Feasibility Studies on Quaternary Reactive Distillation Systems

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Ind. Eng. Chem. Res. 2002, 41, 4632-4642

Feasibility Studies on Quaternary Reactive Distillation Systems Jae W. Lee* Department of Chemical Engineering, City College of New York, New York, New York 10031

A quick and reliable algorithm is proposed to evaluate the feasibility of a double-feed reactive distillation column on the basis of the projection method (AIChE J. 2001, 47, 1333). A critical composition region is introduced where liquid seems to be more volatile than its equilibrated vapor in the projected space. If this critical composition region is spread along the entire binary edge of an upper feed and a top product in the original composition space, then a double-feed column cannot produce desired products. Otherwise, outside the critical composition region, single-stage calculations are carried out under potential pinch points to determine upper and lower bounds of the internal reflux ratio. If the permissible range of the reflux ratio near a pure product vertex overlaps with all the other permissible reflux ranges in one projected space, then a double-feed column can produce specified products. Introduction The most important question when adopting reactive distillation technology is whether the combination of reaction and distillation can achieve a desired reaction conversion and produce pure products. To answer this question, it is very important to understand the interaction between reaction and distillation. Unfavorable phase equilibrium may not allow the removal of products from reactive zones and can cause a significant amount of reverse reaction to take place. Extremely small reaction equilibrium constants cannot lead to desirable reaction conversions with reasonable reflux ratios.2 Graphical design techniques were recently proposed to visualize the reaction-distillation interaction in the composition space.1,3-5 These techniques visualize reaction terms by using reaction difference points6,7 (or poles8) and normalized stoichiometric vectors.9 Several difference points and their reactive operating lines in McCabe-Thiele and Ponchon-Savarit diagrams show the effect of reaction on physical separation. For ternary reactive mixtures, stage calculations are carried out on the reaction equilibrium curve and its equilibrated vapor curve. The visualization of quaternary reactive systems allows us to see the interaction between reaction and separation in several projected triangles cut from the tetrahedral composition space by constant mole fractions of a certain component. On the basis of this understanding of the interaction between reaction and distillation, a new method will be proposed for determining the feasibility of a quaternary reactive distillation system in a double-feed column. First, a feasibility criterion will be discussed for a double-feed reactive distillation column. Then, the concept of a critical composition region will be introduced. The ethyl acetate production system will be taken as an example, and it will be investigated how the critical composition region can affect the possibility of reaching pure products. Then, a quick and reliable * To whom correspondence should be addressed. 140th St and Convent Ave, Department of Chemical Engineering, The City College of City University of New York, New York, NY 10031. Telephone: 1-212-650-6688. Fax: 1-212-650-6660. Email: [email protected].

Figure 1. Schematic of a double-feed reactive distillation column.

algorithm will be proposed for checking the feasibility of quaternary reactive systems by single-stage calculations with potential pinch points. Several acetate production systems will highlight the simplicity and applicability of this method using only chemical and phase equilibrium information. Feasibility Criterion for a Double-Feed Reactive Distillation Column The structure for a double-feed column is shown in Figure 1. This column consists of a nonreactive rectifying section, a reactive stripping section, and a reactive middle section with two feed streams. The following reaction occurs in the column with the assumption that reaction equilibrium is achieved faster than phase equilibrium. The stripping section can be switched to a nonreactive section for solid catalyst systems.

R1 + R2 T P1 + P2

(1)

Here, R1 is less volatile than R2, so R1 and R2 are fed to the upper and lower feed stages in stoichiometric amounts. Product P1 is more volatile than P2 and is recovered at the top. It is assumed that there is no binary azeotrope between pure R1 and pure P1. A double-feed reactive distillation column can produce relatively pure products from a stoichiometric feed if P1 and R1 are dominantly present at the upper feed stage.

10.1021/ie0109859 CCC: $22.00 © 2002 American Chemical Society Published on Web 08/13/2002

Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002 4633

Figure 2. Tetrahedral composition space of the EA production system. The numbers in parentheses are boiling points in degrees Celsius. The hatched surface describes reaction equilibrium in the original composition, and the sliced triangle is a projected space with a constant fraction of AC.

So, the nonreactive rectifying section above the upper feed stage can enrich P1 at the top from the binary mixture of P1 and R1. If this situation is visualized in the composition space, a tray profile between two feed streams approaches the binary edge of R1 and P1 as the composition profile is followed up through the column. In terms of the boundary value method, this composition profile of the reactive middle section connects the P1-R1 binary edge with a stripping profile.10 For example, only methyl acetate and acetic acid are present at the upper feed stage in Eastman’s doublefeed reactive distillation column.11 This means that its composition lies on the binary edge between acetic acid and methyl acetate in the composition space. Reaching this binary edge indicates that methanol is completely consumed by the reaction and that the lowest-boiling methanol-methyl acetate azeotrope will not appear at the top. So, an overall reaction conversion can be almost 100% under identical distillate and upper feed flow rates (D ) E). Pure methyl acetate is separated from the binary mixture of methyl acetate and acetic acid as a top product by the nonreactive rectifying section. Also, all water in the column is extracted by acetic acid and does not appear at the upper feed stage. Therefore, it is recovered at the bottom. Projected Tray-by-Tray Model If a tetrahedral composition space is cut with a constant mole fraction of a component as in Figure 2, material balance equations can be obtained for the sliced triangle. The detailed derivation for the case of multiple reactions can be found in a recent publication.1 Here, the case of reaction 1 occurring in a double-feed column is considered. The material balance envelope in Figure 1 yields the following mole-based equations for the reactive section including the upper feed and top product

Vn+1 ) Ln + D - E - νTξn

(2)

Vn+1yn+1 ) Lnxn + DxD - ExE - ξnν

(3)

The following material balance equations can be derived from eq 3 by the projection method.1

˜ nx˜ n + D ˜ x˜ D - E ˜ x˜ E - ξnν˜ V ˜ n+1y˜ n+1 ) L

(4)

Figure 3. Vapor and liquid equilibrium on the left and right side of the critical compositions at a projected triangle with xAC ) 0.1. Hereafter, the dotted curve represents reaction equilibrium and the solid curve describes the vapor state in phase equilibrium with the reaction equilibrium curve.

where

V ˜ n+1 ) (1 - yn+1,k)Vn+1

(5)

L ˜ n ) (1 - xn,k)Ln

(6)

D ˜ ) (1 - xD,k)D

(7)

E ˜ ) (1 - xE,k)E

(8)

Here, ξn is the sum of reaction extents from the top to stage n. ν˜ is the stoichiometric vector for four projected ternary spaces with a constant liquid mole fraction of each component. For example, if the liquid mole fraction of reactant R1 is constant, the stoichiometric vector in this projected space is [-1, 1, 1]T for R2, P1, and P2. The subscript k is an excluded component in one projected space. The excluded component has a constant liquid mole fraction in that projected triangle. Example: Ethyl Acetate Production System Critical Composition Region. In the ethyl acetate production system, R1, R2, P1, and P2 are acetic acid (AC), ethanol (ET), ethyl acetate (EA), and water (W), respectively, in eq 1. There are four minimum-boiling azeotropes in this reacting mixture. One of them is a ternary minimum-boiling azeotrope among ET, EA, and W. The other three are the binary azeotropes shown in Figure 2. The reaction equilibrium constant (Keq ) KxKγ) is 13.4.12 The phase equilibrium data from Barbosa (1987)10 are used. The simultaneous phase and reaction equilibrium is calculated while considering the vapor phase dimerization of acetic acid. If the tetrahedral composition space is cut by a constant AC mole fraction, then a curved line describes the reaction equilibrium in Figure 2. When xAC is equal to 0.1, Figure 3 shows the reaction equilibrium curve (dotted line) and its vapor curve (solid line). This vapor curve represents the projected compositions in phase equilibrium with the liquid compositions on the reaction equilibrium curve in Figure 3. It should be noted that the liquid compositions on the selected plane have equilibrated vapor compositions that are not on the

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Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002

Figure 4. Possible region for y˜ n+1 when carrying out a stage calculation to the right of the critical composition. *: intersection point of x˜ n and x˜ D ) combined point of y˜ n+1 and ν˜ .

plane and must be projected onto the plane. It is these projections that are drawn as the solid curve in Figure 3. The critical compositions are the projected compositions where the projected liquid composition has the same mole fraction of the most volatile product as the projected vapor composition. The critical vapor and liquid compositions are denoted as y˜ *and x˜ * in Figure 3. To the left of the critical compositions, the vapor composition (y˜ n) lies closer to the EA vertex than its equilibrated liquid composition (x˜ n). So, the vapor contains more EA than the liquid. However, to the right of the critical compositions, the projected liquid composition is more enriched in EA than the projected vapor composition. Thus, when the liquid composition lies to the right of the critical compositions, the EA vertex in the projected space (which is the EA-AC binary edge in the original composition space) is not accessible by simultaneous reaction and separation with this constant acetic acid mole fraction (xAC ) 0.1). This is explained in terms of material balance eq 4 as shown in Figure 4. It should be noted that the stage numbers start from the top. With the constant acetic acid fraction, xAC ) 0.1, eq 4 can be rearranged as follows:

V ˜ n+1y˜ n+1 + ξnν˜ ) 0.9Lnx˜ n + Dx˜ D

(9)

xAC ) 0.1 means that xn,k in eq 6 is 0.1, since all liquid compositions have the constant AC mole fraction, 0.1. Here, the excluded component k is AC. xD,k and xE,k are 0 and 1, since the distillate (D) and upper feed (E) contain pure EA and AC, respectively. The distillate and the upper feed streams (D and E) are assumed to have the same flow rates (because of the stoichiometric feed ratio).

Consider one straight line that connects x˜ n with x˜ D and another straight line that joins yn+1 and ν˜ . To satisfy the material balance constraint in eq 9, these two straight lines should intersect each other. This intersection point (*) lies on the straight line between x˜ n and x˜ D, so its position is determined by one parameter, the internal reflux ratio (Ln/D). The ratio of |x˜ n *|/|* - x˜ D| is 0.9Ln/D. The intersection point (*) is equal to x˜ n under infinite internal reflux and becomes x˜ D under zero internal reflux. For the infinite case, y˜ n+1 is ˜ n+1 are much larger than identical to x˜ n, since Ln and V D and ξn in eq 9. In the same way, y˜ n+1 will be x˜ D for the case of a zero internal reflux ratio. However, under arbitrary internal reflux ratios (between 0 and ∞), y˜ n+1 can lie on the straight line connecting ν˜ and arbitrary intersection points (*) within the valid composition simplex (see the magnified part in Figure 4). The position of y˜ n+1 along the arrow is a function of the reaction extent, ξn. In this way, the shaded region in Figure 4 can be drawn for possible y˜ n+1 over all the internal reflux ratios. Thus, y˜ n+1 is closer to the EA vertex than y˜ n. This clearly means that going up the column from stage n + 1 to n, the composition profile cannot approach the EA vertex (or the binary edge of EA-AC in the original composition space). For any projected liquid composition (x˜ n) lying to the right of the critical liquid composition (x˜ *), the region of y˜ n+1 lies closer to the EA vertex than y˜ n. Thus, it is defined that the critical composition region for this projected space is from the liquid critical composition to the EA vertex along the reaction equilibrium curve. For the entire reaction equilibrium surface, the critical composition region is defined as the liquid composition region at which the projected liquid contains more EA than its equilibrated vapor in all projected spaces. Determination of the Critical Composition Region. Only phase and chemical equilibrium data are needed to calculate the critical composition region. The calculation algorithm is summarized as follows: (1) Calculate phase and chemical equilibrium in projected triangles with constant AC, ET, and W fractions (xk’s). (2) Along the chemical equilibrium curve in each triangle, collect all projected liquid compositions (x˜ n) having larger EA mole fractions than their equilibrated vapor compositions (y˜ n). (3) For the collected projected liquid compositions in step 2, convert them into corresponding original compositions (xn ) (1 - xk)x˜ n) and plot them in the original composition space. This calculation procedure can be extended to any esterification system by changing EA to a new top product. Figure 5 shows the critical composition region for the ethyl acetate production system. The critical composition region along the EA-AC binary edge ends at xAC ) 0.49. It should be noted that this critical composition region does change with different phase and chemical equilibrium data. However, when those data are given, it does not change with different operating conditions such as different reflux ratios or different feed flow rates. So, it is uniquely determined by one set of thermodynamic data. Interpretation of a Critical Composition Region in the Original Composition Space. In reality, the original vapor (y) in phase equilibrium with the liquid lying within the critical composition region is more enriched in EA than the original liquid (x). However, if


Figure 7. Liquid composition profiles calculated from AspenPlus: total number of stages, 150; reactive stages, 2-150; upper and lower feed stages, 20 and 130; external reflux ratio, 5.0. Figure 5. Critical composition region for the EA production system shown as a hatched area.

Figure 6. Projection of original vapor and liquid compositions onto the ET-EA-W triangle. Hatched region in Figures 6-9: critical composition region.

these two original compositions are projected into the triangle of EA, ET, and W in Figure 6, the projected vapor lies further away from the EA vertex than the projected liquid. For example, the original liquid composition (x) within the critical composition region is [0.1, 0.10111, 0.76501, 0.03388] for AC, ET, EA, and W. Its equilibrated original vapor composition (y) is [0.011787, 0.102752, 0.806017, 0.079443]. This original vapor has a larger EA fraction (0.806017) than its corresponding liquid (0.76501). However, if we project the liquid and vapor compositions into the ternary space of EA, ET, and W at xAC ) 0.1, the EA fraction in the projected liquid composition vector (0.76501/(1 - 0.1) ) 0.850011) is greater than that in the projected vapor composition vector (0.806017/(1 - 0.011787) ) 0.815631). This means that once the liquid composition vectors are

within the critical composition region, their equilibrated vapor compositions are digressing from the EA vertex. If this digression is described in the projected space as shown in Figure 4, the composition vectors cannot approach the EA vertex while moving up the column. So, it is impossible to approach the EA-AC binary edge of the original composition space if column profiles pass through this critical composition region. Figure 7 shows a composition profile calculated by AspenPlus in a reactive distillation column when all stages are assumed to be reactive except the condenser. Once the composition profile moves into the critical composition region with the external reflux ratio of 5, it does not hit the EA-AC vertex but digresses from the pure EA vertex. Barbosa (1987)10 also observed this digression from the EA vertex in terms of a saddle-type fixed point when excess AC is provided with a higher reflux ratio of 9.88. In normal situations, we have a nonreactive rectifying section above the upper feed stage to separate EA from the binary mixture of AC and EA. It is shown in Figure 8 that the critical composition region still prevents the composition profile from hitting the binary edge of AC-EA. The nonreactive rectifying profile also digresses from the EA vertex. This movement of composition profiles around the critical composition region provides an important design implication: to obtain pure products, the critical composition region must be avoided. In other words, the composition profile of the reactive middle section should reach the EA-AC binary edge at the upper feed stage before going into the critical composition region. Once it reaches the binary edge, it is possible to separate EA from the mixture of AC and EA in the nonreactive rectifying section. However, from the simulations, the composition profile cannot approach the EA-AC binary edge. Here, it is necessary to more deeply understand why the profile cannot reach this binary edge. On the other hand, the composition trajectory in the methyl acetate (MA) production system hits the MAAC binary edge in Figure 9. In the top nonreactive section, MA can be purified from the MA-AC binary mixture. The critical composition region (hatched area) for the MA production system in this figure is smaller than that for the EA production system in Figure 5. The

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Figure 10. Single-stage calculation under the pinch for the EA production at a projected triangle with the constant AC fraction xAC ) 0.49. *: combined point of x˜ n and x˜ D ) combined point of y˜ n+1 and ν˜ . Figure 8. Simulation results for liquid composition profiles with a nonreactive rectifying section: total number of stages, 150; reactive stages, 20-150; upper and lower feed stages, 20 and 130; external reflux ratio, 5.0.

AC binary edge ends at the AC mole fraction of 0.49, as shown in Figure 5. So, the projected space with this AC fraction is drawn to determine whether composition vectors can move into the EA vertex of the projected composition space. Tray-by-tray calculations in the projected triangle are considered below. If the tetrahedral composition space is cut by the AC fraction of 0.49, the tray-by-tray material balance from eq 4 becomes

V ˜ n+1y˜ n+1 + ξnν˜ ) 0.51Lnx˜ n + Dx˜ D

Figure 9. Critical composition region for the MA production system and liquid composition trajectory (solid line) calculated by using AspenPlus when the external reflux ratio is 2.0.

phase and chemical equilibrium data in ref 13 are used for the MA production system. Feasibility of a Double-Feed Reactive Distillation Column In this section, on the basis of the concept of the critical composition region, a quantitative feasibility evaluation procedure will be derived for double-feed reactive distillation systems. Then the following questions can be answered: (1) For the ethyl acetate production system, why is it difficult to produce pure EA from a stoichiometric feed? (2) How can the methyl acetate production system produce pure MA? Geometric Interpretation of Feasibility for Ethyl Acetate Production. While going up the column, the column composition profile should avoid the critical composition region if pure EA and pure W are to be produced at the top and bottom. Instead, it should approach the binary edge of EA-AC first. The part of the critical composition region that lies near the EA-

(10)

xAC ) 0.49 means that xn,k, xD,k, and xE,k are 0.49, 0, and 1, respectively, in eqs 6-8. Consider an arbitrary liquid composition of x˜ n ) [0.036229, 0.85002, 0.113751] for ET, EA, and W lying on the reaction equilibrium curve. Its original liquid composition is xn ) [0.49, 0.018477, 0.43351, 0.058013] for AC, ET, EA, and W. The ratio 0.51Ln/D can be determined by calculating the ratio |* - x˜ D|/|x˜ n - *|. The ratio of 0.51Ln/D is 9.0 in Figure 10 under a potential pinch condition of y˜ n+1 ) y˜ n ) [0.022218, 0.866367, 0.111416]. The original vapor composition is yn ) [0.156623, 0.018738, 0.730674, 0.093965] for AC, ET, EA, and W. Since the ratio of 0.51Ln/D is 9.0, the internal reflux ratio (Ln/D) is 17.6. To approach the pure EA vertex in this projected space or the binary edge of EA-AC in the original composition space, the internal reflux ratio should be greater than 17.6. Then, with the same original liquid and vapor composition (xn and yn), the single stage calculation is performed in the projected triangle with the constant EA fraction. The mole fraction of EA is 0.43351 in xn and eq 4 is reduced to

V ˜ n+1y˜ n+1 + Ex˜ E ) 0.566 49Lnx˜ n + ξnν˜

(11)

Here, ν˜ is [1, 1, -1] for AC, ET, and W. In eqs 4-8, xD,EA is 1, xE,EA is 0, and xn,EA is 0.43351. Figure 11 shows the construction for a one-stage calculation under the pinch condition. With xn,EA ) 0.43351, x˜ n is [0.864976, 0.032616, 0.102408] for AC, ET, and W. In the same way, y˜ n+1 ) y˜ n ) [0.581537, 0.069573, 0.348890] is calculated with yn,EA ) 0.730674. y˜ n+1 and x˜ E are connected by a straight line, and another straight line connects x˜ n and ν˜ . Then, the combined point (*) of x˜ n and ν˜ (or y˜ n+1 and x˜ E) is determined as being the intersection point of these two straight lines. So, the


Figure 11. Single-stage calculation under the pinch for the EA production at a projected triangle with the constant EA fraction xEA ) 0.433 51.

ratio V ˜ n+1/E is equal to |* - x˜ E|/|* - y˜ n| and is 0.48. In other words, Vn+1/E is calculated as 1.78 when the vapor composition is inserted at stage n + 1 as in V ˜ n+1/E ) (1 - yn+1,EA)Vn+1/E ) (1 - 0.730674)Vn+1/E. One remarkable thing is that the ratio of Vn+1/E should be less than 1.78 if the vapor composition at stage n (y˜ n) is to be placed closer to the AC vertex than the vapor composition at stage n + 1 (y˜ n+1). It should be noted that the ratio of Vn+1/E is equal to the internal reflux ratio of Ln/D, since Vn+1 is equal to Ln under the stoichiometric feed flows of D ) E ) F for the isomolar reaction (νT ) 0) in eq 2. So, the internal reflux ratio should be smaller than 1.78 to approach the AC vertex as going up the column from stage n + 1 to stage n. Approaching the EA or AC vertexes in these two projected triangles means moving into the EA-AC edge in the original tetrahedral composition space. So, the two internal reflux ratios should have a permissible range in order to approach the EA-AC edge, since the two single-stage calculations are performed from the same liquid and vapor compositions (xn and yn). To approach the EA vertex in Figure 10, the internal reflux ratio of Ln/D should be larger than 17.6. However, in Figure 11, Vn+1/E (which equals Ln/D) should be smaller than 1.78 to reach the AC vertex. This is impossible. Thus, the composition trajectory cannot approach the EA-AC edge in the original composition space because the upper bound of the internal reflux ratio, 1.78 in Figure 11, is not greater than the lower bound of the internal reflux ratio, 17.6 in Figure 10. This is why the combined reaction and distillation is not feasible for producing pure ethyl acetate and water from a stoichiometric ratio of acetic acid and ethanol in a double-feed column. Geometric Interpretation of Feasibility for Methyl Acetate Production. If the same analysis is carried out for the methyl acetate production system, it can be easily understood that a double-feed reactive distillation column can produce pure MA. The critical composition region that lies along the MA-AC edge of the tetrahedral composition space ends at the AC fraction of 0.23 in Figure 9. With this fraction, the projected material balance in eq 4 becomes

V ˜ n+1y˜ n+1 + ξnν˜ ) 0.77Lnx˜ n + Dx˜ D

(12)

Consider the arbitrary choice of x˜ n at [0.039256,

Figure 12. Single-stage calculation under the pinch for the MA production at a projected triangle with the constant AC fraction xAC ) 0.23.

Figure 13. Single-stage calculation under the pinch for the MA production a projected triangle with the constant MA fraction xMA ) 0.654 51.

0.850013, 0.110731] for MT, MA, and W. The original liquid composition for this choice is xn ) [0.23, 0.030227, 0.65451, 0.085263] for AC, MT, MA, and W. Its vapor counterpart is y˜ n ) y˜ n+1 ) [0.041118, 0.891156, 0.067727] (for which the original vapor composition is yn ) [0.015869, 0.040466, 0.877014, 0.066652]) in Figure 12. First, x˜ n and x˜ D are connected by a straight line, and then y˜ n+1 and ν˜ are also connected by another straight line. The combined point (*) of each of these pairs of projected composition points lies on the intersection of the two lines. The ratio of 0.77Ln/D is equal to |* - x˜ D|/ |x˜ n - *| and is calculated as 2.53 in Figure 12. So, the lower internal reflux ratio, Ln/D, is equal to 3.30. Figure 13 shows a geometrical construction of a single-stage calculation with the constant MA fraction of 0.65451 with the same original liquid and vapor compositions as in Figure 12. For this projected space, the balance equation becomes

V ˜ n+1y˜ n+1 + Ex˜ E ) 0.34549Lnx˜ n + ξnν˜

(13)

x˜ n and y˜ n ) y˜ n+1 in this projected space are equal to [0.665721, 0.087490, 0.246789] and [0.129030, 0.329029, 0.541947], respectively, for AC, MT, and W. The combined point (*) of x˜ n and ν˜ is equal to that of y˜ n+1 and ˜ n+1/E (which equals |* - x˜ E|/|y˜ n - *|) x˜ E in Figure 13. V is 0.586. If yn+1,MA ) 0.877 014 is substituted into the

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Figure 14. Single-stage calculation in the full composition space with the same liquid and vapor compositions as in Figures 12 and 13.

ratio (V ˜ n+1/E ) (1 - yn+1,MA)Vn+1/E), then the internal reflux ratio of Vn+1/E ()Ln/D) becomes 4.76. So the internal reflux ratio should be smaller than 4.76 to put y˜ n closer to the AC vertex than y˜ n+1. The lower bound for the internal reflux ratio is 3.30 from Figure 12, and the upper bound for the internal reflux ratio is 4.76 from Figure 13. Since there is a permissible range between these two internal reflux ratios (specifically, between 3.30 and 4.76), a composition profile can approach the MA-AC binary edge from xn and yn. Geometric Interpretation of Upper and Lower Bounds on the Reflux Ratio in the Full Tetrahedral Composition Space. In this section, the upper and lower internal reflux ratios (Ln/D and Vn+1/E) are interpreted by switching from the projected space to the original composition space. Figure 14 shows xn and yn from Figures 12 and 13 placed in the original composition space. Rearranging eq 3 yields

Vn+1yn+1 + ExE ) Lnxn + Dδn

(14)

where

δn )

DxD - νξn D

(15)

δn is a difference point that combines the distillate composition and the stoichiometric vector. It lies on the straight line that passes through xD and is parallel to the stoichiometric line.9,14 This straight line is called a difference point trajectory in Figure 14. According to eq 14, one straight line connecting yn+1 and xE intersects the other straight line joining xn and δn. With yn+1 ) yn, this intersection point (or combined point (*) in Figure 14) and δn are determined simultaneously. First, yn+1 and xE are joined with a straight line. Then, a straight line is drawn from xn such that it passes through the difference point trajectory and the line connecting yn+1 and xE. The intersections between this new line and the two given lines are δn and the intersection point (*), respectively. The geometrical verification for the uniqueness of the new line is available in Appendix I. The ratio of |* - δn|/|* - xn| is equal to Ln/D and is calculated as 3.30. Another ratio, |* - xE|/|* - yn|, is equal to Vn+1/E and is 4.76. These values are identical to the calculated values (3.30 and 4.76) from the two projected triangles in Figures 12 and 13. However, these two reflux ratios (Ln/D and Vn+1/E) should be equal to satisfy the material balance constraint in eq 14 if the

Figure 15. Movement of composition vectors: (a) upper internal reflux ratio g lower internal reflux ratio; (b) upper internal reflux ratio e lower internal reflux ratio.

pinch situation of yn+1 ) yn is assumed. So, x˜ n (or xn) can be considered a potential pinch point in Figures 1013 if single-stage calculations with y˜ n ) y˜ n+1 are carried out. To satisfy the material balance in eq 14, Vn+1/E is reduced from 4.74 and Ln/D is increased from 3.30. So, 3.30 and 4.74 become the lower and upper limits on the internal reflux ratio. While increasing and decreasing these two ratios, yn+1 can lie further away from the MA-AC binary edge than yn in Figure 15a. Thus, these vapor composition vectors move into this binary edge as the column composition profiles go up through the reactive middle section of the column. If the upper bound of Vn+1/E is smaller than the lower bound of Ln/ D, as in Figures 10 and 11, the liquid composition vectors move away from the EA-AC edge as stepping up from stage n to n - 1 in Figure 15b. The geometric consideration for Figure 15 is found in Appendix I. Generalization of the Procedure for Determining Feasibility. In this section, a general algorithm will be proposed for determining feasibility by using liquid and equilibrated vapor compositions on the reaction equilibrium curve and the vapor curve. To derive the general algorithm, consider the MA production system. The two single-stage calculations in Figures 12 and 13 at a potential pinch situation cannot guarantee that all composition vectors move into the binary edge of AC and MA while moving up the column. To always approach the binary edge, any single-stage calculation at arbitrary x˜ n and y˜ n should indicate that the upper internal reflux ratio is greater than the lower internal reflux ratio. This should happen in at least one projected triangle over an entire range of a reaction equilibrium curve so that the composition vectors coming from other projected triangles can approach the MA-AC edge. From an operational aspect, to reach this MA-AC binary edge (or MA and AC vertexes in the two projected triangles), the permissible range of the internal reflux ratio at the pure MA vertex should overlap with the permissible ranges of internal reflux ratios evaluated at all other x˜ n’s and y˜ n’s. Figure 16 shows that the upper bounds of internal reflux ratios are larger than the lower bounds of internal reflux ratios over the entire range of liquid compositions in the projected space with the constant AC fraction of 0.25. The permissible internal reflux ratio range near the binary edge of MA-AC (0.16 e rn e 4.42) has a common range with the permissible


Figure 16. Upper and lower bounds of internal reflux ratios for the MA production at a projected triangle with the constant AC fraction xAC ) 0.25. The distillate composition (xD) is [0.001, 0.008, 0.99, 0.001] for AC, MT, MA, and W. Hereafter, solid and dotted lines are for the upper and lower bounds on the internal reflux ratios. A rectangle represents the common range of the two bounds with the internal reflux range at the MA mole fraction of 0.99.

Figure 17. Upper and lower bounds of internal reflux ratios for the EA production at a projected triangle with the constant AC fraction xAC ) 0.49. The distillate composition (xD) is [0.001, 0.008, 0.99, 0.001] for AC, ET, EA, and W.

internal reflux ratio ranges at all other liquid compositions in Figure 16. This happens in many constant-ACprojected triangles with xAC > 0.25. Note that the search for this projected space starts from the projected space with xAC ) 0.23, since the critical composition region ends at xAC ) 0.23 along the MA-AC edge in Figure 9. For the ethyl acetate production system, only the limited overlapped region exists with x˜ EA > 0.98 when the tetrahedral composition space is cut with xAC ) 0.49, as shown in Figure 17. This limited overlapped range is repeated at every projected space from xAC ) 0.49 to 0.99. Thus, the EA-AC edge is not accessible and a double-feed reactive distillation column cannot produce relatively pure EA at the top and W at the bottom. It should be noted that these lower and upper bounds of the internal reflux ratio can be calculated in terms of eq 4 even if a given top product purity is slightly less than 100% of an acetate, as in Figures 16 and 17 (MA or EA mole purity ) 99%). For a fixed feed and product specification with liquid and equilibrated vapor compositions, the internal reflux ratios (Ln/D) and reaction extents (ξn/D) are calculated, since these two unknowns are determined by two independent component-wise balance equations in eq 4. The single-stage calculation is very simple, since only phase equilibrium information is needed and only two algebraic equations need to be solved for two constant acid and acetate projected spaces.

Summary of the Feasibility Evaluation Algorithm. The procedure of evaluating the feasibility of quaternary reactive systems in a double-feed column can be generalized: (1) For given phase and reaction equilibrium data, calculate a critical composition region. If the critical composition region covers all of the binary edge between an upper feeding reactant (acetic acid in the above examples) and a desired top product (ethyl or methyl acetate in the above examples), then the double-feed column cannot produce a pure product. Otherwise, the evaluation procedure proceeds to step 2. (2) At the compositions outside the critical composition region, single-stage calculations are carried out in two projected triangles with the same equilibrated liquid and vapor compositions from the original composition space. These single-stage calculations start with the constant upper feed composition that terminates the critical composition region (xAC ) 0.49 for EA production and xAC ) 0.23 for MA production). Then xAC is increased while the single-stage calculations are repeated. All upper bounds on the internal reflux ratio should be greater than their corresponding lower bounds on the internal reflux ratio over the entire range of the reaction equilibrium curve in the projected triangles. If this is not found up to xAC ) 1, pure products cannot be produced in a double-feed column. Otherwise, go to step 3. (3) An overlapped range between the two bounds of the internal reflux ratios should exist in at least one projected triangle with a constant fraction of the upper feed. If the internal reflux range of the binary edge has a common range with the ranges of all other liquid compositions in the same projected triangle, then a double-feed reactive distillation column can produce the specified pure products. Otherwise, it is not feasible to produce the desired products. Application of the Feasibility Algorithm to Methyl Formate and Isopropyl Acetate Reactive Distillation Systems Methyl formate (MF) is produced from formic acid (FA) and methanol (MT). Water (W) is a byproduct. The reaction equilibrium constant (Kx) is 5 on the basis of the mole fractions of the reacting species.15 There are two binary azeotropes at 3 bar. One is a minimum boiling azeotrope between MF and MT, and the other is a maximum boiling azeotrope between FA and W. Here, FA, MT, MF, and W are R1, R2, P1, and P2, respectively, in eq 1. The critical composition region for this reactive mixture shown in Figure 18 (the very small solid black area near the MF vertex) is much smaller than that for the MA production system in Figure 9. The critical composition region ends at the FA fraction of 0.08 along the binary edge of FA-MF. The rigorously calculated liquid profile hits the FA-MF binary edge and then terminates at the pure MF vertex. Starting from the projected space with xFA ) 0.08, the lower and upper bounds of the internal reflux ratio are calculated over the entire range of the reaction equilibrium and vapor curves. At the projected space with xFA ) 0.11, the upper reflux ratio bounds are greater than the lower reflux ratio bounds all over the range of the projected mole fraction of MF in Figure 19. Moreover, the permissible range of the reflux ratio at the MF vertex has part of its range in common with other

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Table 1. Wilson Binary Parameters (1) (2) (3) (4) (5) (6)

AC (i)-IPA (j): aij ) 0.0, aji ) 0.0, bij ) -445.5432, bji ) 422.6946 AC (i)-IPAC (j): aij ) 9.3827, aji ) -7.2101, bij ) -3463.8979, bji ) 2499.3523 AC (i)-W (j): aij ) 0.0, aji ) 0.0, bij ) -470.315687, bji ) 83.4398593 IPA (i)-IPAC (j): aij ) 0.0, aji ) 0.0, bij ) -123.3902, bji ) -99.2611 IPA (i)-W (j): aij ) -1.4485, aji ) 1.4485, bij ) -543.86, bji ) -602.5917 IPAC (i)-W (j): aij ) 0.0, aji ) 0.0, bij ) -1418.00376, bji ) -439.581928

Figure 18. Critical composition region for the methyl formate production system (hatched region near the MF vertex) and liquid composition trajectory (solid line) calculated by using AspenPlus when the external reflux ratio is 2.5.

Figure 19. Upper and lower bounds of internal reflux ratios for the MF production at the projected triangles with xFA ) 0.11. The distillate composition (xD) is [0.0, 0.0, 1.0, 0.0] for FA, MT, MF, and W. Rectangles represent the common range with the internal reflux range at the MF mole fraction of 0.99.

permissible reflux ranges at all other liquid compositions. This happens continuously at many projected triangles with xFA > 0.11. Thus, the MF-FA binary edge can be approached and pure MF can be produced at the top and W at the bottom. Isopropyl acetate (IPAC) is produced by the dehydration of acetic acid (AC) with 2-propanol (IPA) under an acid catalyst. So, R1 is AC, R2 is IPA, P1 is IPAC, and P2 is W. The reaction equilibrium constant (Keq) is 8.7.16 There are four minimum-boiling azeotropes, one of which is a ternary azeotrope among IPA, IPAC, and W and has the lowest boiling point. The other three azeotropes are binary. The phase equilibrium data are summarized in Appendix II. The critical composition region in Figure 20 is very large compared to that of the EA production system in Figure 5. The critical composition region is spread all along the AC-IPAC edge, so it can be easily determined that a double-feed reactive distillation column cannot produce pure IPAC from a stoichiometric feed of AC and IPA.

Figure 20. Critical composition region for the IPAC production system.

Conclusions In this work, a feasibility method was proposed for double-feed reactive distillation systems on the basis of the understanding of the interaction between reaction and distillation. So, a critical composition region is introduced to help us understand this reaction-distillation interaction by using phase and reaction equilibrium data. Once the composition trajectories move into the critical composition region, it is impossible to produce a pure product at the top of the column. Instead, to obtain desired reaction conversion and product purity, the composition profiles should approach the binary edge of an upper feed reactant and a pure top product as the column composition profile moves up through the reactive middle section of the column. The feasibility of a double-feed reactive distillation column is quickly evaluated by carrying out single-stage calculations with potentially pinched liquid and vapor compositions in two projected triangles before trying numerous experiments and rigorous calculations. It is feasible to produce pure products in a double-feed column: (1) if the lower bounds of the internal reflux ratios are smaller than the upper bounds of the internal reflux ratios over the entire range of projected liquid compositions in a certain projected space and (2) if the permissible range of the internal reflux ratio at the binary edge has a common range with the permissible ranges at all other liquid compositions. For future work, this feasibility evaluation procedure will be extended into single non-isomolar reactions and multireaction systems.

Acknowledgment Part of this work was done during the author’s stay at RWTH Aachen as an Alexander von Humboldt


Fellow. The author is grateful for the support of the Alexander von Humboldt Foundation and thankful to Prof. Marquardt for his permission to use the DREIECK visualization code. The author is also thankful to the first reviewer of this article for his valuable comments and James Chin for his geometric interpretation. Nomenclature D ) distillate molar flow rate D ˜ ) distillate molar flow rate in a projected space as defined in eq 7 E ) upper feed molar flow rate E ˜ ) upper feed molar flow rate in a projected space as defined in eq 8 Keq ) reaction equilibrium constant ()KxKγ) Ln ) liquid flow rate at stage n L ˜ n ) liquid flow rate at stage n in a projected space as defined in eq 6 P1, P2 ) reaction products in eq 1 R1, R2 ) reaction reactants in eq 1 rn ) internal reflux ratio (Ln/D) Vn+1 ) vapor flow rate at stage n + 1 V ˜ n+1 ) vapor flow rate at stage n + 1 in a projected space as defined in eq 5 xD ) distillate composition vector xD,k ) mole fraction of excluded component k in xD x˜ D ) distillate composition vector in a projected space [xD/ (1 - xD,k)] xE ) upper feed composition vector xE,k ) the fraction of excluded component k in xE x˜ E ) upper feed composition vector in a projected space [xE/(1 - xE,k)] xk ) liquid mole fraction of excluded component k xn ) liquid composition vector at stage n xn,k ) liquid mole fraction of excluded component k at stage n x˜ n ) liquid composition at stage n in a projected space [xn/ (1 - xn,k)] yn+1 ) vapor composition vector at stage n + 1 yn+1,k ) vapor mole fraction of excluded component k at stage n + 1 y˜ n+1 ) vapor composition vector at stage n + 1 in a projected space [yn+1/(1 - yn+1,k)] Greek Letters δn ) difference point at stage n ν ) stoichiometric coefficient vector ν˜ ) stoichiometric vector in a projected space νT ) sum of stoichiometric coefficients ξn ) accumulated molar reaction extent from the top to stage n Abbreviations AC ) acetic acid ET ) ethanol EA ) ethyl acetate W ) water MT ) methanol MA ) methyl acetate FA ) formic acid MF ) methyl formate IPA ) isopropyl alcohol IPAC ) isopropyl acetate

Appendix I: Uniqueness for the Line Intersecting Two Lines in a Skew Position In Figure 14, the difference point trajectory and the balance line between xE and yn are skew to one another. If a plane is drawn by including xn and the line

Figure 21. Uniqueness for the line intersecting two lines in a skew position. Table 2. Antoine Equation Coefficients and Dimerization Constant17 AC IPA IPAC W

A

B

C

23.3618 25.3358 21.7798 23.4776

-4457.8300 -4628.9600 -3307.730 -3984.920

-14.699 -20.514 -39.485 -39.724

log10(KD) ) -12.5454 + 3166.0/T (T in kelvin)

connecting xE and yn (L1), this plane (P1) intersects another plane (P2) including xn and the difference point trajectory (L2) at a single line. In other words, these two planes form only one line at their intersection, as shown in Figure 21. This intersection line (L3) connects the intersection point (*) and δn. So, the intersection point (*) and δn are uniquely determined. If these two lines, L1 and L2, are not skew but parallel to each other, two cases can occur: (1) an infinite number of intersection lines if L1 and L2 are parallel to each other and lie on the same plane as xn and (2) no intersection line if L1 and L2 are parallel but do not lie on the same plane as xn. In Figure 15a, yn+1 was taken from the same plane (P1) as yn lies on. So, the position of the difference point (δn) is not changed. But if yn+1 is chosen outside plane P1, the difference point is placed at a different location along L2 and its corresponding reaction extent (ξn) changes. During this operation, it is also observed that Vn+1/E decreases and Ln/D increases. In the same way, xn-1 was chosen from the plane of P1 in Figure 15b. So, δn does not change. Otherwise, its position will be changed along L2. Appendix II: Physical Properties Data for the Isopropyl Production System The Wilson equation is used for the calculations of the liquid activity coefficients (Table 1), and they are also estimated by AspenPlus. The Wilson parameters are estimated as Λij ) exp(aij + bij/T), where Λii ) 1. The Antoine equation is ln(Psat) ) A + B/(C + T), where pressure is in pascal and temperature is in kelvin (Table 2). The vapor phase dimerization of acetic acid is included. Literature Cited (1) Lee, J. W.; Westerberg, A. W. Graphical Design Applied to the MTBE and Methyl Acetate Reactive Distillation Processes. AIChE J. 2001, 47, 1333. (2) Okasinski, M. J.; Doherty, M. F. Effects of Unfavorable Thermodynamics on Reactive Distillation Column Design. Inst. Chem. Eng. Symp. Ser. 1997, 2 (142), 695. (3) Lee, J. W.; Hauan, S.; Lien, K. M.; Westerberg, A. W. A Graphical Method for Designing Reactive Distillation Columns Is The Ponchon-Savarit Diagram. Proc. R. Soc. London, Ser. A 2000, 456, 1953.

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(4) Lee, J. W.; Hauan, S.; Lien, K. M.; Westerberg, A. W. A Graphical Method for Designing Reactive Distillation Columns IIs The McCabe-Thiele Diagram. Proc. R. Soc. London, Ser. A 2000, 456, 1965. (5) Lee, J. W.; Westerberg, A. W. Visualization of Stage Calculations in Ternary Reacting Mixtures. Comput. Chem. Eng. 2000, 24, 639. (6) Hauan, S.; Omtveit, T.; Lien, K. M. Analysis of Reactive Separation Systems. AIChE Annual Meeting 1996, Chicago, Nov; AIChE: 1996; paper 5f. (7) Hauan, S.; Westerberg, A. W.; Lien, K. M. Phenomena Based Analysis of Fixed Points in Reactive Separation Systems. Chem. Eng. Sci., 2000, 55, 1053. (8) Stichlmair, J. G.; Fair, J. R. Distillation: Principle and Practice; Wiley-VCH: New York, 1998. (9) Lee, J. W.; Hauan, S.; Lien, K. M.; Westerberg, A. W. Difference Points in Extractive and Reactive Cascade IIs Generating Design alternatives by lever rule for reactive systems. Chem. Eng. Sci. 2000, 55, 3161. (10) Barbosa, D. Distillation of Reactive Mixtures. Ph.D. Dissertation, University of Massachusetts, Amherst, MA, 1987. (11) Agreda, V. H.; Partin, L. R.; Heise, W. H. High Purity Methyl Acetate via Reactive Distillation. Chem. Eng. Prog. 1990, 86, 40.

(12) Kang, Y. W.; Lee, Y. Y.; Lee, W. K. Vapor-Liquid Equilibria with Chemical Reaction EquilibriumsSystems Containing Acetic Acid, Ethyl Alcohol, Water and Ethyl Acetate. J. Chem. Eng. Jpn. 1992, 25, 649. (13) Song, W.; Venimadhavan, G.; Manning, J. M.; Malone, M. F.; Doherty, M. F. Measurement of Residue Curve Maps and Heterogeneous Kinetics in Methyl Acetate Synthesis. Ind. Eng. Chem. Res. 1998, 37, 1917. (14) Hauan, S.; Ciric, A. R.; Westerberg, A. W.; Lien, K. M. Difference Points in Extractive and Reactive Cascades. IsBasic Properties and Analysis. Chem. Eng. Sci. 2000, 55, 3145. (15) Bessling, B.; Schembecker, G.; Shimmrock, K. H. Design of Processes with Reactive Distillation Line Diagrams. Ind. Eng. Chem. Res. 1997, 36, 3032. (16) Lee, L.; Kuo, M. Phase and Reaction Equilibria of the Acetic Acid-2-Propanol-Isopropyl Acetate-Water System at 760 mmHg. Fluid Phase Equilib. 1996, 123, 147. (17) Venimadhavan, G.; Malone, M. F.; Doherty, M. F. Bifurcation Study of Kinetic Effects in Reactive Distillation. AIChE J. 1999, 45, 546.

Received for review December 6, 2001 Revised manuscript received April 19, 2002 Accepted June 19, 2002 IE0109859

Feasibility Studies on Quaternary Reactive Distillation Systems

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