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Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2010, Article ID 738482, 12 pagesdoi:10.1155/2010/738482

Research Article

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

B. Godongwana,1 D. Solomons,2 and M. S. Sheldon1

1 Department of Chemical Engineering, Cape Peninsula University of Technology, P.O. Box 652, Cape Town 8000, South Africa2 Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag, Ronderbosch 7700, South Africa

Correspondence should be addressed to B. Godongwana, [email protected]

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 AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper presents an analytical model of substrate mass transfer through the lumen of a membrane bioreactor. The model is asolution of the convective-diffusion equation in two dimensions using a regular perturbation technique. The analysis accounts forradial-convective flow as well as axial diffusion of the substrate specie. The model is applicable to the different modes of operationof membrane bioreactor (MBR) systems (e.g., dead-end, open-shell, or closed-shell mode), as well as the vertical or horizontalorientation. The first-order limit of the Michaelis-Menten equation for substrate consumption was used to test the developedmodel against available analytical results. The results obtained from the application of this model, along with a biofilm growthkinetic 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 theearly 1970s, this technology has been used in as wideranging applications as enzyme production to bone tissueengineering. One of the current research areas of interestinto biofilm-attached membrane bioreactors (MBRs) is thedevelopment of cost-effective and environmentally friendlymethods of producing various primary and secondarymetabolites from bacterial, fungal, and yeast cells. Theseinclude: manganese and lignin peroxidase, secreted by thefungus Phanerochaete chrysosporium [1, 2]; actinorhodin,a noncommercial antibiotic produced by the filamentousbacterium Streptomyces coelicolor [3]; glutamic acid, aningredient 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 ofintermediate value [6]. It has been reported that bioreactorproductivity, in the production of these types of products,greatly impacts on the product cost [7].

The productivity of biofilm-attached MBRs is deter-mined in large by the biomass growth, and one of themost important factors that influence biomass growth is theavailability and transport of nutrients through the bioreactor[8, 9]. The momentum transfer of solutes through MBRshas been thoroughly studied, from a theoretical and exper-imental perspective, for a number of configurations [10–15]. Similarly, the mass transfer has received considerableattention [8, 9, 16–22]. With the exception of the modelsdeveloped by Heath and Belfort [17]; Li and Tan [19]; andWillaert et al. [22], the mass transfer models were solvedusing numerical procedures such as finite difference schemesand control volumes. A difficulty in implementing suchschemes is the choice of the appropriate technique for aspecific MBR system [21], and these techniques are subjectto discretization errors and stringent stability criterion. Inthe models presented by Heath and Belfort [17]; Willaertet al. [22]; and Li and Tan [19], the convective-diffusionequation governing mass transfer was solved analytically.These authors, however, neglected the effects of axial dif-fusion and radial convective flow in their models. Boththese assumptions may not be justified in all cases. Anumber of theoretical and experimental investigations have

2 International Journal of Chemical Engineering

Permeate collection and air

Nutrientsolution

Peristaltic pump

Glass bioreactor

Outlet pressure transducerHumidified air

Capillary membrane

Inlet pressure transducer

Figure 1: A schematic diagram of the single capillary membranegradostat reactor (MGR).

demonstrated the significance of radial convective flowsin improving MBR efficiencies [9, 15, 21]. In the dead-end ultrafiltration mode, particularly, the assumption ofnegligible radial convective flow is not justifiable. At axialPeclet numbers (Peu) smaller than unity large concentrationgradients exist, and under these circumstances ignoring axialdiffusion is also not justified.

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

2. Model Development

2.1. The Membrane Gradostat Reactor. The models devel-oped in this study are applicable to a hollow-fiber MBRsystem, consisting of either a single fibre or a bundle of fibres;with nutrient flowing on the lumen side of the membraneand the micro-organism immobilised either on the lumenside or on the shell side. The notation used, however, isspecifically for a single hollow fiber membrane gradostatreactor (MGR). The construction of the MGR, as patentedby Edwards et al. [23], is illustrated schematically in Figure 1.It consists of a single hollow-fibre, made of surface modifiedpolysulphone, encased in a glass bioreactor. The membranesare asymmetric and characterized by an internally skinnedand externally unskinned region of microvoids; approxi-mately 0.15 mm long and 0.015 mm thick. These membraneshave inner and outer diameters of approximately 1.395 mmand 1.925 mm, respectively. The nutrient solution permeatesfrom the lumen side to the shell side of the MGR due tothe transmembrane pressure gradient. The micro-organismis immobilised on the shell side of the MGR. Humidified air

is supplied on the shell side, and two pressure transducers arefitted at the inlet and outlet of the MGR as shown in Figure 1.

2.2. Model Assumptions. The theoretical models to be devel-oped will be based on the following conditions of operationand assumptions: (1) the system is isothermal, meaningthe energy equation has been decoupled from the massand momentum transfer; (2) the flow regime within themembrane 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) inthe dense and spongy layers of the membrane matrix the flowis only one dimensional (i.e., there are no axial componentsof the velocity profiles in the membrane matrix); and (5) theaspect ratio of the membrane is much smaller than unity. Theaspect ratio, ϕ, is the ratio of the membrane inner radius tothe effective membrane length (i.e., RL/L), and if it is muchsmaller than unity then normal stress effects are negligible inthe momentum transfer analysis.

3. Mathematical Formulation

The starting point of the analysis is the convective-diffusionequation [24]:

Dc

Dt= DAB∇2c + rA, (1)

where c is the local substrate concentration; t is time; DAB isthe substrate diffusivity, assumed to be constant; and rA, therate of substrate production (or consumption), is a functionof the local biofilm density, and time. Equation (1), for steadystate, two-dimensional flow, without reaction, in cylindricalco ordinates may be written as

u∂c

∂z−DAB

∂2c

∂z2= DAB

r

(∂c

∂r+ r

∂2c

∂r2

)− v ∂c

∂r. (2)

It is convenient to express this equation in dimensionlessform by introducing the following dimensionless variables:

U = u

u0, V = v

v0, C = c

c0,

Z = z

L, R = r

RL, ϕ = RL

L.

(3)

The expressions of U and V in (3) are obtained from themomentum transfer analysis given in Appendix A. Substitut-ing the dimensionless variables in (3) into (2) results in

ϕPeuU∂C

∂Z− ϕ2 ∂

2C

∂Z2= 1R

(∂C

∂R+ R

∂2C

∂R2

)− PevV ∂C

∂R, (4)

where the axial and radial Peclet numbers (Peu,v) are definedas

Peu = u0RLDAB

, Pev = v0RLDAB

. (5)

The boundary conditions which match the imposed operat-ing conditions of the MBR system are presented in Table 1.

International Journal of Chemical Engineering 3

3. shell-side

2. matrix

RL R 2

1. lumen-side

R3

Figure 2: A cross section of the single fibre MGR.

Boundary condition 1 (B.C.1) corresponds to a uniforminlet substrate concentration; B.C.2 and B.C.8 corresponds tocylindrical symmetry at the centre of the membrane lumen;B.C.3 indicates that the substrate concentration is only afunction of the axial spatial coordinate at the membrane wall;B.C.4 corresponds to continuity of the substrate flux at thelumen-matrix interface; B.C.5 indicates a distance along thelength of the membrane above which the axial concentrationgradient becomes zero. The boundary conditions B.C.7–11are employed in the solution of the velocity profiles given inAppendix A. A cross section of the MGR illustrating the threeregions of the reactor, with the notation used in the modeldevelopment, is shown in Figure 2.

If U in (4) is radially averaged, to become Uav, then theleft-hand side (LHS) of (4) is only a function of Z and theright-hand side (RHS) only a function of R. This can onlybe true if both the LHS and RHS are independent of thevariables R and Z. Equation (4) may therefore be solved byseparation of variables to give a solution of the form

C = F(Z)T(R). (6)

Substituting (6) into (4) gives

ϕPeuUav

F

dF

dZ− ϕ2

F

d2F

dZ2

= 1RT

(dT

dR+ R

d2T

dR2

)− PevV

T

dT

dR= −λ2.

(7)

The equating of the two ordinary differential equations(ODEs) to the arbitrary constant −λ2 in (7) is due to the factthat the two ODEs are independent of the variables R and Z.

3.1. Solution of the Axial Concentration Function F(Z). Tosolve for the axial function F(Z) of the substrate concentra-tion profile, the ODE on the LHS of (7) is considered

d2F

dZ2− PeuUav

ϕ

dF

dZ− λ2

ϕ2F = 0. (8)

The radially-averaged axial velocity Uav in (8) is defined as

Uav = 2∫ 1

0URdR = −1

8

(dP

dZ− Re

Fr

), (9)

where Fr and Re are the Froude and Reynolds number,respectively, given by

Fr = u20

gRL,

Re = ρu0RLμ

,

(10)

where g is the gravitational acceleration; ρ is the solutiondensity; and μ is the solution dynamic viscosity. The solutionof the axial velocity U in (9) is given in Appendix A as

U = −14

(1− R2)(dP

dZ− Re

Fr

), (11)

with the axial pressure gradient given by

dP

dZ= 4

√ϕ−1κβ sinh

(4√ϕ−1κ

)Z + ϕa cosh

(4√ϕ−1κ

)Z,

(12)

where P is the dimensionless hydrostatic pressure; β is thedimensionless transmembrane pressure; a is the dimension-less entrance pressure drop; and κ is the dimensionlessmembrane hydraulic permeability. The entrance pressuredrop a in (12) is given by

a =4√ϕ−1κβ sinh

(4√ϕ−1κ

)− Re Fr−1

(1− f

)ϕ[f − cosh

(4√ϕ−1κ

)] , (13)

where f is the fraction retentate ( f = 0 for the dead-endmode and f = 1 for the closed-shell mode). The membranehydraulic permeability κ in (12) is much smaller than unity(κ� 1), therefore, this equation can be approximated by thefollowing expression:

dP

dZ≈ ϕa + 16βϕ−1κZ. (14)

This approximation makes (8) a confluent hypergeometrictype differential equation. This is more evident if thefollowing sequential substitutions are made.

Substitution 1.

ξ =

⎧⎪⎪⎨⎪⎪⎩−Peu

8ϕ

(aϕ +

16βκZϕ

− ReFr

), 0 ≤ Z < Z0,

0, Z0 ≤ Z ≤ 1.(15)


Table 1: The boundary conditions of the MBR.

B.C. R, Z C, U , V , PL Range Equation

B.C.1 Z = 0 C = 1 0 ≤R ≤ 1 (6a)

B.C. 2 R = 0∂C

∂R= 0 0 ≤Z ≤ 1 (6b)

B.C. 3 R = 1 C = C(Z) 0 ≤Z ≤ 1 (6c)

B.C. 4 R = 1 −∂C∂R

+ PevVC = Sh[C(Z, 1)− C(Z,R2)] 0 ≤Z ≤ 1 (6d)

B.C. 5 Z ≥ ϕ

16βκ

(ReFr− aϕ

)C(Z) = constant 0 ≤R ≤ 1 (6e)

B.C. 6 R = 0 C = finite 0 ≤Z ≤ 1 (6f)

B.C. 7 R = 1 U = 0 0 ≤Z ≤ 1 (6g)

B.C. 8 R = 0∂U

∂R= 0 0 ≤Z ≤ 1 (6h)

B.C. 9 R = 0 V = 0 0 ≤Z ≤ 1 (6i)

B.C.10 R = 1 V = VM 0 ≤Z ≤ 1 (6j)

B.C.11 Z = 0 P = P0; P′ = ϕa 0 ≤R ≤ 1 (6k)

The substitution ξ in (15) represents the axial gradient (ordriving force) of the substrate concentration profile. Theaxial distance along the membrane length at which thisgradient is zero represents the point at which the axialfunction of the concentration profile, F(Z), is constant. Thisaxial distance, Z0, is obtained by equating (15) to zero asfollows:

∴ ξ = 0 at Z0 =ϕ

16βκ

(ReFr− ϕa

). (16)

The substitution in (15) transforms (8) to

d2F

dξ2− Aξ dF

dξ− A2λ2

ϕ2F = 0, (17)

where

A = −ϕ2(2Peuβκ)−1

. (18)

Substitution 2.

θ = 12Aξ2. (19)

This substitution transforms (17) to

θd2F

dθ2+(

12− θ

)dF

dθ− Aλ2

2ϕ2F = 0. (20)

Equation (20) is the standard Kummer hypergeometricequation and has two solutions, the Kummer function ofthe first kind M(α, γ, θ) and the Tricomi function Φ(α, γ, θ),respectively [25]:

M(α, γ, θ

) = 1 +α

γθ +

(α)2(γ)

22!θ2 + · · · +

(α)n(γ)nn!

θn, (21)

where

(α)n = α(α + 1)(α + 2) + · · · (α + n− 1), (α)0 = 1,

Φ(α, γ, θ

) = π

sinπγ

{M(α, γ, θ

)Γ(1 + α− γ)Γ(γ)

−θ1−γ M(1 + α− γ, 2− γ, θ

)Γ(α)Γ

(2− γ)

},

(22)

where Γ(n) is the gamma function. Therefore, the solution of(20) becomes

F(θ) = F0M

(Aλ2

2ϕ2,

12

, θ

)+ F1Φ

(Aλ2

2ϕ2,

12

, θ

), (23)

where F0and F1 are coefficients obtained from the inletcondition ((6a) in Table 1). The Tricomi function approachesinfinity as values of θ approach zero [26], therefore, thecoefficient F1 in (23) must be zero for this equation to satisfy((6a) in Table 1). The coefficient F0 is obtained from the inletcondition ((6a) in Table 1):

C = F0M

(Aλ2

2ϕ2,

12

, θ0

)T(R) = 1 at Z = 0, (24)

where

θ0 = − Peu256βκ

(aϕ− Re

Fr

)2

. (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 γ /= 0).The two piecewise solutions of the axial function of thedimensionless concentration profile, F(θ), are therefore

F(θ) =

⎧⎪⎪⎨⎪⎪⎩F0M

(Aλ2

2ϕ2,

12

, θ

), 0 ≤ Z < Z0,

F0, Z0 ≤ Z ≤ 1.(26)


3.2. Solution of the Radial Concentration Function T(R)

3.2.1. Zero-Order Approximation. To solve for the radialfunction T(R) of the substrate concentration profile, theODE on the RHS of (7) is solved

d2T

dR2+(

1R− PevV

)dT

dR+ λ2T = 0. (27)

The radial velocity V in (27) is given in Appendix A as

V = ϕ(u0

v0

)[R

8

(1− R2

2

)]d2P

dZ2. (28)

From the approximation in(14)

d2P

dZ2≈ 16βϕ−1κ, κ� 1. (29)

Equation (27) is solved by a regular perturbation technique:

T(R) =∞∑n=0

κnTn. (30)

The magnitude of the membrane hydraulic permeability κis very small; hence, validity of the perturbation method isassured. The equations to solve for the zero-order approxi-mation, T0, of (27) are

d2T0

dR2+

1R

dT0

dR+ λ2T0 = 0; (31)

T0(0)− B1 = 0;dT0(0)dR

− B2 = 0 (32)

Equation (31) is Bessel’s differential equation and has astandard solution of the form

T0 = B1J0(λR) + B2Y0(λR). (33)

As R approaches zero in (33), the function Y0 tends to minusinfinity; and therefore, B2 must be zero for the equation tosatisfy B.C.6 at R is equal to zero.

∴ T0 = B1J0(λR). (34)

Imposing B.C.3, the dimensionless concentration C is onlya function of the axial coordinate Z at R = 1; therefore, thefunction T0 should be equal to the constant B1J0(λ) at R = 1,but J0(λ) is an oscillating function that can have a number ofroots that satisfy the condition of B.C.3:

∴ T0 =∞∑m=1

B1mJ0(λmR). (35)

The solution for the coefficient B1m in (35) is given inAppendix B.1; the eigenvalues λm are derived from B.C.4 andare given in Appendix B.2.

3.3. First-Order and Second-Order Approximations. Theequations to solve for the first-order approximation, T1, of(27) are

d2T1

dR2+

1R

dT1

dR+ λ2

mT1 = δ

[R

(1− R2

2

)]dJ0(λmR)

dR, (36)

T1(0) = 0,dT1(0)dR

= 0, (37)

where

δ = 2Pevβ(u0

v0

) ∞∑m=1

B1m. (38)

Equation (36) is evaluated by making use of the followingidentity of Bessel functions [25]:

dJ0(λmR)dR

= −λmJ1(λmR). (39)

Substituting (39) into (36) results in the following inhomo-geneous O.D.E:

d2T1

dR2+

1R

dT1

dR+ λ2

mT1 = −δλm[R

(1− R2

2

)]J1(λmR).

(40)

Equation (40) is further simplified by making use of thefollowing substitution:

x = λmR. (41)

This substitution simplifies (40) to

xd2T1

dx2+dT1

dx+ xT1 = −δx

2

λ2m

(1− x2

2λ2m

)J1(x). (42)

Some mathematical architecture is required to solve (42)and this is described in Appendix C.1. The solution of thisequation is given in Appendix C.2 as

T1(x) = i3

[x2J2(x)

3!!+ i1

x3J3(x)5!!

+ i2x4J4(x)

7!!

], (43)

where

i1 = − 203λ2

m, i2 = 35

4λ2m

, i3 = − 3δ4λ2

m. (44)

The differential equations to solve for the second-orderapproximation, T2, of (27) are

xd2T2

dx2+dT2

dx+ xT2 = −δx

2

λ2m

(1− x2

2λ2m

)T′1(x), (45)

T2(0) = 0,dT2(0)dx

= 0. (46)


The mathematical architecture required for the solutionof these equations is also given in Appendix C.1, and thesolution is given in Appendix C.3:

T2(x)

=(

3δ2

2λ6m

){203λ2mx3J3(x)

5!!

−[

105− 73λ2m(6i1 − 5)

]x4J4(x)

7!!

+9

10

[7(60− 48i1) + 2λ2

m(8i2 − 7i1)]x5J5(x)

9!!

− 9912

[(35− 112i1 + 80i2) + 2λ2

mi2]x6J6(x)

11!!

− 9 · 11 · 1314

(7i1 − 20i2)x7J7(x)

13!!

−9 · 11 · 13 · 1516

· i2 x8J8(x)15!!

}.

(47)

Substituting (26) and (30) into (6), the solution of thedimensionless luminal concentration profile is therefore

C(θ, x) =∞∑m=1

∞∑n=0

Fm(θ)Tmn(x)κn. (48)

The perturbation solution is only extended up to second-order approximation; therefore, (48) reduces to

C(θ, x) =∞∑m=1

Fm(θ)[Tm0(x)κ0 + Tm1(x)κ1 + Tm2(x)κ2].

(49)

4. Results

4.1. Model Validation. The small diameters of the capillarymembranes used in the construction of the MBRs render it avery difficult task to validate the accuracy of the developedmodels experimentally, and thus the predictive power ofthe models will be compared to currently available resultsfor similar MBR systems; in particular the model developedby Heath and Belfort [17]. The approach used by theseauthors in solving the convective-diffusion equation wasto assume a constant cross-sectional concentration profilein the membrane lumen and matrix regions, and to solvethe simplified form of the equation. The lumen side axialconcentration profile was then obtained from a mass fluxbalance. This approach is equivalent to assuming that thesubstrate consumption takes place in the lumen side of theMBR; in such case the convective-diffusion equation given in(1) becomes

u∂c

∂z−DAB

∂2c

∂z2= DAB

r

(∂c

∂r+ r

∂2c

∂r2

)− v ∂c

∂r− Vmc

Km + c,

(50)

where Vm is the maximum rate of reaction and Km is theMichaelis constant. This equation is made dimensionless byintroducing the variables in (3) and (5), and two-additionalvariables the Thiele modulus φ and the dimensionlessMichaelis constant K∗m , respectively:

K∗m =Kmc0

, φ =√VmR

2L

c0DAB. (51)

Equation (50) then becomes

ϕPeuU∂C

∂Z− ϕ2 ∂

2C

∂Z2

= 1R

(∂C

∂R+ R

∂2C

∂R2

)− PevV ∂C

∂R− φ2C

K∗m + C.

(52)

Assuming the first-order limit of the Michaelis-Mentenequation (i.e., K∗m C) the solution of (52) takes the samegeneral approach as that of (4). The solution of this equationis identical to (49) with an adjustment of the axial functionFm(θ) and the coefficient of the perturbation solution B1m toaccount for the reaction rate:

C(θ, x) =∞∑m=1

Fm(θ)[Tm0(x)κ0 + Tm1(x)κ1 + Tm2(x)κ2],

(53)

where

Fm(θ) = F0M

(A(λ2mK

∗m + φ2

)2ϕ2K∗m

,12

, θ

),

B1m = 2λmF0M

(A(λ2mK∗m + φ2

)/2ϕ2K∗m , 1/2, θ0

)

×[

J1(λm)J20 (λm) + J2

1 (λm)

].

(54)

The input variables used in validating (53) are obtained fromHeath and Belfort [17] and are given in Table 2. The valuesof the membrane hydraulic permeability (km), the fractionretentate ( 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 parameterswill be used. A transmembrane (TMP) pressure of 5 kPa isassumed across the MBR, and the solution properties areassumed to be those of water at 30◦C.

In Figure 3, it can be seen that (53) compares satisfac-torily with the model of Heath and Belfort [17] for theparameter values listed in Table 2. At the centre of the MBR(R = 0) Equation (53) predicts that the concentrationdecreases axially to 34% of its original value when theapplied TMP is 5kPa, and at the membrane wall (R =1) the concentration decreases up to 24% of its originalvalue as shown in Figure 4. The model of Heath and Belfort[17] predicts a 44% decrease for all values of R for thecorresponding conditions. It is important to note, however,that only the resulting trends from the two models can be


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

Model parameter Symbol Unit Basic measured value

Membrane hydraulic permeabilitya km m/Pas 3.82 × 10−11

Fraction retentatea F 0.70

Membrane inner radius RL m 1.30 × 10−4

Annulus radius r3 m 4.08 × 10−4

Effective membrane length L m 5.7 × 10−2

Lumen-side entrance axial velocity u0 ms−1 1.67 × 10−3

Permeation velocity v0 ms−1 3.82 × 10−7

Number of fibres N 150

Lumen-side inlet hydrostatic pressurea p0 Pa 106 325

Shell-side hydrostatic pressurea pS Pa 101 325

Glucose diffusivity DAB m2s 1.0 × 10−10

solution densitya ρ kg m−3 998

solution viscositya μ Pas 9.7 × 10−4

Glucose inlet concentration c0 g dm−3 2.00

Kinetic constants Vm/Km s−1 1.00aTypical operational values are used.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

c/c 0

Equation (53)Heath and Belfort, 1987

z/L

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

compared, since (53) requires a more detailed description ofthe MBR system than that required for the model of Heathand Belfort [17].

5. Conclusion

A rigour analytical mathematical model for substrate con-centration profiles in the lumen of a hollow fiber MBRwas developed. The model was based on the solution ofthe convective-diffusion equation in dimensionless form.The model allows evaluation of the influence of the general

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1c/c 0

R = 0R = 0.5R = 1

z/L

Figure 4: Substrate concentration profiles assuming a first-orderlimit for substrate consumption at different radial positions.

operating parameters of a MBR on the concentrationprofiles. These parameters are the fraction retentate ( f );the membrane hydraulic permeability (κ); the axial andradial Peclet numbers (Peu,v); the Thiele modulus (φ); thefluid properties; and the dimensions of the MBR. Thedeveloped model can be further used to evaluate reactorperformance from basic principles, since it allows analyticalevaluation of performance parameters (e.g., the performanceindex and the effectiveness factor). The current paper is aprecursor to a paper by the same authors on bioreactorperformance. The models developed in the current paper willbe used to develop expressions for the effectiveness factor and


performance index (PI) of a capillary membrane gradostatreactor.

Appendices

A. Momentum Transfer Analysis

A.1. Momentum Transfer Analysis. The z-component of thenonconservative form of the Navier-Stokes equation incylindrical coordinates is made dimensionless by introducingthe variables in (4) and the following additional variables:

P = pR2L

μu0L, τ = tu0

RL, Fr = u2

0

gRL; (A.1)

where p and P are the hydrostatic and dimensionlesshydrostatic pressures, respectively; t and τ are the time anddimensionless time, respectively; μ is the solution dynamicviscosity; and Fr is the Froude number. The dimensionlessform of the Navier-Stokes thus becomes

∂U

∂τ= 1

Re

[1R

∂

∂R

(R∂U

∂R

)+ ϕ2 ∂

2U

∂Z2− dP

dZ

]+

1Fr

, (A.2)

and the continuity equation

1R

∂(RV)∂R

= −ϕ(u0

v0

)∂U

∂Z. (A.3)

Equation (A.2) is solved following the procedure of Godong-wana et al. [13], and the boundary conditions listed inTable 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 by

U = −14

(1− R2) (

dP

dZ− Re

Fr

). (A.4)

The dimensionless radial velocity profile V is obtained bysubstituting (A.4) into (A.3) and imposing B.C.9:

V = ϕ(u0

v0

)[R

8

(1− R2

2

)]d2P

dZ2. (A.5)

The dimensionless pressure profile P is obtained by imposingB.C.10, where the matrix velocity VM is governed by Darcy’slaw:

VM = −(u0

v0

)κ(PS − P0), (A.6)

where the dimensionless membrane hydraulic permeabilityκ is given by

κ = μkmL

R2L

. (A.7)

Substituting (A.5) and (A.6) into (6j) results in

ϕ

16d2P

dZ2= −κ(PS − P0). (A.8)

Equation (A.8) is solved by applying B.C.11 to give:

P = β cosh(

4√ϕ−1κ

)Z +

ϕa

4√ϕ−1κ

sinh(

4√ϕ−1κ

)Z + PS,

(A.9)

where the dimensionless entrance pressure drop a in (A.9) isobtained by defining a fraction retentate, f , as the ratio ofthe exit to the inlet velocity, that is, f = U1/U0:

a =4√ϕ−1κβ sinh

(4√ϕ−1κ

)− Re Fr−1

(1− f

)ϕ[f − cosh

(4√ϕ−1κ

)] . (A.10)

B. Solution of Constants

B.1. Solution of the Coefficient B1m. Imposing the inletcondition (6a) into (48) gives

C(θ0,R) =∞∑m=1

∞∑n=0

Fm(θ0)Tmn(λmR)κn = 1, at Z = 0.

(B.1)

For the zero-order approximation of the radial functionT(R), (B.1) becomes

F0

∞∑m=1

M

(Aλ2

m

2ϕ2,

12

, θ0

)B1mJ0(λmR) = 1, at Z = 0. (B.2)

To solve for B1n in (B.2) both the LHS and the RHS aremultiplied by J0(λmxR)R and integrated with respect to Rover the interval 0-1,

∫ 1

0J0(λmxR)R

⎧⎨⎩

∞∑m=1

M

(Aλ2

m

2ϕ2,

12

, θ0

)B1mJ0(λmR)

⎫⎬⎭dR

= 1F0

∫ 1

0J0(λmxR)RdR.

(B.3)

The RHS of (B.3) is evaluated by making use of the followingproperty of Bessel functions:

∫ z

0rvJv−1(r)dr = zvJv(z). (B.4)

The R.H.S of (B.3) then becomes

R.H.S = J1(λmx)λmxF0

. (B.5)

The L.H.S of (B.3) may be evaluated by making use ofLommel integrals:

∫ x

0Jn(αkr)Jn(αmr)r dr = 0 (k /=m),

∫ x

0rJ2n(αmr)dr = x2

2

[J ′n(αmx)2 +

(1− n2

α2mx2

)J2n(αmx)

].

(B.6)


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

m λm

1 4.0 × 10−08

2 3.701

3 6.945

4 10.125

5 13.286

6 16.440

7 19.591

8 22.738

9 25.885

10 29.030

To give the following equation:

LHS = B1m

2M

(Aλ2

m

2ϕ2,

12

, θ0

)[J20 (λm) + J2

1 (λm)]. (B.7)

Substituting (B.5) and (B.7) back into the RHS and LHS of(B.3), respectively

B1m = 2

λmF0M

(Aλ2

m

2ϕ2,

12

, θ0

)[

J1(λm)J20 (λm) + J2

1 (λm)

]. (B.8)

B.2. Solution of the Eigenvalues λm. The solution of theeigenvalues is obtained from B.C.4 in Table 1. When theenzyme is immobilized on the lumen side of the membranethis equation reduces to

∂C

∂R− PevVC = 0, at R = 1. (B.9)

Substituting the expression for C, (49) into (B.9) and limitingthe perturbation to first-order approximation:

(PevVT0 − dT0

dR

)κ0 −

(PevVT1 − dT1

dR

)κ1 = 0, at R = 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 theparameter values in Table 2.

C. Laplace Transform Solutions

C.1. Mathematical Architecture for the Solution of T(x). Wedefine a parameter p in Laplace space as

p = 1√1 + s2

. (C.1)

The following properties of p are all noteworthy, differentia-tion:

p′ = −sp3. (C.2)

Integration:

∫pns ds = pn−2

2− n , n /= 2. (C.3)

Inverse Laplace Transform:

L−1{p} = J0(x),

L−1{p2n+1} = xnJn(x)(2n− 1)!!

.(C.4)

Laplace Transforms involving the First-Order Bessel func-tion:

L{J1(x)} = 1− sp. (C.5)

More relations involving p:

s2 = p−2 − 1, (C.6)

d

ds

[1− sp] = −p3,

d2

ds2[1− sp] = 3sp5,

d3

ds3[1− sp] = 15p7 − 12p5,

d4

ds4[1− sp] = 60sp7 − 105sp9.

(C.7)

We also define the following polynomial in p, which issignificant for the first- and second-order approximation ofT(x) in Sections C.2 and C.3 , respectively,

u(p) = p5 + i1p

7 + i2p9. (C.8)

Then it is easy to show that the second derivative of theproduct su(p) with respect to the Laplace variable s is

d2

ds2[su(p)]

= s[20p7 + 7(6i1 − 5)p9 + 9(8i2 − 7i1)p11 − 99i2p13].

(C.9)

The fourth derivative of the product su(p) with respect to theLaplace variable s is

d4

ds4[su(p)] = s

[840p9 − 63(60− 48i1)p11

+ 99(35− 112i1 + 80i2)p13

− 9 · 11 · 13(20i2 − 7i1)p15

+ 9 · 11 · 13 · 15 · i2p17].

(C.10)

Both (C.9) and (C.10) will become important in AppendixC.3.


C.2. Solution of the First-Order Approximation Function,T1(x), in (42) . If the function g(s) is taken as the Laplacetransform of the function T1(x), that is, L{T1(x)} = g(s),then the Laplace transform of (42) yields

− s2 ddsg(s)− sg(s)− d

dsg(s)

= δ

2λ4m

{L[x4J1(x)

]− 2λ2mL

[x2J1(x)

]}.

(C.11)

In terms of the parameter p, defined in Appendix C.1, thisequation may be written as

p−2 d

dsg(s) + sg(s)

= − δ

2λ4m

{d4

ds4(1− sp)− 2λ2

md2

ds2(1− sp)

}.

(C.12)

The right-hand side of (C.12) is evaluated from the relationsinvolving p in Appendix C.1 (C.7). Multiplying through by p(C.12) becomes

d

ds

[p−1g(s)

] = − δ

2λ4m

{−60sp8 + 105sp10 + 6λ2msp

6}.(C.13)

Integrating (C.13) with respect to s, to find the Laplace spacesolution:

g(s) = − 3δ4λ2

m

(p5 − 20

3λ2mp7 +

354λ2

mp9

). (C.14)

For convenience of expressing the solutions of the first- andsecond-order approximations, T1(x) and T2(x), respectively,the following constants and function are defined:

i1 = − 203λ2

m, (C.15a)

i2 = 354λ2

m, (C.15b)

g(s) = i3u(p), (C.15c)

where

i3 = − 3δ4λ2

m. (C.16)

The polynomial u(p) in (C.15c) was defined in (C.8). Theinverse Laplace transform of the function g(s) in (C.14) isthe solution of T1(x):

T1(x) = L−1{g(s)}

= i3L−1{p5 + i1p

7 + i2p9}

= i3

[x2J2(x)

3!!+ i1

x3J3(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, theLaplace transform of the second-order approximation, forL{T2(x)} = h(s), yields

− s2 ddsh(s)− sh(s)− d

dsh(s)

= δ

2λ4m

{L[x4T′1(x)

]− 2λ2mL

[x2T′1(x)

]}.

(C.18)

In terms of the parameter p, defined in Appendix C.1, thisequation may be written as

p−2 d

dsh(s) + sh(s) = − 2i23

3λ2m

{d4

ds4su− 2λ2

md2

ds2su

}. (C.19)

The expressions for the fourth derivative and second deriva-tive of su(p) are given in Section C.1, (C.9)-(C.10). The first-order differential equation in h(s) has an integrating factorp−1, so again (C.19) is multiplied by p to obtain

d

ds

[p−1h(s)

] = 2i23s3λ2

m

{[840p10 − 63(60− 48i1)p12

+ 99(35− 112i1 + 80i2)p14

+ 9 · 11 · 13(7i1 − 20i2)p16

+9 · 11 · 13 · 15 · i2p18]− 2λ2

m

[20p8 + 7(6i1 − 5)p10

+9(8i2 − 7i1)p12 − 99i2p14]}.(C.20)

The integral of (C.20), after simplifying by grouping termswith like powers of p, is

p−1h(s) = 2i233λ2

m

{203λ2mp

6 −[

105− 73λ2m(6i1 − 5)

]p8

+9

10

[7(60− 48i1) + 2λ2

m(8i2 − 7i1)]p10

− 9912

[(35− 112i1 + 80i2) + 2λ2

mi2]p12

− 9 · 11 · 1314

(7i1 − 20i2)p14

−9 · 11 · 13 · 1516

· i2p16}.

(C.21)


Recalling that L{T2(x)} = h(s), the second-order approx-imation of (27) is simply the inverse Laplace transform of(C.21):

T2(x)

=(

3δ2

2λ6m

){203λ2mx3J3(x)

5!!

−[

105− 73λ2m(6i1 − 5)

]x4J4(x)

7!!

+9

10

[7(60− 48i1) + 2λ2

m(8i2 − 7i1)]x5J5(x)

9!!

− 9912

[(35− 112i1 + 80i2) + 2λ2

mi2]x6J6(x)

11!!

− 9 · 11 · 1314

(7i1 − 20i2)x7J7(x)

13!!

−9 · 11 · 13 · 1516

· i2 x8J8(x)15!!

}.

(C.22)

Nomenclature

a: Dimensionless entrance pressure dropBn: Constants of integration of Bessel’s

equation, n = 1, 2c: Substrate concentration (g dm−3)c0: Substrate feed concentration (g dm−3)C = c/c0: Dimensionless substrate concentrationDAB: Substrate diffusivity (m2 s−1)f = u1/u0: Fraction retentateFn: Coefficients of the solution of

Kummer’s confluent hypergeometricequation, n = 1, 2

Fr = u20/(gRL): Froude number

F(Z): Dimensionless axial concentrationfunction

g: Gravitational acceleration (m s−2)g(s): Laplace transform of the first-order

approximation of the function T(x)h(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 firstkind

ka: Mass transfer coefficient (m s−1)km: Membrane hydraulic permeability

(m Pa−1 s−1)Km: Michaelis constant (g dm−3)K∗m = Km/c0 : Dimensionless Michaelis constantL: Membrane effective length (m)M(α, γ, θ): Kummer function of the first kindp: Hydrostatic pressure (Pa)

P = p/(ρu20): Dimensionless hydrostatic pressure

Peu = u0RL/DAB: Axial Peclet numberPev = v0RL/DAB: Radial Peclet numberrA: Rate of substrate

production/consumption (g dm−3 s−1)r: Radial spatial coordinate (m)R = r/RL: Dimensionless radial spatial coordinateRL: Membrane lumen radius (m)Re = ρu0RL/μ: Reynolds numberSh = kaRL/DAB: Sherwood numbert: Time (s)T(R): Dimensionless radial concentration

functionu: Axial velocity (m s−1)u0: Feed axial velocity (m s−1)U = u/u0: Dimensionless axial velocityv: Radial velocity (m s−1)v0 = km(p0 − pS): Permeation velocity (m s−1)V = v/v0: Dimensionless radial velocityVm: Maximum rate of reaction (g dm−3 s−1)x = λmR: Substitution variableYn(λ): Bessel function of order n of the second

kindz: Axial spatial coordinate (m)Z = z/L: Dimensionless axial spatial coordinateZ0: Dimensionless axial distance at which

the concentration gradient is zero.

Greek Letters

α: First parameter in the Kummerfunctions of the first and second kind

β = P − PS: Dimensionless transmembrane pressureδ: Lumped parameter in(38)ε: Membrane porosityφ: Thiele modulusΦ(α, γ, θ): Tricomi function/Kummer function of

the second kindγ: Second parameter in the Kummer

functions of the first and second kindΓ(n): Gamma function, n = 1, 2, . . .ϕ = RL/L: Aspect ratioκ = μkmL/R

2L: Dimensionless membrane hydraulic

permeabilityλm: Eigen values, m = 1, 2, . . .μ: Solution dynamic viscosity (Pa s)θ: Substitution variableρ: Solution density (kg m−3)τ = tu0/RL: Dimensionless timeξ: Axial gradient/driving force of substrate

concentration profile.

Subscripts

0: Membrane entrance1: Membrane exitS: Shell-side of the MBR.


Acknowledgments

The authors would like to thank the Deutscher AkademischerAustausch Dienst (DAAD) for financial support, Dr. Alettavan der Merwe for her contribution in validating some of themathematical principles.

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