Exploring Conversion of Biphasic Catalytic Reactions: Analytical Solution andParameter Study

Martina Peters,† Marrit F. Eckstein, † Gert Hartjen, ‡ Antje C. Spiess,§ Walter Leitner, † andLasse 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 ingeneral and are particularly important for industrial application of reactions since they are the measure ofoverall reaction efficiency. Biphasic systems are commonly applied in the chemical industry, especially incatalytic reactions. Because of the number of physically important parameters and their interdependencies, itis difficult to predict thermodynamic conversion and yield for biphasic systems. The results of this workallow 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 parametersof interest are discussed. The theoretical considerations are compared to experimental results. Even thoughsimplifications were applied, the predictions fit well with the experimental outcome.

1. Introduction

The use of biphasic systems for the retention of homogeneousor soluble catalysts is a very attractive approach to their large-scale industrial application.1 The ecological and economicbenefits are demonstrated by the Shell higher-olefin process(SHOP)2,3 and the Ruhrchemie/Rhoˆne Poulenc process.4 Bi-phasic catalytic reactions are, therefore, investigated extensivelyin experimental studies for the application of homogeneouschemo- and biocatalysts. Especially the application of syntheti-cally interesting biocatalysts in such systems is arousing interestin industry and academia.5-11 The choice of phases is a keyaspect for the efficiency of such systems, especially if limitationsby the thermodynamic equilibrium have to be taken into accountas for reversible reactions.12-14 However, there are only a fewstudies attempting to develop rational and analytic measures toarrive 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 practicallyquantitative.15 This was predicted by analysis of the thermo-dynamic equilibrium.16 Martinek and co-workers extended theapplicability and were able to take the generation of a secondphase into account.17,18

The analysis of such systems for the thermodynamic bound-aries is important because it allows one to distinguish thermo-dynamic from other influences such as kinetic limitations, e.g.,catalyst deactivation or mass transfer limitations. In our previousstudies, an analytical solution was derived describing thethermodynamically limiting conversion in a monophasic systemconsidering the initial substrate ratioS.19 This approach wasthen extended for the biphasic case, and analytical equationsfor the thermodynamic conversion and yield in biphasic systemswere derived and verified experimentally.20,21

Here, results concerning the exploration of the general trendsutilizing the analytical equation, interdependencies, and sensitiv-ity toward the parameters for the thermodynamic boundariesof conversion and yield in such systems are presented. Especiallythe interdependencies of partition coefficients and targets foroptimization were the aim of our exploration.

2. Results and Discussion

A general reaction scheme including the vast majority ofreactions is the bimolecular reversible reaction with partitioningof all reactants (Figure 1). The catalyst is restricted quantitativelyto the reactive phase (index) R). The second (e.g., organic)phase acts as a reservoir for the reactants and is, thus, definedas the nonreactive phase (index) N) (see the Symbols section).The system was idealized in order to simplify and exemplifythe general underlying trends. This was carried out in regard toconversion and yield as measures for general efficiency of thesystem and guidance for synthetic purposes.

The reactant’s affinity for the nonreactive phase is expressedby the partition coefficientR of the reactant A and is given bythe ratio of concentrations in the reactive [A]R and thenonreactive phases [A]N asR ) ([A] N)/([A] R), with â, γ, andδdefined accordingly. Partition coefficients were regarded aslinear and independent of each other. Activity coefficients andselectivity were set to unity. Initially, no products are present.Selectivity toward the products was regarded as unity. The initialratio of excess and limiting reactantS ) (n(B)0)/(n(A)0) andthe phase volume ratioV ) (VR)/(VN) are influencing theequilibrium position.

For the idealized system, an analytical expression describingthe thermodynamic conversionX of a bimolecular reaction withpartitioning of all reactants in a biphasic system as a functionof the parameters as given in Figure 1 was derived. A closedsolution for the recoverable, and thus technically relevant,amount of product in the nonreactive phase expressed as yieldη was also found. By this, the values can be calculated directly,avoiding numerical simulations. Furthermore, with the analytical

* Corresponding author. Phone:+49(241)8026484. Fax:+49(241)-80626484. E-mail: greiner@itmc.rwth-aachen.de.

† Institute for Technical and Macromolecular Chemistry, RWTHAachen University.

‡ Chair for Mathematics B, RWTH Aachen University.§ Institute for Biochemical Engineering, RWTH Aachen University.

7073Ind. Eng. Chem. Res.2007,46, 7073-7078

10.1021/ie070402g CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 10/02/2007

solutions at hand, derivatives with respect to all the variablesare, in principle, available.

The analytical solution was derived by expressing theconversion of the limiting substrate A by the molar amountsn(A) and n(C)

with the initial total molar amount of An(A)0.The equilibrium constantK is given by mass action law and,

in the idealized system, by concentrations of all reactants inthe reactive phase

Applying the relevant mass balance equations yields

The molar amounts of the reactants in the reactive phase aregiven by the following mass balances:

By this, the expression for the equilibrium constant of thesystem can be written as

We definedm as a factor for the effective concentrations in thereactive phase as

For single-phase systems (V ) 0, thusm ) 1), eq 11 has beensolved numerically22 as well as analytically.19 Solving eq 11with respect toX for two-phase systems withmK * 1 gives

Note that, formK ) 1, it can be shown thatX ) S/(1 + S) isthe continual completion in analogy to the monophasic case.19

Practically, the amount of product in the nonreactive phaseis of particular interest because it gives the recoverable product.With the mass balance for the desired product C (eq 9), thelimiting yield η can be defined as

where (γV/(γV + 1)) is the selectivity factor for the extractionefficiency.

With respect toS with 1 > X > 0 and 1> η > 0, eq 11gives

and with rearrangement of eq 14 gives

This givesS as a function of the desired target variables, thusenabling the calculation of the initial amounts of substratesneeded at a minimum to reach a desired value ofX or η. Asimilar separation of variables to obtain either a solution forVor the partition coefficients is not possible.

The first- and second-order derivatives of the equations arecomplex and reveal no physically reasonable extreme points,neither by direct calculation nor by numerical inspection.Consequently, a parameter study with the analytical expressionswas carried out to show their influences and interdependencies.The boundaries were chosen to exemplify the general trendswithin physically reasonable limits. In the following, theinfluences ofK, S, V, and the partition coefficientsR, â, γ, andδ on X andη will be discussed.

2.1. Equilibrium Constant K. The equilibrium constantKis an intrinsic property of the reaction system and is not easilyaccessible for optimization. The influence ofK follows theintuitional trend that risingK leads to an increase inX andη(Figure 2). The sensitivity ofX and η on other reactionparameters increases with decreasingK. This mirrors thatreactions with smaller thermodynamic driving forces can beinfluenced more easily.

With increasing equilibrium constantK, the conversionXrises. However, at different levels ofK, the influences of phasevolume ratioV and substrate excessSare different. For example,at low K, it is obvious that increasingS leads to a monotonousbut convex increase ofX. For increasingV, a monotonous butconcave behavior may be observed. The bigger the value ofK,the less significant these influences become.

Figure 1. Bimolecular reaction with partitioning of all reactants: A andB are the substrates; C and D are the products; indicesR andN indicatereactive and nonreactive phases, respectively;K is the equilibrium constant;V is the phase volume ratio;R, â, γ, andδ represent the partition coefficients.

X )n(A)0 - n(A)

n(A)0

)n(C)

n(A)0

(1)

K )[C]R[D]R

[A] R[B]R(2)

n(A)R + n(A)N ) n(A)0(1 - X) (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(A)R )n(A)N

RV)

(n(A)0)(1 - X)

(RV + 1)(7)

n(B)R )n(B)N

âV)

(n(A)0)(S- X)

(âV + 1)(8)

n(C)R )n(C)N

γV)

(n(A)0) X

(γV + 1)(9)

n(D)R )n(D)N

δV)

(n(A)0)X

(δV + 1)(10)

K(γV + 1)

(RV + 1)

(δV + 1)

(âV + 1)) X2

(1 - X)(S- X)(11)

m )(γV + 1)

(RV + 1)

(δV + 1)

(âV + 1)(12)

X )mK((S+ 1) - x(1 - S)2 + 4S(mK)-1)

2(mK - 1)(13)

η )n(C)N

n(A)0

) γV(γV + 1)

X

)

γV (δV+1)

(RV+1)(âV+1)K((S+ 1) - x(1 - S)2 + 4S(mK)-1)

2(mK - 1)(14)

S(X) ) X[1 + ( 1

(X-1 - 1)mK)] (15)

S(η) ) η(1 + (γV)-1)(1 + 1

η(1 + (γV)-1)mK) (16)

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2.2. Cosubstrate/Substrate RatioS. The influence of thecosubstrate/substrate ratioSon the conversion is similar to theinfluence ofK as risingS, and thus higher excess of cosubstrate,leads to increasedX (Figure 3, upper row). The same holds truefor the influence ofSonη (Figure 3, lower row). The sensitivityof X andη as a function ofS is mainly influenced byK and theeffective ratio in the reactive phase is given by the partitioncoefficients. Sensitivity with respect toS is greater if the affinityof the products to the nonreactive phase is low. The influenceon η is accordingly.

2.3. Partition Coefficients r, â, γ, δ. In general, there aremany different combinations of partition coefficients. Theinfluence of partition coefficients depends on the species. Forthe variation of one at a time, the influence onX andη dependson whether substrates or products are concerned. Whereas anincrease ofR or â decreases bothX and η, a high affinity ofthe products to the organic phase, as expressed by an increaseof γ or δ, leads to increasingX andη.

However, the independence of partition coefficients is anunlikely scenario. In the majority of catalytic reactions, changesin the molecules are small and partition properties are intercon-

nected. The ratio and absolute values of the partition coefficientsdominate the influence ofV. Practically relevant cases werechosen that reflect the interconnection of the chemical natureof the reactants. This was done in order to demonstrate generaltrends. As shown in Figure 4, the three cases are (i) that mostof the substrates are contained in the nonreactive phase, (ii) thatthere is equal distribution of all substrates and products, and(iii) that most of the products are contained in the nonreactivephase. In the following, we will study the influences of the phasevolume ratioV and the cosubstrate/substrate ratioS on theconversionX and on the yieldη for these different combinations.Prediction of solubility and thereby partition coefficients orliquid/liquid equilibria, respectively, is the target of the currentresearch.21,20

2.4. Phase Volume RatioV. The influence ofV is interde-pendent with the partition coefficients. A well-behaved systemwith respect toX and η can be expected if the products aresoluble in the reactive phase and the products are easily extracted(see above). In practice, such systems are rarely found, becausethe partition coefficients for substrate and product will be related,especially if overall changes in the molecule are small.

Figure 2. ConversionX and yieldη as a function ofS andV for different K; R ) â ) 1, γ ) δ ) 0.01.

Figure 3. (a)-(c) Influence ofSon X; (d)-(f) influence ofSon η (note the difference in ordinate scaling). For all sections, forV ) 10: blue,S) 1; red,S ) 2; black,S ) 10; green,S ) 100.

Figure 4. Dependency of conversionX and yieldη on the partition coefficients.

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There is a turning point for the influence ofV on X withregard to the partition coefficients. For equal partitioning of allreactants,V has no influence onX and marks a turning point ofthe general behavior (Figure 5, upper row). Furthermore, whenR ) γ andâ ) δ or R ) δ andâ ) γ, V does not affect theconversionX. With regard toη, no such reversal of trends isapparent (Figure 5, lower row). For high affinity of the substratesto the nonreactive phase, increasingV will lead to a decreaseof X, whereas for good extraction of the products, an increasein V will increaseX. Generally, three cases can be identifiedwith practical relevance: (i) IfR ) â is smaller thanγ ) δ, Xand η rise with increasingV. (ii) If all partition coefficientsequal each other, the phase volume ratioV does not influenceX, but theη rises with increasingV. (iii) If R ) â is biggerthanγ ) δ, X andη will react contrary onV: with rising V, Xwill decrease, whereasη increases.

So, even thoughX may decrease with risingV because oflower availability of the products in the reactive phase,ηgenerally increases with increasingV. As m approaches unitywhenV takes much greater values than the partition coefficients,X andη converge for great values ofV. This can be either bysimultaneous increase ofη and decrease ofX or by bothX andη approaching unity for high values ofV.

The influence ofV on η for fixed values ofS is shown inFigure 5. The trends described above for the influence ofV onX reverse for the influence ofV on η. This difference dependsstrongly on the value forγ when comparing eqs 15 and 14.

2.5. Experimental Verification. To show the accuracy ofthe developed mathematical expression, the reduction of aceto-phenone to (R)-1-phenylethanol with 2-propanol, catalyzed byalcohol dehydrogenase fromLactobacillus breVis (LB-ADH),was examined with respect to its equilibrium conversionX andequilibrium yieldη in several biphasic reaction media.20,21Theresults are depicted in Figure 6. CalculatedX andη are in allcases lower than experimentally obtained values, but the generaltrends are predicted correctly and the errors are in a reasonablerange.

Whereas the partition coefficients are intrinsically linked tothe solvents and reactants,SandV are independent parameters.

For methyltert-butyl ether (MTBE) as the nonreactive phase,K ) 0.426 and the partition coefficients for acetophenone,2-propanol, phenylethanol, and acetone were determined asR) 66.7,â ) 3.5, γ ) 32.3, andδ ) 1.0, respectively.20,21 Fora given reaction system, conversionX and yieldη are the keytargets, especially in terms of downstream processing. Followingthe general guidelines valid for this model system, conversioncan be maximized with an increase ofS and a decrease ofV.For desiredX and η, isoconversion and -yield lines can becalculated according to eqs 16 and 17, respectively. With fixedX andη, S andV are dependent on each other, following thegeneral trends (Figure 7). Generally,S must be increased toachieve the same conversion whenV is increased. Furthermore,for higher values ofV, the excess of cosubstrate (S) needed tomaintain the same values ofX andη converges. Each particularsystem has to be judged on a individual basis because thisdependency is dependent on both the absolute values for thepartition coefficients and their combination. For well-behavedsystems with higher extraction efficiency of the products, the

Figure 5. (a)-(c) Influence ofV on X; (d)-(f) influence ofV on η (note the difference in ordinate scaling). For all sections, forS) 1: blue,V ) 0.1; red,V ) 1; black,V ) 10; green,V ) 100.

Figure 6. Comparison of calculated and experimentally measured20,21

equilibrium conversionX and equilibrium yieldη for different solvents forthe reduction of acetophenone to (R)-1-phenylethanol at 30°C.

7076 Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007

behavior is reversed so that, with increasingV, S is decreasedat the same level ofX andη.

3. Conclusion

An exact mathematical derivation of an analytical expressionfor conversion and yield in a general ideal catalyzed biphasicsystem is discussed. With this exact expression, it is possibleto generalize the influence of different parameters. Overall, aprediction of conversion and yield is possible. Because of theinterdependencies of the different parameters, case-by-caseconsideration is necessary.

For a given system, the degrees of freedom are reduced tosubstrate ratioSand phase volume ratioV. Here, isoconversionlines for substrate ratioSand phase volume ratioV are helpfultools for experimental planning. Because a given system canonly be influenced within the thermodynamically imposedboundaries, knowledge and prediction of this experimental frameenables one to reveal and adequately discuss other relevantinfluences.

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 describingthe analytical details, the determination of partition coefficients,and the measurement of equilibrium conversion and yield atfull length.20,21

Acknowledgment

The work is financially supported by the DFG Graduierten-kolleg GK 1166 BioNoCo (“Biocatalysis in non-conven-tional media”, www.bionoco.org) and CRC 540 (www.sfb540.rwth-aachen.de). M.F.E. would like to thank the Ministry ofInnovation, Science, Research and Technology of the State ofNorth-Rhine-Westphalia, Germany (MIWFT), for a Lise-Meit-ner-Scholarship. We thank Christian Steffens (Bayer MaterialScience, Germany) and Claas Michalik (RWTH Aachen Uni-versity, Germany) for fruitful discussion.

Symbols

A ) limiting substrate AB ) excess substrate BC ) desired product CD ) coupled product DK ) equilibrium constantm ) factor for the effective reactive concentrationsn ) molar amount

S ) initial ratio S ) (n(B)0)/(n(A)0)V ) phase volume ratioV ) (VR)/(VN)X ) conversionη ) yieldR ) partition coefficient for limiting substrate Aâ ) partition coefficient for excess substrate Bδ ) partition coefficient for coupled product Dγ ) partition coefficient for desired product CN ) index for nonreactive phaseR ) index for reactive phase0 ) index for initial conditions

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Figure 7. (Left) ConversionX and yieldη as functions ofS andV at 30°C; (Right) isoconversion and -yield lines ofSandV for the experimentalsystem 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, â ) 3.5, γ )32.3,δ ) 1.0, K ) 0.426).

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ReceiVed for reView March 16, 2007ReVised manuscript receiVed July 26, 2007

AcceptedAugust 1, 2007

IE070402G

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