Description of the diffusive–convective mass transport in a hollow-fiber biphasic biocatalytic membrane reactor

Journal of Membrane Science 482 (2015) 144–157

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Description of the diffusive–convective mass transport in a hollow-fiber biphasic biocatalytic membrane reactor Endre Nagy a,n, Jozsef Dudás a, Rosalinda Mazzei b, Enrico Drioli b,c, Lidietta Giorno b a

University of Pannonia, FIT, Research Institute of Chemical and Process Engineering, P.O. Box 158, 8201 Veszprém, Hungary Institute on Membrane Technology, National Research Council of Italy, CNR-ITM, C/o University of Calabria, Via P. Bucci, Cubo-17/C, 87036 Rende, CS, Italy c Department of Chemical Engineering and Materials, University of Calabria, Via P. Bucci, Cubo-17/C, 87030 Rende, CS, Italy b

art ic l e i nf o

a b s t r a c t

Article history: Received 14 May 2014 Received in revised form 10 November 2014 Accepted 16 November 2014 Available online 22 January 2015

The substrate transport in a biphasic, biocatalytic, capillary membrane layer has been investigated. The measured data of oleuropein hydrolysis, in olive mill wastewater, have been evaluated in both a wellmixed tank reactor and a polysulphone, biocatalytic, capillary membrane reactor. The β-glucosidase enzyme was immobilized in the sponge layer of the asymmetric, hydrophilic membrane layer. Strong, competitive product inhibition, applying Michaelis–Menten kinetics with product inhibition for evaluation of the measured data, has been obtained in the mixed tank reactor while the reaction did not show inhibition in the biocatalytic membrane layer. Applying the kinetic data for the oleuropein hydrolysis, the performance of a biocatalytic membrane reactor has been discussed under different operating modes. The effect of the lumen radius, membrane thickness, location of the inlet of the substrate, the inlet concentration and Peclet number as well as the effect of the external mass transfer resistances have been discussed and illustrated. It has been shown that all parameters mentioned above can have a strong effect on membrane performance. The model and its presented, so-called forward sweep numerical solution method, where concentration of the first sublayer is given by an explicit expression of closed form, can essentially help the reader estimate the effect of the operating parameters on the performance of a biocatalytic, capillary membrane reactor. The simulation results enable the user to select the right choice between operating conditions, providing a high efficiency membrane reactor. & 2015 Elsevier B.V. All rights reserved.

Keywords: Membrane bioreactor Capillary catalytic membrane Diffusive plus convective mass transport Immobilized enzyme Enzymatic hydrolysis

1. Introduction Enzymatic bioconversion processes are of increasing use in the production, transformation and valorization of raw materials [1]. Their important applications have been developed in the field of food industries, manufacture of fine chemicals, particularly pharmaceuticals or application for environmental purposes, e.g. decomposition of toxic chemicals and organic dyes [2,3]. Membranes, especially the asymmetric ones, are thought to be good supports for immobilization of enzymes by physical or chemical interaction in or on the membrane, where enzymes can be covalently or non-covalently linked to internal and/or external interfaces of the membrane or entrapped in the membrane [4–6]. The enzyme is more often immobilized in the sponge layer [7–10] of the membrane (covalently linked, entrapped, non-covalently linked) or on the skin modified (grafting, etching, coating) membrane surface [4,11]. Various membrane materials, hydrophobic, hydrophilic or organic and inorganic, can be used as bioreactor n

Corresponding author. Tel.: þ 36 88 623511; fax: þ36 88 623795. E-mail addresses: [email protected], [email protected] (E. Nagy).

http://dx.doi.org/10.1016/j.memsci.2014.11.060 0376-7388/& 2015 Elsevier B.V. All rights reserved.

while the membrane layer can be used either in flat sheet or in fibrous form. Depending on the solubility of the substrate(s) and reaction product(s), the bioreactor can be monophasic or biphasic regarding the flowing fluid phases. As monophasic reaction can be mentioned the oxidation/reduction reactions using, e.g., peroxidase, glucose oxidase, laccase [4,12], or removal of toxic chemicals from the environment taking place in aqueous phase [13–15]. In this case, the substrate is often forced to flow through the biocatalytic membrane; thus one of the interesting advantage of this operating mode is the diffusion flow along with mostly the more intensive convection flow makes the reaction more effective [4]. When the substrate is a hydrophobic compound, the biphasic system is often applied for the bioreaction, especially when the product is soluble in aqueous phase. Lipase enzymes are mostly used in organic synthesis as hydrolysis [7,9,16–18] and esterification [7,19,20], or kinetic resolution of racemates for production of optically pure isomers [7,10,11,21,22]. The organic and aqueous phases are separately recirculated on the lumen and shell sides of the membrane, following the criteria that the phase containing the substrate should always be recirculated along the side of the membrane containing the immobilized enzyme [7].

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

An appropriate mathematical model is indispensable in order to be able to estimate the effect of the reaction parameters, as enzyme concentration, reaction kinetic constant, conversion, and the effectiveness of the reaction. The Michaelis–Menten kinetics is used in most cases, but this kinetic model does not involve the diffusive and convective transport of the reactant inside of the membrane. If there is diffusion limitation then the differential mass balance equation should take into account the effect of the bioreaction on the mass transport. Applying capillary or plane biocatalytic membrane, three regions can be distinguished, namely the two sides of the membrane (the lumen and the shell sides) and the membrane itself [23–26]. Several authors analyzed the diffusive–convective mass transport in the lumen, and shell as well as in the membrane matrix [23–25]. These studies have predicted the concentration distribution in every layer in the case of single-phase mass transport, only. Our paper focuses on the mass transport through a capillary biocatalytic membrane layer in two-phase systems, namely when the substrate cannot enter the second immiscible phase. As it will be shown, the difference between these operating modes is caused by the different boundary conditions at the outlet side of the membrane reactor. The fluid phases are often recirculated; thus, their concentration change in a capillary module is not generally important. Otherwise, the mass transfer rates defined for both sides of the membrane should be replaced in the mass balance equation of the fluid phases. The biocatalyst (enzyme) is mostly immobilized (either physical adsorption methods or by chemical linkage [27]) into the internal interface of the porous sponge layer or onto the dense membrane interface in a hydrogel layer [28–30]. The complete description of the catalytic membrane layer needs the solution of the complex Navier–Stokes flow models applying the component mass balance and/or momentum balance equations [23–26,31–34]. The momentum balance equation can often be neglected; thus suitable mass balance equations are often used and recommended for the biocatalytic membrane layer [10,11,23,25,33,34]. The diffusion plus biochemical reaction model is the most often applied mass transport description independently that the biocatalyst is immobilized onto the membrane surface in a hydrogel layer [16,30] or in the internal interface of the porous support layer [7–9,22,33]. This mass balance equation has mostly been solved numerically applying the general Michaelis–Menten kinetics. Often the reactant phase is forced to pass through the membrane pores, inducing additionally convective flow through the membrane [10,11,15,32,34]; accordingly the mass balance equation should be extended by a convective term, as well. Recently Nagy [26] and Nagy et al. [35] analyzed the diffusive–convective mass transport in enzyme membrane bioreactors applying ultrafiltration (i.e. without sweep phase) and recirculation modes (i.e. with sweep phase). They defined the mass transfer rate through the biocatalytic membrane in closed mathematical forms, in limiting cases of the Michaelis–Menten kinetics, in the presence of diffusive and convective flows. A detailed analysis of diffusion–convection mass transport through a biocatalytic, capillary membrane, applying biphasic systems, is missing in the literature, which analyzes the mass transfer properties, namely mass transfer rates, concentration distribution depending on the cylindrical effect of a hollow fiber membrane and effect of the reaction rate as well as operating modes and location of the inlet fluid phase. The diffusion–convection mass transport process, with zero concentration gradient on the outlet side of a capillary membrane bioreactor, is a hardly known process. On the other hand, the effect of the polarization layer is mostly neglected during the calculations. In the diffusive–convective mass transfer rate, the concentration distribution will be analyzed in a cylindrical membrane bioreactor in the presence of a reaction with substrate hindered Michaelis–Menten kinetics. Additionally, the role of the fluid boundary (other saying polarization) layer will also briefly be discussed. A forward numerical solution methodology was developed, the usage of which is much easier than that of the often-recommended Thomas method.

145

According to the solution developed, the mass transfer rate is expressed in a single, closed, explicit mathematical form. All results presented will be shown through an example of the production of isomer of oleuropein aglycon from oleupropein by means of immobilized β-glucosidase in the sponge layer of polysulphone capillary membrane [36–38]. This solution can be recommended for evaluation of biocatalytic processes in the membrane reactor adapting it to the operating modes.

2. Theory A simplified physical model and the mathematical model of biocatalytic reactors will shortly be discussed with membrane as supporters (catalyst is immobilized on its skin surface) or is an asymmetric biocatalytic membrane layer where the catalyst (enzyme) is immobilized in the sponge matrix. 2.1. Physical models of these systems The biocatalyst can be immobilized onto the skin-layer surface of the membrane as a hydrogel or into the porous sponge layer, onto its internal surface. In this latter case the membrane is used as a carrier or matrix for immobilization as well as a selective barrier. Substrate molecule should reach and come into contact with the immobilized enzyme molecule and this substrate transport can be governed by the hydrogel or the polymeric membrane properties depending on the immobilization mode. One of the important advantages of the membrane bioreactor is that the diffusive flow can be combined by transverse flow (convective flow through the membrane matrix) and thus the transport can be more intense than that in the case of diffusive flow, only [4,26,35,39]. A simplified physical model of these systems is illustrated in Fig. 1. Let us look briefly at the important operating modes of these two immobilization cases. A single biocatalytic reaction occurs in the hydrogel layer or in the sponge matrix [AþB2Cþ D]. Often one of the reactant and products is hydrophilic while the other one is hydrophobic. 2.1.1. Enzyme is immobilized into its porous sponge layer (Fig. 1A) The biocatalyst is immobilized into the internal surface of the mostly modified membranes, to make them suitable for immobilization. Three categories are distinguished in Fig. 1A, namely i) the two phases are in contact on the external interface of the sponge layer (A1); ii) due to the higher pressure on the shell side, the substrate-containing phase (aqueous or organic) partly enters the porous layer with a given penetration depth (A2) as well as iii) the substrate-containing phase is pressed by given transmembrane pressure through the biocatalyst membrane (A3) inducing given value of convective flow. The biochemical reaction takes place at the interface of the phases where enzyme is also present, in the biphasic system. The aqueous microenvironment (water molecules form an aqueous layer near the active site of the enzyme) of immobilized enzyme should provide the necessary ternary or quaternary enzyme structure for biocatalytic reaction. The reactant(s) (e.g. hydrolysis, epoxidation, peroxidation, esterification, transesterification, etc. reactions have reactants in both phases) should be transported from the bulk phase(s) to the interface between the phases where the biocatalytic reaction mainly takes place. Note that the reaction can partly take place in the aqueous phase when the organic reactant can dissolve in this phase. As Fig. 1 illustrates the reactant should diffuse from the bulk shell phase through the pores to reach the biocatalyst and the reaction partner, (if there is it at all) in the other (organic or aqueous) phase. When this phase is moving from the shell side through the biocatalytic membrane into the lumen fluid phase as drops then the convective velocity should also be involved in the mass

146

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

Fig. 1. Illustration of the different operating modes of the biocatalytic membrane reactor.

balance equation. It can be assumed that laminar flow exists in the pores; accordingly the diffusion has two directions in this phase, namely in the axial direction in the pores, perpendicular to the membrane interface, and in the traversed direction, perpendicular the pore surface assuming cylindrical pores. Note that thus the axial direction in the pores means perpendicular direction to the membrane interface, i.e. radial direction in the membrane matrix. The pore size of mostly used ultrafiltration membranes ranges between 0.1 and 0.01 μm; thus diffusive transport can be regarded as instantaneous in this direction comparing it to the about 100–300 μm or more longer axial direction or, otherwise saying, membrane thickness. Thus, a one-dimensional mass balance equation can be suitable for describing the mass transport in the membrane pores for any reagent in the shell side phase. Mazzuca et al. [40] published a microscope picture of the insoluble, colored, in situ product distribution during enzymatic hydrolysis. The uniform spreading of the product molecules in the sponge layer of the porous biocatalyst polysulphone membrane indirectly proves similar, closely uniform distribution of the active sites of the β-glucosidase enzyme and pore structure. Accordingly, the diffusive (and the convective) flow can be averaged over the whole volume of the sponge layer. The other reactant (e.g. water, hydrogen peroxide, alcohol, etc.) which is involved in the biochemical reaction, mostly dissolved, in the fluid phase on the lumen side of the cylindrical membrane should also be transported to the aqueous/organic interface. Assuming that the membrane matrix is wetted by this phase, accordingly this compound can diffuse to the interface continuously from the membrane matrix. Thus, this compound can be assumed to have constant concentration in the reaction zone. The product compounds can often affect the bioreactions by inhibition. In this case the product concentration should also be known in order to be able to predict the reaction rate. If the diffusion coefficient of the product essentially differs from that of the substrate then a mass balance equation should also be taken and solved for the product as well [25,32]. In our model we assume that the sum of the product and the substrate concentrations (assuming that the stoichiometric factor is one) will be constant in every point of the diffusion and reaction zone. This assumption should mean the diffusion coefficient of the product which can cause inhibition does not differ essentially from that of the substrate.

2.1.2. Enzyme is immobilized onto the membrane cylindrical surface (Fig. 1B) In the case of the two-phase system the enzyme containing gel layer separates the aqueous and organic phases [29,30]. Both the substrate and product can be hydrophobic or hydrophilic, e.g. lipase can catalyze both organic and aqueous substrates in the organic/aqueous interface [8]. The substrate molecules should pass through the enzyme layer mostly by diffusion (or in special cases by diffusion plus convection). The product will be dissolved in the other phase, which transports it away. Accordingly, the solutiondiffusion-(convection) model, accompanied by biochemical reaction, can be used for description of the substrate transport through the enzyme layer. The biocatalytic function of the enzyme molecules in the hydrogel can be hindered depending on its thickness and loading density [7,39] and thus the variable enzymatic reaction rate (e.g. the values of vmax and/or Km can be varied as a function of the space coordinate) should also be taken into account in the mass balance equation in a given case.

2.2. Mathematical model of mass transport through a biocatalytic sponge layer in the biphasic system Several assumptions should be made to get a simplified, onedimensional model for description of the mass transport in the sponge layer of a biocatalytic membrane according to its physical model:

 uniform distribution of the pores and thus that of the two

  

phase interface as well as the immobilized enzyme in the pore volume of the membrane layer; the diffusion in radial direction in the cylindrical pores is instantaneous due to the small pore size; thus the concentration gradient is negligible in this direction; the diffusion coefficient is assumed to be constant; process is at steady state; penetration depth of the shell side phase is much larger than the pore size, its value is equal to or less than the membrane layer thickness; product inhibition can occur during the reaction for that product which dissolves only in the reactant containing phase; the product compound which dissolves in the phase on the

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157







lumen side can easily be transported from the reaction zone through the membrane matrix; the sum of the product and the substrate concentrations is assumed to be equal to the initial substrate concentration (stochiometric factors of compounds should also be taken into account) in every point of the reaction zone. In other cases, e.g. when the product diffusion coefficient essentially differs from that of the substrate, mass balance equation(s) should be given for the product(s) as well; the second substrate (if there is) supplied by the membrane matrix is assumed to be in excess in the reaction zone; thus its concentration is regarded to be constant during the biocatalytic process; the Michaelis–Menten kinetics can be applied for the reaction with or without inhibition; the reaction rate can change as a function of the space coordinate considering that in a hydrogel layer.

Accordingly, the differential mass balance equation for components, (e.g. in the case of competitive product inhibition), following the simplified Navier–Stokes equations, is given by Eqs. (1) and (3) for both the substrate and product, in the presence of biochemical/chemical reaction [see expression (A7)]. This, for the actual substrate compound, is as
 D d dC r o υo dC vmax C r 7  ¼0 ð1Þ r dr dr r dr K m þC þC p K m =K p As mentioned vmax and Km can often depend on the r coordinate. In the second term of this equation the change of the convective velocity as a function of the radius is also taken into account by a factor of ro/r due to the cylindrical space. The υo value which enters the membrane on the shell side is related to the lumen side interface of the membrane, i.e. it denotes the convective flow rate in dimension of m/s. Accordingly, the convective velocity, entering the membrane on the shell side, will be as r o υo =r e where re ¼ ro þ δm. The transmembrane convective velocity is proportional to the transmembrane pressure gradient according to the Darcy law [23–26]. Accordingly, the υo convective velocity can be given as υo ¼ Lp TMP, where Lp is the hydraulic permeability, m/(sPa), and TMP is the transmembrane pressure, Pa [25]. If the axial pressure gradient is significant then the radial pressure gradient in the biocatalytic membrane can also change in the axial direction. In this case the υo value is dependent on the axial direction. If you neglect it, then the average value of the radial convective velocity is considered in Eqs. (1) and/or (3). Note if the substrate fluid enters the shell side, then the sign of the second term will be positive and if it is reversed direction, inlet of substrate is on the lumen side, then the sign of the convective term will be negative. Assuming that the substrate does not dissolve in the lumen (not miscible) phase and the external mass transfer resistance is negligible in the shell fluid phase (Cn ¼Co), the boundary conditions will be as (if the penetration depth of reactant containing phase is δ ¼ δm then re  δ  ro) at r ¼ r e at r ¼ r e  δ

C ¼ Co dC ¼0 dr

ð2aÞ ð2bÞ

According to the boundary condition given by Eq. (2b) there is no outlet diffusive flow from the membrane, only the convective flow transports the reactant/product compound into the nonmiscible fluid phase on the outlet (lumen/shell) side of the membrane reactor. Note that the same equations can be used when the whole biocatalytic process is taking place in a singlephase system, when the substrate is fed on the given side of the membrane while the product leaves the biocatalytic membrane on its other side [32,41]. However, the second boundary will not be valid in this case while the concentration gradient can be larger

147

than zero due to the diffusion flow into the lumen fluid caused by the flowing sweep phase. The dimensionless form of Eq. (1) is as (c ¼ C=C o , R¼ r/ro) 2

d c dR2

1 dc 2 7 ð1 þ PeÞ  ϑ c ¼ 0 R dR

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vmax =C o r 2o ; ϑ¼ o K m =C þ c þ ð1  cÞK m =K p D

ð3Þ

Pe ¼

r o υo D

ð4Þ

For the actual product compound, if its concentration in the membrane phase cannot be predicted from that of the substrate, due to the large difference in the diffusion coefficients between the reactant and product and assuming that the stochiometric factor is equal to one, the mass balance equation [Eq. (5)] should be given and solved simultaneously with Eq. (1):
 Dp d dC p r o υo dC p vmax C r 7 þ ¼0 ð5Þ r dr dr r dr K m þ C þ C p K m =K p Eq. (5) assumes that only one molecule is produced from one molecule of substrate. The boundary conditions, assuming that the product is dissolving in the non-miscible second fluid phase, will differ from Eq. (2b). In this case the sweeping phase continuously transfers the product compound from the biocatalytic membrane layer, due to which the lower sweep phase concentration causes permanent concentration gradient on the outlet membrane interface. Thus, the boundary condition will be as dCp/dr Z0 at r ¼ro. Note that δ means here the penetration depth of the shell phase in the sponge layer, which can be equal to or less than the membrane thickness (δ r δm). If the organic product can dissolve in the lumen fluid phase then the mass transport with a differential mass balance equation in this phase should also be taken into account.

2.2.1. External (shell side) mass transfer resistance The role of the fluid boundary layer or often called as the polarization layer in the presence of convective velocity is an intensively discussed theme in the literature at different membrane processes, e.g., pervaporation, gas separation [26]. The external mass transfer resistance of the outer laminar boundary layer with thickness of δf in the shell side (Fig. 2) or in the lumen phase can strongly affect the mass transfer rate in the membrane. An important question is whether the role of the fluid boundary layer may have essential effect on the substrate diffusive plus convective transfer rate. If you want to describe the concentration change in the fluid phases on the shell and/ or lumen sides, the mass transfer rates should be determined on both sides of the biocatalytic membrane taking into account the external mass transfer resistances as well, i.e. the mass transport from the bulk phase to the membrane or from the membrane interface into the bulk fluid phase. The mass transfer rate of the boundary layer in the presence of diffusive and convective flows is practically hardly discussed in the literature. That is why it will be analyzed here in a few aspects of this problem. The external, specific mass transfer rate through cylindrical space of the boundary layer, at the membrane outer interface, namely at r¼re, can be expressed as (see details in book [26]), for case when there is no transverse convective velocity, i.e. υ ¼0:   ð6aÞ Jj r ¼ re ¼ β f C of  C nf with

βf ¼

Df 1    r e ln r e þ δf =r e

ð6bÞ

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E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

Fig. 2. Important notations with external boundary layer on the shell side of the membrane.

Note the specific mass transfer rate which leaves the bulk fluid phase and enters the boundary layer at r ¼re þ δf is Jj r ¼ re þ δf ¼ Jj r ¼ re

re r e þ δf

ð7Þ

The superscripts o and n denote the bulk and interface concentration of the fluid phase on the shell side, while the subscript f means the fluid phase. It is easy to see that in limiting case, if re-1 then βf ¼Df/δf. The differential mass balance equation given for the cylindrical boundary layer cannot be solved analytically in the presence of convective velocity in cylindrical coordinate system. If transverse convective velocity exists, then the differential mass balance equation to be solved will be the first two terms of Eq. (1) relating it to the external boundary layer (see Fig. 2). This means a diffusive–convective mass balance equation without biochemical reaction. Accordingly, the differential mass balance equation to be solved will be in dimensionless form, assuming the mass transport through the shell side boundary layer is 2

d C dR2

þ

Pe þ 1 dC ¼0 R dR

D

Diffusion flow can also exist between the internal interface of the boundary layer and membrane layer, namely at r ¼re, causing concentration gradient on it, in this case, thus dC/dR Z0; accordingly the boundary conditions of the external boundary layer are

at r ¼ r e

C ¼ Co C¼C

n

J ¼ β ðC o  ePe C n Þ

ð11Þ

with

β¼

Df

δf

Pef

ePef ; 1

ePef

ð12Þ

with Pef ¼

υo δf Df

ð13Þ

The concentration distribution will briefly be discussed in a cylindrical transport layer depending on the location of the inlet flow, in Section 3.1. Note, the boundary conditions given by Eqs. (9a) and (9b) are valid for the fluid boundary layer independently, where the reactant(s) is supplied on the shell or on the lumen side. 2.3. Mathematical models of fluid phase on the shell (or lumen) side

υo r o

at r ¼ r e þ δf

The sum of the convective and diffusive mass transfer rate can be obtained for the plane interface, r-1, as [26]

ð8Þ

with Pe ¼

thin compared to the ro þ δm value due to the relatively large thickness of the sponge layer in order to have enough amount of enzyme on its internal surface. Accordingly the cylindrical effect is practically often negligible (it can be seen from Eq. (6) that if δf/re o0.1 then the cylindrical effect on the mass transfer rate can be neglected from practical points of view). The effect of the boundary layer on the concentration distribution will be discussed later (see Figs. 8 and 9 to it). This effect can be important, e.g. on the lumen side mass transport from (or to) the fluid phase where the ratio of δf/ro can be relatively high. For the sake of completeness a numerical solution will also be given taking into account the cylindrical effect as well. (A special solution methodology was developed whose results are given in Appendix A2 by Eqs. (A7)–(A11). Applying a finite-difference method [47, p. 101] a special forward method has been developed which can be used much easier than the often-recommended Thomas method [47, p. 108]). The essentiality of this method is that the value of C1 (and due to it the inlet mass transfer rate as well) is expressed in an explicit, closed form, which involves the effect of all mass transfer parameters and, thus, C1 can firstly be predicted and then the other concentration values, namely Ci with i¼ 2N 1 using the internal boundary conditions. The differential mass balance expression, defined by Eq. (1), was applied for this calculation applying the trial-error method to get the correct concentration value. About three–five self-adjusted iteration steps are needed to get the correct concentration distribution, taking into account the non-linearity of concentration dependency of ϑi parameter due the Michaelis–Menten kinetics. Obviously, the values n of the reaction modulus, namely ϑ [or ϑ , see Eq. (19)], are equal to zero in the fluid boundary layer. The specific mass transfer rate can be given as a sum of the diffusive [first term of Eq. (10)] and convective [second term of Eq. (10)] flows, at r¼re is (see [26] for details) ( ) D r e þ δf C o  C 1 þ PeC 1 J¼ f ð10Þ ΔR ro re

ð9aÞ ð9bÞ

In reality the thickness of the boundary layer on the outer surface of a biocatalytic membrane reactor, at r¼re (Fig. 2), is generally very

The fluid phase concentration distribution depends strongly on the operating conditions. The inlet fluid phase(s) are very often recirculated through a reservoir. The concentration change of the shell phase in the capillary membrane can often be neglected due to the large circulation velocity in this case. Three sub-cases are worth to be mentioned for this case: A) Owing to the high shell axial convective fluid velocity the concentration fall of substrate in the fluid phase is very low inside of the membrane module; thus it can be regarded that its concentration fall is negligible compared to the inlet concentration. In this case the concentration change in the reservoir as a function of time can

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

dC V m v C  max  ¼  dt V r K m þ C þ C o C K m =K p

ð14Þ

where Vm and Vr denote the membrane reactor and the reservoir volumes, respectively; Co denotes the starting concentration. B) When the radial concentration gradient is negligible in the shell fluid phase due to the quasi-turbulent conditions and both the diffusion and reaction have to be taken into account in the membrane layer, then the mass balance equation for the shell fluid can be given as uF ðC in  C out Þ ¼ aJ

ð15Þ

where F means the cross-section of the shell side, a denotes the membrane interface at r ¼re, and J means the inlet mass transfer rate into the membrane sponge layer, at r ¼ re; u denotes the inlet axial convective velocity on the shell side fluid. The J value should be determined by means of the solution of Eq. (1) given for the membrane reactor, for the substrate compound. C) When the concentration gradient is not negligible in axial direction of the shell fluid phase, then differential mass balance equations should be given for the fluid phase and to be solved in order to be given the transversal mass transfer rate at the outer membrane surface, namely r¼ro þ δm. Considering an ultrafiltration membrane, the ratio of the convective permeation rate to the convective axial rate, that is Pe/Peax (Peax ¼uL/Df), can orient us. If this ratio is much less than one, the transverse term can be neglected. The balance equation systems to be solved for the fluid phase on the shell side are given in Appendix A, by Eqs. (A1)–(A6). Sum of the mass transfer rates, namely diffusive plus convective flows, at r¼re, should be determined by the solution of e.g. Eqs. (A1)–(A3) or Eqs. (A4)–(A6), taking into account that the outlet (from the shell fluid phase) rate and the inlet (into the catalytic membrane phase) transfer rate should be equal to each other. Thus, one can use this transfer rate as the inlet rate for the membrane reactor and as the outlet rate for the shell fluid phase. Note, when the axial transmembrane pressure drop is significant, then the simplified Navier–Stokes equations given for the fluid and membrane phases should be solved simultaneously as an equation system. 3. Results and discussion 3.1. Mass transport through the fluid boundary layer Investigation of the polarization layer and discussion of its effect on the substrate mass transport into the membrane bioreactor are practically entirely missing in the literature. Note that the outlet boundary condition for the boundary layer can differ from that of the two-phase membrane bioreactor, namely there can be always diffusive outlet stream in the boundary layer, irrespective of whether the internal interface is on the shell side or on the lumen side. Thus, the boundary conditions Eqs. (9a) and (9b) are valid for the polarization layer; accordingly the mass transfer rates can strongly depend on the outlet diffusive flow in the boundary layer, as well. Note, the outlet diffusive flow of the membrane reactor is equal to zero in case of the two-phase system for the entering, non-soluble compound in the nonmiscible fluid and the same is true also for the inlet diffusive flow when there is no biochemical reaction in the membrane. In this latter case, in the lack of the outlet diffusive flow no concentration gradient can form inside of the membrane matrix. For calculation of mass transport in the boundary layer, the solution methodology written in Appendix A2 was used applying Eqs. (A11) and (16) for determination of the value of Jin at r¼re þ δf (Δr¼0.2 μm was chosen for the boundary layer with a thickness of e.g. δf ¼30μ; thus the value of

N¼ 150):
 D Co  C1 J in ¼ f þ PeC o r o j ΔRj

ð16Þ

Normally, the diffusive mass transfer rate in the boundary layer can be higher, due to its thinner layer thickness and often its higher diffusion coefficient, than that in the catalytic membrane layer. But the reacted amount of substrate can essentially be increased due to the reaction and due to it the inlet mass transfer rate into the membrane as well (see Fig. 10). Thus, the role of the polarization layer on the mass transport can also be more and more important, depending on the reaction rate. At higher Peclet number, Pe4about 3, the convective flow can dominate the mass transport rate when the mass transport is not accompanied by biochemical reaction. In this regime, the negative effect of the polarization layer can probably be negligible. However, in the intermediate Pe-number range the role of the polarization layer can be essential. Look at first how the cylindrical space can influence the concentration distribution inside the boundary layer at Cn ¼ 0 value at the outlet membrane interface. Fig. 3 illustrates the effect of the inlet location of the substrate containing fluid phase, namely it can enter either on the shell side or on the lumen side, on the concentration distribution. The value of δm was changed; thus the Peclet number remained constant during this calculation (Pe¼ 2). This figure clearly shows that the cylindrical space can essentially affect the concentration distribution. Curvature of the curves is much higher when the mass transport started from the shell side, because the component is transported in decreasing space in the direction of the lumen side. The opposite direction in the mass transport means increasing space, which decreases the curvature of the concentration in the membrane matrix. Note that the sum of the diffusive and convective flow is practically equal to each other in the two cases, i.e. the mass transfer rate, corrected by the inlet membrane interface, is independent of the location of the substrate entering the membrane. How the mass transfer rate can be varied in the boundary layer, at different values of the outlet concentration, more exactly as a function of the polarization modulus, I (I¼ Cn/Co) is plotted in Fig. 4. The inlet mass transfer rate (Jo) is related to that obtained when the outlet concentration is zero, i.e. Cn ¼ 0, (J oC n ¼ 0 ). Note that the real value of Cn depends on the sweep fluid properties, e.g. on the axial velocity and on the membrane properties, e.g. diffusive membrane mass transfer coefficient, transversal convective velocity, etc. As can be seen the value of J o =J oC n ¼ 0 increases rapidly with the increase of the Peclet number. When Pe4 3 the effect of the polarization modulus can practically be neglected, i.e. the boundary layer cannot hinder the inlet mass transfer rate of the catalytic membrane layer, practically. Some remarks for prediction of the cylindrical mass transfer rate in the case if Cn ¼0: the

Concentration distribution, -

be described as that in a well-mixed reactor. Thus, one can obtain

149

1.0

4 δm /r o =1

0.8

3 0.6

2

0.4

δm /r o =1

0.2 0.0 1.0

Pe=2 1.2

1.4

1.6

1.8

2.0

Boundary layer, R , − Fig. 3. Concentration distribution in the fluid boundary layer in capillary membrane depending on the location of the substrate containing fluid phase, at different values of δm/ro (ro ¼ constant, Pe ¼2, Co ¼ 1, Cn ¼0).

150

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

1.0

3.2.1.1. Results and evaluation of the stirred tank results. The hydrolysis was carried out by free β-glucosidase in a stirred tank reactor. The measured concentration change as a function of time is plotted in Fig. 5 (the points on this figure) at 5 mol/m3 starting concentration of oleuropein with 0.035 kg/m3 enzyme concentration. Parameters of the Km and the vmax Michaelis–Menten constants were calculated by the Lineweaver–Burk method applying the starting reaction rate at different starting substrate concentrations. Parameters obtained are Km ¼4.8 mol/m3, vmax ¼9.6  10  2 mol/m3s [36]

3 2 1

0.6

0.5 Pe=0.01

0.4 0.2 0.0



0.2 0.4 0.6 0.8 Polarisation modulus, I, -

ð18Þ

1.0

Fig. 4. The relative value of the specific mass transfer rate as a function of the polarization modulus in the fluid boundary layer at different values of the outlet concentration, Cn (δm/ro ¼1, I¼ Cn/Co).

cylindrical diffusive transfer rate can be estimated by Eq. (6). The mass transfer rate related to that obtained by diffusive one varies between 1 and 3.21 in the Peclet number range of Pe ¼0–3 (not shown here). The inlet rate changes linearly with the Peclet number starting from this upper value. Knowing the mass transport properties of both the polarization and membrane layers, one can predict the effect of the boundary or polarization layer on the inlet membrane mass transfer rate as will be shown later. 3.2. Biochemical reaction and mass transfer through biocatalytic membrane reactor

β-glucosidase enzyme is immobilized on the internal membrane surfaces, in the porous sponge layer of the biocatalytic membrane reactor. This reactor is the subject of this paper focusing on the mass transport applying hydrolysis of oleuropein as biphasic bioreaction [7,8]. The mass transport through the biocatalytic layers strongly depends on the operating conditions. Several papers measured the biochemical reactions in biphasic membrane systems mostly with lipase enzyme as biocatalyst immobilized into the sponge layer of a porous, asymmetric membrane as is surveyed in this paper. To improve the reactor performance, a suitable mathematical description of the biocatalytic membrane reactor is indispensable in order to access the influence of various parameters, e.g. enzyme activity, diffusive and convective flows, reaction kinetics, etc. This section is divided into two main parts: firstly results of the experimental hydrolysis of oleuropein, using mixed tank and membrane bioreactors, are evaluated applying the approaching solution of Eqs. (14) and (1) as the substrate mass balance expression. Then a few typical curves will be shown for illustrating the effect of operating parameters such as inlet concentration, Peclet number and capillary radius on the reactor performance applying parameter values obtained by the evaluation of the experimental data. Furthermore it will also be investigated how the location of the substrate inlet, namely on the lumen or on the shell side, can affect the biochemical reaction process. 3.2.1. Hydrolysis experiments of oleuropein The hydrolysis measurements were carried out in stirred tank reactor and membrane bioreactor, for comparison of the experimental results obtained by these two operating modes (for details see paper of Mazzei et al. [36,38]). The reaction taking place is Oleuropein þ Water-Algycon þ Glucose

dC v C  max  ¼ dt K m þ C þ C o  C K m =K p

ð17Þ

The substrate was fed as a pure oleuropein in buffer solution (and verified also with olive mill wastewater), the strongly hydrophobic product, aglycon, was extracted by organic phase [36]. The conversion vs. time course was measured at pH¼6.5 and at a temperature of 25 1C at different inlet concentration as a function of the reaction time.

Applying Eq. (18), the concentration change has been predicted as a function of time, without competitive product inhibition, i.e. Kp-1 (solid line in Fig. 5). According to these results, a strong inhibition should take place during the reaction. Adjusting the calculated and measured data, it was obtained that Kp/Km ¼0.005, that is Kp ¼0.019 mol/m3(dotted line in Fig. 5). As you can see the competitive product inhibition model describes very well the experimental data obtained in the stirred tank reactor. 3.2.1.2. Results in biocatalytic membrane reactor [38]. The β-glucosidase enzyme was immobilized in the porous sponge layer of an asymmetric, hydrophilic polysulphone, capillary membrane. The selective layer of the membrane is located on the lumen side and the sponge layer on the shell side. Its internal and external diameters were 1.08  10  3 and 1.75  10  3 m, respectively; thus, the membrane thickness is 335  10  6 m while the lumen radius was 540  10  6 m (Table 1). Taking into account that the surface of the membrane changes with the radius, the mean value of the liquid volume is about 1.54  10  6 m3 in the membrane layer, which is practically equal to the membrane geometrical volume. Enzyme immobilization was carried out by entrapment by pressing the enzyme solution from shell to lumen side. The substrate containing olive mill wastewater with substrate concentration of 2.5 mol/m3 was fed on the shell side of the membrane and applying transmembrane pressure difference the fed solution with the product was pressed through the membrane in transverse direction which then left the membrane on its lumen side. Simultaneously, an organic phase, namely limonene, was recirculated in the lumen side; thus, the aqueous product phase entered as an emulsion the organic phase [38]. The measured data (points) and the predicted data (line) are plotted in Fig. 6, applying the numerical solution of Eq. (1) without biochemical reaction (ϑ ¼ 0) for the boundary layer, and with reaction in the membrane layer. The solution of this system is given in Appendix A2 for both the mass transport layers. The inlet concentration value of the catalytic membrane layer was equal to its outlet

4.3 C/Co,-

J o /JC*=0,-

0.8

3.3 2.3 1.3 0.3 0

30

60

90

120

150

reaction time, min Fig. 5. Concentration vs. reaction time Km ¼ 4.8 mol/m3; vmax ¼ 0.097 mol/m3s; Co ¼ 5 mM (continuous line: without inhibition; dotted line: with product inhibition; points are measured data [36], 1 mM ¼ 1mol/m3).

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

151

Table 1 Data for prediction of the mass transport through the biocatalytic membrane. Membrane

Boundary layer

Simulation data

ro ¼ 540  10  6 m δm ¼335  10  6 m D¼3.7  10  10 m2/s L ¼0.105 m; V ¼ 1.54  10  6 m3 MM-constants: Km ¼ 3.8 mM; Co ¼2.5 mol/m3 vmax ¼ 8.1  10  2 mol/m3s Kp-1

re ¼875  10  6 m δf ¼30  10  6 m D¼ 3.7 10 10m2/s

Co ¼1–50 mM υ ¼ 3–32  10  6 m/s Pe ¼ 3.2–39.4 t¼ 22–200 s

ro ¼ (50–1)  10  6 m δm ¼ (335–1000)  10  6 m vmax ¼8.1  10  2 or 5.8  10  2 mol/m3s

100.0

Conversion, %

• measured

80.0 60.0 40.0 20.0 0.0

0

50

21

8.4

100 time, s

150

4

200

(continuous line in Fig. 6). Looking at the measured and predicted conversion results, the agreement between them is rather good. The predicted conversion permanently, and with increasing gradient, decreases with the decrease of the residence time, i.e. with the increase of the Peclet-number in the investigated regime. Note that the effect of the Peclet number on the conversion is low in the Peclet regime of 0 to 1 (see e.g. Fig. 8) when the diffusion dominates the mass transport. At higher values of Peclet number, the convective flow will more strongly affect the concentration change in the membrane as it is shown in Fig. 6. Besides that, the measured and the predicted data are in good harmony; thus it can be stated that the model is suitable for describing the mass transport through a biocatalytic membrane layer present in both the diffusive and convective flows.

Pe 3.0

Fig. 6. Conversion as a function of the residence time (and Peclet number: Pe¼ υro/D) in biocatalytic membrane reactor; points are measured values [38] while the continuous \line represents the predicted data; (co ¼ 2.5 mol/m3; Km ¼ 3.8 mM; vmax ¼ 0.081 mol/(m3s); Kp-1; ro ¼ 540  10  6 m; δm ¼335  10  6 m; D¼3.7  10  10 m2/s; Df/δf ¼ 1.2  10  5 m/s).

value of the boundary layer. It was assumed that the thickness of this layer is 30 μm, which can be acceptable in the fluid phase mixed by moderate intensity, by means of the recirculating liquid [38]. That means a mass transfer coefficient of 1.2  10  5 m/s. Comparing it to the membrane thickness (δm ¼ 335 μm), the effect on the boundary layer is rather low [this will be discussed later by Figs. 8 and 9]. The conversion was measured at different transverse convective velocities obtained from the volumetric permeation velocity, namely (7, 6, 5, 4, 3, 1.1)  10  9 m3/s. Accordingly, the transverse convective velocity in the biocatalytic membrane layer related to the total membrane interface was changed between 26.7  10  6 and 4.2  10  6 m/s(Pe¼6.1-39.4). The residence time was calculated using the average value of the convective velocity and thus it was varied between 20 and 140 s (Table 1) (These values are somewhat less than those given by Mazzei et al. [38]). The diffusion coefficient of the substrate molecule was estimated by the Stokes–Einstein equation [43–45] and it was obtained to be 3.7  10  10 m2/s in liquid phase and this value was accepted to the porous membrane phase, as well. This value was used for calculation of the Peclet number in the transverse direction. Taking into account the obtained Km and vmax values using the Lineweaver– Burk method (Km ¼3.8 mol/m3, vmax ¼1.28  10  2 mol/m3s), the concentration change was predicted in the biocatalytic membrane and compared to that measured. In order to adjust the measured and the predicted concentrations, the vmax ¼8.1  10  2 mol/m3s has to be taken, while the Kp-1 was used. This means that the product inhibition is negligible during the reaction in membrane reactor. This can be caused by the fact that the product is continuously removed from the reaction zone, decreasing its inhibition effect as well. The conversion was then recalculated as a function of the Peclet number or residence time by the Michaelis–Menten parameters obtained

3.2.1.3. Estimation of the mass transfer rates in both the boundary and membrane layers. Comparison of the mass transfer rates is important to predict the role of the boundary layer on the membrane mass transfer rate. Taking into account that the Peclet number in the membrane can change between 6.1 and 39.4 (Table 1) during our experiments, the convective velocity varies between (4.2 and 26.7)  10  6 m/s. The specific mass transfer rate for the boundary layer is practically equal to the convective flow (this can be stated according to results in Fig. 4 independent of the value of the polarization modulus). Thus, J oin E(Df/δf)PeCo ¼ υoCo ¼ 4.2  10  6 m/s  2.5 mol/m3 ¼ 10.5  10  6 mol/(m2s) at Pe¼6.1 for the boundary layer. The mass transfer rate into the membrane reactor is enhanced by the n biochemical reaction. The value of ϑ will be about 6.46 with vmax obtained. Enhancement of the mass transfer rate was calculated to be 1.19 (see also Fig. 10). Accordingly the specific, inlet mass transfer rate of the membrane reactor is about: J oin ¼1.19  10.5  10  6 ¼ 12.5  10  6 mol/(m2s). (Note that the effective diffusion coefficient in the catalytic membrane layer is regarded to be equal to that in the fluid phase, i.e. the effect of tortuosity and porosity is not taken into account during our calculation.) Accordingly, the diffusive flow should compensate the increase of the reaction enhanced membrane mass transfer rate. It means that the diffusion flow in the fluid phase should be 12.5-10.5  10  6 ¼2.0  10  6 mol/(m2s). Assuming high circulation velocity, the boundary layer thickness can be estimated to be about 30 μm [45–47,48]. Accordingly, the diffusive flux in the boundary layer is J odiff ¼(3.7  10  10/30  10  6)  2.5  Δc. From that it follows that the dimensionless concentration difference between the two sides of the boundary layer will be equal to about 0.065. Thus, the effect of the polarization layer on the membrane inlet mass transfer rate is rather low in the case of oleuropein hydrolysis (similar concentration decrease of the boundary layer is seen in Figs. 8 and 9). According to Fig. 10, enhancement can be much higher on lowering the convective flow. Accordingly, the role of the polarization layer can be even more important at lower Peclet numbers. Taking into account the diffusive flow rate difference between the membrane and the boundary layer,

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

ν

C/Co,-

0.8

μ

0.6

μ

0.4 1.0

1.2

μ

1.4

1.6

1.8

2.0

biocatalytic membrane layer, Fig. 7. The effect of the internal radius on the concentration distribution feeding on the shell side (continuous lines) or the lumen side (dotted lines); (vmax ¼0.057 mol/m3s; Kp-1; other parameters see the left column in Table 1).

this effect can be significant at a relatively large value of the reaction modulus, only.

3.3. Some predicted data on the behavior of the biocatalytic, capillary reactor Applying the Michaelis–Menten reaction equation with the parameter values predicted during the oleuropein hydrolysis, some important effect, namely the effect of the lumen radius, the Pe-number, the inlet concentration on the concentration distribution and the effect of membrane thickness on the effectiveness factor, will be shown in order to get additional information on the working properties of a capillary catalytic membrane bioreactor.

3.3.1. Effect of the lumen radius Cylindrical space of the membrane matrix means continuously changing the mass transfer surface (increasing or decreasing one, depending on the inlet side of the substrate(s) fluid) as a function of the radius. This fact can strongly affect the concentration distribution and accordingly the concentration-dependent reaction rate. That is why the analysis of this effect seems to be important. Fig. 7 illustrates the effect of the lumen radius between 50  10  6 m and 1000  10  6 m on the concentration distribution in the membrane layer, with constant membrane thickness (δm ¼ δ ¼335  10  6 m). Feeding of the substrate phase takes place either on the shell side (continuous lines) or on the lumen side (dotted lines). It is also assumed that the substrate-containing phase is penetrated through the whole cross-section of the sponge layer; accordingly the penetration depth is assumed to be equal to the membrane thickness. In both cases the convective velocity at r ¼ro was the same, namely 8.4  10  6 m/s (Pe ¼12.3). Obviously, the diffusive flow was equal to zero on the outlet surface in both cases, i.e. either at r ¼ro or at r¼ r þ δm, depending on the operating mode, according to the boundary condition given by Eq. (2b). The two feeding modes can serve essentially different concentration distributions in the membrane. This is the consequence of the variable space in the cylindrical membrane. As can be seen, the mass transport in the direction toward the shell side can give much less outlet concentration than in the reversed direction, i.e. when the inlet of substrate is at the shell side, at a given value of lumen radius. The difference between the concentration distributions decreases with the increase of the internal radius, i.e. with the decrease of the cylindrical effect. At the value of ro ¼1000  10  6 m, the two working modes give practically the

same concentration distribution. Generally, it can be stated that the cylindrical effect is negligible if the ratio of δm/ro r0.1. On the other hand, it is obvious that the measure of the inner and the outlet surfaces of the membrane can be very different. Let us look at the value of diffusive flows at the inlet surface of the membrane as a function of the operating modes and the lumen radius. The thickness of the membrane remains constant (δm ¼ 335  10  6 m), i.e. the reactor volume decreases with the decrease of the lumen radius, related to the length of the capillary membrane. The ratios of the diffusive mass transfer rates, namely that obtained on the lumen side (if the substrate is entered on the lumen side) related to that obtained when substrate is feeding on the shell side, are 1.13, 1.45, 1.84, and 3.3 at lumen radius of (2000, 540, 300 and 100)  10  6 m, respectively, at Pe ¼0 and at vmax ¼ 0.057 mol/m3s. On the other hand, the mass transfer surface, related to the length of the membrane, is larger by a factor of 1þ δm/ro, which means 1.17, 1.62, 2.12, and 4.35 respectively. Thus, by correcting the transfer rate on the shell side by these factors, the diffusive mass transfer rate on the lumen side will remain somewhat lower, namely 0.97, 0.91, 0.88, and 0.77, with decrease of the lumen membrane radius. That should mean that the sum of the diffusive and convective mass transfer rate will be somewhat higher when the substrate is feeding on the shell side of a capillary membrane reactor, strongly depending on the lumen radius. 3.3.2. Effect of Peclet number It is obvious that the convective flow has a strong effect on the concentration distribution. Applying the measured convective flow values and inlet substrate concentration, the concentration distribution is plotted in Fig. 8. In this case, the effect of the external boundary

concentration distribution, -

1.0

1.0

Pe=25

0.8 0.6

* =6.5

10

δf =30 μm k f = 1.2 x10-5m/s

5

0.4

1

←

0

0.2 0.0 0.0

ro

membrane layer 0.4

0.8

boundary layer 1.2

1.6 re membrane layer + boundary, -

2.0

δf +re

Fig. 8. The effect of the convective flow on the concentration distribution; vmax ¼ 0.081 mol/(m3s); Kp-1; for other parameters see the left column in Table 1).

concentration distribution, -

152

1.0

0.8

δ

μ

0.6

0.4 0.0

membrane layer 0.4

0.8

boundary layer 1.2

1.6

δ

2.0

membrane layer + boundary Fig. 9. The effect of the inlet concentration (Pe ¼12.3; other parameters are in the left column of Table 1).

layer is also taken into account with Df/δf ¼3.7  10  10/30  10  6 ¼ 1.23  10  5 m/s. According to previous analysis in subsection 3.2.1.3, the mass transfer resistance of the fluid boundary layer has little effect on the mass transport in this bioreactor system, only. This can clearly be seen in this figure, where the concentration fall in the boundary layer changes between about 0.02 and 0.06, depending on the Peclet number. (Note the dimensionless thickness of the boundary layer is equal to that of the membrane layer in Fig. 8 [and Fig. 9] for sake of better illustration. In order to get it, the value of the horizontal axis was calculated by expression of [1þ (1/δf)  (r re)] with re rrr [re þ δf]). As it was mentioned previously, the concentration distribution changes only a little bit in the range of Pe¼0.1 and 1. But above this Pe-range, the change of the outlet concentration can be much higher, as it is illustrated in Fig. 8, as well. That also means that the increasing convective velocity is followed by decreasing inlet diffusive flow and the change in the biochemical reaction rate will also change according to the Michaelis–Menten kinetics. The reacted amount of the substrate increases up to about Pe¼ 10 and then starts to decrease very slowly with increase of the convective velocity. The reacted amount of the substrate changes (2.3 to 10.4)  10  3 mol/(m3s), between Pe¼ 0.1 and 10. The concentration change in the membrane depends on both the inlet concentration and Peclet number. Accordingly, the effectiveness of the membrane also changes as will be shown later. The “optimum” value of the Peclet number depends on the inlet concentration as well.

3.3.3. Effect of the inlet substrate concentration The inlet substrate concentration can significantly affect the biochemical reaction rate and, consequently, the concentration distribution as well (Fig. 9). The value of vmax/(Km þ C) gradually decreases with the increase of inlet concentration of the substrate; accordingly the value of the reaction modulus, ϑi , also decreases. Thus, the value of C/Co increases with increase of Co. At lower value of Co the reaction rate reduces practically linearly with the concentration decrease due to the gradual domination of the Km value in the denominator of the Michaelis–Menten kinetics (this is the case of the first-order reaction). At higher value of the inlet concentration (Co⪢Km), the reaction rate is practically independent of the concentration. This is known as saturation effect. It can be stated the reacted amount of the substrate gradually increases with increase of the inlet substrate concentration at a given value of Peclet number (Pe¼ 12.3). It changes between (0.6 and 6.4)  10  3 mol/m3s in the inlet concentration range investigated. The effect of the boundary layer also increases with decreasing value of the inlet concentration. The right choice of the Co value can significantly raise the reaction efficiency of the enzyme membrane reactor.

Enhancement, E,-

5.0 Pe=0.1

0.5

1

4.0 2

Pe=0.1

0.6

0.5 1 2 5

0.4 0.2

1.00 Reaction modulus, ϑ*, -

10.00

3.3.4. Effect of the reaction rate The biochemical reaction has an essential contribution in the mass transport through biocatalytic membrane due to zero outlet concentration gradient, i.e. there is no diffusive mass transport through this membrane reactor without biochemical reaction. The outlet concentration will be equal to the inlet one in this case; the mass transfer rate is determined by the convective velocity taking into account the ratio of the inlet and outlet membrane surfaces (see Eq. (A13) for plane sheet membrane layer). In order to get information on the mass transfer rates in the presence of convective flow, the mass transfer rate obtained with dC/dx ¼0 at the outlet side [Eq. (A12)] and that obtained with dC/dx 40 at the outlet side [Eqs. (A14) and (A15)] are given in Appendix A3 for plane membrane layer. The next two figures, namely Figs. 10 and 11, illustrate the effect of the reaction rate. Considering that the value of the reaction rate varies as a function of substrate n concentration, the value of ϑ was chosen as the parameter, i.e. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v r 2 =D ϑn ¼ max o ð19Þ Km Obviously, the real value of the reaction modulus, ϑ, will change as a function of C and Km/Kp according to Eq. (4). Enhancement of the inlet mass transfer rate is plotted as a function of the reaction modulus in Fig. 10 at different values of the Pe number. As can be seen the E value significantly increases as n a function of the ϑ values strongly depending on the Peclet number. The increasing convective velocity lowers the effect of the reaction rate constant. This tendency of E value is similar to that of a first-order reaction (not shown here). How the reacted amount of the inlet stream depends on the n values of ϑ and Pe number is illustrated in Fig. 11. The reacted amount of the substrate was calculated by means of difference of the inlet and the outlet mass transfer rate. Accordingly J in  J out J in

J in ¼

5

0.8

Fig. 11. Reacted amount, related to the inlet mass transfer rate, as a function of the reaction modulus (other parameters are in the left column of Table 1).

with

2.0

153

1.0

0.0 0.10

Jr ¼

3.0

1.0 0.10

Reacted amount, Jr /Jin,-

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

 D r e þ δf C o  C 1 þ PeC 1 ro ro ΔR

ð20Þ

ð21Þ

and

1.00 Reaction modulus, *, -

10.00

Fig. 10. Mass transfer rate enhancement (E¼ J in =J oin ; ratio of mass transfer rates with and without biochemical reaction) as a function of the reaction modulus (other parameters are in the left column of Table 1).

J out ¼

D PeC N  1 ro

ð22Þ

This figure clearly shows the necessary reaction rate, which is necessary to reach the full conversion, at different values of the Pe number. Relatively large reaction rate is needed to reach the zero

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

Effectiveness factor, %

154

100 80 60

δm=335μm 500

40 1000

20 0 0.1

2000

1.0 10.0 Peclet-number, -

100.0

Fig. 12. The effectiveness factor as a function of the Peclet number at different thicknesses of the membrane reactor (δm ¼ δ, vmax ¼0.057 mol/m3s, other parameters are in the left column of Table 1).

outlet substrate concentration with membrane thickness of 335 μm. It is obvious that the membrane thickness will essentially affect the value of the outlet concentration. 3.3.5. Effectiveness factor as a function of the Peclet number The effectiveness factor, η, is defined as the observed substrate consumption rate, qob, divided by the maximum rate, qmax, which would be obtained when the dimensionless substrate concentration, namely C/Co, is close to one throughout the catalytic membrane layer:   PN qob i ¼ 1 ðvmax C i =ðK m þC i ÞÞ ðr e =r o Þ  iðδm =r o NÞ   ð23Þ η¼ ¼ qmax ðvmax C o =ðK m þ C o ÞÞ ðr e =r o Þ iðδm =r o NÞ The value of Ci means the concentration of substrate of the ith subsection (i¼ 1  N) applied for the numerical solution. Obviously, the value of η E 1 can be reached in a special case, namely when the membrane is working in reaction-limited regime, only. Probably, reaching a high effectiveness factor is not the real aim during the reaction processes in a biochemical membrane reactor. But it can be a target to reach close to zero substrate concentration in the outlet fluid in a special case. Accordingly, the effectiveness factor should be rather low when one tries to reach zero outlet concentration. Fig. 12 shows typical curves on the change of η as a function of the Peclet number at different values of membrane thickness. Membrane thickness is one of the important factors that can strongly influence the concentration distribution and thus the effectiveness factor. The concentration distribution as a function of the Pe is discussed in Fig. 8. As can be seen the effectiveness factor increases with increase of the Peclet number, as well. But the membrane thickness can significantly affect the value of the effectiveness factor, η. With its increase the η value can significantly decrease. Note that the reaction rate is relatively low during the oleuropein hydrolysis. That is why the outlet concentration is relatively high at even δ( ¼ δm) ¼1000  10  6 m, and low values of Peclet number. But the outlet concentration can be decreased by further increase of the reactor volume, namely the membrane thickness. This model and its special solution enable the user to predict the outlet concentration of the substrate, the mass transfer rate on the two sides of the catalytic membrane layer under any operating conditions if the reaction is taking place in the sponge layer in a biphasic membrane reactor.

4. Conclusion A biphasic membrane bioreactor has been investigated by means of the hydrolysis of oleuropein. It has been stated that the reaction is strongly product-inhibited in the well-mixed tank

reactor while the inhibition effect is negligible in the biocatalytic membrane reactor. Applying the kinetic parameters of the Michaelis–Menten kinetics of this reaction, the effect of operating conditions on the performance of the membrane reactor was simulated and illustrated. It has been stated that the dimension data of the capillary, catalytic membrane layer, the location of the inlet surface (on lumen or shell side of the membrane reactor), operating parameters as inlet concentration, transverse convective velocity, i.e. Pe-number, and external boundary layer can strongly alter the outlet concentration, and the effectiveness of the membrane reactor. The special numerical solution developed can easily be used to predict the mass transfer rates and the concentration distributions in the membrane bioreactor. Then, the inlet and the outlet mass transfer rates of the substrate can be used to describe the concentration change in the flowing fluid phases on both sides of the catalytic membrane layer.

Acknowledgment The National Development Agency Grants TAMOP-4.2.2.A-11-1/ KONV-2012-0072 and TÁMOP-4.2.2/B-10/1-2010-0025 are greatly acknowledged for the financial support.

Appendix A1 Regarding the modeling of the substrate concentration in the fluid phase in the membrane lumen or shell, the Navier–Stokes equations [43] should be simplified, according to the operating conditions, in order to get relatively simple and easily treatable expressions [23]. In the case of ultrafiltration the permeation velocity falls between about 10  4 and 10  6 m/s, while the axial convective velocity may be between 1  10  3 and 3  10  1 m/s depending strongly on the necessary operating conditions, that is, depending on the membrane process itself. As you see, the axial velocity often can be much larger than that of the permeation rate. Look at the value of the axial (Peax ¼uL/Df where L is the capillary length, Df is diffusivity in fluid) and radial Peclet numbers Pe¼ υro/D where ro is the internal radius of capillary and D the diffusion coefficient in the membrane pores; their values can help in further simplification. Let the radius of the capillary be 150 μm, while the diffusion coefficient in the flowing liquid changes between 10  8 and 10  9 m2/s. Thus, we can get the axial Peclet number between about 10 and 4.5  104; thus the axial diffusion term can be neglected. On the other hand, the radial one can change between about 10  5 and 1; thus, the radial concentration gradient is rather low. The radial Peclet number of the fluid can change between about 10  5 and 1, which can also be qualified as low; accordingly, momentum equation of the transverse velocity can be neglected. The equations to be solved for steady state and isothermal conditions where both the radial and axial convective velocities have to be taken into account will be as [23,40] Continuity equation: 1∂ ∂u ðr υ Þ þ ¼ 0 r ∂r ∂z Momentum equation: 



 ∂u ∂u ∂P ∂ 1 ∂ðruÞ ¼  þμ ρ υ þu ∂r ∂z ∂z ∂r r ∂r

ðA1Þ




Mass balance:

∂c ∂c ∂ 1 ∂ðrcÞ υ þu ¼ D ∂r ∂z ∂r r ∂r

ðA2Þ

ðA3Þ

When the ratio of the volumetric permeation rate to the volumetric axial rate, that is υro/uL⪡1, then the transverse term

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

can be neglected. Thus, one can obtain the following simplified equation system to be solved [41,42]: Continuity equation:

and

1∂ ∂u ðr υÞ þ ¼ 0 r ∂r ∂z

for 4 r i r N  1

ðA4Þ

Momentum equation:



 ∂P ∂ 1 ∂ðruÞ 0 ¼  þμ ∂z ∂r r ∂r Mass balance:

∂c ∂ 1 ∂ðrcÞ u ¼D ∂z ∂r r ∂r

ξn1 ¼ b1 ;

ξn2 ¼ b2 ;

155

ξn3 ¼

b3 n ξ a3 ; e2 2

ξni ¼

bi n a ξ  i ξn ei  1 i  1 ei  2 i  2

ðA5Þ

Knowing the value of c1, the value of ci can then be obtained applying Eqs. (A7) and/or (A8): The inlet, specific mass transfer rate of diffusion–convection mass transport at r ¼ro þ δm þ δf is defined by Eq. (16), while the specific outlet mass transfer rate can be expressed as

ðA6Þ

J out ¼

If the change in axial velocity is rather low, then the momentum equation can often be neglected and the second term of the continuity equation will be zero. Taking into account the Darcy equation, namely that the convective velocity is proportional to the transmembrane pressure difference, then the above equation system can be solved analytically. For details see e.g. [23], [24] or [26].

D PeC N ro

ðA10Þ

b) Solution with boundary conditions Eqs. (9a) and (9b). For the sake of completeness, let us give the solution for boundary conditions of Eqs. (9a) and (9b) when there is a miscible, flowing liquid phase on the lumen side of the membrane. Value of c1 is in this case C 1  ξN  1  eN  1 C n =C o ¼ ξN  1 Co n

c1 

ðA11Þ

with

Appendix A2 Let us look for the solution when the substrate is feeding on the shell side; accordingly the sign of the convective term will be positive. The mass balance equation of the substrate for the membrane layer is given by Eq. (3) for constant parameters, in dimensionless form. Rewriting Eq. (3) by finite differences:     2 c i  1 þ  2  α i ΔR  ϑ i ΔR 2 c i þ 1 þ α i ΔR c i þ 1 ¼ 0 ðA7Þ

ξ1 ¼ b1 ;

ξ2 ¼

3 r ir N 1

b2 ξ  a2 ; e1 1

ξi ¼

bi a ξ  i ξ ei  1 i  1 ei  2 i  2

for

and

ξn1 ¼  C n ; 

b e1

ξn2 ¼  2 ;

ai n ξ ei  2 i  2

b e2

a e1

ξn3 ¼  3 ξn2  3 ;

ξni ¼ 

bi n ξ ei  1 i  1

for 4 r ir N  1

or generally expressed ai ci  1 þ bi ci þ ei ci þ 1 ¼ di

ðA8Þ

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vmax =C o r 2o ; ϑi ¼ K m =C o þ ci þ ð1  ci ÞK m =K p D

αi ¼

1 þ Pe ; Ri

Ri ¼ ðRi  1 þRi Þ=2 It is obvious from Eq. (A8) that a trial-error methodology should be used for determination of the c value. The values of ai, bi, ei (1 rirN  1) can be expressed by means of Eqs. (7) and (8) while d1 ¼ a1Co, di ¼0 (2 rir N  1) from boundary condition by Eq. (2b), and di ¼0 (2 r irN 2), dN  1 ¼ eN  1cn by means of Eq. (9b). Note that the thickness of every sublayer is chosen to be small enough (the value of N should be several hundreds); thus the concentration/space coordinatedependent parameters, e.g. ϑi, Ri can be regarded to be constants in these layers. a) Solution with boundary conditions Eqs. (2a) and (2b). The socalled forward solution of the problem means that firstly the value of c1 will be determined and then values of ci (2 rir N 1). Value of c1 will be (cN  1 ¼cN) n

c1 

C1 ξ ¼ N1 C o ξN  1 e1

ðA9Þ

with b ξ1 ¼ b1 ; ξ2 ¼ 2 ξ1  a2 ; e1 3rirN 1

The inlet, specific mass transfer rate of diffusion–convection mass transport at r¼ ro þ δm þ δf can be given by Eq. (16) for this system as well. When the mass transport is simultaneously taken into account for both the boundary and membrane layers then Eq. (A9) was applied for both layers. Obviously, ϑ ¼0 for the boundary layer. The thickness of the boundary layer was chosen to be 30 μm with Δr ¼0.2 μm, while the membrane thickness is 335 μm and the value of Δr was chosen to be 1 μm.

Appendix A3 The specific mass transfer rate can be given as a result of the analytical solution for first-order reaction, for a plane interface [26,35]. The diffusion plus convection inlet flows are When dC/dx¼0 at the outlet side:   2 ðPe2 =4Þ þ Θ tanh Θ þ PeΘ oD   ðA12Þ J in ¼ C δ ðPe=2Þtanh Θ þ Θ The inlet flow without reaction will be lim J in ¼

D

δ

PeC o

ϑ-0

ðA13Þ n

ξi ¼

bi ei  1

ξi  1 

ai ei  2

ξi  2

for

When dC/dx4 0 at the outlet side (C ¼C at the outlet surface): ! J in ¼ β C o 

Θ

 C n  ePe=2 Pe=2 sinh Θ þ Θ cosh Θ

ðA14Þ

156

E. Nagy et al. / Journal of Membrane Science 482 (2015) 144–157

with

β¼

D



δ

  Pe=2 tanh Θ þ Θ tanh Θ

with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe2 2 þϑ ; Θ¼ 4

ϑ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ2 k1 =D;

ðA15Þ

Pe ¼

υδ

μ ϑn ϑ υo

viscosity, kg/ms reaction modulus by Eq. (19), – real value of the reaction modulus, [Eq. (4)] radial convective velocity in the membrane matrix related to its surface on the lumen side, m/s

Subscripts

D

Mass transfer rate of the two operating modes can be essentially different due to the different boundary conditions. Ratio of these mass transfer rates starts from one (at very low reaction modulus, if ϑ is close to zero) and tends again to one at the fast reaction rate regime, namely ϑ 43. In this regime the outlet boundary cannot affect the mass transfer rate due to the fast reaction rate. It has minimum point in the intermediate reaction regime. Its value and location inside the rate regime depends strongly on the Peclet number. Obviously the mass transfer rate with dC/dx 40, at the outlet side, has larger values.

f i in m out o p

fluid ith sub-layer inlet membrane outlet without reaction product

References Notation a C Co Cn Ci c D F Jo J Jr J oC n ¼ 0 Km Kp L N P Pe r ro re R Ri u vmax z

2

interfacial mass transfer area, m concentration, mol/m3 bulk phase concentration, mol/m3 interface concentration, mol/m3 concentration in the ith sub-layer, mol/m3 C/Co effective diffusion coefficient in the porous membrane matrix, m2/s cross section of the inlet portion of the membrane, m2 diffusive plus convective mass transfer rate without reaction, mmol/(m2s) diffusive plus convective mass transfer rate with chemical reaction, mmol/(m2s) reacted amount of the substrate, mmol/(m2s) mass transfer rate at zero outlet concentration, mmol/(m2s) Michaelis–Menten coefficient, mol/m3 inhibition coefficient, mol/m3 length of the capillary membrane, m number of subsection of the membrane layer or boundary layer, – pressure, Pa υoro/DPeax ¼uL/Df radial space coordinate in the membrane matrix, m tube radius of a capillary membrane, m external membrane radius, (re ¼ro þ δm), m dimensionless radius, (r/ro), – average value of R in the ith sub-layer in the membrane [Eq. (A8)], – the axial inlet velocity of the substrate containing fluid, m/s maximum value of reaction rate, mol/m3s space coordinate in axial direction, m

Greek letters

β δ δm

fluid mass transfer coefficient defined by Eqs. (5) and (9), m/s layer thickness of the external boundary layer, m layer thickness of the catalytic membrane bioreactor, m

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