Ind. Eng. Chem. Res. 1993,32, 149-159 Udovic, T. J.; Dumesic, J. A. Adsorptive properties of magnetite surfaces as studied by temperature programmed desorption NO, C02and CO adsorption. J. Catal. 1984,89,314. studies of 02, van Doorn, J. Carbon deposition on hydrotreating catalysts. Ph.D. Thesis, University of Amsterdam, Sept 1989. van Doom, J.; Moulijn, J. A. Carbon deposition on catalysts. Catal. Today 1990,7 , 257-266. Wakker, J. P. Development of a high temperature steam regenerative H a removal process based on alumina supported MnO and FeO. Ph.D. Dissertation, Delft University of Technology, Delft, The Netherlands, 1992. Wakker, J. P.;Gerritsen, A. W. High temperature H$ removal from process gases in a steam regenerative process using MnO or FeO
149
on alumina acceptors. Prepr. Pap.-Am. Chem. SOC.,Diu. Fuel Chern. 1990a,35, 170-178. Wakker, J. P.; Gerritsen, A. W. Coal gasification: high temperature H2S removal in a steam regenerative process under realistic conditions. Prepr. Pap.-Am. Chem. SOC.,Diu. Fuel Chem. 1990b, 35, 179-187. Wal, W. J. J. van der. Desulfurization of process gas by means of iron oxide-on-silica sorbents. Ph.D. Thesis, University of Utrecht, Utrecht, The Netherlands, January 1987.
Received for review April 6 , 1992 Revised manuscript received August 26, 1992 Accepted October 15,1992
Analysis of Nonequilibrium Adsorption/Desorption Kinetics and Implications for Analytical and Preparative Chromatography Roger D. Whitley,*?+Kevin E. Van Cott, and N.-H. Linda Wang School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283
The assumption of local equilibrium in liquid chromatography can give misleading results when applied to high-affinity solutes, such as proteins, or other macromolecules. When adsorption or desorption becomes the rate-limiting step in a system, a number of symptoms will be present: symmetric and asymmetric spreading of waves, slow development of constant pattern, apparent loss of capacity, loss of coherence, and changes in interference patterns for multiple solutes. These results can reduce product purity and introduce serious errors into parameter estimation. These effects are predictable for any affinity and capacity combination if the dimensionless group approach developed herein is applied. By using a rate model of chromatography, we show how a system with nonequilibrium adsorption/desorption kinetics behaves when the controlling rate changes from kinetic to one or more mass-transfer rates and we give guidelines on scaling up processes efficiently by minimizing the nonequilibrium effects.
Introduction For chromatographic processes one can usually assume that the rates of adsorption and desorption are fast enough for the solid phase and adjacent solution concentrations to be in equilibrium. Such an assumption is not always valid for high-affinity solutes such as proteins (Horstmann and Chase, 1989; Skidmore et al., 1990; Mao et al., 1991). Some causes for slow adsorption include steric hindrance, low substrate concentration, and low ligand density; causes for slow desorption include strong binding and multiple binding sites. When slow relative to rates of mass transfer, the nonequilibrium adsorption/desorption (NAD) kinetics give rise to unusual nonequilibrium phenomena. As such, NAD is dependent both on the operating conditions of a separation and on the solute/adsorbent system (Kasche et al., 1981; Aptel et aL, 1988; Horstmann and Chase, 1989). The growing dependence on affinity chromatography for extremely specific selectivity and the development and exploitation of perfusible adsorbenta (Afeyan et al., 1990, 1991) require a better understanding of adsorption/desorption steps in order to optimize separations. Significant Prior Work. Experimental systems reported in the literature show that NAD can reduce product concentration, lessen resolution, and prevent the development of displacement trains (Kasche et al., 1981; Muller and Cam, 1984, Anspach et al., 1990). Several groups have developed experimental procedures to measure these rate
* Author
to whom correspondence should be addressed. 'Current address: Air Products and Chemicals, Inc., 7201 Hamilton Blvd., Allentown, PA 18195-1501.
constants for large molecules: Aptel et al. (1988) obtained parameters which represented lumped mass transfer and adsorption/desorption behavior; Horstmann and Chase (1989) lumped the film and pore mass-transfer resistances together; Liapis et al. (1989) studied nonporous systems and accounted for a separate film mass-transfer resistance; Onyegbado and Susu (1990) were able to determine nonlinear, Langmuir isotherm parameters if they neglected axial dispersion and mass-transfer resistances for their system. Efforts have been made to model this behavior; Wade et al. (1987) and Wade and Carr (1988) used a lumped kinetic rate expression for film mass transfer, pore diffusion, and adsorption. That model was able to explain peak splitting associated with NAD. Wade et al. (1987) also observed that band broadening due to NAD could be decreased by higher column loading for pNp-mannoside on immobilized concanavalin A. Skidmore et al. (1990) found that a model which considered film mass transfer and intraparticle diffusion separately from adsorption kinetics was able to represent slow adsorption in ion exchange chromatography much better than a lumped model. However, their study was limited to frontal analysis, and comparisons between kinetic rates and mass-transfer rates were not quantified. If pore diffusion resistance is negligible, the effects of NAD can be more readily observed. Mao et al. (1991) have reported such effects for affinity chromatography of lysozyme on nonporous particles. While Mao et al. also present a model of such systems, the required approximations for an analytical solution limit them to frontal analysis on an initially clean bed and give only fair agreement as ligand density increases.
0 1993 American Chemical Society ~aaa-5aa5/93/2632-oi49~0~.00/~
150 Ind. Eng. Chem. Res., Vol. 32, NO. 1, 1993
In a series of papers, F. H. Arnold and co-workers (Arnold, 1985;Arnold et al., 1985a,b; Arnold and Blanch, 1986) have contributed a significant amount to the modeling of NAD in affmity chromatography. In Arnold et al. (1985b), techniques are discussed for estimation of a linear isotherm parameter and mass-transfer rates for a single component. Arnold et al. (1985a) and Arnold and Blanch (1986) also defined dimensionless groups relating the adsorptionldesorption rates and mass-transfer rates. Their model, however, invokes the constant-pattern assumption which states that, after a front has traversed a certain length of column, the solid- and solution-phase concentrations at the wave front will cease changing. This assumption precludes frontal analysis of two or more solutes (interference phenomena). They do not calculate a concentration profile for the particle which is necessary for accurate modeling of slow kinetics in the particle. A patchwork of solutions for special cases of one rate or mass-transfer mechanism controlling,even though many have analytical solutions (Arnold et al., 1985a), can be insufficient in the case of a change in controlling mechanism or two mechanisms controlling (Johnston et al., 1991). Arve and Liapis (1988) modeled film diffusion, pore diffusion, and axial dispersion (neglected in all the models mentioned above) and also investigated the effects of monovalent and bivalent interaction between the adsorbate and ligand. Their work only addresses frontal loading and desorption modes-no elution nor displacement work was done. Arve and Liapis (1988) analyzed their results in terms of the Sherwood number and the Porath parameter to give general guidelines for operating conditions. Liapis (1990) summarizes his and others’ work on modeling affinity separations, which is limited to examination of frontal analysis and elution. Also, little is done to describe the impact of changes in design or process parameters on separations. Interference effects are not discussed because of the assumed perfect specificity of affinity chromatography. In the current study, our scope allows for NAD which can exist in many realistic affinity chromatography or ion exchange systems where several high-affinitysolutes can compete for adsorption sites. In a more recent modeling effort, Liapis and McCoy (1992) present a theory for the newer area of perfusion chromatography, limited to single component (lysozyme) frontal analysis. Reflection on Prior Work. While work has been done to study certain aspects of NAD in affinity separations, a detailed study of interference phenomenon and a dimensionless group analysis that considers the effects of parameters such as particle size, concentration, flow rate, and adsorption/desorption rates and provides definite design guidelines have not been presented for general, high-affinity separations. Preliminary studies into the nature of new systems with generalized models that lump kinetic and mass-transfer rates into one term are certainly useful because of the relative ease of solution. However, once the general characteristics of a separation scheme have been established, it is clearly advantageous to use a rigorous model for scaling and optimization because, as will be shown, change in a system parameter such as flow rate or particle size can change the controlling rates in a separation. The “constants” in lumped models often change with these parameters, limiting lumped model use in process scale-up. A model should also be able to simulate the major operating modes of chromatography: frontal (a step increase of concentration into the column, often used for parameter estimation), elution (a pulse of solution, often used in the final separation step), and step change (a step increase of one species into a column loaded with
one or more lower affinity species, used for removing impurities from large volumes of fluid). Objectives and Method. The VErsatile Reaction SEparation in Liquid Chromatography (VERSE-LC) model will be used in this work (Berninger et al., 1991). This model accounts for convection, axial dispersion, film diffusion, pore diffusion, gradient operation, NAD, and both solution- and solid-phase reactions. Previous studies (Whitley et al., 1991a; Va? Cott et al., 1991) verified the model with experimental data from aggregating systems. In this work, the VERSE-LC model will be verified with data exhibiting NAD from the literature. This study will then investigate one- and two-component systems, primarily looking at two questions: (1)under what conditions is NAD likely to be significant and (2) what are the characteristic effects of NAD on a chromatogram? The results will be analyzed with pertinent dimensionless groups, comparing mass transfer, adsorption/desorption rates, capacity, and degree of loading. Guidelines for scale-up and for minimizing the deleterious effects of NAD will be given. Highlights of Results. VERSE-LC was verified with the breakthrough of lysozyme from Skidmore et al. (1990). The guidelines established by Whitley et al. (1991a) for solution-phase reactions, where equilibrium is closely approximated by dimensionless group values 110, were also followed for adsorption/desorption kinetics. Characteristics observed for NAD include symmetric and asymmetric broadening, apparent loss of capacity, failure to develop constant pattern and interference patterns, and slow approach to saturation. Higher loading increases the apparent adsorption rate while lower loading favors desorption, as expected. We have found that attainment of constant pattern is greatly delayed for systems where the intrinsic kinetics are rate limiting. For two-component simulations, the slowest adsorption/desorption rate determines the characteristics of the elution history (assuming the slowest adsorption/desorption rate is also slow relative to the slowest mass-transfer rate). It was also observed that coherence, the assumption that concentrations of different components in the column change simultaneously (Helfferichand Klein, 19701,which is valid for systems at local equilibrium, was not followed when the intrinsic kinetics are relatively slow. If dimensionless loading is considered, one can predict the slow approach to saturation for any affinitylcapacity system. The power of the dimensionless group approach is that a few simple calculations are sufficient to determine the dominating characteristics of a system.
Theory A detailed description of the VERSE-LC model and its numerical solution has been provided by Berninger et al. (1991). The primary differential equations, their boundary conditions, and their initial conditions are presented here (eqs 1-3) to show the origins of the dimensionless groups. mobile phase:
x=l,
-acb,c ax
-0
Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993 151
0 150
200
250 t
300
bin1
Figure 1. Experimental verification of VERSE-LCwith NAD using data of Skidmore et al. (1990).
Scaling of the axial, radial, and temporal variables is by characteristic values:
and the mobile, pore, and solid-phase concentrations are scaled by their respective expected maximum values: Cb.i
-
CD. C D. .. i ..i E,i = Ce,i' CT,i Loading is considered by an inverse loading factor, = (1 - tp)[CT,i/Ce,i],and the impact of ~ $ ~value ~ ' s will be dependent upon the isotherm being used. However, as a multiplier to Ylj in eq 2a, it is clear that 4Limodulates the effective adsorption and desorption rates, with desorption fqvored for low loading. More details on the scaling, assumptions, and numerical solution of these equations are covered by Berninger et al. (1991).The Chung and Wen (1968) correlation is used for axial dispersion in all simulations, except for comparison to the data of Skidmore et al. (1990) for which no axial dispersion was considered. Since Reynolds numbers are low for the simulations (order of 0.01), the film masstransfer coefficient was estimated by the Wilson and Geankoplis (1966) correlation. Choice of NAD Kinetic Expression. The analysis of adsorption/desorption kinetic effects is based primarily upon the choice of isotherm and dimensionless groups. Given the wide range and complexity of chromatographic separations, one should not be surprised to find a plethora of isotherms to represent the binding behavior of these systems (Andrade, 1985; Antia and Horvlth, 1989). Of all these isotherms, the simplest and the most widely used nonlinear isotherm is the equilibrium multicomponent Langmuir (eq 4). For the case of unequal saturation cp,i =
-*
aiCp,i C,i
=
N.
(4)
1 + cbjCpj j=l
capacities, the Langmuir isotherm is thermodynamically inconsistent and thus must be viewed as merely a useful model (Ruthven, 1984). The Langmuir isotherm has a correspondingly simple nonequilibrium form (eq 5). The Nc
Yli = Nl+,icp,i[l- JE~p.jj1 - Nl-,iCp,i =1
(5)
current study employs this NAD expression because of ita general usefulness in protein systems (Skidmore et al., 1990). The simple form of eq 5 is also desirable for illustration of NAD behavior. Dimensiodess Group Approach. When the rate of one kinetic step is 10 times slower than all others, that slowest rate determines system behavior (Whitley, 1990; Van Cott et al., 1991; Whitley et al., 1991b). Ratios of these rates are presented in terms of dimensionleas groups (Table I) in the governing equations (eqs 1-3,5). For analytical systems, convection is usually a controlling mass-transfer rate; as such the groups in the convection column of Table I should be used for comparison to other rates. For preparative scale (large diameter particle systems), intraparticle diffusion is usually the slowest rate. Therefore the last column in Table I should be used. The inverse loading term, 4L,i,represents the effects of capacity and loading on apparent adsorption and desorption rates. It is most useful for breakthrough and step change analysis since Cei is actually reached in the entire column for those modes. In cases of small pulse volumes or high masstransfer resistance, band spreading can significantlyreduce peak concentration from ita injection value. As such, the dimensionless pulse size, = Ce,iuoAti/[(l- ep)(l tb)CT,iL], is important because it considers both pulse size and concentration. This group comes from t_he time-dependent term in eq lb. One should note that CTi has units of moles per liter of solid phase, exclusive of pore volume. Consideration of these groups is shown to be crucial in comparing the behavior of systems with widely different capacities and loadings.
Rssults and Discussion Model Verification: Lysozyme Breakthrough. Before undertaking an extensive study, we validated the basic tenets of the VERSE-LC model for NAD by comparison to the model and experimental data of a lysozyme breakthrough curve on S-Sepharose FF from Skidmore et al. (1990). The results are shown in Figure 1,with simulation parameters given in Table 11. Adsorbent capacity and adsorption rate were given in terms of packed bed concentration and had to be converted to the solidphase-concentration basis of VERSE-LC (multiplication by 1/[ (1- fb) (1- t,)]. The option in VERSE-LC to "turn off" axial dispersion was also used. The relatively high affinity of the system (a = 1.657 X lo5)resulted in a long retention time and a steep particle profile. One can see the convergence of the breakthrough front as the number of particle discretization points is increased. Finally, six particle points were required for a converged simulation (four- and five-point simulations omitted for clarity) because of the sharp particle concentration profile. In the
152 Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993 Table I. Rates for Dimensionless Groups numerator axial dispersion
denominator film mass transfer
convection
intraparticle diffusion
-- -peb,i
u& 1
film mass transfer
1
intraparticle diffusion" adsorptionb desorptionb
u,,R/KeicJJp,i. Shown for nonequilibrium Langmuir isotherm. Table 11. Parameter Values for Model Langmuir kinet const 1, (M-' min-') 1- (mixi') CT (M) system params €b
(Ke)% p eb pep Re
2d (cm)
R (cm) L (cm) C, (M) uo (cm min-')
D' (cm2 min-I)
D, (cm2 min-)
Validation 3.67 x 104 1.86 x 10-1 0.2154 0.35 1 X 0.94 m
5.805 x 102 2.14 X 0.8805 7.76 1.00 4.5 x 10-3 2.30 6.897 X 3.638 7.20 x 10-5 3.00 x 10-5
axial direction, 25 elements of 4 collocation points each were used (no difference was seen as compared to 50 elementa). VERSE-LC agrees with the simulation curve presented by Skidmore et al. (1990)and thus accurately predicts the characteristics of the breakthrough curve. The resulting value of (PI: is 1.82, indicating that the adsorption rate is of the same order as intraparticle diffusion rate for this system. The data show a slower approach to saturation than predicted by either model. Such disagreement has also been observed for several other systems, and it can be due to nonspecific adsorption, multivalent adsorption, diffusion in small pores, or the inaccuracy of the Langmuir kinetics at high surface coverage (Arve and Liapis, 1988; Liapis et al., 1989; Mao et al., 1991). Single-Component Studies. For all subsequent simulations, the affinities are low enough so that two collocation points in the particle element were sufficient for a converged solution (no changes were noted in going to three particle points). Additionally, 50 elements of 4 collocation points each were used in the axial direction, although fewer elements were sufficient for most simulations. Unless otherwise stated, the isotherm parameters and the analytical system conditions of Table 111are used in the simulations. Specific values for 1, can be derived from the dimensionlea groups and the physical parameters shown in Table 111. The ratio l + / L is never varied in a series of simulations; if one rate is changed, the other is changed proportionately. That is not necessarily true for the ratio of N I + / N I -since , some variables, such as con-
Table 111. Base Simulation Parameters equilib isotherm params a 2.23 b_ (M-9 5574.0 Cr" (M) 1.231 X system params analytical preparative tb 0.35 0.35 Ke,% 0.63 X 0.95 = 0.60 0.63 X 0.95 = 0.60 Pe b 2.30 x 103 2.88 X lo2 5.63 X lo2 1.19 x 103 Pep Re 1.12 x 10-2 2.38 X 14.2 0.840 2.01 x 102 ;p 2.58 X 10' 0.46 1.0 d (cm) R (cm) 5.0 x 10-4 5.0 x 10-3 L (cm) 4.0 5.0 Ce (M) 2.239 X 2.239 X uo (cm min-') 17.19 3.64 D' (cm2 min-') 3.56 X 3.56 x 10-5 D p (cm2min-') 2.545 X 2.545 X
CT = 2.051 X
OExcept for Figures 5B and 7B where on a solid-phase basis.
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,
,
,
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,
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Figure 2. Breakthrough curves for a series of adsorption rates in analytical scale chromatography. N , , = 10 (a), 1 (b), 0.1 (c), 0.001 (d).
centration, do not affect the two groups equally. For all step changes in concentration, h = 0.344unless otherwise stated, and the pulse volume for elution simulations is 0.2 mL. 1. Frontal Analysis. (a) Adsorption Rate Series. With the dimensionless groups in mind, one can begin to address the characteristics of slow adsorption and desorption rates. In order to first focus on the effects of slow intrinsic kinetics on the effluent history, mass-transfer
Ind. Eng. Chem. Res., Vol. 32,No. 1, 1993 153 a.w 10"
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effects will be minimized by simulation of an analytical system (Table 111). Since convection is a controlling mass-transfer rate, one can observe how slow adsorption is, relative to convection, by noting the value of Nl+.A series of breakthrough curves for decreasing adsorption rates, represented by N,+, is shown in Figure 2. For N,+ = 10, the breakthrough curve is almost identical to the local equilibrium w e ; desorption is slightly slower at 8.01. As Nl+decreases to 1,the front begins to show asymmetric broadening. When Nl has decreased to 0.1, partial loss of capacity is evident. h e adsorption process has become so slow that little adsorption can occur for the given residence time. By the time N,+reaches 0.001, there is no adsorption evident and the breakthrough curve is identical to that of a nonadsorbing species (which is not a perfectly verticle line because of mass-transfer resistance). Physid y , a decrease in temperature could cause such decreases in adsorption and desorption rates. In preparative scale separations columns are much larger, giving greater processing capacity. To limit pressure drop, particle size is also increased. The parameters for such a system are shown in Table 111. Increasing the particle size often causes the intraparticle diffusion to become the limiting mass-transfer rate (larger Biot numbers). Goto et al. (1990)demonstrated that linear driving force approximations for mass transfer, such as used in many lumped models, become inaccurate as Bi increases beyond 1. One must thus consider the value of (Pl+2 for quantifying the relative rates of adsorption to mass transfer. When the preparative scale has the same value as Nl+has for the analytical simulations, the front shapes are similar, albeit more rounded because of the increased intraparticle diffusion reaistance. This additional resistance causes an increased time lag of the pore-phase concentration behind the mobile-phase concentration, as seen for the column profile (Figure 3). The C, shown is the innermost collocation point (two are used) in the particle. The time lag of c, relative to Cb is mainly due to intraparticle diffusion resistance. Negligible time lag was observed for the analytical curves of Figure 2. Even with the time lag, the Cb and c, histories have an almost parallel shape. The c, has a much more elongated shape, which minora that of the analytical column profile and is due to the NAD kinetics. Thus when NAD is present, mobile-phase- and stationary-phase-concentrationchangea do not occur at the same time, as they would for local equilibrium cases. Modeling with simple lumped mass transfer could give misleading results or parameter estimates for some of these systems (Johnston et al., 1991). (b) Flow Rate Series. If one wishes to reduce the effecta of slow intrinsic kinetics, the definition of N,+shows that a slower flow rate can be used to compensate for slow
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adsorption kinetics. Figure 4 shows a series of flow rates for the same system as in Figure 2 (curve 4b is identical to curve 2b for reference). The corresponding effect on N,+is analogous to that for changing the adsorption rate. Comparing curve 4c to 4a shows that raising the flow rate can enhance the effect of slow kinetics, if one wishes to study such effects. However, as the flow rate is increased, N p decreases toward 1. Intraparticle diffusion then becomes controlling and contributes to additional spreading. Such complex flow rate effects and the transition from curve 4a to curve 4c cannot be accurately predicted by a lumped parameter model. (c) Affinity and Concentration Effects. The dimensionless adsorption rate (N,+)also depends on porephase concentration (Figure 5A). The concentrations and values of 1, (which remain constant as concentration is changed) have been selected such that NLis equal between
154 Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993
c
0
5
ln
15
a,
0
5
O
Figure 6. Analytical scale step-down curves for a series of desorption rates. For Nl- = 10 (a), 1 (b), 0.1 (c), 0.001 (d).
Figure 5A and 5B at 4.5,and the product &,N,+is greater than 10 for all simulations of Figure 5 (15.4for (A) and 207 for (B); they cannot be equal given the other constraints). Curve a overlaps with curve a*, which is not shown, indicating that curve a is close to local equilibrium. M (curve a) to As concentration decreases from 4 X 4 X lo* M (curve c, c*) in Figure 5A, the breakthrough front becomes less sharp as slow adsorption becomes controlling, a change represented by N,+decreasing from 10 to 0.1. Curve c* shows the equilibrium breakthrough at the concentration of 4 X lo* M. A higher affinity system is presented in Figure 5B (a = 30.0and b = 45000.0 M-l; analytical system conditions of Table 111). Concentration is decreased from 4.95 X M (curve a) to 4.95 X lo-' M (curve c, c*). One can see right away that a concentration change has an analogous effect on the high-affinity system if the controlling dimemionless group, N,+,is equal for the two systems. Concentration change has a different effect from changes in adsorption rate and flow rate because as one increases concentration, one also changes the dimensionless loading, and thus the breakthrough time. The solution depends on dimensionless loading because our dimensionless time is not normalized by column capacity. 2. Step Elution Mode. (a) Adsorption and Flow Rate Series. Using the same basic system parameters as for the breakthrough simulations, Figure 6 shows a series of increasing desorption rates for a step decrease in concentration. Because of the shape of the Langmuir isotherm, a step decrease in concentration yields a diffuse concentration front. Regardless of the desorption kinetics, there is some solute in the liquid phase which will be washed out first, delaying the concentration drop beyond 1 bed volume of flow. The slower desorption systems show the earliest and sharpest drop in outlet concentration followed by extreme tailing since the solute is most resistant to desorption. For desorption rates on the order of N,- = (d), the solute is irreversibly bound on a practical time scale. Increasing Nl- by increasing desorption rate causes the concentration drop to be delayed (note sequence (d) to (a) of Figure 6). The higher desorption rates also show smoother decreases in exit concentration because more solute comes off the column earlier-thus delaying the concentrationdrop. Slowing the flow rate also gives a higher N,- and once again similar delay in the initial concentration drop. As was the case in frontal analysis, increasing the flow rate has a similar effect as decreasing the desorption rate for analytical scale systems. (b) Affinity and Concentration Effects. For a step decrease in inlet concentration from Cf to 0 (Figure 7A), changing the initial loaded concentration also changes N,+, but not NI-, which was held at 0.1 for all the simulations
P
10
0
0
a0
el
40
m
im
E
Figure 7. Analytical scale step-down curves for a series of concentration values a t (A) low-affinity and (B)high-affinity conditions. For (A), @L = 0.275 (a), 2.75 (b), 27.5 (c), 275 (d), and for (B),@L = 0.466 (a), 4.66 (b), 46.6 (c), 466 (d); Nl- = 0.1 for all simulations; for (A), Nl+= 1.25 (a), 0.125 (b), 0.0125 (c), 0.00125 (d), and for (B),N,, = 10 (a), 1 (b), 0.1 (c), 0.01 (d).
in Figure 7. Since the values of N,. and of 4JVl+ (0.344 (A) and 4.66 (B))are quite small for the curves, both kinetic rates and loading will have to be considered to predict system behavior. In Figure 7A, curve a shows typical long tailing due to both slow adsorption and desorption. As concentration decreases, the dimensionleas adsorption rate, N,+,decreases even more and contributes to additional spreading. However, there is no change in the elution time of the large vertical drop in concentration (curves 7A,b and 6c are the same for reference). This interesting behavior is a consequence of the small and constant N,- value. One can see how a decrease in concentration reduces the dimensionless adsorption rate (Ni+) in this system by comparing the NAD curves to their local equilibrium counterparts in Figure 7A. The local equilibrium versions show a much later elution time with decreasing concentration and lack the extreme tailing of the NAD simulations. In Figure 7B a higher affinity system in which r#q,N[+is more than 10 times greater than for Figure 7A is studied. The Figure 7B series shows a much smoother increase in retention time with decreasing concentration. However, Ni.is still only 0.1, resulting in the desorption front being = 10.0 but, again, decreases very broad. For curve a Ni+ with decreasing concentration; N , eventually becomes small enough to contribute to the observed spreading for curves b-d. The equilibrium curve a* almost perfectly overlays curve a. Note that nonequilibrium curve d is fairly symmetrical around the corresponding equilibrium curve d; this was not the case for Figure 7A, where the loading is higher (smaller &,) and thus at more nonlinear conditions. 3. Elution Mode. (a) Rate Series. Changing to elution mode, we will show that both adsorption and desorption rates have significant effects on the peak shape. For a relatively large pulse the elution peak shape can be
Ind. Eng. Chem. Res., Vol. 32, No. 1,1993 155 2 . 6 1 D 4 t ' '
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Figure 8. Elution of a single component for a series of (A) adsorption rates, (B)flow rates, and (C) concentrations. For (A), AT, = 0.481, Nl- = 8 (a), 0.8 (b), 0.08 (c), 0.0008 (d). For (B),AT, = 0.481; N L = 8 (a), 0.8 (b), 0.16 (c). For (C), ATo = 4.81 (a), 0.481 (b),0.0481 (c); Nl- = 0.8 (all curves); N,, = 10 (a), 1 (b), 0.1 (c).
explained as a composite of step-up and step-down inlet concentrations. For a relatively small pulse (AT,, much less than lo), elution peak shape is not a simple composite of breakthrough and step elution curves. Figure 8A shows the elution of a pulse (ATo = 0.48) of solute (Table 111, analytical system) for a series of adsorption rates. For curve a, N,- = 8 and falls short of curve a* because of both slow adsorption and slow desorption. As N,- decreases, the peak first begins to broaden (Figure 8A, curve b, which is the same for Figure 8A,B,C). For N,- = 0.8, the peak is very broad and shows splitting because of partial loss of capacity. As N1-continues to decrease below 1, the peak splitting becomes more clear, with the first part due to an apparent loas of capacity. For this case there is still some adsorption at very slow adsorption/desorption rates, and what solute does get on tends to stay on, as evidenced by the extreme tailing (curve c in 8A). Such split peaks due to slow kinetics have been reported in the literature (Hage et al., 1986; Place et al., 1991). For very small values of N1- (O.oooS), the corresponding value of Nl+is also so small (0.001) that practically no solute is able to adsorb (Figure 8A, curve d). An apparent total loss of capacity results, with no noticeable tailing; curve d overlays exactly with a zero-capacity simulation. (a) Flow Rate Series. As was previously noted for frontal mode, flow rate also has an opposite effect on elution peaks from that of adsorption rate. Curve b of caused by a Figure 8B shows a 10-fold decrease in N110-fold increase in the flow rate. All three curves look similar to their counterparts in Figure 8A because their controlling dimensionless group, Nl-, is similar. Curve c is broader and shows lees pronounced NAD behavior since
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Figure 9. Effect of NAD on development of constant pattern for two sets of Z* (denoted by a and b; subscript denotes column length). Nl+ = 10 (a4),2.5 (al), 1.0 (b&,0.25 (bJ.
N,- is twice that in Figure 8A. Additional broadening of curve c occurs because intraparticle diffusion is relatively S ~ O W( N = 2.8). (c) Affinity and Concentration Effects. Even under equilibrium isotherm conditions, changes in concentration can alter elution position and curve shape. Figure 8C, curve a* elutes quickly and has a large tail because of the high concentration and nonlinear isotherm. When concentration is dropped 100-fold (curve c*), the peak elutes much later and is more symmetrical. The peak position of c* is also close to the linear isotherm limit because of the low loading (ATo = 0.048). Since Nl- is only 0.8, (a) is not as sharp as ita equilibrium counterpart, (a*). Figure 8C shows that the increase in concentration going from (c) to (a) gives a Corresponding sharpening of the leading front: Nl increases, while Nl-remains unchanged (by definitions in ?'able I). Such behavior is similar to the experimental results reported by Wade et al. (1987). Curve c is much more spread out than curve c* since the dimensionless adsorption rate has become slow at low concentration (very small N,+),and both slow adsorption and desorption contribute to spreading. Additionally, (c) elutes earlier than (c*) because of a partial loss of capacity and lower loading. Clearly,' both dimensionless pulse size and effective adsorption rate have an impact on this series of simulations, and the dimensionless groups represent that impact. NAD Effects on Development of Constant Pattern. Even though chromatography is an inherently dynamic process, a concentration front can attain a constant pattern if the column is sufficientlylong. The development of this constant pattern requires that the inlet concentration be in the nonlinear portion of the isotherm (high CJ. Since the front will span the range of concentrations from 0 to the inlet concentration, there is a differential rate of movement of solute. Solute at lower concentrations (the leading edge of the front) moves slowly because it "sees" more open sites for binding. Solute at higher concentrations (trailing edge of the front) moves faster since it encounters adsorbent which has already been bound by solute. The result is a self-sharpeningthermodynamic effect which can counterbalance the band spreading due to mass-transfer resistance. Changes in column length and flow rate alter the mass-transfer resistance which must be overcome by thermodynamic sharpening. The presence of NAD can have a strong impact on the development of a constant pattern. To illustrate this point, two curves, denoted as (a) and (b) in order of decreasing adsorption rate, are shown in Figure 9 for two column lengths (1and 4 cm; see Table 111, analytical system). For the equilibrium isotherm constant pattern has been achieved in leas than 1cm of column length because curve a*l has the same shape as curve a*& At the slower absolute adsorption and desorption rates (1, and 1-, respectively), the corresponding dimensionless rates will always be
166 Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993
smaller for the 1-cm column, since N,, is directly proportional to column length. Consequently, the shorter column always exhibits greater NAD behavior (Figure 9). Thus, the shape of (b,) indicates slower apparent kinetics than (bJ. When NAD is present, a constant pattern is not achieved unless the column length is long enough for the values of N,,to be greater than 10. If adsorption kinetics are only slightly slower than mass-transfer rates, spreading can be symmetrical and easily mistaken for slow mass transfer. One can differentiate between the two by changing feed concentration,which will alter NAD-induced spreading. Models which rely on a constant-pattern assumption for their solution lose predictive capability under NAD conditions. Two-Component Studies. 1. Frontal Mode of Two Components. The previous sections show the various effecta that slow intrinsic kinetics can have on individual solutes. Since chromatography usually involves the separation of multiple components, interference effects (or lack thereof) are of great importance. As may be expected, an additional component increases the complexity of the elution history and also brings up the question of what is the overall controlling rate in a multiple-component system. The key idea of the dimensionless group approach developed for single components will be extended in this section to answer that question. The simplest two-component simulation consists of frontal analysis of a mixture of two solutes fed to an initially empty column. Component A has the Langmuir isotherm and kinetic parameters as shown in Table III,and component B has the same parameters as previously in the high-affinity single-component simulations (a = 30.0 and b = 45OOO.0 M-l); inlet concentration of B is 3.0 X lo4 M. Taken together, these two components have different capacities, which, of course, makes the multicomponent Langmuir isotherm thermodynamically inconsistent (Le Van and Vermeulen, 1981). However, widely varying saturation capacitiesare obaerved experimentally (Whitley et al., 1989; Katti et al., 1990) and will be used here for realistic simulations. A series of breakthrough curves for this system is shown in Figure 10. Figure 1OA is the equilibrium case, which shows a leading block of component A being pushed ahead of the equilibrium distribution. The dotted line of Figure 10A is the breakthrough curve of component B in the absence of component A. The earlier breakthrough of component B in the presence of A, as opposed to B alone (dotted curve of Figure lOA), gives a measure of the interference between the two solutes. Figure 10B shows the case of A (lower affinity solute) adsorbing at an effective rate 100 times slower than that of B (Nit,*= 0.1 and Nl,,B = lo), resulting in a loss of capacity for component A. The slight drop in A concentration after the B front passes through indicates that there is still some displacement of A for these relative rates. When the rates are reversed such that N,,,, = 10 and N,+3 = 0.1 (Figure lOC), the breakthrough curve of A appears similar in shape as for Figure 10B. Since B now adsorbs too slowly to be effective in competing for sites, breakthrough occurs sooner and B shows a small degree of displacement by A. If both Nlt,Aand N,+B = 0.1 as in Figure 10D, there is no separation in the breakthrough times of the two components and both are slow to climb to the feed concentration. The major conclusion from this series is that displacement of A by B is reduced as either adsorption rate becomes nonequilibrium. Thus, the slowest adsorption rate controls chromatographic behavior among a group of competing solutes. Moreover,
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"coherence", which is observed for local equilibrium systems (Helfferich and Klein, 1970), does not apply when adsorption/desorption kinetics are slow. Simple multicomponent models which rely on an assumption of coherence will give poor predictions of product quality under these conditions. 2. Displacement of a Presaturated Column. To look more closely at the effect of NAD on displacement,the case of a presaturated column of A being displaced by a step increase of B is studied (Figure 11). The equilibrium simulation, shown as Figure 11A, has sharp concentration waves indicating coherence. For Figure 11B, Nu = 10 and Nl,,B= 0.1. As a consequence of displacer B's slow adsorption rate, it is unable to displace A to any great extent. When the rates are reversed (NIA = 0.1 and NltB= 10.0, Figure llC), A shows the typical slow desorption behavior: a characteristic quick drop in concentration, followed by long tailing, as shown in Figure 6A, curve b. Additionally, the displacement train cannot be developed because of slow adsorption kinetics by one of the solutes. Finally, if both N l - , A and N[,,B equal 0.1 (Figure llD), both solutes lose capacity and the concentration waves are sharp, but it is not from interference. The extreme tailing of solute A serves as a clue that interference is lost. These differences are expected since, for NAD, solutes do not adsorb rapidly enough to be effective competitors for adsorbent capacity. Once again, the use of a profile view may help one in interpreting the effluent history. Figure 12 shows a profile is sharp of Figure 11C at t = 0.40 min. The front of since the corresponding-adsorption rate is high. However, , is very elongated, in agreement the stepdown curve of C with its slow desorption rate. Hwang and Helfferich (1990) present column profiles of solutes with equilibrium nonlinear isotherms which contrast with the nonequilibrium profiles of Figure 12. In their equilibrium study, the concentration waves are coherent, all changing concen-
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Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993 157
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