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Ind. Eng. Chem. Res. 1999, 38, 4044-4050
Analysis of Protein Purification Using Ion-Exchange Membranes Heewon Yang,† Matthias Bitzer,† and Mark R. Etzel*,†,‡ Department of Chemical Engineering, 1415 Engineering Drive, and Department of Food Science, University of Wisconsin, Madison, Wisconsin 53706-1565
The performance of ion-exchange membranes for protein purification is analyzed using numerical solutions of different mathematical models. The models incorporate nonlinear sorption isotherms and mass-transfer coefficients based on either the overall or local solid-phase and liquid-phase driving forces. The numerical solutions are compared to analytical solutions which use overall mass-transfer coefficients only and, in general, are theoretically incorrect for nonlinear isotherms. The numerical solutions are fit to experimental breakthrough curves from the literature. The models allow the determination of the rate-controlling mass-transfer phenomena and solid-phase concentration, and prediction of the operating and membrane-design parameters needed to obtain sharp breakthrough curves. 1. Introduction Rapid developments in biotechnology are fueling demand for reliable, efficient methods to purify labile biological molecules like proteins, peptides, and nucleic acids. The cost of recovering biomolecules often dominates total product manufacturing costs; therefore, a vital interest exists in enhancing the quality and overcoming the limitations of current bioseparation techniques. The recovery of biomolecules typically involves several consecutive separation steps.1 For example, a cellular extract harvested from a fermentation broth may first be homogenized to render the product accessible, then clarified by filtration or centrifugation, and then purified by ion-exchange chromatography using beads packed into a column. Integration of these different separation steps into one step can increase the process yield, compactness, and economics and reduce the time to market, labor costs, and product variability, all while attaining targets for product purity. Ion-exchange membranes, which are made from microporous membranes by chemically attaching charged moieties to the internal pore structure, can achieve process integration by combining into one step clarification, concentration, fractionation, and purification of homogenized cell extracts. Process chromatography using ion-exchange membranes uses one of three different operating modes. In the first mode, a particleladen protein solution is pumped into a membrane cartridge having fine pores. Particulates (i.e., debris, cells, and lipids) are rejected by the microporous membrane and flow out of the cartridge into the retentate stream, while the protein solution flows through the membrane toward the permeate stream. The target protein binds to the immobilized moieties as the permeate flows through the membrane. In the second operating mode, a particle-laden protein solution is pumped into a membrane cartridge having coarse pores, through which the particulates pass. The target protein is captured by adsorption to the immobilized moieties. In * Corresponding author. E-mail:
[email protected]. Telephone: (608) 263-2083. Fax: (608) 262-6872. † Department of Chemical Engineering. ‡ Department of Food Science.
the third operating mode, the protein solution is first clarified, and then pumped, in dead end flow, into a membrane cartridge having fine pores. In this mode, integration of multiple separation steps is not achieved. Still, ion-exchange membranes having fine pores can attain shorter cycle times and higher flow rates (e.g., 9000 column volumes (CV)/h) than packed columns because linear velocity is not limited by pressure drop or by diffusion of protein into the beads. Pressure drop limitations are negligible for thin membranes (e.g., 1 bar at 9000 CV/h). Diffusional limitations are negligible because convective transport of the solution through the porous structure reduces the time for protein to reach the pore wall when the membranes have sufficiently fine pores. The result is that the dynamic capacity can be virtually independent of flow rate. Proper design, selection, operation, and analysis of ion-exchange membrane systems would be greatly enhanced by the development of a mathematical model that can be used to predict the precise influence of system parameters on performance. System parameters of interest include pore size, thickness, surface area, void fraction, static binding capacity, binding affinity, flow rate, concentration, mixing in the flow system, and protein molecular weight. For example, a very important design issue when selecting ion-exchange membranes for process chromatography is to determine the appropriate pore size for a given application that balances slow mass transfer to the immobilized binding site for pores that are too coarse against plugging for pores that are too fine. Although ion-exchange membranes are gaining popularity for use in the biotechnology industry, theoretical analysis of these systems is not as commonplace. Modeling efforts for ion-exchange membrane chromatography have typically used the Langmuir-type secondorder reversible rate equation to describe the kinetics of adsorption.2 These models were developed and work well for affinity membrane chromatography where the intrinsic association kinetics between the soluble protein and the immobilized ligand typically limit the binding rate.3-5 However, for ion-exchange membrane chromatography, association kinetics are much faster than diffusive mass transport and are not rate-limiting steps.6 Therefore, models developed for affinity mem-
10.1021/ie990084o CCC: $18.00 © 1999 American Chemical Society Published on Web 09/16/1999
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brane chromatography are generally inappropriate for ion-exchange membrane chromatography. Gebauer et al.6 developed mathematical models of ion-exchange membrane chromatography systems, wherein different limiting analytical models were used to examine separately the different model cases of a single rate-limiting step, such as intrinsic association kinetics, liquid-phase mass transfer, solid-phase mass transfer, or surface diffusion. To obtain an analytical solution, several assumptions were made, e.g., irreversible adsorption, constant solid-phase solute concentration, and constant mass-transfer coefficients. In the work of Sarfert and Etzel,7 to obtain an analytical solution, both the dimensionless solute concentration in the solid phase (as suggested by Hiester and Vermeulen8) and the overall mass-transfer coefficient were assumed constant. As a result, the model was incapable of solving explicitly for the solute concentration in the solid phase and generated the same breakthrough curves using either the overall liquidphase or solid-phase mass-transfer coefficients. Furthermore, the model was not applicable to separation processes operating in the nonlinear portion of the adsorption isotherm and where both solid-phase and liquid-phase mass-transfer resistances are significant, because in these cases the overall mass-transfer coefficient is not constant. The objective of this work was to improve the model of Sarfert and Etzel and to better simulate the performance of ion-exchange membranes. Ion-exchange membrane separations are described using a numerical solution of the continuity equation and constitutive relations, wherein the governing equations are solved without the assumption of a constant solute concentration or constant overall mass-transfer coefficients. The model is extended to incorporate local mass-transfer coefficients and the slope of the adsorption isotherm. The improved model can be used to predict the solute concentration profile in the solid phase and applies to cases involving nonlinear isotherms where both solidand liquid-phase mass transfer is important. 2. Model with Overall Mass-Transfer Coefficients Membrane chromatography systems share many useful properties with traditional packed-bed systems and can be modeled similarly. Examples of packed-bed system models can be found in the literature.9,10 Adsorption of protein in the membrane can be modeled using a mass balance for the flowing liquid phase. In the dimensionless form, it can be expressed as3
∂Cs ∂C ∂C 1 ∂2C -m + ) ∂τ ∂ζ Pe ∂ζ2 ∂τ
C-
1 ∂C )1 Pe ∂ζ
∂C )0 ∂ζ
at ζ g 0, τ ) 0 at ζ ) 0, τ > 0 at ζ ) 1, τ > 0
( (
∂cs Kdcs ) Kolam(c - c*) ) Kolam c ∂t cl - cs ) Kosam(c/s - cs) ) Kosam
)
clc - cs Kd + c
(5)
)
(6)
where c* is the fictitious concentration of the solute in the liquid phase in equilibrium with cs given by the Langmuir isotherm:
cs )
clc* Kd + c*
(7)
Equations 5 and 6 represent the classic liquid film linear driving force (LFLDF) and solid film linear driving force (SFLDF) models, respectively. Using dimensionless variables, eqs 5 and 6 simplify to
(
Cs ∂Cs nL ) C∂τ m r - (r - 1)Cs
(
)
∂Cs rC - Cs ) nS ∂τ 1 - 1 + (r -1)C
(8)
)
(9)
All practical experimental systems also contain mixing and dead volumes in other than the membrane itself. A simplified but sufficiently accurate method to describe this behavior is the serial combination of the model of the membrane with a continuously stirred tank reactor (CSTR) and an ideal plug flow reactor (PFR)12,13 as shown in Figure 1. Thus, the system volume is
Vsys ) VCSTR + VPFR
(10)
The CSTR model with the initial condition cc(t ) 0) ) 0 is
(1) dcc Q (c - cc) ) dt VCSTR
with the initial condition and boundary conditions
C ) Cs ) 0
A solution to eq 1 requires information regarding the rate of solute (protein) transfer to the solid phase. Transport of solute from the liquid phase to the solid phase involves mass transfer through the liquid phase, mass transfer through the solid phase, and sorption to the binding sites. In the ion-exchange membrane, the morphology of the membrane corresponds to macropore and micropore regions that can be considered as the liquid and solid phase.7 The transfer of solute can be described using overall mass-transfer coefficients in a manner similar to that shown by Lightfoot et al.:9,10
(2) (3) (4)
Here, the Danckwerts’ boundary conditions11 for frontal analysis were used.
(11)
The PFR model shifts the response by the delay time (tdelay). The first temporal moment method may be used to determine VCSTR and VPFR.14 Because the PFR volume includes the void volume (Vvoid ) 7.5 mL) in the membrane, the pure delay time is due to the remaining PFR volume (VPFR - Vvoid). Analytic Solution. By combining and rearranging eqs 1 and 5 (or eq 6) with initial and boundary conditions, without the axial dispersion term in eq 1,
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Figure 1. Components of the experimental system and its simplified model. Splitting the CSTR and PFR elements into inlet and outlet elements gives the same result as incorporating the elements at either the inlet or the outlet because the mass-transfer rate equations are first-order.
an analytical solution can be derived in the form of the classic Thomas solution,15,16
Z J ,T r C) Z 1 T J , T + 1 - J Z, exp - 1 (T - Z) r r r
( ) [
( ) ( )] [(
)
]
(12)
with
J(R,β) ) 1 -
where I0 is the modified Bessel function of zeroth-order. The dimensionless variables are defined for the LFLDF model by
nL r m r + Cs - rCs
r Z ) ZL ) ζnL r + Cs - rCs
cl (M) 2.35 × 10-3 L (cm) 2.04 Q (mL/min) v (cm/s) VPFR (mL)
∫0Rexp(-β - η)I0(2xβη) dη
T ) TL ) (τ - ζ)
Table 1. Values of the Parameters Used in Simulating the Breakthrough Curves (Taken from Sarfert and Etzel7)
(13) (14)
and for the SFLDF model by
r n T ) TS ) (τ - ζ) 1 - S1 - C + rC
(15)
r n m Z ) ZS ) ζ 1 - S 1 - C + rC
(16)
Sarfert and Etzel used a simplification of Cs ) 0.5 in eqs 13 and 14 for the LFLDF model7 and C ) 0.5 in eqs 15 and 16 for the SFLDF model17 as suggested by Hiester and Vermeulen8 to convert the Thomas solution into an explicit form. They fit the model to experimental breakthrough curves using the general regression (GREG18) software package. The overall liquid-phase (Kol in eq 5)7 or solid-phase (Kos in eq 6)17 mass-transfer
Common Parameters Kd (M) dp (cm) 5.45 × 10-7 1.5 × 10-2 D (cm2/s) c0 (M) 6.7 × 10-7 4.15 × 10-5
0.75 Vsys (mL) 11.3
Parameters Depending on Flow Rate 0.2 1.0 5.0 9.05 × 10-4 4.53 × 10-3 2.26 × 10-2 9.5 8.2 7.1
coefficient was used as the sole fitting parameter. This result will be compared with the more exact numerical solutions of the LFLDF and SFLDF models calculated without the simplification and including the axial dispersion term. Numerical Solution. In this work, we used the PDASAC (partial-differential-algebraic sensitivity analysis code) software package19 to solve the equations for the model of ion-exchange membrane performance. Two separate rate equations were used: LFLDF (eq 5) and SFLDF (eq 6). The model equations were fitted to the data of Sarfert and Etzel7 and the mass-transfer coefficients Kol or Kos were used as the fitting parameters in GREG. The fitting results were in terms of the most likely parameter value with a (95% confidence interval. Values of the membrane and flow system parameters used in simulating breakthrough curves, which were taken from Sarfert and Etzel,7 are shown in Table 1. Mass-transfer coefficients obtained using the new numerical solution were compared with those from the analytical solution of Sarfert and Etzel (Table 2). Figure 2 contains the fitted breakthrough curves, for flow rates of 0.2, 1.0, and mL/min. Comparing the numerical results for the LFLDF and SFLDF models, the SFLDF model gave the closest fit to the experimental data. Analytical model results were usually located between the two numerical results. Although Sarfert and Etzel
Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 4047 Table 2. Results of Breakthrough Curve Fitting with Overall Mass-Transfer Coefficientsa analytical
numerical
Q (mL/min)
Kol (cm/s)
Kos (cm/s)
Kol (cm/s)
Kos (cm/s)
0.2 1.0 5.0
9.9 ( 0.6 × 10-7 1.9 ( 0.1 × 10-6 2.8 ( 0.3 × 10-6
1.8 ( 0.1 × 10-8 3.3 ( 0.3 × 10-8 4.9 ( 0.5 × 10-8
1.3 ( 0.1 × 10-6 1.28 ( 0.02 × 10-6 3.9 ( 0.4 × 10-6
2.16 ( 0.08 × 10-8 6.9 ( 0.5 × 10-8 9.5 ( 0.9 × 10-8
a K and K are obtained with the LFLDF and SFLDF model, respectively. In analytical solutions, K is taken from Sarfert and Etzel7 ol os ol and Kos, from Sarfert.17
Figure 3. Concentration driving forces in the Langmuir isotherm.
where ci is the interfacial concentration of the solute in the liquid phase and csi is the interfacial concentration of the solute in the solid phase that is in equilibrium with ci by the Langmuir isotherm (see Figure 3). The relationships between the local and overall masstransfer coefficients are given by
1 1 1 ) + Kol kl m′ks
(18)
1 m′′ 1 ) + Kos kl ks
(19)
and
Figure 2. Experimental breakthrough curves and fitted curves using overall mass-transfer coefficients.
obtained different values for Kol versus Kos for the analytical model, both approaches gave the same fitted curves. All the fitted results for the models failed to give a precise fit on a point-by-point basis, but the analytical and SFLDF numerical models were able to fit the general trend of the experiment. 3. Model with Local Mass-Transfer Coefficients If mass transfer of the solute is controlled by both the liquid film (LF) and solid film (SF), then the overall coefficients depend not only on both local coefficients but also on the slope of the isotherm. Because the slope of the nonlinear portion of the isotherm depends on bulk concentration c or cs, the overall mass-transfer coefficient is not a constant and is dependent on the bulk concentration. If the isotherm is linear, i.e., the slope remains constant, then the overall coefficients are also constant. However, in a usual membrane system, the operating conditions span over the entire isotherm and the overall coefficients are not constant. Thus, unless either the LF or SF is rate-controlling, using constant overall mass-transfer coefficients is theoretically incorrect. The rate equation with local mass-transfer coefficients is expressed as9,10
∂cs ) klam(c - ci) ) ksam(csi - cs) ∂t
(17)
where m′ and m′′ are the slopes of the equilibrium curve shown in Figure 3. When the LF is the rate-controlling step, i.e., kl , m′ks, csi ≈ cs and Kol ≈ kl. On the other hand, when the SF is the rate-controlling step, i.e., ks , kl/m′′, ci ≈ c and Kos ≈ ks. Because ci and csi are in equilibrium in eq 17, both values satisfy the Langmuir isotherm:
csi )
clci K d + ci
(20)
Combining the above equation with eq 17 to eliminate the interfacial concentrations gives
∂cs am ) [kl(Kd + c) + ks(cl - cs) ∂t 2
x(kl(Kd + c) + ks(cl - cs))2 + 4klks(csKd - clc + csc)]
(21)
PDASAC was used to solve the equations for the model of the ion-exchange membrane with eq 21 as the rate equation instead of LFLDF (eq 5) or SFLDF (eq 6). Both local mass-transfer coefficients kl and ks were used as fitting parameters with GREG. Table 3 contains the kl and ks values obtained from the fitting procedure. On the basis of comparison of ks in Table 3 and numerical Kos in Table 2 (Kos ≈ ks), the SF is the rate-controlling step. In agreement, a comparison of kl and numerical Kol did not reveal a
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Table 3. Numerical Results of Breakthrough Curve Fitting with the Model Combining Two Local Mass-Transfer Coefficients
a
Q (mL/min)
kl (cm/s)a
ks (cm/s)
0.2 1.0 5.0
8 × 10-6 5 × 10-5 2 × 10-5
2.2 ( 0.1 × 10-8 6.9 ( 0.5 × 10-8 9.5 ( 0.1 × 10-8
Model is insensitive to the value of kl.
Figure 4. Concentration profiles of the solute in the solid phase.
significant relationship, and values of kl did not have a significant impact on the predictions from the model. The breakthrough curves generated with eq 21 coincided with the predictions from the numerical SFLDF model, not the numerical LFLDF model, and were not replotted in Figure 2. This coincidence will not occur in general and applies to the analysis of the experimental observations used as an example in this experiment only because the rate-controlling step will not always be in the solid phase. Because the numerical solution could solve also for Cs, we calculated the distribution of Cs in the membrane using the model combining two local mass-transfer coefficients (Figure 4). At Q ) 5.0 mL/min, the concentration of Cs was almost constant over the thickness of the membrane at a given time because the adsorption rate was very slow and the protein in the liquid phase was never consumed completely. Most of the protein loaded into the membrane passed through so fast that it was not captured by the membrane and simply passed through to the effluent solution. After τ ) 60, Cs was still below 0.5. At a slower flow rate of Q ) 0.2 mL/ min, more of the protein in the liquid phase was adsorbed to the membrane. On the basis of these calculations, the membrane was not saturated at τ ) 60. For comparison, for sharp breakthrough curves, τ ≈ 1 + (r - 1)m/r at the breakthrough point.3 For this system, the membrane would saturate at τ ≈ 20 if the
breakthrough curve was sharp. However, even at the lowest flow rate, the calculated solid-phase concentration was nearly independent of the distance and slow mass transfer prevented attaining sharp breakthrough curves. 4. Discussion The morphology of the microporous ion-exchange membranes shows the membrane consists of a continuous spongelike structure sparsely interspersed by solid support fibers (refer to Figure 1 in Sarfert and Etzel7). The membrane matrix itself consists of numerous lamellae of random orientation. Macropores result from holes in the lamellae and from platelike sheets that abut at angles. Although at low magnification, the lamellae appear to consist of smooth tight walls; upon closer examination, a structure is revealed wherein the regenerated cellulose base matrix is permeated by fine micropores. Thus, macropore and micropore regions can be considered as the liquid and solid phase, respectively. When the transfer of solute is expressed as eq 5 or 6, the form of the final equation is the same whether the packed-bed system or membrane system is selected. However, in packed-bed systems, the solid phase has regular spherical shape and the local mass-transfer coefficients can be predicted using well-known correlations and theories for solute transport around and into the solid particle.10 In membrane systems, however, the solid phase (micropore regions) has irregular shape and well-established correlations and theories are not available to predict the local mass-transfer coefficients. Thus, the mass-transfer coefficients were used as fitting parameters. Broadly disperse breakthrough curves correspond to low membrane capacity utilization, a delay in the saturation time, and a waste of protein in the feed solution.20 From Figure 2, the breakthrough curves investigated in this work were so broad that the loss of protein in the feed solution was severe. Even at the lowest flow rate of Q ) 0.2 mL/min, saturation of the membrane did not occur (cout/c0 < 0.9) during the experiment (τ ) 54), whereas saturation would have occurred at τ ) 20 if the breakthrough curve was sharp. It was of interest to find the maximum flow rate which would generate a sharp breakthrough curve. We used the 70% sharpness criteria of Sarfert and Etzel7 which states a breakthrough curve is considered to be sharp if the concentration of the solute near the exit of the membrane decreases from 70% to 10% of the feed solution concentration over a distance equal to 10% of the membrane thickness. For convective mass transfer, the mass-transfer coefficients depend on the flow velocity.10 For creeping flow, film theory predicts the following form:7
ks ) b1vb2
(22)
where b1 and b2 are constant. A least squares fit of the ks values in Table 3 yields b1 ) 3.6 × 10-7 cm/s and b2 ) 0.34 where v is in cm/s. Figure 5 contains the result of the simulation with the SFLDF model. In these calculations, VPFR corresponding to Q ) 0.2 mL/min was used. The breakthrough curve at Q ) 0.0095 mL/min meets the 70% sharpness criteria. The corresponding concentration profile of the solute in the solid phase is presented in Figure 6. At this flow rate, both the solid-phase and liquid-phase breakthrough curves are sharp and mass
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Figure 5. Effect of flow rate on the sharpness of breakthrough curves. The SFLDF model with Kos is used.
Figure 6. Concentration profile of the solute in the solid phase at Q ) 0.0095 mL/min.
transfer is not a substantial limit to performance. However, this flow rate is too low to be practical, giving a residence time of the liquid in the membrane of 13 h. A membrane having finer pores (3-5 µm) would result in a sharper breakthrough curve at more practical residence times of 1-100 s.7 When the isotherm is linear, the overall mass-transfer coefficients are constant and are adequate to represent the rate equation. In this case, using the overall masstransfer coefficient in the liquid phase or solid phase will give the same predicted breakthrough curve. The rate equation with constant overall mass-transfer coefficients, however, cannot correctly describe the breakthrough curve when the isotherm is nonlinear. In this case, the rate equation must incorporate the two local mass-transfer coefficients and the nonconstant slope of the isotherm. The exceptional case occurs if one of the two mass-transfer processes is the rate-limiting step because, then, the overall mass-transfer coefficient equals the local mass-transfer coefficient of the same phase and the constant overall mass transfer is adequate for the rate expression. Even when one masstransfer process is dominant, using the rate equation incorporating the two local mass-transfer coefficients will help to identify the dominant mass-transfer process. The analytical solution gives the same fitted breakthrough curves using Kol or Kos. Correspondingly, the ratio of Kol to Kos for the analytical solution is constant (≈57) in Table 2. The reason for this is that the LFLDF and SFLDF models have the same form of analytical solution: the Thomas solution (eq 12). After the models are fitted to the experimental breakthrough curves, both models will have the same solute concentration at a given value of the dimensionless time T ) TL ) TS and dimensionless distance Z ) ZL ) ZS (see eqs 13-16). When the Hiester-Vermeulen assumptions are applied, TL/TS ) ZL/ZS ) c0Kol/(c∞s Kos) ) 1. Therefore, at a given solute concentration, Kol/Kos ) c∞s /c0 ) 57 and the LFLDF and SFLDF model will generate the same fitted breakthrough curves. Because the two analytical solu-
tions yield the same results, no information is obtained to determine the rate-controlling mass-transfer phenomena. For the experimental data analyzed in this work, the models were able to fit the general trend of the observations, but failed to give a precise point-by-point fit. The numerical solution using local mass-transfer coefficients was the most theoretically appropriate model and gave a closer fit to the data than the previous analytical model of Sarfert and Etzel wherein the dimensionless solute concentration in the solid phase and overall masstransfer coefficient were assumed constant. Nevertheless, at a flow rate of 0.2 mL/min the predicted breakthrough curve from the numerical solution was always somewhat above the experimental breakthrough curve (Figure 2). At flow rates of 1 and 5 mL/min, the predicted breakthrough curves initially rose faster, and later more slowly than the experimental breakthrough curves. In both the experimental observations and the numerical solutions, the sudden increase in the concentration at the point of breakthrough was the result of protein that was not captured because the residence time of the liquid in the membrane was too short for protein mass transfer to the binding site. As the flow rate increased, the fraction of protein that was not captured increased, which increased the extent of the sudden initial rise in concentration, and slowed the rate of approach to saturation. The large average pore size (150 µm) of the membrane used to collect the experimental data was the cause of the poor capture of protein. For a membrane with such large pores, a flow rate of 0.0095 mL/min (5% of the lowest experimental flow rate) would have been needed to attain a sharp breakthrough curve. In addition, increasing the membrane pore size generally increases the pore size distribution, an effect not accounted for in the models. A closer fit of the model to the experimental data may have resulted from a system where the membrane pore size was smaller, as is found in current commercial products. 5. Conclusions In this work, mathematical models of mass-transfer limitations in protein separations using ion-exchange membranes were solved numerically and compared to experimental breakthrough curves measured at three different flow rates. The numerical solutions were mathematical models which describe the binding behavior of the protein including convection, diffusion mass transfer, and Langmuir sorption. The numerical solutions were compared to classic analytical solutions using the Hiester-Vermeulen assumption. The analytical solutions gave the same results for the SFLDF and LFLDF models, incorporating constant overall masstransfer coefficients. Conversely, the numerical simulations incorporating constant overall mass-transfer coefficients gave different fittings wherein the SFLDF model fit better to the experimental data than the LFLDF model. The numerical model was extended by incorporating the two local mass-transfer coefficients and the slope of the sorption isotherm. The model extension was important because, except for limiting cases, it is theoretically incorrect to use models incorporating constant overall mass-transfer coefficients. This model is useful in deciding which phase dominates the mass-transfer rate for an ion-exchange membrane system. For the limiting case where the isotherm is linear, the analytical solution, which incorporates a constant
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overall mass-transfer coefficient, can provide a theoretically correct description of protein transport in the membrane. For general cases of the membrane system, the model incorporating the two local mass-transfer coefficients will provide the best analysis tool because the valid region spans over the entire nonlinear isotherm and the overall mass-transfer coefficients do not have to be constant. Acknowledgment Funding for this work was provided by the National Science Foundation (BES-9631962), the German Academic Exchange Service (Deutscher Akademischer Austauschdienst), and the College of Agricultural and Life Sciences. Symbols Latin Letters am ) ratio of surface area to solid volume, cm-1 ()6/dp(1 - )) c ) solute concentration in the liquid phase, M c* ) solute concentration in the liquid phase in equilibrium with cs, M ci ) interfacial solute concentration in the liquid phase, M cc ) output of CSTR, M cout ) system output, M cl ) membrane capacity based on the solid volume, M cs ) solute concentration in the solid phase based on the solid volume, M c/s ) solute concentration in the solid phase in equilibrium with c, M c∞s ) solute concentration in the solid phase in equilibrium with c0, M ()clc0/(Kd + c0)) csi ) interfacial solute concentration in the solid phase, M c0 ) feed solute concentration, M C ) dimensionless solute concentration in the liquid phase ()c/c0) Cs ) dimensionless solute concentration in the solid phase ()cs/c∞s ) D ) diffusion coefficient of the solute, cm2/s dp ) average pore size of the membrane, cm kl ) local liquid-phase mass-transfer coefficient, cm/s ks ) local solid-phase mass-transfer coefficient, cm/s Kol ) overall liquid-phase mass transfer coefficient, cm/s Kos ) overall solid-phase mass transfer coefficient, cm/s Kd ) desorption equilibrium constant, M L ) length of membrane, cm m ) dimensionless saturation capacity ()(1 - )c∞s /(c0)) m′, m′′ ) dimensionless slope of sorption isotherm nL ) dimensionless number of transfer units in the LFLDF model ()(1 - )KolamL/v) nS ) dimensionless number of transfer units in the SFLDF model ()(1 - )KosamL/v) Pe ) axial dispersion Peclet number()vL/ξ) Q ) volumeric flow rate, mL/s or mL/min r ) dimensionless separation factor ()1 + c0/Kd) t ) time, s tdelay ) delay time due to the PFR volume, s T ) dimensionless time TL ) dimensionless time in the LFLDF model ()(τ - ζ)nLr/m(r + Cs - rCs)) TS ) dimensionless time in the SFLDF model ()(τ - ζ)nSr/ (1 - )(1 - C + rC)) v ) interstitial liquid velocity, cm/s Vsys ) system volume, mL VCSTR ) CSTR volume, mL VPFR ) PFR volume, mL Vvoid ) void volume, mL
z ) spatial coordinate, cm Z ) dimensionless spatial coordinate ZL ) dimensionless spatial coordinate in the LFLDF model ()ζnLr/(r + Cs - rCs)) ZS ) dimensionless spatial coordinate in the SFLDF model ()ζnSmr/(1 - )(1 - C + rC)) Greek Letters ) membrane void fraction ζ ) dimensionless spatial coordinate ()z/L) ξ ) axial dispersion coefficient, cm2/s τ ) dimensionless time ()vt/L)
Literature Cited (1) Roper, D. K.; Lightfoot, E. N. Separation of Biomolecules Using Adsorptive Membranes. J. Chromatogr. A 1995, 702, 3. (2) Shiosaki, A.; Goto, M.; Hirose T. Frontal Analysis of Protein Adsorption on a Membrane Adsorber. J. Chromatogr. A 1994, 679, 1. (3) Suen, S.-Y.; Etzel, M. R. A Mathematical Analysis of Affinity Membrane Bioseparations. Chem. Eng. Sci. 1992, 47, 1355. (4) Tho¨mmes, J.; Kula, M. R. Membrane ChromatographysAn Integrative Concept in the Downstream Processing of Proteins. Biotechnol. Prog. 1995, 11, 357. (5) Tejeda-Mansir, A.; Juvera, J. M.; Magan˜a, I.; Guzman, R. Design of Affinity Membrane Chromatographic Columns. Bioprocess Eng. 1998, 19, 115. (6) Gebauer, K. H.; Tho¨mmes, J.; Kula, M. R. Breakthrough Performance of High Capacity Membrane Adsorbers in Protein Chromatography. Chem. Eng. Sci. 1997, 52, 405. (7) Sarfert, F. T.; Etzel, M. R. Mass Transfer Limitations in Protein Separations Using Ion-Exchange Membranes. J. Chromatogr. A 1997, 764, 3. (8) Hiester, N. K.; Vermeulen, T. Saturation Performance of Ion-Exchange and Adsorption Columns. Chem. Eng. Prog. 1952, 48, 505. (9) Lightfoot, E. N.; Sanchez-Palma, R. J.; Edwards, D. O. Chromatography and Allied Fixed-Bed Separations Processes. In New Chemical Engineering Separation Techniques; Schoen, H. M., Ed.; Wiley Interscience: New York, 1962; Chapter 2. (10) Hines, A. L.; Maddox, R. N. Mass Transfer. Fundamentals and Applications; Prentice Hall: Englewood Cliffs, NJ, 1985. (11) Danckwerts, P. V. Continuous Flow Systems. Distribution of Residence Times. Chem. Eng. Sci. 1953, 2, 1. (12) Raths, K.-R. Flow and Diffusion in Adsorptive Membrane Systems. M.S. Thesis, University of Wisconsin, Madison, WI, 1992. (13) Suen, S.-Y.; Etzel, M. R. Sorption Kinetics and Breakthrough Curves for Pepsin and Chymosin Using Pepstatin A Affinity Membranes. J. Chromatogr. A 1994, 686, 179. (14) Lightfoot, E. N.; Lenhoff, A. M.; Rodriguez, R. L. Use of Moments to Characterize Mass Transport in Steady Flows of Arbitrary Complexity. Chem. Eng. Sci. 1982, 37, 954. (15) Thomas, H. C. Heterogeneous Ion Exchange in a Flowing System. J. Am. Chem. Soc. 1944, 66, 1664. (16) Thomas, H. C. Chromatography: a Problem in Kinetics. Ann. N. Y. Acad. Sci. 1948, 49, 161. (17) Sarfert, F. T. Mass Transfer Limitations in Protein Separations Using Ion Exchange Membranes. M.S. Thesis, University of Wisconsin, Madison, WI, 1995. (18) Stewart, W. E.; Caracotsios, M.; Sørensen, J. P. General Regression Software Package for Non-Linear Parameter Estimation, GREG.; University of Wisconsin: Madison, WI, 1994. (19) Caracotsios, M.; Stewart, W. E. Sensitivity Analysis of Initial-Boundary-Value Problems with Mixed PDEs and Algebraic Equations. Comput. Chem. Eng. 1995, 19, 1019. (20) Arnold, F. H.; Blanch, H. W.; Wilke, C. R. Analysis of Affinity Separations. Part I: Predicting the Performance of Affinity Adsorbers. Chem. Eng. J. 1985, 30, B9.
Received for review February 3, 1999 Revised manuscript received July 7, 1999 Accepted August 4, 1999 IE990084O
