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Ind. Eng. Chem. Res. 1997, 36, 506-509
RESEARCH NOTES Shock Layer Theory and Concentration Dependence of the Axial Dispersion and the Mass-Transfer Rate Coefficients Guoming Zhong, Peter Sajonz, and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6120
Independent determinations of the coefficients of axial dispersion and of the mass-transfer kinetics have been found to be concentration dependent in protein chromatography. The consequences of this observation are studied on a theoretical basis. Numerical calculations of breakthrough curves were carried out, assuming a linear dependence of these coefficients on the solute concentration. The results obtained illustrate the influence of the concentration dependence on chromatographic band profiles. The classical shock layer theory, which assumes constant coefficients, was extended to the case in which they depend linearly on the concentration. The analytical solution of the shock layer profile agrees with all the results of the numerical calculations. Consideration of the shock layer thickness provides an easy access to the kinetic information contained in the breakthrough profiles. Introduction The concentration dependence of the diffusivity and mass-transfer rate coefficients has been reported by different authors (Lederer et al., 1990; Gallagher and Woodward, 1989; Gibbs et al., 1991; Al-Duri and McKay, 1992). This dependence has also been confirmed in preparative chromatography with small and large molecules. Seidel-Morgenstern et al. (1993) have reported such an observation for the apparent axial dispersion coefficient (Da) of Tro¨ger base on microcrystalline cellulose triacetate derived from the profiles of breakthrough curves. Guan-Sajonz et al. (1996) have obtained a similar result for the rate coefficient of the mass-transfer kinetics (kf) of bovine serum albumin on an ion-exchange resin. The dependence found is important since these kinetic parameters may increase several fold in the concentration range of interest in preparative chromatography. Although the amplitude of the variation of these kinetic parameters may vary widely from case to case, this dependence has important consequences because band profiles may be very different from those calculated with constant coefficients (Sajonz et al., 1996a,b). For example, concentrationdependent breakthrough curves are strongly asymmetric around the half concentration point. However, there is no complete theoretical study of this problem yet. The present paper attempts to remedy this situation. The most powerful theoretical tool available for the study of the propagation of self-sharpening fronts in chromatography is the shock layer theory suggested originally by Rosen (1952), used by Glueckauf (1955), and developed by Rhee et al. (1971), Rhee and Amundson (1972), and Gorius et al. (1991). The practical relevance of this theory in nonlinear chromatography has been discussed recently by Guiochon et al. (1994). Rhee et al. (1971, 1972) have shown that the lumped kinetic model of chromatography (transport-dispersive * Author to whom correspondence should be addressed, at the University of Tennessee. S0888-5885(96)00457-5 CCC: $14.00
model) has an asymptotic solution which can be given in closed form and is valid in most practical cases. The shock layer theory has been successfully applied to preparative chromatography (Zhu and Guiochon, 1993; Zhu et al., 1993). In view of the practical importance of the shock layer concept and of its usefulness as a model of breakthrough curves, it is worthwhile to extend it to cases in which the parameters of the mass-transfer properties are functions of the concentration. This is the other goal of this paper. Theoretical Section I. Transport-Dispersive Model. The transportdispersive model of chromatography was used in this work (Guiochon et al., 1994). The partial differential equation for this model is written as
[
]
∂ ∂C ∂C ∂C + Fkf(q* - q) + u ) D ∂t ∂z ∂z L ∂z
(1)
where C and q are the sample concentrations in the mobile and stationary phase, u ) L/t0 is the linear velocity of the mobile phase (L, column length; t0, holdup time of the column), F is the ratio of stationary phase over mobile phase volume, t is the time, and z is a location inside the column. It combines the mass balance equation and the linear driving force kinetic model. In eq 1 two coefficients are used in order to account for band broadening in the column, an axial dispersion coefficient DL and a mass-transfer rate constant kf. These coefficients are assumed to have a linear concentration dependence which is expressed as follows:
DL(C) ) DL0 + DL1C
(2a)
kf(C) ) kf0 + kf1C
(2b)
where DL0, DL1, kf0, and kf1 are numerical coefficients denoting the extent of the concentration dependence. © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 507
1+ ∆τ(C) )
1 k′0Rl
{ [
( )
1 1-θ (1 + Rl + Pe1Cl) ln + 0 θ λ(1 - R ) Pe l
] )( )
Pe1Cl(1 - 2θ)(1 - Rl) + λ(1 - λ)
[(
1 1-θ ln + 1 l θ St 1 + St C 1 1 + St1Clθ Rl ln 1 l 1 + St C 1 + St1Cl(1 - θ)
(
0
Rl +
)(
)]}
(6)
In this equation Rl ) 1/(1 + bCl) and k′0 ) Fa. St and Pe are the Stanton and Peclet numbers, respectively Figure 1. Illustration of the definition of the shock layer thickness.
The Langmuir isotherm is used to account for the adsorption equilibrium
q* )
aC 1 + bC
(3)
Its numerical coefficients, a and b, are independent of the concentration. The initial and boundary conditions necessary in order to solve eq 1 are those of the frontal chromatography mode. The latter includes the classical Danckwerts condition as applied classically in chromatography (Guiochon et al., 1994). These conditions are written as
C(z,0) ) 0 uC(0,t) - DL(C)
∂C (0,t) ) uC0 ∂z
for t g 0
∂C (L,t) ) 0 ∂z
(4)
Numerical calculations were carried out on the system of eqs 1-4 using a finite difference method (Guiochon et al., 1994; Zhong and Guiochon, 1995; Sajonz et al., 1996a,b). The results of these calculations are presented in the Illustrations and Discussion section. II. Shock Layer Theory. The classical shock layer theory (Rhee and Amundson, 1972) was extended for the case where kf and Da are concentration dependent, as expressed in eq 2. Figure 1 illustrates the definition of the shock layer thickness. The concentrations at the beginning, Cls, and at the end, Crs, of the shock layer are defined through the parameter θ as follows:
Cls ) Cl + θ(Cr - Cl) rs
r
r
l
C ) C - θ(C - C )
(5a) (5b)
where Cl and Cr are the initial and final concentrations, respectively, of the step injection. Following the work of Rhee and Amundson (1972), we can derive an analytical solution of the thickness of the shock layer in reduced time units:
St )
kfL ) St0(1 + St1C) u
(7a)
Pe0 uL ) DL 1 + Pe1C
(7b)
Pe )
The parameter λ in eq 6 is the reduced shock layer velocity. It relates the linear velocity of the mobile phase u and the velocity of the concentration shock us:
λ)
us ) u
1 q*l - q*r 1+F l C - Cr
(8)
Illustrations and Discussion The following numerical values of the parameters were used in the calculations: phase ratio, F ) 0.45; concentration step jump, Cr ) 0, Cl ) 25 mg/mL (kg/ m3); Langmuir isotherm coefficients, a ) 12 and b ) 0.024 mL/mg (kg/m3). In the figures, the breakthrough curves are plotted as concentration versus the dimensionless time, τ ) t/t0, where t0 ) L/u is the holdup time of the column. In the following discussion, we consider separately the contributions of axial dispersion and mass-transfer kinetics. As seen in eqs 1 and 6, they are additive (provided the product PeSt is very large; Rhee et al., 1971). Shock Layer Thickness Caused by Axial Dispersion (St ) ∞). Three breakthrough curves calculated with the same average Peclet number (Pe ) 320) are shown in Figure 2a. The average value of Pe during the experiment is given by 2(Pei-1 + Pef-1)-1, with Pei and Pef the initial and final values, respectively. The two curves corresponding to values of Pe increasing and decreasing by the same amount during the elution of the breakthrough are approximately symmetrical with respect to the center of the breakthrough curve obtained for a constant value of Pe. However, a closer examination of the dependence of the shock layer thickness on the amplitude of the change of Pe during the experiment (Figure 2b) shows that the shock layer thickness is slightly smaller when Pe increases during the elution of the breakthrough curve than when it decreases (at constant average value). Shock Layer Thickness Caused by Mass-Transfer Kinetics (Pe ) ∞). We consider now the influence on the shock layer thickness of a concentration dependence of the sole rate coefficient of the mass-transfer kinetics, as characterized by the Stanton number, St. The results obtained are quite similar to those calcu-
508 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997
Figure 2. Breakthrough curves obtained for the same average value, Pe(C) ) 2(Pei-1 + Pef-1)-1. (a) Breakthrough curves from numerical calculations. Constant Pe ()320), solid line; increasing Pe (200 f 800), dotted line; decreasing Pe (800 f 200), dashed line. (b) Comparison of the values of the shock layer thickness derived from the analytical solution and from numerical calculations. ∆Pe ) Pef - Pei. From top to bottom: θ ) 0.001, 0.01, 0.05, 0.2.
Figure 3. Breakthrough curves obtained for the same average St (a) Breakthrough curves from numerical calculations. Constant St ()50), solid line; increasing St (5 f 95), dotted line; decreasing St (95 f 5), dashed line. (c) Comparison of the values of the shock layer thickness derived from the analytical solution and from numerical calculations. ∆St ) Stf - Sti. Lines and symbols as in Figure 2b.
lated above, in the study of the influence of the concentration on the dispersion coefficient. Figure 3a shows three breakthrough curves calculated with the same average Stanton number, 50 (average St ) (Sti + Stf)/2). The two breakthrough curves corresponding to concentration-dependent Stanton numbers with the same excursion are nearly symmetrical around the center of the curve obtained for St ) 50. Figure 3b shows the dependence of the shock layer thickness of breakthrough curves having the same average Stanton number on the amplitude of the total St excursion during the elution. By contrast with the result obtained in the study of the influence of the Peclet number (Figure 3b), there is a well-defined minimum of the shock layer thickness for ∆St ) 0, and the plots show a markedly thicker shock layer when St decreases with increasing concentration than when it increases (Figure 3b). The change observed in the shock layer thickness with increasing value of St1 comes from the increasingly long and diffuse front or tail of the profiles which is caused by the low values of the Stanton number experienced during the runs, either at their beginning or at their end. This, in turn, originates from the condition that the average value of St remains constant. There is an excellent agreement between the values of the shock layer thickness derived from eq 6 (lines) and
those calculated from the numerical solutions of the system of eqs 1-4. Conclusion The concentration dependence of the axial dispersion and mass-transfer kinetic coefficients appears to be important in preparative biochromatography. Our results show that, when such a dependence arises, the profiles of the breakthrough curves are less symmetrical than those obtained in the constant coefficient case. As a consequence, the retention times of the half concentration point and of the inflection point of the breakthrough curves may be quite different from the true retention time of the breakthrough curve. This result has important practical consequences for the determination of isotherms as shown by Sajonz et al. (1996b). The classical shock layer theory developed in the constant coefficient case was easily extended to this new situation. The analytical, asymptotic expression of the shock layer thickness obtained in the case of a linear dependence was found to be in excellent agreement with the values derived from numerical solutions. This confirms that the asymptotic solution is well representative of the results obtained in most practical situations. Taking into account the concentration dependence of the parameters controlling mass transfer in
Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 509
chromatographic columns should widen the scope of application of the shock layer theory, especially in the investigations of problems involved in the preparative separation of large molecules such as proteins. Acknowledgment This work has been supported in part by Grant CHE9201663 of the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. We acknowledge the support of our computational effort by the University of Tennessee Computing Center. Notation a, b ) first and second parameters of the Langmuir isotherm, m3/kg C ) liquid phase concentration of the component, kg/m3 Cl ) left boundary concentration ()C0), kg/m3 Cr ) right boundary concentration, kg/m3 DL ) axial dispersion coefficient, m2/s DL0 ) constant part of the dispersion coefficient, m2/s DL1 ) constant concentration dependence coefficient, m5/ (s kg) F ) phase ratio kf ) mass-transfer coefficient, 1/s kf0 ) constant part of the rate coefficient, 1/s kf1 ) constant concentration dependence coefficient, m3/ (skg) k′0 ) Fa retention factor at infinite dilution L ) column length, m Pe ) uL/DL Peclet number Pei ) initial Peclet number ()Pstart) Pef ) final Peclet number ()Pend) Pe0 ) constant part of the Peclet number Pe1 ) constant concentration dependence coefficient, m3/ kg q ) solid phase concentration, kg/m3 q* ) solid phase concentration at equilibrium, kg/m2 R ) 1/(1 + bC) St ) kfL/u Stanton number St0 ) constant part St1 ) constant concentration dependence coefficient, m3/ kg t ) time, s u ) liquid phase flow velocity, m/s us ) shock layer propagation velocity, m/s z ) axial position in the column, m Greek Symbols λ ) reduced propagation velocity of the shock layer τ ) ut/L dimensionless time θ ) parameter defining the shock layer thickness (eq 5) ∆τ ) SLT in reduced time units
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Received for review July 29, 1996 Revised manuscript received November 19, 1996 Accepted November 26, 1996X IE960457A X Abstract published in Advance ACS Abstracts, January 1, 1997.
