Chern. Anal. (Warsaw), 41, 139 (1996) REVIEW

Mass Overloading in Elution Liquid Chromatography

by J.S. Kowalczyk and T. Wrobel

Department ofAnalytical Chemistry, Technical.University ofGdaflSk, 80-952 Gdansk, Poland

Key words: preparative liquid chromatography, optimization of production rate, mass

or concentration overloaded systems

The problems associated with description of mass overloaded chromatography systemsare presented. It shoul4 be stressed that up to date, because of complexity of discussedproblems, applicability of all existing models is limited to single compound or binarymixtures, when Langmuir isotherms are valid.As the first, a theoretical model, assuming the eXistence ofa col umn of infini te efficiencyis described. The prediction of separation conditions for real column, by employing theideal model, is not possible. TIle next semi-ideal model, considering finite efficiency ofa column, is discussed. The model enables to determine separation conditions in realcolumn, but, asa result of some simplifications, only for selected cases. Next presented,the equilibrium-dispersive Guiochon and coworkers model assumes that all thecontribu­tions to band broadening can be lumped into a single apparent dispersion coefficient.In all cases of practical importance, despite some simplifying approximations, a pro­posed procedure of numerical calculations gives satisfactory results. Finally, a semi-em­pirical model wi th the concept of"column blockage" proposed by ~nyder and coworkersis presented. The practical usefullness of the models should be verified on the basis ofcomparison of parameters obtained for its practical ranges.

przedstawiono problemy zwi'\zane z opisem masowo przeladowanych uklad6w duo­matografii cieczowej. Stwierdzono, ze dotychczas, na skutek zlozonosci dyskutowanychproblem6w, wykorzystanie istniej,\cych modeli jest ograniczonedo uklad6w maksymal­nie dwuskladnikowych charakteryzowanych izotermami Langmuira. Jako pierwszyopisano teoretyczny model idealny, opracowany przez Guiochona, zaktadaj,\cy istnieniekolumny 0 niesk06czonej sprawnosci. Przewidywanie, na podstawie tego modelu,warunk6w rozdzielania w realnej kolumnie nie jest mozliwe. Jako nastetpny om6wionomodel semi-idealny, zaktadaj,\cy skollczon,\ sprawnosc kolumny. Na· skutek zatoze6upraszczaj,\cych ma on ograniczon,\ przydatnosc praktyczn,\. Trzecim z kolei jest modelr6wnowagowo-dyspersyjny, Guiochona i wsp6lpracownikow, najbardziej przydatny doopisu omawiao.ych tu procesow, w ktorym zatozono,.ze ksztattowaniepasm rozdziela-

140 J.S. Kowalczyk and T. Wrobel

nych substancji mozna charakteryzowae poprzez wspolczynnik dyspersji osiowej. Wy­niki obliczen numerycznych s,\ dla tego modelu zadowalaj'\ce.Jako ostatni opisano polempiryczny model Snydera i wsp6lpracownik6w, kt6rzy opra­cowali koncepcj<e tzw. "kolumny blokowanej".Ocena przydatnosci opisanych modeli powinna bye dokonana na podstawie porownanotrzymanych wartosci parametr6w charakteryzuj,\cych warunki rozdzielania.\'

The preparative liquid chromatography is the method of separation and purifica­tion of chemical substances on a large scale [1,2]. It is a very attractive luethod ofobtaining analytical standards too. Whereas in the analytical chromatography theproduct of separation is infonnation, in the preparative liquid chromatography theamount of material of required purity, isolated per unit of time, called productionrate, is a criterion parameter [1-"3]

Q,.Pr·=-···

1 tcA (1)

where A =rrd~/4 denotes column cross section, de - column diameter, Qr - amountof obtained substance i of required purity, te - separation time.

In order to achieve a maximum production rate the volume or the concentrationof the sample injected may be increased. The former method is called "volumeoverload" and the latter - "mass overload" or "concentration overload". Then theproduction rate may be described as:

C9v?Prj = Rj -'_I (2)

tc

where R; is the recovery of injected substance i, C? - concentration of injectedsubstance i, v? - volume of injected solution of substance i.

Let us recall the shape ofpeaks for analytical injections and for ,both kinds ofoverloading (Fig. 1). In volume overload, as in the analytical injection method, thesample concentration is kept constant and is confined to the linear range of theadsorption isotherm. Effects of sample volume increase have been well recognizedand theoretical assumptions have been experimentally verified [2,4-6]. But Knox andPyper [7] have Shown that concentration overload always allows a higher productionrate than that achieved by means of volume overload, sometimes 5-10 titnes [2,8].Therefore, apart froln papers of theoretical character, the papers dealing with practi­cal aspects of concentration overloading have been published in recent years [9,10].Conder and Purnell [11] were engaged in topics dealing with throughput of chroma­tographic system. Scott and Kucera discussed various factors that affect the loadingcapacity [12]. Hupe and Lauer [13] wondered which, and in what way parameters ofchromatographic system influence productivity (production rate). Poppe and Kraak[14] discussed the loadability term of chromatographic system. However, the papersby Gareil et al. [15] seem specially interesting. They considered the strongly nonli­near behaviour of the process and a set of its characteristic properties derived throughexperiments. Of practical significance is a test based on the additivity of independentcontributions to the statistical parameters of chromatographic peaks. It permits

Mass overloading in elution LC 141

further optimization depending on the preparative objects: size, volume and concen­tration of the sample according to the efficiency of the column and stationary phasecapacity.

ANAL. SAMPLE

Vi

VOLUME OVERLOADING v.

v.

Figure 1. Development of peak profiles during migration along a column for analytical and overloadedsamples

There are two basic limitations of preparative chromatography with concentra­tion (mass) overloading. The first of them is, obviously, too low solute solubility ina mobile phase. Even more importanmt difficulties result from the complexity ofchromatographic processes in most overloaded systems. This often makes the pre­diction of separation parameters impossible and "trial-and-error" method must beapplied.

In this paper the present state of knowledge about concentration overloading inbi-phase systems has been described and the conclusions with practical significancefor the users of preparative liquid chromatography have been reported.

Theoretical principles

An interest in the theoretical description of the above mentioned bi-phase systemsstarted with papers by Wicke [16], Wilson [17], De Vault [18], Glueckauf [19-21]and others who undertook the task of describing the ideal model taking the Langmuir

142 J.S. Kowalczyk and T. Wrobel

(3)

isothenninto account. In general, for complete description of chromatographicprocesses, mass balance equations for all lnixture components should be used.Simultaneously one should consider thermodynamics of phase equilibrium or thedistribution of a mixture components between mobile and stationary phases (equa­tions for isotherms) and kinetics of mass transfer between these phases and alsokinetics ofsorption-desoption processes. For linear range ofsorption isothenns, eachcomponent can be treated independently. Equilibrium equations are then quite simpleand the problem has an analytical solution. In mass overloaded chromatography, i.e.in so called nonlinear chromatography, description of processes is extremely com­plex. It is because retention thennodynamics and mass transfer kinetics cannot bedealt with separately. As a result, behaviour of any mixture component cannot beconsidered separately. Elution of a whole mixture should be considered as a problemwhich all aspects, including kinetics and thennodynamics of phase equilibrium,aretreated simultaneously. In other words, the complexity of nonlinear chromatographyresults from

a) isothenn nonlinearity,b) possible influence of aII other components of the mixture on the isothe~n of

a given component,c) interdependence of mass transfer kinetics of all components,d) interdependence of thermodynamic effects.Due to mentioned complexity, the general solution of a model of processes in

concentration overloaded chromatography, characterized by nonlinear isotherms hasnot been found. Also, one should bear in mind that difficulty would mount since everysingle solution requires many factors not easy to find and even more difficult tomeasure in the course of a chromatographic process. To solve the problems, varioussimplified models were proposed. Specially useful from theoretical point of view hasbeen the ideal model developed by Guiochon and coworkers [3,22-30].

The theoretical ideal model

The model assumes the existence of an ideal chromatographic column of infiniteefficiency, in which mass transfer is infinitely fast, and, hence, the system is atequilibrium at any moment. In such a column the axial dispersion as weUas resistanceto mass transfer do not occur and, hence, the flow has a plug profile. In this model,the mass balance equation for a binary mixture is following:

aCi aqi aCi-+F-+u-=Oat at az

where qi and C i are the concentrations of the component i in the stationary andmobile phases at equilibrium [q = fCC) isotherm], respectively, t and z are time andlength, respectively, F is the phase ratio [F =(1 - E)/E], Ebeing total porosity of thebed packing, u is mobile phase flow velocity.

Mass overloading in elution LC 143

(5)qi= --n---

Due to ignoring the mass transfer kinetics, the conditions are satisfied forequilibrium thermodynamics, which under these conditions plays the major role inshaping the bands ofsubstances at higher concentrations.

However, in nonlinear chromatography, components compete for adsorption sitesof the stationary phase. Therefore, to describe band profiles it is necessary to employcompetitive isotherm equations for multicomponent mixture. For binary mixture thecompetitive isotherm equation is as follows

qs,i bi Ci~= ~1 + bi Ci + bj Cj

If saturation capacities of the column for selected components are equal the compe­titiveLangmuir isotherms are described in the following way:

arCi

1+~ bjCj1

However, for a binary mixture, we obta)in a competitive bi-Langmuir isothermequation which is the sum of two Langmui,r equations

qi=ai,ICi

+ai,ZCi

(6).n n

1 + ~bolCo 1 +~ bozC·j, J j, J1 1

where aj , bj,jare the isothenn coefficients.For the Langmuir isotherm we can calculate derivatives from·· mass balance

equations, and then, by rearrangements the next equation can be obtained

abl~,.z.- (a - 1 +abi c? - b2~)r - b2C~ =0 (7)

when C1 = C~ and C2 =C~.In this equation a =k',f/kP =ava}, r =dC1/dCz, C~ is the concentration of com­

ponent i in a sample pulse, a denotes selectivity, and (0 - column capacity factor atinfinite dilution for component i.

The isotherms for a given system are obtained by measuring parameter· ofisothenn for single component and inserting this parameter into the model of amulticomponent Inixture isothenn.

When Gibbs adsorption equations apply and when saturation capacities of thecolumn for components are different, then the description of tbe system by theLangmuir isotherm is not satisfactory. If isotherms for two components are differentthen the ideal adsorbed solution theory predicts the procedure of numerical calcula­tions. When saturation capacities are different but each component is described by aLangmuir isotherm then the Levan and Vermeulen isotherm as atrial of the correctionthe competitive Langmuir isotherms is used. By using the ideal adsorbed solution

144 J.S. Kowalczyk and T. Wrobel

theory and Gibbs adsorption equation,Levan and Vermeulen [27] introduced thecompetitive binary Langmuir isothenn,which is transformed into competitive Lang­muir isothenn, when saturation capacities for both components are equal. Thus theequation (7) become valid. Therefore, the exact solution for the ideal model can beobtained in a compact form for a single compound for any isotherm and a binarymixture for the' competitive Langmuir isotherms. The correctness of predictionsdepends only on the reliability of isotherms determined for given system. In thismodel, the asymptotic solutions are triangular profiles in substance zones. However,this assumption is valid when the so called loading factor Lf (ratio of componentamount in a sample to column capacity) is not greater than 1% (Lfs 1%) [27].

The significance of the ideal model can be underestimated. It is simple and hasanalytical solutions in some theoretically important cases. The mechanism of pro­cesses described thermodynamically and determining band profiles of a substance inmass overloaded columns can be understood by employing the ideal model to itsdescription. Unfortunately, the prediction of all separation parameters for a realcolumn by employing the ideal model is, in principle, not possible. Therefore, manyauthors have undertaken studies on column process models, which could, at least toa limited range, be useful in practical applications.

The Knox and Pyper model

The simplest model is proposed by Knox and Pyper [7]. According to this modelband profiles are triangular, and band shape,S are not influenced by interactionsbetween substances. The equations used for the description of this model are offundamental significance and found application in the description of the model byseveral authors. The considerations by Knox and Pyper were significant for thesuccessive niodels.

Semi-ideal model

It is known that thermodynamic and kinetic interactions affect the band profileof·separated substances. The second type of interactions has been l!eglected in thepreviously discussed ideal model and it is the main reason of impossibility ofpredicting the separation conditions on the basis of such a model in real columns.Modification of the solutions obtained for the ideal model, considering finite effi­ciency of a column, is possible but solutions can be realized only for selected cases,using long and complicated numerical calculations. It is the result of multiparameteroptimization without a complete theoretical model.

More practical solutions, conducted in an analytical way are only possible asapproximated solutions for selected conditions defined by a semi-ideal model[7,23,24,28,30,31]. First simplifications of such a model can. be done assuming acolumn with finite efficiency and finite sample amount as a column in whichthennodynamic and kinetic interactions are independent of each other. It can be alsoassumed that the mass transfer kinetics between phases is independent of increasingconcentration of sample components. The validity of such an assumption for most

Mass overloading in elution LC 145

real cases results from the conviction that diffusion coefficients of diluted solutions(up to 5%) occurring in chromatography are nearly constant. Consequently, theintensity of nonlinear effects, depending on the amount of sorbed components,prevails in the substance zone, in which the concentration of components is high. Butthe influence of diffusion and mass transfer resistance dominate forsmaH concentra­tions of mixture components. On the basis of the above realtionships the practicalrules of selection of optimal separation conditions for nonlinear chromatographicsystems in real columns have been derived. So the optimalopcning times of thefraction collector for collecting pure components (so called "cutting times") have .been established. For example, the collection of the second component starts afterthe time [32]

2

tc,2 - tp +P+ Y(t~,2 - P)[l~~:] + () (8)

where the first three terms represent the optimal cutting time for P'maX' previouslygiven for the ideal model, and the last parameter (6) determines an additional bandwidth, resulting from kinetic effects assumed for a semi-ideal model. Otherwise 6 isequal to the difference between real and thermodynamic ,bandwidth of a sample ofthe pure second compOllent," 'equal to Lf,pp loading factor for the second band tail(ratio between the. sample size and the column s~t.urationcapacity) (32)

L (1-1/«)2 (9)'j,2p=, . (t+btTt/bz)2

The remaining symbols of equation (8) are as follow: tc,2 - second cutting titne whencomponent 2 begins to be collected, tp - width of the injected band, to ..... retentiontime of an unretained compound (dead time), tR,2 - retention t~me of component 2 atinfinite dilution, x - parameter related to the required degree of purity [3].

(1 ... Puz) (10)x=----PU2'a. '1

where PU2 is purity of component 2, '1 is positive root ofequation (7).Since the thermodynamic and kinetic contr~butions to band broadening are

assumed to be independent, the width of the real profile is given by [32]

,wr= wth + H1in (11)

The ideal model gives the thennodynamic band width [32]

W2,th = (t~,2 - P)(Uj!2 -Lt) (11a)

where WZ,th is contribution of the thermodynamics to the band width of the compo­nent 2, t~,2 is retention time of the component 2 at infinite dilution,Lt -loading factorcorresponding to the mixed zone [3]

Lf= (1 + b~:1)Lf'2 (9.)

Optimum loading factor of the second component [32) is

146 J.S. Kowalczyk andT. Wrobel

(12)

2

Lp. = (l+bl~l/bzf~~~a) (9b)

Band width for the kinetic band contributions, assumed for semi-ideal model (for asample ofextremely small size) is equal [32]

WZ,kin =4t~,z/(NJ)tl2 (lIb)

Consequently, the last parameter in eq.(8) is equal [32]

(0 0) [( (l+ko2)2 )t/2 ~() =WZ,T - WZ,th = tR,Z - t WZ,th 1 + 16kONJ~2 - 1. 2 2,th

where:

W2,thWZ,th =-00 =2Lf,Zp- Lf,Zp (lie)

tR,2-t

NJ is number of theoretical plates ofthe column at infinite dilution.For calculation of () the dependence of column efficiency on saltlple amount is

used. Due to finite efficiency of the column, the elution of the first component isslightly longer than for the ideal model.

The value of maximum production rate of the second component under new, realconditions is defined by the equation [32]

p,f3X= bZ~:lrl {I - [C~~;xr + ~rr (P)

where £ is volume void fraction of the column, S - column cross section area, U ­

average li~ear velocity of the mobile phase, <p =()Jft~,2-tO), () is defined by eq.(12),y is ratio (abtrt + b2)/(btrt + bz) used to simplify the writting of equations.

The determining of the optimal value ofLI is very important forthe real columns.It was established, on the basis of numerical calculations for these columns, that Lfvalue decreases only insignificantly in relation to the assumed and previously givenvalue for the ideallnodel. Let us consider the following empirical equation for thecorrected plate height [7]

(14)

where h denotes reduced plate height of column at infinite dilution, L - columnlength, dp - average particle size of packing material, B - molecular diffusionparameter, v - reduced mobile phase velocity, A - eddy diffusion parameter, and C- mass transfer resistance parameter.

For high values of v eq.(14) simplifies to the form [7]

h =Cv (14a)

Mass overloading in elutionLC 147

(16)

(15)

Then the equation for a plate number is as follows [32]

NJ _ 1]Dm [£]2-IIJAPC d;

where 11 denotes viscosity of the mobile phase, Dm - solute diffusion coefficient, kO-specific column permeability,AP-pressuredrop ofthemobilephase in the column.

Taking into account eq.(15) and eq.(13) together with the development express­ion for l) [according to eq.(12)], the derive4 relationship between Pr21 and v, fe' handcolumn characteristics, determined byL, dp , AP- the following equation is obtained [32]

1 22

It should be stressed that the production rate determined by the eq.(16) can beconsidered as properly calculated when v is respectively high, so the/column effi­ciency depends altnost only on resistance mass transfer (C), such u value can beobtained inpractlce, when the columns are filled with big particles in order to reachthe working column pressure (AP) without any problem. For preparative columnssuch value equals P max =20 MPa. For smallerparticles, dp< 10t-tm, at the mentionedpressure a smaller value of v is reached. Under such conditions, the parameter A[eq.(14)] describing eddy diffusion should be taken into account. At that case, thedependence for Pr is mor~ complicated than according to 'eq.(16). The solution ofthis new complicated relationship can be obtained using numerical calculations. Itcan be noticed that Pr, according to eq.(16), depends also on the ratio dp

2/L, but notseparately on dp and L. The above considerations lead to the following evaluation ofthe applicability of the semi-ideal model:

- the model enables programming of the separation conditions in a real, concen­tration overloaded column for binary mixture, when Langmuir isothenns are validfor them and therefore such a model has an advantage over the ideal model;

- conditions programming is quite precise, when difficult and time consumingnumerical calculations are applied;

- analytical solutions give approximated results, but valuable for practice, whenthe separation conditions given aboveare kept, especiallythe v value so high that theefficiency can be described by equation h ;.: Cv.

The applicability of semi-ideal model can be ilustrated by comparison of calcu­lated results with those verified experimentally. Figures 2-7 show the variation ofthe production rate of the second component as the functions of the fundamentalseparation parameters.

It should be stressed for the second time that a realistic model should take intoaccount the effects of the columns with finite efficien~y. The semi-ideal model

148 J.S. Kowalczyk and T. Wrobel

Pr2IFr0.025

r 98%

0.02

99%0.015

IT 99.S%

0.01

a=J.20.005 L=2Scm

d,= 10 JU1'N=SOOO

0

0 0.1 0.2 G.3 0.4 In

Figure 2. Production rate per unit mobile phase. flow rate versus loading factor (i.e. sample size) forvarious purity requirements. The lines are plots derived from eqs.(9), (12), (13). The pointsare determined from elution profiles of the two components obtained by calculating thesolutions of semi-ideal model, for C1-0.5 mol, ~=4.5 mol, width oCthe injected band = 1 s,isotherms: hI = 2.5; b2= 3.0; capacity factor k~o=6.0, feed composition C 1/C2 = 1:9. (Reprintedwith permission from reference [32]) .

=2.0

L=20cmN =1370n=lOO

o +o---10.G5f----O'+-J--O'-+JS---1cuf----Icus ( a-Ifa )2

G.OI

0Jl6

Figure 3. Maximum production rate per unit cross-section area of the column versus selectivity of thephase system. The points are determined from elution profiles of the two components obtainedby calculating the solutions of the semi-ideal model. The solid line is derived from eqs.(9),(12)-(14), Isotherm and feed composition as for Fig. 2. Purity 99%. (Reprinted with per­mission from reference [32])

PrieS0.014

Mass overloading in elution LC 149

Figure 4. Production rate per unit cross-section area of the column versus the feed composition forvarious column efficiency. The points are determined from elution profiles of two componentsobtained by calculating the solutions of the semi-ideal model. TIle solid lines are derived fromeqs.(9), (12)-(14). Same experimental conditions as for Fig. 3, except feed l;9mposition anda = 1.20. (~eprinted with permission from reference [32])

4000 6000 8000 10000

N

Figure 5. Production rate, per unit of mobile phase flow rate, versus the column efficiency, for variouspurity requirements. Same experimental conditions as for Fig. 2, except variable columnefficiency. (Reprinted with permission from reference [32])

considers simple relationships resulting from a real column work but under condi­tions when kinetic effects are negligible in comparison with the thermodynamicalones. Such theoretical and empirical Guiochon's and coworkers model was propa­gated earlier. Presently, that model has been rather disregarded by the authors whenthe new, disprsive-equilibrium model was elaborated by them. The previous modelwas even not mentioned in their reviews [29,30]. But the practitioners assume thesemi-ideal model still useful for typical labs, considering the actual qualifications of

150 J.S. Kowalczyk and T. Wrobel

the working personel first of all there, where the separation conditions are oftenchanged for various substances.

7

6

4

2

4.25

dp2JL (in 10"'cm)

o+----+----+---+--+---to ~ ~ ~ - ~ g

Figure 6. Maximum production rate, per unit of cross-section area,which can be achieved with a certainvalue of the inlet pressure versus the inlet pressure. Corresponding values of the ratio if/L (in10-8 cm): 1) 21.1 (P=2 MPa), 2) 13.1 (P=5 MPa), 3) 9.3 (P=lO MPa), 4) 7.65 (P=15 MPa),5) 6.65 (P=20 MPa), 6) 5.4 (P=30 MPa), 7) 4.75 (P=40 MPa), 8) 4.25 (P=50 MPa) (Reprintedwith permission from reference [32])

PriES

..1l10-a

3

2

O~-+--+---+--f---1I---+--+---t

o ~ ~ ~ ~ ~ ~ m ~

v

Figure 7. Production rate, per unit of cross-section area, versus the reduced mobile phase velocity. Thepoints are determined from elution profiles of two components obtained by calculating thesolutions of the semi-ideal model. The lines are derived from eqs. (9), (12)-(14). Plateequation:A =1,B =2, C =0.10. Isotherm coefficients and feed composition as in Fig. 2, excepta = 1.10, I1P = 20 MPa for the curves end. (Reprinted with permissionfrom reference [32])

Mass overloading in elution LC 151

Equlibrium-dispersive model of chromatography

Guiochon, Golshan-Shirazi and coworkers [26,29,30] assumed that mixturecomponents are in equilibrium between both phases and that all contributions to bandbroadening can be characterized by simple expression with apparent dispersioncoefficient. The mass equation has consequently the following form:

aCi aCi aqi a2Ci ,(17)u-+-+F-=D . -- i= I,ll

az at at a,t az2

where Da,; denotes apparent dispersion coefficient related to colu~n HETP, and fora single component

D _Hua- 2 (17a)

and F = VJVm = (1- £)/£ - phase ratio.In linear chromatography the variances ofthe contributions of the various sources

of band broadening are additive. Thus an equilibrium-dispersive model in the caseof linear. chromatography gives the correct solutions describing band shapes(profiles). But in non-linearchromatography an additivity, mentioned above, dependson solute concentration and can be described by eq.(I2)

H = 2DL + 2(~)2 -E- {I8)u I+k kkm

where H is the "height equivalent to a theoretical plate, DL - the axial dispersioncoefficient, k =F, !:J.q/!:J.c is the slope of the isotherm chord, km - the mass transfercoefficient, related to the film mass transfer coefficient and the pore diffusioncoefficient. Then, cand hence k ill eq.(18) change along the elution band profile,which makes the situation more complicated. The essential simplification for anequilibrium-dispersive model can be done, if one assumes that Da,; is independentof the concentration and has the same value as in linear chromatography. Suchsimplification is permissible when the column efficiency is not too low (30). Further­more, the solution will be satisfactory in practice because D a,; changes are nones­sential in the range of the actual concentration in chromatography. Then, the effectof Da,; for band profiles is small in comparison with the influence ofthennodynamiceffects. So, it should be remembered that band profiles are very "sensitive" for theisothenn courses. The necessity for their precise measuring is an essential hindranceto the practical application of this model.

In order to solve the mass equilibrium equation (17) the values of C, do u, Vo(column dead volume), the equilibrium isothenns qi(Cl",Ci"') and coefficient Da,;

should be known. The last value can be calculated from eq.(I8) Da,; =0.5(Hxu), ifHETP is known for linear chromatography. This is the next simplification, whichmakes the use of a model much easier. But, in spite of takingsuch simplifications ananalytical fonn of tnass equilibrium equation is not available for the described model,even for individual components [30]. Guiochol1 and coworkers proposed numerical

152 J.S. Kowalczyk and T. Wrobel

calculations in that case [29,30]. For such calculations the commonly used methodsare applied for solution of partial differential equations and their systems as forexample orthogonal collocation [33] or finite difference method [29-31]. Compli­cated, untypical algorithms are time consuming and good results are not easilyobtained. Additionally, due to instability of numerical calculations, the accumulationof errors with not properly defined parameters frequently occurs. The authors ofcomputer programmes, for such calculations, usually try to minimize the numericalerrors by modification of general algorithms for specific cases, but that approachcauses many limitations during detennination of the general properties of the ob­tained solutions.

Another approach for solution of equations set in the equilibriUln-dispersivemodel assumes ignoring the expression for axial dispersion. The differences obtainedin solutions in that approach are compensated by numerically estimated corrections.Ignoring the expression for axial dispersion simplifies the problem of solution of afirst-order differential equation. Then, using computer the task is simple and easy.But two problems arise during that approach. Firstly, the approximation error can bedetermined and precisely compensated only in the case of linear isotherm. Theaccurate determining of the transfering error is not possible in non-linear chromato­graphy. Secondly, there is no method for error compensation for a multicomponentmixture. So, such an approach is limited only to binary mixtures [29,30]. Thecorrectness of the equilibrium-dispersive model can be illustrated by SOlne examples(Fig. 8).

Gmglml ~a) c,mglml I b)

"20 40

IS 30

10 20

5 10

0 0

0 2 4 6 8 \ min 0 2 4 6 8 t/min

Figure 8. Comparison of the profiles calculated with forward-backward difference using the actualinjection profile (solid lines) and a rectangular pulse injection (dashed lines), and the ex­perimental (symbols) elution profiles of a mixture of2-phenylethanol and 3-phenylpropanol.Experimental conditions: de = 4.6 0101, L = 25 Col, dp = 10 !Am Vydac C18, methanol-waterv = 1011 min-I, sample sizes: a) Ci/C2 = 10:10 mg (Lf =7 and 7%), b) CdC2 = 30:10 mg (Lf=21 and 7%). (Reprinted with permission from reference [30])

In spite of the examples given by Guiochon and coworkers for forecasting theseparation conditions checked in practice, many users of preparative chromatographybelieve the method has limited application due to sofisticated algorithms.

Mass overloading in elution LC 153

Kaczmarski and Zapala [34] have publi.shed the paper suggesting that anyliterature analysis does not allow to state which model of mass transport is the mostcorrect for description of chromatographic mass overloading systems. After theanalysis of the usefulness of four models the authors have ascertained that the mostprecise and requiring the' least time for calculations was model, in which masstransport resistance is ignored, but exial dispersion and resistance of sorption­desorption processes were taken into account. The results were verified by compari­son with experimental data obtained for overloaded systems.

.A noteworthy semi-empirical model for mass-overloaded separations has beenpresented by Snyder and coworkers [35-47]. They have studied HPLC systems, usingCraig distribution model, assuming Langmuir isotherms and introducing reformu­lated Knox and Pyper equations. They have determined the elution curves as afunction of separation conditions, taking into account sample sizes, capacity factorsanfl COlUlllll platenlllnberS. Several silllpleequationshavebeen proposed for predict~ing the nonlinear behaviour of the compound bands, for two-solute sample.

The above overloaded model for different solute-columncompositions, underreverse-phase conditions, also for gradient elution, has been tested. Because of thestrong influence of the solutes ona deformatiOII of their band profiles a. graphicalcorrelation between the retention time of the band maxima, the apparent columnefficiency and the loading factors have been established. On the basis of the ex­perimental data of various origin, including deviations from Langmuir adsorptionbehaviour, initially formulated equations were subsequently Inodified. This modifi­cation has been modeled by the concept of semi-empirical "column blockage model"[37]. It appears that each of two solutes effectively "blocks" the front of the column,thus "reducing" the column length available for sorption of the other solute. In thesimplest case, the second solute is strongly adsorbed, preventing the first solute frombeing sorbed on that place. This reduces the effective column efficiency and capacityand results in reducing the retention of the first band. "Blockage effect" of the secondpeak by the first one is usually negligible.

Finally, Snyder and coworkers have proposed the procedure for calculation (jfthe predicted chromatogram at any. desired load from two initial runs: one atanalytical load and the other at higher load. The correctness of Snyder and coworkersseparation conditions prediction method is illustrated in Fig. 9.

It is obvious that the assumptions of Snyder and coworkers model are not fullytheoretically justified. Although Snyder gives some examples of the agreement ofexpected separation results with experimental data, Guiochon [32] asserts that thismodel is doomed to failure as it neglects the two fundamental aspects of nonlinearchromatography: the interaction between the band profiles and the occurrence ofshock or shock layers between the bands. Moreover according to Guiochon [28]"Snyder and Cox reformulated the equations derived by Knoxand Pyper,multiplyingthe number of non-independent equations and obfuscating the issue without contribu­tion to the solution".

We believe that the practical usefulness of the Snyder and coworkers model instandard labs may be checked by the errors of the expected separation parameters for

154 J.S. Kowalczyk and T. Wrobel

practical ranges of their values, and by the degree of the complication in comparisonwith models recommended by Guiochon or other authors.

I1

HET

I2 t,min

Figure 9. Comparison of Snyder's model with experiment: reversed-phase separation of two xanthines(HET- hydroxyethyltheophylline, 2.5 mg; HPT - hydroxypropyl theophylline, 25 mg). Dottedlines - preparative simulation (obtained according to the data from [38]). Solid line ­experiment (reprinted with permission from reference [38]). Experimental conditions: de =4.6 mm,L = 15 cm, dp= 5 I-tmZorbax C8; mobile phase: aqueous 0.1 mol I-I sodium phosphatemonobasic + mathanol + acetonitrile (75+20+5, V+V+V), v = 1 ml min-I, t = 1.43 min

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ReceivedJune 1995Accepted December 1995