Ind. Eng. Chem. Res. 2007, 46, 7073-7078
7073
Exploring Conversion of Biphasic Catalytic Reactions: Analytical Solution and Parameter Study Martina Peters,† Marrit F. Eckstein,† Gert Hartjen,‡ Antje C. Spiess,§ Walter Leitner,† and Lasse Greiner*,† Institute for Technical and Macromolecular Chemistry, RWTH Aachen UniVersity, Worringerweg 1, 52074 Aachen, Germany; Chair for Mathematics B, RWTH Aachen UniVersity, Templergraben 64, 52062 Aachen, Germany; and Institute for Biochemical Engineering, RWTH Aachen UniVersity, Worringerweg 1, 52074 Aachen, Germany
Conversion, yield, selectivity, and catalyst consumption are among the key targets of chemical reactions in general and are particularly important for industrial application of reactions since they are the measure of overall reaction efficiency. Biphasic systems are commonly applied in the chemical industry, especially in catalytic reactions. Because of the number of physically important parameters and their interdependencies, it is difficult to predict thermodynamic conversion and yield for biphasic systems. The results of this work allow the prediction of the maximum conversion as well as the yield of any catalyzed biphasic reaction. Analytical solutions for both conversion and yield as a function of substrate ratio, phase volume ratio, equilibrium constant, and partition coefficients are given. General trends and interdependencies of the parameters of interest are discussed. The theoretical considerations are compared to experimental results. Even though simplifications were applied, the predictions fit well with the experimental outcome. 1. Introduction The use of biphasic systems for the retention of homogeneous or soluble catalysts is a very attractive approach to their largescale industrial application.1 The ecological and economic benefits are demonstrated by the Shell higher-olefin process (SHOP)2,3 and the Ruhrchemie/Rhoˆne Poulenc process.4 Biphasic catalytic reactions are, therefore, investigated extensively in experimental studies for the application of homogeneous chemo- and biocatalysts. Especially the application of synthetically interesting biocatalysts in such systems is arousing interest in industry and academia.5-11 The choice of phases is a key aspect for the efficiency of such systems, especially if limitations by the thermodynamic equilibrium have to be taken into account as for reversible reactions.12-14 However, there are only a few studies attempting to develop rational and analytic measures to arrive at generic solutions for this problem. Klibanov et al. showed that the yield of an ester synthesis is ∼0.01% in water, whereas, in a biphasic system, it is practically quantitative.15 This was predicted by analysis of the thermodynamic equilibrium.16 Martinek and co-workers extended the applicability and were able to take the generation of a second phase into account.17,18 The analysis of such systems for the thermodynamic boundaries is important because it allows one to distinguish thermodynamic from other influences such as kinetic limitations, e.g., catalyst deactivation or mass transfer limitations. In our previous studies, an analytical solution was derived describing the thermodynamically limiting conversion in a monophasic system considering the initial substrate ratio S.19 This approach was then extended for the biphasic case, and analytical equations for the thermodynamic conversion and yield in biphasic systems were derived and verified experimentally.20,21 * Corresponding author. Phone: +49(241)8026484. Fax: +49(241)80626484. E-mail:
[email protected]. † Institute for Technical and Macromolecular Chemistry, RWTH Aachen University. ‡ Chair for Mathematics B, RWTH Aachen University. § Institute for Biochemical Engineering, RWTH Aachen University.
Here, results concerning the exploration of the general trends utilizing the analytical equation, interdependencies, and sensitivity toward the parameters for the thermodynamic boundaries of conversion and yield in such systems are presented. Especially the interdependencies of partition coefficients and targets for optimization were the aim of our exploration. 2. Results and Discussion A general reaction scheme including the vast majority of reactions is the bimolecular reversible reaction with partitioning of all reactants (Figure 1). The catalyst is restricted quantitatively to the reactive phase (index ) R). The second (e.g., organic) phase acts as a reservoir for the reactants and is, thus, defined as the nonreactive phase (index ) N) (see the Symbols section). The system was idealized in order to simplify and exemplify the general underlying trends. This was carried out in regard to conversion and yield as measures for general efficiency of the system and guidance for synthetic purposes. The reactant’s affinity for the nonreactive phase is expressed by the partition coefficient R of the reactant A and is given by the ratio of concentrations in the reactive [A]R and the nonreactive phases [A]N as R ) ([A]N)/([A]R), with β, γ, and δ defined accordingly. Partition coefficients were regarded as linear and independent of each other. Activity coefficients and selectivity were set to unity. Initially, no products are present. Selectivity toward the products was regarded as unity. The initial ratio of excess and limiting reactant S ) (n(B)0)/(n(A)0) and the phase volume ratio V ) (VR)/(VN) are influencing the equilibrium position. For the idealized system, an analytical expression describing the thermodynamic conversion X of a bimolecular reaction with partitioning of all reactants in a biphasic system as a function of the parameters as given in Figure 1 was derived. A closed solution for the recoverable, and thus technically relevant, amount of product in the nonreactive phase expressed as yield η was also found. By this, the values can be calculated directly, avoiding numerical simulations. Furthermore, with the analytical
10.1021/ie070402g CCC: $37.00 © 2007 American Chemical Society Published on Web 10/02/2007
7074
Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007
For single-phase systems (V ) 0, thus m ) 1), eq 11 has been solved numerically22 as well as analytically.19 Solving eq 11 with respect to X for two-phase systems with mK * 1 gives
X) Figure 1. Bimolecular reaction with partitioning of all reactants: A and B are the substrates; C and D are the products; indices R and N indicate reactive and nonreactive phases, respectively; K is the equilibrium constant; V is the phase volume ratio; R, β, γ, and δ represent the partition coefficients.
solutions at hand, derivatives with respect to all the variables are, in principle, available. The analytical solution was derived by expressing the conversion of the limiting substrate A by the molar amounts n(A) and n(C)
X)
n(A)0 - n(A) n(A)0
n(C) n(A)0
(1)
with the initial total molar amount of A n(A)0. The equilibrium constant K is given by mass action law and, in the idealized system, by concentrations of all reactants in the reactive phase R
K)
[C] [D]
R
[A]R[B]R
(2)
Applying the relevant mass balance equations yields
(3)
n(B) ) n(A)0(S - X)
(4)
n(C)R + n(C)N ) n(A)0X
(5)
n(D) ) n(A)0X
(6)
N
The molar amounts of the reactants in the reactive phase are given by the following mass balances:
n(A)R )
n(A)N (n(A)0)(1 - X) ) RV (RV + 1)
(7)
n(B)R )
n(B)N (n(A)0)(S - X) ) βV (βV + 1)
(8)
n(C)R )
n(C)N (n(A)0) X ) γV (γV + 1)
(9)
n(D)R )
n(D)N (n(A)0)X ) δV (δV + 1)
(10)
By this, the expression for the equilibrium constant of the system can be written as
K
(γV + 1) (δV + 1) X2 ) (RV + 1) (βV + 1) (1 - X)(S - X)
(11)
We defined m as a factor for the effective concentrations in the reactive phase as
m)
(γV + 1) (δV + 1) (RV + 1) (βV + 1)
n(C)N γV ) X n(A)0 (γV + 1)
γV (δV+1) K((S + 1) - x(1 - S)2 + 4S(mK)-1) (RV+1)(βV+1) ) 2(mK - 1) (14) where (γV/(γV + 1)) is the selectivity factor for the extraction efficiency. With respect to S with 1 > X > 0 and 1 > η > 0, eq 11 gives
[ (
S(X) ) X 1 +
n(A) + n(A) ) n(A)0(1 - X) R
(12)
(13)
Note that, for mK ) 1, it can be shown that X ) S/(1 + S) is the continual completion in analogy to the monophasic case.19 Practically, the amount of product in the nonreactive phase is of particular interest because it gives the recoverable product. With the mass balance for the desired product C (eq 9), the limiting yield η can be defined as
η) )
mK((S + 1) - x(1 - S)2 + 4S(mK)-1) 2(mK - 1)
(X
-1
1 - 1)mK
)]
(15)
and with rearrangement of eq 14 gives
(
S(η) ) η(1 + (γV)-1) 1 +
1 η(1 + (γV)-1)mK
)
(16)
This gives S as a function of the desired target variables, thus enabling the calculation of the initial amounts of substrates needed at a minimum to reach a desired value of X or η. A similar separation of variables to obtain either a solution for V or the partition coefficients is not possible. The first- and second-order derivatives of the equations are complex and reveal no physically reasonable extreme points, neither by direct calculation nor by numerical inspection. Consequently, a parameter study with the analytical expressions was carried out to show their influences and interdependencies. The boundaries were chosen to exemplify the general trends within physically reasonable limits. In the following, the influences of K, S, V, and the partition coefficients R, β, γ, and δ on X and η will be discussed. 2.1. Equilibrium Constant K. The equilibrium constant K is an intrinsic property of the reaction system and is not easily accessible for optimization. The influence of K follows the intuitional trend that rising K leads to an increase in X and η (Figure 2). The sensitivity of X and η on other reaction parameters increases with decreasing K. This mirrors that reactions with smaller thermodynamic driving forces can be influenced more easily. With increasing equilibrium constant K, the conversion X rises. However, at different levels of K, the influences of phase volume ratio V and substrate excess S are different. For example, at low K, it is obvious that increasing S leads to a monotonous but convex increase of X. For increasing V, a monotonous but concave behavior may be observed. The bigger the value of K, the less significant these influences become.
Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7075
Figure 2. Conversion X and yield η as a function of S and V for different K; R ) β ) 1, γ ) δ ) 0.01.
Figure 3. (a)-(c) Influence of S on X; (d)-(f) influence of S on η (note the difference in ordinate scaling). For all sections, for V ) 10: blue, S ) 1; red, S ) 2; black, S ) 10; green, S ) 100.
Figure 4. Dependency of conversion X and yield η on the partition coefficients.
2.2. Cosubstrate/Substrate Ratio S. The influence of the cosubstrate/substrate ratio S on the conversion is similar to the influence of K as rising S, and thus higher excess of cosubstrate, leads to increased X (Figure 3, upper row). The same holds true for the influence of S on η (Figure 3, lower row). The sensitivity of X and η as a function of S is mainly influenced by K and the effective ratio in the reactive phase is given by the partition coefficients. Sensitivity with respect to S is greater if the affinity of the products to the nonreactive phase is low. The influence on η is accordingly. 2.3. Partition Coefficients r, β, γ, δ. In general, there are many different combinations of partition coefficients. The influence of partition coefficients depends on the species. For the variation of one at a time, the influence on X and η depends on whether substrates or products are concerned. Whereas an increase of R or β decreases both X and η, a high affinity of the products to the organic phase, as expressed by an increase of γ or δ, leads to increasing X and η. However, the independence of partition coefficients is an unlikely scenario. In the majority of catalytic reactions, changes in the molecules are small and partition properties are intercon-
nected. The ratio and absolute values of the partition coefficients dominate the influence of V. Practically relevant cases were chosen that reflect the interconnection of the chemical nature of the reactants. This was done in order to demonstrate general trends. As shown in Figure 4, the three cases are (i) that most of the substrates are contained in the nonreactive phase, (ii) that there is equal distribution of all substrates and products, and (iii) that most of the products are contained in the nonreactive phase. In the following, we will study the influences of the phase volume ratio V and the cosubstrate/substrate ratio S on the conversion X and on the yield η for these different combinations. Prediction of solubility and thereby partition coefficients or liquid/liquid equilibria, respectively, is the target of the current research.21,20 2.4. Phase Volume Ratio V. The influence of V is interdependent with the partition coefficients. A well-behaved system with respect to X and η can be expected if the products are soluble in the reactive phase and the products are easily extracted (see above). In practice, such systems are rarely found, because the partition coefficients for substrate and product will be related, especially if overall changes in the molecule are small.
7076
Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007
Figure 5. (a)-(c) Influence of V on X; (d)-(f) influence of V on η (note the difference in ordinate scaling). For all sections, for S ) 1: blue, V ) 0.1; red, V ) 1; black, V ) 10; green, V ) 100.
There is a turning point for the influence of V on X with regard to the partition coefficients. For equal partitioning of all reactants, V has no influence on X and marks a turning point of the general behavior (Figure 5, upper row). Furthermore, when R ) γ and β ) δ or R ) δ and β ) γ, V does not affect the conversion X. With regard to η, no such reversal of trends is apparent (Figure 5, lower row). For high affinity of the substrates to the nonreactive phase, increasing V will lead to a decrease of X, whereas for good extraction of the products, an increase in V will increase X. Generally, three cases can be identified with practical relevance: (i) If R ) β is smaller than γ ) δ, X and η rise with increasing V. (ii) If all partition coefficients equal each other, the phase volume ratio V does not influence X, but the η rises with increasing V. (iii) If R ) β is bigger than γ ) δ, X and η will react contrary on V: with rising V, X will decrease, whereas η increases. So, even though X may decrease with rising V because of lower availability of the products in the reactive phase, η generally increases with increasing V. As m approaches unity when V takes much greater values than the partition coefficients, X and η converge for great values of V. This can be either by simultaneous increase of η and decrease of X or by both X and η approaching unity for high values of V. The influence of V on η for fixed values of S is shown in Figure 5. The trends described above for the influence of V on X reverse for the influence of V on η. This difference depends strongly on the value for γ when comparing eqs 15 and 14. 2.5. Experimental Verification. To show the accuracy of the developed mathematical expression, the reduction of acetophenone to (R)-1-phenylethanol with 2-propanol, catalyzed by alcohol dehydrogenase from Lactobacillus breVis (LB-ADH), was examined with respect to its equilibrium conversion X and equilibrium yield η in several biphasic reaction media.20,21 The results are depicted in Figure 6. Calculated X and η are in all cases lower than experimentally obtained values, but the general trends are predicted correctly and the errors are in a reasonable range. Whereas the partition coefficients are intrinsically linked to the solvents and reactants, S and V are independent parameters.
Figure 6. Comparison of calculated and experimentally measured20,21 equilibrium conversion X and equilibrium yield η for different solvents for the reduction of acetophenone to (R)-1-phenylethanol at 30 °C.
For methyl tert-butyl ether (MTBE) as the nonreactive phase, K ) 0.426 and the partition coefficients for acetophenone, 2-propanol, phenylethanol, and acetone were determined as R ) 66.7, β ) 3.5, γ ) 32.3, and δ ) 1.0, respectively.20,21 For a given reaction system, conversion X and yield η are the key targets, especially in terms of downstream processing. Following the general guidelines valid for this model system, conversion can be maximized with an increase of S and a decrease of V. For desired X and η, isoconversion and -yield lines can be calculated according to eqs 16 and 17, respectively. With fixed X and η, S and V are dependent on each other, following the general trends (Figure 7). Generally, S must be increased to achieve the same conversion when V is increased. Furthermore, for higher values of V, the excess of cosubstrate (S) needed to maintain the same values of X and η converges. Each particular system has to be judged on a individual basis because this dependency is dependent on both the absolute values for the partition coefficients and their combination. For well-behaved systems with higher extraction efficiency of the products, the
Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7077
Figure 7. (Left) Conversion X and yield η as functions of S and V at 30 °C; (Right) isoconversion and -yield lines of S and V for the experimental system from refs 20 and 21 (solid ) X, dashed ) η; in top-down order: blue, 0.85; red, 0.75; black, 0.65; green, 0.55; R ) 66.7, β ) 3.5, γ ) 32.3, δ ) 1.0, K ) 0.426).
behavior is reversed so that, with increasing V, S is decreased at the same level of X and η. 3. Conclusion An exact mathematical derivation of an analytical expression for conversion and yield in a general ideal catalyzed biphasic system is discussed. With this exact expression, it is possible to generalize the influence of different parameters. Overall, a prediction of conversion and yield is possible. Because of the interdependencies of the different parameters, case-by-case consideration is necessary. For a given system, the degrees of freedom are reduced to substrate ratio S and phase volume ratio V. Here, isoconversion lines for substrate ratio S and phase volume ratio V are helpful tools for experimental planning. Because a given system can only be influenced within the thermodynamically imposed boundaries, knowledge and prediction of this experimental frame enables one to reveal and adequately discuss other relevant influences. 4. Experimental Section Algebraic transformations were carried out with Maple 10 (The Mathworks). Graphs have been created using PSTricks. We have previously reported an experimental section describing the analytical details, the determination of partition coefficients, and the measurement of equilibrium conversion and yield at full length.20,21 Acknowledgment The work is financially supported by the DFG Graduiertenkolleg GK 1166 BioNoCo (“Biocatalysis in non-conventional media”, www.bionoco.org) and CRC 540 (www.sfb540. rwth-aachen.de). M.F.E. would like to thank the Ministry of Innovation, Science, Research and Technology of the State of North-Rhine-Westphalia, Germany (MIWFT), for a Lise-Meitner-Scholarship. We thank Christian Steffens (Bayer Material Science, Germany) and Claas Michalik (RWTH Aachen University, Germany) for fruitful discussion. Symbols A ) limiting substrate A B ) excess substrate B C ) desired product C D ) coupled product D K ) equilibrium constant m ) factor for the effective reactive concentrations n ) molar amount
S ) initial ratio S ) (n(B)0)/(n(A)0) V ) phase volume ratio V ) (VR)/(VN) X ) conversion η ) yield R ) partition coefficient for limiting substrate A β ) partition coefficient for excess substrate B δ ) partition coefficient for coupled product D γ ) partition coefficient for desired product C N ) index for nonreactive phase R ) index for reactive phase 0 ) index for initial conditions Literature Cited (1) Cornils, B.; Herrmann, W. A.; Horvath, I. T.; Leitner, W.; Mecking, S.; Olivier-Bourbigou, H.; Vogt, D. (Hrsg.) Multiphase Homogeneous Catalysis; Wiley-VCH: Weinheim, Germany, 2005. (2) Keim, W. Multiphase catalysis and its potential in catalytic processes: The story of biphasic homogeneous catalysis. Green Chem. 2003, 5, 105-111. (3) Keim, W. Nickel: An Element with Wide Application in Industrial Homogenous Catalysis. Angew. Chem., Int. Ed. Engl. 1990, 29, 235. (4) Kohlpaintner, C. W.; Fischer, R. W.; Cornils, B. Aqueous Biphasic Catalysis: Ruhrchemie/Rhoˆne-Poulenc Oxo Process. Appl. Catal., A. 2001, 221, 219-225. (5) Sheldon, R. A. Green solvents for sustainable organic synthesis: State of the art. Green Chem. 2005, 7, 267. (6) Ansorge-Schumacher, M. B.; Greiner, L.; Schroeper, F.; Mirtschin, S.; Hischer, T. Operational concept for the improved synthesis of (R)-3,3′furoin and related hydrophobic compounds with benzaldehyde lyase. Biotechnol. J. 2006, 1, 564. (7) Cull, S. G.; Holbrey, J. D.; Vargas-Mora, V.; Seddon, K. R.; Lye, G. J. Room-temperature ionic liquids as replacements for organic solvents in multiphase bioprocess operations. Biotechnol. Bioeng. 2000, 69, 227. (8) Reetz, M. T.; Wiesenho¨fer, W.; Francio`, G.; Leitner, W. Continuous flow enzymatic kinetic resolution and enantiomer separation using ionic liquid/supercritical carbon dioxide media. AdV. Synth. Catal. 2003, 345, 1221. (9) Faber, K. Biotransformations in Organic Chemistry. A Textbook; Springer: Berlin, 2004. (10) Faber, K. Industrial Biotransformations; Springer: Wiley-VCH, Weinheim, Germany, 2006. (11) Carrea, G. Biocatalysis in water-organic solvent two phase systems. Trends Biotechnol. 1984, 2, 102-106. (12) Halling, P. J. Thermodynamic predictions for biocatalysis in nonconventional media: Theory, tests, and recommendations for experimental design and analysis. Enzyme Microb. Technol. 1994, 16, 178206. (13) Ulijn, R. V.; Halling, P. J. Solid-to-solid biocatalysis: Thermodynamic feasibility and energy efficiency. Green Chem. 2004, 6, 488496. (14) Uma, R.; Cre´visy, C.; Gre´e, R. Transposition of allylic alcohols into carbonyl compounds mediated by transition metal complexes. Chem. ReV. 2003, 103, 27-51. (15) Klibanov, A. M.; Samokhin, G. P.; Martinek, K.; Berezin, I. V. A new approach to preparative enzymic engineering. Biotechnol. Bioeng. 1977, 19, 1351-1361. (16) Antczak, T.; Hiler, D.; Krystynowicz, A.; Bielecki, S.; Galas, E. Mathematical modelling of ester synthesis by lipase in biphasic system. J. Mol. Catal. B: Enzym. 2001, 11, 1043-1050. (17) Martinek, K.; Semenov, A. N.; Berezin, I. V. Enzymatic synthesis in bipasic aqueous-organic systems. I. Chemical equilibrium shift. Biochim. Biophys. Acta 1981, 658, 76-89. (18) Martinek, K.; Semenov, A. N. Enzymatic synthesis in bipasic aqueous-organic systems. II. Shift of ionic equilibrium. Biochim. Biophys. Acta 1981, 658, 90-101. (19) Greiner, L.; Laue, S.; Liese, A.; Wandrey, C. Continuous homogeneos asymmetric transfer hydrogenation of ketones: Lessons from kinetics. Chemistry 2006, 12, 1818-1823. (20) Eckstein, M. F.; Peters, M.; Lembrecht, J.; Spiess, A. C.; Greiner. L. Maximise Equilibrium Conversion in Biphasic Catalysed Reactions: Mathematical Description and Practical Guideline. AdV. Synth. Catal. 2006, 348, 1591-1596. (21) Eckstein, M. F.; Lembrecht, J.; Schumacher, J.; Peters, M.; Roosen, C.; Greiner, L.; Eberhard, W.; Spiess, A.; Leitner, W.; Kragl, U. Maximize
7078
Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007
your equilibrium conversion in biphasic catalysed reactions: How to obtain equilibrium constants for reactions of practical relevance? AdV. Synth. Catal. 2006, 348, 1597-1604. (22) Wisman, R. V.; de Vries, J. G.; Deelman, B. J.; Heeres, H. J. Kinetic studies on the asymmetric transfer hydrogenation of acetophenone using a homogeneous ruthenium catalyst with a chiral amino-alcohol ligand. Org. Process Res. DeV. 2006, 10, 423-429.
ReceiVed for reView March 16, 2007 ReVised manuscript receiVed July 26, 2007 Accepted August 1, 2007
IE070402G
