Journal of Membrane Science 206 (2002) 217–241

A theoretical analysis of transport phenomena in a hollow fibermembrane bioreactor with immobilized biocatalyst

Vincenza Calabrò, Stefano Curcio, Gabriele Iorio∗Department of Chemical Engineering and Materials, University of Calabria, I-87030 Arcavacata di Rende (CS), Italy

Received 7 March 2001; received in revised form 12 July 2001; accepted 13 July 2001

Abstract

The behavior of a hollow fiber (HF) membrane bioreactor with entrapped biocatalyst is analyzed and characterized froma theoretical point of view. The model is based on the numerical solution of the dimensionless balance equations governingmass transfer within the regions that can be defined for this reacting system, namely the fiber lumen, the membrane denseand spongy layers. The chemical reaction is supposed to take place solely in the last two regions (skin and sponge) where thebiocatalyst is confined by entrapment and distributed according to the ratioE3/E2. The evaluation of the significant transportmechanisms of the substrate consumption rates, and more generally of the reactor performance have been carried out byanalyzing two representative parameters, i.e. the effectiveness factor and the performance index, as functions of the operatingconditions, expressed in terms of a set of characteristic dimensionless groups. The enzyme distribution ratio,E3/E2, its overallloading, the applied trans-membrane pressure (TMP) difference, the substrate feed concentration were investigated as thesystem key parameters. The reaction kinetics have been modeled with reference to Michaelis–Menten rate equation, modifiedto account for possible substrate and product inhibition. The results of the theoretical analysis enable to predict an optimalvalue of TMP and, therefore, of the permeate flux, as a trade-off between the necessity of reducing the transfer resistances inthe substrate access to enzyme and that of allowing suitable residence time. The bioreactor performances are no longer affectedby the distribution ratioE3/E2, once that this parameter has reached almost unity value : this means that the enzyme can beeffectively immobilized just in the membrane spongy region, with the advantage of a significant system simplification fromboth a technological and an economic standpoint. The model, however, is quite general and can be applied also to differentreacting systems, in order to evaluate how the catalyst position within a specific support can affect their performances. Acomparison between the theoretical model predictions and some literature experimental data has been also attempted, showinga good agreement for some typical operating conditions. © 2002 Elsevier Science B.V. All rights reserved.

Keywords: Fiber membranes; Membrane reactors; Membrane-bound enzymes; Membrane mass transfer

1. Introduction

Synthetic membranes are well-assessed supportfor the immobilization of biological catalysts, such

∗ Corresponding author. Tel.:+39-0984-49-2118x2035/2038;fax: +39-0984-49-2058/2038.E-mail addresses: vincenza.calabro@unical.it (V. Calabro),stefano.curcio@unical.it (S. Curcio), gabriele.iorio@unical.it(G. Iorio).

as enzymes or whole cells, under milder conditionsas compared with techniques implying the establish-ment of a chemical-bound. A synthetic membrane,with a suitable molecular weight cut-off (MWCO),somehow replicates artificially the cell membranefunction, ensuring the protection of purified enzymeagainst contaminants and inhibitors. In a membranebioreactor biocatalyst is confined in a well-defined re-gion of the space by means of a selective membrane,

0376-7388/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0376-7388(01)00766-9

218 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

Nomenclature

ai mean specific surface area of membrane dense and spongy layer (m−1)Ai surface exposed to the contact with the substrate (m2)C substrate concentration (mol/m3)C∗ = C/C0 dimensionless substrate concentrationCiE substrate concentration at the interface with the enzyme in the dense

or in the spongy layer (mol/m3)C∗iE dimensionless substrate concentration at the interface with the enzyme

in the dense or in the spongy layerC0 substrate feed concentration (mol/m3)DAB substrate diffusivity (m2/s)DPB product diffusivity (m2/s)Dp generic pore diameter (m)Dp mean diameter of a pore (m)E total enzyme concentration immobilized in membrane (Kg/m3)Ei enzyme concentration immobilized in each membrane region (Kg/m3)f(Dp) function of pore diameterjD = Sh Re−1Sc1/3 Colburn factorKf mass transport coefficient in the liquid film (m/s)ki product inhibition constant (mol/m3)kp substrate inhibition constantk1, k3, k4, k5, k6 constants for the semiempirical correlationk2 kinetic constant (mol/(Kg s))Km Michaelis constant (mol/m3)L fiber lumen length (m)Lp membrane hydraulic permeability (m/(s Pa))Nf number of hollow fibersNpi number of pores of membrane dense and spongy layerP product concentration (mol/m3)P ∗ = P/C0 dimensionless product concentrationPatm atmospheric pressure (Pa)Peax = v0L/DAB axial Peclet numberPe′

ax = Peax(R1/L)2 modified axial Peclet number (see Eq. (B.1))

P ∗iE dimensionless product concentration at the interface with the enzyme

in the dense or in the spongy layerPL module outlet pressure (Pa)Perad = vpR1/DAB radial Peclet numberPI Performance indexP0 module inlet pressure (Pa)Qp total permeate flow rate (m3/s)Q1,p permeate flow rate referred to a single fiber (m3/s)Q0 total feed flow rate in the axial direction at system inlet (m3/s)Q1,0 feed flow rate in axial direction referred to the inlet of a single fiber (m3/s)r radial generic coordinate (m)R1 fiber lumen radius (m)Rei = γ vp/aiµψ Reynolds number

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 219

Sc = µ/γDAB Schmidt numberSh Sherwood numberTMP trans-membrane pressure (Pa)v reaction rate (mol/(m3 s))vr(r,z) radial velocity profile (m/s)v∗

ri = vr(r, z)/vp dimensionless radial velocity profile in each fiber regionvz(r,z) velocity profile in the axial direction (m/s)v∗

zi = vz(r, z)/v0 dimensionless axial velocity profile in each fiber regionv0 = Q0πR

21 feed velocity in axial direction (m/s)

Vi total volume of each membrane region (m3)Vmax limiting reaction rate (mol/(m3 s))V p mean volume of a pore (m3)vp = Lp TMP permeation velocity in radial direction (m/s)V2 dense layer volume (m3)V3 spongy layer volume (m3)z axial generic coordinate (m)

Greek lettersδ membrane layer thickness (m)ε porosity of each membrane regionsγ solution density (Kg/m3)η effectiveness factorλ2 = k2ER2

1/C0DAB squared Thiele modulusµ solution viscosity (Pa s)ϑ = km/C0 dimensionless Michaelis constantρ = r/R1 dimensionless radial generic coordinateψ shape factorζ = z/L dimensionless axial generic coordinate

Subscripts and superscriptsi = 1, 2, 3(membrane region) 1: membrane lumen; 2: membrane dense layer; 3: membrane spongy layer.

or immobilized by absorption or entrapment withinthe polymeric matrix of a membrane. Enzymes areprecluded, due to their molecular size, from diffusingout or permeating through the membrane, while sub-strates and products can readily permeate through themembrane.

The hollow fiber (HF) configuration is moreadvantageously used for membrane reactors be-cause its high surface-to-volume ratio permits toobtain high biocatalyst density in a small reac-tor volume. A typical device of this kind consistsof a bundle of parallel polymeric hollow fibers

assembled in a cylindrical shell. The reactor isdivided into the luminal and the shell side by theporous membrane wall that acts as a selective bar-rier with respect to the transport of the involvedspecies.

Although diffusion is the primary mass transfermechanism, the membrane bioreactors may be evenoperated with ultrafiltration fluxes through the mem-brane wall. These fluxes are generally promoted bythe application of a trans-membrane pressure (TMP)difference. If significant TMP values are achieved,the mass transfer rate is enhanced by the addition of

220 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

a convective component to the diffusive mass transfermechanism.

The utilization of bioreactors with enzymes orwhole cells immobilized is diffused in several researchareas. The analysis of a hybrid bioartificial membranepancreas (HBMP) in which porcine islets of Langer-hans were segregated in the shell side of a hollowfiber module [1], showed the importance of convec-tive fluxes in determining the reactor performances.A distributed parameter model, taking into accountthe radial convective fluxes was formulated to achievea better description of mass transfer and kinetic phe-nomena occurring in this kind of bioreactor. Balcaoet al. [2] present a review of literature studies per-taining to reactors containing lipases in immobilizedforms. Highlights are given to industrial applicationsof lipases (including control and economic consider-ations), as well as to methods of immobilization andconfigurations of reactors in which lipases are used.Features associated with immobilized lipase kinet-ics such as enzyme activities, adsorption properties,optimum operating conditions, and estimates of thelumped parameters in classical kinetic formulationsare presented. The hydrolysis of triolein by lipase in ahollow fiber reactor can be modeled and explained bya Michaelis–Menten mechanism at the interface be-tween the aqueous and organic phases, together withcompetitive inhibition by oleic acid, diffusion of allchemical species and adsorption of lipase at the inter-face [3]. A mathematical formulation was developedto simulate the behavior of a multiphase hollow fibermembrane reactor [4]. The study shows the effects ofmass transport limitation on the system performances.In particular, an increase in enzyme loading resultingin a higher Thiele modulus, causes a more elevatedrate constant as well as a concomitant decrease in theeffective diffusivity.

Different researchers have investigated the hy-drolysis of lactose by means of immobilized�-galactosidase. Arica et al. [5] report that theactivity of �-galactosidase immobilized into apoly(2-hydroxyethyl methacrylate) (pHEMA) mem-brane increased upon increase in enzyme loading.Shin et al. [6] present the continuous production ofGalactooligosaccharides (GalOS) using lactose andimmobilized �-galactosidase fromBullera singu-laris ATCC 24193 in a packed bed reactor. Partiallypurified �-galactosidase was immobilized in Chito-

pearl BCW 3510 bead (970 GU/g resin) by simpleadsorption. A total of 55% (w/w) oligosaccharideswere obtained continuously with a productivity of4.4 g/(l h) from 100 g/l lactose solution during a15-day operation. The lactose hydrolysis by meansof �-galactosidase is inhibited by both substrate andproduct [7]; Portaccio et al. [8] studied the inhibitionof �-galactosidase, fromAspergillus oryzae, immobi-lized on chirosan beads or a nylon membrane (immun-odyne). In both cases, competitive inhibition has beenobserved. The value of the inhibition constant Ki,for the �-galactosidase/chitosan system was higherthan the one for the�-galactosidase/immunodyne,showing that the former system is more appropriateto perform lactose hydrolysis. The authors suggest anew technology, based on the use of nonisothermalbioreactors, to overcome the inhibition problems.

The behavior of hollow fiber membrane bioreac-tors have been widely investigated from a theoreticalpoint of view. Kim and Cooney [9] presented a the-oretical model in which hollow fiber enzyme reactorsperformances were discussed in several situationscorresponding to very low and high values of Thielemodulus. Schonberg and Belfort [10] showed that ra-dial convection of the feed in a shell and tube hollowfiber reactor is useful to increase the catalyst efficiencyand the production rates per unit volume. The studyis based on the solution of governing equations andassuming Darcy flow, first order kinetics, steady statetransport and irreversible reaction. Davis and Watson[11] reported a comparative study between a wholecell hollow fiber reactor and an annular bed reactor;the authors developed a novel numerical technique tosolve the governing equations and trend-lines showingthe concentration profiles in some typical situationswere obtained. Salzman et al. [12] studied hollow fiberenzyme reactors for continuous, single-pass opera-tion. The enzyme was physically confined within theshell section of the reactor and the substrate solutionflowed through the lumen section of it; both substratesand reaction products diffused through the fiber wall.The developed analytical model allowed calculatingthe expected conversion as a function of volumetricflow rate, kinetic constants, diffusion coefficients, ge-ometric dimensions of the reactor, the flow conditionsin the apparatus and the substrates concentrations.Tervo [13] obtained a system of nonlinear partial dif-ferential equations modeling the bioreactor behavior

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 221

and presented an application related to a spatiallyone-dimensional fixed-bed bioreactor. Basing on thework of Kobayashi and Laidler, Rusu [14] reported amathematical approach for the modeling of reactionscatalyzed by enzymes attached to the inner surfaceof a tube through which the substrate solution passes.In the paper, the analysis is presented for differenttransport situations and for reactions with classicalMichaelis–Menten kinetics and with different typesof inhibition. Explicit power series representationsof the solutions are established for several reactionvelocity models. The consistency of the mathematicalmodel used is established by proving the existenceand the uniqueness of the continuous solution of thebasic integral equation.

In the specific application of lactose hydrolysis, thekinetic behavior of a cross-flow bioreactor equippedwith a bundle of capillary ultrafiltration bioactivemembranes was discussed in order to show the effectsof trans-membrane pressure and of substrate feedconcentration on reactor performance [15]. Moreover,Carrara and Rubiolo [16] developed a model for eval-uating lactose hydrolysis with immobilized enzymein a packed bed reactor. Plug flow conditions were as-sumed, with kinetics governed by Michaelis–Mentenrate equation accounting for competitive productinhibition. The effects on the reaction rate of theexternal mass transfer resistance were consideredby estimating the overall mass transfer coefficient.Papayannakos et al. [17] analyzed the kinetics oflactose hydrolysis reaction, by both free and immo-bilized �-galactosidase, in a batch reactor systemin the absence of any mass transfer limitations. Inboth cases the totally competitive galactose inhibitionkinetic model was used to predict the experimentaldata. Simulation of the performance of a labora-tory continuous flow immobilized�-galactosidasereactor system showed that experimental resultscould be predicted by the ideal plug flow modelwhen an apparent effectiveness factorη = 0.65 isused to take into account the external mass transferlimitations.

In the present work, the behavior of a hollow fibermembrane bioreactor with immobilized biocatalysthas been investigated from a theoretical point ofview. A transport model, based on the momentumand on the substrate mass balance equations, wasdeveloped for hollow fibers polymeric membranes

with biocatalyst entrapped within the pores. The ef-fects of convective fluxes through the capillary fibers,realized by means of an applied trans-membrane pres-sure difference, have been also accounted for. Theresulting equations, written in dimensionless form,were solved by a numerical algorithm. The substrateconsumption, the product(s) formation and the over-all bioreactor performance were expressed in termsof a set of dimensionless characteristic parameters,combining all the different operating, physical andgeometric variables. Aim of this paper is to char-acterize the transport phenomena in a hollow fiberreactor and to determine the operating conditions thatmaximize the system productivity when a chemicalreaction, inhibited by the product and the substrate,occurs.

The innovations introduced by this study are repre-sented by:

• the evaluation of the role played by the enzymedistribution function, expressed in terms of theenzyme concentration ratio (E3/E2), on systemperformances. TheE3/E2 ratio accounts for therelative amount of enzyme immobilized in the dif-ferent regions of an asymmetric hollow fiber and,particularly, the enzyme entrapped in the spongylayer (region 3) as compared to that immobilizedin the membrane “skin” (region 2);

• the estimation, via the definition of a local Sher-wood number, of inner transport resistance at theinterface between the enzyme, the substrate andthe product; this, in order to accomplish a correctevaluation of substrate and product interfacial con-centrations, necessary to quantify the productionterms within the membrane regions;

• the geometrical characterization of membraneporous structure (mean diameter of a pore, specificsurface area, etc.) by introducing a semiempiricalcorrelation that expresses the membrane pores dis-tribution as a function of five model parameters.Also the correct estimation of membrane specificarea is needed to evaluate the interfacial concen-trations.

The theoretical results have been, finally, comparedto the experimental data from a previous experimentalwork by the same authors’ research group on the char-acterization of a membrane reactor with biocatalystimmobilized within the fiber wall. A good agreement

222 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

between theoretical predictions and experimentalresults has been observed.

2. Theory

The membrane bioreactor under investigation con-sists of a bundle of ultrafiltration hollow fibers as-sembled in a cylindrical cartridge according to atube and shell configuration. The fibers are asym-metric membranes composed by a thin skin sup-

Fig. 1. System configuration and description of the different fiber zones.

ported on a porous matrix, which provides the fiberstheir structural integrity. The model of the systemused for the theoretical analysis refers to a sin-gle fiber, in which three regions, see Fig. 1, canbe distinguished: the membrane lumen (region 1),where substrate continuously flows mainly in theaxial direction and partially in the radial direction;the membrane dense layer or “skin” (region 2) andthe membrane spongy layer (region 3). Substrateis forced by ultrafiltration to permeate through themembrane wall; here it comes in contact with the

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 223

Fig. 2. Notation used for the model development.

immobilized enzyme and its bioconversion takesplace. Products and unconverted substrate are contin-uously removed in radial direction from the permeatestream.

The solution of the problem consists in the deter-mination of velocity and of concentration profiles ineach fiber region. The presented theoretical model isbased on the correct evaluation of the transport resis-tances that are localized in a liquid film surroundingeach immobilized enzyme molecule, both in the denseand in the spongy layers, (Fig. 2). This is achievedby defining a local Sherwood number, modified to ac-count for the pores size distribution of the differentfiber regions and the by the estimation of the mem-brane specific surface area.

The hypotheses used to formulate the theoreticalmodel are the following:

1. the system is isothermal;2. the system operates in steady state conditions;3. the diffusion mechanism is governed by the Fick’s

law;4. axial diffusion, as well as back diffusion of prod-

ucts, are negligible in all the fiber regions;5. the flow regime within the fiber lumen is laminar;

6. physical and transport parameters (density, viscos-ity and diffusivity) are constant;

7. no chemical reaction occurs in the membrane lu-men;

8. the substrate flows in axial and in radial direc-tions only in the lumen, whereas in the dense andin the spongy layers the flux is only in radial di-rection;

9. the trans-membrane pressure (TMP) is assumed tobe constant

TMP = P0 + PL

2− Patm (1.1)

wherePatm is the atmospheric pressure andP0 andPLare, respectively, the pressures measured at the fiberinlet and outlet sections. In laminar flow regime, andfor 10 cm long fibers having an internal diameter of5.5 × 10−4 m, the pressure drop can be estimated as65 Pa (less than 0.05% ofP0) and, therefore, the as-sumption is absolutely reasonable.

The reaction rate can be expressed by a modifiedMichaelis–Menten equation that accounts for bothsubstrate and product inhibition effects [7]

v = VmaxC

Km(1 + P/ki)+ (1 + kp)C(1.2)

whereC andP represent the substrate and the productconcentration,Vmax = k2 × E and Km are, respec-tively, the reaction limiting velocity and the Michaelisconstant,ki and kp are the product and the substrateinhibition constants.

Starting from these hypotheses, the momentum andmass transport equations were obtained and put in adimensionless form, introducing a set of characteristicdimensionless numbers.

2.1. Momentum analysis

A preliminary fluid-dynamic analysis of the systemhas been carried out. The axial and the radial velocityprofiles in the fiber lumen were derived from the ana-lytical solution of momentum and continuity equationsformulated by Apelblat et al. [18] (see Appendix A).In regions 2 and 3, the porous matrix severely hindersfluid transport in the axial direction therefore only theradial velocity component was considered. In the fol-lowing the velocity profiles for each region of a singlefiber are reported.

224 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

2.1.1. Lumen side

vz(r, z) = 2Q0

πR21Nf

(1 − Qp

Q0

z

L

)(1 − r2

R21

)(1.3)

wherevz(r, z) is the velocity profile in the axial direc-tion, r andz are, respectively, the radial and the axialgeneric coordinate;Q0 is the total feed flow rate inthe axial direction;Qp is the permeate flow rate;R1andL represent, respectively, the lumen radius and itslength; Nf is the number of hollow fibers assembledin the cylindrical cartridge.

The above solution represents a modified form ofthe Poiseuille equation, where a dependence on the ax-ial coordinate is superimposed to the radial parabolicprofile. This dependence is expressed by a term con-taining the permeation ratioQp/Q0, that is constantonce the trans-membrane pressure and the axial flowrate have been fixed.

As far as the radial velocity profile,vr1(r,z), is con-cerned, the application of continuity equation gives

vr1(r, z) = Qp

2πR1LNf

(2r

R1− r3

R31

)(1.4)

2.1.2. Dense and spongy layersThe velocity profiles in both the dense and spongy

layer are obtained after the utilization of the conti-nuity conditions among the different regions of themembrane

vr,i (r) = 1

εiLp × TMP

R1

r

i = 2,3 andR1 < r < R3 (1.5)

in the last equation the termLp is the membrane hy-draulic permeability and the product betweenLp andTMP represents the permeation velocity,vp, in the ra-dial direction,εi is the porosity of each of the tworegions.

2.2. Mass transfer analysis

The contemporary formulation of continuity andmass balance equations, allows obtaining, for the threeregions of the hollow fiber, a system of differentialequations for both the substrate and the product con-centrations.

The mass balance equations, put in dimensionlessform, are the following (see Appendix B).

1. Membrane lumen

substrate :∂2C∗

1

∂ρ2+ 1

ρ

∂C∗1

∂ρ

−Peradv∗r1∂C∗

1

∂ρPeax

(R2

1

L2

)v∗z

∂C∗1

∂ζ= 0 (1.6)

product :∂2P ∗

1

∂ρ2+ 1

ρ

∂P ∗1

∂ρ− Perad

(DAB

DPB

)

×v∗r1∂P ∗

1

∂ρ− Peax

(R2

1

L2

)(DAB

DPB

)v∗z

∂P ∗1

∂ζ= 0

(1.7)

2. Dense layer

substrate :∂2C∗

2

∂ρ2+ 1

ρ

∂C∗2

∂ρ− v∗

r2Perad∂C∗

2

∂ρ

− λ2(E2/E)C∗2E

θ + kmkiP ∗

2E + (1 + kp)C∗2E

= 0 (1.8)

product :∂2P ∗

2

∂ρ2+ 1

ρ

∂P ∗2

∂ρ− v∗

r2Perad

(DAB

DPB

)

×∂P∗2

∂ρ+λ

2(DAB/DPB)(E2/E)C∗2E

θ + kmkiP ∗

2E + (1 + kp)C∗2E

= 0 (1.9)

3. Spongy layer

substrate :∂2C∗

3

∂ρ2+ 1

ρ

∂C∗3

∂ρ− v∗

r3Perad∂C∗

3

∂ρ

− λ2(E3/E)C∗3E

θ + kmkiP ∗

3E + (1 + kp)C∗3E

= 0 (1.10)

product :∂2P ∗

3

∂ρ2+ 1

ρ

∂P ∗3

∂ρ− v∗

r3Perad

(DAB

DPB

)

×∂P∗3

∂ρ+λ

2(DAB/DPB)(E3/E)C∗3E

θ + kmkiP ∗

3E + (1 + kp)C∗3E

= 0 (1.11)

whereC∗i andP ∗

i are, respectively, the dimensionlessconcentration for both substrate and product in eachof the three fiber regions (i = 1, 2, 3);ρ = r/R1isthe dimensionless radial coordinate, andζ = z/L is

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 225

the dimensionless axial coordinate. The mass trans-fer equations have been obtained, by introducing anappropriate set of dimensionless parameters, such asthe axial Peclet number,Peax = v0L/DAB, the ra-dial Peclet number,Perad = vpR1/DAB, the squaredThiele modulus,λ2 = k2ER2

1/C0DAB, the dimension-less Michaelis constant,ϑ = km/C0, the ratiosE2/Eand E3/E, that, respectively, represent the amount ofbiocatalyst immobilized in the membrane dense layer(skin) and in the porous layer (sponge), compared tothe total amount of immobilized enzyme. In the aboveequations, the termsC∗

2E, C∗3E, P ∗

2E andP ∗3E are, re-

spectively, the dimensionless forms of substrate andproduct concentrations at the enzyme–substrate inter-face and at the enzyme–product interface in the denseand in the spongy layers.

The system of partial differential Eqs. (1.6)–(1.11)is completed with the appropriate set of boundary con-ditions

B.C.1 ζ = 0 ∀ρ C∗1 = 1 (1.12.a)

B.C.1′ ζ = 0 ∀ρ P ∗1 = 0 (1.12.b)

B.C.2 ρ = 0 ∀ζ ∂C∗1

∂ρ= 0 (1.12.c)

B.C.2′ ρ = 0 ∀ζ ∂P ∗1

∂ρ= 0 (1.12.d)

B.C.3 ρ = 1 ∀ζ C∗1 = C∗

2 (1.12.e)

B.C.3′ ρ = 1 ∀ζ P ∗1 = P ∗

2 (1.12.f)

B.C.4 ρ = 1 ∀ζ − ∂C∗1

∂ρ+ PeradC

∗1

= −ε2∂C∗

2

∂ρ+ PeradC

∗2 (1.12.g)

B.C.4′ ρ = 1 ∀ζ − ∂P ∗1

∂ρ+ Perad

(DAB

DPB

)P ∗

1

= −ε2∂P ∗

2

∂ρ+ Perad

(DAB

DPB

)P ∗

2 (1.12.h)

B.C.5 ρ = R2

R1∀ζ C∗

2 = C∗3 (1.12.i)

B.C.5′ ρ = R2

R1∀ζ P ∗

2 = P ∗3 (1.12.j)

B.C.6 ρ = R2

R1∀ ζ − ε2

∂C∗2

∂ρ+ R1

R2PeradC

∗2

= −ε3∂C∗

3

∂ρ+ R1

R2PeradC

∗3 (1.12.k)

B.C.6′ ρ = R2

R1∀ζ − ε2

∂P ∗2

∂ρ+ R1

R2Perad

×(DAB

DPB

)P ∗

2

= −ε3∂P ∗

3

∂ρ+ R1

R2Perad

(DAB

DPB

)P ∗

3

(1.12.l)

B.C.7 ρ = R3

R1∀ζ ∂C∗

3

∂ρ= 0 (1.12.m)

B.C.7′ ρ = R3

R1∀ζ ∂P ∗

3

∂ρ= 0 (1.12.n)

The Eqs. (1.12.e)–(1.12.l) express the continuity ofsubstrate and product concentrations and of molarfluxes at the interfaces between the three regionsdefined in this theoretical model. Eqs. (1.12.m) and(1.12.n) are two Danckwerts-type conditions, whichexpress the vanishing of substrate and product con-centrations gradients at the interface between thespongy layer and the bioreactor shell side.

The substrate and the product interface concentra-tions,C∗

iE andP ∗iE (i = 2, 3), are generally different

from the bulk concentration values,C∗i andP ∗

1 (i = 2,3), owing to the mass transfer diffusive resistances ex-ternal to the immobilized biocatalyst molecules (seeFig. 2). The interfacial concentrations may be evalu-ated supposing that all these resistances are localizedin a liquid film close to the biocatalyst particle, bymeans of a mass balance at the enzyme catalytic site.In steady state conditions

amount of substrate reaching the biocatalyst surface

= amount of substrate reacted owing to the

chemical reaction

i.e. for both the membrane regions

KfAi(Ci − CiE)

= k2E(Ei/E)CiE

km(1 + (PiE/ki))+ (1 + kp)CiEεiVi (1.13)

226 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

where, Kf is the mass transport coefficient, eval-uated in the liquid film,Vi is total available vol-ume of each of two membrane regions andAi isthe surface actually exposed (wetted) to the con-tact with the liquid containing the substrate. Defin-ing a mean specific surface for both the dense andthe spongy layers (ai), as the ratio between the to-tal surface of membrane region where the reactiontakes place and the total volume of the consideredregion, i.e.

ai = total wetted surface

total volume= Ai

Vi= Np,iπδiDp,i

Np,i (Vp,i/εi)

(1.14)

where the subscripti is referred either to the skin orto the membrane spongy layer,Np is the number ofpores,V p is the mean volume of a pore,Dp is themean diameter of a pore,ε is the layer porosity andδ is the layer thickness, after some manipulations, thedimensionless form of the above equation (Eq. (1.13))is obtained for both the skin and the dense layers(i = 2, 3)

Sh a∗i (C

∗i − C∗

iE) = λ2(Ei/E)C∗iE

θ + (km/ki)P∗iE + (1 + kp)C

∗iE

(1.15)

whereSh = KfR1/DAB is the Sherwood number anda∗i = (ai/εi)R1 is the dimensionless specific area of

each membrane region.A good estimate of the Sherwood number (Sh) and

of the dimensionless mean specific surface area (a∗2,

a∗3) either for the dense or the spongy layer is needed

to correctly evaluate the termsC∗2E,C∗

3E, P ∗2E andP ∗

3E,used in mass balance equations. A semiempirical cor-relation [19], obtained for porous media is used to es-timate the Sherwood number by means of the Colburnfactor

jD = Sh

Re Sc1/3= 0.91ψRe−0.51 (1.16)

where Sc = µ/γDAB is the Schimdt number andψ is a shape factor characteristic of the porousmedium equal to 0.86 for irregular shapes, like mem-branes.

In the Eq. (1.16), the Reynolds number,Re, isdefined through a characteristic length that is the

reciprocal of the mean specific surface areaai , andnot the fiber pore diameter

Rei = γ vp

aiµψ(1.17)

As far as the estimation of skin and spongy layer spe-cific area is concerned (see Eq. (1.14)), the definitionof a pore size distribution function that may allow eval-uating the microscopic characteristics of membrane,Dp,i andVp,i is necessary. The analysis of a severaldata regarding ultrafiltration membranes suggested theutilization of a pore diameter distribution function,f(Dp), according to the following five parameters re-lationship

f (Dp) = k1Dk2p exp(−k3D

k4p )k

−Dp

5 (1.18)

The estimation of the unknown empirical parametershas been carried out by solving a system of five inte-gral equations, obtained imposing the conditions thatthe probability of finding a pore diameter between aminimum and a maximum value can be expressed by adefinite integral off(Dp) function. It is, therefore, pos-sible to calculate the mean diameter and the mean vol-ume of a pore for each of the two membrane regions,estimating, by Eq. (1.14), the unknown parameter,ai .Once the interfacial concentration has been obtained,it is possible to solve the mass balance equations andto determine the bioreactor behavior in different oper-ating conditions.

2.3. Numerical method of solution of masstransfer differential equations

The non-linear form of modified Michaelis–Mentenrate equation and the particular boundary conditions(both the product and the substrate interfacial concen-trations are initially unknown) do not allow to reachan analytical solution of the problem. The partial dif-ferential Eqs. (1.6) and (1.7) have been solved bymeans of a modified Crank–Nicholson method leadingto a tridiagonal system of algebraic equations solved,in turn, according to the Thomas algorithm, [20]. Apreliminary Von Neumann stability analysis [21] hadbeen demonstrating that the solution is always stable.The assumption of hindered solute transport in theaxial direction, in both regions 2 and 3, makes pos-sible to transform the Eqs. (1.8)-(1.11) into ordinary

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 227

Fig. 3. Block diagram of the Calculus algorithm.

228 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

differential equations that were solved via a fourth-order Runge–Kutta method.

As reported in Fig. 3, an iterative scheme has to beformulated; the numerical algorithm starts, for eachaxial step, with two guess values of interfacial con-centrationsC∗

1 = C∗2 andP ∗

1 = P ∗2 . Based on this as-

sumption the lumen partial differential equations canbe, firstly, solved and, subsequently, the dense andthe spongy layers mass balance equations. The iter-ation procedure is carried on successively with newguess values for the interfacial concentrations until theseventh boundary condition—vanishing concentrationgradient at sponge-shell interface—is fulfilled.

2.4. Relevant parameters for bioreactorperformances analysis

On the basis of theoretical results, in terms ofsubstrate and product dimensionless concentrationprofiles, two characteristic parameters have been de-fined to describe the overall bioreactor performancesas functions of the operating conditions: the perfor-mance index (PI) and the effectiveness factor (η).

The performance index is defined as the ratiobetween the amount of substrate reacted in the mem-brane and that of the substrate fed to the fibers (Eq.(1.19)) and represents a measure of relative signif-icance of mass transfer mechanisms and chemicalreaction. An effectiveness factor has been defined toevaluate how the diffusion resistance in the liquid filmclose to the biocatalyst surface affects the overall con-version process. It is calculated as the ratio betweenthe actual reaction rate and the reaction rate that wouldbe observed in absence of mass transfer limitations,(Eq. (1.20)).

Performance index(PI)

= substrate reacted in membrane

substrate fed to the fibers(1.19)

Effectiveness factor(η)

= actual reaction rate

reaction rate in absence of mass transfer limitations(1.20)

The expressions for the calculations of PI andη arereported in the Appendix C (Eqs. (C.1) and (C.2)).An alternative definition of PI was also considered in

terms of the substrate flowing across the membranerather than the substrate fed to the fibers. For this work,the definition used in Eq. (1.19) is preferred for tworeasons: (1) as defined, PI provides a measure of theoverall system performance, and (2) the definition inEq. (1.19) corresponds to our earlier experimental datareported in the reference article (15) and allow for easycomparison between these data and this theoreticalstudy.

3. Results

The results of the theoretical analysis have beenfirstly examined, reporting the substrate and productconcentration profiles as functions of the enzymedistribution ratio (E3/E2) and of the squared Thielemodulus (λ2). Then, the attention is focused on theinfluence of significant dimensionless parameters,related to the system operating conditions, on the ef-fectiveness factor and the performance index, in orderto predict, in any situation, the controlling transporton reaction mechanisms and, therefore, the conditionsthat may optimize the bioreactor performances. Thevalues of fiber geometrical characteristic and of somefixed dimensional quantities, necessary to performthe simulations, are reported in Table 1.

Fig. 4a and b show some typical dimensionless con-centration profiles, either for substrate or for product,versus the generic dimensionless radial coordinate,with λ2/θ ratio as parameter. It is interesting to outlinethe effect of theλ2/θ ratio which, for fixed values of all

Table 1Dimensional parameters used to perform the numerical simulations

Fiber length (L) (m) 0.1Lumen radius (R1) (m) 0.00055Lumen+ dense layer radius (R1 + R2) (m) 0.00056Lumen+ dense+ spongy layer radius

(R1 + R2 + R3) (m)0.00082

Dense layer porosity,ε2 0.113Spongy layer porosity,ε3 0.696Number of fibers 30Dense layer volume,V2 (m3) 0.01182× 10−6

Spongy layer volume,V3 (m3) 2.3553× 10−6

V2/V3 0.005Axial Peclet numberPeax = v0L/DAB 5 × 106

Feed velocity in axial directionv0 (m s−1) 0.052Feed flow rate in the axial direction

for a single fiber (m3 s−1)4.94 × 10−8

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 229

Fig. 4. (a) Substrate concentration profiles in membrane at different Thiele modulus. (b) Product concentration profiles in membrane atdifferent Thiele modulus.

230 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

Fig. 5. (a) Substrate concentration profiles in membrane at different values of enzyme distribution function. (b) Product concentrationprofiles in membrane at different values of enzyme distribution function.

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 231

the other physical and operating parameters and of thefeed substrate concentration, expresses a direct mea-sure of the total amount of entrapped enzyme. An in-crease in substrate consumption and a greater amountof product formed are in agreement with a higherλ2/θratio. This situation is represented by a series of steeperand steeper curves and in portions of lumen region ofincreasing thickness, affected by profiles variations.

Fig. 5a and b show the fundamental role playedby E3/E2 ratio in the reacting system characteriza-tion. This parameter represents, by definition, the ra-tio between the concentration of enzyme immobilizedin the spongy layer and that of the enzyme immo-bilized in the skin. In practice, an increase inE3/E2ratio is consistent with a higher substrate conversion.Due to the different physical and geometric character-istics of dense and spongy layers (the dense layer ismuch thinner than the spongy layer, see Table 1), nofurther conversion increase is observed asE3/E2 ratiotends to unity values. Moreover, curves correspondingto E3/E2 1 practically overlap those correspondingto E3/E2 = 1. This kind of behavior can be explainedexamining the following Table 2, where the values ofE2/E and E3/E are calculated by Eq. (1.20) and re-ported as a function of the chosen parameter,E3/E2

E2

E= 1 + (V2/V3)

(E3/E2)+ (V2/V3), and

E3

E= 1 + (V3/V2)

(E2/E3)+ (V3/V2)= E3

E2

E2

E(1.21)

It can be observed that the ratioE3/E2 has no furthereffect on the reactor performance, once it reachesvalues of unity order of magnitude or greater; in suchcases, in fact, the total amount of enzyme resultspractically confined only in the spongy region of thefiber. The previous conclusions are also confirmed

Table 2Enzyme distribution in the dense and in the spongy layers

E3/E2 m3/m2 E2/E E3/E

1.00 × 10−6 1.99 × 10−4 200.2 0.000201.00 × 10−4 1.99 × 10−2 196.3 0.01961.00 × 10−2 1.99 66.9 0.66901.00 199 1.0 1.0000100 1.99× 104 0.010 1.0050104 1.99 × 106 1.01 × 10−4 1.0050106 1.99 × 108 1.01 × 10−6 1.0050

by an analysis of Fig. 6 that represents a typical di-agram of biocatalyst effectiveness factor versus thesquared Thiele modulus. It can be noticed that allthe curves tend to reach unity when the transportresistances, outside the enzyme, are less significantand that all the curves tend to rapidly decrease whenkinetic contribution becomes predominant. When theenzyme is mostly immobilized in the fiber denselayer,E3/E2 = 10−5, and the reaction rate is veryfast,λ2∼10, the biocatalyst efficiency is much lowerthan that calculated whenE3/E2 ratio is unitary.

The outside diffusive resistances can be overcome ifthe prevailing contribution to mass transfer is the con-vection in radial direction and, therefore, if the radialPeclet number is sufficiently high. However, in thiscase, the contact time between the substrate and theenzyme may be too low to achieve a significant sub-strate conversion. A trade-off between the high con-versions typical of lowPerad values and the loweringof outside diffusive resistances typical of highPeradvalues is, consequently, to be searched for. Figs. 7 and8 show, respectively, a plot of biocatalyst efficiencyversus the squared Thiele modulus and a plot of theoutlet substrate dimensionless concentration versusthe radial Peclet number. The two diagrams, obtainedwith reference to lactose hydrolysis by means of im-mobilized �-galactosidase, confirm the importanceof all the involved transport phenomena (diffusion,convection and reaction rate) on substrate conversion.In particular, the role played by thePerad on both en-zyme efficiency and on the substrate concentration iscrucial. A comparison of trend-lines shown in Figs. 7and 8 allows to successfully utilize the simulationresults as a powerful tool for the bioreactor optimiza-tion. Fig. 8 shows that, actually, an optimal value interms of radial Peclet number can be individuated.Indeed, a minimum in the outlet substrate concentra-tion, testifies that it is necessary to choose a systemtrans-membrane pressure that may, contemporarily,lower the outside diffusive resistances allowing a suf-ficient contact time between the substrate and the im-mobilized enzyme. This optimal value belongs to therange ofPerad between 0.1 and 1. It can be seen thatthe highest substrate conversion is attained forPeradequal to 0.18 whenλ2/θ = 0.1 (not shown in figure),for Perad = 0.31 whenλ2/θ = 1, for Perad = 0.40whenλ2/θ = 10 and, finally, forPerad = 0.57 whenλ2/θ = 100. This shift towards higher radial Peclet

232 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

Fig. 6. Effectiveness factor vs. Thiele modulus at different values of enzyme distribution function.

Fig. 7. Effectiveness factor vs. Thiele modulus at different values of radial Peclet number.

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 233

Fig. 8. Outlet substrate concentration vs. radial Peclet number at different Thiele modulus.

number values is explained in terms of the higherresistances that the system has to overcome in order toreach the biocatalyst molecules. Indeed, an increaseof the radial Peclet number has a benefic effect on thereduction of the external resistances (see Fig. 7), butdetermines, forPerad higher than 1, a too fast passageof substrate within the membrane regions where theenzyme has been actually immobilized.

The determination of the optimal operating condi-tions implies, therefore, an accurate analysis of bothefficiency versusλ2 and of substrate concentration(conversion) versusPerad plots, coupling, in this way,a significant substrate conversion to a good utilizationof the immobilized biocatalyst.

The previous discussion shows that a set of oper-ating conditions that offer the best results in terms ofconversion exists. However, it is necessary to verifyif these conditions can maximize the system perfor-mances, in terms of the amount of formed product.As it was observed, substrate conversion increaseswhen high values of square Thiele modulus are used;conversely, if this parameter is too high, the enzymeeffectiveness factor dramatically decays. Moreover,the external diffusive resistances are lowered by asignificant convective contribution in radial direction

(see Fig. 7) and by aE3/E2 ratio at least equal to 1(see Fig. 6). In the following, the effects ofE3/E2 ratioon the bioreactor performance index will be shown insome general and typical applications, where the ef-fects of the radial Peclet number and of the inhibitionconstants will be taken into account.

Fig. 9 shows that it is possible to choose a set of op-erating conditions in order to obtain a significant yieldof desired product and, then, high values of the perfor-mance index. This figure clearly explains that perfor-mance index is practically the same forE3/E2 rangingfrom 0.1 to 104. Having reached a particular value,bioreactor behavior is no more affected by enzyme dis-tribution ratio. Indeed, it would be more advantageousto immobilize all the biocatalyst in the spongy layer ofthe membrane because the bioreactor productivity, forall the Perad values, reaches a maximum either whenthe enzyme is equally distributed in the membrane re-gions or when it is mostly entrapped in the fiber spongylayer. The latter conclusion is very important for thedesign of membrane bioreactors because the enzymeimmobilization, only in the spongy layer, is simplerfrom a technological point of view and cheaper.

Figs. 10 and 11 show the bioreactor behavior, ver-sus theE3/E2 ratio, in some general cases. These

234 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

Fig. 9. Performance Index vs. enzyme distribution function at different values of radial Peclet number.

were obtained for different combinations of kineticconstants and using values of both radial Peclet num-ber and of squared Thiele modulus that ensure highsubstrate conversions corresponding to effectivenessfactors equal to about 10% whenE3/E2 is unity (seeFigs. 6 and 7). It can be observed that, also in thesediagrams, enzyme distribution ratio influence is irrel-evant for values higher than 0.1. When inhibition phe-nomena become predominant (decrease of constantkiand increase of constantkp) a gradual diminution ofsystem performance index can be observed. This sit-uation can be counterbalanced only by an increase ofthe amount of immobilized biocatalyst that, however,determines a decrease in the biocatalyst effectivenessfactor.

3.1. Model validation

In order to verify the capability of the proposedmodel to predict the behavior of a real membrane

bioreactor with immobilized enzyme, it was decidedto perform some computer simulations that mightreproduce some experimental data reported in anexperimental work carried out by the same authors’research group on the characterization of the behaviorof a membrane reactor with biocatalyst immobilizedwithin the fiber wall [15]. The reference data wererelative to experiments carried out on a membranebioreactor in which whole cells ofSulpholobus sul-fataricus, containing�-galactosidase, were immobi-lized during the membrane preparation phase. The�-galactosidase in the cells catalyzes the hydrolysisof lactose. The bioreactor performances were re-ported in the paper versus the radial Peclet number,for different feed compositions.

Starting from the dense and the spongy layers SEMreported in the paper, the pore size distribution hasbeen evaluated by means of Eq. (1.17). This allowedestimating the average pore size, the porosity and thespecific surface area for both the fiber skin and the

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 235

Fig. 10. Performance index vs. enzyme distribution function at different values of product inhibition constant.

spongy layers. These calculations are reported in thefollowing Table 3.

The kinetic and the inhibition constants for lactosehydrolysis were evaluated [7] at the same temperatureof 70◦C, in which the experiments were performed;they are reported in the following Table 4.

The amount of the immobilized whole cells wasequal to 1 g/ml and the initial enzyme activity wasof 0.218�molglucose/min; this corresponds to a calcu-lated enzyme concentration of 0.0052 mg/ml. Owing

Table 3Estimation of capillary membrane characteristics [15]

Skin Spongy layer

Average pore size,D (�m) 0.159 0.9243Average volume,V (�m3) 0.371 331.74Porosity,ε 0.113 0.6965Specific area (cm−1) 15210 15850

to the immobilization technique described in the arti-cle, it is assumed that the amount of enzyme immobi-lized in the skin is, accounting for the volumes of thetwo layers, the same as that entrapped in the sponge.For this reason, anE3/E2 ratio equal to the volume ra-tio between dense and spongy layer,V2/V3 = 0.005,has been used. Fig. 12 shows the comparison betweenthe theoretical model and the experimental results, interms of bioreactor performance index. The agreementbetween the experimental data and the model predic-tions is rather satisfactory.

Table 4Kinetic and inhibition constants for the hydrolysis of lactose [7]

k2 (ml M/(mg min)) 0.122Km (mM) 77.5ki (mM) 0.626kp 0.0728

236 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

Fig. 11. Performance index vs. enzyme distribution function at different values of substrate inhibition constant.

Fig. 12. Comparison among theoretical prediction and experimental data.

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 237

4. Conclusions

A theoretical analysis of substrate and product in-hibited chemical reactions in a hollow fiber membranebioreactor, with immobilized enzyme, was presented.The numerical solution of momentum and mass bal-ance equations, expressed in terms of a set of dimen-sionless parameters, gave the following results:

• the system productivity is strongly affected by theradial Peclet number,Perad, the squared Thiele mod-ulus,λ2, and the enzyme distribution ratio,E3/E2;

• an increase in squared Thiele modulus determinesa continuous improvement of substrate conversion,against a significant decrease of biocatalyst effi-ciency;

• an optimal value of radial Peclet number can beindividuated. It represents a trade-off between theovercoming of diffusive resistances in the liquidfilm outside the biocatalyst particle, typical of highPerad values and a sufficient contact time betweenthe substrate and the enzyme, typical of lowPerad;

• it is better to immobilize the enzyme prevalently inthe membrane spongy layer; this procedure is easierfrom a technological point of view and cheaper froman economical one;

• the model validation with literature data gave satis-factory results.

Appendix A

Apelblat’s solution (in the case of only one fiber)[18]

Pressure profile

P(z) = Patm + (PL − Patm)sinh(αz)− (P0 − Patm)sinh(α(z− L))

sinh(αL)(A.1)

Axial velocity profile

vz(r, z) = 1

4

α(r2 − R21)[(PL − Patm)cosh(αz)− (P0 − Patm)cosh(α(z− L))]

µ sinh(αL)(A.2)

Radial velocity profile

vr(r, z)=Lp[P(z)− Patm]

(2r

R1− r3

R31

)=Lp[(PL − Patm)sinh(αz)−(P0 − Patm)sinh(α(z− L))]

(2 rR1

− r3

R31

)sinh(αL)

(A.3)

Feed flow rate at fiber inlet section

Q1,0 = 1

8

[(P0 − Patm)cosh(αL)− (PL − Patm)]

µ sinh(αL)(A.4)

Permeate flow rate

Q1,p = 2πR1Lpcosh(αL)− 1

α sinh(αL)(P0 + PL − 2Patm)

(A.5)

with αL (permeation parameter) given by

αL = 4

õLp

R31

L (A.6)

The trend for low permeation rate can be achievedby expanding the above expression in Taylor series,retaining at most first order powers ofαL.

After some algebraic manipulations the followingrelationships, for low permeability rate, can be found.

Pressure profile

P(z) = P0 − (P0 − PL)z

L(A.7)

Radial velocity profile

vr(r, z)=Lp

[P0 − (P0 − PL)

z

L− Patm

]

×(

2r

R1− r3

R31

)(A.8)

For a 10 cm long fiber under laminar flow regime,the pressure drop is negligible and a mean value of

238 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

pressure profile can be considered

P = P0 + PL

2(A.9)

and, therefore,

vr(r, z) = Lp

(P0 + PL

2− Patm

)(2r

R1− r3

R31

)

(A.10)

Feed flow rate at fiber inlet section

Q1,0 = πR41

8µ

P0 − PL

L(A.11)

Permeate flow rate

Q1,p = πR1LLp(P0 + PL − 2Patm) (A.12)

Appendix B. Mass balance equations

Lumen side (no generation, no accumulation)

substrate : −DAB

(1

r

∂C1

∂r+ ∂2C1

∂r2+ ∂2C1

∂z2

)+ vr

∂C1

∂r+ vz

∂C1

∂z= 0 (B.1)

product : −DPB

(1

r

∂P1

∂r+ ∂2P1

∂r2+ ∂2P1

∂z2

)+ vr

∂P1

∂r+ vz

∂P1

∂z= 0 (B.2)

Dense layer (no accumulation)

substrate : −DAB

[∂2C2

∂r2+ 1

r

∂C2

∂r

]+ vr2

∂C2

∂r+ k2E(E2/E)C2E

km(1 + P2E/ki)+ (1 + kp)C2E= 0 (B.3)

product : −DPB

[∂2P2

∂r2+ 1

r

∂P2

∂r

]+ vr2

∂P2

∂r− k2E(E2/E)C2E

km(1 + P2E/ki)+ (1 + kp)C2E= 0 (B.4)

Spongy layer (no accumulation)

substrate : −DAB

[∂2C3

∂r2+ 1

r

∂C3

∂r

]+ vr3

∂C3

∂r+ k2E(E3/E)C3E

km(1 + P3E/ki)+ (1 + kp)C3E= 0 (B.5)

product : −DPB

[∂2P3

∂r2+ 1

r

∂P3

∂r

]+ vr3

∂P3

∂r− k2E(E3/E)C3E

km(1 + P3E/ki)+ (1 + kp)C3E= 0 (B.6)

Introducing the following dimensionless quantities:

C∗1,2,3 = C1,2,3

C0, P ∗

1,2,3 = P1,2,3

C0, C∗

2E,3E = C2E,3E

C0, P ∗

2E,3E = P2E,3E

C0, ρ = r

R1,

ζ = z

L, v∗

r1,r2,r3 = vr1,r2,r3

vp, v∗

z = vz

v0

Axial velocity profile

vz(r, z)= (P0 − PL)(R21 − r2)

(1

4µL− α2L2

8

z

L

)

−2Q1,p

πR41

z

L(R2

1 − r2) (A.13)

which, in the same degree of approximation, i.e. re-taining at most first order powers ofαL, becomes

vz(r, z) = 2Q1,0

πR21

(1 − Q1,p

Q1,0

z

L

)(1 − r2

R21

)(A.14)

in the case ofNf fibers the total flow rates can beconsidered and Eq. (A.14) is (see Eq. (1.3))

vz(r, z) = 2Q0

NfπR21

(1 − Qp

Q0

z

L

)(1 − r2

R21

)(A.15)

V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241 239

Membrane lumen

substrate :∂2C∗

1

∂ρ2+ 1

ρ

∂C∗1

∂ρ+(R2

1

L2

)∂2C∗

1

∂ζ 2− vpR1

DABv∗

r1∂C∗

1

∂ρ− v0R

21

DABLv∗z

∂C∗1

∂ζ= 0 (B.7)

product :∂2P ∗

1

∂ρ2+ 1

ρ

∂P ∗1

∂ρ+(R2

1

L2

)∂2P ∗

1

∂ζ 2− vpR1

DPBv∗

r1∂P ∗

1

∂ρ− v0R

21

DPBLv∗z

∂P ∗1

∂ζ= 0 (B.8)

Dense layer

substrate :∂2C∗

2

∂ρ2+ 1

ρ

∂C∗2

∂ρ− v∗

r2vpR1

DAB

∂C∗2

∂ρ− k2E(E2/E)C

∗2E(R

21/DAB)

km + (km/ki)C0P∗2E + (1 + kp)C0C

∗2E

= 0 (B.9)

product :∂2P ∗

2

∂ρ2+ 1

ρ

∂P ∗2

∂ρ− v∗

r2vpR1

DPB

∂P ∗2

∂ρ+ k2E(E2/E)C

∗2E(R

21/DPB)

km + (km/ki)C0P∗2E + (1 + kp)C0C

∗2E

= 0 (B.10)

Spongy layer

substrate :∂2C∗

3

∂ρ2+ 1

ρ

∂C∗3

∂ρ− v∗

r3vpR1

DAB

∂C∗3

∂ρ− k2E(E3/E)C

∗3E(R

21/DAB)

km + (km/ki)C0P∗3E + (1 + kp)C0C

∗3E

= 0 (B.11)

product :∂2P ∗

3

∂ρ2+ 1

ρ

∂P ∗3

∂ρ− v∗

r3vpR1

DPB

∂P ∗3

∂ρ+ k2E(E3/E)C

∗3E(R

21/DPB)

km + (km/ki)C0P∗3E + (1 + kp)C0C

∗3E

= 0 (B.12)

By means of the following dimensionless numbers

Perad = vpR1

DAB, Peax = v0L

DAB, λ2 = k2ER2

1

C0DAB, θ = km

C0

and being theR1/L ratio 1 and the axial Peclet number,Peax 1.The following equations, in non-dimensional form, are obtained.Membrane lumen

substrate :∂2C∗

1

∂ρ2+ 1

ρ

∂C∗1

∂ρ− Peradv

∗r1∂C∗

1

∂ρ− Peax

(R2

1

L2

)v∗z

∂C∗1

∂ζ= 0 (B.13)

product :∂2P ∗

1

∂ρ2+ 1

ρ

∂P ∗1

∂ρ− Perad

(DAB

DPB

)v∗

r1∂P ∗

1

∂ρ− Peax

(R2

1

L2

)(DAB

DPB

)v∗z

∂P ∗1

∂ζ= 0 (B.14)

Dense layer

substrate :∂2C∗

2

∂ρ2+ 1

ρ

∂C∗2

∂ρ− v∗

r2Perad∂C∗

2

∂ρ− λ2(E2/E)C

∗2E

θ + (km/ki)P∗2E + (1 + kp)C

∗2E

= 0 (B.15)

product :∂2P ∗

2

∂ρ2+ 1

ρ

∂P ∗2

∂ρ− v∗

r2Perad

(DAB

DPB

)∂P ∗

2

∂ρ+ λ2(DAB/DPB)(E2/E)C

∗2E

θ + (km/ki)P∗2E + (1 + kp)C

∗2E

= 0 (B.16)

Spongy layer

substrate :∂2C∗

3

∂ρ2+ 1

ρ

∂C∗3

∂ρ− v∗

r3Perad∂C∗

3

∂ρ− λ2(E3/E)C

∗3E

θ + (km/ki)P∗3E + (1 + kp)C

∗3E

= 0 (B.17)

product :∂2P ∗

3

∂ρ2+ 1

ρ

∂P ∗3

∂ρ− v∗

r3Perad

(DAB

DPB

)∂P ∗

3

∂ρ+ λ2(DAB/DPB)(E3/E)C

∗3E

θ + (km/ki)P∗3E + (1 + kp)C

∗3E

= 0 (B.18)

240 V. Calabro et al. / Journal of Membrane Science 206 (2002) 217–241

Appendix C

Performance index

PI = 2λ2

Pe′ax

(E2

Eε2

∫ 1

0

∫ R2/R1

1

C∗2E

θ + (1 + kp)C∗2E + (km/ki)P

∗2Eρ dρ dξ

+E3

Eε3

∫ 1

0

∫ R3/R1

R2/R1

C∗3E

θ + (1 + kp)C∗3E + (km/ki)P

∗3Eρ dρ dξ

)(C.1)

Effectiveness factor

η= 2(ϑ + 1)R21

ε2(R22 − R2

1)+ ε3(R23 − R2

2)

(E2

Eε2

∫ 1

0

∫ R2/R1

1

C∗2E

θ + (1 + kp)C∗2E + (km/ki)P

∗2Eρ dρ dξ

+E3

Eε3

∫ 1

0

∫ R3/R1

R2/R1

C∗3E

θ + (1 + kp)C∗3E + (km/ki)P

∗3Eρ dρ dξ

)(C.2)

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