Ind. Eng. Chem. Res. 1996, 35, 1173-1179
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Isotherm Measurement by Frontal Chromatography in the Presence of an Adsorbing Mobile Phase Modulator Ajoy Velayudhan* Department of Bioresource Engineering, Oregon State University, Corvallis, Oregon 97331-3906
Csaba Horva´ th Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-8286
Frontal chromatography, in which the feed mixture is continuously fed into a chromatographic column, is commonly used to measure adsorption isotherms. Ion-exchange frontal systems with multiple feed components have been widely studied for the case where the total concentration (in equivalents) is constant, both in the mobile phase and on the stationary phase. In the present work, the assumption of constancy of the total mobile phase concentration is relaxed, which allows the frontal measurement of single feed components in the presence of a buffer salt (modulator) at any composition in ion exchange. Since isotherms have to be measured in the presence of the modulator, it may be better to use a range of modulator concentrations as well as a range of feed concentrations, for both ion-exchange and reversed-phase systems. It is shown that isotherms measured in the presence of modulators can provide two data points toward the feed-modulator multicomponent isotherm surface. An alternative approach gives one data point toward the feed-modulator isotherm and one data point toward the single-component isotherm of the modulator for reversed-phase chromatography. The selectivity of a single product and a modulator is discussed for ion-exchange chromatography, and an explicit result is given for the curve of unit selectivity. Simulations are presented to show that selectivity reversal, which plays a crucial role in the column dynamics of the elution and displacement modes, is less of an issue in the frontal measurement of multicomponent isotherms. However, due to geometric distortion, selectivity reversal may be erroneously thought to occur in certain representations of adsorption data as line isotherms. Various line isotherm representations are discussed, and one that is free of distortion is recommended. Introduction In frontal chromatography, a sample mixture dissolved in the mobile phase with which the column has been preequilibrated is fed continuously into the column. The method, in addition to being the process of feed introduction in other modes of chromatography, such as displacement and preparative elution, can also be used to recover the least-retained component of a sample mixture at relatively high concentration and purity (Lee et al., 1988; Liao and Horva´th, 1990). Frontal chromatography has also been used to measure the single-component and multicomponent adsorption isotherms of sample mixtures (Conder and Young, 1979; Huang and Horva´th, 1985a,b; Jacobson et al., 1987; Guan and Guiochon, 1994). In this work, the frontal chromatography of a single feed component dissolved in a binary mobile phase is examined. Attention will be restricted to a mobile phase consisting of a nonadsorbing component, the carrier, and an adsorbing component, the modulator. Frontal chromatography is arguably the simplest nonlinear operational mode of chromatography in the sense that intermediate transitions between the initial and the feed states are generated instantaneously, and no further transitions are produced as the feed components move down the column. It is therefore immediately and for all later time a coherent system (Helfferich and Klein, 1970). Considerable work has been done on frontal systems under the assumption that the total concentration (in terms of equivalents) of all the adsorbable species is constant, both in the mobile phase and on the stationary phase (Tondeur et al., 1967; 0888-5885/96/2635-1173$12.00/0
Tondeur and Klein, 1967; Helfferich, 1967; Helfferich and Klein, 1970). However, the system that we shall treat in this work, one or more feed components and a mobile phase modulator that adsorbs nonlinearly, does not lend itself to the assumption that the total mobile phase concentration of all adsorbable species is constant. When a single feed is involved, the concentrations of the feed component (or “product”) and the modulator can be varied independently, making it a bivariant system, i.e., one with 2 degrees of freedom. It is in this respect analogous to the ternary systems under the assumption of constancy of the total mobile phase concentration discussed by Tondeur et al. (1967). Transitions can be abrupt, gradual, or composite. Abrupt transitions, or shock layers, tend to retain their shape as they move down the column, and are preferable in frontal isotherm measurement, because they preserve the intermediate plateau (Rhee et al., 1970; Helfferich and Klein, 1970; Courant and Friedrichs, 1976). (Strictly speaking, it is only after a shock layer has achieved a dynamic balance between the offsetting effects of the isothermswhich tends to sharpen the shock layersand nonequilibrium effectsswhich tend to smoothen the shock layersand no longer changes its shape as it moves down the column that it should be called a constant pattern.) A gradual, or proportionate-pattern transition, on the other hand, continues to spread as it moves down the column due to diffusive and mass-transfer effects (Rhee et al., 1970; Helfferich and Klein, 1970; Courant and Friedrichs, 1976). Composite transitions contain both abrupt and gradual portions (Amundson et al., 1965; Rhee et al., 1989). © 1996 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996
Measurement of Adsorption Isotherms The present work examines the determination of feed isotherms in the presence of a modulator. The modulator is a component of the mobile phase that serves to alter the retention of the feed components. Typical examples are salts in ion-exchange chromatography and organic modifiers in reversed-phase chromatography. In ion exchange, a modulator is always present, in order to satisfy the electroneutrality condition (when there is no feed component in the column, there must still be one counterion species to bind to the sorbent and preserve its electrical neutrality). Although there is no generally valid analog of electroneutrality in reversedphase chromatography, a modulator is very frequently used in practice for a variety of reasons including complete solvation of the stationary phase. In addition, a modulator is necessary in the separation of macromolecules to induce them to desorb and thereby prevent irreversible sorption. Thus modulators are used in many practical systems, and the determination of adsorbate isotherms in their presence is useful in understanding and predicting the behavior of such separations. In the measurement of isotherms by frontal chromatography, the column is first equilibrated with the mobile phase. Then a step of the product component in the presence of the modulator is introduced into the column. Its breakthrough curve is used to determine the amount of product adsorbed corresponding to the concentration of that step. The process is repeated for a series of steps of successively higher product concentrations, thus building up the adsorption isotherm, which can be viewed as the functional relationship between the equilibrium compositions of the adsorbed and mobile phases. The calculations by which the isotherm is determined from such frontal experiments are outlined below. It is assumed that the adsorption process is isothermal and isochoric and has reached equilibrium by the column outlet. The equations below are in terms of velocities and therefore assume constant flow rate and constant column cross-sectional area. They can be recast in terms of effluent volumes in order to relax these last assumptions (Claesson, 1949). The times at which the two fronts emerge from the column are denoted by tA for the feed-intermediate front and tB for the intermediate-initial front. Direct measurement is assumed to give the concentrations of the product and the modulator in the intermediate plateau. The standard multicomponent frontal approach for liquid chromatography (Jacobson et al., 1987) based on the results of Claesson (1949) gives the following result from an overall mass balance:
φqfeed ) j
(
)
tA - t0 feed (cj - cinter )+ j t0 tA - tB inter (cj - cinitial ), j t0
(
)
j ) p, m (1)
This equation is valid regardless of whether the transitions are gradual or abrupt, but only one data point is generated from each experiment. Note that the composition of the intermediate plateau and the phase ratio are needed to evaluate the adsorbed concentrations. Our approach in this paper is based on the observation that, if both the transitions are abrupt, then two data points can be generated from each experiment. The
velocity of an abrupt front is given by (DeVault, 1943; Claesson, 1949)
vfront )
v0 v0 ) ∆qp ∆qm 1+φ 1+φ ∆cp ∆cm
(2)
where the ∆’s stand for the difference in concentrations across the front. Since each abrupt transition gives rise to such an equation, it is, in principle, possible to obtain the adsorbed concentrations in both the intermediate and the feed states simultaneously. We now turn to the issue of how to maximize the possibility that both the transitions are abrupt. In a bivariant system, there are two characteristic directions associated with any point in cm-cp space (the hodograph plane). Gradual transitions follow these characteristics; an abrupt transition need not but will reduce to the characteristic as the compositions between which that transition occurs become more similar. Each characteristic direction has a velocity associated with it at each point; the two possible transitions through a point are therefore called the “slow” and “fast” transitions according to their velocities (Vermeulen et al., 1984). The transition from the feed state F to the intermediate state I must follow the slow transition, and that from I to the initial state O must follow the fast transition, in order to maintain the stability of the intermediate plateau (Glueckauf, 1949; Rhee et al., 1989). In the absence of selectivity reversal, the fast transition from the feed state will be one of negative slope in the hodograph plane, and the slow transition from the intermediate to the initial state will have positive slope. Abrupt transitions do not, in general, fall on the characteristics, but the fast transition must continue to have negative slope and the slow transition positive slope. Of course, we cannot know the disposition of the characteristics beforehand, since these derive from the isotherm that we intend to measure. Nevertheless, it is possible to use the qualitative features mentioned above to come to useful conclusions regarding how the experiments should be run. First let us consider the feed state F. The initial state O must lie below and to the left of the shock through F (which has negative slope); otherwise, the transition from I to O will involve an increase in the product concentration, which, in turn, implies that the transition will be gradual, assuming that the product is the more retained component (Tondeur and Klein, 1967). Thus, F must lie to the northeast of O. Similarly, on considering O, we can say that F must lie above and to the left of the shock from the intermediate state I to O (which has positive slope); otherwise, the transition from F to I will involve an increase in product concentration and will therefore be gradual. Thus, F must lie to the northwest of O. These two constraints together require that F lie due north of O, i.e., cfeed > cinitial ; cfeed ) p p m initial cm . The above argument may no longer be valid if selectivity reversal should arise; then it may not be possible to provide a general rule to choose F and O so as to make both transitions abrupt. However, some simulations involving selectivity reversal will be shown later in the paper that follow the “due north” criterion above and generate two abrupt transitions. This approach may therefore be applicable even when selectivity reversal occurs. Another approach, applicable when one of the two transitions is abrupt, is to use the
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1175
appropriate shock condition coupled with the overall mass balance (eq 1); it again becomes possible to generate two data points from each frontal experiment. To the extent that the difference between abrupt and gradual transitions is minimized as the adjoining plateaus approach each other in the hodograph plane, it may be worthwhile to minimize the size of each step. Then even gradual and composite transitions may be approximated as abrupt for the purposes of using both transitions. On the other hand, it is necessary that the various plateaus be well-defined and easily measured, and this argues for large step sizes. A trade-off is clearly necessary, and it is possible that an effective step size can be found experimentally from a few runs and then used for all the remaining runs. A set of experiments should therefore lie along a vertical line in the hodograph plane. The first step is a special case, in that the column is initially equilibrated with only modulator. Then the intermediate transition will only contain modulator, and the product will only be found in the feed state (Figure 1a). In this case, the adsorbed product concentration can be found directly from the feed-intermediate shock, in a manner analogous to a single-component frontal run. The velocity of the feed-intermediate transition is
v0
vA ) 1+
qfeed p φ feed cp
) 1+
v0 feed qm φ feed cm
(3)
- qinter m - cinter m
This can easily be rewritten in terms of the time of emergence, tA, of the front:
[
tA ) t0 1 + φ
] [
qfeed p cfeed p
) t0 1 + φ
]
inter qfeed m - qm inter cfeed m - cm
(4)
Since tA is easily measured, the first part of the above result directly gives the adsorbed product concentration in equilibrium with the feed state, without knowledge of the composition of the intermediate plateau. This composition is needed, however, in order to calculate the corresponding adsorbed concentration of the modulator in the feed state. In reversed phase chromatography, the intermediateinitial transition gives
[
]
- qinter qinitial m m
tB ) t0 1 + φ
cinitial - cinter m m
Figure 1. Frontal chromatographic profiles for a single product in the presence of an adsorbing modulator in the mobile phase. Figure 1a depicts the situation when the column is initially free of product and Figure 1b the situation when product is present both in the initial and in the feed states. In both cases, the modulator level is the same for feed and initial states. The product is depicted with a solid line and the modulator with a dotted line. The feed state is represented by F, the intermediate state by I, and the initial state by O.
front. However, we can directly set qinter to the satum ration concentration Λ in eq 4 and thus obtain qfeed m . In all subsequent steps, the initial and the feed states will contain product and modulator, and so will the intermediate state (Figure 1b). Then two equations are available from each front, for both reversed-phase and ion-exchange systems:
(5)
qinitial m
It is assumed that is already known (for instance, it could be measured by the breakthrough of the modulator that was fed into the column during preequilibration). Then, since cinter is known by direct m measurement, eq 5 gives qinter . This can then be m substituted into eq 4 to get qfeed . Note that qinter is a m m data point toward the single-component isotherm of the modulator. Such an approach could not be used in ion-exchange chromatography, where electroneutrality implies that the adsorbent is already saturated with modulator prior to feed introduction. The intermediate plateau consists of only modulator and therefore moves at the mobile phase velocity. In this case, eq 5 cannot be used, since there is no change in adsorption level across the I-O
v0 feed qp φ feed cp
vA ) 1+
vB ) 1+
v0 initial qp φ initial cp
-
-
qinter p cinter p
)
qinter p cinter p
)
1+
1+
v0 feed qm φ feed cm
(6)
- qinter m - cinter m
v0 initial qm φ initial cm
(7)
- qinter m - cinter m
The corresponding equations in time units are:
[
] [
- qinter qfeed p p
tA ) t0 1 + φ
cfeed - cinter p p
]
inter qfeed m - qm
) t0 1 + φ
inter cfeed m - cm
(8)
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Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996
[
] [
- qinter qinitial p p
tB ) t0 1 + φ
cinitial - cinter p p
]
- qinter qinitial m m
) t0 1 + φ
cinitial - cinter m m (9)
inter , and Since tA and tB are measured, as are cinter m , cp initial initial qm , qp are known, since they were measured in the previous stepsthe feed state in the (k - 1)th step becomes the initial state in the kth stepsit is clear that qinter and qinter can be calculated from eq 9 and substim p tuted into eq 8 to obtain qfeed and qfeed m p . Thus the stationary phase concentrations corresponding to both the intermediate and feed mobile phase compositions are determined in a single step. This process is repeated as desired for various sets of runs, varying the product concentration in the mobile phase but keeping the modulator concentration constant in each set. Since the adsorbed concentrations in each step can be seen to depend on adsorbed concentrations calculated in previous steps, the error incurred in each measurement includes the error in all previous measurements. This implies that measurements at higher concentrations have greater variability than those at lower concentrations. However, these errors only increase arithmetically and may be acceptable, assuming that the error in a single frontal run is quite small. Frontal Measurement When the Initial State Is Always Product-Free. If the product(s) of interest is available in reasonable amounts, it may be worthwhile to regenerate the column after each frontal step, so that each step is introduced into a column that is free of product. This could be particularly useful when there are impurities in the product that could build up on the column if successive steps were introduced without regeneration. If regeneration is used after each step, every experiment will yield a chromatographic profile consisting of a band of enriched modulator ahead of a plateau of product and modulator at their feed concentrations, as in Figure 1a. Then the adsorbed concentration of the product can be determined directly, as in a single-component isotherm measurement, from eq 3. Note that, if only the product isotherm, as a function of both product and modulator mobile phase concentration, is wanted, this is a direct method involving no measurement of intermediate compositions. This is an important difference between the multicomponent isotherm measurement of two product components and of one product component in the presence of the modulator. Since the modulator is already in the column prior to feed introduction, keeping the modulator level constant in the feed and initial states generates an intermediate plateau that is free of product, as discussed above. In addition, the concentrations of the enriched modulator fronts in each run can simultaneously be used to arrive at the single-component isotherm of the modulator for modes such as reversed-phase chromatography. We note that this method has the additional advantage that the measurement of the adsorbed composition does not depend on those previously measured, and the error associated with a measurement is independent of mobile phase composition.
Selectivity Reversal in Ion-Exchange Systems Selectivity reversal introduces considerable complexity into elution or displacement chromatography; complete separation may never be reached in some cases. Its role in the frontal measurement of multicomponent
isotherms is far less severe, particularly if the plateaus and transitions are well demarcated. The central difficulty is that the “due north” criterion suggested earlier may not suffice to ensure that both transitions are abrupt, and it does become more likely that composite or gradual transitions may occur. If one transition remains abrupt, the generalization suggested earlier may be used. It is only when both transitions become gradual that the present method becomes not generally applicable; even in this case, recourse may be had to the classical method of the overall mass balance (eq 1) and one data point generated per experiment. Below, we discuss selectivity reversal for the case of ion-exchange isotherms for macromolecules, with shielding effects accounted for (Velayudhan, 1990; Brooks and Cramer, 1992; Gadam et al., 1993). Cramer and coworkers have shown that shielding effects are important in the ion-exchange chromatography of proteins and have theoretically and experimentally demonstrated the implications of such shielding for displacement separations. Here, we discuss the selectivity reversal for a product and a modulator under this isotherm formalism and discuss its consequences for frontal isotherm measurement. The classical mass-action formalism, in the absence of shielding effects, gives rise to selectivity reversal. The curve of unit selectivity has been shown to be a straight line of negative slope for a bivariant system (Tondeur et al., 1967; Tondeur and Klein, 1967). Such a negative straight line of unit selectivity has also been shown to hold for the Langmuir isotherms derived under the ideal adsorbed solution approach (Antia and Horva´th, 1991). Here we show that the mass-action isotherm with shielding gives rise to curves of unit selectivity (between the product and the modulator) that have positive slope in the hodograph plane. The process of stoichiometric displacement (Boardman and Partridge, 1955; Helfferich, 1962) gives
cp + Rqm,exch a qp + Rcm
(10)
where R is the binding charge on the product, and qm,exch represents the concentration of exchangeable bound counterions (Brooks and Cramer, 1992). The process of binding the product sterically shields a certain number of already bound counterions from interacting with any other large molecule. This number is characteristic of a given product, assuming that it binds in only one configuration, and is called the shielded charge, σ (Brooks and Cramer, 1992). We therefore have
qm,unexch ) σqp
(11)
The equilibrium is represented by
K)
qp(cm)R cp(qm,exch)R
(12)
Ideality is assumed, so that activities may be replaced by concentrations. Electroneutrality of the stationary phase gives
Rqp + qm,exch + qm,unexch ) Λ
(13)
where Λ is the concentration of fixed charges on the adsorbent. Equations 11-13 can be solved simultaneously to arrive at the stationary phase concentrations corresponding to a given mobile phase composition. Here
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1177
Figure 2. Curve of unit selectivity for a single product with the modulator. Mass-action kinetics with shielding are assumed. Isotherm parameters are given in the text.
we are interested in selectivity reversal between the product and the modulator and therefore define the selectivity as
Spm )
cm qp/cp qp ) qm,total/cm cp (qm,exch + qm,unexch)
(14)
This can be evaluated in terms of the previous equations, and the condition for unit selectivity obtained as
(
Kp 1 -
) (
σcp cm
R
)
)
Rcp + cm Λ
R-1
(15)
Figure 2 shows that eq 15 gives a curve in the hodograph plane that is very close to a straight line, which has positive slope (the parameters used are R ) 3, σ ) 5, Kp ) 2000, and Λ ) 0.5 M). The effect of selectivity reversal on frontal runs is exemplified in the simulations of Figure 3. Figure 3a shows a run where the product is more retained than the modulator (Spm > 1), and Figure 3b exemplifies the opposite situation (Spm < 1). In both cases, a band of pure modulator is formed ahead of the feed plateau, and the data can be used to arrive at isotherm information in the standard way. If truly multicomponent feeds (i.e., more than one product component) are used and their competitive isotherms are to be measured in the presence of the modulator, the situation is obviously even more complex. The likelihood of selectivity reversal between any pair of adsorbable components is increased, and several additional plateaus may be formed (Tondeur, 1970, 1971; Rhee and Amundson, 1972). However, the curve of unit selectivity between any two products again becomes a straight line of negative slope, given by
Λ ) (R + σA)KAcA
( ) KB KA
R/(R-β)
(β + σB)KBcB
+
( ) KB KA
β/(R-β)
( )
+ cm
KB KA
1/(R-β)
(16)
where SAB refers to the selectivity factor for two feed components, component A with binding charge R, shielding charge σA, and equilibrium constant KA and com-
Figure 3. Simulations of frontal chromatography of a single product. In Figure 3a, the selectivity Spm is 1.56 for the feed composition, so that the product is the more retained compound. In Figure 3b, Spm is 0.79 for the feed composition, so that the modulator is the more retained compound.
ponent B with binding charge β, shielding charge σB, and equilibrium constant KB. For a constant modulator concentration in the mobile phase, cm, eq 16 represents a straight line of negative slope in the cA-cB plane. Representation of Multicomponent Line Isotherms One complication with multicomponent isotherms is that they represent surfaces in hyperspace and are, in general, not easy to visualize. Even the binary system treated here, which gives rise to two-dimensional isotherm surfaces in three-dimensional space (the mobile phase concentrations of the product and the modulator being the two independent variable axes and the stationary phase concentration of either species being the dependent variable axis), is not always easy to interpret, and sectional plots in two-dimensional space are frequently used instead. One such representation of binary isotherms plots the line isotherms corresponding to varying the mobile phase composition along rays, so that the ratio of the two mobile phase concentrations remains constant. Here, each stationary phase concentration is expressed as a curve in a two-dimensional plot with respect to its corresponding mobile phase concentration. This allows the two line isotherms to be directly compared. However, there is a geometric distortion implicit in this
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stationary phase compositions. Values on both sides of unity unambiguously indicate selectivity reversal. Conclusions
Figure 4. Plane representation of the multicomponent Langmuirian isotherm for a binary system. Sections of the isotherm surfaces (line isotherms) corresponding to a constant ratio of mobile phase concentrations (c1 ) 0.3c2) are shown to cross.
representation. Along a ray in the plane of mobile phase concentrations, the ratio of the two concentrations is constant; this means that the concentrations are unequal, except for the special case where this ratio is unity. Hence drawing both these concentration variations on a single scale necessarily expands the smaller, or compresses the larger, mobile phase concentration and consequently distorts the disposition of the corresponding stationary phase concentrations. This is exemplified in Figure 4 for empirical Langmuirian isotherms, which for a binary system are of the form
aici qi ) , 1 + b1c1 + b2c2
i ) 1, 2
(17)
The isotherm parameters used are a1 ) 5.0, b1 ) 1.0 mM-1, a2 ) 7.2, and b2 ) 1.2 mM-1. The line isotherms shown in Figure 4 are for the ray c1 ) 0.3c2 and are seen to cross, which is usually indicative of selectivity reversal. However, the Langmuirian isotherms are known to have constant separation factors and are therefore incapable of selectivity reversal. The representation in Figure 4 must therefore be used with caution. (In the special case where the two mobile phase concentrations are equal, no distortion occurs, and crossing of the line isotherms is likely to represent selectivity reversal.) A line representation that does not suffer from distortion is possible if the two mobile phase concentrations are read off different scales, the ratio which is equal to the concentration ratio along the specified ray. In this form, whenever the surface representation does not undergo crossing, the lines representing the stationary phase concentrations are also uncrossed. If Figure 4 were to be redrawn in this fashion, the empirical Langmuirian isotherms would be seen not to cross. However, it is not easy to see when selectivity reversal occurs, precisely because of the different scales for the mobile phase concentration. Another approach that does not generate geometric distortion is to use perpendicular sections of the isotherm surfaces, i.e., sections where one of the two mobile phase concentrations is held constant. In general, however, the question of selectivity reversal is best determined by direct calculation from the measured
Frontal chromatography is a standard method for the measurement of multicomponent adsorption isotherms. The present work outlines how such measurements could be made in the presence of a mobile phase modulator that achieves its modulation of adsorbate retention by itself binding to the sorbent. A method is suggested for which the resulting transitions are likely to be abrupt, thereby generating two equilibrium data points per run. Even when one abrupt transition occurs, the present method may be adapted to obtain two data points per run. Only when both transitions are not abrupt is it necessary to return to the classical massbalance method and get one data point per run. Selectivity reversal is analyzed for ion-exchange isotherms incorporating shielding effects, and its consequences for frontal measurement of isotherms are discussed. It is shown that representation of multicomponent isotherms as line diagrams corresponding to a ray of constant composition ratio can result in apparent crossing of the resulting line isotherms even when selectivity reversal does not occur. Acknowledgment This work was supported by Grant No. 20993 from the National Institutes of Health, U.S. Public Health Service. Nomenclature c ) mobile phase concentration, M q ) stationary phase concentration, M t ) elapsed time, s t0 ) column hold-up time, s v0 ) chromatographic velocity, m/s x ) distance into column, m Greek Symbols φ ) phase ratio, dimensionless Λ ) concentration of binding sites on the adsorbent, M Subscripts exch ) exchangeable adsorbed counterions p ) product m ) modulator unexch ) nonexchangeable adsorbed counterions Superscripts initial ) corresponding to the initial state inter ) corresponding to the intermediate state feed ) corresponding to the feed state
Literature Cited Amundson, N. R.; Aris, R.; Swanson, R. On Simple Exchange Waves in Fixed Beds. Proc. R. Soc. (London) 1965, A286, 129139. Antia, F.; Horva´th, Cs. Analysis of Isotachic Patterns in Displacement Chromatography. J. Chromatogr. 1990, 556, 119-143. Boardman, N. K.; Partridge, S. M. Separation of Neutral Proteins on Ion-Exchange Resins. Biochem. J. 1955, 59, 543-552. Brooks, C. A.; Cramer, S. M. Steric Mass-Action Ion Exchange: Displacement Profiles and Induced Salt Gradients. AIChE J. 1992, 38, 1969. Claesson, S. Theory of Frontal Analysis and Displacement Development. Discuss. Faraday Soc. 1949, 7, 34-38.
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1179 Conder, J. R.; Young, C. L. Physicochemical Measurements by Gas Chromatography; Wiley: New York, 1979. Courant, R.; Friedrichs, K. O. Supersonic Flow and Shock Waves; Springer-Verlag: New York, 1976. DeVault, D. The Theory of Chromatography. J. Am. Chem. Soc. 1943, 65, 532-540. Gadam, S. D.; Jayaraman, G.; Cramer, S. M. Characterization of Non-Linear Adsorption Properties of Dextran-Based Polyelectrolyte Displacers in Ion-Exchange Systems. J. Chromatogr. 1993, 630, 37. Glueckauf, E. Theory of Chromatography: VII. The General Theory of Two Solutes Following Nonlinear Isotherms. Discuss. Faraday Soc. 1949, 7, 12-25. Guan, H.; Guiochon, G. Properties of Some C18 Stationary Phases for Preparative Liquid Chromatography. I. Equilibrium Isotherms. J. Chromatogr. A 1994, 687, 179-200. Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962. Helfferich, F. G. Multicomponent Ion-Exchange in Fixed Beds. Generalized Equilibrium Theory for Systems with Constant Separation Factors. Ind. Eng. Chem. Fundam. 1967, 6, 362364. Helfferich, F.; Klein, G. Multicomponent Chromatography. Theory of Interference; University Microfilms: Ann Arbor (originally, Marcel Dekker: New York, 1970). Huang, J.-X., Horva´th, Cs. Adsorption Isotherms on HPLC Sorbents. I. Peptides and Nucleic Acid Constituents on OctadecylSilica. J. Chromatogr. 1985a, 406, 275-284. Huang, J.-X., Horva´th, Cs. Adsorption Isotherms on HPLC Sorbents. II. Proteins on Cation Exchangers with Silica Support. J. Chromatogr. 1985b, 406, 285-294. Jacobson, J. M.; Frenz, J. H.; Horva´th, Cs. Measurement of Competitive Adsorption Isotherms by Frontal Chromatography. Ind. Eng. Chem. Res. 1987, 26, 43-50. Lee, A. L.; Liao, A. W.; Horva´th, Cs. Tandem Separation Schemes for Preparative High-Performance Liquid Chromatography of Proteins. J. Chromatogr. 1988, 443, 31-43. Liao, A.; Horva´th, Cs. Purification of β-Galactosidase by Combined Frontal and Displacement Chromatography. Ann. NY Acad. Sci. 1990, 589, 182-191.
Rhee, H.-K.; Amundson, N. R. An Analysis of an Adiabatic Adsorption Column: Part IV. Adsorption in the High Temperature Range. Chem. Eng. J. 1972, 3, 121-135. Rhee, H.-K.; Aris, R.; Amundson, N. R. On the Theory of Multicomponent Chromatography. Philos. Trans. R. Soc. (London) 1970, 267, 419-455. Rhee, H.-K.; Aris, R.; Amundson, N. R. First-Order Partial Differential Equations: Volume II; Prentice-Hall: Englewood Cliffs, NJ, 1989. Tondeur, D. Theory of Ion-Exchange Columns. Chem. Eng. J. 1970, 1, 337-346. Tondeur, D. Dynamique des Transfers en Lit Fixe. J. Chim. Phys. 1971, 68, 311-323. Tondeur, D.; Klein, G. Multicomponent Ion-Exchange in Fixed Beds. Constant-Separation-Factor Equilibrium. Ind. Eng. Chem. Fundam. 1967, 6, 351-361. Tondeur, D; Klein, G.; Vermeulen, T. Multicomponent IonExchange in Fixed Beds. General Properties of Equilibrium Systems. Ind. Eng. Chem. Fundam., 1967, 6, 339-351. Velayudhan, A. Studies in Nonlinear Chromatography. Ph.D. Dissertation, Yale University, New Haven, CT, 1990. Vermeulen, T.; LeVan, M. D.; Hiester, N. K.; Klein, G. Adsorption and Ion-Exchange. In Perry’s Chemical Engineers’ Handbook; Perry, R. H., Green, D., Eds.; McGraw-Hill: New York, 1984; Chapter 16.
Received for review July 11, 1995 Revised manuscript received November 20, 1995 Accepted December 12, 1995X IE950420A
X Abstract published in Advance ACS Abstracts, February 15, 1996.
