Effectiveness Factors and Conversion in a Biocatalytic Membrane Reactor

Analytical expressions of the effectiveness factor of a biocatalytic membrane reactor, and its asymptote as the Thiele modulus becomes large, are presented. The evaluation of the effectiveness factor is based on the solution of the governing equations for solute transport in the two regions of the reactor, i.e. the lumen and the matrix (with the biofilm immobilized in the matrix). The lumen solution accounts for both axial diffusion and radial convective flow, while the matrix solution is based on Robin-type boundary conditions. The effectiveness factor is shown to be a function of the Thiele modulus, the partition coefficient, the Sherwood number, the Peclet number, and membrane thickness. Three regions of Thiele moduli are defined in the effectiveness factor graphs. These correspond with reaction rate limited, internal-diffusion limited, and external mass transfer limited solute transport. Radial convective flows were shown to only improve the effectiveness factor in the region of internal diffusion limitation. The assumption of first order kinetics is shown to be applicable only in the Thiele modulus regions of internal and external mass transfer limitation. An iteration scheme is also presented for estimating the effectiveness factor when the solute fractional conversion is known. The model is validated with experimental data from a membrane gradostat reactor immobilised with Phanerochaete chrysosporium for the production of lignin and manganese peroxidases. The developed model and experimental data allow for the determination of the Thiele modulus at which the effectiveness factor and fractional conversion are optimal.


Introduction
Membrane bioreactors (MBR's) offer a number of advantages over traditional bioreactors and their use for various bioconversions have been extensively reported [1][2][3]. The main challenge in the use of MBR's remains the diffusional resistance of the membrane which adversely affects their performance [4,5]. The effectiveness factor (η), defined as the ratio of the observed rate of reaction to the hypothetical rate in the absence of mass transfer limitations [6], is generally used to evaluate the performance of a catalytic reactor. A thorough review of mathematical methods employed in evaluating exact solutions of this parameter was given by Aris [6]. This study presented effectiveness factors for single and multiple reactions taking place in various shapes of porous catalysts. Webster and co-workers [7,8] presented analytical models for a membrane bioreactor immobilized with whole cells, based on both Robin-type and Dirichlettype boundary conditions. The former boundary type accounts for external mass transfer limitations, while the latter assumes the concentration at the membrane wall is known. Willaert et al. [9] obtained identical effectiveness factor expressions to Webster and Shuler [7] based on Dirichlet boundary conditions. In these studies, as well as in the majority of available exact solutions [10][11][12], axial diffusion and radial convective flows are neglected and the kinetics are generally considered linear. These assumptions are not always justified [13] and are imposed with the intention of attaining closed-form expression of the transport equation. The analytical solution of the mass balance equation is not always feasible, and a number of numerical schemes have been developed for this purpose [14][15][16][17][18][19][20]. Analytical models however are preferred for their simplicity.
The current analysis is aimed at developing expressions of the effectiveness factor for an MBR immobilized with biofilm, based on the model developed by Godongwana et al. [13]. The asymptotic behaviour as the Thiele-modulus becomes large will be considered. The models are based on the MBR system shown in Fig 1, and the following conditions of operation are assumed: (1) the system is isothermal; (2) the flow regime within the membrane lumen is fully developed, laminar, and homogeneous; (3) the physical and transport parameters are constant; (4) in the membrane matrix the flow is only one dimensional (i.e. there are no axial components of the velocity in the membrane matrix).

Governing equations
The governing equations for solute transport in the lumen and matrix of the MBR are respectively: The MBR is considered axisymmetric and the associated boundary conditions to Eqs (1) and (2) are: B:C:5 at r ¼ R 2 8z where u and v are the axial and radial velocity components, respectively; c 1 and c 2 are the local Pe v = v 0 R 1 /D AB , radial Peclet number; r, radial spatial coordinate (m); R = r/R 1 , dimensionless radial spatial coordinate; R 1 , membrane lumen radius (m); Re = ρu 0 R 1 /μ, Reynolds number; Sc = μ/ρD AB , Schmidt number; Sh = k a R 1 /D AB , Sherwood number; u, axial velocity (m s -1 ); u 0 , feed axial velocity (m s -1 ); U = u/u 0 , dimensionless axial velocity; v, radial velocity (m s -1 ); v 0 = k m (p 0 -p S ), permeation velocity (m s -1 ); V = v/v 0 , dimensionless radial velocity; V M , maximum rate of reaction (g dm -3 s -1 ); X, average biofilm density (g dm -3 ); Y x/s , yield of biofim per unit substrate; z, axial spatial coordinate (m); Z = z/L, dimensionless axial spatial coordinate; β, dimensionless transmembrane pressure; γ, membrane partition coefficient; δ = K m /c 0 C b , modified dimensionless Monod constant; =1/ϕ, substitution variable; η, effectiveness factor for general kinetics; η 0 , effectiveness factor for zero-order kinetics; η 1 , effectiveness factor for first-order kinetics; θ, substitution variable; κ=μk m L/R 1 substrate concentrations in the lumen and fiber matrix, respectively; c 1b is the bulk lumen concentration; c 1E is the concentration on the internal surface of the membrane; D 1 and D 2 are the substrate diffusion coefficients in the lumen and matrix, respectively; k a is the mass transfer coefficient; K m is the saturation constant; and V M is the maximum rate of reaction. Boundary condition 1 (B.C.1) corresponds to a uniform inlet substrate concentration; B.C.2 corresponds to cylindrical symmetry at the centre of the membrane lumen; B.C.3 and B.C.4 corresponds to continuity of the substrate flux at the lumen-matrix interface. The partition coefficient for the transfer from outside the film layer to inside the film is assumed to be unity. Only the partition for the transfer from the film layer to the matrix is considered, as shown in Fig 1. B.C.5 implies there is no diffusion across the matrix-shell interface. In single-substrate limited biofilms, V M in Eq (2) is given by [21]: where X is the average biofilm density, μ max is the maximum specific growth rate, and Y x/s is the yield of biofilm per unit substrate.

MBR lumen (Region 1)
In the lumen-side of the MBR, Eq (1) in dimensionless form becomes: where: Effectiveness Factors in a MBR The solution of Eq (5) was given by Godongwana et al. [13] as an asymptotic expansion in terms of the membrane hydraulic permeability κ: Where and F m (θ) in Eq (7) is the Kummer function: The zero-order and first-order approximations of T n (x) in Eq (7) are, respectively: where λ m are the eigenvalues, J n is the Bessel function of the first kind of order n.
The eigenvalues are obtained from B.C.3 in Eq (3c), and are roots of the equation [20]: The coefficient B m is obtained by imposing the inlet condition B.C.1 of Eq (3a):

MBR Matrix (Region 2)
First-order Kinetics. The rate of solute consumption inside the membrane matrix is governed by Monod kinetics. Assuming the first-order limit, i.e. K m >> c, Eq (2) for the matrix in dimensionless form becomes: where the first-order Thiele modulus ϕ is defined as: Eq (15) is amenable to an analytical solution by regular perturbation only when the hydraulic permeability is much smaller than unity κ << 1. For brevity only the zero-order approximation will be considered, the first order perturbation approximation is given in Appendix A following the procedure of Godongwana et al. [13]. Eq (15) then reduces to: Eq (17) is evaluated subject to B.C.4, which in dimensionless form becomes: Where γ is the partition coefficient and Sh is the Sherwood number. A good estimate of Sh for hollow fiber membranes is given by Wickramasinghe et al. [22]: where Sc = μ/ρD AB is the Schmidt number and Re = ρvR 1 /μ is the Reynolds number. The dimensionless bulk lumen concentration is defined as: Eq (17) is the modified Bessel's equation and has a solution of the form [23]: where I 0 and K 0 are the modified Bessel functions of the first kind and second kind, respectively. The constants B 1 and B 2 are obtained with the use of B.C.4 and B.C.5 as: and where The effectiveness factor is defined as: In dimensionless form: where d ¼ K Ã m C b and ϕ 0 is the zero-order Thiele modulus defined as: Assuming first-order kinetics (δ >> 1) Eq (26) reduces to: Substituting Eqs (20)(21)(22)(23)(24) into Eq (28) gives: The reciprocal of the effectiveness factor is generally considered a mass transfer resistance [6,8,24]. Thus, the reciprocal of Eq (29) is the sum of the internal resistance and the external resistance (ψ) to mass transfer. This is explicit in the asymptotic form of Eq (29) given in the Appendix B: The first and second terms inside the curly brackets in Eq (30) represent the external resistance and internal resistance to mass transfer, respectively. The series-of-resistances nature of Eqs (29) and (30) is a result of using the Robin-type boundary condition, B.C.4, in the evaluation of Eq (17). In both equations the parameters with the greatest influence on the effectiveness factor are: the Thiele modulus, partition coefficient, Sherwood number, and membrane thickness. The influence of the Peclet (Pe u ) number on the effectiveness factor is presented in Appendix A. By definition η = 1 when the Thiele modulus ϕ becomes zero since this value of the Thiele modulus corresponds with a reaction rate-controlled transfer with no mass transfer limitations.
Zero-order Kinetics. Assuming the zero-order limit, i.e. K m << c, the dimensionless form of Eq (2) becomes: Eq (31), subject to B.C. 4 and B.C.5, has a solution of the form: The dimensionless zero-order effectiveness factor from Eq (26) is: Non-linear Kinetics. The effectiveness factor allows for the determination of the overall reaction rate in terms of the Thiele modulus. However, when the reaction kinetics are not linear as was assumed in the above analysis Eq (26) is not amenable to an analytical solution. A practical measure of evaluating the effectiveness factor is attained by making the following approximation: Substituting Eq (34) into Eq (26) gives: where Y is the fractional conversion. Eq (35) allows for empirical determination of the effectiveness factor when the fractional conversion is known, from the following procedure: (i) guess the wall concentration (C 2 j R¼R 1 ) and obtain the concentration gradient from Eq (34), (ii) substitute the concentration gradient dC 2 dR j R¼1 into Eq (26) to obtain the effectiveness factor, (iii) substitute the effectiveness factor η into Eq (35) and compare the experimental conversion to the attained value, and (iv) repeat the procedure until the experimental conversion is equal to the value obtained from the iteration. (29) and (30) respectively, as functions of the normalized Thiele modulus F for different values of the Sherwood number. The normalized modulus is defined as: Eq (30) provides a simple mathematical approximation to Eq (29) and for F > 1 gives exact values for the effectiveness factor, as shown in Fig 2. Three regions of Thiele moduli may be defined from Fig 2, as characterised by the effectiveness factor. In the first region (F < 0.01) the effectiveness factor is unity, and the rate of solute transport in the MBR is controlled by the rate of reaction. When the MBR is operated in this region the diffusional resistance offered by the membrane is negligible. In the second region (0.01 < F < 0.1) the rate of solute transport is limited only by internal diffusion through the membrane, and hence the effectiveness factor is not a function of the Sherwood number. In the third region (F > 0.1) external mass transfer limitations control the rate of solute transport through the MBR, and the effectiveness factor is greatly influenced by the Sherwood number. This result is consistent with the Robin-type boundary condition.  Table 1. From Fig 3 an operating Thiele modulus may be found at which both substrate conversion and the effectiveness factor are optimal. This point corresponds with low effectiveness factors when the objective is to maximise solute conversion [19].

Fig 2 is a plot of effectiveness factors and corresponding asymptotes, from Eqs
The experimental effectiveness factor in Fig 3 is obtained from Eq (35) and is plotted against the first-order model of Eq (29). The two plots exhibit the same trend, with the model underestimating the effectiveness factor at values of F < 0.5. This is because at low values of the Thiele modulus solute transport is reaction rate controlled and the first-order kinetics premise assumes a lower rate of reaction than the maximum. At higher values of the Thiele modulus solute transport is limited by internal and external diffusion, and the first-order model approximately matches the experimental effectiveness factor.
In the region of internal diffusional limitation (0.01 < F < 0.1) radial convective flows can significantly improve the effectiveness factor, as illustrated in The increase in η with Pe v is only restricted to the region of internal diffusional limitation. The maximum relative increase in the effectiveness factor is obtained in the transitional region from kinetic to internal-diffusional control (F % 0.01), and minimal in the boundary region between internal-diffusional control and external mass transfer limitation. Increasing Pe v outside this region may drastically reduce the contact time between the substrate and the biocatalyst, and hence lead to reduced substrate conversions as was shown by Calabro et al. [17]. In this region (F > 0.1), as previously discussed the effectiveness factor can be improved by increased Sherwood numbers.

Conclusion
Mathematical models were developed for solute concentration profiles and effectiveness factors in an MBR, assuming the zero-order and first-order limits of the Michaelis-Menten (or Monod) equation. The first-order kinetic model was shown to be applicable only when the MBR is operated at high Thiele moduli. Experimental results show that the effectiveness factor Appendix B Asymptotic solution of the Effectiveness factor (ϕ ! 1) Eq (17) may be written as: where: The solution of Eq (17) may be approximated by an asymptotic expansion when ( 1 as: In order to keep the second-order derivative in the solution of the coefficient b 0 in Eq (B3), the following variable is defined: Eq (B1) then becomes: The leading order term sub-problem is: The corresponding boundary conditions are B.C.4 and B.C.5 of Eq (3): and db 0 do j o¼ð1ÀR 2 Þ= ¼ 0 ðB7bÞ The solution of Eq (B6), subject to the boundary conditions of Eq (B7) is: The effectiveness factor is obtained by taking the derivative of Eq (B8) and substituting into Eq (28) to obtain: