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International Journal of Chemical Engineering
Volume 2010, Article ID 738482, 12 pages
Research Article

A Solution of the Convective-Diffusion Equation for Solute Mass Transfer inside a Capillary Membrane Bioreactor

1Department of Chemical Engineering, Cape Peninsula University of Technology, P.O. Box 652, Cape Town 8000, South Africa
2Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag, Ronderbosch 7700, South Africa

Received 5 November 2009; Revised 23 March 2010; Accepted 27 April 2010

Academic Editor: Jose C. Merchuk

Copyright © 2010 B. Godongwana et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper presents an analytical model of substrate mass transfer through the lumen of a membrane bioreactor. The model is a solution of the convective-diffusion equation in two dimensions using a regular perturbation technique. The analysis accounts for radial-convective flow as well as axial diffusion of the substrate specie. The model is applicable to the different modes of operation of membrane bioreactor (MBR) systems (e.g., dead-end, open-shell, or closed-shell mode), as well as the vertical or horizontal orientation. The first-order limit of the Michaelis-Menten equation for substrate consumption was used to test the developed model against available analytical results. The results obtained from the application of this model, along with a biofilm growth kinetic model, will be useful in the derivation of an efficiency expression for enzyme production in an MBR.

1. Introduction

Since the first uses of hollow-fiber membrane bioreactors (MBRs) to immobilize whole cells were reported in the early 1970s, this technology has been used in as wide ranging applications as enzyme production to bone tissue engineering. One of the current research areas of interest into biofilm-attached membrane bioreactors (MBRs) is the development of cost-effective and environmentally friendly methods of producing various primary and secondary metabolites from bacterial, fungal, and yeast cells. These include: manganese and lignin peroxidase, secreted by the fungus Phanerochaete chrysosporium [1, 2]; actinorhodin, a noncommercial antibiotic produced by the filamentous bacterium Streptomyces coelicolor [3]; glutamic acid, an ingredient in flavour enhancers of meats and vegetables, secreted by the bacterium Corynebacterium glutamicum [4]; ethanol, extracted from the yeast Saccharomyces cerevisiae [5]; and many others. With the exception of ethanol, these bioproducts are generally classified as products of intermediate value [6]. It has been reported that bioreactor productivity, in the production of these types of products, greatly impacts on the product cost [7].

The productivity of biofilm-attached MBRs is determined in large by the biomass growth, and one of the most important factors that influence biomass growth is the availability and transport of nutrients through the bioreactor [8, 9]. The momentum transfer of solutes through MBRs has been thoroughly studied, from a theoretical and experimental perspective, for a number of configurations [1015]. Similarly, the mass transfer has received considerable attention [8, 9, 1622]. With the exception of the models developed by Heath and Belfort [17]; Li and Tan [19]; and Willaert et al. [22], the mass transfer models were solved using numerical procedures such as finite difference schemes and control volumes. A difficulty in implementing such schemes is the choice of the appropriate technique for a specific MBR system [21], and these techniques are subject to discretization errors and stringent stability criterion. In the models presented by Heath and Belfort [17]; Willaert et al. [22]; and Li and Tan [19], the convective-diffusion equation governing mass transfer was solved analytically. These authors, however, neglected the effects of axial diffusion and radial convective flow in their models. Both these assumptions may not be justified in all cases. A number of theoretical and experimental investigations have demonstrated the significance of radial convective flows in improving MBR efficiencies [9, 15, 21]. In the dead-end ultrafiltration mode, particularly, the assumption of negligible radial convective flow is not justifiable. At axial Peclet numbers (Peu) smaller than unity large concentration gradients exist, and under these circumstances ignoring axial diffusion is also not justified.

The current study presents an analytical solution of the convective-diffusion equation, for solute transport trough a single fiber isotropic capillary membrane, in two dimensions. This study will not include the growth kinetics of the microorganism, as conversion is assumed to take place in the shell side of the MBR; the current analysis is restricted to the lumen side. For comparison with literature models, however, the first-order limit of the Michaelis-Menten equation will be superimposed on the developed model in the results section.

2. Model Development

2.1. The Membrane Gradostat Reactor

The models developed in this study are applicable to a hollow-fiber MBR system, consisting of either a single fibre or a bundle of fibres; with nutrient flowing on the lumen side of the membrane and the micro-organism immobilised either on the lumen side or on the shell side. The notation used, however, is specifically for a single hollow fiber membrane gradostat reactor (MGR). The construction of the MGR, as patented by Edwards et al. [23], is illustrated schematically in Figure 1. It consists of a single hollow-fibre, made of surface modified polysulphone, encased in a glass bioreactor. The membranes are asymmetric and characterized by an internally skinned and externally unskinned region of microvoids; approximately 0.15 mm long and 0.015 mm thick. These membranes have inner and outer diameters of approximately 1.395 mm and 1.925 mm, respectively. The nutrient solution permeates from the lumen side to the shell side of the MGR due to the transmembrane pressure gradient. The micro-organism is immobilised on the shell side of the MGR. Humidified air is supplied on the shell side, and two pressure transducers are fitted at the inlet and outlet of the MGR as shown in Figure 1.

Figure 1: A schematic diagram of the single capillary membrane gradostat reactor (MGR).
2.2. Model Assumptions

The theoretical models to be developed will be based on the following conditions of operation and assumptions: (1) the system is isothermal, meaning the energy equation has been decoupled from the mass and momentum transfer; (2) the flow regime within the membrane lumen is fully developed, laminar, homogeneous, and at steady state; (3) the physical and transport parameters (e.g., density, viscosity, and diffusivity) are constant; (4) in the dense and spongy layers of the membrane matrix the flow is only one dimensional (i.e., there are no axial components of the velocity profiles in the membrane matrix); and (5) the aspect ratio of the membrane is much smaller than unity. The aspect ratio, 𝜑, is the ratio of the membrane inner radius to the effective membrane length (i.e., 𝑅𝐿/𝐿), and if it is much smaller than unity then normal stress effects are negligible in the momentum transfer analysis.

3. Mathematical Formulation

The starting point of the analysis is the convective-diffusion equation [24]: 𝐷𝑐𝐷𝑡=𝐷𝐴𝐵2𝑐+𝑟𝐴,(1) where 𝑐 is the local substrate concentration; 𝑡 is time; 𝐷𝐴𝐵 is the substrate diffusivity, assumed to be constant; and 𝑟𝐴, the rate of substrate production (or consumption), is a function of the local biofilm density, and time. Equation (1), for steady state, two-dimensional flow, without reaction, in cylindrical co ordinates may be written as𝑢𝜕𝑐𝜕𝑧𝐷𝐴𝐵𝜕2𝑐𝜕𝑧2=𝐷𝐴𝐵𝑟𝜕𝑐𝜕𝜕𝑟+𝑟2𝑐𝜕𝑟2𝑣𝜕𝑐𝜕𝑟.(2) It is convenient to express this equation in dimensionless form by introducing the following dimensionless variables:𝑢𝑈=𝑢0𝑣,𝑉=𝑣0𝑐,𝐶=𝑐0,𝑧𝑍=𝐿𝑟,𝑅=𝑅𝐿𝑅,𝜑=𝐿𝐿.(3) The expressions of 𝑈 and 𝑉 in (3) are obtained from the momentum transfer analysis given in Appendix A. Substituting the dimensionless variables in (3) into (2) results in𝜑𝑃𝑒𝑢𝑈𝜕𝐶𝜕𝑍𝜑2𝜕2𝐶𝜕𝑍2=1𝑅𝜕𝐶𝜕𝜕𝑅+𝑅2𝐶𝜕𝑅2𝑃𝑒𝑣𝑉𝜕𝐶𝜕𝑅,(4) where the axial and radial Peclet numbers (𝑃𝑒𝑢,𝑣) are defined as𝑃𝑒𝑢=𝑢0𝑅𝐿𝐷𝐴𝐵,𝑃𝑒𝑣=𝑣0𝑅𝐿𝐷𝐴𝐵.(5) The boundary conditions which match the imposed operating conditions of the MBR system are presented in Table 1. Boundary condition 1 (B.C.1) corresponds to a uniform inlet substrate concentration; B.C.2 and B.C.8 corresponds to cylindrical symmetry at the centre of the membrane lumen; B.C.3 indicates that the substrate concentration is only a function of the axial spatial coordinate at the membrane wall; B.C.4 corresponds to continuity of the substrate flux at the lumen-matrix interface; B.C.5 indicates a distance along the length of the membrane above which the axial concentration gradient becomes zero. The boundary conditions B.C.7–11 are employed in the solution of the velocity profiles given in Appendix A. A cross section of the MGR illustrating the three regions of the reactor, with the notation used in the model development, is shown in Figure 2.

Table 1: The boundary conditions of the MBR.
Figure 2: A cross section of the single fibre MGR.

If 𝑈 in (4) is radially averaged, to become 𝑈av, then the left-hand side (LHS) of (4) is only a function of Z and the right-hand side (RHS) only a function of 𝑅. This can only be true if both the LHS and RHS are independent of the variables 𝑅 and 𝑍. Equation (4) may therefore be solved by separation of variables to give a solution of the form𝐶=𝐹(𝑍)𝑇(𝑅).(6) Substituting (6) into (4) gives𝜑𝑃𝑒𝑢𝑈av𝐹𝑑𝐹𝜑𝑑𝑍2𝐹𝑑2𝐹𝑑𝑍2=1𝑅𝑇𝑑𝑇𝑑𝑑𝑅+𝑅2𝑇𝑑𝑅2𝑃𝑒𝑣𝑉𝑇𝑑𝑇𝑑𝑅=𝜆2.(7) The equating of the two ordinary differential equations (ODEs) to the arbitrary constant 𝜆2 in (7) is due to the fact that the two ODEs are independent of the variables 𝑅 and 𝑍.

3.1. Solution of the Axial Concentration Function F(Z)

To solve for the axial function 𝐹(𝑍) of the substrate concentration profile, the ODE on the LHS of (7) is considered𝑑2𝐹𝑑𝑍2𝑃𝑒𝑢𝑈av𝜑𝑑𝐹𝜆𝑑𝑍2𝜑2𝐹=0.(8) The radially-averaged axial velocity 𝑈av in (8) is defined as𝑈av=2101𝑈𝑅𝑑𝑅=8𝑑𝑃𝑑𝑍ReFr,(9) where Fr and Re are the Froude and Reynolds number, respectively, given by𝑢Fr=20𝑔𝑅𝐿,Re=𝜌𝑢0𝑅𝐿𝜇,(10) where 𝑔 is the gravitational acceleration; 𝜌 is the solution density; and 𝜇 is the solution dynamic viscosity. The solution of the axial velocity 𝑈 in (9) is given in Appendix A as1𝑈=41𝑅2𝑑𝑃𝑑𝑍ReFr,(11) with the axial pressure gradient given by𝑑𝑃𝑑𝑍=4𝜑14𝜅𝛽sinh𝜑1𝜅4𝑍+𝜑𝑎cosh𝜑1𝜅𝑍,(12) where 𝑃 is the dimensionless hydrostatic pressure; 𝛽 is the dimensionless transmembrane pressure; 𝑎 is the dimensionless entrance pressure drop; and 𝜅 is the dimensionless membrane hydraulic permeability. The entrance pressure drop 𝑎 in (12) is given by𝑎=4𝜑1𝜅𝛽sinh4𝜑1𝜅ReFr1(1𝑓)𝜑𝑓cosh4𝜑1𝜅,(13) where 𝑓 is the fraction retentate (𝑓=0 for the dead-end mode and 𝑓=1 for the closed-shell mode). The membrane hydraulic permeability 𝜅 in (12) is much smaller than unity (𝜅1), therefore, this equation can be approximated by the following expression:𝑑𝑃𝑑𝑍𝜑𝑎+16𝛽𝜑1𝜅𝑍.(14) This approximation makes (8) a confluent hypergeometric type differential equation. This is more evident if the following sequential substitutions are made.

Substitution 1
𝜉=𝑃𝑒𝑢8𝜑𝑎𝜑+16𝛽𝜅𝑍𝜑ReFr,0𝑍<𝑍0,0,𝑍0𝑍1.(15) The substitution 𝜉 in (15) represents the axial gradient (or driving force) of the substrate concentration profile. The axial distance along the membrane length at which this gradient is zero represents the point at which the axial function of the concentration profile, 𝐹(𝑍), is constant. This axial distance, 𝑍0, is obtained by equating (15) to zero as follows: 𝜉=0at𝑍0=𝜑16𝛽𝜅ReFr𝜑𝑎.(16) The substitution in (15) transforms (8) to 𝑑2𝐹𝑑𝜉2𝐴𝜉𝑑𝐹𝐴𝑑𝜉2𝜆2𝜑2𝐹=0,(17) where 𝐴=𝜑22𝑃𝑒𝑢𝛽𝜅1.(18)

Substitution 2
1𝜃=2𝐴𝜉2.(19) This substitution transforms (17) to 𝜃𝑑2𝐹𝑑𝜃2+12𝜃𝑑𝐹𝑑𝜃𝐴𝜆22𝜑2𝐹=0.(20) Equation (20) is the standard Kummer hypergeometric equation and has two solutions, the Kummer function of the first kind 𝑀(𝛼,𝛾,𝜃) and the Tricomi function Φ(𝛼,𝛾,𝜃), respectively [25]: 𝛼𝑀(𝛼,𝛾,𝜃)=1+𝛾𝜃+(𝛼)2(𝛾)2𝜃2!2++(𝛼)𝑛(𝛾)𝑛𝜃𝑛!𝑛,(21) where (𝛼)𝑛=𝛼(𝛼+1)(𝛼+2)+(𝛼+𝑛1),(𝛼)0𝜋=1,Φ(𝛼,𝛾,𝜃)=sin𝜋𝛾𝑀(𝛼,𝛾,𝜃)Γ(1+𝛼𝛾)Γ(𝛾)𝜃1𝛾𝑀(1+𝛼𝛾,2𝛾,𝜃),Γ(𝛼)Γ(2𝛾)(22) where Γ(𝑛) is the gamma function. Therefore, the solution of (20) becomes 𝐹(𝜃)=𝐹0𝑀𝐴𝜆22𝜑2,12,𝜃+𝐹1Φ𝐴𝜆22𝜑2,12,𝜃,(23) where 𝐹0and 𝐹1 are coefficients obtained from the inlet condition ((6a) in Table 1). The Tricomi function approaches infinity as values of 𝜃 approach zero [26], therefore, the coefficient 𝐹1 in (23) must be zero for this equation to satisfy ((6a) in Table 1). The coefficient 𝐹0 is obtained from the inlet condition ((6a) in Table 1): 𝐶=𝐹0𝑀𝐴𝜆22𝜑2,12,𝜃0𝑇(𝑅)=1at𝑍=0,(24) where 𝜃0=𝑃𝑒𝑢256𝛽𝜅𝑎𝜑ReFr2.(25) From the definition in (21), the Kummer function 𝑀(𝛼,𝛾,0) is equal to 1 for all real values of 𝛼 and 𝛾 (where 𝛾0). The two piecewise solutions of the axial function of the dimensionless concentration profile, 𝐹(𝜃), are therefore 𝐹𝐹(𝜃)=0𝑀𝐴𝜆22𝜑2,12,𝜃,0𝑍<𝑍0,𝐹0,𝑍0𝑍1.(26)

3.2. Solution of the Radial Concentration Function 𝑇(𝑅)
3.2.1. Zero-Order Approximation

To solve for the radial function 𝑇(𝑅) of the substrate concentration profile, the ODE on the RHS of (7) is solved𝑑2𝑇𝑑𝑅2+1𝑅𝑃𝑒𝑣𝑉𝑑𝑇𝑑𝑅+𝜆2𝑇=0.(27) The radial velocity 𝑉 in (27) is given in Appendix A as𝑢𝑉=𝜑0𝑣0𝑅8𝑅122𝑑2𝑃𝑑𝑍2.(28) From the approximation in(14)𝑑2𝑃𝑑𝑍216𝛽𝜑1𝜅,𝜅1.(29) Equation (27) is solved by a regular perturbation technique:𝑇(𝑅)=𝑛=0𝜅𝑛𝑇𝑛.(30) The magnitude of the membrane hydraulic permeability 𝜅 is very small; hence, validity of the perturbation method is assured. The equations to solve for the zero-order approximation, 𝑇0, of (27) are𝑑2𝑇0𝑑𝑅2+1𝑅𝑑𝑇0𝑑𝑅+𝜆2𝑇0=0;(31)𝑇0(0)𝐵1=0;𝑑𝑇0(0)𝑑𝑅𝐵2=0(32) Equation (31) is Bessel’s differential equation and has a standard solution of the form𝑇0=𝐵1𝐽0(𝜆𝑅)+𝐵2𝑌0(𝜆𝑅).(33) As 𝑅 approaches zero in (33), the function 𝑌0 tends to minus infinity; and therefore, 𝐵2 must be zero for the equation to satisfy B.C.6 at 𝑅 is equal to zero.𝑇0=𝐵1𝐽0(𝜆𝑅).(34) Imposing B.C.3, the dimensionless concentration 𝐶 is only a function of the axial coordinate 𝑍 at 𝑅=1; therefore, the function 𝑇0 should be equal to the constant 𝐵1𝐽0(𝜆) at 𝑅=1, but 𝐽0(𝜆) is an oscillating function that can have a number of roots that satisfy the condition of B.C.3:𝑇0=𝑚=1𝐵1𝑚𝐽0𝜆𝑚𝑅.(35) The solution for the coefficient 𝐵1𝑚 in (35) is given in Appendix B.1; the eigenvalues 𝜆𝑚 are derived from B.C.4 and are given in Appendix B.2.

3.3. First-Order and Second-Order Approximations

The equations to solve for the first-order approximation, 𝑇1, of (27) are𝑑2𝑇1𝑑𝑅2+1𝑅𝑑𝑇1𝑑𝑅+𝜆2𝑚𝑇1𝑅𝑅=𝛿122𝑑𝐽0𝜆𝑚𝑅,𝑇𝑑𝑅(36)1(0)=0,𝑑𝑇1(0)𝑑𝑅=0,(37) where𝛿=2𝑃𝑒𝑣𝛽𝑢0𝑣0𝑚=1𝐵1𝑚.(38) Equation (36) is evaluated by making use of the following identity of Bessel functions [25]:𝑑𝐽0𝜆𝑚𝑅𝑑𝑅=𝜆𝑚𝐽1𝜆𝑚𝑅.(39) Substituting (39) into (36) results in the following inhomogeneous O.D.E:𝑑2𝑇1𝑑𝑅2+1𝑅𝑑𝑇1𝑑𝑅+𝜆2𝑚𝑇1=𝛿𝜆𝑚𝑅𝑅122𝐽1𝜆𝑚𝑅.(40) Equation (40) is further simplified by making use of the following substitution:𝑥=𝜆𝑚𝑅.(41) This substitution simplifies (40) to𝑥𝑑2𝑇1𝑑𝑥2+𝑑𝑇1𝑑𝑥+𝑥𝑇1=𝛿𝑥2𝜆2𝑚𝑥122𝜆2𝑚𝐽1(𝑥).(42) Some mathematical architecture is required to solve (42) and this is described in Appendix C.1. The solution of this equation is given in Appendix C.2 as𝑇1(𝑥)=𝑖3𝑥2𝐽2(𝑥)3!!+𝑖1𝑥3𝐽3(𝑥)5!!+𝑖2𝑥4𝐽4(𝑥)7!!,(43) where𝑖1=203𝜆2𝑚,𝑖2=354𝜆2𝑚,𝑖3=3𝛿4𝜆2𝑚.(44) The differential equations to solve for the second-order approximation, 𝑇2, of (27) are𝑥𝑑2𝑇2𝑑𝑥2+𝑑𝑇2𝑑𝑥+𝑥𝑇2=𝛿𝑥2𝜆2𝑚𝑥122𝜆2𝑚𝑇1𝑇(𝑥),(45)2(0)=0,𝑑𝑇2(0)𝑑𝑥=0.(46) The mathematical architecture required for the solution of these equations is also given in Appendix C.1, and the solution is given in Appendix C.3:𝑇2=(𝑥)3𝛿22𝜆6𝑚203𝜆2𝑚𝑥3𝐽3(𝑥)75!!1053𝜆2𝑚6𝑖1𝑥54𝐽4(𝑥)+97!!7106048𝑖1+2𝜆2𝑚8𝑖27𝑖1𝑥5𝐽5(𝑥)9!!991235112𝑖1+80𝑖2+2𝜆2𝑚𝑖2𝑥6𝐽6(𝑥)11!!91113147𝑖120𝑖2𝑥7𝐽7(𝑥)13!!911131516𝑖2𝑥8𝐽8(𝑥).15!!(47) Substituting (26) and (30) into (6), the solution of the dimensionless luminal concentration profile is therefore𝐶(𝜃,𝑥)=𝑚=1𝑛=0𝐹𝑚(𝜃)𝑇𝑚𝑛(𝑥)𝜅𝑛.(48) The perturbation solution is only extended up to second-order approximation; therefore, (48) reduces to𝐶(𝜃,𝑥)=𝑚=1𝐹𝑚𝑇(𝜃)𝑚0(𝑥)𝜅0+𝑇𝑚1(𝑥)𝜅1+𝑇𝑚2(𝑥)𝜅2.(49)

4. Results

4.1. Model Validation

The small diameters of the capillary membranes used in the construction of the MBRs render it a very difficult task to validate the accuracy of the developed models experimentally, and thus the predictive power of the models will be compared to currently available results for similar MBR systems; in particular the model developed by Heath and Belfort [17]. The approach used by these authors in solving the convective-diffusion equation was to assume a constant cross-sectional concentration profile in the membrane lumen and matrix regions, and to solve the simplified form of the equation. The lumen side axial concentration profile was then obtained from a mass flux balance. This approach is equivalent to assuming that the substrate consumption takes place in the lumen side of the MBR; in such case the convective-diffusion equation given in (1) becomes𝑢𝜕𝑐𝜕𝑧𝐷𝐴𝐵𝜕2𝑐𝜕𝑧2=𝐷𝐴𝐵𝑟𝜕𝑐𝜕𝜕𝑟+𝑟2𝑐𝜕𝑟2𝑣𝜕𝑐𝑉𝜕𝑟𝑚𝑐𝐾𝑚+𝑐,(50) where 𝑉𝑚 is the maximum rate of reaction and 𝐾𝑚 is the Michaelis constant. This equation is made dimensionless by introducing the variables in (3) and (5), and two-additional variables the Thiele modulus 𝜙 and the dimensionless Michaelis constant 𝐾𝑚, respectively:𝐾𝑚=𝐾𝑚𝑐0,𝜙=𝑉𝑚𝑅2𝐿𝑐0𝐷𝐴𝐵.(51) Equation (50) then becomes𝜑𝑃𝑒𝑢𝑈𝜕𝐶𝜕𝑍𝜑2𝜕2𝐶𝜕𝑍2=1𝑅𝜕𝐶𝜕𝜕𝑅+𝑅2𝐶𝜕𝑅2𝑃𝑒𝑣𝑉𝜕𝐶𝜙𝜕𝑅2𝐶𝐾𝑚.+𝐶(52) Assuming the first-order limit of the Michaelis-Menten equation (i.e., 𝐾𝑚𝐶) the solution of (52) takes the same general approach as that of (4). The solution of this equation is identical to (49) with an adjustment of the axial function 𝐹𝑚(𝜃) and the coefficient of the perturbation solution 𝐵1𝑚 to account for the reaction rate:𝐶(𝜃,𝑥)=𝑚=1𝐹𝑚𝑇(𝜃)𝑚0(𝑥)𝜅0+𝑇𝑚1(𝑥)𝜅1+𝑇𝑚2(𝑥)𝜅2,(53) where𝐹𝑚(𝜃)=𝐹0𝑀𝐴𝜆2𝑚𝐾𝑚+𝜙22𝜑2𝐾𝑚,12,𝐵,𝜃1𝑚=2𝜆𝑚𝐹0𝑀𝐴𝜆2𝑚𝐾𝑚+𝜙2/2𝜑2𝐾𝑚,1/2,𝜃0×𝐽1𝜆𝑚𝐽20𝜆𝑚+𝐽21𝜆𝑚.(54) The input variables used in validating (53) are obtained from Heath and Belfort [17] and are given in Table 2. The values of the membrane hydraulic permeability (𝑘𝑚), the fraction retentate (𝑓), the lumen-side inlet hydrostatic pressure (𝑝0), the shell-side hydrostatic pressure (𝑝𝑆), the solution density (𝜌), and viscosity (𝜇) are not specified by these authors; therefore, typical operational values of these parameters will be used. A transmembrane (TMP) pressure of 5 kPa is assumed across the MBR, and the solution properties are assumed to be those of water at 30°C.

Table 2: Parameter values used to determine the concentration profile [17].

In Figure 3, it can be seen that (53) compares satisfactorily with the model of Heath and Belfort [17] for the parameter values listed in Table 2. At the centre of the MBR (𝑅=0) Equation (53) predicts that the concentration decreases axially to 34% of its original value when the applied TMP is 5kPa, and at the membrane wall (𝑅=1) the concentration decreases up to 24% of its original value as shown in Figure 4. The model of Heath and Belfort [17] predicts a 44% decrease for all values of 𝑅 for the corresponding conditions. It is important to note, however, that only the resulting trends from the two models can be compared, since (53) requires a more detailed description of the MBR system than that required for the model of Heath and Belfort [17].

Figure 3: A comparison of the concentration profiles resulting from (53) with Heath and Belfort [17] assuming a first-order limit for substrate consumption (at 𝑅=0).
Figure 4: Substrate concentration profiles assuming a first-order limit for substrate consumption at different radial positions.

5. Conclusion

A rigour analytical mathematical model for substrate concentration profiles in the lumen of a hollow fiber MBR was developed. The model was based on the solution of the convective-diffusion equation in dimensionless form. The model allows evaluation of the influence of the general operating parameters of a MBR on the concentration profiles. These parameters are the fraction retentate (𝑓); the membrane hydraulic permeability (𝜅); the axial and radial Peclet numbers (𝑃𝑒𝑢,𝑣); the Thiele modulus (𝜙); the fluid properties; and the dimensions of the MBR. The developed model can be further used to evaluate reactor performance from basic principles, since it allows analytical evaluation of performance parameters (e.g., the performance index and the effectiveness factor). The current paper is a precursor to a paper by the same authors on bioreactor performance. The models developed in the current paper will be used to develop expressions for the effectiveness factor and performance index (PI) of a capillary membrane gradostat reactor.


A. Momentum Transfer Analysis

A.1. Momentum Transfer Analysis

The z-component of the nonconservative form of the Navier-Stokes equation in cylindrical coordinates is made dimensionless by introducing the variables in (4) and the following additional variables: 𝑃=𝑝𝑅2𝐿𝜇𝑢0𝐿,𝜏=𝑡𝑢0𝑅𝐿𝑢,Fr=20𝑔𝑅𝐿;(A.1) where 𝑝 and 𝑃 are the hydrostatic and dimensionless hydrostatic pressures, respectively; 𝑡 and 𝜏 are the time and dimensionless time, respectively; 𝜇 is the solution dynamic viscosity; and Fr is the Froude number. The dimensionless form of the Navier-Stokes thus becomes𝜕𝑈=1𝜕𝜏1Re𝑅𝜕𝑅𝜕𝑅𝜕𝑈𝜕𝑅+𝜑2𝜕2𝑈𝜕𝑍2𝑑𝑃+1𝑑𝑍Fr,(A.2) and the continuity equation1𝑅𝜕(𝑅𝑉)𝑢𝜕𝑅=𝜑0𝑣0𝜕𝑈𝜕𝑍.(A.3) Equation (A.2) is solved following the procedure of Godongwana et al. [13], and the boundary conditions listed in Table 1 (B.C.7 and B.C.8). Ignoring normal stresses 𝜕2𝑈/𝜕𝑍2 and considering steady-state conditions, the solution of (A.2) is given by1𝑈=41𝑅2𝑑𝑃𝑑𝑍ReFr.(A.4) The dimensionless radial velocity profile 𝑉 is obtained by substituting (A.4) into (A.3) and imposing B.C.9:𝑢𝑉=𝜑0𝑣0𝑅8𝑅122𝑑2𝑃𝑑𝑍2.(A.5) The dimensionless pressure profile 𝑃 is obtained by imposing B.C.10, where the matrix velocity 𝑉𝑀 is governed by Darcy’s law:𝑉𝑀𝑢=0𝑣0𝜅𝑃𝑆𝑃0,(A.6) where the dimensionless membrane hydraulic permeability 𝜅 is given by 𝜅=𝜇𝑘𝑚𝐿𝑅2𝐿.(A.7)

Substituting (A.5) and (A.6) into (6j) results in𝜑𝑑162𝑃𝑑𝑍2𝑃=𝜅𝑆𝑃0.(A.8) Equation (A.8) is solved by applying B.C.11 to give:4𝑃=𝛽cosh𝜑1𝜅𝑍+𝜑𝑎4𝜑1𝜅4sinh𝜑1𝜅𝑍+𝑃𝑆,(A.9) where the dimensionless entrance pressure drop 𝑎 in (A.9) is obtained by defining a fraction retentate, 𝑓, as the ratio of the exit to the inlet velocity, that is, 𝑓=𝑈1/𝑈0:𝑎=4𝜑1𝜅𝛽sinh4𝜑1𝜅ReFr1(1𝑓)𝜑𝑓cosh4𝜑1𝜅.(A.10)

B. Solution of Constants

B.1. Solution of the Coefficient 𝐵1𝑚

Imposing the inlet condition (6a) into (48) gives𝐶𝜃0=,𝑅𝑚=1𝑛=0𝐹𝑚𝜃0𝑇𝑚𝑛𝜆𝑚𝑅𝜅𝑛=1,at𝑍=0.(B.1) For the zero-order approximation of the radial function 𝑇(𝑅), (B.1) becomes𝐹0𝑚=1𝑀𝐴𝜆2𝑚2𝜑2,12,𝜃0𝐵1𝑚𝐽0𝜆𝑚𝑅=1,at𝑍=0.(B.2) To solve for 𝐵1𝑛 in (B.2) both the LHS and the RHS are multiplied by 𝐽0(𝜆𝑚𝑥𝑅)𝑅 and integrated with respect to 𝑅 over the interval 0-1,10𝐽0𝜆𝑚𝑥𝑅𝑅𝑚=1𝑀𝐴𝜆2𝑚2𝜑2,12,𝜃0𝐵1𝑚𝐽0𝜆𝑚𝑅=1𝑑𝑅𝐹010𝐽0𝜆𝑚𝑥𝑅𝑅𝑑𝑅.(B.3) The RHS of (B.3) is evaluated by making use of the following property of Bessel functions:𝑧0𝑟𝑣𝐽𝑣1(𝑟)𝑑𝑟=𝑧𝑣𝐽𝑣(𝑧).(B.4) The R.H.S of (B.3) then becomes𝐽R.H.S=1𝜆𝑚𝑥𝜆𝑚𝑥𝐹0.(B.5) The L.H.S of (B.3) may be evaluated by making use of Lommel integrals:𝑥0𝐽𝑛𝛼𝑘𝑟𝐽𝑛𝛼𝑚𝑟𝑟𝑑𝑟=0(𝑘𝑚),𝑥0𝑟𝐽2𝑛𝛼𝑚𝑟𝑥𝑑𝑟=22𝐽𝑛𝛼𝑚𝑥2+𝑛12𝛼2𝑚𝑥2𝐽2𝑛𝛼𝑚𝑥.(B.6) To give the following equation:𝐵LHS=1𝑚2𝑀𝐴𝜆2𝑚2𝜑2,12,𝜃0𝐽20𝜆𝑚+𝐽21𝜆𝑚.(B.7) Substituting (B.5) and (B.7) back into the RHS and LHS of (B.3), respectively𝐵1𝑚=2𝜆𝑚𝐹0𝑀𝐴𝜆2𝑚2𝜑2,12,𝜃0𝐽1𝜆𝑚𝐽20𝜆𝑚+𝐽21𝜆𝑚.(B.8)

B.2. Solution of the Eigenvalues 𝜆𝑚

The solution of the eigenvalues is obtained from B.C.4 in Table 1. When the enzyme is immobilized on the lumen side of the membrane this equation reduces to𝜕𝐶𝜕𝑅𝑃𝑒𝑣𝑉𝐶=0,at𝑅=1.(B.9) Substituting the expression for C, (49) into (B.9) and limiting the perturbation to first-order approximation:𝑃𝑒𝑣𝑉𝑇0𝑑𝑇0𝜅𝑑𝑅0𝑃𝑒𝑣𝑉𝑇1𝑑𝑇1𝜅𝑑𝑅1=0,at𝑅=1.(B.10) The eigenvalues of (B.8) are values of 𝜆𝑚 that satisfy (B.10). The first 10 of these values are listed in Table 3 for the parameter values in Table 2.

Table 3: Positive roots of (B.10).

C. Laplace Transform Solutions

C.1. Mathematical Architecture for the Solution of 𝑇(𝑥)

We define a parameter 𝑝 in Laplace space as1𝑝=1+𝑠2.(C.1) The following properties of 𝑝 are all noteworthy, differentiation:𝑝=𝑠𝑝3.(C.2) Integration:𝑝𝑛𝑝𝑠𝑑𝑠=𝑛22𝑛,𝑛2.(C.3) Inverse Laplace Transform:1{𝑝}=𝐽0(𝑥),1𝑝2𝑛+1=𝑥𝑛𝐽𝑛(𝑥).(2𝑛1)!!(C.4) Laplace Transforms involving the First-Order Bessel function:𝐽1(𝑥)=1𝑠𝑝.(C.5) More relations involving 𝑝:𝑠2=𝑝2𝑑1,(C.6)[]𝑑𝑠1𝑠𝑝=𝑝3,𝑑2𝑑𝑠2[]1𝑠𝑝=3𝑠𝑝5,𝑑3𝑑𝑠3[]1𝑠𝑝=15𝑝712𝑝5,𝑑4𝑑𝑠4[]1𝑠𝑝=60𝑠𝑝7105𝑠𝑝9.(C.7) We also define the following polynomial in 𝑝, which is significant for the first- and second-order approximation of 𝑇(𝑥) in Sections C.2 and C.3, respectively, 𝑢(𝑝)=𝑝5+𝑖1𝑝7+𝑖2𝑝9.(C.8) Then it is easy to show that the second derivative of the product 𝑠𝑢(𝑝) with respect to the Laplace variable 𝑠 is𝑑2𝑑𝑠2[]𝑠𝑢(𝑝)=𝑠20𝑝7+76𝑖1𝑝59+98𝑖27𝑖1𝑝1199𝑖2𝑝13.(C.9) The fourth derivative of the product 𝑠𝑢(𝑝) with respect to the Laplace variable 𝑠 is𝑑4𝑑𝑠4[]𝑠𝑢(𝑝)=𝑠840𝑝9636048𝑖1𝑝11+9935112𝑖1+80𝑖2𝑝139111320𝑖27𝑖1𝑝15+9111315𝑖2𝑝17.(C.10) Both (C.9) and (C.10) will become important in Appendix C.3.

C.2. Solution of the First-Order Approximation Function, 𝑇1(𝑥), in (42)

If the function 𝑔(𝑠) is taken as the Laplace transform of the function 𝑇1(𝑥), that is, {𝑇1(𝑥)}=𝑔(𝑠), then the Laplace transform of (42) yields 𝑠2𝑑𝑑𝑑𝑠𝑔(𝑠)𝑠𝑔(𝑠)=𝛿𝑑𝑠𝑔(𝑠)2𝜆4𝑚𝑥4𝐽1(𝑥)2𝜆2𝑚𝑥2𝐽1.(𝑥)(C.11) In terms of the parameter 𝑝, defined in Appendix C.1, this equation may be written as𝑝2𝑑𝛿𝑑𝑠𝑔(𝑠)+𝑠𝑔(𝑠)=2𝜆4𝑚𝑑4𝑑𝑠4(1𝑠𝑝)2𝜆2𝑚𝑑2𝑑𝑠2.(1𝑠𝑝)(C.12) The right-hand side of (C.12) is evaluated from the relations involving 𝑝 in Appendix C.1 (C.7). Multiplying through by 𝑝 (C.12) becomes𝑑𝑝𝑑𝑠1𝛿𝑔(𝑠)=2𝜆4𝑚60𝑠𝑝8+105𝑠𝑝10+6𝜆2𝑚𝑠𝑝6.(C.13) Integrating (C.13) with respect to 𝑠, to find the Laplace space solution:𝑔(𝑠)=3𝛿4𝜆2𝑚𝑝5203𝜆2𝑚𝑝7+354𝜆2𝑚𝑝9.(C.14) For convenience of expressing the solutions of the first- and second-order approximations, 𝑇1(𝑥) and 𝑇2(𝑥), respectively, the following constants and function are defined:𝑖1=203𝜆2𝑚,(C.15a)𝑖2=354𝜆2𝑚,(C.15b)𝑔(𝑠)=𝑖3𝑢(𝑝),(C.15c)where𝑖3=3𝛿4𝜆2𝑚.(C.16) The polynomial 𝑢(𝑝) in (C.15c) was defined in (C.8). The inverse Laplace transform of the function 𝑔(𝑠) in (C.14) is the solution of 𝑇1(𝑥):𝑇1(𝑥)=1{𝑔(𝑠)}=𝑖31𝑝5+𝑖1𝑝7+𝑖2𝑝9=𝑖3𝑥2𝐽2(𝑥)3!!+𝑖1𝑥3𝐽3(𝑥)5!!+𝑖2𝑥4𝐽4(𝑥).7!!(C.17)

C.3. Solution of the Second-Order Approximation Function, 𝑇2(𝑥), in (45)

Similar to the first-order approximation, the Laplace transform of the second-order approximation, for {𝑇2(𝑥)}=(𝑠), yields𝑠2𝑑𝑑𝑑𝑠(𝑠)𝑠(𝑠)=𝛿𝑑𝑠(𝑠)2𝜆4𝑚𝑥4𝑇1(𝑥)2𝜆2𝑚𝑥2𝑇1.(𝑥)(C.18) In terms of the parameter 𝑝, defined in Appendix C.1, this equation may be written as𝑝2𝑑𝑑𝑠(𝑠)+𝑠(𝑠)=2𝑖233𝜆2𝑚𝑑4𝑑𝑠4𝑠𝑢2𝜆2𝑚𝑑2𝑑𝑠2𝑠𝑢.(C.19) The expressions for the fourth derivative and second derivative of 𝑠𝑢(𝑝) are given in Section C.1, (C.9)-(C.10). The first-order differential equation in (𝑠) has an integrating factor 𝑝1, so again (C.19) is multiplied by 𝑝 to obtain𝑑𝑝𝑑𝑠1=(𝑠)2𝑖23𝑠3𝜆2𝑚840𝑝10636048𝑖1𝑝12+9935112𝑖1+80𝑖2𝑝14+911137𝑖120𝑖2𝑝16+9111315𝑖2𝑝182𝜆2𝑚20𝑝8+76𝑖1𝑝510+98𝑖27𝑖1𝑝1299𝑖2𝑝14.(C.20) The integral of (C.20), after simplifying by grouping terms with like powers of 𝑝, is𝑝1(𝑠)=2𝑖233𝜆2𝑚203𝜆2𝑚𝑝671053𝜆2𝑚6𝑖1𝑝58+97106048𝑖1+2𝜆2𝑚8𝑖27𝑖1𝑝10991235112𝑖1+80𝑖2+2𝜆2𝑚𝑖2𝑝1291113147𝑖120𝑖2𝑝14911131516𝑖2𝑝16.(C.21) Recalling that {𝑇2(𝑥)}=(𝑠), the second-order approximation of (27) is simply the inverse Laplace transform of (C.21): 𝑇2=(𝑥)3𝛿22𝜆6𝑚203𝜆2𝑚𝑥3𝐽3(𝑥)75!!1053𝜆2𝑚6𝑖1𝑥54𝐽4(𝑥)+97!!7106048𝑖1+2𝜆2𝑚8𝑖27𝑖1𝑥5𝐽5(𝑥)9!!991235112𝑖1+80𝑖2+2𝜆2𝑚𝑖2𝑥6𝐽6(𝑥)11!!91113147𝑖120𝑖2𝑥7𝐽7(𝑥)13!!911131516𝑖2𝑥8𝐽8(𝑥).15!!(C.22)


𝑎:Dimensionless entrance pressure drop
𝐵𝑛:Constants of integration of Bessel’s equation, 𝑛=1, 2
𝑐:Substrate concentration (g dm−3)
𝑐0:Substrate feed concentration (g dm−3)
𝐶=𝑐/𝑐0:Dimensionless substrate concentration
𝐷𝐴𝐵:Substrate diffusivity (m2 s−1)
𝑓=𝑢1/𝑢0:Fraction retentate
𝐹𝑛:Coefficients of the solution of Kummer’s confluent hypergeometric equation, 𝑛=1, 2
Fr=𝑢20/(𝑔𝑅𝐿):Froude number
𝐹(𝑍):Dimensionless axial concentration function
𝑔:Gravitational acceleration (m s−2)
𝑔(𝑠):Laplace transform of the first-order approximation of the function 𝑇(𝑥)
(𝑠):Laplace transform of the second-order approximation of the function 𝑇(𝑥)
𝑖𝑛:Constants in the first and second-order approximations of the function 𝑇(𝑥), 𝑛=1, 2, 3
𝐽𝑛(𝜆):Bessel function of order 𝑛 of the first kind
𝑘𝑎:Mass transfer coefficient (m s−1)
𝑘𝑚:Membrane hydraulic permeability (m Pa−1 s−1)
𝐾𝑚:Michaelis constant (g dm−3)
𝐾𝑚=𝐾𝑚/𝑐0:Dimensionless Michaelis constant
𝐿:Membrane effective length (m)
𝑀(𝛼,𝛾,𝜃):Kummer function of the first kind
𝑝:Hydrostatic pressure (Pa)
𝑃=𝑝/(𝜌𝑢20):Dimensionless hydrostatic pressure
𝑃𝑒𝑢=𝑢0𝑅𝐿/𝐷𝐴𝐵:Axial Peclet number
𝑃𝑒𝑣=𝑣0𝑅𝐿/𝐷𝐴𝐵:Radial Peclet number
𝑟𝐴:Rate of substrate production/consumption (g dm−3 s−1)
𝑟:Radial spatial coordinate (m)
𝑅=𝑟/𝑅𝐿:Dimensionless radial spatial coordinate
𝑅𝐿:Membrane lumen radius (m)
Re=𝜌𝑢0𝑅𝐿/𝜇:Reynolds number
Sh=𝑘𝑎𝑅𝐿/𝐷𝐴𝐵:Sherwood number
𝑡:Time (s)
𝑇(𝑅):Dimensionless radial concentration function
𝑢:Axial velocity (m s−1)
𝑢0:Feed axial velocity (m s−1)
𝑈=𝑢/𝑢0:Dimensionless axial velocity
𝑣:Radial velocity (m s−1)
𝑣0=𝑘𝑚(𝑝0𝑝𝑆):Permeation velocity (m s−1)
𝑉=𝑣/𝑣0:Dimensionless radial velocity
𝑉𝑚:Maximum rate of reaction (g dm−3 s−1)
𝑥=𝜆𝑚𝑅:Substitution variable
𝑌𝑛(𝜆):Bessel function of order 𝑛 of the second kind
𝑧:Axial spatial coordinate (m)
𝑍=𝑧/𝐿:Dimensionless axial spatial coordinate
𝑍0:Dimensionless axial distance at which the concentration gradient is zero.
Greek Letters
𝛼:First parameter in the Kummer functions of the first and second kind
𝛽=𝑃𝑃𝑆:Dimensionless transmembrane pressure
𝛿:Lumped parameter in(38)
𝜀:Membrane porosity
𝜙:Thiele modulus
Φ(𝛼,𝛾,𝜃):Tricomi function/Kummer function of the second kind
𝛾:Second parameter in the Kummer functions of the first and second kind
Γ(𝑛):Gamma function, 𝑛=1,2,
𝜑=𝑅𝐿/𝐿:Aspect ratio
𝜅=𝜇𝑘𝑚𝐿/𝑅2𝐿:Dimensionless membrane hydraulic permeability
𝜆𝑚:Eigen values, 𝑚=1,2,
𝜇:Solution dynamic viscosity (Pa s)
𝜃:Substitution variable
𝜌:Solution density (kg m−3)
𝜏=𝑡𝑢0/𝑅𝐿:Dimensionless time
𝜉:Axial gradient/driving force of substrate concentration profile.
0:Membrane entrance
1:Membrane exit
𝑆:Shell-side of the MBR.


The authors would like to thank the Deutscher Akademischer Austausch Dienst (DAAD) for financial support, Dr. Aletta van der Merwe for her contribution in validating some of the mathematical principles.


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