Ind. Eng. Chem. Res. 2007, 46, 3255-3264
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‘Feasible Products for Double-Feed Reactive Distillation Columns† Sagar B. Gadewar Gas Reaction Technologies (GRT), Inc., 861 Ward Dr., Santa Barbara, California 93111
Michael F. Malone Department of Chemical Engineering, UniVersity of Massachusetts, Amherst, Massachusetts 01003
Michael F. Doherty* Department of Chemical Engineering, UniVersity of California, Santa Barbara, California 93106
We have developed a quick and effective method to predict the feasible products from a double-feed reactive distillation column. The method relies on the prediction of the pinch point for the middle section, using a cross-flow arrangement of vapor-liquid continuously stirred tank reactors (CSTRs). The cross-flow model substantially simplifies the prediction of the compositions at both ends of the middle section of a two-feed column, which are then used to predict the feasible distillate and bottoms compositions, using existing methods. The cross-flow arrangement can also be used to predict minimum and maximum flows for double-feed reactive distillation columns. Introduction Separation accompanied by chemical reaction, has been a constant theme of research throughout M. M. Sharma’s career. His first published paper, coauthored with his doctoral thesis advisor, P. V. Danckwerts, on absorption with chemical reaction1 remains a classic in the field. During the 1980s and 1990s, he published many papers on the themes of “separations through reactions” (for example, see Gaikar and Sharma2) and reactive distillation.3 The theme continued into the new century with his encyclopedic description of potential applications of reactive distillation in practice, which was coauthored by Mahajani.4 Many of these applications are practiced in two-feed, countercurrent cascade reactive distillation columns, which are notoriously difficult to design. In particular, it is difficult to rapidly assess the feasible products that can be obtained from such devices. In this paper, we develop an approximate, but accurate, solution to the feasibility problem. Double-feed columns are often used in extractive distillation. Knapp and Doherty5 published an approach based on bifurcation theory to determine minimum entrainer flows for double-feed extractive distillation columns. They found that, in addition to a minimum reflux, every extractive distillation exhibits a maximum reflux, above which the desired separation is impossible, and a minimum entrainer flow rate, below which the separation is also impossible. Petlyuk and Danilov6 described a study of the fixed points in the middle section of a doublefeed column. They also developed a method to determine the minimum entrainer flow rate and the minimum reflux ratio. A double-feed reactive distillation column is used to achieve good contact between light-boiling reactants and heavy-boiling reactants. A heuristic by Bessling et al.7 suggests that a doublefeed column is desirable when one or both products are saddles in the concentration profile (usually determined from residue * To whom correspondence should be addressed. Tel.: (805) 8935309. Fax: (805) 893-4731. E-mail address:
[email protected]. † This paper is dedicated to Professor M. M. Sharma, in recognition of his many outstanding contributions to chemical engineering research, education, and practice.
curve maps for simple reactive distillation). Barbosa and Doherty8 described a method for calculating minimum reflux ratios for double-feed reactive distillation columns in the limit of phase and reaction equilibrium. Methods for double-feed nonreactive distillation columns were adapted for reactive distillation columns, using reaction invariant compositions. Chin and Lee9 published a method for the design of double-feed reactive distillation columns to determine feasible reflux ratios and entrainer flow rates for a specified reaction extent and product purity. A method to determine feasible product compositions from a single-feed reactive distillation column was developed by Chadda et al.10 They determined that the product composition region, at finite rates of reaction, are bounded by the limits of minimum reflux on one side of the region and minimum number of stages at the other. They exemplified the methodology on a ternary reaction mixture undergoing a single reversible reaction. Although such an approach is useful for mapping the feasible products for a double-feed reactive distillation column, the methodology is cumbersome and difficult to extend to more than three components or to multiple reactions. A feasibility method described by Lee11 determines if pure products can be achieved in a double-feed reactive distillation column. A bifurcation-based approach can be used to track the sharp split products from a single-feed reactive distillation column.12 The approach is based on the hypothesis that the compositions achieved in an isobaric flash cascade lie in the feasible product regions for continuous reactive distillation. The fixed points for an isobaric flash cascade are hypothesized to correspond to the products from a single-feed reactive distillation column. This method is applicable to multiple reactions with many components. Another important advantage of the method is the noniterative calculations needed to find the feasible products. A model that calculates the profile for the middle section of a two-feed reactive distillation column was published by Gadewar et al.13 Chadda et al.14 extended the isobaric flash cascade approach to double-feed reactive distillation columns, and also to hybrid reactive distillation columns (consisting intermixed sections of reaction and no-reaction within the column), using
10.1021/ie060867r CCC: $37.00 © 2007 American Chemical Society Published on Web 03/08/2007
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kf xi,j-1 - xi,j + φ(yi,j+1 - yi,j) + Da (ν - νTxi)r(xj) ) 0 kf,ref i (for i ) 1, ..., c - 1; j ) 1, ..., N) (1) At a pinch point in the cascade, (xi,j-1 - xi,j) f 0 as j f ∞, which implies that (yi,j-1 - yi,j) f 0, and, in turn, means that (yi,j - yi,j+1) f 0. At a fixed point xˆ , therefore, eq 1 gives
Da(νi - νTxˆ i)r(xˆ ) ) 0
Figure 1. Schematic of a double-feed reactive distillation column.
(for i ) 1, ... , c - 1)
(2)
From eq 2, for all 0 < Da < ∞, one of the fixed points (the pinch point, if it is on the profile) always lies at the pole point xˆ i ) νi/νT (for i ) 1, ..., c - 1). All other fixed points (or pinch points) lie on the reaction equilibrium surface:
r(xˆ ) ) 0
(3)
This means that the pinch point is independent of the Damko¨hler number (Da) and lies on the equilibrium surface, even when the cascade operates in the kinetic regime. The trajectory, of course, will be dependent on the specific value of Da. Equation 1 can be formulated in terms of reaction invariants. For example, consider the reaction A + B h C + D. Choosing C as the reference component, the reaction invariant liquid mole fractions are X1 ) xA + xC and X2 ) xB + xC. Using eq 1, we get the following:
Xi,j-1 - Xi,j + φ(Yi,j+1 - Yi,j) ) 0
(for i ) 1, ..., c - 2) (4)
Writing overall material balances for the cascade between the first stage and the stage where the pinch point occurs (as shown by the bounding box in Figure 3a), we get Figure 2. Schematic of a counter-current cascade of two-phase continuously stirred tank reactors (CSTRs).
a counter-current cascade of vapor-liquid continuously stirred tank reactors (CSTRs) to represent the middle section of a double-feed column and isobaric flash cascade to represent the stripping and rectifying sections. Unlike the isobaric flash cascade, which requires sequential stage-to-stage calculation, the counter-current cascade equations must be solved simultaneously, making it difficult to converge to solutions.13 Therefore, there is a need to develop a simple configuration that represents the middle section of a double-feed column. The objective of this article is to describe a cross-flow arrangement that simplifies the determination of feasible products from a counter-current cascade of vapor-liquid CSTRs. The method described here can be used to predict feasible splits for continuous double-feed reactive distillation that is not limited by the number of reactions or components. The method uses minimal information on the phase equilibrium between the components in the mixture, reaction kinetics, and feed specification. Middle Section of a Double-Feed Column A schematic for a double-feed column is shown in Figure 1. The middle section is equivalent to a counter-current cascade of vapor-liquid CSTRs, as shown in Figure 2. The model equations for a constant molar overflow counter-current cascade are given by Gadewar et al.,13 which, at steady state, for any stage j, are given as
Xi,0 - φYi,1 ) Xˆ i - φYˆ i
(for i ) 1, ..., c - 2)
(5)
Now, writing overall material balances for the cascade between the stage where the pinch point occurs and the last stage of the cascade (as shown by the bounding box in Figure 3b), we get
Xi,N - φYi,0 ) Xˆ i - φYˆ i
(for i ) 1, ..., c - 2)
(6)
From eqs 5 and 6, we write
Xi,0 - φYi,1 ) Xi,N - φYi,0 ) Xˆ i - φYˆ i (for i ) 1, ..., c - 2) (7) Equation 7 shows how the pinch point composition is related to the liquid and vapor product composition for the cascade. If the pinch point composition is known, the reaction-invariant product compositions can be determined for given feed compositions. Pinch point calculations for nonreactive distillation require solving a set of algebraic equations, as described by Julka and Doherty.15 These methods require product specifications and the objective is to find minimum flows (e.g., minimum reflux ratio). In feasibility studies, however, the product specifications are not known and the purpose of the study is to determine these product specifications. To determine the pinch points in a counter-current cascade, the model (eq 1) must be solved for a large number of stages (N > 50). The solutions to eq 1 are time-consuming and often do not converge. Therefore, we need a simple device configuration that can predict the pinch point compositions by sequential calculation from stage to stage,
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Figure 3. Schematic diagrams for (a) the section of counter-current cascade between the top stage and the pinch point and (b) the section of counter-current cascade between the pinch point and the bottom stage.
and the temperature is equal to the boiling point of the mixture. The overall mass balance for the jth stage is
Lj ) Lj-1
(8)
The material balance for the ith component on the jth stage is
Ljxi,j ) Lj-1xi,j-1 + V′j(yi,0 - yi,j) + νikfHjr(xj) (for i ) 1, ..., c - 1; j ) 1, ..., N) (9) where L and V′ are liquid and vapor molar flows; xi and yi are the liquid and vapor mole fractions for component i; νi is the stoichiometric coefficient for component i; kf is the forward rate constant with the dimensions of reciprocal time; Hj is the molar liquid holdup in the jth stage; and r(x) is the reaction driving force:
r(x) ) Figure 4. Schematic of the cross-flow arrangement.
without solving the rigorous counter-current cascade model for all stages simultaneously. Cross-flow Arrangement We use the cross-flow arrangement shown in Figure 4 to estimate the products from the middle section of a two-feed column. The heavy reactant is the liquid feed to the first stage in the cross-flow device. The light reactant is added to each stage of the device to achieve contact between the reactants. The objective of this device is to predict the pinch point in the middle section. The underlying assumption of the analysis is that the pinch point for the device in Figure 4 is an estimate of the pinch point in a counter-current cascade. Each flash device in Figure 4 is a two-phase CSTR with a liquid inlet and a vapor inlet; the reaction occurs only in the liquid phase. We consider a single reaction with equimolar chemistry and steady-state conditions, therefore, a vapor stream is removed from each stage at the same rate as the vapor inlet. The vapor inlet composition for each stage in the cross-flow device has the same composition (y0) and each stage is isobaric
(
1
a-ν aνi ∏ ∏ i K products reactants i
eq
i
)
(10)
where Keq is the chemical equilibrium constant and ai is the activity for component i. For liquid-phase reactions, activities are represented by the product of the activity coefficient γi and the liquid-phase composition xi: ai ) γixi. Substituting eq 8 in eq 9, we get
( )
kf Dajr(xj) kf,ref (i ) 1, ..., c - 1; j ) 1, ..., N) (11)
xi,j ) xi,j-1 + φ′j(yi,0 - yi,j) + νi
We have defined two dimensionless parameters: (i) φ′j ) V′j/Lj-1), which is the ratio of the vapor feed to the liquid feed in the jth stage. (ii) Daj ) [(Hj/Lj-1)/(1/kf,ref)], which is the Damkohler number for stage j. This is the ratio of the characteristic liquid residence time to a characteristic reaction time; kf,ref is the forward rate constant at a reference temperature Tref. No reaction occurs in the limit of Daj f 0, and reaction equilibrium is achieved as Daj f ∞. At intermediate values of Daj, the stage operates in the kinetically controlled regime. Assuming that each stage has the same residence time,
Da1 ) Da2 ) ... ) Daj ) ... ) DaN ) Da
(12)
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The liquid feed composition to the cross-flow scheme is the same as the composition of the liquid feed to the corresponding counter-current cascade. We use a vapor feed composition on the reaction equilibrium surface, because the pinch point that we want to estimate lies on the equilibrium surface (according to eq 3). The same vapor feed composition is specified for each stage in the cross-flow device. A convenient choice is to specify the vapor feed based on the feed to the corresponding countercurrent cascade. We choose the vapor feed in the cross-flow device to have the dew-point composition of a liquid mixture at reaction equilibrium that corresponds to the overall feed (of the corresponding counter-current cascade). For example, if the liquid feed of the counter-current cascade is acetic acid and the vapor feed is isopropanol with equal flow rates (i.e., φ ) 1), the vapor feed for the cross-flow device is in phase equilibrium with a liquid mixture obtained by reacting a mixture of 50 mol % acetic acid and 50 mol % isopropanol without vapor removal until it reaches reaction equilibrium (for instance, in a plugflow reactor (PFR)). An important decision in this configuration is the vapor addition policy. It is desirable that the total amount of vapor added in the cross-flow device be equal to that for the corresponding counter-current cascade (given that the liquid flow rate is the same in both devices). Vapor addition policies can be constructed such that (1) vapor addition rate increases with the stage number, (2) vapor addition rate is constant with increasing stage number, and (3) vapor addition rate decreases with increasing stage number. The cross-flow arrangement is solved until a fixed point is reached; therefore, the number of stages is not known a priori. If a constant value of vapor feed flow rate per stage is used in the model, it will correspond to a very large total vapor feed, which is unrealistic, compared to the counter-current cascade with a finite vapor feed rate. Furthermore, for a large number of stages, the pinch point for the cross-flow device will have a composition that is identical to the vapor feed composition. For the same reasons, a increasing vapor rate policy cannot be used. Therefore, only the decreasing vapor rate policy can be used. To construct a decreasing vapor addition policy, it must be dependent on the stage number. We use a decreasing vapor rate policy, according to the following relation:
φ′j )
j
(φ +φ 1)
(for j ) 1, ..., N)
(13)
Here, φ is the ratio of the molar vapor feed rate to the molar liquid feed rate in the corresponding counter-current cascade of vapor-liquid CSTRs. The vapor flow rate decreases with stage number, because φ/(φ + 1) < 1. Adding the vapor feed using this policy in the cross-flow device gives ∞
φ′j ) φ ∑ j)1
(14)
This policy ensures that the amount of vapor feed is equivalent to the corresponding counter-current cascade. The cross-flow device calculations are sequential, compared to the simultaneous calculations for a counter-current cascade. The iterative calculations for a counter-current cascade are time-consuming and often difficult to converge for a large number of stages. A single calculation for either the cross-flow device or the corresponding counter-current cascade gives a pinch point that is independent of Da. It is useful to predict the product compositions for a range of Da values, as shown in the following example.
Example 1: Production of Isopropyl Acetate Isopropyl acetate can be produced by the esterification of acetic acid with isopropanol. The reaction is
acetic acid (HOAc) + isopropanol (IPOH) h isopropyl acetate (IPOAc) + water (H2O) (15) The thermodynamic equilibrium constant has a value of Keq ) 8.7 in the boiling temperature range of interest.16 The kinetic model was taken from Manning17 for experiments performed on a heterogeneous catalyst (Amberlyst 15W, Rohm and Haas Company). A heterogeneous Langmuir-Hinshelwood/HougenWatson kinetic model used by Manning17 is
R ) Wk′s(aHOAcaIPOH - aIPOAcaH2O/Keq)/(1 + KHOAcaHOAc + KIPOHaIPOH + KIPOAcaIPOAc + KH2OaH2O)2 (16) where k′s is the apparent forward rate constant, given by
ln k′s ) 23.81 -
8253.6 T
(17)
Here, T is the temperature (given in Kelvin), k′s has units of moles reacted per mole of H+ ions per minute, W is the catalyst concentration (given in moles of H+ per mole of liquid mixture), and the adsorption equilibrium constants are assumed to be independent of temperature:
KHOAc ) 0.1976
(18a)
KIPOH ) 0.2396
(18b)
KIPOAc ) 0.147
(18c)
KH2O ) 0.5079
(18d)
The vapor-liquid equilibrium (VLE) is modeled using the Antoine vapor-pressure equation, the nonrandom two-liquid (NRTL) equation for activity coefficients, and including vaporphase dimerization of acetic acid. The physical property models are given by Gadewar et al.13 Consider a counter-current cascade (Figure 2) with pure acetic acid as the liquid feed and pure isopropanol as the vapor feed. If the vapor feed rate is equal to the liquid feed rate, φ ) 1. Our intention is to use the cross-flow device to predict the product compositions from such a counter-current cascade, because the pinch point in the cross-flow device is hypothesized to be an estimate of the pinch point in the counter-current cascade. The liquid feed to the cross-flow device is pure acetic acid, and the vapor feed has a composition at phase equilibrium with a equimolar mixture of acetic acid and isopropanol reacted until it reaches reaction equilibrium. Using φ′j ) (0.5)j and Da ) 10, we solve eq 11 to a pinch point, as shown in Figure 5. Because the pinch point is at reaction equilibrium, we use reaction-invariant compositions using the transformations published by Ung and Doherty:18
X1 ) xHOAc + xIPOAc
(19a)
X2 ) xIPA + xIPOAc
(19b)
From eqs 5 and 6, the corresponding vapor and liquid product compositions are calculated as shown in Figure 5. Although the product compositions are dependent on the value of Da, their projection in the reaction invariant composition space is independent of Da. We use this projection to simplify the view
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Figure 6. Predicted liquid and vapor products for the middle section in the X1-X2-xIPOAc space for the isopropyl acetate system.
Figure 5. (4) Predicted pinch point, (b) predicted liquid products, (O) predicted vapor products, and (×) a calculated trajectory for a countercurrent cascade in the reaction invariant space for the isopropyl acetate system (Da ) 1, φ ) 1, and N ) 50).
of the composition profile. The pinch point is also independent of the Damko¨hler number Da, and, therefore, the same pinch point is obtained by solving eqs 11 for any finite value of Da. A liquid composition profile for a counter-current cascade with 50 stages, φ ) 1, and Da ) 1, with acetic acid as the liquid feed and isopropanol as the vapor feed, is shown in Figure 5. The product compositions predicted from the cross-flow model give a very good estimate of the calculated values. There is a one-to-one correspondence between the reaction invariant space (X1-X2) and the mole fraction space at reaction equilibrium. However, this is not the case in the kinetic regime; therefore, we use the three-dimensional space given by X1, X2, and xIPOAc, which is equivalent to the three-dimensional mole fraction space. The liquid product composition and vapor product composition in the reaction invariant space are given by XL and YV, respectively. A range of xIPOAc values that satisfies the reaction invariant product compositions XL and YV, respectively, is determined using a simple linear model that satisfies the following equations and constraints:
(for i ) 1, ..., c -1)
0 e xi,L e 1
(20)
c
xi,L ) 1 ∑ i)1
(21)
xHOAc,L + xIPOAc,L ) X1,L
(22)
xIPOH,L + xIPOAc,L ) X2,L
(23)
One way to solve this model is to use a linear programming method (e.g., Simplex method). In the solution procedure, a guess value for xIPOAc,L is chosen, and the values for xHOAc,L and xIPOH,L are varied until the equations and constraints (eqs 20-23) are satisfied. We can choose the difference between the right-hand side and the left-hand side of eq 22 as the objective function and minimize it in a linear program. If the equations and constraints are not satisfied, the guessed value of xIPOAc is considered infeasible. An equivalent linear model can be arranged for the vapor product composition, YV. The range of compositions that solve the aforementioned linear
Figure 7. Calculated counter-current cascade trajectory at Da ) 1 and φ ) 1 for the isopropyl acetate system.
program is shown in Figure 6; the liquid products and vapor products each lie in a straight line in the X1-X2-xIPOAc composite space. We now check how well our predictions match with the calculations for a counter-current cascade with a large number of stages. Figure 7 shows a liquid composition profile for a counter-current cascade with 50 stages, φ ) 1, and Da ) 1. The liquid feed is pure acetic acid, and the vapor feed is pure isopropanol. Figure 7 shows that the liquid product from a counter-current cascade lies very close to the predicted value. Although the predicted vapor composition lies very close to the calculated value, it cannot be clearly observed in Figure 7 (the projection in Figure 5 gives a better view). Figure 8 shows the liquid composition profile and the liquid and vapor product compositions for a counter-current cascade for Da ) 100; all other parameters are the same as mentioned previously. The calculated products lie very close to the predicted compositions. Figures 7 and 8 also show that the predicted products are swept out by a counter-current cascade as Da varies between the limits of no reaction (Da ) 0) and reaction at equilibrium (Da ) ∞). However, note that the predicted products cannot be directly linked to a specific value of Da. Therefore, we must pick product compositions from the predicted feasible range without exactly knowing the Da value for the middle section that produced those products.
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Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 Table 1. Thermodynamic Data for the Methyl Acetate Examplea Antoine Equation ln(Psat) ) a +
b b T+c
Psat in Pa and T in K. Antoine Coefficients
component
molar normal volume boiling (cm3/g-mol) point (°C)
acetic acid (1) methanol (2) methyl acetate (3) water (4)
a
c
57.24 40.50 79.31
118.0 64.7 57.08
22.1001 -3654.62 -45.392 23.499 -3643.31 -33.434 21.1520 -2662.78 -53.460
18.051
100.1
23.2256 -3835.18 -45.343
Wilson Equationc j)n
Figure 8. Calculated counter-current cascade trajectory at Da ) 100 and φ ) 1 for the isopropyl acetate system.
b
ln(γi) ) 1.0 - ln
k)n
∑ xA - ∑ j ij
j)1
k)1
( ) xkAki
j)n
∑x A
j kj
where Aij )
(
j)1
)
Vj aij + bijT exp Vi RT
Binary Interaction Parameters, Aij (cal/g-mol) A11 ) 0.0 A12 ) 2535.2019 A13 ) 1123.1444 A14 ) 237.5248
A21 ) -547.5248 A22 ) 0.0 A23 ) 813.1843 A24 ) 107.3832
A31 ) -696.5031 A32 ) -31.1932 A33 ) 0.0 A34 ) 645.7225
A41 ) 658.0266 A42 ) 469.5509 A43 ) 1918.232 A44 ) 0.0
Dimerization in the Vapor Phased kij )
( ){
Zij 1 Zi Zj P
exp[BFijP/(RT)]
exp[BFiiP/(RT)] exp[BFjjP/(RT)]
kij )
}
-BDij (2 - δij) RT
a Data taken from ref 18. b Saturation pressure Psat given in units of Pa and temperature T given in Kelvin. c Dimerization of acetic acid in the vapor phase is also included. In this equation, γi is the activity coefficient of component i, n is the number of components, aij is the non-temperaturedependent energy parameter between components i and j (cal/g-mol), bij is the temperature-dependent energy parameter between components i and j (cal/g-mol), and Vi is the molar volume of pure liquid component i (L/gmol). d In this equation, Z represents the true mole fractions of the species in phase equilibrium, P is the system pressure, and kij is the dimerization reaction equilibrium constant for acetic acid (kij ) 10.0-12.5454+(3166/T)). Also, δij ) 0 if i * j, and δij ) 1 if i ) j, BF is the contribution to the second virial coefficient from physical forces, and BDij is the contribution of dimerization to the second virial coefficient.
Figure 9. Nonreactive stripping and rectifying cascade trajectories, starting from the predicted products for the middle section of a doublefeed hybrid column: (a) trajectories in the X1-X2-xIPOAc space and (b) projection in the reaction invariant space.
To determine the feasibility of a double-feed column, the vapor product from the counter-current cascade (middle section), V, is fed to an isobaric flash cascade that represents the rectifying section, and the liquid product, L, is fed to an isobaric flash cascade that represents the stripping section.12 These rectifying and stripping sections can be reactive or nonreactive, as desired. The cross-flow device predicts the products from the middle section; however, the composition profile for the middle section cannot be predicted. Because the objective of this feasibility analysis is to determine the product splits from a double-feed column, the profile for the middle section need not be known. We choose a set of
compositions from the predicted (using the cross-flow device) range of feasible products as a feed to the isobaric flash cascade, and the feasible splits for a double-feed column are then determined. Figure 9a shows the feasible product splits for a double-feed column with nonreactive stripping and rectifying sections. A projection of the feasible products splits in the reaction invariant space is shown in Figure 9b. Pure acetic acid can be obtained as a bottoms product, whereas isopropyl acetate cannot be obtained as a pure product from a doublefeed hybrid reactive distillation column. The stripping and rectifying cascade trajectories are used to select the feasible distillate and bottoms composition from a double-feed column, such that the overall material balances between feed and products are satisfied.
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Figure 10. (4) Predicted pinch point, (b) predicted liquid products, (O) predicted vapor products, and (×) a calculated trajectory for a countercurrent cascade in the reaction invariant space for the methyl acetate system (Da ) 1, φ ) 1, and N ) 50).
Figure 12. Nonreactive stripping and rectifying cascade trajectories starting from the predicted products for the middle section of a double-feed hybrid column: (a) trajectories in the X1-X2-xMeOAc space and (b) projection in the reaction invariant space.
Figure 11. Calculated counter-current cascade trajectory at Da ) 1 and φ ) 1 for the methyl acetate system.
Algorithm for Feasible Splits The following algorithm determines the feasible splits for a double-feed reactive distillation column. (i) For a given reaction chemistry and column pressure, choose the heavy reactant as the upper feed and the light reactant as the lower feed. (ii) Based on the reaction stoichiometry, determine the upper feed and lower feed flow rates (e.g., equal flow rates for a heavy acetic acid upper feed and the light isopropanol lower feed in the case of producing isopropyl acetate). (iii) Use the cross-flow arrangement shown in Figure 4 to predict the pinch point in the middle section. Based on the lower feed to upper feed ratio in step (ii), determine the vapor addition policy for the cross-flow device using eq 13. (iv) Determine the reaction invariants for the liquid and vapor products from the middle section using eq 7. (v) Determine the range of reference components for the liquid and vapor product compositions in a three-dimensional (3-D) composition space (consisting of the reaction invariant
space plus the reference component mole fraction composition), using a linear model equivalent to eqs 20-23. (vi) Choose a liquid and a vapor composition from the product compositions in the 3-D composition space and determine the corresponding compositions in the mole fraction space (using transformations for reaction invariant compositions). (vii) The liquid product composition from the middle section is the feed to the stripping section and the vapor product composition is the feed to the rectifying section. (viii) Use isobaric flash cascade method12 to determine both reactive and nonreactive stripping and rectifying cascade profiles. (ix) Choose a desirable composition on the rectifying cascade trajectory (potential distillate) and calculate the corresponding composition on the stripping cascade trajectory (potential bottoms) by solving the overall mole balances for the column. Alternatively, the potential bottoms product can be chosen and the potential distillate is calculated using mole balances. Example 2: Production of Methyl Acetate Methyl acetate is produced via the esterification of methanol with acetic acid. The reaction is given as
acetic acid (HOAc) + methanol (MeOH) h methyl acetate (MeOAc) + water (H2O) (24) A pseudo-homogeneous rate model is given as
(
r ) kf aHOAcaMeOH -
)
aMeOAcaH2O Keq
(25)
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The reaction equilibrium constant and the rate constant are given as19
(782.98 T ) -6287.7 ) ) 9.732 × 10 exp( T ) Keq ) 2.32 exp
kf (h-1
8
(26) (27)
where T is given in Kelvin. The rate constant was obtained by fitting the pseudo-homogeneous rate equation to predictions from a more-complex heterogeneous rate model.20 The heat of reaction is -3.01 kJ/mol, which is slightly exothermic. We neglect heat effects in this example; however, in some examples where the heat of reaction is large, heat effects must be considered. Therefore, the liquid and vapor flow rates are constant from stage to stage. The liquid-phase activity coefficients are well-represented by the Wilson equation, using the parameters listed in Table 1. For a counter-current cascade with a liquid feed of acetic acid and a vapor feed of methanol, the pinch point at φ ) 1 is estimated using the cross-flow device. The cross-flow device follows the vapor rate policy in eq 13. The predicted pinch point and the corresponding liquid and vapor products in a reaction invariant space are shown in Figure 10. A reaction invariant liquid composition profile for a counter-current cascade with 50 stages, φ ) 1, and Da ) 1, with acetic acid as the liquid feed and methanol as the vapor feed, is also shown in Figure 10. A linear program equivalent to eqs 20-23 determines the predicted product compositions in the X1-X2-xMeOAc space, as shown in Figure 11. A liquid composition profile for the countercurrent cascade in Figure 11 shows that the predictions are very similar to the calculated values of the product compositions. We choose a composition predicted from the cross-flow device as a feed to the isobaric flash cascade to determine feasible splits for a double-feed column. Figure 12a shows the splits for a double-feed column with nonreactive stripping and rectifying sections. A projection of the feasible products splits in the reaction invariant space is shown in Figure 12b. The figure shows that pure methyl acetate can be obtained as a distillate and pure water can be obtained as a bottoms product from a double-feed hybrid reactive distillation column. The doublefeed column configuration described here does not have a distinct nonreactive extractive section at the top of the reactive zone. Chadda et al.14 have shown, using column simulations, that such a configuration is indeed feasible. Effect of Vapor Rate on the Middle Section The cross-flow arrangement predicts the pinch point in the middle section. The vapor addition policy for the cross-flow device is related to the ratio of the lower feed to the upper feed by eq 13. Extractive distillation columns are known to exhibit minimum entrainer flows (or a minimum ratio of upper feed flow rate to lower feed flow rate), below which the desired products cannot be obtained, even in a column with an infinite number of stages. Therefore, double-feed reactive distillation columns can be expected to exhibit such a minimum ratio. The predictions for the liquid products from the middle section in the reaction invariant space at different values of the ratio of lower feed rate to the upper feed rate are shown in Figure 13a. For a value of φ > 1.3, the cross-flow device predicts product compositions that lie outside the feasible mole fraction space (although the predicted pinch point lies in the positive mole fraction space at all values of φ). The vapor products in the reaction invariant space are shown in Figure 13b. Figure
Figure 13. (a) Predicted liquid products and (b) predicted vapor products, each at varying feed ratios in the reaction invariant space.
13b shows that, for φ < 0.6, the cross-flow arrangement predicts vapor products that do not lie in the positive mole fraction space. Comparing these results with the calculations for a countercurrent cascade with 50 stages for different feed ratios, we observe that, for φ > 1.5, the liquid product from the countercurrent cascade (middle section) consists of pure isopropanol (lower feed). Therefore, even by adding a stripping section, the reactant isopropanol will emerge as a bottoms product. Also, from the counter-current cascade calculations, we observe that, for φ < 0.5, the vapor product from the counter-current cascade (middle section) consists of pure acetic acid. Hence, even after adding a rectifying section, the reactant acetic acid will emerge as a distillate in a double-feed column. This happens because the pinch occurs at the lower feed stage for φ > 1.5 (the upper feed flow is possibly below the minimum feasible value), and
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point composition allows the prediction of the composition of liquid and vapor products from a counter-current cascade using material balances. Also, a single calculation coupled with a simple linear model predicts the range of product compositions for different Damko¨hler number (Da) values. These product compositions are used as feed compositions to an isobaric flash cascade model that represents the stripping and rectifying sections. We find that a double-feed reactive distillation column cannot give isopropyl acetate as a pure product; however, methyl acetate can be obtained as a pure product in a double-feed hybrid reactive distillation column. Nomenclature
Figure 14. Calculated counter-current cascade trajectory at Da ) 1 and φ ) 1 for the methyl acetate system. Predicted products for the vapor addition policy given by eq 28.
at the upper feed stage for φ < 0.5 (the lower feed flow is possibly below the minimum feasible value). An important utility of the cross-flow arrangement is to estimate the upper and lower limits of the feed ratio for a desirable double-feed column, because it is usually not desirable to get pure reactants as a distillate or bottoms. Vapor Addition Policy The vapor feed addition policy in the cross-flow device follows a geometric series given by eq 13. This is not the only vapor addition policy that can be used to predict the pinch point in the middle section. For example, an alternative vapor addition policy is given as
φ′j )
4φ 3j(j + 2)
(28)
Using this policy, eq 14 is satisfied; therefore, the net vapor added to the cross-flow device is equal to that in the corresponding counter-current cascade. Figure 14 shows a liquid composition profile for a counter-current cascade with acetic acid as the liquid feed and methanol as the vapor feed, Da ) 4, and φ ) 1. The figure shows that the calculated product compositions lie very close to the feasible products predicted by the cross-flow device using eq 28. In fact, the policy in eq 28 gives a slightly better prediction of the products, compared to the policy in eq 13, and also gives a better estimate of the vapor-to-liquid ratios at which the pinch moves to either end of the middle section. However, a simpler policy (eq 13) is desirable at the expense of a small amount of inaccuracy in the predictions. Other policies can be devised for the vapor feed addition policy; however, we know of no simpler expression than eq 13. Conclusions We have developed a method that does not require repetitive simulations to determine feasible products from a double-feed reactive distillation column. The tool developed in this work complements the method for single-feed reactive distillation columns described by Chadda et al.12 A cross-flow arrangement is used to predict the pinch point for the middle section of a double-feed column. Use of a cross-flow device simplifies the calculations for the middle section. Knowledge of the pinch
A, B, C ) generic chemical species ai ) activity of component i c ) total number of reacting and inert components Da ) Damko¨hler number H ) liquid-phase molar holdup (mol) Hj ) liquid-phase molar holdup on stage j (mol) kf ) forward reaction rate constant (1/time) kf,ref ) forward reaction rate constant at the reference temperature (1/time) Lj ) liquid molar flow rate from stage j (mol/s) N ) number of two-phase CSTRs in the cascade r(x) ) mole fraction based on the rate of reaction (1/time) Tref ) reference temperature (K) V′ ) vapor flow rate (mol/s) Vj ) vapor molar flow rate from stage j (mol/s) x ) column vector of c mole fractions xi,j ) mole fraction of component i in the liquid phase for the jth stage xˆ ) vector of liquid mole fractions at the pinch point Xi,j ) reaction invariant liquid mole fraction of component i in stage j XL ) vector of reaction invariant compositions of the liquid product from the middle section of a double-feed column yi,j ) mole fraction of component i in the vapor phase for the jth stage Yi,j ) reaction invariant vapor mole fraction of component i in stage j YV ) vector of reaction invariant compositions of the vapor product from the middle section of a double-feed column Greek Symbols νi,r ) stoichiometric coefficient of component i in reaction r νTj ) sum of stoichiometric coefficients of reaction j φ ) ratio of the vapor rate to the liquid inlet rate (dimensionless) φj ) ratio of the vapor rate to the liquid inlet rate for stage j (dimensionless) φ′j ) ratio of the vapor rate to the liquid inlet rate for stage j in the cross-flow arrangement (dimensionless) Subscripts, Superscripts, and Accents 0 ) initial i ) component i Ref ) indicates reference components T ) total ∧ ) pinch Literature Cited (1) Sharma, M. M.; Danckwerts, P. V. Fast Reactions of CO2 in Alkaline Solutions. Chem. Eng. Sci. 1963, 18, 729. (2) Gaikar, V. G.; Sharma, M. M. Separations through Reactions and Other Novel Strategies. Sep. Purif. Methods 1989, 18, 111.
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(3) Saha, B.; Sharma, M. M. Esterification of Formic Acid, Acrylic Acid and Methacrylic Acid with Cyclohexene in Batch and Distillation Column Reactors: Ion-exchange Resins as Catalysts. React. Funct. Polym. 1996, 28, 263. (4) Sharma, M. M.; Mahajani, S. M. Industrial Applications of Reactive Distillation. In ReactiVe Distillation; Sundmacher, K., Kienle, A., Eds.; Wiley-VCH Verlag GmbH: Weinheim, Germany, 2003; p 3. (5) Knapp, J. P.; Doherty, M. F. Minimum Entrainer Flows for Extractive Distillation: A Bifurcation Theoretic Approach. AIChE J. 1994, 40, 243. (6) Petlyuk, F. B.; Danilov, R. Yu. Sharp Distillation of Azeotropic Mixtures in a Two-Feed Column. Theoret. Found. Chem. Eng. 1999, 33, 233. (7) Bessling, B.; Schembecker, G.; Simmrock, K. H. Design of Processes with Reactive Distillation Line Diagrams. Ind. Eng. Chem. Res. 1997, 36, 3032. (8) Barbosa, D.; Doherty, M. F. Design and Minimum-Reflux Calculations for Double-Feed Multicomponent Reactive Distillation Columns. Chem. Eng. Sci. 1988, 43, 2377. (9) Chin, J.; Lee, J. W. Rapid Generation of Composition Profiles for Reactive and Extractive Cascades. AIChE J. 2005, 51, 922. (10) Chadda, N.; Malone, M. F.; Doherty, M. F. Feasible Products for Kinetically Controlled Reactive Distillation of Ternary Mixtures. AIChE J. 2000, 46, 923. (11) Lee, J. W. Feasibility Studies on Quaternary Reactive Distillation Systems. Ind. Eng. Chem. Res. 2002, 41, 4632. (12) Chadda, N.; Malone, M. F.; Doherty, M. F. Effect of Chemical Kinetics on Feasible Splits for Kinetically Controlled Reactive Distillation. AIChE J. 2001, 47, 590.
(13) Gadewar, S. B.; Malone, M. F.; Doherty, M. F. Feasible Region for a Counter-current Cascade of Vapor-liquid CSTR’s. AIChE J. 2002, 48, 800. (14) Chadda, N.; Malone, M. F.; Doherty, M. F. Feasibility and Synthesis of Hybrid Reactive Distillation. AIChE J. 2002, 48, 2754. (15) Julka, V.; Doherty, M. F. Geometric Behavior and Minimum Flows for Nonideal Multicomponent Distillation. Chem. Eng. Sci. 1990, 45, 1801. (16) Lee, L-S.; Kuo, M-Z. Phase and Reaction Equilibria of the Acetic Acid-Isopropanol-Isopropyl Acetate-Water System at 760 mm Hg. Fluid Phase Equilib. 1996, 123, 147. (17) Manning, J. M. Kinetics and Feasibility of Reactive Distillation in Isopropyl Acetate Synthesis. M.S. Thesis, University of Massachusetts, Amherst, MA, 1999. (18) Ung, S.; Doherty, M. F. Theory of Phase Equilibria in Multireaction Systems. Chem. Eng. Sci. 1995, 50, 3201. (19) Song, W. R.; Huss, R.; Doherty, M. F.; Malone, M. F. Measurement of Residue Curve Maps and Heterogeneous Kinetics in Methyl Acetate Synthesis. Ind. Eng. Chem. Res. 1998, 37, 1917. (20) Doherty, M. F.; Malone, M. F. Conceptual Design of Distillation Systems, McGraw-Hill: New York, 2001; p 479.
ReceiVed for reView July 5, 2006 ReVised manuscript receiVed January 4, 2007 Accepted January 9, 2007 IE060867R
