (6.10), one obtains

Given the product specification (xb), flowrates (Vn, B, Ln+i), and x«+j, now jn can be computed directly from the overall material balance [Eq. (6.6)]. Then, yn can be computed from Eq. (6.7). Using the ternary mixing rule, the variables x«+j, u, and 8n+! lie on a straight line in the ternary space. Applying the lever arm rule, 8n+! can be located according to Eq. (6.12) as shown in the dashed straight line in Figure 6.15b. Next, yn can be found graphically using the lever arm rule according to Eq. (6.11). If the column is operated under total boilup (Vn/B ! 1), 8n+! coincides with yn. Thus, the points u, yn, and x«+i form the straight line shown in Figure 6.15c. This is useful because we can perform the material balance at the limiting condition. Once yn becomes available, we can find the corresponding xn from the tie line (vapor-liquid equilibrium). Hence, one can step down the column all the way to XB, if it is feasible. Figure 6.15d shows how one performs the tray-by-tray calculation toward the column base. However, when the direction of the reaction direction is the same as the separation direction (tie line), we have a pinch point and the desired product specification (xb) will not be reached (Fig. 6.15d).

It is interesting to note that the pinch point in Figure 6.15d is exactly the "reactive azeotrope" that was discovered by Doherty and Malone and discussed in their book.8 A mathematical approach is taken and the framework for the analysis is based on the "transformed composition" that is invariant from the standpoint of reaction. Let us

8M. F. Doherty and M. F. Malone, Conceptual Design of Disii'Haiion Systems, McGraw-Hill, New York, 2001.

use the HK , LK + IK system to illustrate the approach. The transformation can be expressed as

where x7 and xH are liquid mole fractions and y and yH are vapor mole fractions of the IK and HK, respectively. Because mole fractions add up to unity (xL + x/ + xH = 1), Eq. (6.13) can be rewritten as

It is clear that X/ falls between zero and one (1 > X/ > 0). When x/ = 1, we have unity for the transformed variable X/ = 1 and, similarly, xL = 1 gives X/ = 0. Consider the ternary space in Figure 6.16a where the solid line represents the chemical equilibrium line and the dashed line is the corresponding vapor line (in vapor liquid equilibrium with liquid composition in the chemical equilibrium line). The transformation of

Figure 6.16 Transformed composition analysis: (a) projecting xl and xL onto L-/ edge as X1, (b) projecting corresponding vapor composition onto L-/ edge as Y1, (c) X1 = Y1 indicating reactive azeotrope, and (d) reactive azeotrope in T-X-Y diagram.

Figure 6.16 Transformed composition analysis: (a) projecting xl and xL onto L-/ edge as X1, (b) projecting corresponding vapor composition onto L-/ edge as Y1, (c) X1 = Y1 indicating reactive azeotrope, and (d) reactive azeotrope in T-X-Y diagram.

Eqs. (6.13) and (6.15) projects a composition in two-dimensional space (xj, xL) onto the L-j edge (X) along the direction of y with the value of Xj. Let us use plane geometry to illustrate this. Assume Xj is the projection of (xj, xL). Figure 6.16a shows that the two rectangles formed by A-y-B-(xJ, xL) and A'-y-B'-XJ are similar, and the two triangles formed by A'-Xj-L and B'-J-Xj are equilateral right triangles. The following condition is satisfied:

1 - Xf r • (1 - xf) 1 - xf where r is the ratio of the two lines formed by A-(x/, xL) and A'-Xf. Rearranging Eq. (6.16) gives exactly the same equation that we used in the transformation [Eq. (6.15)]. In other words, the transformation of Eqs. (6.13) and (6.14) is indeed the projection of (xf, xL) on the L-f edge, and Figure 6.16a shows that projection Xf is simply the intersection of the straight line formed by y and (xf, xL).

Figure 6.16b provides the projections of liquid (Xf) and corresponding equilibrium vapor (Yf) compositions for the transformed variables. The singular point with Xt = Y at chemical equilibrium is the reactive azeotrope, which is shown in Figure 6.16c. The feature of this point is that the direction of the reaction is the same as the direction of the material balance line from a tray-by-tray calculation at an infinite reflux ratio. A more familiar expression for the reactive azeotrope is the maximum (or minimum) value on a T-X-Y diagram, except that X and Y are the transformed composition, as shown in Figure 6.16d. Therefore, given a chemical reaction system and corresponding vapor-liquid equilibrium, we can compute the reactive azeotrope numerically with simple transformation.

Certainly, the reactive azeotrope is the achievable bottoms product purity. Figure 6.17 shows the computed reactive azeotrope, assuming chemical equilibrium. Assuming the temperature is 393 K, the chemical equilibrium has to be >0.48 to reach 98% specification. That implies that a one-column configuration will be feasible for KEq > 0.48.

Figure 6.14 gives a possible process flowsheet. This is a reactive distillation with a rectifying section. To keep the composition near chemical equilibrium, a reactive stripping section is always necessary. The base of the column has 10 times the amount of catalyst used on the reactive trays. Heavy key reactant is fed to the top of the reactive sections and the light product can be easily obtained from the top of the column. The purity of the intermediate key product increases stage-by-stage in the reactive stripping sections all the way to the bottoms.

By setting the product specifications at 98%, light key product in the top, and intermediate key product in the bottoms, the reflux ratio and the boilup rate are manipulated to control product purities. Distillate and bottoms flows are manipulated to control the inventory of reflux-drum and column base.

The design procedure based on the TAC is the following:

1. Place the reactive zone in the lower section of the column and fix the number of reactive trays (N^). Note that the column base is packed with 10 times the amount of catalyst used on a reactive tray.

2. Guess the number of trays in the rectifying section (NR).

3. Fix the reactant feed tray (NF).

4. Perform a dynamic simulation using relaxation via feedback control to meet the product specifications.

5. Return to step 3 and change NF until the TAC is minimized.

6. Return to step 2 and change NR until the TAC is minimized.

7. Return to step 1 and vary NRX until the TAC is minimized.

The optimal design is provided in Table 6.5. Figure 6.18 gives the composition profiles.

Increasing the reactive tray holdup decreases the vapor boilup because the products are going toward the opposite direction as the result of consuming all of reactant A, which is shown in Figure 6.19. There is no counterintuitive effect.

Increasing the number of reactive trays dramatically reduces the vapor boilup, which is illustrated in the upper right graph in Figure 6.19. This is the result of approaching the reactive azeotrope toward the bottoms of the column, which can only be achieved gradually by

TABLE 6.5 Steady-State Conditions and Design Parameters for TAC Optimum Case

Fresh feed flowrate of A F0A (mol/s) Distillate flowrate D (mol/s) Bottom flowrate B (mol/s) Vapor boilup VS (mol/s) Reflux flowrate R (mol/s) Reactive trays NRX Rectifying trays NR Feed tray location NF

(KEQ)366

Liquid holdup on reactive tray MRX (mol) Pressure P (bar)

Product Composition (Mole Fraction)

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