1. Introduction
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1. Introduction Affinity chromatography has been an industrial standard method used to economically purify highvalue biomolecules present at very low concentrations in complex biological fluids because of its simplicity and high degree of specificity [1]. Conventionally, affinity purification is carried out in columns packed with porous beads to which the affinity ligand is immobilized. However, the compressibility, slow mass transfer and adsorption kinetics of the traditional particle media for column chromatography have significantly limited their application in purification of bioproducts on a large scale [2]. In order to overcome these drawbacks, affinitymembrane chromatography, using porous structures with flat sheet and hollow fiber forms, introduces a different approach to exploit the biospecific interactions between a ligate and a ligand for biomedical applications. Affinity membranes are operated in convective mode, which can significantly reduce diffusion and pressure drop limitations commonly encountered in column chromatography [3] and [4]. In our previous work, polylysineattached affinity membranes were prepared and were used to adsorb bilirubin from the bilirubin–phosphate solution and bilirubin–albumin solution [5], [6], [7] and [8]. The results showed that the polylysineattached affinity membranes have a higher capacity for bilirubin. Currently, the affinitymembrane chromatography is well accepted as a powerful technology for separation and purification of biomolecules.
The optimization and scaleup of affinitymembrane operations in the recovery, separation and purification of biochemical components is of major industrial importance [9]. The development of mathematical models to describe affinitymembrane processes and the use of these models in computer programs to predict membrane performance are an engineering approach that can help to attain these bioprocess engineering tasks successfully [1] and [10]. However, in literature, very few models have been developed so far to describe the observed breakthrough behavior. The Thomas model [11], which involves Langmuir reaction kinetics as the ratelimiting step, has been used frequently to describe the process of affinity chromatography in a packed column. In membrane column affinity chromatography, the Thomas solution can be used when working with axial Peclet numbers greater than 40 and radial Peclet numbers smaller than 0.04, and good agreement is observed between experiments and theory, particularly in the early portion of the breakthrough curves. TejedaMansir et al. [12] developed a method, which was based on the analytical solution of the Thomas model for frontal analysis in membrane column adsorption, for affinitymembrane column design. This method permits to find the operating conditions to reach 93.5% of the column capacity as operating capacity, using a sharpness restriction for the system breakthrough curve. Sridhar [13] proposed a model including convection, diffusion and rate kinetics, and used the model to analyze the design and operation of affinitymembrane
bioseparations. The results obtained from model simulation showed that the breakthrough of the solute is significantly influenced by Peclet number, feed protein concentration, ligand number, Damkőhler number, membrane thickness and flow rate. Suen and Etzel [14] extended the affinitymembrane chromatography model to study extracolumn effects in membrane systems. Their results showed that these effects are relevant for a proper description of performance in these types of separations. To evaluate nonlinear chromatographic performance, a multiplate mathematical model for affinitymembrane was proposed through frontal analysis by Hao and Wang [15]. The main advantage of this model is that the parameters can be easily calculated from experimental data. Due to good correlation between it and the equilibriumdispersive or Thomas models, this model can be used to obtain information about bandbroadening effects such as dispersion and sorption effects. The equilibrium adsorption of a solute on an affinity matrix based on the above models is often described simply by the Langmuir equation, with the assumption that single site, homogeneous interaction occurs between the solute and the ligand, and that nonspecific interactions promoted by the support are absent [9] and [16]. However, homogeneity is seldom true in practical cases, which has been shown in some of the recent works on affinity adsorption, ionexchange adsorption and adsorption to polymer surface [17]. Recently, we developed a new affinitymembrane model based
upon Freundlich equation [18]. The model was simulated by the experimental data of bilirubin adsorbed on affinitymembrane. The experimental and modeling results are in good agreement. This model can be used to describe the affinitymembrane processes in which the adsorption mechanism between ligand and ligate is Freundlich adsorption. The key performance criteria for affinitymembrane processes are breakthrough curve sharpness and residence time at the adsorption stage [1]. In fact, not only the feed solute concentration, ligand number, Peclet number, membrane thickness and flow rate but also the nonuniformites in membrane thickness [19] and [20] and poresize [20] may have significant effects on the breakthrough curve sharpness and residence time. In this paper, variation analysis of the membrane thickness and pore size is proposed to predict the effects of nonuniformites on membrane performance by using our affinitymembrane model [18].
2. Theoretical background The assumptions of the Langmuir model are that surface is homogeneous, and adsorption energy is constant over all sites. In fact, as the adsorbent surface is often heterogeneous and/or interaction among adsorbed molecules cannot be neglected, the heat of adsorption varies with the surface coverage.
When adsorption between ligand and ligate can be described by kinetics equation of Freundlich, we propose the following model to predict the breakthrough behavior of affinitymembrane [18]. A mass balance over a section of the membrane gives the following continuity equation:
(1)
The kinetics equation of Freundlich is
(2) where cl is the total adsorbed capacity, c and cs the solute concentration in the fluid phase and concentration of solute–ligand complex in the solid phase, the void porosity, D the axial diffusion coefficient, v the flow velocity of the solute through the membrane, and ka0 and kd0 are association and dissociation rate constants of Freundlich adsorption equation, respectively.
Initial conditions:
c=0 at z≥0,t=0
(3)
cs=0 at z≥0,t=0
(4)
Boundary conditions:
(5)
(6)
For convenience, Eqs. (1), (2), (3), (4), (5) and (6) can be converted to dimensionless groups as follows:
(7)
(8)
Initial conditions:
C=0 at ζ≥0,τ=0
(9)
Cs=0 at ζ≥0,τ=0
(10)
Boundary conditions:
(11)
(12)
The definitions and physical meanings of the dimensionless parameters in Eqs. (7) and (8) are summarized in Table 1.
Table 1.
Definitions and physical meanings of dimensionless parameters
Dimensionless parameters and
their
Physical meanings
definitions τ = vt/L
Dimensionless time
ζ = z/L
Dimensionless spatial variable
C = c/c0
Dimensionless ligate concentration in the fluid phase
Cs = cs/cl
Dimensionless concentration of ligand–ligate complex in the solid phase
Pe = vL/D
Axial Peclet number
m=((1−ε)cl)/εc0
Ratio of total adsorbed capacity to feed solute concentration
n=((1−ε)ka0L)/εv
Dimensionless number of transfer units
r=1+(c0/Kd)
Dimensionless separation factor which determines the maximum ligate loaded onto the membran
We used the finitedifference method to solve the Eqs. (7), (8), (9), (10), (11) and (12) for the model of affinitymembrane performance.
3. Results and discussion The sample is fed continuously into the membrane column during the operation of affinitymembrane column. For a short time, the solute in the feed is taken up almost completely; however, after a while, solute breakthrough occurs and the effluent concentration of solute increases with time. Much of the information of column performance can be evaluated in plots of effluent concentration as a function of time or throughput volume, i.e., breakthrough curves. Fig. 1 shows a typical breakthrough curve for the
adsorption step. Breakthrough time is defined as the time required for the exit solute concentration to reach 10% of the inlet solute concentration (C = 0.1). The breakthrough curve can be used to determine (1) the column capacity used, (2) solute lost in the effluent and (3) the processing time. This is precisely the performance information needed to optimize processing [21]. For quantitative comparison purpose, the following parameters, which include solute recovery efficiency [22], ligand utilization efficiency [22], and thickness of unused membrane [13], are defined.
(7K)
Fig. 1. Breakthrough curve for affinity adsorption.
The values S1, S2, and S3 represent the areas of the shadowed portions in Fig. 1. These values of solute recovery efficiency, ligand utilization efficiency,
thickness of unused membrane and breakthrough time are important in the evaluation of the performance of affinitymembrane columns. For convenience, our affinitymembrane model based upon Freundlich equation [18] is used to investigate the effects of the variations of thickness and poresize on membrane performance under the condition of Pe ≥ 30. Other parameter values used in variation analyses are taken from the literature [18] and are listed in Table 2.
Table 2.
The values of model parameter used in variation analysis
Parameter
Value
m
38.57
n
23.36
r
5.52
b1
0.8
b2
1.5
3.1. Thickness variation model The thickness of affinity membranes not only represents the length of membranepore but also is proportional to the capacity of the membranes. Therefore, it is necessary to discuss the effect of thickness variations on the performance of affinitymembrane columns. We assume that the membrane
thickness increases linearly from a thin region to a thick region [19]. For a case of 10% variation in thickness, the membrane thickness linearly increases from 90 to 110% of the average thickness of L. Fluid flow through a microporous membrane in the laminar region can be expressed by the Hagen–Poiseuille equation [23]:
(13) where ΔP is the pressure drop across the membrane, ra the pore radius, Qv the volumetric flow rate, μ the solution viscosity and L is the membrane thickness.
The flow velocity v through membrane is
(14)
When the pressure drop across the membrane and solution viscosity are kept constant, the flow velocity can be calculated using Eqs. (13) and (14). Eqs. (13) and (14) show that the flow velocity is inversely proportional to the membrane thickness. Furthermore, the capacity of the membranes is proportional to the membrane thickness. Thus, the time required for pores with thickness of L to reach saturation is proportional to L2. The exit concentration in this model is the volumeaveraged value. For a percentage thickness variation, δ, from the mean thickness L, the exit concentration can be calculated as follows:
(15)
The breakthrough curves for affinity membranes with known thickness variation can be obtained by using our affinitymembrane model [18] and carrying out an integration of Eq. (15). Fig. 2 shows the effect of membranethickness variation on the shape of the breakthrough curve. As the percentage thickness variation increases, we can find that (1) the time of total saturation is delayed; (2) the loading capacity at the point of breakthrough is decreased; (3) solute recovery efficiency and ligand utilization efficiency are decreased; (4) the thickness of unused membrane is increased. Therefore, membranethickness variations of affinity membranes are an important factor affecting the overall performance of the membrane when adsorption between ligand and ligate can be described by kinetics equation of Freundlich. For thickness variations less than 3%, the effect of thickness variations is insignificant. This result is consistent with the membranethickness variation analysis based upon Thomas model [19].
(7K)
Fig. 2. The effect of membranethickness variation on the shape of the breakthrough curve.
3.2. Poresize distribution model Poresize variations over the membrane are unavoidable in the manufacturing process. Therefore, it is necessary to discuss the effect of poresize distribution on the performance of affinitymembrane. The flow velocity of the solution through the membrane can also be calculated by Eqs. (13) and (14) by assuming that the pressure drop across the membrane is constant and the membranes have uniform porosity and ligand density. From the Hagen–Poiseuille equation and Eq. (14), we can show that the flow velocity is proportional to
. Therefore, the residence time of fluid through a
membrane with a pore radius ra would be inversely proportional to
. Due to
the hypothesis of uniform ligand density of a membrane, the capacity of the membranes is proportional to the surface area of the membrane pore (2πraL). Thus, the time required for pores with radius ra to reach saturation can be calculated from residence time and membrane capacity [20]. We used normal distribution to describe the poresize variation of microporous membrane and then discussed the effect of poresize distribution on the affinitymembrane system. For membranes of mean pore radius r0 and pore variation expressed in terms of the standard deviation, σ, from the mean value, the probability density function of membrane pores with pore radius ra can be expressed as the following relation [24]:
(16)
The breakthrough curves for affinity membranes with known poresize distribution can then be constructed by integrating the breakthrough for every pore [20]. Fig. 3 shows the effect of poresize distribution on the shape of the breakthrough curve. The breakthrough curves broaden as the poresize distribution increases. A 0.03 standard deviation of poresize will cause a significant broad breakthrough cure. The poresize standard deviation should be less than 0.01 for the effect to be insignificant. This is consistent with the poresize distribution model proposed by Liu [20] based upon Thomas model. As the poresize distribution increases, solute recovery efficiency and ligand utilization efficiency decreases and the thickness of unused membrane increases. So the membranes with a narrow poresize distribution are more likely to be used as affinitymembrane. If poresize of membranes, ra, has a normal distribution with mean pore radius ra and standard deviation σ, 68, 95 and 99% of the membrane pores are within σ, 2σ and 3σ of membrane mean poresize, respectively [24]. For example, with 1.00 μm mean pore radius and 0.03 standard deviation, 95% of the membrane pores are within 6% of the membrane mean poresize. This is a very narrow poresize distribution for phase inversion membranes. Therefore, the poresize of membranes should be strictly controlled during membrane
manufacturing because even small variations will greatly degrade membrane performance.
(7K)
Fig. 3. The effect of membrane poresize distribution on the shape of the breakthrough curve.
4. Conclusions The nonuniformites in membrane thickness and poresize may severely degrade the performance of affinity membranes. The main influences incarnate (1) the time of total saturation is delayed; (2) the loading capacity at the point of breakthrough is decreased; (3) solute recovery efficiency and ligand utilization efficiency are decreased; (4) the thickness of unused membrane is increased. In order to reduce the effect of the variations of thickness and poresize, we should choose the uniform membrane, of which membranethickness variations and poresize distribution are under 3% and 0.01. This variation analysis is carried out by using our affinitymembrane model based on Freundlich equation however, the results are basically consistent with the variation analysis by using Thomas model based on
Langmuir equation. Therefore, the membrane properties (thickness and poresize) should be strictly controlled, no matter the adsorption mechanism of ligand and ligate is Freundlich or Langmuir adsorption. In the scaleup operations of affinitymembrane, the variation effects on membrane performance can be reduced by increasing the amount of membrane disks to average out the dispersion.
5. Nomenclature b1 constant of Freundlich adsorption equation
b2 constant of Freundlich adsorption equation
c solute concentration in the fluid phase (mol L−1)
c0 feed solute concentration in the fluid phase (mol L−1)
cs solute concentration in the solid phase (mol L−1)
C dimensionless feed concentration
Cs
dimensionless concentration of solute–ligand complex in the solid phase
D axial diffusion coefficient (cm2 s−1)
ka0 association rate constant of Freundlich adsorption equation (s−1)
Kd dissociation equilibrium constant of Freundlich adsorption equation (mol L−1)
kd0 dissociation rate constant of Freundlich adsorption equation (mol L−1 s−1)
L membrane thickness (cm)
m ratio of maximum concentration of solute adsorbed to feed solute concentration
n dimensionless number of transfer units ΔP pressure drop across the membrane (dyne cm−2)
Pe
axial Pelect number
Qv volumetric flow rate (cm3 s−1)
r dimensionless separation factor
ra membrane pore radius (cm)
r0 membrane mean pore radius
t time (s)
v flow velocity (cm s−1)
z axial distance along membrane (cm)
Greek letters δ percentage of thickness variation
void porosity of membrane
μ
viscosity of feed solution (g cm−1 s−1)
σ standard deviation
τ dimensionless time
ζ dimensionless spatial variable
Acknowledgements We are deeply indebted to the Natural Science Foundation of Fujian Province (No. C0510005) and the National Nature Science Foundation of China (No. 29776036) for supporting this research.
References [1] R.M. Montesinos, A. TejedaMansir, R. Guzman, J. Ortega and W.E. Schiesser, Sep. Purif. Technol. 42 (2005), p. 75. SummaryPlus  Full Text + Links  PDF (380 K)
[2] L. Yang, W.W. Hsiao and P. Chen, J. Membr. Sci. 197 (2002), p. 185. SummaryPlus  Full Text + Links  PDF (344 K) [3] W. Guo and E. Ruckenstein, J. Membr. Sci. 211 (2003), p. 101. SummaryPlus  Full Text + Links  PDF (186 K) [4] D.K. Roper and E.N. Lightfoot, J. Chromatogr. A 702 (1995), p. 3. Abstract  Abstract + References  PDF (1933 K) [5] W. Shi, F.B. Zhang and G.L. Zhang, J. Chromatogr. B 819 (2005), p. 301. SummaryPlus  Full Text + Links  PDF (208 K) [6] W. Shi, F.B. Zhang, G.L. Zhang, D.T. Ge and Q.Q. Zhang, Polym. Int. 54 (2005), p. 790. AbstractFLUIDEX  AbstractCompendex
 Full Text via
CrossRef [7] W. Shi, F.B. Zhang, G.L. Zhang, L.Q. Jiang, Y.J. Zhao and S.L. Wang,
Mol. Simulat. 29 (2003), p. 787. Full Text via CrossRef [8] W. Shi, F.B. Zhang and G.L. Zhang, Chin. Chem. Lett. 16 (2005), p. 1085. [9] A. TejedaMansir, R.M. Montesinos and R. Guzman, J. Biochem. Biophys.
Methods 49 (2001), p. 1. SummaryPlus  Full Text + Links  PDF (217 K) [10] A.I. Liapis, K.K. Unger and G. Steet, Highly Selective Separations in Biotechnology, Chapman and Hall, Glasgow, NZ (1994). [11] H.C. Thomas, J. Am. Chem. Soc. 66 (1944), p. 1664. Full Text via CrossRef
[12] A. TejedaMansir, J.M. Juvera, I. Magana and R. Guzman, Bioprocess
Eng. 19 (1998), p. 115. AbstractElsevier BIOBASE  AbstractEMBASE

Full Text via CrossRef [13] P. Sridhar, Chem. Eng. Technol. 19 (1996), p. 398. AbstractCompendex  AbstractFLUIDEX
 Full Text via CrossRef
[14] S. Suen and M.R. Etzel, J. Chromatogr. A 686 (1994), p. 179. Abstract [15] W.Q. Hao and J.D. Wang, J. Chromatogr. A 1063 (2005), p. 47. SummaryPlus  Full Text + Links  PDF (179 K) [16] A.I. Liapis and M.A. McCoy, J. Chromatogr. A 660 (1994), p. 85. Abstract [17] S. Suen, J. Chem. Technol. Biotechnol. 70 (1997), p. 278. AbstractEMBASE  AbstractCompendex
 Full Text via CrossRef
[18] W. Shi, F.B. Zhang and G.L. Zhang, J. Chromatogr. A 1081 (2005), p. 156. SummaryPlus  Full Text + Links  PDF (161 K) [19] S. Suen and M.R. Etzel, Chem. Eng. Sci. 47 (1992), p. 1355. Abstract [20] H. Liu and J.R. Fried, AIChE J. 40 (1994), p. 40. AbstractCompendex
 Full Text via CrossRef
[21] F.H. Arnold, H.W. Blanch and C.R. Wilke, Chem. Eng. J. 30 (1985), p. B9. Abstract [22] Y. BoLun, G. Montonobu and G. Shigeo, Colloids Surf. 37 (1989), p. 369.
[23] R.B. Bird, W.F. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York (1960). [24] J.S. Milton and J.C. Arnold, Probability and Statistics in the Engineering and Computing Sciences, McGrawHill, New York (1986).
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