Mass transfer kinetics and breakthrough and elution curves for bovine serum albumin using cibacron blue cellulose membranes

Journal of Chromatography A, 1114 (2006) 123–131

Mass transfer kinetics and breakthrough and elution curves for bovine serum albumin using cibacron blue cellulose membranes Weiqiang Hao a , Junde Wang b,∗ , Xiangmin Zhang a a

b

Department of Chemistry, Fudan University, Shanghai 200433, China Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China

Received 1 December 2005; received in revised form 15 February 2006; accepted 17 February 2006 Available online 20 March 2006

Abstract The mass transfer of bovine serum albumin onto a stack of cibacron blue cellulose membranes in the loading and elution stages were studied. The breakthrough curves obtained in the loading stage were fitted to the pore and the lumped kinetic (LK) models, respectively. Then experimental data obtained in the elution stage were described by using the LK model in which the kinetic equation, the initial and the boundary conditions were rewritten according to the operation. For the breakthrough curves it was found that the contribution of the sorption kinetics to band broadening was significant whereas that of axial dispersion was negligible. In contrast, both of these contributions were significant to the profile of the elution curve. By studying the mass transfer kinetics in the elution stage, information about the influence of the module geometry on the performance of affinity membrane separations may be obtained. © 2006 Elsevier B.V. All rights reserved. Keywords: Membrane affinity chromatography; Mass transfer; Pore model; Lumped kinetic model

1. Introduction Recently membrane affinity chromatography (MAC) has become a promising technique for large-scale bioseparations [1–5]. To effect separation, specific ligands are grafted onto the membranes and then these resulting matrices may be used for the isolation and purification of biomolecules. Compared to other bioseparations that are carried out in packed bed columns, MAC can operate at high flow rate and require only low pressure. This translates in many applications to simple equipment and safe operations. Kinetics studies of mass transfer in chromatography may help to understand the separation mechanisms. In the literature, it is usually assumed that the mass transfer processes consist of axial dispersion and others leading to mass transport resistances in and around the particles for packed column chromatography or the pores for membrane chromatography. In order to accurately account for their contributions to band broadening the so-called general rate model has been suggested [6,7]. However,

∗

Corresponding author. Tel.: +86 41184379510; fax: +86 41184379517. E-mail address: [email protected] (J. Wang).

0021-9673/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2006.02.047

it is not suited for practical use, because the solution of the equations of this model requires sophisticated numerical algorithms and significant computation times. For these reasons some models simplifying the transport processes, such as the pore model and the lumped kinetic (LK) model, have been proposed and frequently applied successfully [8–19]. These models may be applied for kinetic studies of mass transfer in MAC. However, some features of this chromatographic technique should be noticed. Firstly, MAC operates in the nonlinear range of adsorption isotherm. Thus, the conventional chromatographic theories, which assume linear isotherms, are not suited for it. Moreover, many studies also show that the sorption kinetics in affinity membrane separations are usually slow [20,21]. Secondly, some recent reviews have pointed out that the design of appropriate modules to house the membranes is one of the important opportunities for more applied development of affinity membranes [1,3]. Because factors such as the geometry of the membrane module affect the flow regime in the separation, information about such effects may be obtained by studying the contribution of axial dispersion to band broadening (also see Eq. (24)). However, most kinetic studies of the mass transfer in MAC are usually carried out so far by neglecting the effects of axial dispersion at the beginning, because its contribution to the

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Then the LK model is rewritten according to the operation in the elution stage of MAC so as to account for experimental data obtained in this stage. To describe these chromatographic models, it is convenient to introduce the dimensionless parameters. The following dimensionless variables will be used [9]: y=

C , C0

τ=

tu , LεT

yp =

Cp , C0

Q=

q , C0

x=

λ=

Cl , C0

ψ=

Kd , C0

Pe =

kap LεT , u kf,L LεT , StL = u

ka LεT , u ks LεT St  = u

St =

Sta =

St  =

Z , L uL , DL εT

kf LεT , u (1)

All symbols used in Eq. (1) are explained in Section 6. Fig. 1. Diagram for the operation procedure of MAC.

2.1. The pore model profile of the breakthrough curve is usually found to be negligible [22–25]. Therefore, the influence of the geometry of the membrane modules on the performance of affinity membrane separations cannot yet be well understood in these studies. In practice the operation procedure of MAC usually consists of loading, washing and elution stages as depicted in Fig. 1 [26–28]. Firstly, the sample solution prepared in a loading buffer is pumped through the membranes until a plateau of outlet concentration is reached. Then a washing buffer, which is usually the same as the loading buffer, is used to remove the molecules that are not adsorbed by the specific ligands. Finally, the washing buffer is replaced by a strong elution buffer and the bound target biomolecules is consequently eluted from the membranes. In the literature, many kinetic studies for MAC were made by investigating the influence of experimental factors on the profile of the breakthrough curve obtained in the loading and washing stages [5,22–25]. In contrast, the kinetic studies for the elution stage turn out to be much fewer. By considering that the profile of the elution curve is more sensitive to some contributions of the mass transfer processes than the breakthrough curve [29], we still expect to obtain some useful information by studying the mass transfer kinetics in the elution stage as well as the loading stage. In this paper, a stack of cellulose membranes immobilized with cibacron blue F3GA were chosen as the stationary phase and bovine serum albumin (BSA) as the sample. The pore and the LK models of chromatography were used to describe experimental breakthrough curves. The elution curves were fitted to the LK model in which the kinetic equation, the initial and the boundary conditions were rewritten according to the operation in the elution stage. The influence of flow rate on the profile of both the breakthrough and elution curves was discussed. 2. Theoretical The pore and the LK models have been frequently used for the studies of mass transfer in chromatography [7–12]. We first present here briefly the characteristic features of these models used in this study to describe experimental breakthrough curves.

In the pore model there are two mass balance equations for the solute, one in the mobile phase percolating through in the throughout pores of the membranes, the other inside the mesopores [7–9,27]. The latter involves the sorption process of the solute between the stationary and mobile phases. ∂y εT ∂y 1 ∂ 2 y 1 − εe − St(y − y¯ p ) + = ∂τ εe ∂x Pe ∂x2 εe εp

¯ ∂y¯ p ∂Q + (1 − εp ) = St(y − y¯ p ) ∂τ ∂τ

(2) (3)

The sorption kinetic equation is given by [6] ¯ ∂Q ¯ ∗ − Q) ¯ (4) = Sta (Q ∂τ ¯ ∗ denotes the equilibrium average concentration where symbol Q in the stationary phase and it is accounted for in this study by the simple Langmuir model, ¯∗= Q

λ¯yp 1/ψ + y¯ p

(5)

The initial and boundary conditions for the pore model are [8] y(0, x) = 0,

y¯ p (0, x) = 0,

¯ Q(0, x) = 0,

for 0 < x < 1

1 − y(τ, 0) = − ∂y(τ, 1) =0 ∂x

εe 1 ∂y(τ, 0) εT Pe ∂x

(6)

for τ > 0; x = 0

for τ > 0; x = 1

(7) (8)

2.2. The LK model for the loading stage Compared to the pore model, the LK model turns out to be simple and has been widely used in the studies of mass transfer

W. Hao et al. / J. Chromatogr. A 1114 (2006) 123–131

in both column and membrane chromatography [10–25,30,31]. This model uses the intrinsic axial dispersion coefficient and lumps all the remaining mass transport resistances into an overall mass transport coefficient. The mass balance and kinetic equations of this model are usually given by [10] ∂y 1 − εT ∂Q ∂y 1 ∂2 y + + = ∂τ εT ∂τ ∂x Pe ∂x2 ∂Q = St  (Q∗ − Q) ∂τ

(9) (10)

where the Stanton number St of the LK model accounts for the contributions of all the mass transfer processes contributing to band broadening, except for axial dispersion [10]. The initial and boundary conditions are [10,25] y(0, x) = 0,

Q(0, x) = 0

1 − y(τ, 0) = − ∂y(τ, 1) =0 ∂x

1 ∂y(τ, 0) Pe ∂x

for 0 < x < 1

(11)

for τ > 0; x = 0

(12)

for τ > 0; x = 1

(13)

The simplification of the LK model may give the transport model [13,14]. In this model, the contribution of axial dispersion is considered negligible, i.e., Pe = ∞ in Eqs. (9) and (12), and all the contributions of the mass transfer processes to band broadening are accounted for by one parameter StL , which replaces St in Eq. (10). This model is also used in this study as a reference to the LK model. 2.3. The LK model for the elution stage Although there have been many mathematical models applied to describe the elution curves obtained in column chromatography, it should be noted that the operation to obtain those curves is somewhat different from that of MAC (see Fig. 1). In the former, the chromatographic system is usually first equilibrated with the mobile phase and then the sample solution is injected into it. In the latter, however, the protein has already been adsorbed on the stationary phase before the elution begins and no more protein will be introduced into the system in the elution stage. This difference in operation will lead to different expressions of initial and boundary conditions in the mathematical model. Thus, the fitting of experimental data to some known formula for the peak profile deduced according to the conventional operation conditions, such as Gaussian, will not be applicable. Moreover, before the elution stage begins, the system is filled with the washing buffer. This buffer usually has little effect on the interaction between the ligand and the protein. When the washing buffer is replaced by the elution buffer, this interaction will be interrupted and consequently the protein desorbed from the solids. In most cases, the equilibrium solid concentration of the protein in the elution buffer is always found to be very low [20,21,26–28]. This significant difference in the adsorption manners of the solute in the washing and elution buffers also makes it difficult to choose an appropriate adsorption isotherm for the sorption process. Therefore, the chromatographic models presented in the liter-

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ature cannot be directly used for the elution curves obtained in MAC unless some modifications of them are made. In this study the LK model mentioned earlier is rewritten so as to describe experimental elution curves obtained in the elution stage. For simplicity it is assumed that the flow regime is the plug flow mode as taken in most chromatographic models and the replacement of the washing buffer by elution buffer will have a little affect on the axial dispersion of mass transfer, considering that both of the buffers are primarily composed of water. According to these assumptions the mass balance equation is also written as Eq. (9). In the formulation of the kinetic equation, two assumptions are used. First, the equilibrium concentration of the solute on the membranes in the elution buffer, q* , is approximately taken as zero because it is always found to be very low in practice. Second, it is assumed that the desorption of the protein from the solids occurs only when the elution buffer contacts it due to significant different adsorption manners of the solute in the washing and the elution buffers. With these assumptions, the kinetic equation may be written as: ⎧ ∂q ⎪ ⎨ =0 for t < ZεT /u ∂t (14) ⎪ ⎩ ∂q = ks (q∗ − q) = −ks q for t ≥ ZεT /u ∂t where ks is defined as the rate coefficient of mass transfer and ZεT /u is the time when the elution buffer reaches the adsorption site on the  membranes. By introducing the Heaviside function, 0 τ<0 H(τ) = , Eq. (14) is equivalent to 1 τ≥0    ∂q ZεT = −ks q H(t) − H −t (15-a) ∂t u The dimensionless form of Eq. (15-a) is      LεT LεT LεT ∂Q = −St  Q H τ −H x− τ ∂τ u u u = −St  Q[H(τ) − H(x − τ)]

(15-b)

In this paper, the adsorbed protein is assumed to distribute evenly on the membranes with the concentration q¯ ini at the beginning of the elution stage. Thus, the initial and boundary conditions may be written as follows and similar to Eqs. (11)–(13): y(0, x) = 0, y(τ, 0) =

1 ∂y(τ, 0) Pe ∂x

∂y(τ, 1) =0 ∂x

q¯ ini ¯ ini =Q C0

for 0 < x < 1 (16)

for τ > 0; x = 0

(17)

Q(0, x) =

for τ > 0; x = 1

(18)

The value of q¯ ini may be calculated by dividing the amount of eluted protein (i.e., the area under the elution curve) by the membrane solid volume [21]. In order to understand better the LK model here for the elution stage, a special case of it is discussed. In this case, the

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contribution of axial dispersion to band broadening is ignored, which is similar to the simplification of the LK model into the transport model. By substituting Eq. (15-b) into Eq. (9) and letting Pe = ∞, we obtain ∂y ∂y 1 − εT  + = St Q[H(τ) − H(x − τ)] ∂τ ∂x εT

(19)

The analytical solution of the above hyperbolic equation may be obtained by analyzing the characteristics [32] (see Appendix A): ⎧ (τ ≤ x) ⎨0 y(τ, x) = 1 − εT  (20) ¯ ini x exp[−St  (τ − x)] (τ > x) ⎩ St Q εT When x is equal to 1 Eq. (20) gives the outlet concentration ⎧ (τ ≤ 1) ⎨0 y(τ, 1) = 1 − εT  ¯ ini exp[−St  (τ − 1)] (τ > 1) ⎩ St Q εT (21-a) or its dimension form   ⎧ LεT ⎪ ⎪ t≤ ⎨0 u      C(t, L) = 1 − εT LεT LεT ⎪ ⎪ ⎩ ks q¯ ini L exp −ks t − t> u u u (21-b)

3. Experimental 3.1. Equipment and materials The equipment and materials are the same as those reported in reference [33].

3.3. Computational Experimental data were fitted to the chromatographic models by using a nonlinear fitting procedure based on Levenberg– Marquardt (LM) algorithm [34,35]. This program was written in house in Visual Basic code. In the fittings, minor adjustment of the retention time of experimental curves was made by shifting the whole set of the data points so as to bring its mass center at the center of the calculated curve [13,14]. In order to reduce computation time, 100 evenly spaced points on the breakthrough curve within the range from 0.05C0 to 0.95C0 were chosen for the fitting procedure. For the elution curve, 100 points were chosen: 50 evenly spaced points before and 49 points after the peak maximum, as well as the peak maximum itself. The sum of residuals, defined in Eq. (22), was then minimized [29]. 1

(Cex − Cth )2 n − np n

χ2 =

i=1

The pore and the LK models were solved numerically by using the method of orthogonal collocation on finite elements [8,9]. The solution of the LK model for the elution stage was computed by using finite difference methods [32]. The element of the Jacobian matrix in LM iterations, F, was computed numerically. The standard error of the parameter was estimated by [36] σi =

 Cii χ2

(23)

where Cii is the diagonal element of the variance–covariance matrix, C = (F  ⊗ F )−1 . The fitting procedure was used several times on each curve with different initial guesses to assure that false minima were not obtained.

3.2. Procedures More details about the procedures were also reported in reference [33]. The thickness of the membranes was 0.324 cm. The total porosity was calculated as 0.77 by comparing the weights of the soaked and dried membranes with water as the mobile phase [26]. The external porosity was estimated at 0.3. The loading and washing buffers were both 0.2 M NaCl–0.05 M NaAc buffer (pH 4.0). The elution buffer was 0.2 M KSCN–0.05 M phosphate buffer (pH 8.0). In the loading stage 1.0 mg/mL BSA solution flowed through the membranes. Then the washing buffer removed the unadsorbed solutes. Finally the elution buffer was used to desorb the bound protein from the membranes. To investigate the influence of elution flow rate on the profile of the elution curve, the volumes of loading and washing buffers, as well as flow rates in the loading and washing stages, were fixed so that the amount of adsorbed protein on the membranes could be kept at the same values as possible. Every breakthrough or elution curve was determined thrice under the same condition to assure that no significant difference was found.

(22)

Fig. 2. Langmuir isotherm of the phase equilibrium of BSA.

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Table 1 Estimation of the coefficients of mass transfer in the loading stage u (cm/min)

LK model St

0.14 0.28 0.41 0.54 0.67 0.80 a b

0.319 0.272 0.285 0.279 0.271 0.289

Pe ± ± ± ± ± ±

0.004 0.005 0.004 0.004 0.004 0.004

Pore modela

Transport model

5.7 15.0 19.8 22.4 41.3 36.0

± ± ± ± ± ±

1.6 15.7 20.8 30.4 100.9 72.6

χ2 × 104

StL

2.1 3.9 3.0 3.4 3.7 3.7

0.316 0.273 0.287 0.278 0.274 0.287

± ± ± ± ± ±

0.003 (0.315)b 0.004 (0.273)b 0.003 (0.289)b 0.004 (0.280)b 0.003 (0.275)b 0.004 (0.288)b

χ2 × 104

St

2.8 3.8 3.1 3.4 3.6 3.7

10.9 12.0 10.0 11.0 12.0 14.8

χ2 × 104

Sta ± ± ± ± ± ±

4.0 10.9 5.7 8.1 10.4 15.3

0.123 0.106 0.112 0.109 0.107 0.113

± ± ± ± ± ±

0.001 0.002 0.002 0.002 0.002 0.002

2.0 3.8 2.9 3.3 3.6 3.6

The value of Pe is set to be 1000 in the fittings. The values in the parentheses are calculated according to the relationship between the transport and the multi-plate model [33].

4. Results and discussion 4.1. Adsorption isotherm Fig. 2 shows equilibrium isotherm data of BSA on cibacron blue cellulose membranes in the loading buffer. These data are well accounted for by the simple Langmuir isotherm [33]. The isotherm parameters obtained were Cl = 17.38 mg/mL and Kd = 5.17 × 10−3 mg/mL. The adsorption of BSA on the membranes in the elution buffer was also investigated in this study. It was found that little protein was absorbed on the solids. This result validates the assumption made in Section 2.3 that the solid concentration of the solute at equilibrium in the elution stage may be approximately considered as zero.

4.2. Mass transfer kinetics in the loading stage Experimental breakthrough curves obtained at different flow rates were fitted to the pore and the LK models, respectively. The best values of the coefficients of mass transfer for the fittings are presented in Table 1. As expected, the pore model may give a better description to experimental data than the LK model. But the differences between their predicted values are found to be very small. Therefore, the pore model and the LK model may be basically taken equivalent in this study. The Stanton number St of the LK model may be considered as a lumped coefficient, which accounts for the contributions of the mass transfer processes including fluid-to-pore mass transfer, intrapore diffusion, and the adsorption/desorption [10]. In Fig. 3, the differences between theoretical outputs for the best fitting calculated by using the LK and the transport models are also found to be very small. This good agreement is also confirmed by close values of the Stanton numbers (St and StL ) presented in Table 1. These results show that the influence of axial dispersion on the profile of the breakthrough curve may be ignored in this study. This conclusion may also be seen from the large relative estimated error for the Peclet number obtained by the fitting procedure, which turns out to vary in a wide range, from about 30% to 200%. This uncertainty shows some difficulties in obtaining accurate information about axial dispersion from the profile of the breakthrough curve. In Table 1, the values of StL obtained by using the nonlinear fitting procedure are also compared to those calculated according to the relationship

Fig. 3. Fitting of experimental breakthrough curve (symbol) to the LK model (solid line) and the transport model (dashed line). Note that the theoretical curves almost overlap. The value of u was 0.41 cm/min.

between the transport model and the multi-plate (MP) model proposed in our previous studies [33,37–39]. These values are found to be very close and it further validates the applicability of the MP model in practice. Fig. 4 shows the fittings of the pore model to experimental breakthrough curves. In these fittings the value of Pe in Eq. (2) is set to be 1000 so that the contribution of axial dispersion to band broadening is negligible. Moreover, the sorption process of the solute on the stationary phase is not simply accounted for by the adsorption isotherm as done in previous studies [8,9]. In extensive preliminary calculations, not illustrated below, we found that this simplification would lead to a significant discrepancy between the experimental and predicted values. Therefore, the sorption kinetics in this study cannot be considered infinitely fast, and then the expression for the sorption process is written in the form of Eq. (4) as suggested by Kucera [6]. The best fitting results for the pore model are also presented in Table 1. From this table it may be clearly seen that the sorption kinetics have significant effect on the profile of the breakthrough curve, for the estimated error of Sta turns out to be very small. In contrast, the estimated error for St, which is usually assumed to account for the contribution of the mass transfer process to band broadening including fluid-to-pore mass transfer and intrapore diffusion [8],

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Fig. 4. Fitting of experimental breakthrough curve (symbol) to the pore model: solid line, calculated with St = 10.0 and Sta = 0.112 for the best fitting; dashed line, calculated with St = 1.0 and Sta = 0.112; dotted line, calculated with St = 1000 and Sta = 0.112. The inset shows their differences for the initial region of the curve. The value of u was 0.41 cm/min.

is somewhat large but less than those for the Peclet number. This result may be due to its limited effect on the profile of the curve. Fig. 4 also shows the theoretical curves calculated with a much smaller and a much larger value of St compared to the best one for the fitting. When St is much smaller, the calculated curve is distorted and theoretical values deviates from experimental ones significantly. When St is much larger, the profile of the theoretical curve calculated by using the pore model becomes closer to that of the curve calculated by using the transport model. For the best parameters obtained by the fitting procedure, the calculated curve is close to the one calculated by using a much larger value of St. The difference between them only lies in the initial region of the profile. This result shows that the contribution of the external and internal mass transfer processes to band broadening is not as significant as that of the sorption process and explains why the fitting procedure gives a lager estimated error for this parameter than Sta . Finally, results presented in Table 1 also show that flow rate has little effect on the values of the coefficients of mass transfer for these models obtained by comparing theoretical outputs to experimental breakthrough curves. 4.3. Mass transfer kinetics in the elution stage Experimental elution curves obtained at six different elution flow rates are shown in Fig. 5. In this figure the elution peak tails and its height decreases with increasing flow rate. The eluted amount of BSA was calculated by integrating the area under the elution curve. As presented in Table 2, it decreases but varies slightly with increasing flow rate. As mentioned earlier the chromatographic models presented in the literature cannot be used directly for kinetic studies in the elution stage of MAC due to the differences in the operation. Therefore, we rewrote the LK model to account for experimental data. So far we have not yet got the analytical solution of

Fig. 5. Experimental elution curves obtained at different values of u: (a) 0.06 cm/min; (b) 0.12 cm/min; (c) 0.17 cm/min; (d) 0.22 cm/min; (e) 0.27 cm/min and (f) 0.33 cm/min.

this model. However, by dropping the axial dispersion term in the mass balance equation, its analytical solution may be easily obtained as Eq. (21-b). Fig. 6 shows some theoretical curves calculated by using this equation. The front parts of the profiles are steep and the rear parts are diffuse. These profiles are found to be very similar to those calculated by using the ideal chromatographic model with a nonlinear Langmuir type isotherm [7]. However, it should be noted that there are some differences between them. The steep front calculated by using the ideal model is due to the curvature of the isotherm, which leads to the association of the migration velocity of the solute with its concentration and a concentration discontinuity or shock tends to build up [7]. Therefore, the maximum of outlet concentration and its corresponding retention time are determined by the isotherm. In contrast, for the elution stage, the profile is formed by the sweep of the elution buffer on the adsorption site and the desorption of the bound protein from the membranes. Due to high elution capability of the elution buffer, the partition of the desorbed protein between stationary and mobile phases will be negligible. Therefore, the concentration of the protein in the mobile phase accumulates at the front of the elution buffer, and the maximum of outlet concentration will always appear at the hold-up time, LεT /u. Eq. (21-b) also shows that the maximum of outlet concentration is inversely proportional to flow rate. This Table 2 Estimation of the coefficients of mass transfer in the elution stage u (cm/min)

0.06 0.12 0.17 0.22 0.27 0.33

Eluted protein (mg)

LK model for the elution stage

9.34 9.08 8.88 8.67 8.40 8.41

2.58 1.97 1.65 1.71 1.63 1.44

St

χ2 × 103

Pe ± ± ± ± ± ±

0.05 0.04 0.03 0.04 0.04 0.03

35.6 27.8 24.3 17.7 16.1 19.1

± ± ± ± ± ±

1.9 1.9 1.4 1.3 1.2 1.3

4.5 3.4 1.9 2.1 1.7 1.5

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Fig. 6. Elution curves calculated by using the analytical solution (Eq. (21b)) for the LK model which neglects the contribution of axial dispersion to band broadening. The values of u are (a) 0.08 cm/min; (b) 0.16 cm/min and (c) 0.32 cm/min, respectively. Other parameters used are: L = 0.324 cm, q¯ ini = 7.0 mg/mL, εT = 0.77, ks = 1.0 min−1 .

conclusion is well consistent with experimental results shown in Fig. 5. In Fig. 5 the front parts of the profiles are somewhat diffusive. This shows that the mass transfer process of axial dispersion also contributes to band broadening. Therefore, the sets of experimental data were fitted to the complete LK model and some of the fittings are shown in Fig. 7. It is found there is small discrepancy between experimental and theoretical values. This result shows that this model proposed here may be used to describe experimental elution curves. The coefficients of mass transfer for the best fittings are presented in Table 2. In this table these values turn out to decrease with increasing flow rate though the changes are not very significant. It should be noted that the relative estimated error of Pe varies only from about 5% to 7%, which turns out to be much lower than that obtained from the breakthrough curve. This shows that the profile of the elution curve is more sensitive to axial dispersion than the breakthrough curve. The breakthrough curve is often self-sharpened because of the formation of the shock in the migration due to a Langmuir isotherm [7]. This is not the case for the elution profiles. Furthermore, the mode of elution by first adsorbing the sample on the stationary phase and then using a stronger buffer for the desorption process naturally leads to disturbances in the band profile, i.e., band broadening. By fitting DL to u linearly with Origin software (OriginLab, MA, USA) we obtain: DL = 0.022u (r = 0.980), where the interception may be ignored according to the statistics output. Generally it is assumed that axial dispersion in chromatography results from two different mechanisms, axial molecular diffusion and the fluid flow dispersion [10] DL = γ1 Dm +

γ 2 dp u εT

(24)

Fig. 7. Fittings of experimental elution curves (symbol) to the LK model (solid line). The values of u are (a) 0.06 cm/min and (b) 0.33 cm/min, respectively.

According to the regression equation obtained above, the contribution of axial molecular diffusion (the first term in the right hand side of Eq. (24)) to DL turns out to be negligible compared with that of the fluid flow dispersion, which is accounted for by the second term and due to the effects of factors such as the module geometry and membrane pore distribution. Because the diffusion coefficients of large molecules such as protein are always found to be very small [22], the small contribution of axial molecular diffusion to band broadening is reasonable. In this study the slope, γ 2 dp /εT , is 0.022 cm, which may be taken as a “geometry factor” for the membrane module. As mentioned earlier the design of appropriate membrane modules provides important opportunities for more applied development of MAC. The results presented in this study show that the elution curve may provide useful information for such studies, although kinetic studies on the elution stage are still somewhat fewer than those on the loading stage.

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5. Conclusions

yp , y¯ p

The mass transfer kinetics in the loading and elution stages of MAC were studied. In the loading stage, it was found that the contribution of axial dispersion to the profile of the breakthrough curve was negligible compared to that of the sorption kinetics. In contrast, the profile obtained in the elution stage was more sensitive to the contribution of axial dispersion to band broadening. Therefore, kinetic studies of mass transfer in the elution stage will help to design appropriate membrane modules to improve the performance of affinity membrane separations in future work.

Z

6. Nomenclature

ap C C0 Cl ¯p Cp , C dp D Dm H k ka kf kf, L ks Kd L n np Pe q ¯ Q, Q q¯ ini ¯ ini Q r St t u x y

surface area of the pores (cm−1 ) concentration of the solute in the mobile phase (mg/mL) feed concentration (mg/mL) maximum capacity of adsorbent (mg/mL) concentration or average concentration in the mobile phase in the pores (mg/mL) pore diameter (cm) axial dispersion coefficient (cm2 /min) molecular diffusivity (cm2 /min) Heaviside function mass transfer coefficient in the pore model (min−1 ) adsorption rate constant (min−1 ) rate coefficient of mass transfer in the LK model for the loading stage (min−1 ) lumped rate coefficient of mass transfer in the transport model (min−1 ) rate coefficient of mass transfer in the LK model for the elution stage (min−1 ) dissociation equilibrium constant (mg/mL) membrane thickness (cm) number of experimental points number of the model parameters (uL)/(DL εT ) = Peclet number concentration of the solute in the stationary phase (mg/mL) dimensionless or average dimensionless concentration in the solid phase average desorbable protein concentration in the solid phase (mg/mL) dimensionless average desorbable protein concentration in the solid phase correlation coefficient St = kap LεT /u; Sta = ka LεT /u; St  = kf LεT /u; StL = kf,L LεT /u; St  = ks LεT /u = Stanton numbers time (min) superficial velocity (cm/min) dimensionless axial coordinate dimensionless concentration of solute in the mobile phase

dimensionless concentration or average dimensionless concentration in the mobile phase in the pores axial coordinate (cm)

Greek letters χ2 sum of residuals εe , εT external and total porosity γ 1 , γ 2 numerical parameters of Eq. (25) λ dimensionless maximum capacity of adsorbent σ standard error of the coefficient τ dimensionless time ψ dimensionless association equilibrium constant ζ variable denoting the dimensionless time for the integral in Eq. (A.5a) Subscripts i index ex experimental value th theoretical value Superscripts equilibrium value

*

Appendix A The differential equation of the family of characteristic curves for Eq. (19) is [32] dx =1 dτ

(A.1)

The solution along a characteristic curve is given by dy 1 − εT  = St Q[H(τ) − H(x − τ)] dτ εT

(A.2)

As shown in Fig. 8, the Cartesian equation of the characteristic curve through the point P (xR , 0) is x = τ + xR . So the solution

Fig. 8. Diagram for the analysis of the characteristics of the hyperbolic equation Eq. (19).

W. Hao et al. / J. Chromatogr. A 1114 (2006) 123–131

along this curve is given by

Summarizing the work above, we have the analytical solution of the LK model for the elution stage which neglects the contribution of axial dispersion to band broadening, Eq. (20).

dy 1 − εT  = St Q[H(τ) − H(τ + xR − τ)] dτ εT =

1 − εT  St Q[H(τ) − H(xR )] = 0 εT

(A.3)

With the initial condition y(0, x) = 0, it may be concluded that y = 0 under and along the line x = τ. When τ > x, Eq. (15-b) is rewritten as ∂ ln Q = −St  [H(τ) − H(x − τ)] ∂τ

(A.4)

¯ ini , Eq. (A.4) integrates With the initial condition Q(0, x) = Q to Q = −St  ln Qini = −St



τ

x

[14]

[H(ζ) − H(x − ζ)] dζ = −St  (τ − x)

x

[15]

(A.5a)

¯ ini exp[−St  (τ − x)] Q=Q

(A.5b)

The Cartesian equation of the characteristic curve through the point R (0, τ R ) is x = τ − τ R , and the solution along it is given by dy 1 − εT  St Q[H(τ) − H(τ − τR − τ)] = dτ εT 1 − εT  1 − εT  St Q[H(τ) − H(−τR )] = St Q εT εT

(A.6)

Substituting Eq. (A.4) into Eq. (A.5b) and letting x = τ − τ R , we obtain dy 1 − εT  ¯ = St Qini exp(−St  τR ) dτ εT

(A.7)

With the boundary condition y(τ, 0) = 0, Eq. (A.7) integrates to y= τR

1 − εT  ¯ St Qini exp(−St  τR ) dζ εT

1 − εT  ¯ St Qini x exp[−St  (τ − x)] εT

[20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

[33] [34] [35] [36] [37] [38] [39]

1 − εT  ¯ = St Qini (τ − τR ) exp(−St  τR ) εT =

[16] [17] [18] [19]

So we have

τ

[5] [6] [7] [8]

[12] [13]

[H(ζ) − H(x − ζ)] dζ 0

=

[1] [2] [3] [4]

[10] [11]

0

− St 

References

[9]

[H(ζ) − H(x − ζ)] dζ

τ

131

(A.8)

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