Analysis of Ester Hydrolysis Reactions in a Chromatographic Reactor

Ind. Eng. Chem. Res. 2005, 44, 9565-9574

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Analysis of Ester Hydrolysis Reactions in a Chromatographic Reactor Using Equilibrium Theory and a Rate Model T. D. Vu,† A. Seidel-Morgenstern,†,‡ S. Gru 1 ner,§ and A. Kienle*,†,‡ Otto-von-Guericke-Universita¨ t, Universita¨ tsplatz 2, D-39106 Magdeburg, Germany, Max-Planck-Institut fu¨ r Dynamik komplexer technischer Systeme, Sandtorstrasse 1, D-39106 Magdeburg, Germany, and Institut fu¨ r Systemdynamik und Regelungstechnik, Universita¨ t Stuttgart, Pfaffenwaldring 9, D-70550 Stuttgart, Germany

In this paper, a new methodology for the conceptual design of chromatographic reactors is proposed and illustrated for ester hydrolysis reactions in a fixed-bed chromatographic reactor with a pulse injection of the ester. In the first step, an extended equilibrium theory approach for chromatographic reactors is used to study the feasibility, i.e., to clarify whether total conversion of the reactant and total separation of the products are possible. In the second step, the required column length is calculated using a rate based model, which takes into account the finite reaction kinetics and axial dispersion. Finally, the theoretical predictions are compared with the experimental results. 1. Introduction The integration of chemical reaction and separation in the same device can be very attractive compared to conventional processes where reaction and separation are carried out in different devices. The main advantage of this integration for equilibrium limited reversible reactions lies in a possible shift of the chemical equilibrium to the product side by simultaneous separation of the reaction products, which can lead to almost 100% conversion with simultaneous separation of the products. Depending on the separation principle, different types of integrated reaction-separation processes can be distinguished such as reactive distillation or reactive chromatography, for example. A recent overview was given by Sundmacher et al.1 Today, reactive distillation seems fairly well-understood and has several impressive industrial applications.2 Compared to that, reactive chromatography is still in an early stage of development. Powerful theoretical concepts for understanding and designing such processes are a major key to success. Recently, a new unifying approach for analyzing and understanding the dynamics of integrated reaction-separation processes was proposed.3,4 The approach is based on the assumption of simultaneous phase and reaction equilibrium and can be viewed as an extension of the well-known equilibrium theory.5,6 It makes use of transformed concentration variables which were first introduced by Doherty and co-workers for the steady state design of reactive distillation processes.7,8 The concept provides profound insight into the dynamic behavior of integrated reaction-separation processes and reveals inherent bounds of feasible operation caused by reactive azeotropy. In the present contribution, the extended equilibrium theory is used to study the dynamics and feasibility of ester hydrolysis reactions in a chromatographic fixedbed reactor. The focus is on the heterogeneously catalyzed hydrolysis reactions of methyl formate and methyl * To whom all correspondence should be addressed. Fax: +49-391-6110-515. E-mail: [email protected]. † Otto-von-Geuricke-Universita ¨ t. ‡ Max-Planck-Institut fu ¨ r Dynamik Komplexer Technischer Systeme. § Universita ¨ t Stuttgart.

acetate using the anion exchange resin DOWEX 50W-X8.9 The approach is combined in a second step with a rate model to account for the finite reaction kinetics and axial dispersion. The outline of the paper is the following: First, the extended equilibrium theory approach is introduced as a useful tool to study the feasibility of chromatographic reactors. In the context of this paper, it is used to decide whether total conversion and total separation in a chromatographic fixed-bed reactor with a pulse injection of the reactants is possible. Application is demonstrated for systems with reactions of the type 2A h B + C or A h B + C, respectively. It is shown that the difference in stoichiometry has strong consequences for the feasibility. Namely, for the first reaction, total conversion and total separation is only possible if the reactant has intermediate adsorptivity. In contrast to this, for the second reaction, total conversion and total separation is always possible for any order of adsorptivities, provided that the chromatographic column is sufficiently long. Ester hydrolysis reactions with water in excess are considered as a practical application for the second type of reaction. Afterward, the column length required for total conversion and total separation is estimated using a rate based model, which takes into account the finite reaction kinetics and axial dispersion. The theoretical predictions are finally confirmed by the experimental results. 2. Equilibrium Theory Pulse and wave patterns in nonreactive chromatographic processes are easily predicted by means of equilibrium theory.5,6,10 Equilibrium theory is also the basis for triangle theory which was developed by Storti, Mazzotti, and Morbidelli11 for the design of moving-bed chromatographic processes. The theory is based on a simple equilibrium model. For a fixed-bed process, the model reads (for a more detailed derivation, see ref 6)

∂ ∂c (Fq(c) + c) + )0 ∂t ∂z q, c ∈ RNs with dimensionless space and time coordinates

10.1021/ie050256j CCC: $30.25 © 2005 American Chemical Society Published on Web 10/22/2005

(1)

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Ind. Eng. Chem. Res., Vol. 44, No. 25, 2005

z)

zj l

t)

ht vf l

(2)

and the dimensionless quantity

F)

1-

(3)

In contrast to a previous formulation for countercurrent processes,3 the spatial coordinate z is directed in the direction of fluid flow. This model is readily extended to the reactive case by adding the specific reaction rates r.

∂c ∂ (Fq(c) + c) + ) νr ∂t ∂z q, c ∈ RNs

r ∈ RNr

(4)

Therein, ν is a Ns × Nr matrix of stoichiometric coefficients. In addition to phase equilibrium, reaction equilibrium can be assumed for fast reversible chemical reactions. In reaction equilibrium, the reaction rates are unknown and have to be eliminated from eq 4. Since the model equations (eq 4) are linear in terms of the reaction rates, this is always possible. Following the ideas of Ung and Doherty,8 this is achieved by choosing the first Nr equations of (eq 4) as reference. The reference equations are solved for the unknown reaction rates and afterward substituted into the remaining N - Nr equations. The procedure leads to the following reduced set of equations:

∂C ∂ (FQ(C) + C) + )0 ∂t ∂z

(5)

with transformed concentration variables according to

C ) cII - νII(νI)-1cI II

II

Q ) qII - νII(νI)-1qI Ns-Nr

C, Q, c , q ∈ R

I

I

c,q ∈R

(6)

Nr

Therein, cI and qI represent the concentrations corresponding to the Nr reference equations and cII and qII represent the concentrations corresponding to the remaining Ns - Nr equations. Accordingly, the matrix of stoichiometric coefficients is split into two parts νI and νII. For details, the reader is referred to Gru¨ner et al.4 In this reduced set of transformed concentration variables, the reactive problem (eq 5) is completely equivalent to a nonreactive problem (eq 1) with Ns Nr components. Therefore, transformed concentration variables represent an easy way to extend the equilibrium theory to reactive chromatography with fast reversible reactions.3,4 An application to other integrated reaction-separation processes including membrane reactors and sorption enhanced gas-phase reactions is straightforward.4 2.1. Reactions of Type 2A h B + C. Let us first consider reactions of the type 2A h B + C, which have been treated in detail by Gru¨ner and Kienle3 using the extended equilibrium theory described above. Hence, only a brief summary of the main results will be given. Afterward, a comparison with reactions of the type A h B + C will be made. If reactant A is chosen as the reference component, the definitions of the two independent transformed variables in both phases are

c1 2 q1 Q 1 ) q2 + 2 C1 ) c2 +

c1 2 q1 Q2 ) q3 + 2 C2 ) c3 +

(7) (8)

In this definition, c1 is the concentration of component A, c2 is the concentration of component B, and c3 is the concentration of component C. The dynamic behavior is illustrated in terms of characteristic profiles of the transformed concentration variables in Figure 1. Two different scenarios for three different orders of adsorptivities are considered, corresponding to the two rows and three different columns in Figure 1. In the first row, reactant A is supplied to an empty column with a constant concentration and the column is loaded. In the second row, a totally loaded column is regenerated with pure solvent. The reaction takes place in the fluid phase with a chemical equilibrium of the type

K)

c2c3

(9)

c12

The phase equilibrium is described by competitive Langmuir isotherms according to

qi )

Ki ci Ns

1+

i ) 1, ..., Ns

(10)

∑ bkck

k)1

The parameters for the three different cases corresponding to the three different columns in Figure 1 are given in Table 1. In the first case, reactant A has intermediate adsorptivity; in the second case, A has the highest adsorptivity, and in the third case, it has the lowest adsorptivity. Further, it is assumed that product B always has higher adsorptivity than product C. In the first case, the profiles of the transformed concentration variables during the loading of an empty column consist of a first fraction where C1 ) 0, corresponding to pure product C, and a second fraction which contains a mixture of all three components in chemical equilibrium (first row of Figure 1a). The reverse situation is observed during the regeneration in the second row of Figure 1a. Here, the profiles consist of a first fraction containing a mixture of all three components and a second fraction where C2 ) 0, corresponding to pure product B. Analogous concentration profiles can be observed in nonreactive chromatography during the separation of a mixture of B and C. A pulse injection of reactant A can be considered to be a combination of the two scenarios described above. Hence, a feed pulse of reactant A will be resolved into a front pulse of pure product C and a second pulse of pure product B in the rear, provided the column is long enough. Again, analogous patterns of behavior can be observed in nonreactive chromatography during the separation of a feed pulse with a mixture of B and C. A different situation is observed if reactant A has the highest or lowest adsorptivity. If A has the highest adsorptivity, separation fails during regeneration, as illustrated in the second row of Figure 1b, and only during loading is a fraction of pure product C obtained. This leads to a pulse pattern consisting of two pulses: a pulse of pure product C in front and a pulse containing

Ind. Eng. Chem. Res., Vol. 44, No. 25, 2005 9567

Figure 1. Dynamic behavior of a fixed-bed chromatographic reactor with a reaction of the type 2A h B + C in the fluid phase and Langmuir isotherms: (solid line) C1; (dashed line) C2; (first row) loading of an empty bed, feed is reactant A; (second row) regeneration of a totally loaded bed with pure solvent; (a) reactant A has an intermediate adsorptivity; (b) reactant A has the highest adsorptivity; (c) reactant A has the lowest adsorptivity. Table 1. Adsorption Isotherm Parameters and Chemical Equilibrium Constants for the Ternary Model Systems in Figures 1-6 reactant A has parameter

(a) intermediate adsorptivity

(b) highest adsorptivity

(c) lowest adsorptivity

K1 ) b1 K2 ) b2 K3 ) b3 K

3.0 5.0 1.0 1.0

5.0 3.0 1.0

1.0 5.0 3.0

a mixture of all three components in the rear. Thus, total conversion and total separation are not feasible in this case, no matter how long the column is. If A has the lowest adsorptivity, separation fails during loading, as illustrated in the first row of Figure 1c. This leads to a pulse pattern consisting of a first pulse containing a mixture of all three components in front and a second pulse of pure product B in the rear. Again, total conversion and total separation are not feasible in this case, no matter how long the column is. These patterns of behavior can be predicted and analyzed using the extended equilibrium theory, as described in the previous section. The starting point is the so-called hodograph plot, shown in Figure 2 for the three different cases in Figure 1. The hodograph plot is generated from the eigenvectors of dQ/dC, the Jacobian of the transformed equilibrium function in eq 5.3,4 Constant boundary and initial conditions represent points in this hodograph plot. For the loading of an empty bed, the initial condition is represented by the origin. A feed of reactant A and the corresponding equilibrium compositions lie on the bisection line. Boundary and initial conditions are reversed for the regeneration of a totally loaded bed. In Figure 2, the wave patterns of Figure 1 are represented in good approximation by the line segments connecting the given boundary concentration at the inlet with the given initial concentration of the bed starting with the dotted

line from the given boundary concentration.3 In the first case, the wave solutions adjacent to the origin coincide with the axes C1 ) 0 and C2 ) 0, corresponding to a fraction of pure component C in front during loading and a fraction of pure component B in the rear during regeneration of the bed. In the second and the third cases, a line of reactive azeotropy occurs according to

Q1 Q 2 ) C1 C2

(11)

Along this line, reaction and separation compensate each other. Hence, this line cannot be crossed in a chromatographic column. In Figure 2b and c, this line is represented by the sloped boundary of the shaded region. Hence, in Figure 2b, the axis C2 ) 0 is not accessible. This leads to a situation where separation fails during regeneration, as illustrated in the second row of Figure 1b. On the reverse, in Figure 2c, the axis C1 ) 0 is not accessible. This leads to a situation where separation fails during loading, as illustrated in the first row of Figure 1c. From the above analysis the following important geometrical condition for total conversion and total separation can be extracted. Total conversion and total separation in a fixed-bed chromatographic reactor with pulse injection of the reactant is possible if the relevant wave solutions through the origin lie on the pure component axes of the products. It should be noted that up to this point no statement is made about the column length which is required to achieve total conversion and total separation. We will come back to this point later. Finally, it is worth noting that the patterns of behavior described above do not depend on the specific parameter values in Table 1 but only on the order of the adsorptivities. Further, it was shown that analogous patterns of behavior will arise if the reaction takes place in the solid phase.3

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Figure 3. Line of reactive azeotropy in the hodograph space of the transformed concentration variables for a reaction of the type A h B + C in the fluid phase and linear isotherms: (a) component A has the highest adsorptivity; (b) component A has the lowest adsorptivity.

nent, the definition of the transformed variables is now Figure 2. Representation of wave solutions in the hodograph space of the transformed concentration variables corresponding to Figure 1: (a) reactant A has an intermediate adsorptivity; (b) reactant A has the highest adsorptivity; (c) reactant A has the lowest adsorptivity.

In view of the ester hydrolysis reactions to be considered as a practical application in the second part of the paper, next, the focus is on reactions of the type A h B + C with linear adsorption isotherms, which surprisingly show a completely different type of behavior. In particular, it is shown that this is mainly due to the different stoichiometry. 2.2. Reactions of Type A h B + C. Consider a reaction A h B + C in the fluid phase. The reaction equilibrium is given by

K)

c2c3 c1

(12)

If reactant A is chosen again as the reference compo-

C1 ) c2 + c1

C2 ) c3 + c1

(13)

Q 1 ) q2 + q1

Q2 ) q 3 + q 1

(14)

due to the change in stoichiometry. The phase equilibrium is described by linear adsorption isotherms

qi ) Kici

i ) 1, ..., Ns

(15)

The parameters for the reaction and adsorption equilibrium are taken again from Table 1. Under these conditions, again, reactive azeotropy arises if reactant A has the highest or lowest adsorptivity, corresponding to cases b and c in Table 1. However, there is an important difference in the geometry, which will give rise to completely different behavior in a fixedbed chromatographic reactor and which will, therefore, be studied in some detail in the remainder of this work. The line of reactive azeotropy can be calculated from eq 113 and is illustrated in Figure 3 for cases b and c in


a fraction of pure product C in front during loading and a fraction of pure product B in the rear during regeneration in all three cases. This is confirmed using characteristic profiles of the transformed concentration variables during loading and regeneration in Figure 5. So, in contrast to the previous case (2A h B + C), total conversion and total separation for a pulse injection of reactant A is now possible in all three cases, which is in agreement with the geometric condition given above. The same arguments apply for large feed concentrations, e.g., F2 in Figure 4c, since the relevant wave solutions through the origin contain a fraction of pure product C during loading and a fraction of pure product B during regeneration. Hence, the geometric condition also applies to large feed concentrations. Again, it is readily shown that analogous patterns of behavior will arise if the reaction is taking place in the solid phase. Further, separation during loading and regeneration and, hence, total conversion and total separation for a pulse injection of reactant A are also possible in all three cases if the linear isotherm is replaced by the Langmuir isotherm, as illustrated in Figure 6. Consequently, the difference in behavior between the systems considered in the previous section and the systems considered in this section can be clearly attributed to the different stoichiometries. 2.3. Ester Hydrolysis Reactions. As a practical application, in this section, ester hydrolysis reactions are considered. The focus is on heterogeneously catalyzed hydrolysis of methyl formate and methyl acetate using the anion exchange resin DOWEX 50W-X8. The general reaction mechanism is

ester + water h acid + alcohol

(16)

Reactant water is used in large excess as a solvent. Hence, it is a good approximation to assume a constant water concentration. This reduces the ester hydrolysis to a system of the type A h B + C, which was considered in the previous section. The reaction equilibrium, reaction kinetics, and adsorption isotherms of the above ester hydrolysis reactions were studied by Mai et al.9 In terms of fluid phase concentrations, the reaction equilibrium follows from

Khom eq )

calcoholcacid cestercwater

(17)

For a constant water concentration, we may rewrite eq 17 as Figure 4. Representation of wave solutions in the hodograph space of the transformed concentration variables for a reaction of the type A h B + C in the fluid phase and linear isotherms: (a) reactant A has an intermediate adsorptivity; (b) reactant A has the highest adsorptivity; (c) reactant A has the lowest adsorptivity.

Table 1. In contrast to Figure 2, the line of reactive azeotropy is now a parabola. However, it can be shown that only the branch corresponding to the solid line is relevant, since the physical concentrations cB and cC are always negative on the branch through the origin. This has important consequences for the construction of wave solutions in the hodograph space, as illustrated in Figure 4 again for three different orders of adsorptivities. For sufficiently small feed concentrations represented by F1 in Figure 4b and c, the relevant wave solutions through the origin lie completely on the pure component axes of products B and C. This gives rise to

K ) Khom eq cwater )

calcoholcacid cester

(18)

which is equivalent to eq 12. Furthermore, the adsorption isotherms were described by a linear equation

qi ) Kici

i ) 1, ..., Ns

(19)

The parameters were given by Mai et al.9 In particular, the following order of adsorption constants was found for DOWEX 50W-X8:

Kester > Kalcohol > Kacid

(20)

This corresponds to case b in Figure 4, where reactant A has the highest adsorptivity. It is worth noting that the order of adsorptivity may depend on the concentra-

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Figure 5. Dynamic behavior of a fixed-bed chromatographic reactor with a reaction of the type A h B + C in the fluid phase and linear isotherms: (solid line) C1; (dashed line) C2; (first row) loading of an empty bed, feed is reactant A; (second row) regeneration of a totally loaded bed with pure solvent; (a) reactant A has an intermediate adsorptivity; (b) reactant A has the highest adsorptivity; (c) reactant A has the lowest adsorptivity.

Figure 6. Dynamic behavior of a fixed-bed chromatographic reactor with a reaction of the type A h B + C in the fluid phase and Langmuir isotherms: (solid line) C1; (dashed line) C2; (first row) loading of an empty bed, feed is reactant A; (second row) regeneration of a totally loaded bed with pure solvent; (a) reactant A has an intermediate adsorptivity; (b) reactant A has the highest adsorptivity; (c) reactant A has the lowest adsorptivity.

tion range and on the specific ion exchange resin. In previous studies on methyl acetate synthesis by Lode et al.,12 Poepken et al.,13 and Sainio et al.14 performed in a broader range of solvent compositions using ion exchange resin Amberlyst 15, different orders of adsorptivities were observed. A hodograph plot for the hydrolysis of methyl formate is shown in Figure 7a. The hydrolysis of methyl acetate is shown in Figure 7b. In both cases, c1 is the (physical) concentration of the ester, c2 is the concentration of the alcohol, and c3 is the concentration of the acid. In case a, a line of reactive azeotropy is predicted for very high

concentration values. In practice, such high concentrations are not feasible due to the limited solubility of methyl acetate in water. In case b, reactive azeotropy is predicted for much lower concentration values, which lie in the physically relevant range. However, this has no effect on the feasibility. From the above geometric condition, we conclude that in both cases total conversion and total separation is possible for a fixed-bed chromatographic reactor with a pulse injection of ester, provided the chromatographic column is long enough. These results will be validated with a simulation study and experimental results in the next section.


from the theoretical number of plates according to

Dap )

vf l 2N

(23)

The reaction rates of the heterogeneously catalyzed ester hydrolysis reactions considered are given by

(

r ) (1 - )khet qestercwater -

)

qalcoholqacid Khet eq

(24)

Therein, the chemical equilibrium constant Khet eq in terms of solid-phase concentrations can be calculated in eq 17 and the adsorption from the constant Khom eq equilibrium constants (eq 19) according to

Khet eq )

qalcoholqacid KalcoholKacid ) Khom eq qestercwater Kester

(25)

The reaction rate, chemical equilibrium, and adsorption constants are taken from Mai et al.9 In the mentioned paper, two different ion exchange resins with different particle sizes were used as catalysts. In the remainder, the focus is exclusively on catalyst “1” in ref 9. For the simulation studies, the standard Danckwerts boundary conditions for a rectangular injection profile are used. At the inlet, the boundary conditions are

{

|

Dap ∂ci inj vf ∂z˜ z˜ )0,t˜ for 0 e ht e ht ci(zj ) 0, ht ) ) Dap ∂ci + for ht g ˜tinj vf ∂z˜ z˜ )0,t˜ cinj i +

Figure 7. Representation of wave solutions in the hodograph space of the transformed concentration variables for the hydrolysis of methyl formate (a) and methyl acetate (b).

Further, the column length required for total conversion and total separation is determined using a rate model and, thereby, illustrates the influence of the reaction kinetics and axial dispersion.

with injection time

ht inj )

Vinj π vf d2 4

At the outlet, we have

3. Influence of Finite Reaction Kinetics and Axial Dispersion For the purpose of studying the influence of the axial dispersion, the equilibrium model (eq 4) is extended with an appropriate term to describe the axial dispersion in the fluid phase.

∂ ∂c ∂2c ((1 - )q(c) + c) + vf ) +Dap 2 + νr (21) ∂zj ∂th ∂zj In view of the experimental results to be discussed subsequently, a formulation in terms of physical dimensions is used in this section. The quantity Dap is an apparent dispersion coefficient which lumps together all effects causing band broadening. Alternatively, the column efficiency can be described by N, the number of theoretical plates. N can be estimated from the chromatographic peak characteristic using the following relationship15

N ≈ 5.54(tR/w0.5)2

|

(26)

(22)

with retention time tR and the peak width at half-height w0.5. The apparent dispersion coefficient then follows

|

∂ci ∂zj

zj)l,th

)0

(27)

As an initial condition, an empty bed is assumed according to

ci(zj, ht ) 0) ) qi(zj, ht ) 0) ) 0

(28)

For the numerical solution of the equations of the rate model, the PRESTO commercial software program was applied (http://www.cit-wulkow.de/tbbpres.htm). 3.1. Simulation Study. Applying the rate model (eq 21) described above, simulations were carried out to evaluate the influence of the convection, dispersion, and reaction on the conversion and degree of separation. The parameters used were based on a previous study of the hydrolysis of methyl formate (MF) and methyl acetate (MA). The first reaction is relatively fast, whereas the second reaction is more than 1 order of magnitude slower. The parameters are given in Table 3. They were obtained from the experiments using a column of 0.25 m in length.9 Figure 8 illustrates the influence of the finite reaction kinetics and axial dispersion on wave solutions in the

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Figure 8. Influence of the axial dispersion and finite reaction rate on MF hydrolysis in the hodograph space: (solid line) reference case with khet ) 14 × 10-3 L/(mol min) and N ) 1250; (dashed line) khet ) 14 × 10-4 L/(mol min) and N ) 1250; (dotted line) khet ) 14 × 10-3 L/(mol min) and N ) 250.

Figure 9. Eluted concentration profiles of MF, FA, and M with the column length varied from 0.05 to 0.25 m with l/N ) 2 × 10-4 m.

Table 2. Parameters for Figures 8-11a figure

column length (m)

l/N (m)

8, MF 9, MF 10, MF 11, MA 12, MF 13, MA

0.25 0.05 and 0.25 0.125 and 1.0 0.25 and 12.5 0.25 0.25

2 × 10-4 2 × 10-4 2 × 10-4 and 2 × 10-3 2 × 10-4 and 2 × 10-3 2 × 10-4 2 × 10-4

aIn all cases, a column diameter of 8 mm, a water flow rate of 0.3 mL/min, and a temperature of 25 °C were used. Feed concentrations are 0.5 mol/L, and the injection volume is 100 µL (except in Figure 8, where loading of an empty bed followed by the regeneration of a totally loaded bed is considered).

Table 3. Parameters for the Hydrolysis Reactions of MF and MA9 solute

khet (×103 L/(mol min))

Khom eq

Ki

MA MF M FA AA

14.00 0.38

0.22 0.38

0.85 1.085 0.628 0.432 0.520

hodograph space of the transformed concentration variables for MF hydrolysis. The figure shows the loading of an empty bed, followed by a regeneration of a totally loaded bed. For the largest values of khet and N (solid line), the predictions of the rate model almost match those of the equilibrium theory, which were illustrated in Figure 7a. It is worth noting that these values represent the experimental conditions to be discussed subsequently. Larger deviations from the equilibrium theory are observed in Figure 8 for smaller values of khet (slower reaction, dashed line) and N (higher dispersion, dotted line). The case with the slow rate of reaction lies in the range of MA hydrolysis. In Figure 9, the development of the concentration profiles for a pulse injection of the ester is shown for MF hydrolysis for different column lengths. For a relatively short column (l ) 0.05 m), there is obviously no complete conversion and no complete separation of the products. For a longer column (l ) 0.125 m), there is almost no unconverted MF left and a significant separation is achieved. This separation can be quantified in terms of a ratio of B/A, as indicated in Figure 9. If this ratio approaches 0, complete separation is reached. This situation is shown for a column length of 0.25 m in Figure 9. It can also be seen that an increase in the column length leads to an increase in the dispersion of the product peaks.

Figure 10. Influence of the column length on the conversion factor (solid lines) and the ratio of B/A (dashed lines) of a MF hydrolysis reaction. The column length is varied from 0.125 to 1.0 m: (squares) reference case with khet ) 14 × 10-3 L/(mol min) and l/N ) 2 × 10-4 m; (circles) first extended case with khet ) 14 × 10-4 L/(mol min) and l/N ) 2 × 10-4 m; (triangles) second extended case with khet ) 14 × 10-3 L/(mol min) and l/N ) 2 × 10-3 m. The arrow refers to the experimental column in Figure 12.

The results illustrated in Figure 9 are further extended in Figure 10. This figure shows the dependencies of the conversion factor (solid lines) and the ratio of B/A (dashed linessa measure of product separation) as functions of the column length. Obviously, an increase in the column length leads to an improvement in both performance parameters. The squares correspond to the reference parameters (Table 3), whereas the circles and the triangles hold for a 10-fold decrease in the khet and N values, separately. The reference case achieved the desired process goals of total conversion and total separation at a column length of 0.25 m. Obviously, the effects of finite rates of reaction and limited efficiency of separation must be compensated by using longer columns compared to the results available from the equilibrium theory. A similar plot is shown in Figure 11 for the second reaction system considered, i.e., the hydrolysis of MA. The parameters for this system are given in Table 2. In this figure, it can be seen that a much longer column must be used in order to achieve complete conversion and complete separation. The squares correspond to the reference parameters (Table 3), whereas the circles hold for only a 10-fold increase in the khet value and the triangles hold for only a 10-fold decrease in the N value. For the flow rate and column diameter used in the calculations, a column longer than 10 m would be required in order to achieve these process goals. 3.2. Experimental Illustration. Selected simulation results are compared in Figures 12 and 13 with experi-


4. Conclusion

Figure 11. Influence of the column length on the conversion factor (solid lines) and the ratio of B/A (dashed lines) of a MA hydrolysis reaction. The column length is varied from 0.25 to 12.50 m: (squares) reference case with khet ) 0.38 × 10-3 L/(mol min) and l/N ) 2 × 10-4 m; (circles) first extended case with khet ) 0.38 × 10-2 L/(mol min) and l/N ) 2 × 10-4 m; (triangles) second extended case with khet ) 0.38 × 10-3 L/(mol min) and l/N ) 2 × 10-3 m. The arrow refers to the experimental column in Figure 13.

Figure 12. Experimental (solid line) and simulated elution profiles (dashed line) for MF hydrolysis with the parameters in Tables 2 and 3.

In this paper, a two step procedure was proposed for the conceptual design of chromatographic reactors. In the first step, an extended equilibrium theory is applied to verify that total conversion and total separation in a fixed-bed chromatographic reactor can be obtained. It is shown that this crucially depends on the order of the adsorptivities and/or the stoichiometry of the reaction system. In particular, it was shown that for reactions of the type 2A h B + C total conversion and total separation is only possible if reactant A has an intermediate adsorptivity. Instead, for reactions of the type A h B + C total conversion and total separation is possible for any order of adsorptivities. An example for the latter are ester hydrolysis reactions, which are performed with water in excess and which were considered in the second part of this paper. Hence, according to the theory, total conversion and total separation is possible, although the ester has higher affinity to the ion exchange resin than the alcohol and the acid. This result was confirmed by experiments. These findings are somewhat surprising and not easy to anticipate based on physical intuition only. This underlines the usefulness of a good theory for analyzing and predicting possible process limitations correctly. In the second step, a more detailed model with finite reaction kinetics and axial dispersion was used to quantify the design parameters. In particular, the effect of column length on conversion and separation was studied. In the present contribution, the focus was on a fixedbed chromatographic reactor with a pulse injection of the reactant. However, a similar approach can also be used to study other modes of operation of fixed-bed or moving-bed chromatographic reactors or other integrated reaction-separation processes. Acknowledgment The financial support of DFG (Deutsche Forschungsgemeinschaft) within the joint research project SFB 412 on computer aided modeling and simulation of chemical processes for process analysis, synthesis, and operation is gratefully acknowledged by A.K. and S.G.. Furthermore, the financial support provided by the Vietnamese government for T.D.V. and the support of Fonds der Chemischen Industrie are also gratefully acknowledged. Notation

Figure 13. Experimental (solid line) and simulated elution profiles (dashed line) for MA hydrolysis with the parameters in Tables 2 and 3.

mental data published recently.9 In these experiments, diluted samples of MF and MA were injected into a column of 0.25 m in length at a water flow rate of 0.3 mL/min. According to Figure 12, this column length should be almost sufficient to achieve complete conversion of MF but not of MA (Figure 13). The results presented are given in terms experimental and theoretical detector responses. The latter were calculated using calibration factors determined for all of the components. The simulated and experimental detector responses show good agreement. In particular, the expected total conversion of MF and the complete separation of M and FA is confirmed.

ci ) molar concentration in the fluid phase (kmol/m3) Ci ) transformed concentration variable of the fluid phase (kmol/m3) Dap ) apparent dispersion coefficient (m2/s) ) void fraction of the bed F ) volumetric ratio of the solid and fluid phases K ) chemical equilibrium constant Ki ) adsorption constant Khom ) chemical equilibrium constant in terms of the eq fluid phase concentrations Khet eq ) chemical equilibrium constant in terms of the solid phase concentrations khet ) reaction rate constant (m3/(kmol s)) l ) length (m) N ) number of theoretical plates Ns ) number of solutes Nr ) number of reactions νij ) stoichiometric coefficients

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qi ) molar concentration in the solid phase (kmol/m3) Qi ) transformed concentration variable of the solid phase (kmol/m3) rj ) reaction rate (kmol/m3) t ) dimensionless time ht ) time (s) vf ) fluid phase velocity (m/s) z ) dimensionless spatial coordinate zj ) spatial coordinate (m) Abbreviations AA ) acetic acid FA ) formic acid M ) methanol MA ) methyl acetate MF ) methyl formate

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Hyperbolic Systems of Quasilinear Equations; Prentice Hall: Upper Saddle River, NJ, 1989. (7) Barbosa, D.; Doherty, M. F. A new set of composition variables for the representation of reactive phase diagrams. Proc. R. Soc. London 1987, A413, 459-464. (8) Ung, S.; Doherty, M. F. Theory of phase equilibria in multireaction systems. Ind. Eng. Chem. Res. 1995, 34, 3195-3202. (9) Mai, P. T.; Vu, T. D.; Mai, K. X.; Seidel-Morgenstern, A. Analysis of heterogeneously catalyzed ester hydrolysis performed in a chromatographic reactor and in a reaction calorimeter. Ind. Eng. Chem. Res. 2004, 43, 4691-4702. (10) Helfferich, F. G.; Klein, G. Multicomponent Chromatography. Theory of Interference; Marcel Dekker: New York, 1970. (11) Storti, G.; Mazotti, M.; Morbidelli, M.; Carra, S. Robust design of binary countercurrent adsorption separation processes. AIChE J. 1993, 39, 471-492. (12) Lode, F.; Houmard, M.; Migliorini, C.; Mazzotti, M. Continuous reactive chromatography. Chem. Eng. Sci. 2001, 56, 269291. (13) Po¨pken, T.; Steinigeweg, S.; Gmehling, J. Synthesis and hydrolysis of methyl acetate by reactive distillation using structured catalytic packings: Experiments and simulation. Ind. Eng. Chem. Res. 2001, 40, 1566-1574. (14) Sainio, T.; Laatikainen, M.; Paatero, E. Phase equilibria in solvent mixture-ion exchange resin catalyst systems Fluid Phase Equilib. 2004, 218, 269-283. (15) Guiochon, G.; Shirazi, S. G.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: London, 1994.

Received for review February 27, 2005 Revised manuscript received September 16, 2005 Accepted September 16, 2005 IE050256J

Analysis of Ester Hydrolysis Reactions in a Chromatographic Reactor

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