Influence of the mass transfer kinetics on the separation of a binary

1960

Anal. Chem. 1989, 6 1 , 1960-1970

Influence of Mass Transfer Kinetics on the Separation of a Binary Mixture in Displacement Liquid Chromatography Sadroddin Golshan-Shirazi, Bingchang Lin,and Gorges 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

purged out of the column, and only after this regeneration step has been completed, can another separation be attempted on the original column. In displacement chromatography,the elution of the sample out of the column exhibits a series of zones, with thin boundaries, where two neighbor zones interfere, with wide concentration plateaus in between, during which a pure component of the sample is eluted at a constant concentration. These series of zones are called "isotachic displacement trains", because, once formed, they move at constant velocity and profile. Therefore, the optimum conditions for a separation are those for which the isotachic train is just formed when it exits the column, or, possibly, slightly before that moment, if analysis time or production rate are of serious concern. The retention data (time or volume) do not mean much in displacement chromatography: they depend on the composition of the sample. The maximum concentration of the band of a given compound does not depend on its concentration. The width or duration of this band, on the other hand, is proportional to the compound concentration in the sample. In the 1940s, analytical instrumentation was very rudimentary. Analysts would collect fractions at the exit of the column and analyze them by combinations of chemical reactions and colorimetry. A separation process such as displacement chromatography, which delivers fractions more concentrated than elution chromatography, was highly valuable. When analytical instrumentation progressed and the liquid chromatograph was developed in the 1960s, displacement was abandoned because it does not give well-resolved bands, which can be turned into easily handled signals by the very sensitive detectors now available. On-line detection of the isotachic train is not suitable for analytical purposes. Horvath et al. brought displacement chromatography back to the limelight in the early 19809, as a possible implementation of preparative liquid chromatography, when it became obvious that the pharmaceutical industry was in need for a general purpose, highly selective method of separation for its intermediates ( 2 ) . In spite of the publication of a number of important contributions (2-8), the lack of theoretical understanding of the kinetics of the displacementprocess still hinders experimental studies of the optimization of the experimental conditions under which displacement chromatography is run. Most theoretical investigations of displacement chromatographyare based on the use of an equilibrium or ideal model (2-6). The rates of mass transfers of the components of the sample between the two phases of the chromatographic system are assumed to be infinite, while axial dispersion is neglected. This amounts to assuming the column efficiency to be infinite. Under such an assumption, the bands in an isotachic train are completely resolved and merely touch each other. The boundaries of each zone are vertical. Helfferich and Klein have presented an analysis of the phenomenon based on the coherence concept (i.e., isotachic migration of two components) and using the h-transform ( 4 ) . They have shown how to predict the progressive formation of the isotachic train and the characteristics of that train. Rhee et al. have analyzed

The derivation and propertles of the system of equations descrlMng a nonlinear, nonideai model of chromatography are discussed in the case of the separation of the components of a blnary mlxture by dlsplacement chromatography. This Is a three-component problem and the system of equations includes three mass balance equatlons and three klnetlc equations, one mass balance equation and one kinetk equation for each component of the binary mixture and for the displacer. The nature of the speclflc problem (dlsplacement) Is accounted for In the boundary conditlons: a rectangular pulse of the sample is Injected In a column containing only the pure moMk phase. After the InJectlon,a stream of a solution of dlsplacer lo parsed through the column. The procedure for calculating numerical solutions of the system of nonilnear partial dlfferentiai equatlons obtained Is explained and justifled. A serles of simulated chromatograms Is presented, corresponding to various combinations of the mass transfer coefficients of the three compounds Involved, covering a wide range of numerical values. When the values of the three mass transfer coemCients become large, the results approach those derived from a semiequilibrlum model of dlsplacement chromatography. I t is shown, however, that dlsplacement chromatography remains possible when the kinetics of mass transfer between phases Is slow, provided long columns are used at low flow veioclty of the moblle phase. The crltlcal parameters for a displacement separatlon are the displacer concentratlon, the loading factor (ratio of sample size to column saturatlon capacity), and the number of theoretical plates as measured In Hnear elution, for a very small sample size.

INTRODUCTION Displacement chromatography was first introduced by Tiselius in 1943, as a new mode of carrying out chromatographic separations ( I ) . A chromatographic phase system is selected so that the components of the sample have large retention volumes and the proper column is prepared. A volume of sample is introduced in the column. This volume can be rather large compared to the column volume if the sample is dilute, but it should remain moderate compared to the retention volume, so that frontal analysis does not take place. Elution is then carried out with a displacer, i.e., a solution of a compound that is more retained than any component of interest in the sample. The front of the displacer moves along the column, sweeping the sample components in front of it. These components are stacked in neat zones, ordered after their increasing retention in the phase system selected. Under proper conditions, the separation of these zones is complete, and the corresponding components can be recovered at a high degree of purity. The displacer must be *Author to whom correspondence should be addressed at the University of Tennessee. 0003-2700/89/0381-1960$01.50/0

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1989 American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 61, NO. 17, SEPTEMBER 1, 1989

the details of multicomponent ideal displacement chromatography, when the adsorption isotherms are given by the classical competitive Langmuir equations (5, 6 ) . In spite of the very high column efficiencies available today in high-performance liquid chromatography(HPLC), the ideal model permits only fist-order approximations. The nonlinear effects due to the curvature of the equilibrium isotherms and the coupling between the isotherms of the various components are properly taken into account. On the other hand, however, the ideal model predicts razor-sharp boundaries between the displacement zones of an isotachic train and this is not realistic. Using a semiideal model, in which the finite column efficiency is taken into account, Katti and Guiochon have shown that the ideal model assumption grossly overestimates the recovery yield of a separation and its production rate, because it neglects the losses due to the zone boundaries which are significant (7). The semiideal model gives excellent results, which account very well for the experimentalprofiles obtained, when the chromatographic system and the sample used provide for rapid mass transfers between phases, and especially for a fast kinetics of the retention process (7). There are cases of practical importance, however, where mass transfers between phases are not fast, especially when proteins are concerned. In a number of practical applications of bioaffinity chromatography, or even of adsorption chromatography, the kinetics of the retention and/or of the mass transfers between phases are slow, and their effect on the band profiles must be taken into account. This can be done through the use of a kinetic model. Recently, Philips et al. presented a numerical solution of such a model (8). This approach has some limitations, which will be discussed in the present paper. We have recently described a procedure for the computer calculation of numerical solutions of the classical kinetic model, in the case of a single compound (9) and of a binary mixture (IO), in isocratic elution chromatography. The purpose of the present paper is in the extension of our previous results to the solution of a multicomponent problem and in the application of these results to the investigation of the influence of the kinetics of mass transfers between phases on the profiles of zones in displacement chromatography, on the rate of formation of the isotachic train, and on the recovery yields in a separation. The following discussion is limited to the case of a binary mixture, with a single component used as a displacer (three-component problem). Extensions to the case of multicomponent mixtures appear rather straightforward.

MATHEMATICAL MODEL AND NUMERICAL SOLUTION The problem investigated here involves a three-component chromatographicsystem, with particular boundary conditions simulating the experimental conditions under which a displacement run is carried out. In the following, subscripts 1 and 2 refer to the two components of the binary mixture to be separated, in the order of their elution, and subscript d refers to the displacer. 1. The Mass Balance Equations. The mass balance equations for the three components involved are written in exactly the same way as for a single compound

ac,

ac,

at

az

- + U-

+ F-aql = D1-a2c1 at

az2

(la)

In these equations t and z are the time and abscissa along the

1961

column, respectively, C and 4 are the concentrations of the compounds indicated by the subscript in the mobile and stationary phase, respectively, F = (1- e ) / € is the phase ratio of the column, and t is the column porosity, u is the mobile phase velocity, which is constant along the column and during the experiment,and D is the axial dispersion coefficient, which accounts for the effects of the molecular diffusion of the corresponding compound, of the column tortuosity and of the lack of homogeneity of the packing on the axial dispersion of the chromatographic bands. A similar set of equations has been previously discussed (9, IO). It contains two functions for each compound, the concentrations in the mobile and the stationary phases. These functions must be related. In the ideal and the semiideal models, the relationship is derived by assuming that the two phases are always at equilibrium (ideal model) or near equilibrium (semiideal model). In the present work, we employ a kinetic model to account for the rate of mass transfer between the two phases. 2. Kinetic Equations. Several models are conceivable for the kinetic equation relating the concentration of a compound in the mobile and the stationary phase. Thomas (11),Goldstein ( I 2 ) ,and Wade et al. (13) used the Langmuir kinetics of adsorption/desorption in their investigations. Since the Langmuir kinetics cannot account for the mass transfer kinetics, which is the main source for deviation from equilibrium in most chromatographic separations, we have preferred the solid film driving force model (9). The corresponding equation assumes that the rate of mass transfer is proportional to the difference between the actual concentration in the stationary phase and the one that would correspond to thermodynamic equilibrium between the two phases. Accordingly, we have

swat = -kfl(ql

- ql)

(2a)

In these equations kfi stands for the lumped mass transfer coefficient of the compound i and qi is the concentration of the corresponding compound, i, in the stationary phase, at equilibrium with the concentration Ci in the mobile phase. qi is given by the appropriate isotherm equation. 3. Isotherm Equations. In principle any isotherm equation can be used to complete the system of equations. Special difficulties arise, however, when the phenomenon popularly referred to by experimentalists as “isotherm intersection”takes place (14,15). In the present work, we have used the competitive Langmuir isotherms for three compounds

These ternary isotherms account for what is probably the most important single feature of multicomponent nonlinear chromatography, the competition between the different compounds involved for interaction with the stationary phase. More complex isotherm equations are available, which accounts for the nonideal behavior of the mobile and the stationary phases. 4. Boundary and Initial Conditions. The initial conditions correspond to a column initially fided with pure mobile phase and empty of either the binary mixture or the displacer

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C L ( x 0, ) = ql(x, 0) = 0 (i = 1, 2) x > 0 Cd(X,

0) =

qd(X,

0) = 0 x

>0

(4a) (4b)

At time t = 0, a rectangular pulse of the binary mixture is injected. The duration of the pulse is t, and its concentrations, C,O and Czo,and the amounts of components 1and 2 injected are respectively CInt$, and C2OtPv,where F, is the volume flow rate of mobile phase; hence the boundary conditions for the binary mixture are

C,(O, t ) =

cot

C,(O, t ) = 0

0 < t 5 t,

(54

> t,

(5b)

t

After the injection is finished, a step of constant concentration, Cd, of the displacer in the mobile phase is introduced in the column at constant flow velocity, until the experiment ends

Cd(0, t ) = 0 Cd(0, t ) =

Cod

t5

(64

tp

t > tp

(6b)

The problem is now fully determined and ready for solution. 5. Numerical Analysis. The mathematical model involves the three mass balance equations, eq 1, the three kinetic equations, eq 2, the three isotherm equations, eq 3, the initial condition, eq 4, and the boundary conditions, eq 5 and 6. It is impossible at present to obtain an analytical solution for this system. Even the numerical analysis is not simple, because the convergence of a numerical algorithm toward the exact solution of the system has not been demonstrated, so that stability conditions of the numerical procedures are unknown. The convergence, stability, and truncation errors of several numerical procedures used for the calculation of solutions of the kinetic model have been discussed (16, 27). In the present work we have replaced the set of partial differential equations 1 and 2 by the followingequivalent finite difference equations (i = 1, 2, d): U

(C,)J,S1, h

+ (CJA+l - (CJA

(CL)A+l

7

+

pA+'

- (qI)A T

=o (74

where h and 7 are the space and time increments and n and j are the space and time indices, respectively. In order to minimize the impact of the truncation errors on the numerical solution, we have carried out the calculations assuming that the axial dispersion coefficients are all equal to 0. Under such conditions, the truncation errors introduce an artificial dispersion term which can be made to compensate exactly for the lost axial dispersion term, if the space and time integration increments are properly chosen (26). The error analysis shows that, in the linear case, the error made in replacing each eq 1 by a finite difference equation introduces an artificial dispersion term that is equal to D,#C/ax2, where D, is the apparent or artificial dispersion coefficient introduced by the truncation errors. D, is equal to (a - l ) h u o / 2 ,where a is the Courant number, equal to uo7/h(l+ k,?, which must be kept larger than 1 to avoid numerical instabilities and negative values of the concentration. In the present work, we have taken a = 2 and the space integration increment equal to 2D/uo, where D is the axial dispersion coefficient. Under these conditions, T = 4D(1 + ko')/uo2 and the apparent dispersion coefficient, D,, is equal to D , as required. The axial dispersion coefficient includes the axial molecular diffusion, the column tortuosity, and the unevenness of the flow pattern in the packed column. Thus it corresponds to the sum of the first two terms of the classical Knox equation, (2rD,/u) + A u ' / ~(28). The third term of

the HETP equation, Cuo, the resistance to mass transfer in the stationary phase is accounted for here by the kinetic equations (eq 2). The importance of the truncation errors was ignored in ref 8, which may affect the validity of some of the conclusions of that work, and certainly results in the prediction of profiles which are smoother than they should. A computer program implementing this algorithm has been written and numerical calculations have been carried out on the VAX 8750 a t the Computer Center of the University of Tennessee.

RESULTS AND DISCUSSION Using the mathematical model, the calculation procedure, and the computer program described above, we have studied systematically the influence of the displacer concentration, the column length, the mobile phase velocity, the axial dispersion, and the mass transfer coefficients on the characteristics of the separation achieved. The results are described in Figures 1 to 20. 1. Influence of the Displacer Concentration. The influence of the displacer concentration has already been discussed in detail by Rhee et al., using the ideal model (5, 6). They have shown that there is a threshold concentration, below which displacement of a compound band can never take place. More precisely, displacement is possible only if the slope of the chord between the origin and the point on the isotherm of the pure displacer corresponding to its concentration in the mobile phase, c d , is lower than the slope of the origin tangent to the isotherm of the compound studied. This condition can be written (6)

Therefore, the critical concentration of the displacer during a chromatographic development is i..=-(--l) 1 ad bd

ai

and a successful displacement requires that C d >.c , Thus, for a multicomponent mixture, the critical displacer concentration for displacement of the whole band system depends only on the adsorption isotherms of the pure displacer and of the pure lesser retained component of the mixture. It does not depend on the other components of the mixture, except that, of course, the displacer must also be more strongly retained than the more retained component of the mixture. It does not depend on the composition of the feed either. The use of the kinetic model permits a more detailed investigation, since it predicts the exact profile of the component zones at the end of the displacement, when they elute from the column. If needed, it can also predict the progressive changes in the zone profiles which accompany the formation of the isotachic train (7). We can investigate three types of displacement experiments, depending on the value of the displacer concentration. Let 1 be the subscript which stands for the last eluted component of a mixture and f the subscript for the first eluted one. a. ai< qd/Cd. T h e Chord Slope Exceeds the Slope of the Origin Tangent of the Most Retained Component. In this case the displacer concentration is lower than the threshold concentration for the last component

(10) The displacer has no influence on the behavior of any of the mixture components. Their migration along the column is too fast and the displacer front can never take over and in-

ANALYTICAL CHEMISTRY, VOL. 61, NO. 17, SEPTEMBER 1, 1989 N ?

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r

0

0

x

m O

x n

i

O U

L8 0

v 0

0 0 t

2 0 N

x

0

9 0

40

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40

50

52

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50

I

Time ( m i d Figure 1. Simulated zone profile for displacement chromatography.

Experimental conditions: column length, 25 cm; phase ratio, 0.25; flow velocity, 0.05 cm/s. Curve 1, first component; curve 2, second component; curve 3, displacer. Isotherms: first component, q = 18.75C1/(1f 12.5C, f 15C2f lac,);second component, q 2 = 22.5c2/(1 12.512~ 15C2 18Cd);displacer, q d = 27C2/(1+ 12.5C1 15C2 18Cd). Relative retention of the two components of the feed: a = 1.20. Sample size: injection duration, t , = 250 s; concentration in the feed, C I o = 0.01 M, C: = 0.02 M. Loading factors: L ,, = 1.333%, L = 2.666%. Displacer concentration: Cd = 0.01 M. Mass transfer coefficients: k,, = kk = kfd= 1 s-'. Axial dispersion coefficient: D, = 1.25 X lo-' cm /s.

,

+

+

+

+

+

teract with them. A conventional chromatogram, typical of elution or overloaded elution, is recorded. For example, Figure 1 has been obtained with Cd = 10 mM, a concentration for which the inequality 10 is satisfied. Although there is some interference between the bands of the two components of the binary mixture, there is none between the second component and the displacer, as expected. b. af < qd f C d < al. The Chord Slope Is Intermediate between the Slopes of the Origin Tangents of the Two Components. In this case the displacer concentration is higher than the threshold concentration for the last component, but lower than the threshold concentration for the first one

The displacer has no influence on the behavior of the first component but does displace the second component. Figure 2 has been simulated with the same parameters as Figure 1, except that the displacer concentration is now 20 mM. It shows that the zone profile of the second component is very different from the one on Figure 1. A displacement effect is taking place, although the isotachic train has not had time to fully form. In agreement with theory, the formation of an isotachic train is possible for the second component and it is nearly achieved under the conditions of Figure 2. As theory also predicts, it is impossible to ever incorporate the band of the first component in an isotachic train. In fact, the first

Figure 2. Simulated zone profile for displacement chromatography.

Same conditions as for Figure 1, except displacer concentration Cd = 0.02 M. component profile is almost the same as on Figure 1,except that at its very end its tail is shorter and ends 2 min earlier, because of the increased concentration of the second component in the corresponding time zone, enhancing the competition with the first component molecules. Under the experimental conditions simulated by Figures 1and 2, the critical displacer concentration for displacement of the sample mixture is 24.5 mM. c. qd/Cd < ar. The Chord Slope Is Lower Than the Slope of the Origin Tangent of the Lesser Retained Component. In this case the displacer concentration is higher than the threshold concentration for a successful displacement (i.e., Cd > (l/bd)((Ud/Uf) - 1)). Figure 3 corresponds to a displacer concentrationof 40 mM, which satisfies this condition. Rather sharp boundaries appear on both sides of each zone and on the displacer front. An isotachic displacementtrain is formed. The ideal model predicts that a concentration shock should take place on both sides of each profile (5, 6). In actual columns, which have a finite efficiency,the true concentration discontinuities are replaced by shock layers, which have a finite thickness, proportional to the column HETP. The length of the plateau is shorter than that predicted by the ideal model: the actual zone profiles are not rectangular. The ideal model also predicts the height of each plateau, C?, which is given by (5)

The plateau concentration for each component of the mixture is independent of the feed composition, but depends only on the equilibrium isotherm of the corresponding component, as a pure compound (only ai and bi enter in eq 12, not the bis (j# i ) ) ,and on the displacer concentrationand isotherm. For example, eq 12 predicts for the plateau concentrations of the two components of the binary mixture, under the experimental

1964


t 0 0

")

0

0

n

5 E W

N 0

V 0 0

0

-

0

0

0 0 0

18

19

20

Time (mid

Figure 3. Simulated zone profile for displacement chromatography. Same conditions as for Figure 1, except displacer concentration C, = 0.04 M.

Figure 4. Simulated zone profile for displacement chromatography. Same conditions as for Figure 1, except displacer concentration C,

conditions simulated on Figure 3, Cp = 0.015 556 M and CZp = 0.028889 M respectively. The values predicted by the computer calculations are identical, 0.015 556 M and 0.02889 M, respectively. Thus, the finite column efficiency, i.e., the axial dispersion and the finite mass transfer kinetics, narrows the plateaus, smooths their edges, and turns the shocks into shock layers. It does not change the maximum concentration of the zones, if a plateau can be formed. As we show later, when the mass transfer kinetics decreases excessively, the plateau narrows and disappears. As long as the plateau exists, however, its concentration is given by eq 12. d. Optimization of the Displacer Concentration. As we have shown in the previous section, the minimum concentration for the formation of an isotachic train involving the bands of all the mixture components is given by eq 9. However, the ideal model predicts that the best results are obtained if the displacer concentration is chosen so that the ratio qa/Cd is smaller than the smallest root, &, of the equation

the displacer concentration rises above the critical value (eq 9), the plateau at the top of each band narrows and the ratio of the width of the interference regions to the band widths increases. The recovery yield decreases. For a while, the production rate increases, because the cycle time decreases. At still higher displacer concentrations,the plateaus disappear, the separation between the two components degrades, and eventually the two bands are eluted on the front of the displacer without any significant degree of resolution (see Figure 4). The recovery yield and the production rate are negligible. Thus, the optimum displacer concentrationin displacement chromatography with real columns is slightly higher than the critical threshold concentration (eq 9)) but not as large as the optimum value predicted by the ideal model (C,corresponding to the smallest root of eq 13). For example, in the experimental case investigated above (see Figures 1to 4), the critical displacer concentration, for the phenomenon to take place is 24.4 mM. Simulation with a displacer concentrationof 30 mM fails to show an adequate isotachic train developed for columns shorter than 35 cm. This is why we have chosen a concentration of 40 mM for the rest of the investigation. 2. Influence of the Column Length. In order for the isotachic train to fully develop, the sample size should not exceed a value corresponding to a certain, optimum loading factor. The loading factor is inversely proportional to the column length (19). Thus, if the sample is too large or the column length too short, the isotachic train will not have time to form, but with the same sample size, it will develop with another, longer column. Alternatively, if the sample is smaller than the optimum loading factor, the isotachic train will be formed before the end of the column. Increasing the column length will result in no change in the band profiles of the mixture components, their resolution, or the displacer front at the column exit but will decrease the production rate. Figures 5-1 show the concentration profiles observed at the exit of columns having lengths of 10, 25, and 50 cm, respec-

(13) In this case, the shocks between the two components and between the last component and the displacer appear as soon as the displacer is injected and are stable and propagate as such. Thus a short column is needed for the formation of an isotachic displacementtrain of fully separated zones, a column shorter than in the previous cases (6). If we use a displacer concentrationhigher than the one given by eq 9, we can, within the framework of the ideal model, increase the plateau heights (see eq 12). The zone width decreases, which is not detrimental with an ideal column of infinite efficiency, but is with a real column. The actual boundary between displacement zones is always a shock layer, as explained above, and a mixed, interference boundary zone, with a finite thickness, appears between each successive pair of bands. The recovery yield may never approach 100%. As

= 0.5 M.

2NALYTICAL CHEMISTRY, VOL. 61, NO. 17, SEPTEMBER 1, 1989

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1 12

13

14

15

16

17

7

18

Time (rnin) Flgure 5. Simulated zone profile for displacement chromatography. Same conditions as for Figure 3, except (I) mass transfer coefficients k,, = k,, = kfd= 50 s-' and (il) column length L = 10 cm. Since the sample size is the same, the loading factors are now L,, = 3.33% and L = 6.65 % , respectively.

Time (rnin) Flgure 6. Simulated zone profile for displacement chromatography. Same conditions as for Figure 5, except column length L = 25 cm, hence L,, = 1.33%, L,, = 2.66%.

,

tively, for the same sample size. At the end of the first column (Figure 5), the isotachic displacement train is not formed; the loading fador was too large. At the end of the second column (Figure 6), it is formed. The band profiles are identical at the exit of the last two columns (see Figures 6 and 7). This observation confirms the existence of a constant pattern behavior, as demonstrated in the case of convex isotherms, such as the Langmuir isotherm, by Cooney and Lightfoot (20). In the initial region of the column, the finite kinetics of mass transfer spreads the bands as the front progresses. At some distance from the column inlet, however, the band front approaches an asymptotic form and moves as a stable profile thereafter, with no further change in its shape. Such a concentration profile is called a constant pattern profile (20). Finally, the profile shown in Figure 8 has been obtained with the same column and under the same conditions as the one in Figure 5, except that the sample size was 2.5 times smaller. Thus, the loading factor was the same as for Figure 6, and the band profiles of these two figures are identical (constant pattern profiles). This demonstrates that, in order to obtain an isotachic train just fully developed at the column exit with a displacer of constant concentration, the critical parameters are not the column length, or the injection bandwidth, or the mobile phase velocity, or the composition of the feed, but the loading factor.. As shown below, the column efficiency, i.e., the value of the mass transfer coefficients, also has some effect. Since the loading factor is inversely proportional to the column length, and since the residence time of the displacer front is proportional to the column length, the maximum production rate of a chromatographic system in displacement chromatography is independent of the column length, when the column is operated ut its optimum loading factor.

1

I

Figure 7. Simulated zone profile for displacement chromatography. Same conditions as for Figure 5, except column length L = 50 cm; hence L,, = 0.666% and L f 2 = 1.333%.

3. Influence of the Axial Dispersion. In the numerical analysis section, we use the truncation error caused by the finite nature of the space and time integration increments to

1966


3

R 0 0

0 N 0

c

0 0

0 0

39

Time (rnir)

Figwe 8. Simulated zone profile for displacement chromatography. Same conditions as for Figure 5, except sample size and loading factors L f l = 1.333% and L f p = 2.666%. simulate the effect of the axial dispersion in the calculations. This effect smooths the band profiles, as seen in Figures 1 to 8 and in the following ones. The numerical dispersion effect in Figures 5 to 8 corresponds to a space integration increment h of 50 pm (Le., a numerical dispersion coefficient D, = 1.25 X cm2/s) and to a mass transfer coefficient equal to 50 s-l. In classical mass transfer terminology, the height of a mixing stage is equal to 2D/u0 and the height of a mass transfer stage is given by 2uoko'/(l k O 2 k f (IO,21). They are respectively equal to 0.005 and 0.000 29 for these figures. The two curves in Figure 9, on the other hand, have been calculated by using the same mass transfer coefficient ( k f = lOOO), and two values of the numerical dispersion coefficient, the same as before (corresponding to a height of a mixing stage equal to 50 pm) and a second one 10 times smaller (mixing stage height, 5 pm). When the kinetics of mass transfer are not very slow, which is true with the columns discussed so far, the band profiles under linear conditions (small sample sizes) are Gaussian and their variances correspond to a HETP given by (21)

+

(14)

The numerical values corresponding to the two curves on Figure 9 are 50.1 and 5.15 pm, respectively, corresponding to theoretical plate numbers of 4990 and 48 550. The shock layers are much thinner, the plateaus broader and their ends more abrupt on the second curve (Figure 9) than they are on the first one (see captions). Comparison between these two curves illustrates the importance of column efficiency in displacement chromatography. It also illustrates the potentially very important effect of numerical errors if they are not properly accounted for. In this work, we have adjusted the truncation errors so that they account exactly for the axial dispersion effects, and therefore our numerical results are correct. In some previous work the

40

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6

Figure 9. Simulated zone profile for displacement chromatography. Same conditions as for Figure 6, except (I) mass transfer coefficients: kfl = kf2= kfd= 1000 s-'. Curves 1, 2 , and 3: elution profiles of the first and second components and the displacer, respectively, with a mixing stage height of 5 wm. Curves l', 2', 3', same profiles, but with a numerical dispersion coefficient corresponding to a height of a mixing stage equal to 50 pm.

truncation errors have been neglected. Since they cannot be avoided, the numerical results obtained are inaccurate and cannot be used for quantitative predictions of production rates and recovery yields (8). The width of the mixed regions, between the different component zones, is proportional to the thickness of the shock layer, itself a function of the column HETP. The recovery yield and the production rate increase with increasing column efficiency. 4. Influence of the Mass Transfer Kinetics (kf,, = ke2). In this subsection, we assume that the mass transfers proceed at the same rate for the two components of the binary mixture investigated and for the displacer. In the next subsection, we investigate the influence of widely different mass transfer coefficients for these three compounds and the resulting effects of these differences on the profiles of the displaced bands. Figures 9 (curve l), 6, and 3 above, and Figures 10-13 show a series of displacement chromatograms that have been simulated under identical conditions, except for the value of the mass transfer coefficient, which decreases in the following order: 1000 s-l, (Figure 9); 50 s-l (Figure 6); 1t - 9 (Figure 3); 0.5 s-l (Figure 10);0.2 s-l (Figure 11); 0.05 s-l (Figure 12); 0.01 s-l (Figure 13). The corresponding value of the column HETP is given in Table I. The factor that determines the profile shape is the ratio of the column length to the height equivalent to a theoretical plate (i.e., the sum of the height of a mass transfer stage, 2u0k,'/(l + k O 2 k fand the height of a mixing stage, 2D/uo). The heights of the mass transfer stage corresponding to the series of mass transfer coefficients used for Figures 9 , 6 , 3 , and 10-13 range from 1.45 cm (Figure 13) to 1.5 X 10" cm (Figure 9). In the worst case, the column contains only 17 stages. When k f decreases from 1000 to 0.5 s-l (Le., the plate number decreases from 5000 to 733, the plateau at the top


1967

*

9 0

1cI

9 0

h

_1

1

E9

W N

60 0

0

.-

x

0

9 0

25

Figure 10. Simulated zone profile for displacement chromatography. Same conditions as for Figure 6, except mass transfer coefficients: k,, = k,* = k,, = 0.5 s-'.

30

35

45

40

50

55

60

,

Figure 12. Simulated zone profile for displacement chromatography. Same conditions as for Figure 6, except mass transfer coefficients: k,, = k f p= k,, = 0.050 s-'.

*

9 0

Ti

8

i E

- N

.?

0 0

0

0

-9 0

0

9 n 37

#

38

39

,

40

I

41

I

,

I

42

43

44

#

45

l-L46

47

48

0 0

I

8"c---r I

30

40

50

60

70

3

Time ( m i d

Time ( m i d Figure 11. Simulated zone profile for displacement chromatography. Same conditions as for Figure 6, except mass transfer coefficients: k , , = k f 2= k,, = 0.2 s-l.

Figure 13. Simulated zone profile for displacement chromatography. Same conditions as for Figure 6, except mass transfer coefficients: k , , = k f p= k,, = 0.010s-l.

of the displacement zones predicted by the ideal model remains present and its height is constant. Its width decreases steadily, however, while the thickness of the shock layers

increases, and the plateau disappears when kf becomes smaller than 0.5 s-l. The degree of interference between bands increases and the recovery yield decreases. Further decrease

1968


Table I. Correspondence between Column Efficiency and Coefficient of Mass Transfer mass transfer coeff, s-l

height of a mixing stage: cm

height of a mass transfer stage,b cm

column height equiv to a theoretical platec

1000 1000 50 1.0 0.50 0.20 0.050 0.010

0.000 50 0.005 0 0.0050 0.005 0 0.005 0 0.005 0 0.005 0 0.005 0

0.000015 0.000 015 0.000 30 0.014 5 0.029 0.072 5 0.29 1.45

5.1 50.1 53 195 340 775 2 950 14 550

*

2D./u0. 2uokb2/(l+ k’&kf.

t

9-

-4

3

In micrometers.

36

38

40

42

44

46

48

I

50

52

Time (mid Figure 15. Simulated zone profile for displacement chromatography. Same conditions as for Figure 11, except dispiacer concentration Cd = 0.03 M. t

2

c)

9 0

Time (mid Figure 14. Simulated zone profile for displacement chromatography. concentration C, Same conditions as for Figure 11, exceDt . displacer . = 0.10 M.

of the mass transfer coefficient to 0.2 s-l and below gives bell-shaped bands. The plateau has disappeared and the band height decreases with decreasing mass transfer coefficient (see Figures 11to 13). The displacement process just does not take place anymore. The mass transfer kinetics has become too slow. The experimentalist placed in such a difficult situation may try one of three things in order to attempt to improve the experimental results: increase the displacer concentration, increase the column length, or decrease the mobile phase velocity. Figure 14 shows that increasing the displacer concentration has no significant effect on the displaced zone profiles and certainly does not improve the separation. The overdisplacement effect shown on Figure 4 appears (compare Figures 4 and 14), as was expected. Figure 15 shows that a reduction of the displacer concentration (from 0.04 to 0.03 M) very slightly improves the separation. Figures 16 and 17 show that increasing the column length and keeping the same loading factor permits a serious im-

5. 0

E

W N

.?

00 C

0

c)

-

7 0

0

9

n

Time ( m i d Figure 16. Simulated zone profile for displacement chromatography. Same conditions as for Figure 11, except column length L = 50 cm, hence loading factors L = 0.667 % and L f p = 1.333% .

provement of the performance achieved. For Figure 16, the experimental conditions, including the sample size, have been kept constant (asfor Figure ll),except for the column length,


1969

t

x

f-

n

x h _J

1 0 E

W N

.?

0 0

0

0

-

? 0

0

9

2

4

Time (win)


Same conditions as for Figure 16, except sample size and loading factors, L,, = 1.333% and L,, = 2.667%. which has been multiplied by 2, thus, effectively decreasing by half the loading factor. The zone profiles are very comparable to those in Figure 11 and there is no improvement in the separation. The separation time is double, but the zone widths and heights remain nearly the same. For Figure 17, the experimental conditions remain the same as for Figure 16, except the sample size is double, so the loading factor is now the same as for Figure 11. Although there is not yet a detectable plateau, the maximum of the zone profile is now at the plateau concentration predicted by the ideal model and the separation is improved. The displacement train profile is nearly identical with the one obtained on Figure 10, with the same loading factor, a column twice as short but a mass transfer coefficient 2.5 times as large (number of mass transfer stages in the columns, 645 for Figure 17, instead of 735 for Figure 10). Figure 17 shows that, when the mass transfer is slow, the separation and the production rate of displacement chromatography can be improved by increasing the column length, and injecting proportionallymore solute (so as to keep constant the loading factor and the throughput). We note that the improvement observed above (cf. Figure 16 and 17) in the separation performance when doubling the loading factor is contrary to what is expected in ideal chromatography, which predicts that there should be no sample size effect, as long as the loading factor is smaller than the optimum. In real columns, concentrationshocks do not appear on both sides of a large size displacement zone, but shock layers, whose width is proportional to the column HETP. When the sample size decreases, the plateau narrows down, but the shock layer thickness remains constant. Eventually, the plateau disappears when the tops of both shock layers come into contact. Finally, Figure 18 shows the result obtained by decreasing the mobile phase velocity, at constant column length. It has been assumed here that the velocity has no effect on the height of a mixing stage ( 2 D / u 0 ) .The results of the displacement

Figure 18. Simulated zone profile for displacement chromatography. Same conditions as for Figure 11, except mobile phase flow velocity: u,, = 0.02 cm/s, and loading factors L,, = 1.66% and L,, = 3.33%.

*

x

m

x h

i

8. 9

W N

00

0

0

-

0 0

0 0

40

42

44

46

40

Time (mid

Figure 19. Simulated zone profile for displacement chromatography. Same conditions as for Figure 11, except mass transfer coefficients k f 2 = 0.20 s-' and k,, = kt = 1.0 s-'. experiment are much better, because a reduction in the mobile phase velocity results in a proportional decrease of the height of the mass transfer stage (compare Figures 11 and 18). We

1970


r 3-

1

44 Time ( m i d


Same conditions as for Figure 11, except mass transfer coefficients

k,, = 0.20 s-' and k,, = k,l = 1.0 s-'.

must emphasize, however, that this result is valid only if the mobile phase diffusion coefficient of the compounds studied is small (e.g., proteins). Otherwise, the improvement could be only marginal. The critical parameter for the onset of a displacement train appears to be the ratio of the column length to the height equivalent to a theoretical plate; i.e., a minimum value of the column plate number is required for a successful displacement experiment. 5. Influence of the Mass Transfer Kinetics (kf,,# kf2). All the results presented here have been obtained with the assumption that the mass transfer coefficients of the two components of the binary mixture studied and the displacer are equal (Figures 1 to 18). When the kinetics of mass transfers proceed a t different rates for these three compounds, the fronts of the zone profiles of the compounds that transfer rapidly from one phase to the other are much steeper than those of the compounds that exchange more slowly. The compounds that have a high mass transfer rate cannot, however, separate easily from those that have a slow mass transfer kinetics, and this may result in paradoxical situations. These effects are illustrated by Figures 19 and 20. For Figure 3, the mass transfer coefficients were all equal to 1 s-l. For Figure 11,they were all equal to 0.2 s-l. These kfzI, figures should be compared to Figure 19 ( k f , = kfd = 1 SK = 0.2 s-l] and Figure 20 (kfz = kfd = 1 s-l, K , = 0.2 s-l). The

first separation (Figure 3) is the best. Surprisingly enough, the other three separations are almost equally bad, except that, on Figure 20, the first component, which has the slower mass transfer kinetics, can be produced with a somewhat better yield under the conditions of Figure 20 than under the other two sets of conditions. In these three cases, there is an interference between the first component and the displacer band, so the second component cannot be prepared at a high degree of purity with any significant recovery yield. Increasing the mass transfer coefficient from 1 to 50 s-l for the two components which have fast mass transfers does not change the profiles significantly.

CONCLUSION As results from the work described here, high production rates and satisfactory recovery yields in preparative liquid chromatography can be obtained only by using chromatographic systems for which mass transfer between phases is sufficiently fast, and efficient, well-packed columns. The successive development of applications of displacement chromatography, especially in the life sciences, requires some understanding of the factors that influence kinetic phenomena in chromatographic columns. The relationships between the kinetics of mass transfer in chromatographic systems, the nature of the retention mechanisms involved, and the nature of the components of the mixture whose separation is studied are largely unknown. It is also likely that the kinetics depends on the mobile phase concentration of the compounds that are to be separated. A solution to these important problems would help considerably in the development of strategies for the rapid optimization of separation schemes by preparative liquid chromatography. LITERATURE CITED Tiselius. A,, Ark. Kemi, Mineral. Geot. 1943, 18, 1. Horvath, Cs.; Nahum, A.; Frenz, J. H. J. Chromatogr. 1981, 218, 365. Glueckauf, E. Discuss. faraday SOC. 1949, 7 , 12. Helfferich, F.; Klein, G. Multicomponent Chromatography. A Theory of Interference: Marcel Dekker: New York, 1970. (5) Rhee, H. K.; Aris, R.; Amundson, N. R. Proc. R. SOC. London. A 1970, A267, 419. (6) Rhee, H. K.; Amundson. N. R. AZCh€ J. 1982, 28, 423. (7) Katti, A. M.; Guiochon, G. J. Chromatogr. 1988, 449, 25. (8) Philips. M. W.; Subramanian, G.; Cramer, S. M. J. Chromatogr. 1988, 454, 1. (9) Lin, 8.; Golshan-Shirazi, S.; Guiochon, G. J. Phys. Chem. 1989, 9 3 , 3363. (IO) Golshan-Shirazi. S.; Lin, B.; Guiochon, G. J. Phys. Chem., in press. (11) Thomas, H. C. J. Am. Chem. SOC. 1944, 66, 1664. (12) Goldstein, S., Proc. R . SOC.London, A 1953, A219, 151. (13) Wade, J. L.; Bergold, A. F.; Carr, P. W. Anal. Chem. 1987, 59, 1786. (14) Huang, J. X.; Guiochon, G. J. Coltold Interface Sci. 1989, 128, 577. (15) Horvath, Cs.; Lee, A. Communication to HPLC'88, Washington, DC, June 1988. (16) Lin. B.; Guiochon, G. S e p . Sci. Techno/. 1889, 2 4 , 31. (17) Lin, B.; Ma, 2.; Guiochon, G. J. Chromatcgr., In press. (18) Knox, J. H.; Saleem, M. J. Chromatogr. Sci. 1969, 7 , 745. (19) Golshan-Shirazi, S.; Guiochon, G. Anal. Chem. 1988, 8 0 , 2364. (20) Cooney, D. 0.; Lightfoot, E. N. Znd. Eng. Chem. fundam. 1966, 5 , 212. (21) Van Deemter. J. J.; Zuiderweg, F. J.; Klinkenberg. A. Chem. fng. Sci. 1956, 5 , 271. (1) (2) (3) (4)

RECEIVED for review March 6, 1989. Accepted May 23, 1989. This work has been supported in part by Grant CHE-8715211 of the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory.

Influence of the mass transfer kinetics on the separation of a binary

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