Biotechnol. Prog. 1992, 8, 462-464
482
The Solution of Hollow Fiber Bioreactor Design Equations? V. K. Jayaraman Chemical Engineering Division, National Chemical Laboratory, Pune 411008, India
A methodology for simplifying the solution procedure for hollow fiber bioreactor design equations has been described. Such a procedure facilitates decoupling of membrane and spongy matrix equations from the tube side equations. The equivalence between the reduced equations and the hemodialyzer problem has been explicitly obtained. ~
Introduction Hollow fiber bioreactors (HFBR) have recently been recognized as a viable alternative to conventional immobilized enzyme reactors. Due to the inherent advantages, HFBRs have been found to have applications in diverse fields like enzymatic reactions, microbial fermentations, animal cell culture, and plant cell culture and, recently, in integrated fermentation-separations (Belfort, 1989; Shukla et al., 1989). Consequently, a large number of theoretical analyses have been made for modeling and simulation of HFBRs. In their trend-setting analysis, Waterland et al. (1974) solved the first-order reaction case using a Stieltjes integral formulation. Kim and Cooney (1976) in their signal work presented an improved theoretical model involving Kummer functions. Kleinstreuer and Agarwal(1986) adopted a computational approach to simulate the performance of the HFBR for widely varying values of design parameters. Schoenberg and Belfort (1987) analyzed the enhancement in conversion due to convection. Kelsey et al. (1990) recently made a rigorous theoretical analysis for predicting the convective flow profiles in the reactor. In the present analysis, simplification of Kim and Cooney's strategy for solving the HFBR design equations has been outlined. Such a strategy would essentially decouple the fiber and spongy matrix equations from the lumen equation. By virtue of this decoupling, the solution procedure becomes simplified for a variety of situations.
Theoretical Development In a typical HFBR, the enzyme/cells are separated from the substrate by a thin semipermeable membrane. This membrane permits passage of the substrate and product but is impermeable to enzyme/cells. Normally, the enzyme/cells are confined in the outer annular region (known as the spongy matrix). During operation, the substrate, which is continuously fed through the central region (known as the lumen side), diffuses through the fiber and reacts with the enzymelcellsin the spongymatrix. The product diffuses back through the fiber to the lumen. In the present analysis, it is assumed that (1)the reactor geometry is cylindrical; (2) the substrate flow in the lumen is laminar and parabolic; (3) there is no radial convection in the fiber and spongy regions; (4) the quasi steady state prevails in the reactors; (5) the membrane is inert and reaction occurs only in the spongy matrix region; and (6) the reaction rate can be approximated by the first-order t
NCL Communication No. 5374.
~~~~~
limit of Michaelis-Menten kinetics. With these assumptions, the governing equations for the lumen, fiber, and spongy matrix regions can be written as follows:
(3) The associated boundary conditions are dc1 _ -0
at r = O
(4)
_ -- 0 ac3
at r = d
(7)
ar
ar
c1, c2, and cg are the concentrations of the substrate in the lumen, fiber, and spongy matrix regions, respectively. u is the inner radius of the membrane, b is the outer radius of the membrane, and d is the outer radius of the spongy matrix of the fiber. The other symbols are explained in the Notation section. To solve the set of equations, Kim and Cooney (1976) adopted a novel strategy which essentially consists of two steps: (1) they reduced the number of equations to two. This was facilitated by the integration of the membrane equation whose function appeared as a boundary condition in the modified set of equations. (2) They applied Laplace transformations to the remaining equations. After evaluation of the constants, these equations were inverted to get the final solution.
Simplified Strategy Let us first start with two dimensionlessequationswhich were obtained by Kim and Cooney (1976) after integration of the membrane equation:
8756-7938/92/3008-0462$03.00/0 0 1992 Amerlcan Chemical Society and American Institute of Chemical Engineers
Biotechnol. hog., 1992, Vol. 8, No. 5
1 -( a x-) ac3
xax ax
469
the substrate through the membrane and the spongy matrix. An inspection of eq 21 further reveals that the spongy matrix equation can also be decoupled from the set of equations. The combined influence of the membrane and the spongy matrix functions appears in the form of eq 21, which is now the modified boundary condition to the lumen equation. It must also be obvious that we are now left with a set of equations (i.e., tube side equation (eq 8) and the boundary conditions (eqs 10, 11, and 21)) which is exactly equivalent to the hemodialyzer problem (Cooney et al., 1974). Hemodialyzers are artificial kidney devices in which the blood flows through the lumen side of a membrane and is cleansed of low molecular weight impurities via their passage by diffusion through membranes into a flowing dilute electrolyte solution. With the electrolyte concentration (C3) remaining constant everywhere,the hemodialyzerproblem reduces to a singlepartial differential equation (Cooneyet al., 1974). The decoupled HFBR problem is exactly equivalent to the hemodialyzer problem with the overall mass transfer resistance equivalent to l/Nsh, the inverse of the wall Sherwood number. Therefore, the HFBR problem (the lumen equation), like the hemodialyzer problem, can be solved by separation of variables and subsequent transformation into Kummer's equation to obtain the solution as
= X2C3
The relevant boundary conditions are at Z = O
Cl=l
ac, ax
at X = O
-=0
-=
KC^ - c,)
-ac3 - 0
at X = l
at X=j3
ax
We can deduce from eqs 12 and 13 that
where C = cfc,;
K
= K,/K,;
j3 = d f b
m
h = -K,D,/D, In (alb); CY = D3b/D,a X = rla for the substrate side X = rlb for the spongy matrix side Z = z/aPe; Pe = voa/Dl; X2 = b2Vm,/D3K,
C,(X,Z) = E A , exp (-A:Z)
Having eliminated the membrane equation, Kim and Cooney (1976) applied Laplace transforms to the tube side and shell side equations and obtained the solution in the Laplace domain. They then applied the boundary conditions to express the unknowns in terms of known functions before finally inverting to the Z domain. The logic used in the present approach is if the membrane equation can be solved as a ODE (ordinary differential equation) then the same methodology can be applied to the spongy matrix equation also. The general solution to the spongy matrix equation is C3 = a3(Z)Io(XX)+ b3(Z)Ko(XX) (17) where a3 and b3 are some functions of the axial distance. Applying the boundary condition at X = j3 (eq 14), we get C3 =
F3
exp(-A,X2/2)M(a,b,c)
n=2
(18)
where = IoCXX) + I,(XP) Ko(WIK1(Xj3) (19) By applying the boundary condition a t X = 1 (eq 15), the unknown function a3(Z) can be expressed as F3
Now the relation (eq 12) can be used to get (21) where g=--
It must be explicitly clear from eq 21 that Q is indeed the overall mass transfer resistance for the transport of
(23) The definitions of A,, M(a,b,c),and An can be obtained from Kim and Cooney (1976). Once the C1 profile is evaluated, the C3 profile can be readily obtained from eq 18. It must be mentioned here that the modified boundary condition to the lumen equation (eq 21) is still homogeneous and therefore poses no additional difficulties.
Conclusions A simplified strategy to solve the HFBR problem has been presented. Using this strategy, the spongy matrix equation and the membrane equation can be decoupled from the design equations, simplifying the solution procedure considerably. For first-order reactions, the equivalence to the hemodialyzer problem has been explicitly shown. This method could be very useful for multiple reactions because for each component the number of equations is reduced by two. This method would also be useful to obtain asymptoticsolutions to nonlinear reactions occurring in a HFBR.
Notation inner radius of Hollow fiber skin membrane outer radius of hollow fiber skin membrane concentration of substrate at the lumen inlet dimensionless concentration dimensionless mixing cup concentration in the lumen radius of the spongy matrix wall diffusion coefficient of substrate membrane mass transfer coefficient defined by eq 16 modified Bessel function of the first kind, zero order modified Bessel function of the first kind, fist order membraneflumen partition coefficient membrane/spongy matrix partition coefficient modified Bessel function of the second kind, zero order modified Bessel function of the second kind, first order
Bbtechnol. Rog., 1992, Vol. 8,
464
Km
Michaelis-Menten constant
NSh
wall Sherwood number
Pe r
Peclet number defined by eq 16 radial coordinate center line laminar flow velocity for lumen side maximum reaction rate dimensionless radial coordinate defined by eq 16 axial distance coordinate dimensionless axial distance coordinate
VO
Vmax
X 2
2
Greek Letters a coefficient defined by eq 16 B dlb K KdKa x Thiele modulus defined by eq 16 Subscripts lumen, fiber, and spongy matrix regions, respec1, 2, 3 tively
Literature Cited Belfort, G. Membranes and Bioreactors: A Technical Challenge in Biotechnology. Biotechnol. Bioeng. 1989, 33, 1047-1066.
No. 5
Cooney,D. 0.;Kim, S. S.; Davis, E. J. Analysis of Mass Transfer in Hemodialysers for Laminar Blood flow and Homogeneous Dialysate. Chem. Eng. Sci. 1974,29, 1731-1738. Kelsey, J.; Pillarella,M. R.; Zydney, A. L. TheoreticalAnalysis of Convective Flow Profiles in a Hollow Fiber Membrane Bioreactor. Chem. Eng. Sci. 1990,45, 3211-3220. Kim, S. S.; Cooney, D. 0. An Improved Theoretical Model for Hollow Fiber Enzymatic Reactors. Chem. Eng. Sci. 1976,31, 289-294. Kleinstreuer, C.; Agarwal, S. S. Analysis and Simulationof Hollow Fiber Bioreactor Dynamics. Biotechnol. Bioeng. 1986, 28, 1233-1 240. Schoenberg, J. A.; Belfort, G. Enhanced Nutrient Transport in Hollow Fiber Perfusion Reactor. Biotechnol. h o g . 1987, 3, 80-89. Shukla, R.; Kang, W.; Sirkar, K. K. Acetone Butanol Ethanol ( B E )Production in a Novel Hollow Fiber Bioreactor. Biotechnol. Bioeng. 1989, 34, 1158-1166. Waterland, L. R.; Michaels, A. S.; Robertson,C. R. ATheoretical Model for Enzymatic CatalysisusingAsymmetricHollowFiber Membranes. AIChE J. 1974,20, 50-60. Accepted June 29,1992.
482
The Solution of Hollow Fiber Bioreactor Design Equations? V. K. Jayaraman Chemical Engineering Division, National Chemical Laboratory, Pune 411008, India
A methodology for simplifying the solution procedure for hollow fiber bioreactor design equations has been described. Such a procedure facilitates decoupling of membrane and spongy matrix equations from the tube side equations. The equivalence between the reduced equations and the hemodialyzer problem has been explicitly obtained. ~
Introduction Hollow fiber bioreactors (HFBR) have recently been recognized as a viable alternative to conventional immobilized enzyme reactors. Due to the inherent advantages, HFBRs have been found to have applications in diverse fields like enzymatic reactions, microbial fermentations, animal cell culture, and plant cell culture and, recently, in integrated fermentation-separations (Belfort, 1989; Shukla et al., 1989). Consequently, a large number of theoretical analyses have been made for modeling and simulation of HFBRs. In their trend-setting analysis, Waterland et al. (1974) solved the first-order reaction case using a Stieltjes integral formulation. Kim and Cooney (1976) in their signal work presented an improved theoretical model involving Kummer functions. Kleinstreuer and Agarwal(1986) adopted a computational approach to simulate the performance of the HFBR for widely varying values of design parameters. Schoenberg and Belfort (1987) analyzed the enhancement in conversion due to convection. Kelsey et al. (1990) recently made a rigorous theoretical analysis for predicting the convective flow profiles in the reactor. In the present analysis, simplification of Kim and Cooney's strategy for solving the HFBR design equations has been outlined. Such a strategy would essentially decouple the fiber and spongy matrix equations from the lumen equation. By virtue of this decoupling, the solution procedure becomes simplified for a variety of situations.
Theoretical Development In a typical HFBR, the enzyme/cells are separated from the substrate by a thin semipermeable membrane. This membrane permits passage of the substrate and product but is impermeable to enzyme/cells. Normally, the enzyme/cells are confined in the outer annular region (known as the spongy matrix). During operation, the substrate, which is continuously fed through the central region (known as the lumen side), diffuses through the fiber and reacts with the enzymelcellsin the spongymatrix. The product diffuses back through the fiber to the lumen. In the present analysis, it is assumed that (1)the reactor geometry is cylindrical; (2) the substrate flow in the lumen is laminar and parabolic; (3) there is no radial convection in the fiber and spongy regions; (4) the quasi steady state prevails in the reactors; (5) the membrane is inert and reaction occurs only in the spongy matrix region; and (6) the reaction rate can be approximated by the first-order t
NCL Communication No. 5374.
~~~~~
limit of Michaelis-Menten kinetics. With these assumptions, the governing equations for the lumen, fiber, and spongy matrix regions can be written as follows:
(3) The associated boundary conditions are dc1 _ -0
at r = O
(4)
_ -- 0 ac3
at r = d
(7)
ar
ar
c1, c2, and cg are the concentrations of the substrate in the lumen, fiber, and spongy matrix regions, respectively. u is the inner radius of the membrane, b is the outer radius of the membrane, and d is the outer radius of the spongy matrix of the fiber. The other symbols are explained in the Notation section. To solve the set of equations, Kim and Cooney (1976) adopted a novel strategy which essentially consists of two steps: (1) they reduced the number of equations to two. This was facilitated by the integration of the membrane equation whose function appeared as a boundary condition in the modified set of equations. (2) They applied Laplace transformations to the remaining equations. After evaluation of the constants, these equations were inverted to get the final solution.
Simplified Strategy Let us first start with two dimensionlessequationswhich were obtained by Kim and Cooney (1976) after integration of the membrane equation:
8756-7938/92/3008-0462$03.00/0 0 1992 Amerlcan Chemical Society and American Institute of Chemical Engineers
Biotechnol. hog., 1992, Vol. 8, No. 5
1 -( a x-) ac3
xax ax
469
the substrate through the membrane and the spongy matrix. An inspection of eq 21 further reveals that the spongy matrix equation can also be decoupled from the set of equations. The combined influence of the membrane and the spongy matrix functions appears in the form of eq 21, which is now the modified boundary condition to the lumen equation. It must also be obvious that we are now left with a set of equations (i.e., tube side equation (eq 8) and the boundary conditions (eqs 10, 11, and 21)) which is exactly equivalent to the hemodialyzer problem (Cooney et al., 1974). Hemodialyzers are artificial kidney devices in which the blood flows through the lumen side of a membrane and is cleansed of low molecular weight impurities via their passage by diffusion through membranes into a flowing dilute electrolyte solution. With the electrolyte concentration (C3) remaining constant everywhere,the hemodialyzerproblem reduces to a singlepartial differential equation (Cooneyet al., 1974). The decoupled HFBR problem is exactly equivalent to the hemodialyzer problem with the overall mass transfer resistance equivalent to l/Nsh, the inverse of the wall Sherwood number. Therefore, the HFBR problem (the lumen equation), like the hemodialyzer problem, can be solved by separation of variables and subsequent transformation into Kummer's equation to obtain the solution as
= X2C3
The relevant boundary conditions are at Z = O
Cl=l
ac, ax
at X = O
-=0
-=
KC^ - c,)
-ac3 - 0
at X = l
at X=j3
ax
We can deduce from eqs 12 and 13 that
where C = cfc,;
K
= K,/K,;
j3 = d f b
m
h = -K,D,/D, In (alb); CY = D3b/D,a X = rla for the substrate side X = rlb for the spongy matrix side Z = z/aPe; Pe = voa/Dl; X2 = b2Vm,/D3K,
C,(X,Z) = E A , exp (-A:Z)
Having eliminated the membrane equation, Kim and Cooney (1976) applied Laplace transforms to the tube side and shell side equations and obtained the solution in the Laplace domain. They then applied the boundary conditions to express the unknowns in terms of known functions before finally inverting to the Z domain. The logic used in the present approach is if the membrane equation can be solved as a ODE (ordinary differential equation) then the same methodology can be applied to the spongy matrix equation also. The general solution to the spongy matrix equation is C3 = a3(Z)Io(XX)+ b3(Z)Ko(XX) (17) where a3 and b3 are some functions of the axial distance. Applying the boundary condition at X = j3 (eq 14), we get C3 =
F3
exp(-A,X2/2)M(a,b,c)
n=2
(18)
where = IoCXX) + I,(XP) Ko(WIK1(Xj3) (19) By applying the boundary condition a t X = 1 (eq 15), the unknown function a3(Z) can be expressed as F3
Now the relation (eq 12) can be used to get (21) where g=--
It must be explicitly clear from eq 21 that Q is indeed the overall mass transfer resistance for the transport of
(23) The definitions of A,, M(a,b,c),and An can be obtained from Kim and Cooney (1976). Once the C1 profile is evaluated, the C3 profile can be readily obtained from eq 18. It must be mentioned here that the modified boundary condition to the lumen equation (eq 21) is still homogeneous and therefore poses no additional difficulties.
Conclusions A simplified strategy to solve the HFBR problem has been presented. Using this strategy, the spongy matrix equation and the membrane equation can be decoupled from the design equations, simplifying the solution procedure considerably. For first-order reactions, the equivalence to the hemodialyzer problem has been explicitly shown. This method could be very useful for multiple reactions because for each component the number of equations is reduced by two. This method would also be useful to obtain asymptoticsolutions to nonlinear reactions occurring in a HFBR.
Notation inner radius of Hollow fiber skin membrane outer radius of hollow fiber skin membrane concentration of substrate at the lumen inlet dimensionless concentration dimensionless mixing cup concentration in the lumen radius of the spongy matrix wall diffusion coefficient of substrate membrane mass transfer coefficient defined by eq 16 modified Bessel function of the first kind, zero order modified Bessel function of the first kind, fist order membraneflumen partition coefficient membrane/spongy matrix partition coefficient modified Bessel function of the second kind, zero order modified Bessel function of the second kind, first order
Bbtechnol. Rog., 1992, Vol. 8,
464
Km
Michaelis-Menten constant
NSh
wall Sherwood number
Pe r
Peclet number defined by eq 16 radial coordinate center line laminar flow velocity for lumen side maximum reaction rate dimensionless radial coordinate defined by eq 16 axial distance coordinate dimensionless axial distance coordinate
VO
Vmax
X 2
2
Greek Letters a coefficient defined by eq 16 B dlb K KdKa x Thiele modulus defined by eq 16 Subscripts lumen, fiber, and spongy matrix regions, respec1, 2, 3 tively
Literature Cited Belfort, G. Membranes and Bioreactors: A Technical Challenge in Biotechnology. Biotechnol. Bioeng. 1989, 33, 1047-1066.
No. 5
Cooney,D. 0.;Kim, S. S.; Davis, E. J. Analysis of Mass Transfer in Hemodialysers for Laminar Blood flow and Homogeneous Dialysate. Chem. Eng. Sci. 1974,29, 1731-1738. Kelsey, J.; Pillarella,M. R.; Zydney, A. L. TheoreticalAnalysis of Convective Flow Profiles in a Hollow Fiber Membrane Bioreactor. Chem. Eng. Sci. 1990,45, 3211-3220. Kim, S. S.; Cooney, D. 0. An Improved Theoretical Model for Hollow Fiber Enzymatic Reactors. Chem. Eng. Sci. 1976,31, 289-294. Kleinstreuer, C.; Agarwal, S. S. Analysis and Simulationof Hollow Fiber Bioreactor Dynamics. Biotechnol. Bioeng. 1986, 28, 1233-1 240. Schoenberg, J. A.; Belfort, G. Enhanced Nutrient Transport in Hollow Fiber Perfusion Reactor. Biotechnol. h o g . 1987, 3, 80-89. Shukla, R.; Kang, W.; Sirkar, K. K. Acetone Butanol Ethanol ( B E )Production in a Novel Hollow Fiber Bioreactor. Biotechnol. Bioeng. 1989, 34, 1158-1166. Waterland, L. R.; Michaels, A. S.; Robertson,C. R. ATheoretical Model for Enzymatic CatalysisusingAsymmetricHollowFiber Membranes. AIChE J. 1974,20, 50-60. Accepted June 29,1992.
