Feasibility of a Reactive Distillation Column with Ternary Mixtures

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Ind. Eng. Chem. Res. 2001, 40, 2714-2728

Feasibility of a Reactive Distillation Column with Ternary Mixtures Jae W. Lee, Steinar Hauan, and Arthur W. Westerberg* Department of Chemical Engineering and Institute for Complex Engineered Systems, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

We develop feasibility criteria for reactive distillation systems in terms of phase and reaction equilibria. A reactive distillation column with the liquid-phase reaction R1 + R2 T P1 can be feasible if the feed composition, the pseudo-feed composition, and the reaction difference point lie on a single straight line. In addition to this material balance constraint, the liquid composition vectors starting from the given top product composition should lie within or move toward the forward reaction region as well as lie within or move toward the reachable region from the bottom product of the nonreactive stripping section. We also perform feasibility studies on reactive distillation with the reaction 2R1 T P1 + P2 in terms of the normalized product and reactant coefficient vectors. For these ternary reaction systems, the larger the reflux, the greater the reaction conversion. We finally propose several feasible flowsheets for the methyl tert-butyl ether production system. Introduction Reactive distillation has emerged as an economical unit operation since the industrial methyl tert-butyl ether (MTBE) production system1 was developed in the early 1980s. The main reason for employing this process is that reactive distillation overcomes the reaction equilibrium limitation found in membrane reactors.2,3 Reactive distillation can naturally use the heat of reaction as a heating or cooling source. Furthermore, reducing the initial investment and operating costs for an azeotropic system4 has a dramatic economic effect. One can expect such an economic effect for the petroleum hydrotreatment system5 with the elimination of a recycle compressor with huge pipe lines. This promising reactive distillation technology has resulted in many research activities in academia since the 1980s. Barbosa and Doherty6,7 developed the reactive residue curve map. Ung and Doherty8 performed feasibility studies of reactive distillation columns with multiple reactions in this space. To propose feasible flowsheets, Bessling et al.9 and Stichlmair and Fair10 used the reactive distillation line in reactive residue diagrams in a manner that is similar to the use of a distillation curve for nonreactive distillation under total reflux. Okasinski and Doherty11 proposed reasonable operating ranges in reboil ratios for reacting systems with low equilibrium constants. They also developed a design procedure considering different liquid holdups with reaction kinetics and the effect of heat from the reaction.12 Balashov and Serafimov13 proposed the static analysis of a reactive distillation column. Then, Pisarenko et al.14 approximated the composition trajectory of the reactive rectifying section as a nonreactive distillation curve by assuming that the internal vapor and liquid flow rates were much greater than the external feed flow rates and reaction molar turnovers. Giessler et al.15 extended this idea to propose feasible sequences of reactive distillation systems. Nisoli et al.16 employed the attainable region concept of a reactor to propose various reactive systems, * Author to whom correspondence should be addressed. Tel: 412-268-2344. Fax: 412-268-7139. E-mail: a.westerberg@ cmu.edu.

including reactive distillation columns. Craig et al.17 performed a relative cost estimation of reactive systems by using attainable region results. The visualization methods for tray-by-tray calculations such as Ponchon-Savarit and McCabe-Thiele diagrams18-20 have been regarded as a quick paper-andpencil-based approach to gain insights into the behavior of binary distillation columns. Tray-by-tray calculations with three components were visualized on a ternary composition diagram for nonreactive distillation by first assuming an internal reflux ratio.21-23 One steps down a column by starting with a known or assumed vapor composition leaving a tray. One finds the liquid composition in equilibrium with that vapor, i.e., a step to the next tray along a distillation curve. Then, material balance requires the vapor composition opposing this liquid and coming from the tray below to lie along a straight line between this liquid composition and the difference point that is either the distillate composition or, if there is an upper feed above this tray, the distillation composition combined with the upper feed composition. In this paper, we present a new and easy method for evaluating the feasibility of a reactive distillation column by appropriate geometric use of composition vectors. This evaluation method provides necessary conditions for the feasibility of a reactive distillation column for given reaction extents and reflux ratios. To check the feasibility of a reactive distillation column, the minimum required information is reaction and phase equilibria and a residue curve map. We also show how to perform tray-by-tray calculations graphically for a certain reflux ratio. Finally, we illustrate these concepts by analyzing the methyl tert-butyl ether (MTBE) production system. Feasibility of a Reactive Distillation Column with a Finite Reaction Difference Point A reaction difference point is a fixed point in composition space.24 A physical interpretation is that it is a composition point for the “generated flow rate” arising by considering the reaction as a flow in which reactant leaves and product enters the column. A finite reaction

10.1021/ie0005194 CCC: $20.00 © 2001 American Chemical Society Published on Web 04/28/2001

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2715

Figure 1. Material balance constraint in a reactive distillation column with a single-feed stream for the reaction R1 + R2 T P1: (a) reactive lever rule for top and bottom products by introducing a pseudo-feed, (b) total cascade difference point in the reactive rectifying section, (c) total cascade difference point in the reactive stripping section. The values in parentheses represent the relative volatilities of R1, R2, and P1.

difference point results from a reaction in which the overall number of moles changes, whereas an infinite difference point arises when the number of moles does not change. In this section, we develop the feasibility criterion for a reactive distillation column having a finite reaction difference point. We are given the feed composition and top and bottom product compositions. The reaction is the simple liquid-phase reaction

R1 + R2 S P1

(1)

with the stoichiometric coefficient vector, ν ) [-1, -1, 1]T. This reaction gives rise to the reaction difference point, δR ) ν/νT ) [-1, -1, 1]T/-1 ) [1, 1, -1]T. Reactants R1 and R2 enter the column either as a single feed stream or as two separate feed streams in which the heavy reactant, R2, enters above the reaction zone and the light reactant, R1, enters below. Excess R1 exits at the top, and product P1 exits at the bottom. The relative volatilities of R1, R2, and P1 are assumed to be 5, 3, and 1, respectively, for reaction 1 throughout this paper. This feasibility study is based on previous studies of nonreactive distillation22,23 and extended into reactive distillation through the use of the reactive lever rule25 and the movement of composition vectors for a reactive section.26

Feasibility Criteria for a Reactive Distillation Column with a Single Feed Stream. Rearranging the total and component-wise balances of a single-feed reactive distillation column in eqs 2 and 3, we can obtain the reactive lever rule in eq 4.25 This reactive lever rule states that the feed composition must lie between the pseudo-feed composition (xFˆ ) and the reaction difference point (δR) in Figure 1a. Note that the sum of stoichiometric coefficients, νT, is -1 for reaction 1, and ξ is the total molar turnover flow rate.

F ) B + D - νTξ

(2)

FxF ) BxB + DxD - ξν ) BxB + (D - νTξ)δrR (3) (B + D)(xF - xFˆ ) ) -νTξ(δR-xF) DxD + BxB D+B

(5)

DxD - νTξδR D - νTξ

(6)

xFˆ ) δrR )

(4)

In Figure 1a, we note that the pseudo-feed composition (F ˆ ) lies on the straight line joining the top and

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Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001

Figure 2. Infeasible reactive distillation column with a rectifying reaction zone: (a) infeasible top and bottom products, (b) reverse reaction with D greater than νTξn, (c) reverse reaction with D less than νTξn, (d) diagram for reactive distillation column with a rectifying reactive zone. Legend for remaining figures‚ ‚ ‚, reaction equilibrium line (for the reaction R1 + R2 T P1); shaded area, reachable region from bottom product composition; values in parentheses, relative volatilities of R1, R2, and P1; TC, total reflux curve; PC, pinch point curve from the bottom product composition; O, liquid composition; b, vapor, product, and feed compositions; 0, reactive cascade difference point

bottom product compositions, as the pseudo-feed is an imaginary mixture of the top and bottom products in eq 5. Consider the case where the reaction occurs only in the rectifying section of the column. For given feed and bottom product compositions, the total cascade difference point (δrR), which we show geometrically in Figure 1b, corresponds to the overall reaction turnover for a reactive rectifying section. If we connect the given feed and bottom compositions by a straight line, we find that this total cascade difference point lies on the straight line connecting the distillate composition and the reaction difference point, as shown in Figure 1b. Similarly, in Figure 1c, we show the case where the reactive zone is only in the stripping section. Here, we find that the total cascade difference point (δsR), which is the combined point of the bottom product composition (B) and the reaction difference point (δR), corresponds to the overall reaction molar turnover (ξ). The feed

composition lies between the total cascade difference point and the distillate composition. For given top and bottom product compositions, we can easily check the feasibility of a reactive distillation column with a reactive rectifying section. The basic idea is that, even though there can be forward and reverse reactions on each reactive tray, the overall reaction conversion must be feasible to obtain desired top and bottom products. To have a positive overall reaction conversion, the liquid compositions on some reactive stages (at least more than one stage) or preferably all reactive stages should be within a forward reaction zone in composition space. In terms of separation, the final composition of the liquid stream coming down to a nonreactive stripping section must be within the reachable region of the bottom composition. This reachable region is the area between the pinch curve from the bottom product composition and the total reflux curve (distillation curve) going through the bottom product


composition.22 The symmetric argument is valid for a reactive stripping section, i.e., the composition of the vapor stream going up into the nonreactive rectifying section must be within the reachable region defined by the distillate composition. Consider first a reactive rectifying section. In Figure 2a, we have plotted the reaction equilibrium curve as a dotted curve for reaction 1. It represents the compositions for all of the liquids that are simultaneously at their bubble points and at reaction equilibrium. Compositions to the right of this line correspond to compositions where the forward reaction must occur, whereas points to the left correspond to compositions where the reverse reaction occurs. For the given feed (F) and product (D and B) compositions in Figure 2a, the total cascade difference point (δrR), i.e., the difference point that corresponds to the net total reaction occurring in the column, lies on the right extension of the line connecting the feed and bottom product compositions. If the forward reaction is assumed to occur on each reactive stage, then the reactive cascade difference point for each reactive stage in the rectifying section will move from the distillate composition ultimately to this total cascade difference point as we step down the column. We can derive the equations defining the movement for the reactive cascade difference point as we step from tray to tray as follows

Vn+1 ) Ln + D - νTξn

(7)

Vn+1yn+1 ) Lnxn + DxD - νTξnδR

(8)

r Vn+1yn+1 ) Lnxn + (D - νTξn)δR,n

(9)

r δR,n )

DxD - νTξnδR D - νTξn

(10)

r is the reactive cascade difference point for where δR,n tray n in the rectifying section. The stages are numbered from the top. We see in eq 10 that this reactive cascade difference point is a linear combination of the distillate composition, xD, and the reaction difference point, δR. These two quantities are composition vectors in this equation, whereas all of the other terms on the right r lies deside are scalars. Where the composition δR,n pends on the accumulated reaction turnover, ξn, that has occurred from the top to stage n. This reaction turnover (ξn) for each reactive stage can be different according to catalyst volumes and reaction rates. However, the total material balance can fix the position of the final cascade difference point (δrR). Thus, we can choose ξn as positively increased values for the feasible r between the xD and forward reaction by locating δR,n r δR while moving down the column. Start at the distillate composition. In Figure 2b, we plot the composition for L1, the liquid in vapor/liquid equilibrium with the composition of the distillate. For a column with a total condenser, L1 is the liquid that leaves from the top tray. Note that its composition lies to the left of the reaction equilibrium line and thus in the reverse reaction zone. If reaction occurs on the first stage, then the reverse reaction must occur. In Figure 2b, we show the reactive cascade difference point for r , moving downward from D if D is great tray 1, δR,1 than νTξn. According to eq 9, the vapor composition (V2)

Figure 3. Feasibility of a reactive distillation column with a rectifying reaction zone in terms of feasible directions of composition vectors.

lies between the liquid composition (L1) and the reactive r cascade difference point δR,1 . The liquid composition in vapor/liquid equilibrium with the vapor composition at stage 2 moves us toward vertex P1 (the least volatile of the three species). This composition lies even further inside the reverse reaction region. If more reaction occurs on the next stage, we continue to step closer to the lower edge and then toward the vertex P1. Even if no reaction occurs at stage 1, the liquid composition (L2) at stage 2 still moves toward the reverse reaction region. In this case, the vapor composition at stage 2 lies between the distillate (D) and liquid compositions (L1) for all reflux ratios. Therefore, as the reaction will be in the reverse region in all instances, this reactive distillation column is infeasible and cannot produce the desired products (D and B). If νTξ1 is greater than the distillate flow rate (D), the reactive cascade difference point lies on the right extension of the reaction difference point in Figure 2c, and the liquid (L1) composition lies on the middle of the line connecting the vapor composition (V2) and the reactive cascade difference point. Thus, the vapor composition (V2) is near the binary edge of P1-R1, and its equilibrated liquid composition still moves into vertex P1 in Figure 2c. The reverse reaction occurring at the consecutive stages makes the column infeasible for producing the specified products. In contrast, a column having a reactive rectifying section can produce the products, as we show in Figure 3. This column is feasible because the reaction drives the vapor (V2) and liquid (L2) compositions at the second stage to be within the forward reaction area in Figure 3. If stage 3 is the feed tray, then the liquid composition (L3) in vapor/liquid equilibrium with the composition of the vapor (V3) lies within the reachable region from the bottom product composition, i.e., the gray region emanating from the bottom product composition. We can investigate the feasibility of a reactive stripping section in the same way as we have just done for a reactive rectifying section. We determine geometrically whether the reaction will force the liquid composition vectors to lie within the forward reaction zone. As we did in eqs 7-10 above, we write material balances around the reactive stripping section to find the reactive

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Figure 4. Infeasible directions of composition vectors in a reactive distillation column with a stripping reaction zone. Shaded area: reachable region from the top product.

cascade difference point for the stripping section

Ls ) Vs+1 + B - νTξs+1

(11)

Lsxs ) Vs+1ys+1 + BxB - νTξs+1δR

(12)

s Lsxs ) Vs+1ys+1 + (B - νTξs+1)δR,s+1

(13)

s δR,s+1 )

BxB - νTξs+1δR B - νTξs+1

(14)

s where δR,s+1 is a reactive cascade difference point in the stripping section (superscript s). We note that it is a linear combination of the bottom product composition and the reaction difference point in eq 14. Its position varies along the line joining these two compositions depending on the total molar turnover of reaction, ξs+1, that occurs below tray s + 1. Once we have selected the feed and products for our column, we can determine the total cascade difference point δsR geometrically, as shown in Figure 4. The s , will move reactive cascade difference points, δR,s+1 from the bottom composition ultimately to the total cascade difference point, δsR, as one moves through the reaction zone in the column. It will move toward δsR as we go up the column if the forward reaction occurs on each tray. The liquid composition (Ls) will lie between the bottom product composition and the vapor composition (Vs+1) for a nonreactive bottom stage. The vapor composition (Vs) is available using a bubble-point calculation. The liquid composition (Ls-1) will lie between s ) and the the reactive cascade difference point (δR,s vapor composition for reactive stage s (Vs). Here, the liquid composition (Ls-1) is within the forward reaction region, and the reaction can take place at this stage. If stage s - 1 is the feed tray, then the vapor composition (Vs-1) in equilibrium with the liquid composition (Ls-1) moves into the R1-R2 edge and away from the reachable region for the distillate composition. Thus, the reactive distillation column with a stripping reaction zone will be infeasible for the given products in Figure 4 because the vapor composition vectors in the bottom section move to the R1-R2 edge instead of going into the reachable region for the top product composition. However, if the distillate composition lies much nearer

to the R1-R2 edge, its reachable region will move upward, yielding a feasible reactive distillation column. Feasibility Criteria for a Reactive Distillation Column with Double Feed Streams. The total and component-wise material balances around the double feed column in Figure 5 are

Ft ) E + F ) B + D - νTξ

(15)

FtxFt ) BxB + DxD - νξ

(16)

(B + D)(xFt - xFˆ ) ) -νTξ(δR - xFt)

(17)

xFt )

ExF + FxF E+F

(18)

where xFt is the total feed composition, including both the upper and lower feed compositions (E and F, respectively). The total feed composition lies between the straight line connecting the pseudo-feed composition and the reaction difference point in Figure 5. The total cascade difference point, δrR, in eq 6 lies on the right extension of the line connecting the total feed and bottom product compositions. The material balance around the top section including the reaction zone is available in eq 19 by introducing the reactive cascade difference point in eq 10. r Vn+1yn+1 + ExE ) Lnxn + (D - νTξn)δR,n

(19)

The column is feasible in Figure 6 as the liquid composition at stage 1 lies within the forward reaction region and the liquid compositions at the consecutive stages are within the reachable region for the bottom composition. According to eq 19, the combined point marked with an asterisk (*) in Figure 6 lies between the liquid composition and the reactive cascade differr . Also according to eq 19, this ence point for stage 1, δR,1 combined point is the linear combination of the vapor composition at stage 2 and the upper feed composition (E, which is pure reactant R2, the heavier of the reactants). When we connect the given liquid composir ) tion (L1) and the reactive cascade difference point (δR,1 by a straight line, the location of the combined point (*) depends on the reflux ratio. The vapor composition (V2) lies on the right extension of the straight line


Figure 5. Material balance constraint in a double-feed reactive distillation column for the reaction R1 + R2 T P1.

[-2, 1, 1]T/(-2 + 1 + 1) ) [-∞, ∞, ∞]Τ. We decompose the stoichiometric coefficient vector (ν)[-2, 1, 1]Τ) into the normalized reactant (cR ) [1, 0, 0]Τ) and product (cP ) [0, 1/2, 1/2]Τ) coefficient vectors to allow us to carry out our analysis within a finite region in our composition diagram. Most of reactant R1 is supplied to the feed stage and goes through the rectifying section or stripping section with a localized liquid-phase reaction zone. Relatively pure P1 comes out at the top, and P2 at the bottom. The relative volatilities of R1, P1, and P2 are 3, 5, and 1, respectively, in this reaction. The total and component-wise material balances around an entire column are in eqs 2 and 3. Substituting eq 2 into eq 3 and decomposing ν into cP and cR yields eq 21.

(D + B)(xF - xFˆ ) ) (-2ξ)(cP - cR) Figure 6. Feasible directions of composition vectors in a reactive distillation column with double feed streams: *, linear combination of the liquid composition and the reactive cascade difference point at the first stage; #, linear combination of the liquid composition at stage 2 and the reactive cascade difference point at the first stage; shaded area: reachable region from the bottom product. Upper feed stage, 1; reactive stage, 1.

connecting the upper feed composition and this combined point (*). The equilibrated liquid composition at stage 2 is within the reachable region for the bottom product composition. The combined point (#) lies between the liquid composition (L2) and the reactive r ) if stage 1 is reactive and cascade difference point (δR,1 stage 2 is not. The vapor composition at stage 3 also lies on the right extension of the straight line connecting the upper feed composition and the combined point (#), as shown in Figure 6. If the feed stage is stage 3, then the liquid composition at stage 3 lies within the reachable area for the bottom product composition. Feasibility of a Reactive Distillation Column with an Infinite Reaction Difference Point An infinite reaction difference point comes from a reaction that does not change the overall number of moles as in eq 20.

2R1 S P1 + P2

(20)

The reaction difference point for this reaction is

(21)

Represented here are two parallel straight lines.25 The length of the straight line connecting the product and the reactant coefficient vectors (cP and cR, respectively) represents the relative amount of the feed flow rate (F) or the product flow rates (D + B), whereas the distance from the feed to the pseudo-feed composition denotes 2 times the overall reaction molar turnover (2ξ) in Figure 7a. Note that the product flow rate (D + B) is always greater than or equal to the reaction molar turnover 2ξ. If we rewrite eq 3 by decomposing the stoichiometric coefficient vector into the normalized reactant and product coefficient vectors, then the following equations are available for the rectifying reaction section.

FxF ) BxB + DxD - 2ξ(cP - cR) ) BxB + DδrR (22) δrR )

DxD + 2ξcR - 2ξcP D + 2ξ - 2ξ

(23)

The total cascade difference point, δrR, in eq 23 lies at the intersection point of the upper right extension of the straight line that connects the bottom product and feed compositions and the line starting from the distillate composition that runs parallel to the vector (cR cP) in Figure 7a. Where it lies depends on the total reaction turnover in the column, ξ. Equation 24 is the material balance around the reactive rectifying section. We find that the reactive r , combines the distillate cascade difference point, δR,n composition and the reactant and product coefficient

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Figure 7. Total material balance and reactive cascade difference point in a reactive distillation column with an infinite reaction difference point: (a) geometry of the total material balance, (b) reactive cascade difference point of the reactive rectifying section for the reaction 2R1T P1 + P2.

Figure 8. Directions of composition vectors in a rectifying reactive distillation column for the reaction 2R1 T P1 + P2: (a) infeasible direction of composition vectors from product compositions, (b) feasible direction of composition vectors from product compositions. Shaded area: reachable region from the bottom product.

vectors in eq 25. r Vn+1yn+1 ) Lnxn + (D + 2ξn - 2ξn)δR,n r δR,n )

DxD + 2ξncR - 2ξncP D + 2ξn - 2ξn

(24) (25)

This reactive cascade difference point moves parallel to the stoichiometric line (cP-cR), starting away from the distillate composition and ultimately moving to the total cascade difference point, δrR, as one steps down the reactive section of the column in Figure 7b. The length of the line from the distillate composition to the reactive cascade difference point represents 2ξn compared to length of the stoichiometric line, which denotes the amount of D.

For the given top and bottom product compositions in Figure 8a, the reactive distillation column is infeasible because the liquid composition at stage 1 is within the reverse reaction area. Even when stage 1 is not reactive, the liquid composition (L2) will still lie within the reverse reaction region because the equilibrated vapor composition at stage 2 (V2) lies between the distillate composition and the liquid composition (L1) for all reflux ratios. The reverse reaction in consecutive reactive stages moves the liquid and vapor composition vectors to vertex P2, trapping the system away from the desired bottoms product. For the product pair in Figure 8b, the column is feasible, as consecutive liquid compositions in the lower stages of the rectifying section can be within the forward reaction region and ultimately within the reachable


Figure 9. Directions of composition vectors in a stripping reactive distillation column for the reaction 2R1 T P1 + P2: (a) infeasible direction of composition vectors from product compositions, (b) feasible top and bottom products.

region for the bottom product. The vapor composition at stage 2 lies between the liquid composition at stage 1 and the reactive cascade difference point according to eq 24, and the liquid composition in vapor/liquid equilibrium is within the forward reaction part as well as the reachable region for the bottom product composition. If the reaction zone is partially distributed in the stripping section, the material balance equations become s Vs+1ys+1 ) Lsxs - (B + 2ξs+1 - 2ξs+1)δR,s+1 (26) s ) δR,s+1

BxB + 2ξs+1cR - 2ξs+1cP B + 2ξs+1 - 2ξs+1

(27)

With the top and bottom products in Figure 8a, the column remains infeasible even when the reactive zone is switched to the stripping section, as indicated in Figure 9a. The reverse reaction occurs at stage s as the liquid composition (Ls) is within the reverse reaction zone. For any reboil ratio, the liquid composition (Ls-1) is also within the reverse reaction zone as the vapor composition (Vs) already lies within the reverse reaction region. Although its bubble-point vapor composition (Vs-1) can be within the reachable region from the top product in Figure 9a, the reverse reaction in consecutive reactive stages is infeasible for the desired product specification. The second product pair in Figure 8b remains feasible when the reaction zone is moved to the stripping section and if the reaction occurs only in the bottom stage (stage s + 1), as we can discern by examining Figure 9b. The liquid composition (Ls) can lie within the forward reaction zone as it is between and on the straight line s ) connecting the reactive cascade difference point (δR,s+1 and the vapor composition (Vs+1), according to eqs 26 and 27. However, we should have only one reactive stage (s + 1). A second reactive stage (stage s) will cause the composition vector to move toward the R1-P1 edge and away from the reachable region for the top product composition. In terms of the design of this reactive distillation column, the operating line should shift from the stripping to the rectifying section at the sth stage, which will be the feed stage, and the reboil ratio should be large to provide a high overall conversion.

Minimum Reflux Ratio in Reactive Distillation In the process of going from feasibility to design studies of a reactive distillation column with a ternary mixture, we can geometrically determine the minimum reflux needed to obtain an overall reaction conversion. Here, we focus on a single-feed reactive distillation column for reaction 1. This idea can be readily extended to a double-feed reactive distillation column or a singlefeed reactive distillation column with reaction 20, which has its reaction difference point at infinity. If the reaction zone is in the rectifying section, the total cascade difference point, δrR, for the stage directly above the feed stage will lie on the right extension of the straight line connecting the bottom product and feed compositions according to eqs 3 and 6, as illustrated in Figure 10. This final liquid composition (Ln) will be on the reaction equilibrium curve, as we show it to be here. Somewhere along this line between the compositions for Ln and δrR, we should place Vn+1. By the lever rule (and from eq 9), we note that the length of the line between Ln and Vn+1 is proportional to the flow rate (D + ξ), whereas that from Vn+1 to δrR is proportional in length to Ln. Because ξ is fixed for given feed and product specifications, the minimum reflux ratio, R ) Ln/D, corresponds here to placing a line emanating from δrR to the composition along the reaction equilibrium line having the smallest ratio of the length between Vn+1 and δrR to the length between Ln and Vn+1. In addition we should place the vapor composition at the feed stage (Vn+1) so that the liquid composition at the feed stage can be within the reachable region for the bottom product composition. If the heat of the reaction is negligible, Figure 10 shows that the reflux ratio (L0/D) is nearly 3 when the overall reaction molar turnover (ξ) is equal to the distillate flow rate (D). Figure 11 shows that, as the reaction equilibrium constant decreases, the minimum reflux increases for the same reaction conversion. These two diagrams make this behavior evident, a behavior that Okasinski and Doherty11 discovered by using simulation runs. Tray-by-Tray Calculations with a Finite Reaction Difference Point We can perform tray-by-tray calculations for singlefeed and double-feed reactive distillation columns with

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Figure 10. Geometrical interpretation of the minimum reflux in a rectifying reaction zone. The shaded area represents the reachable region from the bottom product.

Figure 11. Minimum reflux for different reaction equilibrium constants. The lower equilibrium constant, the larger the reflux required for the same overall reaction conversion. K1 is greater than K2. For the same ξn value, the ratio Ln′/(D - νTξn) is larger than Ln/(D νTξn).

reaction 1. Every time the reaction “turns over,” we lose 1 mol/s in flow. For simplicity, we compute phase equilibrium using constant relative volatilities

yn,i )

Rixn,i

∑j

,

i,j ) R1,R2,P1

(28)

Rjxn,j

Also for simplicity, we assume that the heat of vaporization is constant (independent of the composition of the mixture) and that the heat of reaction is negligible compared to the heat of vaporization. Single-Feed Column. We mainly focus on tray-bytray calculations for the reactive rectifying section in a single-feed column. With the reactive stripping section, we can also carry out tray-by-tray calculations in the same way. The reactive cascade difference point in eq 10 will move from the distillate composition ultimately to the total cascade difference point because the accumulated reaction molar turnover from the top to stage n, ξn, becomes equal to ξ as we go to the lowest reactive stage. Numbering from the top, the vapor composition at stage n + 1 will lie between the straight line connecting the liquid composition and the reactive

cascade difference point at stage n. Figure 12 shows tray-by-tray calculations for the saturated liquid feed for the case when the feed flow rate is 3 times the distillate flow rate (3D). An explanation of Figure 12 is as follows. Throughout, we assume the location at which and the extent to which the reaction occurs. The distillate composition is the same as the vapor composition at stage 1 when a total condenser is used. From the vapor composition at stage 1, we can calculate the composition of the liquid leaving using a dew-point calculation. If we have plotted distillation curves throughout the composition space, two consecutive points along one such curve are in vapor/ liquid equilibrium with each other, making it easy to estimate where this liquid composition lies. We assume no reaction at stage 1; thus, the vapor composition at stage 2 lies along the straight line connecting the compositions of the distillate and L1. Arbitrarily, we take the reflux ratio, L0/D, to be equal to 4. With no reaction on the top stage, L1 is equal to L0. We use this reflux ratio to partition this line. Again, the liquid composition at stage 2 comes from a dew-point calculation based on the vapor composition at stage 2. We assume that the reaction molar turnover at reactive stage 2 (ξ2) is one-half of the distillate flow rate (0.5D).


Figure 12. Tray-by-tray calculation for the ternary mixture in a single-feed reactive distillation column for the reaction R1 + R2 T P1 taking place in the rectifying section. Feed stage, 4; reactive stages, 2 and 3.

Then the reactive cascade difference point at stage 2 r ) lies one-third of the way along the straight line (δR,2 from the distillate composition to the reaction difference point. The vapor composition at stage 3 lies between the liquid composition and the reactive cascade difference point at stage 2. The liquid flow rate at stage 2, L2, reduces to 3 1/2 times the distillate flow rate (3.5D) because of the reaction on this stage. The liquid composition at stage 3 comes from a dew-point calculation based on the vapor composition at stage 3. For reactive stage 3, the reaction molar turnover at stage 3, we assume ξ3 to be equal to D, which means the net reaction molar turnover on stage 3 (ξ3 - ξ2) is 0.5D. We can locate the vapor composition at stage 4 between the straight line connecting the liquid composition and the reactive cascade difference point at stage 3 in Figure 12. The flow rate of the liquid stream at the feed stage (stage 4) is now 6 times the distillate flow rate (6D) as the flow rate of the saturated liquid feed is assume to be 3D and the liquid flow rate at stage 3 (L3) is 3D. Because the overall reaction molar turnover is equal to the distillate flow rate (D), the bottom product flow rate becomes D according to eq 2. The reboil ratio (V/B) is 5 for the stripping section because the liquid flow rate (L4) at the feed stage is 6D. We can switch the operating lines at the liquid composition of stage 4 into the nonreactive stripping section because the required reaction extent is satisfied at stage 3 in terms r . Tray-by-tray calculaof cascade difference point δR,3 tions proceed with this reboil ratio for the nonreactive bottom section. Double-Feed Column. Equation 19 is used for the tray-by-tray calculations of reactive stages between two feed streams as in Figure 13. We neglect the effects of heat from the reaction as in the previous section. All feed streams are assumed to be saturated liquid. The combined point (*) from the vapor composition at stage n + 1 and the upper feed composition (E) lies between the liquid composition and the reactive cascade difference point at stage n. The upper feed stream going into stage 1 contains pure R2, and its flow rate is D. The lower feed entering stage 3 is most of component R1,

Figure 13. Tray-by-tray calculation for the ternary mixture in a double-feed reactive distillation column for the reaction R1 + R2 T P1 taking place between the feed streams. Upper feed stage, 1; lower feed stage, 3; reactive stage, 2. *, combined point with E and V2 ) combined point with L1 and D; #, combined point with E and V3 ) combined point with L1 and δrR,2.

and its flow rate is 2D. Here, we arbitrarily set the reflux ratio (L0/D) to be 6. The liquid composition at stage 1 is the dew-point composition based on the distillate composition. The combined point (*) with the vapor composition at stage 2 and the upper feed composition lies between the liquid composition at stage 1 and the distillate composition as we assume that no reaction occurs at the first stage. By summing the reflux rate (6D) and the upper feed flow rate (D), the flow rate of the liquid stream at stage 1 is 7D. We can locate the vapor composition at stage 2 (V2) at the lower side of the combined point (*), which is one-seventh of the length of the straight line connecting the upper feed composition and the combined point. A dew-point calculation based on the vapor composition (V2) gives us the liquid composition at stage 2. The flow rate of the liquid stream at reactive stage 2 decreases from 7D to 6D if we assume a reaction molar turnover, ξ2, equal to D. The combined point (#) is located between the reactive cascade difference point and the liquid composition at stage 2. The vapor composition at stage 3 lies on the lower extension of the combined point (#). Tray-by-tray calculations switch into the stripping section at the feed stage (stage 3), where the flow rate of the liquid stream is 8D (combining the influx of the lower feed stream and the liquid stream at stage 2). The reboil ratio (V/B) equals 7 as the bottom flow rate is D from the total balance given by eq 15. In Figure 13, display the results of tray-by-tray calculations for the nonreactive stripping section with this reboil ratio. Tray-by-Tray Calculations with an Infinite Reaction Difference Point If the heat of reaction can be negligible and the heat of vaporization does not change much depending on the compositions of the mixture, the constant molar overflow (CMO) assumption is valid as the isomolar reaction in eq 20 does not change the total number of moles. Consider Figure 14. Let us assume the reflux ratio (L0/D) to be 9. The saturated liquid feed stream is pure

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MTBE (Methyl tert-Butyl Ether) Production System Methyl tert-butyl ether (MTBE) can be produced from isobutene (IBUT) and methanol (MeOH) using an ionexchange resin27 or sulfuric acid28 as the catalyst in reaction 29.

MeOH(CH3OH) + IBUT(i-C4H8) S MTBE(C5H12O) (29) Because IBUT is more volatile, the IBUT feed stream is the lower feed stream, and the MeOH is in the upper feed stream. The reaction takes place essentially within a reaction zone between two feed stages where MeOH and IBUT flow in opposing directions. The top product is excess IBUT, whereas the bottom product is MTBE. We employ the reaction equilibrium constant, which is a function of temperature (T in Kelvin) in eq 30, from Venimadhavan et al.29

ln(K) )

Figure 14. Tray-by-tray calculations for the ternary mixture in a single-feed reactive distillation column for the reaction 2R1 T P1 + P2. Feed stage, 5; reactive stages, 2-4.

R1, whose flow rate is 2D. We assume the overall reaction molar turnover, ξ, to be 0.9D. The bottom flow rate is identical to the top flow rate (D) because the feed flow rate is 2D. Stages 2-4 are reactive. As previously mentioned, the reactive cascade difference points move parallel to the stoichiometric line from the distillate composition, ultimately reaching the total cascade difference point, δrR. They move toward the total cascade difference point, δrR, for the forward reaction and in the reverse direction for the reverse reaction. The feed stage is stage 5, and the reboil ratio is 10 as the feed flow rate is 2D and the liquid flow rate above the feed stage is 9D when constant molar overflow is assumed. We obtain the liquid composition of stage 1 using a dewpoint calculation based on the vapor composition leaving stage 1, which is the same as the distillate composition when a total condenser is used. The vapor composition at stage 2 lies between and on the straight line connecting the distillate composition and the liquid composition at stage 1. It is close to the liquid composition for the given reflux ratio L0/D ) 9. The vapor composition for stage 3 lies between the liquid composition for stage 2 and the reactive cascade difference point in Figure 14. We can determine the vapor compositions of stages 4 and 5 in a similar manner. Tray-by-tray calculations for the nonreactive stripping section in stages 6 and 7 are shown in Figure 14 for the reboil ratio, V/B, equal to 10.

6820.0 - 16.33 T

(30)

We assume that the column operates at 8 atm. We combine bubble-point calculations with reaction equilibrium calculations to determine the reaction equilibrium curve in Figure 15. We calculate the phase equilibrium by using property set KVL in AspenPlus, predicting two minimum-boiling binary azeotropes: MeOH-MTBE (56 mol % of MeOH) and MeOH-IBUT (6.8 mol % of MeOH). We draw the operating line from the material balance in eq 19. We can conceptually generate many design alternatives in terms of the reactive lever rule implied by eqs 4 and 17. We examine whether we can obtain the top and bottom products shown in Figure 15. The distillate composition is almost on the reaction equilibrium curve, and its equilibrated liquid composition (L1) moves toward the MTBE vertex. The consecutive liquid compositions at the lower stages lie within the reverse reaction region and outside the reachable region for the bottom product. The vapor composition at the bottom stage can move into the forward reaction region, but the two composition trajectories from the top and bottom products will not meet for any reflux ratio because the composition vectors of the reactive rectifying section move along the MTBE-IBUT binary edge and finally arrive at the MTBE vertex. Thus, it is not feasible for this reactive distillation column to produce the top and bottom products given in Figure 15. Figure 16 shows a feasible set of top and bottom products. The top product (D) contains relatively pure IBUT rather than the IBUT-MeOH azeotrope, and the bottom product is pure MTBE. In terms of the reactive lever rule implied by eqs 6 and 16, we can determine the total cascade difference point by connecting the given bottom and total feed compositions with a straight line. The vapor composition at stage 2 (V2) lies between the distillate composition and the liquid composition (L1) for all reflux ratios if no reaction occurs at stage 1. Thus, the equilibrated liquid composition (L2) can lie between L1 and L1* in Figure 16. With no reflux, V2 is the same as D, and then L2 becomes L1. Under total reflux, V2 is the same as L1, which means that L2 becomes L1*. One should note that L1 is in phase equilibrium with L1*. In this range, L2 is within the forward reaction region.


Figure 15. Infeasible products for the MTBE production systems. 9, azeotropes; shaded region, reachable region from the bottom product. The values in parentheses represent the boiling points of MeOH, MTBE, and IBUT.

Figure 16. First feasible flowsheet with pure MTBE at the bottom and relatively pure IBUT at the top. The solid line with an arrow on each end denotes the possible composition of L2 for all reflux ratios. Shaded area, reachable region from the bottom product. *, combined point with L2 and δrR,2 ) combined point with E and V3. L1* is the liquid stream equilibrated with L1. Reactive/upper feed/lower stages, 2/2/3.

By connecting an arbitrary liquid composition at stage 2 (L2) and the total cascade difference point, we can obtain the combined point (*) based on a lever rule, whose ratio is L2/(D - νTξ2). Here, the upper feed stream (E) enters stage 2, and the reaction occurs on this stage. We can determine the vapor composition at stage 3 (V3) on the lower right extension of the straight line that connects the upper feed composition (E) and the combined point (*). The liquid in vapor/liquid equilibrium with V3 is L3, and its composition lies within the reachable region for the bottom product. Thus, if the

lower feed (F) enters at stage 3, this column can produce the specified products. We next determine whether the column is feasible when the top product is very close to the IBUT-MeOH azeotrope and we wish to obtain the same bottom product as in Figure 16. Here, we test for a distillate composition of (0.925, 0.069) for IBUT and MeOH. The liquid composition at stage 1 (L1) is in vapor/liquid equilibrium with this distillate composition. For all reflux ratios, V2 lies between L1 and D. L2, in vapor/ liquid equilibrium with V2, ranges from L1 for no reflux

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Figure 17. First feasible flowsheet with pure MTBE at the bottom and IBUT-MeOH azeotrope at the top. Shaded area, reachable region from the bottom product. *, combined point with L2 and D ) combined point with E and V3; #, combined point with L3 and δrR,3 ) combined point with E and V4. Reactive/upper feed/lower stages, 3/2/4.

Figure 18. Feasible flowsheet with a reactor followed by a reactive distillation column for MTBE production: (a) the final top and bottom products in Figure 16, (b) final top and bottom products in Figure 17.

to L1* for total reflux in Figure 17. L2 is within the forward reaction region but far outside the reachable region for the bottom product. There are two ways to place the vapor and liquid compositions at stage 3 near the reachable region for the bottom product: by having

the reaction occur at stage 2 or by providing the upper feed (E) at this stage. In Figure 17, we place the upper feed at stage 2. For an arbitrarily chosen L2 between L1 and L1*, the vapor composition at stage 3 (V3) can lie near the MTBE-IBUT edge. We can determine V3

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2727 Table 1. AspenPlus Simulation Results for the Two Product Pairs in Figures 16 and 17 product pair: top/bottom feed stream upper MeOH feed lower IBUT feed product flow (kmol/h) distillate flow/IBUT purity bottom flow/MeOH purity total number of stages upper/lower feed stages reactive stage reflux ratio molar reaction conversion based on MeOH feed flow rate

IBUT/MTBE

IBUT-MeOH azeotrope/ MTBE

1000 1200

1000 1200

200/96.2% 1003.7/98.9% 8 2/3 2 27.0 99.6%

200/92.8% 400/97.1% 8 2/4 3 25.0 98.5%

on the right extension of the combined point (*) by drawing two straight lines, as in Figure 13. The corresponding liquid composition (L3) lies within the forward reaction region, and then the reaction can occur on stage 3. Again, we can obtain the vapor composition on stage 4 (V4) by connecting two straight lines. The combined r (#) lies between the line joining point of L3 and δR,3 these two points. We can place V4 on the right extension of the straight line connecting the combined point (#) and E. The vapor/liquid equilibrium liquid (L4) lies within the reachable region from the bottom product. Hence, if stage 4 is the lower feed stage, this column is feasible. Table 1 shows the simulation results for the two product pairs in Figures 16 and 17. The total number of stages is fixed at 8 for both cases. With the same reaction trays as in Figures 16 and 17, we can obtain the same product compositions as in these two figures for both cases. Thus, the feasibility criteria developed here can easily generate feasible design alternatives before detailed simulations are performed. We can generate other flowsheets with a reactive distillation column preceded by a reactor, as in Figure 18. The feed composition (F1) of the first reactor lies between the reaction difference point and the product composition (F2), which is the feed composition for the reactive distillation column. The length of the line between the reaction difference point and the feed composition (F1) represents the relative amount of the feed flow rate for the column (F2). The other line connecting the feed composition of the first reactor (F1) and the feed composition of the column (F2) denotes the reaction molar turnover flow rate in the first reactor. If the reaction conversion in the first reactor is obtainable, these flowsheets are also feasible as the top and bottom product compositions of the reactive distillation column are the same as in Figures 16 and 17. The conversion of the first reactor is definitely feasible as the product composition (F2) still lies away from the reaction equilibrium and within the forward reaction region. Therefore, these two flowsheets are also feasible for production of the specified products. Conclusions We present a simple method for assessing the feasibility of a proposed reactive distillation column for given top and bottom products for both single-feed and doublefeed columns. The method is based on an inspection of the directions in which composition vectors move. A main point of the feasibility criteria is that the composi-

tion profile of a reactive section should connect the nonreactive reachable regions for the top and bottom products. If the liquid compositions at the reactive stages are within the forward reaction zone and if the liquid composition at the feed stage lies within the reachable region of the bottom product composition, then the column is feasible for a rectifying reaction zone. If the liquid compositions above the bottom are moving into the forward reaction zone as well as the reachable region from the top product composition, one can have a feasible reactive stripping section. The minimum reflux is geometrically available in terms of the total cascade difference point, the reaction equilibrium curve, and the reachable region of a nonreactive section. This feasibility analysis enables us to see, geometrically, what is happening inside a column. It provides us with a visual tool to aid in the design of a reactive distillation column. For an actual production system such as an MTBE system, we can easily evaluate the feasibility of proposed flowsheets. Acknowledgment The authors are grateful for the support of the NSF through Grant CTS9710303 and the support of the Eastman Chemical Company. Notation B ) bottom molar flow rate cP ) the normalized product coefficient vector cR ) the normalized reactant coefficient vector D ) distillate molar flow rate E ) upper feed molar flow rate Ft ) total feed molar flow rate F ) feed molar flow rate F ˆ ) pseudo-feed molar flow rate K ) reaction equilibrium constant Pi ) product of component i Rk ) reactant of component k Vn ) vapor molar flow rate at stage n xi ) liquid composition vector for stream i ) B, D, E, F, F t, F ˆ yn ) vapor composition vector at stage n Greek Letters Ri ) relative volatility of component i δR ) reaction difference point δrR ) the total cascade difference point in the rectifying section δsR ) the total cascade difference point in the stripping section r δR,n ) reactive cascade difference point at stage n in the rectifying section s δR,s ) reactive cascade difference point at stage s in the stripping section ν ) the stoichiometric coefficient vector νT ) sum of the total stoichiometric coefficients ξ ) total molar turnover flow rate (total reaction extent) ξn ) sum of the total molar turnover flow rate from the top to n stages ξs ) sum of molar turnover flow rate from bottom to s stages in the reactive stripping section ∆ξn) net reaction molar turnover flow rate at stage n

Literature Cited (1) Smith, L. A. Catalytic distillation process. U.S. Patent 4,307,254, Dec 22, 1981.

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Received for review May 25, 2000 Revised manuscript received March 22, 2001 Accepted March 28, 2001 IE0005194

Feasibility of a Reactive Distillation Column with Ternary Mixtures

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