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Mass transport through biocatalytic membrane reactor
Desalination 245 (2009) 422–436
Mass transport through biocatalytic membrane reactor Endre Nagy*, Edina Kulcsár University of Pannonia, FIT, Research Institute of Chemical and Process Engineering, P.O. Box 158, 8201 Veszprém, Hungary Tel. +36-88-624040; Fax +36-88-624038; email: [email protected] Received 30 June 2008; revised 03 February 2009; accepted 09 February 2009
Abstract Mathematical models have been developed to calculate the mass transfer rates through catalytic membrane layer and through the concentration boundary layer by means of explicit, closed expressions for first-order and zeroorder bioreactions as well as even in the case of the nonlinear Michaelis–Menten reaction kinetics and/or in the case of variable mass transport parameters as diffusion coefficient, convective velocity. Some typical examples, applying the Michaelis–Menten kinetics and its limiting cases, namely the first-order kinetic (KM >> C) and zeroorder kinetic (C >> KM), are shown regarding the concentration distribution and the mass transfer rates as a function of the reaction modulus or of the Peclet numbers of the boundary layer and/or membrane layer. It has been shown that the mass transport parameters and the biochemical reaction rate can essentially alter the mass transfer rates and that significant differences of the results can be obtained by the three different reaction orders. Keywords: Membrane reactor; Biochemical reaction; Mass transfer rate; Michaelis–Menten kinetics; First-order reaction; Zero-order reaction
1. Introduction Membrane bioreactor (MBR) technology is advancing rapidly around the world both in research and commercial applications [1–4]. Integrating the properties of membranes with biological catalyst such as cells or enzymes forms the basis of an important new technology called membrane bioreactor. The MBR have been introduced several decades ago and until now it is rec*Corresponding author.
ommended or applied for production of foods, bio-fuels, amino acids, antibiotics, vitamins, proteins, fine chemicals, etc. [2,5]. The experiments are focused primarily on the hollow-fiber bioreactor with biocatalyst, either live cells or enzymes, inoculated into the shell and immobilized within the membrane matrix or in a thin layer at the membrane matrix–shell interface [7,8]. Cells are either grown in the extracapillary space with medium flow through the fibers and supplied with oxygen and nutrients, or grown within the fibers with medium flow outside or
Presented at the conference Engineering with Membranes 2008; Membrane Processes: Development, Monitoring and Modelling – From the Nano to the Macro Scale – (EWM 2008), May 25–28, 2008, Vale do Lobo, Algarve, Portugal. 0011-9164/09/$– See front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.desal.2009.02.005
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across the fibers while wastes and desired products are removed. The main advantages of the hollow-fiber bioreactor are the large specific surface area (internal and external surface of the membrane) for cell adhesion or enzyme immobilization; the ability to grow cells to high density; the possibility for simultaneous reaction and separation; relatively short diffusion path in the membrane layer; the presence of convective velocity through the membrane if it is necessary in order to avoid the nutrient limitation [9,10]. The performance of a hollow-fiber or sheet bioreactor is primarily determined by the momentum and mass transport rate [11,12] of the key nutrients through the bio-catalytic membrane layer. Thus, the operating conditions (trans-membrane pressure, feed velocity), the physical properties of membrane (porosity, wall thickness, lumen radius, matrix structure, etc.) can considerably influence the performance of a bioreactor, the effectiveness of the reaction. Mathematical description of the transport processes enables us to predict the concentration distributions of nutrients in the catalyst membrane layer, and thus, it makes possible to choose correctly the operating conditions which provide sufficient level of nutrient concentration in the membrane layer. The main aim of this study is to give closed, as simple as possible, mathematical equations in order to predict the concentration distribution and the mass transfer rate through plane, biocatalytic membrane layer with constant or varied transport parameters. The mass transport accompanied by first-order reaction kinetics, and zero-order kinetics as a limiting cases of Monod kinetics as well as accompanied by the general Monod or Michaelis–Menten kinetics ([Q = vmaxC/(KM + C)] will be solved and discussed. In every three cases the mass transport process has been treated by special mathematical methods. These equation should then be replaced into the full scale mass transport models [9,10], in order to calculate the concentration of the bulk liquid phase on both sides of the membrane.
2. Mass transfer equations of a membrane bioreactor (porous layer, biofilm) Principle of the mass transport of substrates/nutrients into the immobilized enzyme/cells, through a solid, porous layer (membrane, biofilm) or through a gel layer of enzyme/cells is the same. The structure, the thickness of this mass transport layer can be very different, thus, the mass transport parameters, namely diffusion coefficient, convective velocity, the bio-reaction rate constant, their dependency on the concentration and/or space coordinate is characteristic of the porous layer and of the nature of the biocatalysts. Several investigators modeled the mass transport through this biocatalyst layer, through enzyme membrane layer or cell culture membrane layer [6,8,9,13]. The enzyme or microorganism is immobilized in a gel layer on a membrane layer [14] or in the porous support layer of an asymmetric membrane [15,16]. The permeation rate through the skin layer is estimated to be 10−6 to 10−4 m/s at ΔP = 20–30 kPa (Table 1) [16]. This convective velocity should exist in the support layer, as well. Thus, Pe number of the support layer can be changed between 0.1 and 10 (Table 1). Consequently, both the convective flow and the diffusive flow should be taken into account in the spongy membrane layer. Fig. 1 illustrates the enzyme (or microorganism) membrane reactor with a concentration boundary layer. The effect of the skin layer is not involved in
Table 1 Membrane module characteristics and physical parameters applied for calculation of the mass transfer rates into and out of a sheet membrane [6,9,15] Pressure difference Pe number Thiele-modulus, Φ Diffusion coefficient Membrane thickness
20–30 kPa 0.1–10 0.1–5 10–9–10–10 m2/s (100–1000) × 10–6 m
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expression of the differential mass balance equation to the biocatalytic membrane layer, are
Membrane with biomass layer
Cb Skin layer Φ
C*
C*m Cp
Y=0
δ
δm
Fig. 1. The membrane layer with biomass with thickness of δm and the concentration polarization layer with thickness of δ. C* and Cm* are concentrations on the liquidmembrane interface (→: increasing reaction modulus).
the mass transfer rates defined to the enzyme membrane layer (see Sections 2.1–2.3). Its effect on the overall mass transfer rate, Jsum, is briefly discussed in Section 2.4. Recently Nagy [17] studied the mass transfer rate into a biocatalytic membrane layer with constant mass transport parameters. He defined the mass transfer rates for both side of the membrane surface. The rate equations are expressed as product of the mass transfer coefficient and driving force as it is traditionally applied for the diffusion systems, e.g. in gas–liquid systems. Applying these inlet mass transfer rate, the concentration profile of the two layers, namely that of the boundary layer and biocatalytic membrane layer can be calculated. These will be demonstrated in the case of first- and zeroorder reactions as well as in the case of the general Monod kinetics. Assumptions, made for
• reaction occurs at every position within the biocatalyst layer; • reaction has one rate-limiting substrate/ nutrient; • mass transport through the biocatalyst layer occurs by diffusion and convection; • the partitioning of the components (substrate, product) is negligible (Thus, C* = C*m; see Fig. 1); • the mass transport parameters (diffusion coefficient, convective velocity, bioreaction rate coefficient) are constant or varying; • the effect of the concentration boundary layer should also be taken into account; For the sake of simplification, let us regard a steady-state reaction as well as let us use the Cartesian co-ordinate, thus, the differential mass balance equation, applying the Michaelis– Menten kinetics, can be given as follows (y is here the transverse space coordinate, perpendicular to the membrane interface): ⎧⎪ d ⎛ dC ⎞ d (υ C ) ⎫⎪ vmax C =0 ⎬− ⎨ ⎜D ⎟− dy ⎭⎪ K M + C ⎩⎪ dy ⎝ dy ⎠
(1)
In general case, as it was mentioned, the diffusion coefficient and/or convective velocity can depend on the space coordinate, thus D = D(y), υ(y), (or on the concentration, D = D(C) or both of them, D = D(C, y)]. In the boundary conditions the external mass transfer resistance should also be taken into account. The membrane layer with biomass colony on it and the concentration boundary layer are illustrated, with important notations and concentration profiles, in Fig. 1. The boundary conditions of Eq. (1) (the diffusive mass transfer resistance of the concentration boundary layer is expressed here by k0 value) will be as follows:
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resistance in the boundary layer is neglected here, thus C* = Cb and the place y = 0 means here the inlet interface of the membrane layer):
if y = δ then
υin C * + kL0 ( Cb − C * ) = −D
dC dy
+υ C y =δ +
(2a) y =δ +
υout Cδ* + kδo ( Cδ* − C p ) dC dy
+υ C y =δ
+δ m−
C = Cb
(4a)
at y = δm
C = Cp
(4b)
The general solution of Eq. (3), applying standard methodology (for details of solution see [17]), will be as follows (Y = y/δm):
if y = δ + δm then
= −D
at y = 0
(2b) y =δ +δ m+
* δ
where C*, C denote the concentration of liquid at the membrane interface, on the two sides of membrane, δm is the membrane thickness; kδ0 represents here the liquid downstream side’s mass transfer coefficient (the resistance of the skin layer is neglected in this equation). The membrane concentration, C is given here in a unit of measure of gmol/m3. This can be easily obtained by means of the usually applied in the, e.g. g/g unit of measure with the equation of C = wρ/M, where w concentration in kg/kg, ρ – membrane density, kg/m3, M – molar weight, kg/mol (The subscript m denoting the membrane layer has been left here for the sake of simplified writing). In the next sections, the mass transfer with both constant (Sections 2.1– 2.4) and variable parameters (Section 2.6) will be discussed.
C = Tm eλY + Pm eλY
(5)
The “overall” mass transfer rate, namely the sum of the diffusive and the convective mass flow, can be expressed as: J = − Dm
dC dY
Y =0
(
+ υ C Y = 0 ≡ km0 λTm + λ Pm
)
(6)
According to Eqs. (5) and (6), the J value can be expressed as follows: J = β ( Cb − KCp )
β = km0 K=
(7)
( Pem / 2 ) tanh Θ m + Θ m
cosh Θ m
(8)
tanh Θ m
Θ m e − Pem / 2 ( Pem / 2 ) tanh Θ m + Θ m
(9)
and 2.1. Mass transfer accompanied by first-order reaction for the single membrane layer The differential mass balance equation in the case of constant transport parameters is as follows: d C dC −υ − k1C = 0 2 dy dy
Pem Pe Pe + Θ λ = m − Θ Θ = m 1 + 2ξ 2 2 2 2
Pem =
2
Dm
λ=
(3)
Let us look at the biocatalytic membrane layer without concentration boundary layer. Thus, the boundary conditions are as (the mass transfer
υδ m kδ2 Φ= 1 m Dm Dm
ξ=
k1δ m υ
km0 =
Dm δm
The Φ defined here is the well-known Thiele modulus defined for the catalyst membrane layer while the Peclet number corresponds to the often used Bodenstein number. The ξ dimensionless
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variable is the ratio of the reaction rate and the convective flow. Similarly to Eqs. (5) and (7), the concentration distribution and the mass transfer rate at membrane interface, without chemical reaction can be given as follows, respectively: C = Tm e PemY + Pm
(10)
J 0 = β m0 ( Cb − e − Pem Cp )
(11)
β m0 = km0 Pem
Dm Pem e Pem e Pem / 2 (12) ≡ e Pem − 1 δ m 2 sinh ( Pem / 2 )
Eqs. (10) and (7) should define the mass transfer rate of any membrane process with convective and diffusive mass flows without biochemical or chemical reaction, applying suitable boundary conditions. From Eq. (10) it follows that the concentration distribution in the membrane layer is not linear due to the convective velocity. Thus, the often applied mass transfer rate for membrane separation, namely J0 = k0(Cb – Cp) is not valid generally in the presence of convective velocity in the membrane layer (k0 represents here a general mass transfer coefficient). This mass transfer equation is involved in Eq. (10) as a limiting case, namely if Pem tends to be zero.
⎛ ⎞ Φ 2 ⎞⎛ e − Pem J = β m0 ⎜1 + 2 ⎟ ⎜ Cb − Cp 2 2 ⎟ 1 + Φ / Pem ⎠ ⎝ Pem ⎠ ⎝
(14)
where
Φ=
k0δ m2 Dm Cb
The enhancement of the mass transfer coefficient will be a simple expression, namely: k0δ m β Φ2 ξ2 = 1 + ≡ 1 + ≡ 1 + β m0 Pem2 Cbυ Pem
(15)
with
ξ=
k0δ Cbυ
2.3. The two-layer (boundary- and membrane layer) mass transfer rate
2.2. Mass transfer accompanied by zero-order reaction
In the boundary layer there is no biochemical (or chemical) reaction, thus the mass transfer rate through it can be given according to Eq. (11), as follows (here C* ≤ Cb according to the measure of the mass transfer resistance of the boundary layer; in special cases the interface concentration, C* can also be higher than the bulk concentration, Cb, i.e. C* ≥ Cb, this is also correctly described by Eq. (16) or Eq. (11)):
The differential mass balance equation to be solved is as:
J 0 = β L0 ( Cb − e − PeL C ∗ )
Dm
d 2C dC −υ − k0 = 0 2 dy dy
(13)
The solution of Eq. (13) has been discussed in details in Nagy’s paper [17]. The applied methods for the solution of the above differential equation are standard ones that are well known, thus, it is not discussed here. The mass transfer rate for the membrane layer can be given as:
(16)
with
β L0 = kL0 PeL with kL0 =
DL δL
DL PeL e PeL e PeL / 2 ≡ (17) e PeL − 1 δ L 2 sinh ( PeL / 2 )
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where C* is the concentration on the membrane interface as it is illustrated in Fig. 1 (it was assumed here that the solubility coefficient of the substrate∗ in the porous support layer is unit, thus, C* = Cm). J0 in Eq. (16) and the value of J in Eq. (7) are equal to each other, thus the overall mass transfer rate can be given by means of the resistance-in-series principle. The total, inlet mass transfer rate is, for first-order reaction, as follows: J = β tot ( Cb − Ke − PeL Cp )
(18)
(19)
⎛ ⎞ e −( PeL + Pem ) J = β tot ⎜⎜ Cb − Cp ⎟⎟ 2 2 1 + Φ / Pem ⎠ ⎝
(20)
with 1 e − PeL 1 + β L0 β m0 (1 + Φ 2 / Pem2 )
(21)
Similarly the two-layer mass transfer rate, without biochemical reaction in the layers can be expressed as: J =β 0
0 tot
(C
b
−e
− PeL
This case will not be discussed in this paper, in details. For sake of completeness the total mass transfer rate will be defined in this subsection taking into account all three layers, namely boundary layer, enzyme membrane layer and skin layer. For it the outlet mass transfer rate of substrate should also be defined. For example, for first-order biochemical reaction, the outlet mass transfer rate at y = δ + δm is as follows [17]:
≡ β δ ( Cb − ξ Cp )
1 1 e − PeL + β L0 β
For zero-order reaction, the mass transfer rate, and the mass transfer coefficient, for two layers can be given as:
β tot =
2.4. Mass transfer rate including the skin layer, as well
⎛ Θ tanh Θ m − Pem / 2 ⎞ Cp ⎟ J δ = βδ ⎜ Cb − cosh Θ m m Θ m e Pem / 2 ⎠ (24) ⎝
where
β tot =
427
Cp )
(22)
βδ =
Dm e Pem / 2 δ m sinh Θ m
(25)
The resistance-in series model is applied to describe the overall mass transfer rate, similar to that done in Section 2.3. The mass transfer rate through the skin layer can be given by the same equation as that for the concentration boundary layer, namely by Eq. (16) taking into account the real values of the transport parameters of the skin layer. The effect of the skin layer on the outlet concentration is illustrated by dotted line in Fig. 1. The value of Jδ is equal to the inlets into the skin layer, i.e. Jδ = Js, where Js denotes the mass transfer rate through the skin layer. Thus one can obtain the following expression for the three-layer mass transfer rate (S is the solubility coefficient in the skin layer, related to the spongy layer): ⎛ e − PeL ξ ⎞ J sum = β sum ⎜ Cb − Cp ⎟ S ⎝ ⎠
(26)
where 0 = β tot
1 1 e − PeL + 0 β L0 βm
(23)
βsum =
1 1 ξ + βsk S βδ
(27)
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The value of βsk is similar to that given for the boundary layer by Eq. (17), S denotes the solubility of the substrate in the skin layer; is given in Eq. (24). Effect of the skin layer on the mass transfer rate is not discussed in this paper, in details. Applying the above equations, it is easy to predict the effect of the skin layer on the sum of the substrate mass transfer rate, Jsum. 2.5. The two-layer concentration profile for first- and zero-order reactions Knowing the mass transfer rate, the parameter values of the concentration distribution (TL, PL, Tm, Pm) can easily be obtained using the external boundary conditions (Eqs. (4a) and (4b)) and the internal boundary conditions on the feed membrane interface, that is C ∗ = Cm∗ and the mass transfer rate into the boundary layer, Eq. (16), and that into the membrane layer, Eq. (7), are equal to each other, i.e. J0 = J. Knowing the mass transfer rate, the value of C* can be obtained e.g. from Eq. (16). According to Eq. (10) and boundary condition (4a), the TL and PL parameters of the concentration distribution can be given as C ∗ − Cb e PeL − 1
(28)
PL = Cb – TL
(29)
TL =
The Tm and Pm parameters can be obtained by means of C ∗ = Cm∗ at Y = 1 (here Y = y/δ) and of the external boundary condition, here at Y = 1 + δm/δ, Eq. (4b). Thus, one can get for first-order reaction, as
Tm =
Pm =
(e
C ∗eλm − Cp λm
(e
)
− eλm eλmδ / δ m
Cp − C ∗eλm λm
)
− eλm eλmδ / δ m
(30)
(31)
The Tm and Pm parameters for the zero-order reaction can be given in similar way. Its concentration distribution obtained by general solution of Eq. (13) for the membrane layer is as: C = Tm e( PemY δ / δ m ) + Pm − Cbξ 2δ / δ m (32)
1 ≤ Y ≤ 1 + δm/δ
Applying Eq. (32) at y = δ (Y = 1) and at y = δ + δm, (Y = 1 + δm/δ), one can easily get the necessary equation to determine the value of parameters in the case of zero-order reaction: Tm =
Pm =
C ∗ − Cp − Cbξ 2δ / δ m
(1 − e ) e Pem
(33)
Pemδ / δ m
Cp + Cbξ 2 (1 + δ / δ m ) − ( C ∗ + Cbξ 2δ / δ m ) e Pem
(1 − e ) Pem
(34)
2.6. Approaching analytical solution of the mass transport with variable parameters and/or with Monod kinetics for single membrane layer For the solution of Eq. (1) with boundary conditions Eqs. (2a) and (2b) an analytical approach has been developed that makes possible to express the mass transfer rate and the concentration distribution in explicit, closed mathematical equations. In essentials, this solution methodology serves the mass transfer rate and the concentration distribution in closed, explicit mathematical expression. The method can be applied for Cartesian coordinate and cylindrical coordinate as will be shown. For the solution of Eq. (1), the biocatalytic membrane should be divided into N sublayer, in the direction of the mass transport, that is perpendicular to the membrane interface (for details see e.g. Nagy’s papers [18,19]), with thickness of Δδ (Δδ = δm/N) and with constant transport parameters in every sub-layer. Thus, for the nth
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sub-layer of the membrane layer, using dimensionless quantities, it can be obtained:
Cn Y =Y + = Cn +1 Y =Y −
d 2 Cn dC Dn − υn n − kn Cn = 0 yn–1 < y < yn, (35) 2 dy dy
The solution of the algebraic equation system obtained by means of the internal ((39a) and (39b)) conditions with n = 1,2,...,N and external boundary conditions ((2a) and (2b)) can be given by numerically or by analytically. The analytical approach of the solution is discussed in Nagy’s papers [19,20]. The mass transfer rate on the upstream side of the membrane can be given, for that case, as follows:
where the value of kn can be obtained according e.g. to the Michaelis–Menten kinetics as follows: kn =
vmax K M + Cn
(36)
where Cn denotes the average value of C in the nth membrane sub-layer. In dimensionless form one can get the following equation: 2
d Cn dC − Pen n − Φ 2n Cn = 0 dY 2 dY Φ n = δ m2 kn / Dn ;
where
(37) Pen =
υ nδ m Dn
The solution of Eq. (36) can be easily obtained by well-known mathematical methods as it follows (Y = y/δm): Cn = Tn e
( λnY )
+ Pn e(
λnY )
Yn–1< Y < Yn
(38)
with
λn =
Pen − Θn 2
Pe λn = n + Θ n 2
Tn and Pn parameters of Eq. (37) can be determined by means of the boundary conditions for the nth sub-layer (with 1 ≤ n ≤ N). The boundary conditions at the internal interfaces of the sublayers (1 ≤ n ≤ N–1; Ym = nΔY; ΔY = 1/N) can be obtained from the following two equations ([Eqs. (39a) and (39b)): −
dCn D ⎛ dC ⎞ + Pen Cn = n +1 ⎜ − n +1 + Pen +1Cn +1 ⎟ dY Dn ⎝ dY ⎠
at Y = Yn
n
J=
n +1
(39b)
at Y = Yn
DC10 D ⎛ dC ⎞ − + PeC λ1T1 + λ1 P1 = ⎜ ⎟ δ ⎝ dY δ ⎠ Y =0
(
)
(40)
The outlet mass transfer rate can be similarly given. This value should be as less as possible to avoid the loss of the substrate during the process. In the next section some typical figures will be shown to illustrate the effect of the parameters on the mass transport through the catalytic membrane layer. 2.7. Full-scale model The full-scale models describe the concentration distribution in the feed bulk phase [9,10]. This will not be discussed in this paper. The mass transfer rates defined in Eqs. (6), (14), (20)–(23) could essentially affect the bulk phase concentration. For example, in the case of a capillary membrane, the concentration distribution should be given in both axial and radial directions. For it, the mass transfer rate equation developed in this paper serve as boundary conditions at the membrane interface [9,10]. Applying equations developed, it is easy to predict the effect of the biochemical reaction on the feed phase concentration. This effect will be the object of a separate paper. 3. Results and discussion
(39a)
The axial and radial depletion of substrate, e.g. oxygen, nutrient, can be often critical scale-
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limiting factor in cell culture hollow fiber reactor [12–14]. In order to increase the substrate concentration in the membrane bioreactor, sufficient diffusion rate and/or convective flow has to be provided through the lumen, in axial direction, and through the membrane layer, in radial direction, of a hollow fiber. Typical operating conditions of a hollow-fiber bioreactor were applied (Table 1) to calculate the inlet and outlet mass transfer rates of a substrate. From that, the effectiveness of the bio-catalytic reaction as well as sufficiency of the nutrient supply could be estimated. The biochemical reaction rate depends on the amount of catalyst immobilized in the membrane or on the density of cells in the membrane structure. 3.1. Two-layer concentration profiles and mass transfer rates 3.1.1. Mass transfer without biochemical reaction Eqs. (28)–(31) enable us to simulate the concentration profiles in the boundary layer and membrane layer under different mass transport conditions. Thus, one can select the desired mass transfer conditions as the diffusion resistance in the boundary layer and membrane layer, the trans-membrane pressure to obtain the necessary convective velocity of the substrate or nutrient in order to get sufficient activity of the microbe in the biomass layer on the membrane feed interface and/ or in the membrane matrix., etc., depending on the biochemical reaction rate. In the case of the first- and zero-order reactions, simple, explicit transfer equations have been developed for the calculation. The more complex intermediate reaction rate regime of the Monod or Michaelis–Menten kinetics lies between these two limiting cases, thus, these equations can be used for general prediction of the effect of Monod’s reaction kinetics. The equations developed make also possible to predict the concentration profile without chemical
1.0 Pem=10 Concentration distribution, –
430
0.8 2
0.6
1
PeL=1
0.4
0.5
k0L =1×10–4 δL/δm=1
0.2
0.1
0.0 0.0
0.4
0.8
1.2
1.6
2.0
Boundary layer and membrane thickness, –
Fig. 2. Concentration related to that of the value of Cb as function space coordinate, in the boundary layer and membrane layer without biochemical reaction at different values of the membrane Pe number (Cp/Cb = 0.1, km0 Pem = kL0 PeL).
reaction. Let us look at first some typical profiles at different membrane Pe numbers (Fig. 2). As you know, the convective velocity should be the same in both layers, according to the continuity law. Thus, when the Pem value changes and PeL value remains constant (PeL = 1), it should mean that the diffusive mass transfer coefficient ( km0 = Dm / δ m) should also be changed. In this case, because the ratio of the membrane and the boundary layer thickness was chosen to be constant, the diffusion coefficient in the membrane should be varied with the variation of the value of Pem. Fig. 2 illustrates well that the value of Pem has strong effect on the concentration distribution in both layers and thus, on the value of concentration of the membrane interface. That means that the well-known parameters as concentration polarization modulus, I (I = C*/Cb, [21–23]), enrichment factor, E0 (E0 = Cp/C*) are essentially dependent on the mass transport properties of the biocatalytic membrane layer. Recently Nagy and Borbély [24] studied how the membrane properties alter the concentration polarization modulus and the intrinsic enrichment factor. The equation developed can easily be used to describe the role of both layers on the separation of transported components.
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ϕ = 0.01
Pem = 0.1
17.0
0.8
0.1
1 PeL=Pem = 1
0.6
Enhancement, β/βm, –
Concentration distribution, –
1.0
2
o
δL/δm = 1 k0L =k0m = 1×10–4
0.4
5
10
0.2 Boundary layer
1 13.0
4
9.0 10
5.0
Membrane layer
0.0
25
1.0 0.0
0.4
0.8
1.2
1.6
2.0
Boundary layer + membrane, –
Fig. 3. The effect of first-order chemical reaction on the relative value of concentration profiles in the two layers at different values of the reaction modulus (Cp/Cb = 0.1).
3.1.2. The effect of the first-order reaction Similarly to the case of the gas–liquid or liquid–liquid mass transfer, the chemical reaction could essentially influence the concentration profile in the membrane phase as a consequence, the concentration profile in the boundary layer and the feed mass transfer rate. Typical concentration profiles are shown in Fig. 3 at different values of the reaction modulus. The Φ reaction modulus can significantly lower the concentration profile not only in the membrane layer but also in the concentration boundary layer, as well. At larger Φ values the concentration curves have inflexion points, at Y = 1, at the feed membrane interface, where the curves change their concave trends to convex one. The relative value of the outlet concentration (= Cp/Cb) was chosen to be 0.1. The concentration inside the membrane matrix, depending on the reaction rate, can decrease below its outlet value, at high Φ values. These curves have minimum, after the minimum value, with increasing space coordinate, the concentration gradient will be positive which means that the direction of the diffusion stream will be turned into direction of the feed interface. That should mean that to maintain the obtained concentration additional feeding of the substrate on the outlet side of the membrane is needed. Obviously, it is not a real demand. The model equation serves a simple means to calculate
2
1.0
10.0
100.0
Reaction modulus, Φ, −
Fig. 4. Enhancement of the inlet mass transfer coefficient given for the biocatalytic membrane layer, without mass transfer resistance in the concentration boundary layer (Cp/Cb = 0.1).
the mass transport through biocatalytic membrane layer under all important mass transport and reaction kinetics parameters. Let us look typical examples for the enhancement as a function of the biochemical reaction rate. Regarding single layer mass transfer (mass flow without mass transfer resistance in the boundary layer), the enhancement increases without limit as a function of the reaction rate (Fig. 4). This behavior is similar to that of gas– liquid diffusive mass transfer. According to parameters Φ and Θ (Eq. (9)), the value of Pem and reaction rate constant, Φ, can be varied, independently. With the increase of the Peclet number, the effect of the reaction rate strongly lowers. It should be noted that with increasing Pem number, the convective mass flow increases compared that to the diffusive flow. The decrease of the 0 value of β / β m, with increasing value of Pem, at a given Φ value, can be a consequence of the increasing convective mass flow. With the change 0 of the diffusive mass transfer coefficient, km , the value of Φ also changes. If one plots the function of β / β m0 vs. ξ function this change reduces essentially but the above tendency remains valid (not shown here). How the concentration boundary layer can influence the overall enhancement is illustrated
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E. Nagy and E. Kulcsár / Desalination 245 (2009) 422–436 1.5
1.0 Φ = 0.01
1.4
δL = δm
1.3
kL = 1×10−4
Concentration distribution, –
0 Enhancement, βtot/βtot ,–
Pe L =1
0
1.2
Pem=0.1
10
25
1.1 0.1 1.0
0.8
δ L/ δ m = 1 0.4
1.0
10.0
100.0
Reaction modulus, Φ,−
Fig. 5. Enhancement of the inlet mass transfer coefficient for two-layer mass transport process (Cp/Cb = 0.1, km0 Pem = kL0 PeL).
in Fig. 5. The essential difference between results obtained for single layer (membrane layer, Fig. 4) and two layers (concentration boundary layer and membrane layer) is that the curves tends to liming values with increasing reaction rate constant, due to the effect of the boundary layer. At high reaction rate, the concentration at liquid-membrane interface can decrease close to zero, this determines the highest value of the enhancement. With increasing value of Pem, the curves are shifted to higher reaction rate regime and the maximum value of the enhancement also increases. The figure clearly shows that the role of the concentration boundary layer is essential regarding the enhancement, or the inlet substrate mass transfer rate. 3.1.3. The effect of the zero-order reaction The effect of zero-order reaction can essentially differ from that of first-order reaction. This reaction as limiting case of Monod kinetics is important since it very often can occur in case of bioctalytic reactions. Because the zero-order reaction rate is independent of the concentration, this effect, on the concentration profile (Fig. 6), will be much higher. With the increase of the Φ value, the concentration in the membrane layer, lowers quickly, between Φ = 2 and Φ = 3, practically
2
0 k0L = km = 1×10−4
3
0.2 5 Boundary layer
0.0 0.1
1
PeL= Pem =1
0.6
0.0
0.4
Membrane layer
10 0.8
1.2
1.6
2.0
Boundary layer + membrane, –
Fig. 6. The effect of the zero-order reaction on the twolayer concentration profile (Cp/Cb = 0.1).
linearly, to zero. That means that much higher convective velocity is needed in order to keep the concentration above the necessary critical value during the reaction than in the case of first-order reaction. The enhancement can be very easy to obtain in the case of zero-order reaction according to Eq. (15) for single layer mass transfer. Practically, the value of β / β m0 increases linearly, with the increase of the reaction modulus. The enhancement of the two-layer mass transfer can be also easily obtained by means of Eq. (21). 3.1.4. Single layer mass transfer with Monod or Michaelis–Menten kinetics The inlet mass transfer rate was calculated by Eq. (40) applying the solution of Eqs. (35)–(39b) [19,20]. The outlet mass transfer rate can similarly be obtained. For the sake of completeness, the mass transfer rates for limiting cases, namely for first- and zero-order reactions will also be given. The results obtained for the limiting cases are identical with that obtained by analytical solutions. How the relative value of the outlet mass transfer rate changes as a function of Thiele modulus is illustrated in Fig. 7 at Pem = 1 applying the general Michaelis–Menten kinetics (line 2), and as limiting cases, the first-order kinetic (KM >> Cm, line 1) as well as the zero-order kinetics (Cm
E. Nagy and E. Kulcsár / Desalination 245 (2009) 422–436 1.0 Pem =1
0.8
Jout/J−
0.6 0.4
1
0.2
2
0.0
3
−0.2 −0.4 0.1
1.0
10.0
Reaction modulus, Φ–
Fig. 7. The relative value of the outlet mass transfer rate as a function of reaction modulus applying the Michaelis–Menten kinetics (line 2) and its limiting cases, namely first-order (line 1) and zero-order (line 3) kinetics. (δm = 100 μm, Dm = 5.4 × 10–10 m2/s, kL0 → ∞, kδ0 → ∞, K M / C10 = 1, C20 / Cb = 0.2).
>> KM, line 3). Similar to the concentration distribution, the zero-order kinetics (line 3) has essentially higher conversion than the other reaction types (line 2: Michaelis–Menten kinetics; line 1: first-order kinetic). How the starting value of Φ was calculated is given in Appendix. The starting value of Φ was the same in every three cases. Its value was changed, inside the membrane with increasing diffusion path, according to the reaction order simulated. The dimensionless outlet
433
concentration was chosen here to be 0.2 which was considered as critical concentration. Below this value the microorganism was supposed to work insufficiently. With the increase of the reaction modulus, the outlet mass transfer rate strongly decreases. The value of Jout/J can also have negative value. That is the case when the direction the diffusion flow is contrary to the convective velocity due to its concentration gradient. In the membrane reactors, the desired value of the outlet mass transfer rate of the substrate should be equal to zero. With the model developed, the necessary convective velocity can be easily predicted at which the outlet concentration will be desired one, while the outlet mass transfer rate on the downstream side of the membrane will be approximately zero. It is obvious that the Peclet number also has strong effect on the mass transfer rates. The effect of the Peclet number is illustrated in Fig. 8, applying the nonlinear Monod kinetics. With increase of the Pe number the value of Jout also increases. For instance at Φ = 1, the value of Jout/J is about 50% higher at Pem = 10 than that at Pem = 1. The inlet mass transfer enhancement was shown in Figs. 4 and 5. Applying the solution of the model, the effect of all important parameters can be predicted and thus, the desired reaction conditions can be chosen.
1.0 0.8
Jout/J,–
4. Conclusion
Pem = 10
0.6 0.4
1 0.2 0.1
0.0 −0.2
MM-kinetics
−0.4 0.1
1.0
10.0
Reaction modulus, Φ, –
Fig. 8. The change of the outlet mass transfer rate as a function of reaction rate at different values of membrane Peclet number applying the Michaelis–Menten (MM) reaction kinetics (δm = 100 μm, Dm = 5.4 × 10–10 m2/s, k 0 → ∞, k 0 → ∞, K / C 0 = 1, C 0 / C = 0.2). L
δ
M
1
2
b
Modeling of membrane bioreactor is now in its starting stage. Exact explicit mathematical equations in order to predict how the mass transport parameters, diffusion coefficients, convective velocity, the bioreaction rate parameters could alter the concentration distribution and the mass transfer rate in a biolayer (enzyme/microorganism membrane layer) is very important. It has been proved that the simultaneous diffusive and convective mass transfer rate can also be expressed as product of a mass transfer coefficient and the driving force, similarly to that of the diffusive mass transfer coefficient given for gas–liquid or
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liquid–liquid systems. Mass transfer coefficients have been defined for mass transfer without and with biochemical reactions. For the first-order and zero-order reactions, these mass transfer coefficients can be given in closed mathematical form. Applying the mass transfer equations given for single layers, the so-called overall mass transfer rate and mass transfer coefficient can be given making possible to predict the simultaneous effect of the liquid boundary layer and of the biocatalytic membrane layers, on the mass transfer rate. Acknowledgment This work was supported by the Hungarian Research Foundation under Grants OTKA 63615/2006.
Greek letters
β0 β βδ βtot δL δm υ Φ
convective+diffusive mass transfer coefficient without chemical reaction, m/s mass transfer coefficient with biochemical reaction, m/s mass transfer coefficient of the downstream side of the membrane, m/s two-layer mass transfer coefficient with reaction in the membrane layer, m/s thickness of the concentration boundary layer, m thickness of the membrane layer, m convective velocity, m/s 2 = k1δ m2 / Dm or = k0δ m / Dm Cb
Subscripts Nomenclature C Cb Cp C* Cδ* D J0 J k1 k0 k0 kδ0 KM Pem vmax y Y
concentration, gmol/dm3 inlet concentration, gmol/dm3 outlet concentration, gmol/dm3 liquid concentration on the feed membrane interface, gmol/dm3 liquid concentration on the downstream membrane interface, gmol/dm3 diffusion coefficient mass transfer rate without chemical reaction, gmol/m2s mass transfer rate accompanied by chemical reaction, gmol/m2s first-order reaction rate constant, 1/s zero-order reaction rate constant, gmol/sdm3 physical mass transfer coefficient, m/s physical mass transfer coefficient on the downstream side, m/s Michelis–Menten constant = υδm/Dm maximum reaction rate space co-ordinate perpendicular to the membrane interface, m dimensionless space coordinate, (= y/δ or = y/δm)
1 2 n N L m
upstream phase downstream phase nth sub-layer number of the sub-layers in the membrane (N was chosen to be 100) boundary layer membrane layer
Superscripts *
at interface
Appendix Thiele modulus (Φ) was calculated by the following equation for the nth sub-layer of the membrane (the number of sub-layer was chosen to be 100) according to Michaelis–Menten kinetics:
Φn = δ 2
kn Dn
with (A.1)
vmax vmax / Cb kn = ≡ K M + Cn K M / Cb + Cn / Cb
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The value of kn was chosen according to the reaction order in such a way that the starting value of Φ, at Y = 0, denoted by Φ(0), should be of the desired and same value, independently of the limiting case: For the first-order reaction (KM >> Cn, or in dimensionless form: KM/Cb >> 1): kn =
vmax KM
(A.2)
For the zero-order reaction (Cn >> KM): kn =
vmax 1 K M Cn
(A.3)
The starting value of kn was identical in every three cases considering that C(Y = 0) =1. In the case of first-order reaction, the value of kn remains constant inside the membrane, in the case of zero-order reaction the product knCn thus, the value Φ2 remains constant inside the membrane. In the general case the value of kn will change according to Eq. (A.1) in the catalytic membrane. References [1] H. Strathmann, L. Giorno and E. Drioli, An Introduction to Membrane Science and Technology, Institute on Membrane Technology, Italy, 2006. [2] W. Yang, N. Cicek and J. Ilg, State-of-the-art of membrane bioreactors: Worldwide research and commercial applications in North America, J. Membr. Sci., 270 (2006) 201–211. [3] L. Giorno and E. Drioli, Biocatalytic membrane reactors: applications and perspectives, Trends Biotechnol., 18 (2000) 339–349. [4] J.G.S. Marcano and T.T. Tsotsis, Catalytic Membranes and Membrane Reactors, Wiley-VCH, Veinheim, 2002. [5] G. Belfort, Membranes and bioreactors: a technical challenge in biotechnology, Biotechnol. Bioeng., 33 (1989) 1047–1066. [6] J.D. Brotherton and P.C. Chau, Modeling analysis of an intercalated-spiral alternate-dead-ended hollow fiber bioreactor for mammalian cell cultures, Biotechnol. Bioeng., 35 (1990) 375–394.
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[7] L.J. Kelsey, M.R. Pillarella and A.L. Zydney, Theoretical analysis of convective flow profiles in a hollow fiber membrane bioreactor, Chem. Eng. Sci., 45 (11) (1990) 3211–3220. [8] J.A. Schonberg and G. Belfort, Enhanced nutrient transport in hollow fiber perfusion bioreactors: a theoretical analysis, Biotechnol. Prog., 3 (1987) 81–89. [9] J.M. Piret and C.L. Cooney, Model of oxygen transport limitations in hollow fiber bioreactors, Biotechnol. Bioeng., 37 (1991) 80–92. [10] C.A. Sardonini and D. DiBiasio, An investigation of the diffusion-limited growth of animal cells around single hollow fibers, Biotechnol. Bioeng., 40 (1992) 1233–1242. [11] B. Godongwana, M.S. Sheldon and D.M. Solomons, Momentum transfer inside a vertically orientated capillary membrane bioreactor, J. Membr. Sci., 303 (2007) 86–99. [12] V. Calabro, S. Curcio and G. Iorio, A theoretical analysis of transport phenomena in a hollow fiber membrane bioreactor with immobilized biocatalyst, J. Membr. Sci., 206 (2002) 217–241. [13] B.S. Ferreira, P. Fernandes and J.M.S. Cabral, Design and modeling of immobilized biocatalytic reactors, in: J.M.S. Cabral, M. Mota and J. Tramper (Eds.), Multiphase Bioreactor Design, Taylor and Francis, London, UK, 2001, pp. 85–180. [14] T. Gumi, J.F.-D. Albacete, D. Paolucci-Jeanjean, M.-P. Belleville and G.M. Rios, Study of the influence of the hydrodynamic parameters on the performance of an enzymatic membrane reactor, J. Membr. Sci., 311 (2008) 147–152. [15] S.W. Long., S. Bhatia and A. Kamaruddin, Modeling and simulation of enzymatic membrane reactor for kinetic resolution of ibuprofen ester, J. Membr. Sci., 219 (2003) 69–88. [16] L. Giorno, J. Zhang and E. Drioili, Study of mass transfer performance of naproxen acid and ester through a multiphase enzyme-loaded membrane system, J. Membr. Sci., 276 (2006) 59–67. [17] E. Nagy, Equations of mass transfer rates through biocatalytic membrane layer, Asia-Pacific J. Chem. Eng., Doi: 10.1002/apj.242. [18] E. Nagy, Binary, coupled mass transfer with variable diffusivity through cylindrical dense membrane, J. Membr. Sci., 274 (2006) 159–168. [19] E. Nagy, Mass transfer with varying diffusionand solubility coefficient through a catalytic membrane layer, Chem. Eng. Res. Des., 86 (2008) 723–730.
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[20] E. Nagy and G. Borbély, Mass transport through anisotropic membrane layer, Desalination, 240 (2009) 54–63. [21] S. Bhattacharya and S.-T. Hwang, Concentration polarization, separation factor, and Peclet number in membrane processes, J. Membr. Sci., 132 (1997) 73–90. [22] R.W. Baker, J.G. Wijmans, A.L. Athayde, R. Daniels, J.H. Ly and M. Le, The effect of concentration polarization on the separation of volatile
organic compounds from water by pervaporation, J. Membr. Sci., 137 (1997) 159–172. [23] S. De and P.K. Bhattacharya, Modeling of ultrafiltration process for a two-component aqueous solution of low and high (ge-forming) molecular weight solutes, J. Membr. Sci., 136 (1997) 57–69. [24] E. Nagy and G. Borbély, The effect of the concentration polarization and the membrane layer mass transport on the membrane separation, Appl. Membr. Sci. Technol., in press.
Mass transport through biocatalytic membrane reactor Endre Nagy*, Edina Kulcsár University of Pannonia, FIT, Research Institute of Chemical and Process Engineering, P.O. Box 158, 8201 Veszprém, Hungary Tel. +36-88-624040; Fax +36-88-624038; email: [email protected] Received 30 June 2008; revised 03 February 2009; accepted 09 February 2009
Abstract Mathematical models have been developed to calculate the mass transfer rates through catalytic membrane layer and through the concentration boundary layer by means of explicit, closed expressions for first-order and zeroorder bioreactions as well as even in the case of the nonlinear Michaelis–Menten reaction kinetics and/or in the case of variable mass transport parameters as diffusion coefficient, convective velocity. Some typical examples, applying the Michaelis–Menten kinetics and its limiting cases, namely the first-order kinetic (KM >> C) and zeroorder kinetic (C >> KM), are shown regarding the concentration distribution and the mass transfer rates as a function of the reaction modulus or of the Peclet numbers of the boundary layer and/or membrane layer. It has been shown that the mass transport parameters and the biochemical reaction rate can essentially alter the mass transfer rates and that significant differences of the results can be obtained by the three different reaction orders. Keywords: Membrane reactor; Biochemical reaction; Mass transfer rate; Michaelis–Menten kinetics; First-order reaction; Zero-order reaction
1. Introduction Membrane bioreactor (MBR) technology is advancing rapidly around the world both in research and commercial applications [1–4]. Integrating the properties of membranes with biological catalyst such as cells or enzymes forms the basis of an important new technology called membrane bioreactor. The MBR have been introduced several decades ago and until now it is rec*Corresponding author.
ommended or applied for production of foods, bio-fuels, amino acids, antibiotics, vitamins, proteins, fine chemicals, etc. [2,5]. The experiments are focused primarily on the hollow-fiber bioreactor with biocatalyst, either live cells or enzymes, inoculated into the shell and immobilized within the membrane matrix or in a thin layer at the membrane matrix–shell interface [7,8]. Cells are either grown in the extracapillary space with medium flow through the fibers and supplied with oxygen and nutrients, or grown within the fibers with medium flow outside or
Presented at the conference Engineering with Membranes 2008; Membrane Processes: Development, Monitoring and Modelling – From the Nano to the Macro Scale – (EWM 2008), May 25–28, 2008, Vale do Lobo, Algarve, Portugal. 0011-9164/09/$– See front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.desal.2009.02.005
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across the fibers while wastes and desired products are removed. The main advantages of the hollow-fiber bioreactor are the large specific surface area (internal and external surface of the membrane) for cell adhesion or enzyme immobilization; the ability to grow cells to high density; the possibility for simultaneous reaction and separation; relatively short diffusion path in the membrane layer; the presence of convective velocity through the membrane if it is necessary in order to avoid the nutrient limitation [9,10]. The performance of a hollow-fiber or sheet bioreactor is primarily determined by the momentum and mass transport rate [11,12] of the key nutrients through the bio-catalytic membrane layer. Thus, the operating conditions (trans-membrane pressure, feed velocity), the physical properties of membrane (porosity, wall thickness, lumen radius, matrix structure, etc.) can considerably influence the performance of a bioreactor, the effectiveness of the reaction. Mathematical description of the transport processes enables us to predict the concentration distributions of nutrients in the catalyst membrane layer, and thus, it makes possible to choose correctly the operating conditions which provide sufficient level of nutrient concentration in the membrane layer. The main aim of this study is to give closed, as simple as possible, mathematical equations in order to predict the concentration distribution and the mass transfer rate through plane, biocatalytic membrane layer with constant or varied transport parameters. The mass transport accompanied by first-order reaction kinetics, and zero-order kinetics as a limiting cases of Monod kinetics as well as accompanied by the general Monod or Michaelis–Menten kinetics ([Q = vmaxC/(KM + C)] will be solved and discussed. In every three cases the mass transport process has been treated by special mathematical methods. These equation should then be replaced into the full scale mass transport models [9,10], in order to calculate the concentration of the bulk liquid phase on both sides of the membrane.
2. Mass transfer equations of a membrane bioreactor (porous layer, biofilm) Principle of the mass transport of substrates/nutrients into the immobilized enzyme/cells, through a solid, porous layer (membrane, biofilm) or through a gel layer of enzyme/cells is the same. The structure, the thickness of this mass transport layer can be very different, thus, the mass transport parameters, namely diffusion coefficient, convective velocity, the bio-reaction rate constant, their dependency on the concentration and/or space coordinate is characteristic of the porous layer and of the nature of the biocatalysts. Several investigators modeled the mass transport through this biocatalyst layer, through enzyme membrane layer or cell culture membrane layer [6,8,9,13]. The enzyme or microorganism is immobilized in a gel layer on a membrane layer [14] or in the porous support layer of an asymmetric membrane [15,16]. The permeation rate through the skin layer is estimated to be 10−6 to 10−4 m/s at ΔP = 20–30 kPa (Table 1) [16]. This convective velocity should exist in the support layer, as well. Thus, Pe number of the support layer can be changed between 0.1 and 10 (Table 1). Consequently, both the convective flow and the diffusive flow should be taken into account in the spongy membrane layer. Fig. 1 illustrates the enzyme (or microorganism) membrane reactor with a concentration boundary layer. The effect of the skin layer is not involved in
Table 1 Membrane module characteristics and physical parameters applied for calculation of the mass transfer rates into and out of a sheet membrane [6,9,15] Pressure difference Pe number Thiele-modulus, Φ Diffusion coefficient Membrane thickness
20–30 kPa 0.1–10 0.1–5 10–9–10–10 m2/s (100–1000) × 10–6 m
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expression of the differential mass balance equation to the biocatalytic membrane layer, are
Membrane with biomass layer
Cb Skin layer Φ
C*
C*m Cp
Y=0
δ
δm
Fig. 1. The membrane layer with biomass with thickness of δm and the concentration polarization layer with thickness of δ. C* and Cm* are concentrations on the liquidmembrane interface (→: increasing reaction modulus).
the mass transfer rates defined to the enzyme membrane layer (see Sections 2.1–2.3). Its effect on the overall mass transfer rate, Jsum, is briefly discussed in Section 2.4. Recently Nagy [17] studied the mass transfer rate into a biocatalytic membrane layer with constant mass transport parameters. He defined the mass transfer rates for both side of the membrane surface. The rate equations are expressed as product of the mass transfer coefficient and driving force as it is traditionally applied for the diffusion systems, e.g. in gas–liquid systems. Applying these inlet mass transfer rate, the concentration profile of the two layers, namely that of the boundary layer and biocatalytic membrane layer can be calculated. These will be demonstrated in the case of first- and zeroorder reactions as well as in the case of the general Monod kinetics. Assumptions, made for
• reaction occurs at every position within the biocatalyst layer; • reaction has one rate-limiting substrate/ nutrient; • mass transport through the biocatalyst layer occurs by diffusion and convection; • the partitioning of the components (substrate, product) is negligible (Thus, C* = C*m; see Fig. 1); • the mass transport parameters (diffusion coefficient, convective velocity, bioreaction rate coefficient) are constant or varying; • the effect of the concentration boundary layer should also be taken into account; For the sake of simplification, let us regard a steady-state reaction as well as let us use the Cartesian co-ordinate, thus, the differential mass balance equation, applying the Michaelis– Menten kinetics, can be given as follows (y is here the transverse space coordinate, perpendicular to the membrane interface): ⎧⎪ d ⎛ dC ⎞ d (υ C ) ⎫⎪ vmax C =0 ⎬− ⎨ ⎜D ⎟− dy ⎭⎪ K M + C ⎩⎪ dy ⎝ dy ⎠
(1)
In general case, as it was mentioned, the diffusion coefficient and/or convective velocity can depend on the space coordinate, thus D = D(y), υ(y), (or on the concentration, D = D(C) or both of them, D = D(C, y)]. In the boundary conditions the external mass transfer resistance should also be taken into account. The membrane layer with biomass colony on it and the concentration boundary layer are illustrated, with important notations and concentration profiles, in Fig. 1. The boundary conditions of Eq. (1) (the diffusive mass transfer resistance of the concentration boundary layer is expressed here by k0 value) will be as follows:
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resistance in the boundary layer is neglected here, thus C* = Cb and the place y = 0 means here the inlet interface of the membrane layer):
if y = δ then
υin C * + kL0 ( Cb − C * ) = −D
dC dy
+υ C y =δ +
(2a) y =δ +
υout Cδ* + kδo ( Cδ* − C p ) dC dy
+υ C y =δ
+δ m−
C = Cb
(4a)
at y = δm
C = Cp
(4b)
The general solution of Eq. (3), applying standard methodology (for details of solution see [17]), will be as follows (Y = y/δm):
if y = δ + δm then
= −D
at y = 0
(2b) y =δ +δ m+
* δ
where C*, C denote the concentration of liquid at the membrane interface, on the two sides of membrane, δm is the membrane thickness; kδ0 represents here the liquid downstream side’s mass transfer coefficient (the resistance of the skin layer is neglected in this equation). The membrane concentration, C is given here in a unit of measure of gmol/m3. This can be easily obtained by means of the usually applied in the, e.g. g/g unit of measure with the equation of C = wρ/M, where w concentration in kg/kg, ρ – membrane density, kg/m3, M – molar weight, kg/mol (The subscript m denoting the membrane layer has been left here for the sake of simplified writing). In the next sections, the mass transfer with both constant (Sections 2.1– 2.4) and variable parameters (Section 2.6) will be discussed.
C = Tm eλY + Pm eλY
(5)
The “overall” mass transfer rate, namely the sum of the diffusive and the convective mass flow, can be expressed as: J = − Dm
dC dY
Y =0
(
+ υ C Y = 0 ≡ km0 λTm + λ Pm
)
(6)
According to Eqs. (5) and (6), the J value can be expressed as follows: J = β ( Cb − KCp )
β = km0 K=
(7)
( Pem / 2 ) tanh Θ m + Θ m
cosh Θ m
(8)
tanh Θ m
Θ m e − Pem / 2 ( Pem / 2 ) tanh Θ m + Θ m
(9)
and 2.1. Mass transfer accompanied by first-order reaction for the single membrane layer The differential mass balance equation in the case of constant transport parameters is as follows: d C dC −υ − k1C = 0 2 dy dy
Pem Pe Pe + Θ λ = m − Θ Θ = m 1 + 2ξ 2 2 2 2
Pem =
2
Dm
λ=
(3)
Let us look at the biocatalytic membrane layer without concentration boundary layer. Thus, the boundary conditions are as (the mass transfer
υδ m kδ2 Φ= 1 m Dm Dm
ξ=
k1δ m υ
km0 =
Dm δm
The Φ defined here is the well-known Thiele modulus defined for the catalyst membrane layer while the Peclet number corresponds to the often used Bodenstein number. The ξ dimensionless
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variable is the ratio of the reaction rate and the convective flow. Similarly to Eqs. (5) and (7), the concentration distribution and the mass transfer rate at membrane interface, without chemical reaction can be given as follows, respectively: C = Tm e PemY + Pm
(10)
J 0 = β m0 ( Cb − e − Pem Cp )
(11)
β m0 = km0 Pem
Dm Pem e Pem e Pem / 2 (12) ≡ e Pem − 1 δ m 2 sinh ( Pem / 2 )
Eqs. (10) and (7) should define the mass transfer rate of any membrane process with convective and diffusive mass flows without biochemical or chemical reaction, applying suitable boundary conditions. From Eq. (10) it follows that the concentration distribution in the membrane layer is not linear due to the convective velocity. Thus, the often applied mass transfer rate for membrane separation, namely J0 = k0(Cb – Cp) is not valid generally in the presence of convective velocity in the membrane layer (k0 represents here a general mass transfer coefficient). This mass transfer equation is involved in Eq. (10) as a limiting case, namely if Pem tends to be zero.
⎛ ⎞ Φ 2 ⎞⎛ e − Pem J = β m0 ⎜1 + 2 ⎟ ⎜ Cb − Cp 2 2 ⎟ 1 + Φ / Pem ⎠ ⎝ Pem ⎠ ⎝
(14)
where
Φ=
k0δ m2 Dm Cb
The enhancement of the mass transfer coefficient will be a simple expression, namely: k0δ m β Φ2 ξ2 = 1 + ≡ 1 + ≡ 1 + β m0 Pem2 Cbυ Pem
(15)
with
ξ=
k0δ Cbυ
2.3. The two-layer (boundary- and membrane layer) mass transfer rate
2.2. Mass transfer accompanied by zero-order reaction
In the boundary layer there is no biochemical (or chemical) reaction, thus the mass transfer rate through it can be given according to Eq. (11), as follows (here C* ≤ Cb according to the measure of the mass transfer resistance of the boundary layer; in special cases the interface concentration, C* can also be higher than the bulk concentration, Cb, i.e. C* ≥ Cb, this is also correctly described by Eq. (16) or Eq. (11)):
The differential mass balance equation to be solved is as:
J 0 = β L0 ( Cb − e − PeL C ∗ )
Dm
d 2C dC −υ − k0 = 0 2 dy dy
(13)
The solution of Eq. (13) has been discussed in details in Nagy’s paper [17]. The applied methods for the solution of the above differential equation are standard ones that are well known, thus, it is not discussed here. The mass transfer rate for the membrane layer can be given as:
(16)
with
β L0 = kL0 PeL with kL0 =
DL δL
DL PeL e PeL e PeL / 2 ≡ (17) e PeL − 1 δ L 2 sinh ( PeL / 2 )
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where C* is the concentration on the membrane interface as it is illustrated in Fig. 1 (it was assumed here that the solubility coefficient of the substrate∗ in the porous support layer is unit, thus, C* = Cm). J0 in Eq. (16) and the value of J in Eq. (7) are equal to each other, thus the overall mass transfer rate can be given by means of the resistance-in-series principle. The total, inlet mass transfer rate is, for first-order reaction, as follows: J = β tot ( Cb − Ke − PeL Cp )
(18)
(19)
⎛ ⎞ e −( PeL + Pem ) J = β tot ⎜⎜ Cb − Cp ⎟⎟ 2 2 1 + Φ / Pem ⎠ ⎝
(20)
with 1 e − PeL 1 + β L0 β m0 (1 + Φ 2 / Pem2 )
(21)
Similarly the two-layer mass transfer rate, without biochemical reaction in the layers can be expressed as: J =β 0
0 tot
(C
b
−e
− PeL
This case will not be discussed in this paper, in details. For sake of completeness the total mass transfer rate will be defined in this subsection taking into account all three layers, namely boundary layer, enzyme membrane layer and skin layer. For it the outlet mass transfer rate of substrate should also be defined. For example, for first-order biochemical reaction, the outlet mass transfer rate at y = δ + δm is as follows [17]:
≡ β δ ( Cb − ξ Cp )
1 1 e − PeL + β L0 β
For zero-order reaction, the mass transfer rate, and the mass transfer coefficient, for two layers can be given as:
β tot =
2.4. Mass transfer rate including the skin layer, as well
⎛ Θ tanh Θ m − Pem / 2 ⎞ Cp ⎟ J δ = βδ ⎜ Cb − cosh Θ m m Θ m e Pem / 2 ⎠ (24) ⎝
where
β tot =
427
Cp )
(22)
βδ =
Dm e Pem / 2 δ m sinh Θ m
(25)
The resistance-in series model is applied to describe the overall mass transfer rate, similar to that done in Section 2.3. The mass transfer rate through the skin layer can be given by the same equation as that for the concentration boundary layer, namely by Eq. (16) taking into account the real values of the transport parameters of the skin layer. The effect of the skin layer on the outlet concentration is illustrated by dotted line in Fig. 1. The value of Jδ is equal to the inlets into the skin layer, i.e. Jδ = Js, where Js denotes the mass transfer rate through the skin layer. Thus one can obtain the following expression for the three-layer mass transfer rate (S is the solubility coefficient in the skin layer, related to the spongy layer): ⎛ e − PeL ξ ⎞ J sum = β sum ⎜ Cb − Cp ⎟ S ⎝ ⎠
(26)
where 0 = β tot
1 1 e − PeL + 0 β L0 βm
(23)
βsum =
1 1 ξ + βsk S βδ
(27)
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The value of βsk is similar to that given for the boundary layer by Eq. (17), S denotes the solubility of the substrate in the skin layer; is given in Eq. (24). Effect of the skin layer on the mass transfer rate is not discussed in this paper, in details. Applying the above equations, it is easy to predict the effect of the skin layer on the sum of the substrate mass transfer rate, Jsum. 2.5. The two-layer concentration profile for first- and zero-order reactions Knowing the mass transfer rate, the parameter values of the concentration distribution (TL, PL, Tm, Pm) can easily be obtained using the external boundary conditions (Eqs. (4a) and (4b)) and the internal boundary conditions on the feed membrane interface, that is C ∗ = Cm∗ and the mass transfer rate into the boundary layer, Eq. (16), and that into the membrane layer, Eq. (7), are equal to each other, i.e. J0 = J. Knowing the mass transfer rate, the value of C* can be obtained e.g. from Eq. (16). According to Eq. (10) and boundary condition (4a), the TL and PL parameters of the concentration distribution can be given as C ∗ − Cb e PeL − 1
(28)
PL = Cb – TL
(29)
TL =
The Tm and Pm parameters can be obtained by means of C ∗ = Cm∗ at Y = 1 (here Y = y/δ) and of the external boundary condition, here at Y = 1 + δm/δ, Eq. (4b). Thus, one can get for first-order reaction, as
Tm =
Pm =
(e
C ∗eλm − Cp λm
(e
)
− eλm eλmδ / δ m
Cp − C ∗eλm λm
)
− eλm eλmδ / δ m
(30)
(31)
The Tm and Pm parameters for the zero-order reaction can be given in similar way. Its concentration distribution obtained by general solution of Eq. (13) for the membrane layer is as: C = Tm e( PemY δ / δ m ) + Pm − Cbξ 2δ / δ m (32)
1 ≤ Y ≤ 1 + δm/δ
Applying Eq. (32) at y = δ (Y = 1) and at y = δ + δm, (Y = 1 + δm/δ), one can easily get the necessary equation to determine the value of parameters in the case of zero-order reaction: Tm =
Pm =
C ∗ − Cp − Cbξ 2δ / δ m
(1 − e ) e Pem
(33)
Pemδ / δ m
Cp + Cbξ 2 (1 + δ / δ m ) − ( C ∗ + Cbξ 2δ / δ m ) e Pem
(1 − e ) Pem
(34)
2.6. Approaching analytical solution of the mass transport with variable parameters and/or with Monod kinetics for single membrane layer For the solution of Eq. (1) with boundary conditions Eqs. (2a) and (2b) an analytical approach has been developed that makes possible to express the mass transfer rate and the concentration distribution in explicit, closed mathematical equations. In essentials, this solution methodology serves the mass transfer rate and the concentration distribution in closed, explicit mathematical expression. The method can be applied for Cartesian coordinate and cylindrical coordinate as will be shown. For the solution of Eq. (1), the biocatalytic membrane should be divided into N sublayer, in the direction of the mass transport, that is perpendicular to the membrane interface (for details see e.g. Nagy’s papers [18,19]), with thickness of Δδ (Δδ = δm/N) and with constant transport parameters in every sub-layer. Thus, for the nth
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sub-layer of the membrane layer, using dimensionless quantities, it can be obtained:
Cn Y =Y + = Cn +1 Y =Y −
d 2 Cn dC Dn − υn n − kn Cn = 0 yn–1 < y < yn, (35) 2 dy dy
The solution of the algebraic equation system obtained by means of the internal ((39a) and (39b)) conditions with n = 1,2,...,N and external boundary conditions ((2a) and (2b)) can be given by numerically or by analytically. The analytical approach of the solution is discussed in Nagy’s papers [19,20]. The mass transfer rate on the upstream side of the membrane can be given, for that case, as follows:
where the value of kn can be obtained according e.g. to the Michaelis–Menten kinetics as follows: kn =
vmax K M + Cn
(36)
where Cn denotes the average value of C in the nth membrane sub-layer. In dimensionless form one can get the following equation: 2
d Cn dC − Pen n − Φ 2n Cn = 0 dY 2 dY Φ n = δ m2 kn / Dn ;
where
(37) Pen =
υ nδ m Dn
The solution of Eq. (36) can be easily obtained by well-known mathematical methods as it follows (Y = y/δm): Cn = Tn e
( λnY )
+ Pn e(
λnY )
Yn–1< Y < Yn
(38)
with
λn =
Pen − Θn 2
Pe λn = n + Θ n 2
Tn and Pn parameters of Eq. (37) can be determined by means of the boundary conditions for the nth sub-layer (with 1 ≤ n ≤ N). The boundary conditions at the internal interfaces of the sublayers (1 ≤ n ≤ N–1; Ym = nΔY; ΔY = 1/N) can be obtained from the following two equations ([Eqs. (39a) and (39b)): −
dCn D ⎛ dC ⎞ + Pen Cn = n +1 ⎜ − n +1 + Pen +1Cn +1 ⎟ dY Dn ⎝ dY ⎠
at Y = Yn
n
J=
n +1
(39b)
at Y = Yn
DC10 D ⎛ dC ⎞ − + PeC λ1T1 + λ1 P1 = ⎜ ⎟ δ ⎝ dY δ ⎠ Y =0
(
)
(40)
The outlet mass transfer rate can be similarly given. This value should be as less as possible to avoid the loss of the substrate during the process. In the next section some typical figures will be shown to illustrate the effect of the parameters on the mass transport through the catalytic membrane layer. 2.7. Full-scale model The full-scale models describe the concentration distribution in the feed bulk phase [9,10]. This will not be discussed in this paper. The mass transfer rates defined in Eqs. (6), (14), (20)–(23) could essentially affect the bulk phase concentration. For example, in the case of a capillary membrane, the concentration distribution should be given in both axial and radial directions. For it, the mass transfer rate equation developed in this paper serve as boundary conditions at the membrane interface [9,10]. Applying equations developed, it is easy to predict the effect of the biochemical reaction on the feed phase concentration. This effect will be the object of a separate paper. 3. Results and discussion
(39a)
The axial and radial depletion of substrate, e.g. oxygen, nutrient, can be often critical scale-
E. Nagy and E. Kulcsár / Desalination 245 (2009) 422–436
limiting factor in cell culture hollow fiber reactor [12–14]. In order to increase the substrate concentration in the membrane bioreactor, sufficient diffusion rate and/or convective flow has to be provided through the lumen, in axial direction, and through the membrane layer, in radial direction, of a hollow fiber. Typical operating conditions of a hollow-fiber bioreactor were applied (Table 1) to calculate the inlet and outlet mass transfer rates of a substrate. From that, the effectiveness of the bio-catalytic reaction as well as sufficiency of the nutrient supply could be estimated. The biochemical reaction rate depends on the amount of catalyst immobilized in the membrane or on the density of cells in the membrane structure. 3.1. Two-layer concentration profiles and mass transfer rates 3.1.1. Mass transfer without biochemical reaction Eqs. (28)–(31) enable us to simulate the concentration profiles in the boundary layer and membrane layer under different mass transport conditions. Thus, one can select the desired mass transfer conditions as the diffusion resistance in the boundary layer and membrane layer, the trans-membrane pressure to obtain the necessary convective velocity of the substrate or nutrient in order to get sufficient activity of the microbe in the biomass layer on the membrane feed interface and/ or in the membrane matrix., etc., depending on the biochemical reaction rate. In the case of the first- and zero-order reactions, simple, explicit transfer equations have been developed for the calculation. The more complex intermediate reaction rate regime of the Monod or Michaelis–Menten kinetics lies between these two limiting cases, thus, these equations can be used for general prediction of the effect of Monod’s reaction kinetics. The equations developed make also possible to predict the concentration profile without chemical
1.0 Pem=10 Concentration distribution, –
430
0.8 2
0.6
1
PeL=1
0.4
0.5
k0L =1×10–4 δL/δm=1
0.2
0.1
0.0 0.0
0.4
0.8
1.2
1.6
2.0
Boundary layer and membrane thickness, –
Fig. 2. Concentration related to that of the value of Cb as function space coordinate, in the boundary layer and membrane layer without biochemical reaction at different values of the membrane Pe number (Cp/Cb = 0.1, km0 Pem = kL0 PeL).
reaction. Let us look at first some typical profiles at different membrane Pe numbers (Fig. 2). As you know, the convective velocity should be the same in both layers, according to the continuity law. Thus, when the Pem value changes and PeL value remains constant (PeL = 1), it should mean that the diffusive mass transfer coefficient ( km0 = Dm / δ m) should also be changed. In this case, because the ratio of the membrane and the boundary layer thickness was chosen to be constant, the diffusion coefficient in the membrane should be varied with the variation of the value of Pem. Fig. 2 illustrates well that the value of Pem has strong effect on the concentration distribution in both layers and thus, on the value of concentration of the membrane interface. That means that the well-known parameters as concentration polarization modulus, I (I = C*/Cb, [21–23]), enrichment factor, E0 (E0 = Cp/C*) are essentially dependent on the mass transport properties of the biocatalytic membrane layer. Recently Nagy and Borbély [24] studied how the membrane properties alter the concentration polarization modulus and the intrinsic enrichment factor. The equation developed can easily be used to describe the role of both layers on the separation of transported components.
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ϕ = 0.01
Pem = 0.1
17.0
0.8
0.1
1 PeL=Pem = 1
0.6
Enhancement, β/βm, –
Concentration distribution, –
1.0
2
o
δL/δm = 1 k0L =k0m = 1×10–4
0.4
5
10
0.2 Boundary layer
1 13.0
4
9.0 10
5.0
Membrane layer
0.0
25
1.0 0.0
0.4
0.8
1.2
1.6
2.0
Boundary layer + membrane, –
Fig. 3. The effect of first-order chemical reaction on the relative value of concentration profiles in the two layers at different values of the reaction modulus (Cp/Cb = 0.1).
3.1.2. The effect of the first-order reaction Similarly to the case of the gas–liquid or liquid–liquid mass transfer, the chemical reaction could essentially influence the concentration profile in the membrane phase as a consequence, the concentration profile in the boundary layer and the feed mass transfer rate. Typical concentration profiles are shown in Fig. 3 at different values of the reaction modulus. The Φ reaction modulus can significantly lower the concentration profile not only in the membrane layer but also in the concentration boundary layer, as well. At larger Φ values the concentration curves have inflexion points, at Y = 1, at the feed membrane interface, where the curves change their concave trends to convex one. The relative value of the outlet concentration (= Cp/Cb) was chosen to be 0.1. The concentration inside the membrane matrix, depending on the reaction rate, can decrease below its outlet value, at high Φ values. These curves have minimum, after the minimum value, with increasing space coordinate, the concentration gradient will be positive which means that the direction of the diffusion stream will be turned into direction of the feed interface. That should mean that to maintain the obtained concentration additional feeding of the substrate on the outlet side of the membrane is needed. Obviously, it is not a real demand. The model equation serves a simple means to calculate
2
1.0
10.0
100.0
Reaction modulus, Φ, −
Fig. 4. Enhancement of the inlet mass transfer coefficient given for the biocatalytic membrane layer, without mass transfer resistance in the concentration boundary layer (Cp/Cb = 0.1).
the mass transport through biocatalytic membrane layer under all important mass transport and reaction kinetics parameters. Let us look typical examples for the enhancement as a function of the biochemical reaction rate. Regarding single layer mass transfer (mass flow without mass transfer resistance in the boundary layer), the enhancement increases without limit as a function of the reaction rate (Fig. 4). This behavior is similar to that of gas– liquid diffusive mass transfer. According to parameters Φ and Θ (Eq. (9)), the value of Pem and reaction rate constant, Φ, can be varied, independently. With the increase of the Peclet number, the effect of the reaction rate strongly lowers. It should be noted that with increasing Pem number, the convective mass flow increases compared that to the diffusive flow. The decrease of the 0 value of β / β m, with increasing value of Pem, at a given Φ value, can be a consequence of the increasing convective mass flow. With the change 0 of the diffusive mass transfer coefficient, km , the value of Φ also changes. If one plots the function of β / β m0 vs. ξ function this change reduces essentially but the above tendency remains valid (not shown here). How the concentration boundary layer can influence the overall enhancement is illustrated
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E. Nagy and E. Kulcsár / Desalination 245 (2009) 422–436 1.5
1.0 Φ = 0.01
1.4
δL = δm
1.3
kL = 1×10−4
Concentration distribution, –
0 Enhancement, βtot/βtot ,–
Pe L =1
0
1.2
Pem=0.1
10
25
1.1 0.1 1.0
0.8
δ L/ δ m = 1 0.4
1.0
10.0
100.0
Reaction modulus, Φ,−
Fig. 5. Enhancement of the inlet mass transfer coefficient for two-layer mass transport process (Cp/Cb = 0.1, km0 Pem = kL0 PeL).
in Fig. 5. The essential difference between results obtained for single layer (membrane layer, Fig. 4) and two layers (concentration boundary layer and membrane layer) is that the curves tends to liming values with increasing reaction rate constant, due to the effect of the boundary layer. At high reaction rate, the concentration at liquid-membrane interface can decrease close to zero, this determines the highest value of the enhancement. With increasing value of Pem, the curves are shifted to higher reaction rate regime and the maximum value of the enhancement also increases. The figure clearly shows that the role of the concentration boundary layer is essential regarding the enhancement, or the inlet substrate mass transfer rate. 3.1.3. The effect of the zero-order reaction The effect of zero-order reaction can essentially differ from that of first-order reaction. This reaction as limiting case of Monod kinetics is important since it very often can occur in case of bioctalytic reactions. Because the zero-order reaction rate is independent of the concentration, this effect, on the concentration profile (Fig. 6), will be much higher. With the increase of the Φ value, the concentration in the membrane layer, lowers quickly, between Φ = 2 and Φ = 3, practically
2
0 k0L = km = 1×10−4
3
0.2 5 Boundary layer
0.0 0.1
1
PeL= Pem =1
0.6
0.0
0.4
Membrane layer
10 0.8
1.2
1.6
2.0
Boundary layer + membrane, –
Fig. 6. The effect of the zero-order reaction on the twolayer concentration profile (Cp/Cb = 0.1).
linearly, to zero. That means that much higher convective velocity is needed in order to keep the concentration above the necessary critical value during the reaction than in the case of first-order reaction. The enhancement can be very easy to obtain in the case of zero-order reaction according to Eq. (15) for single layer mass transfer. Practically, the value of β / β m0 increases linearly, with the increase of the reaction modulus. The enhancement of the two-layer mass transfer can be also easily obtained by means of Eq. (21). 3.1.4. Single layer mass transfer with Monod or Michaelis–Menten kinetics The inlet mass transfer rate was calculated by Eq. (40) applying the solution of Eqs. (35)–(39b) [19,20]. The outlet mass transfer rate can similarly be obtained. For the sake of completeness, the mass transfer rates for limiting cases, namely for first- and zero-order reactions will also be given. The results obtained for the limiting cases are identical with that obtained by analytical solutions. How the relative value of the outlet mass transfer rate changes as a function of Thiele modulus is illustrated in Fig. 7 at Pem = 1 applying the general Michaelis–Menten kinetics (line 2), and as limiting cases, the first-order kinetic (KM >> Cm, line 1) as well as the zero-order kinetics (Cm
E. Nagy and E. Kulcsár / Desalination 245 (2009) 422–436 1.0 Pem =1
0.8
Jout/J−
0.6 0.4
1
0.2
2
0.0
3
−0.2 −0.4 0.1
1.0
10.0
Reaction modulus, Φ–
Fig. 7. The relative value of the outlet mass transfer rate as a function of reaction modulus applying the Michaelis–Menten kinetics (line 2) and its limiting cases, namely first-order (line 1) and zero-order (line 3) kinetics. (δm = 100 μm, Dm = 5.4 × 10–10 m2/s, kL0 → ∞, kδ0 → ∞, K M / C10 = 1, C20 / Cb = 0.2).
>> KM, line 3). Similar to the concentration distribution, the zero-order kinetics (line 3) has essentially higher conversion than the other reaction types (line 2: Michaelis–Menten kinetics; line 1: first-order kinetic). How the starting value of Φ was calculated is given in Appendix. The starting value of Φ was the same in every three cases. Its value was changed, inside the membrane with increasing diffusion path, according to the reaction order simulated. The dimensionless outlet
433
concentration was chosen here to be 0.2 which was considered as critical concentration. Below this value the microorganism was supposed to work insufficiently. With the increase of the reaction modulus, the outlet mass transfer rate strongly decreases. The value of Jout/J can also have negative value. That is the case when the direction the diffusion flow is contrary to the convective velocity due to its concentration gradient. In the membrane reactors, the desired value of the outlet mass transfer rate of the substrate should be equal to zero. With the model developed, the necessary convective velocity can be easily predicted at which the outlet concentration will be desired one, while the outlet mass transfer rate on the downstream side of the membrane will be approximately zero. It is obvious that the Peclet number also has strong effect on the mass transfer rates. The effect of the Peclet number is illustrated in Fig. 8, applying the nonlinear Monod kinetics. With increase of the Pe number the value of Jout also increases. For instance at Φ = 1, the value of Jout/J is about 50% higher at Pem = 10 than that at Pem = 1. The inlet mass transfer enhancement was shown in Figs. 4 and 5. Applying the solution of the model, the effect of all important parameters can be predicted and thus, the desired reaction conditions can be chosen.
1.0 0.8
Jout/J,–
4. Conclusion
Pem = 10
0.6 0.4
1 0.2 0.1
0.0 −0.2
MM-kinetics
−0.4 0.1
1.0
10.0
Reaction modulus, Φ, –
Fig. 8. The change of the outlet mass transfer rate as a function of reaction rate at different values of membrane Peclet number applying the Michaelis–Menten (MM) reaction kinetics (δm = 100 μm, Dm = 5.4 × 10–10 m2/s, k 0 → ∞, k 0 → ∞, K / C 0 = 1, C 0 / C = 0.2). L
δ
M
1
2
b
Modeling of membrane bioreactor is now in its starting stage. Exact explicit mathematical equations in order to predict how the mass transport parameters, diffusion coefficients, convective velocity, the bioreaction rate parameters could alter the concentration distribution and the mass transfer rate in a biolayer (enzyme/microorganism membrane layer) is very important. It has been proved that the simultaneous diffusive and convective mass transfer rate can also be expressed as product of a mass transfer coefficient and the driving force, similarly to that of the diffusive mass transfer coefficient given for gas–liquid or
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liquid–liquid systems. Mass transfer coefficients have been defined for mass transfer without and with biochemical reactions. For the first-order and zero-order reactions, these mass transfer coefficients can be given in closed mathematical form. Applying the mass transfer equations given for single layers, the so-called overall mass transfer rate and mass transfer coefficient can be given making possible to predict the simultaneous effect of the liquid boundary layer and of the biocatalytic membrane layers, on the mass transfer rate. Acknowledgment This work was supported by the Hungarian Research Foundation under Grants OTKA 63615/2006.
Greek letters
β0 β βδ βtot δL δm υ Φ
convective+diffusive mass transfer coefficient without chemical reaction, m/s mass transfer coefficient with biochemical reaction, m/s mass transfer coefficient of the downstream side of the membrane, m/s two-layer mass transfer coefficient with reaction in the membrane layer, m/s thickness of the concentration boundary layer, m thickness of the membrane layer, m convective velocity, m/s 2 = k1δ m2 / Dm or = k0δ m / Dm Cb
Subscripts Nomenclature C Cb Cp C* Cδ* D J0 J k1 k0 k0 kδ0 KM Pem vmax y Y
concentration, gmol/dm3 inlet concentration, gmol/dm3 outlet concentration, gmol/dm3 liquid concentration on the feed membrane interface, gmol/dm3 liquid concentration on the downstream membrane interface, gmol/dm3 diffusion coefficient mass transfer rate without chemical reaction, gmol/m2s mass transfer rate accompanied by chemical reaction, gmol/m2s first-order reaction rate constant, 1/s zero-order reaction rate constant, gmol/sdm3 physical mass transfer coefficient, m/s physical mass transfer coefficient on the downstream side, m/s Michelis–Menten constant = υδm/Dm maximum reaction rate space co-ordinate perpendicular to the membrane interface, m dimensionless space coordinate, (= y/δ or = y/δm)
1 2 n N L m
upstream phase downstream phase nth sub-layer number of the sub-layers in the membrane (N was chosen to be 100) boundary layer membrane layer
Superscripts *
at interface
Appendix Thiele modulus (Φ) was calculated by the following equation for the nth sub-layer of the membrane (the number of sub-layer was chosen to be 100) according to Michaelis–Menten kinetics:
Φn = δ 2
kn Dn
with (A.1)
vmax vmax / Cb kn = ≡ K M + Cn K M / Cb + Cn / Cb
E. Nagy and E. Kulcsár / Desalination 245 (2009) 422–436
The value of kn was chosen according to the reaction order in such a way that the starting value of Φ, at Y = 0, denoted by Φ(0), should be of the desired and same value, independently of the limiting case: For the first-order reaction (KM >> Cn, or in dimensionless form: KM/Cb >> 1): kn =
vmax KM
(A.2)
For the zero-order reaction (Cn >> KM): kn =
vmax 1 K M Cn
(A.3)
The starting value of kn was identical in every three cases considering that C(Y = 0) =1. In the case of first-order reaction, the value of kn remains constant inside the membrane, in the case of zero-order reaction the product knCn thus, the value Φ2 remains constant inside the membrane. In the general case the value of kn will change according to Eq. (A.1) in the catalytic membrane. References [1] H. Strathmann, L. Giorno and E. Drioli, An Introduction to Membrane Science and Technology, Institute on Membrane Technology, Italy, 2006. [2] W. Yang, N. Cicek and J. Ilg, State-of-the-art of membrane bioreactors: Worldwide research and commercial applications in North America, J. Membr. Sci., 270 (2006) 201–211. [3] L. Giorno and E. Drioli, Biocatalytic membrane reactors: applications and perspectives, Trends Biotechnol., 18 (2000) 339–349. [4] J.G.S. Marcano and T.T. Tsotsis, Catalytic Membranes and Membrane Reactors, Wiley-VCH, Veinheim, 2002. [5] G. Belfort, Membranes and bioreactors: a technical challenge in biotechnology, Biotechnol. Bioeng., 33 (1989) 1047–1066. [6] J.D. Brotherton and P.C. Chau, Modeling analysis of an intercalated-spiral alternate-dead-ended hollow fiber bioreactor for mammalian cell cultures, Biotechnol. Bioeng., 35 (1990) 375–394.
435
[7] L.J. Kelsey, M.R. Pillarella and A.L. Zydney, Theoretical analysis of convective flow profiles in a hollow fiber membrane bioreactor, Chem. Eng. Sci., 45 (11) (1990) 3211–3220. [8] J.A. Schonberg and G. Belfort, Enhanced nutrient transport in hollow fiber perfusion bioreactors: a theoretical analysis, Biotechnol. Prog., 3 (1987) 81–89. [9] J.M. Piret and C.L. Cooney, Model of oxygen transport limitations in hollow fiber bioreactors, Biotechnol. Bioeng., 37 (1991) 80–92. [10] C.A. Sardonini and D. DiBiasio, An investigation of the diffusion-limited growth of animal cells around single hollow fibers, Biotechnol. Bioeng., 40 (1992) 1233–1242. [11] B. Godongwana, M.S. Sheldon and D.M. Solomons, Momentum transfer inside a vertically orientated capillary membrane bioreactor, J. Membr. Sci., 303 (2007) 86–99. [12] V. Calabro, S. Curcio and G. Iorio, A theoretical analysis of transport phenomena in a hollow fiber membrane bioreactor with immobilized biocatalyst, J. Membr. Sci., 206 (2002) 217–241. [13] B.S. Ferreira, P. Fernandes and J.M.S. Cabral, Design and modeling of immobilized biocatalytic reactors, in: J.M.S. Cabral, M. Mota and J. Tramper (Eds.), Multiphase Bioreactor Design, Taylor and Francis, London, UK, 2001, pp. 85–180. [14] T. Gumi, J.F.-D. Albacete, D. Paolucci-Jeanjean, M.-P. Belleville and G.M. Rios, Study of the influence of the hydrodynamic parameters on the performance of an enzymatic membrane reactor, J. Membr. Sci., 311 (2008) 147–152. [15] S.W. Long., S. Bhatia and A. Kamaruddin, Modeling and simulation of enzymatic membrane reactor for kinetic resolution of ibuprofen ester, J. Membr. Sci., 219 (2003) 69–88. [16] L. Giorno, J. Zhang and E. Drioili, Study of mass transfer performance of naproxen acid and ester through a multiphase enzyme-loaded membrane system, J. Membr. Sci., 276 (2006) 59–67. [17] E. Nagy, Equations of mass transfer rates through biocatalytic membrane layer, Asia-Pacific J. Chem. Eng., Doi: 10.1002/apj.242. [18] E. Nagy, Binary, coupled mass transfer with variable diffusivity through cylindrical dense membrane, J. Membr. Sci., 274 (2006) 159–168. [19] E. Nagy, Mass transfer with varying diffusionand solubility coefficient through a catalytic membrane layer, Chem. Eng. Res. Des., 86 (2008) 723–730.
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E. Nagy and E. Kulcsár / Desalination 245 (2009) 422–436
[20] E. Nagy and G. Borbély, Mass transport through anisotropic membrane layer, Desalination, 240 (2009) 54–63. [21] S. Bhattacharya and S.-T. Hwang, Concentration polarization, separation factor, and Peclet number in membrane processes, J. Membr. Sci., 132 (1997) 73–90. [22] R.W. Baker, J.G. Wijmans, A.L. Athayde, R. Daniels, J.H. Ly and M. Le, The effect of concentration polarization on the separation of volatile
organic compounds from water by pervaporation, J. Membr. Sci., 137 (1997) 159–172. [23] S. De and P.K. Bhattacharya, Modeling of ultrafiltration process for a two-component aqueous solution of low and high (ge-forming) molecular weight solutes, J. Membr. Sci., 136 (1997) 57–69. [24] E. Nagy and G. Borbély, The effect of the concentration polarization and the membrane layer mass transport on the membrane separation, Appl. Membr. Sci. Technol., in press.