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Mass Transfer through a Biocatalytic Membrane Reactor Endre Nagy,* Anita Lepossa, and Zsolt Prettl University of Pannonia, FIT, Research Institute of Chemical and Process Engineering, P.O. Box 158, 8201 Veszprem, Hungary ABSTRACT: The mass transport through biocatalytic membrane reactor has been investigated. The enzyme or living cells were immobilized in the porous membrane matrix, and the substrate(s) was fed by convective and diffusive flows through the membrane layer. The MichaelisMenten kinetics, and its limiting cases—namely, the first-order and zero-order reactions—were investigated. The inlet and outlet mass-transfer rates were defined for all three cases in two operating modes, namely, without mass transport into the immiscible sweep phase (Model A) and with mass transport into the sweep phase (Model B). The difference between these modes is in the boundary condition at the outlet side of membrane reactor. In case of Model A, the concentration gradient is equal to zero, while that is larger than zero for Model B. In the latter case, the outlet concentration is determined not only by the reaction conditions in the membrane but by the flow conditions in the sweep phase on the permeate side. Analytical approaches of the solution were developed to evaluate the MichaelisMenten kinetics with a single substrate and with two substrates. Concentration distributions and mass-transfer rates are compared by applying the different reaction orders and operating modes. As a case study, the model equations have been applied to calculate the kinetic resolution of the (S)-ibuprofen.
’ INTRODUCTION Membrane reactors1,2 and membrane bioreactors3,4 are promising unit operations with many industrial application possibilities.1,3 The membrane reactor has two important advantages: (i) proper adjustment of the local concentration of the reactants is possible, which can essentially improve the selectivity,1 and (ii) in situ separation of the product(s). This work is focused primarily on the mathematical description of the mass transport in the hollow-fiber bioreactor with biocatalyst, either live cells or enzymes, inoculated into the shell and immobilized within the membrane matrix. The main advantages of the hollow-fiber bioreactor are these: a large specific surface area (internal and external surface of the membrane) for cell adhesion or enzyme immobilization; the ability to grow cells to high density; the possibility for simultaneous reaction and separation; a relatively short diffusion path in the membrane layer; the presence of convective velocity through the membrane if it is necessary in order to avoid the nutrient limitation.57 A membrane layer is especially useful for immobilizing entire cells (bacteria, yeast, mammalian and plant cells,7,8 and bioactive molecules such as enzymes9,10) to produce a wide variety of chemicals and substances. The enzymatic bioconversion processes are of increasing use in the production, transformation, and valorization of raw materials.9,1116 Important applications have been developed in the food industry, in the field of fine chemicals, or even for environmental purposes.911,17,18 The operating modes of bioreactors can be different.20,21 In every case, the substrate is transported by convective (depending on the transmembrane pressure) and diffusive flows through the biocatalytic membrane layer. In immobilized cell bioreactors, nutrients should be maintained above limiting levels. The limited transport of nutrients can cause serious damage in production.6,19 The introduction of convective transport is crucial in overcoming the diffusive mass-transport limitation of nutrients,5,22 especially that of the sparingly soluble oxygen. The flow through this membrane is proportional to the transmembrane pressure drop and is inversely proportional to r 2011 American Chemical Society
the fluid viscosity. Several investigators predicted the radial velocity, by applying the Darcy equation and the continuity equations.7,20,21,23,24 Recently, Nagy25 analyzed the equations for the radial velocity through the membrane under different operating conditions. A considerable amount of mathematical modeling analysis has been developed to describe the mass-transport limitations in a hollow-fiber bioreactor applying the zero-order, first-order kinetics or even the MichaelisMenten kinetics.7,14,15,2529 These papers calculated the axial or/and radial substrate profiles in the fiber lumen, on the shell side or in the catalytic membrane layer, applying different simplifications. The complete description of all three regions—namely, the lumen, shell, and the catalytic membrane layer—needs the solution of the complex Navier Stokes flow models completing them with the component mass balance and/or energy balance equations.30 In this paper, we analyze the mass transport in the catalytic membrane layer, taking into account both the diffusive and convective flows inside the membrane reactor. The main objective of this study is to give closed, as simple as possible, mathematical equations in order to predict the concentration distribution and the mass-transfer rate through a biocatalyst membrane layer. As bioreaction kinetics, the zero- and first-order reactions with analytical solution, as well as the general MichaelisMenten or Monod kinetics with an analytical approach for its solution, were investigated, by applying one or two substrate components. This latter case is important for aerobic cell culture where both oxygen and carbon source are needed for the bioreaction. Applying the solutions obtained by different operating modes, the inlet and outlet mass-transfer rates have been Special Issue: Nigam Issue Received: April 4, 2011 Accepted: November 11, 2011 Revised: August 23, 2011 Published: November 11, 2011 1635
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Figure 1. Enzyme membrane bioreactors: (A) Model A, ultrafiltration mode, and (B) Model B, recirculation mode.
given. These mass-transfer rates then can be incorporated into the mass balance equations given for the lumen and for the extracapillary space. Obviously, the convective velocity into the membrane layer should be known. Based on the methodology applied, it may essentially be simplified as the description of the hollow-fiber membrane reactor separating the solution of the lumen and the shell sides. On the other hand, the substrate limitation can be predicted by simple mathematical equation given by closed forms for, e.g., the MichaelisMenten kinetics or, in the case of a microbial bioreactor, the Monod kinetics for micro-organism growth. The analytical approach can easily be applied for substrate inhibition and product inhibition kinetics, and, as will be shown for two substrates bioreactions as well as for processes in cylindrical space, namely, when the effect of the cylindrical coordinates cannot be neglected. On the other hand, these equations can easily be adapted to the variable kinetics parameters of membrane reactors.
’ THEORY Simple MichaelisMenten kinetics for enzymatic reaction or Monod kinetics can be applied to describe a bioreaction. Cells or enzyme are generally immobilized in the porous membrane matrix in the case of a membrane bioreactor. The substrate (or substrates) is transported by diffusion or by diffusion plus convection, depending on the transmembrane pressure. The operating mode of enzyme membrane reactor could be different.17,20,21 Here, two operating modes will be discussed, namely, the case when, during the bioreaction in the catalytic membrane layer, the unreacted substrate cannot enter the immiscible sweep phase (Model A) from the catalytic membrane layer and the case when the reacting component can enter the sweep phase (Model B) from the catalytic membrane layer. The difference between these modes is in the boundary condition at the outlet side of membrane reactor. In case of Model A, the concentration gradient is equal to zero, while, for Model B, it is larger than zero. In the first mode, the outlet concentration of the reactant (concentration at R = 1 + δ/r0) is determined by the membrane transport parameters, while in the latter case, the outlet concentration is determined by the flow conditions of the sweep phase. The differential mass balance equation for a substrate component, for a polymeric or macroporous ceramic catalytic membrane layer, for steady state, taking both diffusive and convective flow into account, applying the MichaelisMenten kinetics, can
be given as follows: d dϕ 1 dðDϕÞ dðυϕÞ vmax ϕ ¼0 D þ dr dr r dr dr KM þ ϕ
ð1Þ
In the general case, the diffusion coefficient and/or convective velocity can be dependent on the space coordinate, thus, D = D(r), υ(r) [or on the concentration, D = D(ϕ) or both of them, D = D(ϕ,r)]. The concentration in the biocatalytic membrane is denoted by ϕ as it made usually, in order to distinguish it from the fluid phase concentration (usually denoted as c). From the value of ϕ with units of kmol/m3 or kg/m3, the concentration, which is equal to ϕi/F (in weight fraction) or ϕiVi/Mi (in volume fraction, where Vi is the molar volume (m3/kmol) and Mi is the molar weight (kg/kmol)) can easily be obtained. Biochemical Reaction with One Substrate. Two operating modes will be analyzed, according to Figures 1A and 1B. Regarding the mass transport through the catalytic membrane layer, these two modes involve the other operating modes, as given by Bruining20 and Giorno and Drioli.17 The primary difference between the two models is that their boundary conditions at the outlet membrane interface are different. In the case of Model A (Figure 1A), there is no sweep phase or the sweep phase is not miscible with the permeate component; accordingly, there is no diffusive transport from the membrane (i.e., dϕ/dr = 0). In case of Model B (Figure 1B), there is a miscible sweep phase on the permeate side, which can determine the outlet concentration and, thus, generally, the diffusive mass transport is larger than zero, i.e., dϕ/dr >0 at r = r0 + δ. It is assumed that the measure of the convective velocity is known. First-Order Reaction, Model A. Let us look at the limiting cases of the MichaelisMenten kinetics, namely, the first-order (KM . ϕ) and zero-order reactions (KM , ϕ), with constant parameters and plane membrane layer (r . δ, thus, the second term in eq 1 will be zero), namely, the differential mass balance equations can be solved analytically for a plane sheet membrane. The boundary conditions of this model will be given as (Y = y/δ; the radial space coordinate in eq 1 is replaced by the y-space coordinate]) at Y ¼ 0
then ϕ ¼ ϕ0
ð2Þ
dϕ ¼0 dY
ð3Þ
and at Y ¼ 1 1636
then
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The general solution of eq 1 for this case is known.31,32 Applying the above boundary conditions, for the concentration function can be obtained as sinh½Θð1 Y ÞPe þ Θ cosh½Θð1 Y Þ PeY 0 2 ϕ ¼ ϕ exp Pe 2 sinh Θ þ Θ cosh Θ 2
ð4Þ with rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe2 þ ϑ2 Θ¼ 4
sffiffiffiffiffiffiffiffiffi δ2 k1 ϑi ¼ D
Pe ¼
with
Pe Θ exp D 2 βδ ¼ sinh Θ δ
Zero-Order Reaction, Model A. The solution methodology of the zero-order reaction with convective flow was also discussed31,32 (not shown here). The concentration distribution is given as ! 2
1 ϑ ϑ2 expðPeY Þ 1 þ Pe expðPeÞ Y Φ¼ Pe Pe expðPeÞ Pe
υδ D
ð13Þ
Note that the ϑi parameter is identical to the well-known Hatta number (Ha) that has been used for liquidgas or liquidliquid systems. The inlet mass-transfer rate, namely, the sum of the diffusive and convective flows, can be given as ! Pe2 2 4 þ Θ tanh Θ þ Pe Θ D ð5Þ ϕ0 J ¼ Pe δ tanh Θ þ Θ 2 The outlet mass-transfer rate (Jδ) is the convective flow; thus, Jδ = υϕδ, because the diffusive flow is zero in this case. First-Order Reaction, Model B. According to Figure 1B, the inlet and outlet concentrations are determined by the flow conditions of the two sides of the membrane. Thus, the boundary conditions will be given as at Y ¼ 0
then
ϕ ¼ ϕ0
ð6Þ
at Y ¼ 1
then
ϕ ¼ ϕδ
ð7Þ
The concentration distribution will be given as PeðY 1Þ exp Pe 2 ϕ0 exp sinh½Θð1 Y Þ þ ϕδ sinhðΘY Þ Φ¼ sinh Θ 2
with sffiffiffiffiffiffiffiffiffi k0 δ2 ϑ¼ Dϕ0 The inlet mass-transfer rate is " # D ϑ2 1 1 Pe ϕ0 J ¼ δ expðPeÞ Pe
9 8 > > > > = < Θ 0 ϕδ J ¼β ϕ > > Pe Pe > ; : sinh Θ þ Θ cosh Θ > exp 2 2
with
ϑ2 Pe=2 Pe e S¼ sinh ðPeY =2Þ Y sinh 2 Pe expð PeÞ ϕδ J ¼ β ϕ0 1 þ T
Pe D Pe 2 ð1 þ T Þ β¼ Pe δ 2 sinh 2 exp
with
The outlet mass transfer rate is given as 2 3 9 8 Pe > > > > Θ tanh Θ = < 6 7 6 7 2 0 6 7 Jδ ¼ βδ ϕ cosh Θ ϕδ 4 5 > > Pe > > ; : Θ exp 2
ð16Þ
where
ð9Þ
ð10Þ
ð14Þ
Zero-Order Reaction, Model B. The concentration distribution obtained by the boundary conditions given by eqs 6 and 7 is given as PeY exp Pe 2 ϕ0 sinh ð1 YÞ þ S Φ¼ Pe 2 sinh 2 Pe PeY þ exp sinh ϕδ ð15Þ 2 2
ð8Þ
Pe tanh Θ þ Θ D 2 β¼ δ tanh Θ
ð12Þ
ð17Þ
and
ϑ2 ð1 þ PeÞexpð PeÞ 1 2 Pe 2 3 Pe 6 exp D Pe6 expð PeÞ 7 2 7 0 7ς ϕ 6 ϕδ Jδ ¼ Pe 5 δ 24 ς sinh 2
T ¼
ð11Þ
1637
ð18Þ
dx.doi.org/10.1021/ie200701f |Ind. Eng. Chem. Res. 2012, 51, 1635–1646
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2
ϑ Pe Pe sinh ς ¼ 1 2 Pe 2 exp 2 2 Pe
The Solution for the General MichaelisMenten Kinetics. Equation 1 should be linearized for its analytical solution. The solution methodology of this type of differential equation was given by Nagy33,34 for diffusive mass transport through a membrane reactor and by Nagy and Borbely35 for diffusive plus convective mass transport with variable parameters. This solution methodology essentially addresses the mass-transfer rate and the concentration distribution in closed, explicit mathematical expressions. The method can be applied for Cartesian coordinates and cylindrical coordinates, as will be shown. For the solution of eq 1, the biocatalytic membrane should be divided into N sublayers, in the direction of the mass transport (which is perpendicular to the membrane interface), with a thickness of ΔR [ΔR = δ/(r0N)] and with constant transport parameters in every sublayer. For the sake of a general solution, the parameters of eq 1 (namely, e.g., the diffusion coefficient and reaction rate constant) can be variable. Thus, for the ith sublayer of the membrane layer, using dimensionless quantities, the following expression can be obtained: d2 ϕ 1 dϕ ki ϕ ¼ 0 υi ri1 < r < ri ð19Þ Di 2 þ dr ri dr
where the value of ki can be obtained, e.g., according to the MichaelisMenten kinetics, as follows: vmax ð20Þ ki ¼ KM þ ϕ̅ i ̅ i denotes the average value of ϕi in the ith membrane where ϕ sublayer. Equation 19 is in dimensionless form as [R = r/r0, Φ = ̅ i = (Φi1 + Φi)/2] ϕ/ϕ0, Φ d2 Φ 1 dΦ ϑ2i Φ ¼ 0 Pe ð21Þ i dR 2 R̅ i dR where R i denotes the average value of Ri in the ith sublayer of the catalytic membrane layer [R i = 1 + (i 0.5)ΔR; ΔR = δ/(Nr0)]. vmax sffiffiffiffiffiffiffi r02 ki υi r0 ϕ0 Pei ¼ ki ¼ ϑi ¼ KM Di Di ̅ i þ Φ ϕ0 The ϑi parameter can be regarded as the Thiele modulus for cylindrical membrane layer that was originally developed for spherical particles. Hereafter, we will call it the reaction modulus. ~: Let us introduce the following variable, Φ ! Pe0 R ~ ð22Þ Φ ¼ Φ exp i 2
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where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 Pei þ ϑ2i Θi ¼ 4
The solution of eq 23 can be easily obtained by well-known mathematical methods as follows:3235
~ Φ ¼ Ti exp λi R þ Si expðλi R Þ ð24Þ with Pe0 ~ λi ¼ i þ Θ i 2
The Ti and Pi parameters of eq 24 can be determined by means of the boundary conditions for the ith sublayer (with 1 e i e N). The boundary conditions at the internal interfaces of the sublayers (1 e i e N 1; Ri = 1 + iΔR; ΔR = δ/[Nr0]) can be obtained from the following two equations (eqs 25a and 25b): Di
dΦi þ Pei Φi dR
Φi
After a few manipulations, one can get the following differential equation to be solved: ~ d2 Φ ~ Θ2i Φ ¼ 0 ð23Þ dR 2
¼ Diþ1
R ¼ Ri
dΦiþ1 þ Peiþ1 Φiþ1 dR
R ¼ Riþ
ð25aÞ ¼ Φiþ1
R ¼ Ri
at R ¼ Ri with i ¼ 1; :::; N 1 R ¼ Riþ
ð25bÞ where Φi is the substrate concentration of the ith sublayer in the biocatalytic membrane at R = Ri. The Solution for Model A. The external boundary conditions can be given by eqs 2 and 3, adapting them for the capillary membrane, namely, at R = 1, Φ = 1 and at R = 1 + δ/r0, dΦ/dR = 0, for Model A. The solution methodology is discussed in Nagy’s papers.25,34 After solution of the algebraic equation system containing 2N equations, the value of the integration parameters for the first sublayer can be expressed as25 h iΨT 1 ~ 0 N T1 ¼ ϕ exp λ1 ð26Þ ΨO N 2 coshðΘ1 ΔRÞ
S1 ¼ ϕ expð λ1 Þ 0
where j ψN
* ¼
ΨSN ΨO N
1 2 coshðΘ1 ΔRÞ
j
BN AN
ψN 1 DN1 Pe0 i j 2 Ω N 1 DN
with j ¼ T, S, O
ð27Þ !+
j
ΩN 1 ξN1 ð28Þ
with AN ¼
1 R̅ i
at R ¼ Ri
with Pe0i ¼ Pei
Pe0i Θi 2
λi ¼
Pe0N tanhðΘN ΔRÞ þ ΘN 2
and Pe0N þ ΘN tanhðΘN ΔRÞ 2 The value of ξi1, Ai, and Bi (with i = 1, ..., N 1) as well as Ωji (with j = T, S, O and i = 1, ..., N 1) and Ψji (with j = T, S, O and i = 1, ..., N) can be calculated using the expressions given BN ¼
1638
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Table 1. Parameters Used for the Calculation of Parameters T1 and S1 (Given by eqs 26 and 27) ξi1 ¼ expÆPei Pe2i1 æRi1
with i ¼ 1; :::; N
j j Ωi 1 j Pe0i i ΔRÞ ψi 1 Di1 Ωi ¼ 1 tanhðΘ j Di 2 ξi1 Θi Ω i1
j
ψi 1 Ai ¼
with j ¼ T, S, O; i ¼ 2; :::; N 1
j j ψ Ωi 1 Pe0 ¼ Bi1 Ai1 Ωij 1 DDi1i 2i ξi1 i1
0 Pei 2
tanhðΘi ΔRÞ Θi
The initial values of
Ωji
and
ψji
and Bi ¼
(namely,
Ωji
Pe0i 2
with j ¼ T, S, O; i ¼ 2; :::; N 1
Θi tanhðΘi ΔRÞ
with i ¼ 1; :::; N 1
and ψji) (for j = T, S, O):
ΩT1 = exp(Θ1ΔR) ~1 exp(Θ1ΔR) ψT1 = λ
ΩS1 = exp(Θ1ΔR)
ΩO 1 = tanh(Θ1ΔR)
ψS1
ψO 1 = A1
= λ1 exp(Θ1ΔR)
in Table 1. It is important to note that calculation of the Ωji and Ψji values for i = 1, ..., N 1 (or i = 1, ..., N) requires a very accurate process. Every calculated variable should be given or calculated very accurately, even the value of ΔR (ΔR = δ/(r0N). Each step of the calculation was carried out using a quick basic computer program with an accuracy to 14 decimal places. This is the maximal accuracy by this program. To get the concentration distribution, the values of Ti and/or Si, with i = 2, ..., N, should be determined (the N value was chosen to be 100 during our calculation; note that, in reality, it is sufficient to predict either the value of T1 or that of S1, because if one of these two parameters is known, the other one can be obtained from the boundary conditions given by eq 2). Expressions for them are given by eqs A1A4 in the Appendix. The mass-transfer rate at Y = 0, as a sum of the diffusive and convective flows, can be given as i
D1 h ~ ~ λ1 T1 exp λ1 þ λ1 S1 expðλ1 Þ ð29Þ J ¼ r0 Solution for Model B. The external boundary conditions can be given by eqs 6 and 7, adapting them for the capillary membrane, namely, at R = 1, Φ = 1 and at R = 1 + δ/r0, dΦ/dR > 0, for Model B. The internal boundary conditions are given by eqs 25a and 25b. The values of T1 and S1 parameters of eq 24, in this case, will be given as follows: " # 9 8 > > Pe0N ðδ=r0 Þ
> > > > exp ~ > = < T> exp λ 2 1 Ω ϕδ T1 ¼ NO ϕ0 N Q >2 coshðΘ1 ΔRÞ ΩN > T > > > ΩN coshðΘi ΔRÞ > > > ; : i¼2 ð30Þ "
#
9 > > > > = expð λ Þ 1 0 N ϕ S1 ¼ O ϕ δ> N Q 2 coshðΘ ΔRÞ ΩN > 1 > > > ΩSN coshðΘi ΔRÞ > > > ; : i¼2 8 > > > S> Ω
