Tc

PD

a

ARRAA

KASE

1

rptmipsooeaumta

firti

(

0h

Journal of Chromatography A, 1333 (2014) 116–123

Contents lists available at ScienceDirect

Journal of Chromatography A

jo ur nal ho me pag e: www.elsev ier .com/ locate /chroma

he modeling of overloaded elution band profiles in supercritical fluidhromatography

éter Vajda, Georges Guiochon ∗

epartment of Chemistry, University of Tennessee, Knoxville, TN 37996-1600, USA

r t i c l e i n f o

rticle history:eceived 30 October 2013eceived in revised form 11 January 2014ccepted 13 January 2014vailable online 31 January 2014

a b s t r a c t

Three methods were used to analyze elution bands of methanol on silica, using pure CO2 as the eluent.The results of these analyses were applied to calculate overloaded elution band profiles in supercriti-cal fluid chromatography. The results obtained are compared. To ensure that the mobile phase densityvaries widely along the column bed, high volumetric flow rates of the mobile phase (CO2) were appliedto two columns packed with neat, porous silica. Then, even a slight error made in the determination of

eywords:verage volumetric flow rateupercritical fluid chromatographyquilibrium-dispersive model

the isotherm parameters or during the numerical calculations should be magnified compared to thoseobtained with a low pressure drop along the column. During the determination of the isotherms ofadsorption of methanol from liquid carbon dioxide onto silica, the inlet and outlet pressure of the col-umn, the mass flow rate and the temperature were monitored continuously. Based on these parameters,overloaded elution bands were calculated numerically using three calculation methods. The results arecompared with experimental ones.

. Introduction

The availability of reliable procedures permitting the accu-ate calculation of the profiles of overloaded elution bands wasroved to be necessary for the computer-assisted optimization ofhe experimental conditions used in preparative-scale liquid chro-

atography [1]. The basis of all the calculation procedures availables a numerical solution of the mass-balance equation of the com-ounds considered in the column. During the last twenty years,everal approaches were introduced to determine the profiles ofverloaded elution bands in large scale separations [1]. The resultsbtained when using several isotherm models were compared withxperimental chromatograms [2–4]. The conclusion was that, when

proper numerical algorithm and a suitable isotherm model aresed, it is possible to design large scale separations that achieveaximum productivity at relatively low cost [5,6]. Our purpose is

o modify the classical procedures used in liquid chromatographynd apply them to supercritical fluid chromatography (SFC).

The foundation of all the procedures used to calculate the pro-les of overloaded elution bands is a deep understanding of the

etention mechanism taking place in the chromatographic sys-em used. A precise and accurate determination of the adsorptionsotherm parameters is also an important requirement for the

∗ Corresponding author. Tel.: +1 865 974 0733; fax: +1 865 974 2667.E-mail addresses: guiochon@utk.edu, guiochon@ion.chem.utk.edu

G. Guiochon).

021-9673/$ – see front matter © 2014 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.chroma.2014.01.034

© 2014 Elsevier B.V. All rights reserved.

success of the calculation process [7–9]. There are more experi-mental parameters to affect the reproducibility and the accuracy ofthe isotherm parameters and the optimization of large scale sep-arations in SFC than in liquid chromatography. These factors werereviewed and discussed by Guiochon and Tarafder [10].

The adsorption properties of small molecules on silica sur-face have been studied by various methods. Scott and Kucera [11]showed that polar molecules able to give hydrogen bonding, suchas isopropyl alcohol or ethyl acetate dissolved in n-heptane, inter-act with silica surface and form a monolayer at low concentrationsor an adsorbed double layer at high concentrations. Lochmüller andMink [12] measured the adsorption isotherm of ethyl-acetate on asilica adsorbent from supercritical carbon dioxide and modeled theadsorption isotherm with the Langmuir isotherm model.

Recently, the nonlinear adsorption characteristics of thenaproxen enantiomers were investigated, and several methodsof isotherm determination were compared to estimate theiradsorption isotherm parameters [13]. The transfer of isothermdetermination methods from liquid chromatography to supercrit-ical fluid chromatography is now actively discussed. The issuesaffecting the transfer into SFC of the Elution by CharacteristicPoints, the Retention Time Method, the Inverse Method and thePerturbation Peak method were investigated under quasi isopycnicconditions [14].

This work discusses the consequences of the choice madebetween different approaches of modeling overloaded elution bandprofiles. To investigate the influence of the pressure drop along thecolumn on the adsorption properties, our experiments were carried

omato

owamdwctcvptac

2

2

uhmkpiagc

wmufpIto

F

Dl

2

mdmibamcffit

P. Vajda, G. Guiochon / J. Chr

ut with high pressure drops along the column. The calculationsere based on the adsorption data of methanol on two silicagel

dsorbents. Such a system is suitable to investigate adsorptionechanisms. All our calculations used the isotherm parameters

etermined by frontal analysis experiments. Three approachesere investigated for these calculations. In all cases, we used the

ontinuous plate model. The calculations were done first by usinghe volumetric flow rate set with fixed isotherm parameters in eacholumn section. Second, this flow rate was replaced by an averageolumetric flow rate calculated from the mass flow of the mobilehase and its average density inside the column and the calcula-ions were carried out using fixed isotherm parameters. In the thirdpproach, the isotherm parameters and the linear velocity werehanged in each column section.

. Theory

.1. The model of supercritical fluid chromatography

The equilibrium-dispersive model of chromatography is widelysed to describe the migration of large volumes of molecules atigh concentrations along a chromatographic column when theass transfer kinetics is fast. The main difference between the well

nown mass-balance equation introduced in liquid chromatogra-hy and the one needed in SFC is the need to incorporate the

nfluence of the significant change of the mobile phase velocitylong the column. The mass-balance equation for a chromato-raphic system that is operated with a compressible mobile phasean be written [1,15]:

∂Ci(z)∂t

+ F∂qi(z)

∂t+ ∂(Ci(z) × u(z))

∂z= Da,i

∂2Ci(z)

∂z2(1)

here Ci(z) and qi(z) are the concentrations of compound i in theobile and the stationary phases at equilibrium, respectively, while

(z) is the mobile phase linear velocity. All of these variables areunction of the position along the column axis, z, since the mobilehase density varies along the column, due to its compressibility.

n Eq. (1) t is the time and F is the phase ratio – independent fromhe density change of the mobile phase – or the ratio of the volumesf the stationary Vs and the mobile phase V0 inside the column:

= Vs

V0(2)

a,i is the apparent dispersion coefficient. All the kinetic effects areumped into this term.

.2. Frontal analysis

The simplest model of nonlinear chromatography is the idealodel. It assumes the column efficiency to be infinite, with no axial

ispersion and constant equilibrium between the two phases. Thisodel neglects completely the influence of the mass transfer kinet-

cs and of axial dispersion on the band profiles. Consistent with theasic assumptions of the ideal model, the band profiles obtaineds solutions of this model agree well with experimental chro-atograms recorded for large samples eluted on highly efficient

olumns, in which case the deviation of the equilibrium isothermrom linear behavior is important and the dispersive effects of anite column efficiency are small. According to the above assump-

ions, Eq. (1) reduces to:

∂Ci(z)∂t

+ F∂qi(z)

∂t+ ∂(Ci(z) × u(z))

∂z= 0 (3)

gr. A 1333 (2014) 116–123 117

The retention time of the breakthrough front, tR,F,i, of componenti during a frontal analysis measurement is provided by a solutionof Eq. (3):

tR,F,i =∫ L

0

1u(z)

[1 + F

qi(z)Ci(z)

]dz (4)

where L is the column length. When several simplifications andassumptions are introduced, Eq. (4) can be transformed into anequation that is practically more relevant.

1. It is possible to accurately estimate the temperature, the pres-sure and the mass flow rate of the mobile phase at any point inthe instrument used, including the column.

2. Based on the known pressure and temperature, the mobile phasedensity can be calculated accurately in any place along the col-umn z axis. Using this calculated density profile, an averagedensity value � can be introduced.

�(z) � � (5)

The value of � reflects properly the density variation along thecolumn.

3. The variable linear velocity along the column can be replaced bythe average linear velocity derived from the average density ofthe mobile phase through the equation

u(z) � u(�) (6)

4. The linear velocity is independent of the adsorption process tak-ing place along the column. In gas chromatography, it was shownthat, when the adsorption of one compound affects the flowrate, a well defined deviation from the ideal band profile can beobserved [16]. Such deviations have never yet been reported totake place in supercritical fluid chromatography. Frontal analysismeasurements provide sharp fronts, as the ideal model suggeststhat they should.

5. The adsorbed concentration of a compound on the adsorbentsurface that is variable along the column can be accuratelyreplaced by an average value derived from the average mobilephase density.

qi(z) � qi(�) (7)

6. The equilibrium mobile phase concentration of component ivaries due with the density of the mobile phase. It can be replacedby an average value, equal to its feed concentration.

Ci(z) � Ci(�) � Ci (8)

With the above assumptions Eq. (4) simplifies to:

tR,F,i = L

u(�)

[1 + F

qi(�)Ci

](9)

By rearranging Eq. (9) and using t0 = L/u(�) we obtain:

qi(�) = Ci

F

tR,F,i − t0

t0= Ci

V0

Vs

tR,F,i − t0

t0(10)

Introducing the average volumetric flow rate of the mobile phase,FV , when the column is connected to the chromatograph and using:

V0 = FV × t0 and VR,F,i = FV × tR,F,i (11)

provides

qi(�) = Ci(VR,F,i − V0)Vs

(12)

To take into account the contribution of the extra-column volume ofthe instrument, particularly those of the mixer and the connectingtubes to the retention volume of the breakthrough front measured,an extra parameter must be introduced, the retention volume of

1 omato

tcat

q

wrn

caiitbidvfl

2

ticst

q

woa

2p

czoocs

cpt

tft

wtl

�

wpt

18 P. Vajda, G. Guiochon / J. Chr

he fronts measured when the column is replace by a zero volumeonnector VR0,F . The final equation allowing the calculation of thedsorbed concentration of component i from the retention times ofhe fronts of the breakthrough curves gives:

i(�) = Ci(VR,F,i − VM − VR0,F )Vs

(13)

here VR0,F = tr0,F × FV,0 where FV,0 is the average volumetric flowate measured when the column is replaced by a zero volume con-ector.

The parameters FV and FV,0 must be carefully estimated byonsidering the density drop of the compressible mobile phaselong the column and through the channels of the chromatographicnstrument. Due to this density drop, the volumetric flow ratencreases along these channels, from the delivery pump head tohe column inlet, to the column outlet, to the detector, and to theack pressure regulator. These variations differ when the column

s replaced with a zero volume connector. A detailed analysis of theensity drop in both cases is required to estimate how the averageolumetric flow rate incorporates the changes of local volumetricow rate along the whole instrument.

.3. Isotherm equation

A modification of the classical Langmuir equation [17] was usedo describe the adsorption of methanol from liquid carbon diox-de onto the silica surface. This modified equation assumes theo-existence of two types of independent adsorption sites on theurface. Since qi(�) is an average value, the parameters of this equa-ion will be related to the average mobile phase density:

i(�) = qs,1(�)bs,1(�)Ci

1 + bs,1(�)Ci

+ qs,2(�)bs,2(�)Ci

1 + bs,2(�)Ci

(14)

here qs,1(�) and qs,2(�) are the monolayer saturation capacitiesf the two types of sites and their adsorption equilibrium constantsre bs,1(�) and bs,2(�).

.4. The Martin–Synge algorithm for modeling overloaded elutionrofiles

During the numerical calculations, the column was divided in Nontinuous sections that were considered as flow mixers of length

(with N = 500). This number of column sections was chosen inrder for the calculation to provide sharp fronts but also to requirenly a reasonable calculation time. Increasing the number of theolumn sections above 500 does not improve the peak shapes butignificantly increases the calculation time.

Eq. (3) was solved for each of these small sections assumed to beontinuously operated in the countercurrent mode, with a mobilehase linear velocity u. In each column section, constant values ofhe linear velocity and the isotherm parameters were assumed.

Eq. (3) was solved for each column section by using the concen-ration profile eluted from the previous section as the inlet profileor the next section. The resultant chromatogram is the solution ofhe simplified mass balance equation for the last column section:

dcn[t]dt

+ Fdqn[t]

dt+ u

Cn[t] − Cn−1[t]�z

= 0 (15)

here n is the sequential number of the column section. Assuminghat the column is divided into N sections, we have 1 ≤ n ≤ N. Theength of a section is �z, given by:

z = L(16)

N

here L is the column length. Cn−1 and Cn are the concentrationrofiles of the solute in the n − 1th and (n)th sections. The ini-ial condition is Cn[t = 0] = C0, which was assumed to be a modified

gr. A 1333 (2014) 116–123

EMG function. A detailed description of the calculation method waspublished by Horváth et al. [18].

Eq. (15) was solved using two basic approaches assuming in eachsmall section constant isotherm parameters and linear velocity:

1. The equation was solved in all column sections using the averageisotherm parameters – qs,1(�), qs,2(�), bs,1(�) and bs,2(�) – andlinear velocity u(�).

2. The solution was carried out with different isotherm parametersand linear velocity in each sections. The change of all parametershas been estimated for the whole column length.

3. Materials and reagents

A 4.6 mm × 100 mm Phenomenex (Torrence, CA, USA) Kine-tex silica column packed with 2.6 �m core-shell particles and a4.6 mm × 100 mm Waters Viridis (Milford, MA, USA) silica columnpacked with 5.0 �m porous particles were used in this study. Thethermodynamic void volumes of these columns were determinedby the weight difference method [19]. Their total porosities were�T,kinetex = 0.61 and �T,viridis = 0.78 cm3.

Carbon dioxide was purchased from Airgas (Knoxville, TN, USA)and methanol from Fisher Scientific (Fair Lawn, NJ, USA).

The measurements were carried out using a Waters UPC2 super-critical fluid chromatograph. This instrument includes a binaryhead pump with an accumulator for carbon dioxide, a organic mod-ifier pump, an automated injector with a 10 �L sample loop, acolumn thermostat, a back pressure regulator, a diode array UV/VISdetector and a data station.

The column was placed in the thermostated oven of the instru-ment kept at 40 ◦C. The flow rate was set on the carbon dioxidepump at 4.0 or 2.0 cm3/min. All measurements were carried out atthe constant back pressure of 104 bar. The signal of the UV-detectorwas recorded between 190 and 280 nm. The temperatures at thecolumn inlet and outlet were measured and those values were usedto estimate the temperature profile along the column.

The mass flow rate of the CO2 stream was measured with a miniCORI-FLOW instrument from Bronkhorst High-Tech B.V. (Ruurlo,NL), Model No. M13-ABD-11-0-S, Serial No. B11200776A. Thismodel provides an accuracy of ±(0.2% of the read value + 0.5 g/h).The mobile phase densities were calculated using the NIST REFPROPdatabase.

All the numerical calculations were made using the WolframMathematica 8 software.

4. Results and discussion

4.1. Determination of the average volumetric flow rates

In Eq. (13) two different volumetric flow rates were used. Onewas used when the instrument was connected to the column, theother when the column was replaced with a zero volume connector.This difference would be unimportant in liquid chromatographybut it has a great importance in SFC. Liquid carbon dioxide is highlycompressible. A pressure change in the system causes a densitychange. Replacing the column with a zero volume connector butkeeping the same mass flow rate and the same back pressure wouldresult in a strong decrease of the inlet pressure, which would cause

an important density change. Then a change of the volumetric flowrate could not be avoided.

Generally, these two volumetric flow rates can be calculatedknowing the mass flow rates Fm and Fm,0 of the mobile phase and

P. Vajda, G. Guiochon / J. Chromatogr. A 1333 (2014) 116–123 119

0.66

0.68

0.70

0.72

0.74

0.76

0.78

0.80

0.82

0.84

0.86

2 4 6 8 10

ρ CO

2

column lenght [cm ]

kinetexviridis

Fig. 1. Variations of the density of carbon dioxide along the column in the case ofthe two columns were applied in our study. In both cases during the calculation ofttt

ts

F

lccs

pdatouw

uhtotcmc

icwctvt

Table 2Best fitted parameters of the modified EMG to the experimental inlet profiles ofmethanol injected with an uniform injection time of 0.5 min to the pure carbondioxide with a setted flow rate of 4 cm3/min when the column was replaced by azero volume connector.

CMeOH v/v% � [min] � [min] tp [min] m [min]

5 0.0173 0.0288 0.4864 0.1285

TC

he density profiles temperature and pressure are assumed to vary linearly alonghe column. The applied volumetric flow rate was 4.0 cm3/min in both cases, withhe same column outlet pressures of 104 bar.

he corresponding average densities � and �0 that represent theystem properties:

V = Fm

�FV,0 = Fm

�0(17)

The mass flow rate was measured for a volumetric flow rate ofiquid carbon dioxide set at 4.0 cm3/min. When the zero volumeonnector replaced the column, the mass flow rate of supercriticalarbon dioxide was 3.77 g/min at 4.0 cm3/min and 1.89 g/min at aet 2.0 cm3/min volumetric flow rate.

During the determination of the average density of the mobilehase, it was assumed that both the pressure and the temperatureecrease linearly along the column. This linear variation is accept-ble for the temperature but it may cause a slight error in the case ofhe pressure. The temperature drop between the column inlet andutlet was only 1 ◦C. As explained above, the column was dividedp into five hundred sections and the temperature and pressureere assumed to be constant in each of these sections.

Fig. 1 shows the variation of the density as a function of the col-mn length. It is clear that the column packed with smaller particlesas a higher inlet density and experiences a steeper density changehan the one packed with larger particles. The nonlinear variationf the density profile along the column should also be noted. Due tohis non-linear behavior, the arithmetic average density betweenolumn inlet and outlet density does not provide an accurate esti-ate of the average density in the column, especially when the

olumn is operated with a high pressure drop.To calculate the average mobile phase density when the column

s replaced by a zero volume connector, knowledge of the operatingonditions of the instrument is needed. The instrument used in thisork uses a delivery pump head operated at 14◦C. The total extra-

olumn volume of the instrument was 200 �L from the pump heado the detector and 50 �L from the injector to the detector cell, botholumes measured using pure methanol. Because the mobile phaseemperature is controlled only along the last 25% of the tubing

able 1haracteristics of the liquid carbon dioxide pumped through the chromatographic colum

Column pin [bar] pout [bar] Fm [g/min]

Viridis 167.8 104.0 3.80

Kinetex 218.1 104.0 3.89

143.6 104.0 1.86

2 0.0141 0.0230 0.5094 0.11471 0.0095 0.0228 0.5403 0.1053

volume, a weighted average of the density had to be calculated, tak-ing into account the ratio of the volumes in which the temperaturewas controlled.

Key physical properties of liquid carbon dioxide, and partic-ularly its average density are listed in Table 1. They were usedto estimate the average volumetric flow rates under the differentexperimental conditions used.

4.2. The inlet profiles

The importance of making proper measurements of the extracolumn volume contributions was discussed in detail by Fornstedtet al. [20,21]. The importance of a correct interpretation of thesedata is particularly critical when a compressible mobile phase isused, as it is in SFC. As was discussed in the previous section, theelution times measured on the same instrument with and withoutthe column are not additive, due to a change of the mobile phaseproperties.

The inlet profiles of the sample plug were recorded when thecolumn was replaced by a zero volume connector and the gathereddata were modeled using an exponentially modified-EMG function[1,9]:

C(t) = 12a

erfct0 − t√

2�− erfc

tp + t0 − t√2�

+ exp

(�2

2�2+ t0 − t

t

)

×[

etp/�erfc(

�√2�

+ tp + t0 − t√2�

)− erfc

(�√2�

+ t0 − t√2�

)](18)

where � accounts for the dispersion taking place in the connectingtubes, � is the time constant of the section-type extra-column vol-ume, tp is the time of the injection and m is the residence time ofthe band in the system.

The profiles recorded upon the injection for 0.5 min of solutionsof 1, 2 and 5 v/v% methanol in CO2 and their best fit to Eq. (18)are shown in Fig. 2. The values of the parameters of Eq. (18) bestfitted to the experimental chromatograms are listed in Table 2. Therear part of the experimental inlet profiles are steep, suggestingthe lack of effective mixing of cold liquid carbon dioxide and theorganic modifier.

In the interpretation of the results of frontal analysis experi-ments, the retention time of the highest concentration injection

was used. It was corrected for the average volumetric flow rateaccording to Eq. (13). In the calculations made for the overloadedelution bands, the difference between the experimental parametersmeasured with or without the column can be taken into account by

ns at a given flow rate and at 40 ◦C.

� [g/cm3] FV,set [cm3/min] FV [cm3/min]

0.75 4.00 5.050.79 4.00 4.930.73 2.00 2.55

120 P. Vajda, G. Guiochon / J. Chromatogr. A 1333 (2014) 116–123

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1

Abs

[AU

]

time [min]

Fig. 2. Inlet profiles of methanol plugs injected in supercritical carbon dioxide, whenthe column was replaced by a zero volume connector. The injection times were0.5 min and the methanol concentrations in carbon dioxide were 1 (red), 2 (green),and 5 (black) %. The modifier pump was used as the injector. (For interpretation ofthe references to color in this figure legend, the reader is referred to the web versionof the article.)

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25

Abs

[AU

]

time [min]

Fig. 3. Selected breakthrough curves of solutions of methanol in pure liquid carbondioxide on 202 nm. Experimental conditions: Viridis column, 4 mL/min CO . Thers

mv

4

cbet

0

50

100

150

200

250

0 5 10 15 20 25 30 35 40

q [g

/cm

3 ]

CMeOH [q/cm3]

viridis FV=4.0 cm3/minkinetex FV=2.0 cm3/minkinetex FV=4.0 cm3/min

Fig. 4. Adsorption isotherm data points of methanol dilute in liquid carbon dioxideon two silicagel columns. On the Kinetex column, the adsorption isotherms were

TB

2

etention times of the fronts decrease with increasing injected concentrations. Thisuggests a Langmuir-type adsorption.

ultiplying the best fit m parameter with the ratio of the averageolumetric flow rates with and without the column.

.3. Adsorption isotherms of methanol

Breakthrough curves of methanol were recorded in the con-entration range between 0.1 and 6.0 v/v%. In this range twenty

reakthrough curves were recorded, using a method describedlsewhere [22]. Examples of the breakthrough curves recorded onhe Viridis column are shown in Fig. 3. The retention times of the

able 3est bi-Langmuir parameters obtained by fitting the adsorption isotherm data points of m

Column FV,set [cm3/min] qs,1 [g/cm3] bs,1 [c

Viridis 4.00 132.6 1.193Kinetex 4.00 30.43 1.204

2.00 37.59 0.917

determined at two set volumetric flow rates, 2.0 and 4.0 cm3/min while on the Viridiscolumn only the higher set volumetric flow rate was applied. The curves show thebest fit of the data points to the bi-Langmuir isotherm.

fronts were taken as those for which the derivative of the chro-matogram was maximum.

The calculation of the experimental concentration of methanolin the mobile phase was done by assuming that the volumetric flowof methanol is given by the set instrumental parameters and thatthe volumetric flow rate of carbon dioxide is its average volumetricflow rate.

The retention times and the equilibrium concentrations of theisotherm data points were determined using Eq. (13). Fig. 4 givesan overview of the adsorption isotherm data points and the bestfit to the isotherm model corresponding to the experimental con-ditions applied. The best parameters obtained from the nonlinearfit of Eq. (14) to the experimental data points are listed in Table 3.The approach used permits these numerical parameters to be con-sidered as the thermodynamically correct values that describe theadsorption of methanol from its solution in carbon dioxide.

The retention factors calculated from the isotherm parametersshow that methanol is strongly adsorbed on both adsorbents. Thisstrong retention permits an estimate of the errors made in the cal-culation processes. The errors made are much larger in this casethan they are for analytes that are moderately retained.

4.4. First approach: using the set volumetric flow rate

To record these overloaded elution bands, methanol wasinjected with the modifier pump into a stream of pure liquid car-bon dioxide, with a set volumetric flow rate of 4.00 cm3/min in allcases. The elution profiles of plugs of solutions of 1, 2 and 5 v/v%methanol injected for 0.5 min in a stream of liquid carbon diox-ide were recorded at 202 nm. In each case, a constant phase ratio,constant isotherm parameters (as listed in Table 3), and a constantlinear velocity were set for the two columns. The average linear

velocity was estimated as:

u(�) = FV

AcoleT(19)

ethanol measured from liquid carbon dioxide onto silicagel adsorbents.

m3/g] qs,2 [g/cm3] bs,2 [cm3/g] k

242.8 0.0167 45.20 23.35 0.1061 20.994 105.8 0.0127 22.81

P. Vajda, G. Guiochon / J. Chromatogr. A 1333 (2014) 116–123 121

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8

C [g

/dm

3 ]

time [min ]

0

5

10

15

20

25

30

35

0 1 2 3 4 5

C [g

/dm

3 ]

time [min ]

Fig. 5. Comparisons between experimental (dots) and calculated (solid lines) bandprofiles measured on the Viridis (upper) and the Kinetex (below) columns. The injec-tion times were 0.5 min, the concentrations 1, 2 and 5 v/v% of methanol in CO2, andtet

w

flTdtfitog

4

rdtdt

u

m

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8

C [g

/dm

3 ]

time [min ]

0

5

10

15

20

25

30

35

0 1 2 3 4 5

C [g

/dm

3 ]

time [min ]

he set flow rate 4.0 cm3/min. Uniform average linear velocity and isotherm param-ters along the column were assumed; the average linear velocity was derived fromhe set volumetric flow rate.

here Acol is the column cross sectional area.As discussed earlier, the difference between the set volumetric

ow rate and the actual average volumetric flow rate is significant.his difference can be neglected only in special cases [23] but it isefinitely not negligible at high pressure drops. As shown in Fig. 5,he difference between the calculated and the recorded band pro-les is significant: the higher the retention time of the peak front,he more pronounced this difference. The numerical determinationf the isotherm parameters using the set volumetric flow rate mayive overestimates of these values.

.5. Second approach: using the mean volumetric flow rate

Replacing the set volumetric flow rate by the average volumet-ic flow rate causes a change in the linear velocity of liquid carbonioxide. In this case, all the parameters of the mass balance equa-ion were assumed to be constant in every column section. Theifference between this second approach and the previous one ishe estimate of the linear velocity:

˜

(�) = FV

AcoleT(20)

This change in the flow rate results in a significant improve-ent of the agreement between the calculated and recorded band

Fig. 6. Same as Fig. 5, but the linear velocity was calculated based on the averagevolumetric flow rate.

profiles, as shown in Fig. 6. In theory, this approach could permitan accurate modeling of the chromatographic band profiles elutedby sub- or supercritical fluids, even when there is a relatively highdensity drop along the column. The advantage of using the aver-age volumetric flow rates is that this allows the application of themethods of isotherm determination and high concentration bandmodeling algorithms that were developed for high performanceliquid chromatography. With this approach, these well establishedand tested algorithms may give accurate results for supercriticalfluid chromatography as well.

4.6. Third approach: consider the change of the isothermparameters and mobile phase velocity along the column

To take into account a possible change of the isotherm param-eters and the linear velocity along the column, it is necessary toassume a function that describes the variations of these parametersalong the column. This is necessary and sufficient to implementin the numerical calculations the assumption that the isothermparameters and the mobile phase velocity vary along the column.

If linear changes of the pressure and the temperature along thecolumn are assumed, it is easy to calculate the linear velocity ineach column section knowing the temperature and the pressure.

The variation of the linear velocity along the column was describedby a fourth order polynomial:

u(�)n = p0 + p1 × n + p2 × n2 + p3 × n3 + p4 × n4 (21)

122 P. Vajda, G. Guiochon / J. Chromatogr. A 1333 (2014) 116–123

44

46

48

50

52

54

56

58

2 4 6 8 10

u [c

m/m

in]

column lenght [cm ]

Fo

wsaF

cttpatbfl

puce

2arcpphtcfltw

ictst

icilfior

32

33

34

35

36

37

38

39

40

41

0 2 4 6 8 10 80

90

100

110

120

130

140

150

q s,1

q s,2

column lenght [cm]

qs,1qs,2

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

0 2 4 6 8 100.008

0.010

0.012

0.014

0.016

0.018

0.020

0.022

b s,1

b s,2

column lenght [cm]

bs,1bs,2

Fig. 8. Variations of the bi-Langmuir isotherm parameters along the Kinetex columnas functions of the column length.

0

5

10

15

20

25

30

35

0 1 2 3 4 5

C [g

/dm

3 ]

time [min]

Fig. 9. Comparison between calculated and experimental band profiles for the varia-

ig. 7. Variation of the linear velocity along the Kinetex column with a set flow ratef 4 cm3/min.

here p0,p1, p2, p3, p4 are the best fit parameters and n is theequential number of the column section, (1 ≤ n ≤ 500). The vari-tion of the linear velocity along the column length is shown byig. 7.

Modeling the variations of the isotherm parameters is a moreomplex issue. There is no detailed mapping of these properties inhe literature. Since our goal is to review all the methods availableo calculate overloaded elution band profiles, a rough estimate isrovided here to permit the calculations. One way to map the vari-tions of the isotherm parameters along the column is to measurehe adsorption isotherm of a compound for a constant back pressureut with different inlet pressures or for the same set volumetricow rate and for different back pressures.

The same back pressure was used to determine the isothermarameters on the Kinetex column at the two volumetric flow ratessed. Each average value of the isotherm parameters determinedan be assigned to a section inside the column under the givenxperimental conditions.

In this particular case, the pressure of the mobile phase was18.1 bar in the first column section and 104 bar in the last one,ccording to the pump head pressure and the set back pressure,espectively. Assuming a linear decrease of the pressure along theolumn provides the pressure in each column section. The isothermarameters determined at a given experimental average mobilehase pressure can be assigned to the section of the column whichas the average pressure value under which the isotherm parame-ers were derived. For a 4 cm3/min set flow rate, this was the 249tholumn section. Similarly, the parameters determined with a setow rate of 2 cm3/min can be assigned to the 412th column sec-ion. These column sections correspond to the experimental set-upith the Kinetex column at 4 cm3/min set volumetric flow rate.

To describe the variations of the parameters of the bi-Langmuirsotherm model in each case, it was assumed that no parameterhange took place between the 1st and the 249th sections and thatheir variations were exponential between the 250th and 499thections. The resulting variations of the isotherm parameter alonghe column are shown in Fig. 8.

Assuming these variations of the linear velocity and thesotherm parameters, the concentration profiles in each section wasalculated along the column. The band profiles obtained are shownn Fig. 9. The main difference between Figs. 6 and 9 is the slightly

esser agreement between the recorded and calculated band pro-les for low injected concentrations. Obviously, the rough estimatef the isotherm parameters is responsible for this decrease in accu-acy.

tions of the linear velocity and the isotherm parameters assumed along the Kinetexcolumn as provided in the text. The experimental parameters are the same as inFigs. 5 and 6.

5. Conclusions

These results strengthen the conclusions of our previous analy-sis of the disturbance peaks of methanol and of its excess isothermon silicagel. The very high retention factor of this compound shows

omato

taoawpo

bramtab

vpccc

mdfiodda

btmdEaopusd

ittc

aiuifl

[[[[

[

[

[

[[

[[

P. Vajda, G. Guiochon / J. Chr

hat is very important for a study of the thermodynamics of thedsorption process from supercritical fluids [24]. The parametersf the methanol isotherm that were measured showed that theffinity for the stationary phase of small polar analytes increaseshen the average column pressure decreases toward the criticalressure. A similar trend was observed during the investigation ofther excess isotherms [23].

Overloaded band profiles of methanol eluted with liquid car-on dioxide on columns packed with silicagel particles wereecorded and calculated. The calculations were based on thedsorption isotherm parameters determined from the averageobile phase properties calculated along the columns. Based on

hese parameters, three calculation methods were used, compared,nd discussed. Their results are compared with the experimentaland profiles.

The use of the set volumetric flow rate in the calculations pro-ides profiles that agree with the recorded ones only when theressure drops along the column is low, providing nearly isopy-nic conditions, and when the column temperature is close to theritical temperature of carbon dioxide. Then, the error made in thealculations is acceptable.

Replacing the set volumetric flow rate with the average volu-etric flow rate derived from the mass-flow rate and the average

ensity of the liquid carbon dioxide along the column provides pro-les that are in a significantly better agreement with the recordedverloaded bands profiles in a wider range of experimental con-itions. The band profiles calculated and those experimentallyetermined are in good agreement, even when the pressure droplong the column is high.

The variations of the isotherm parameters along the column cane estimated by measuring the adsorption isotherm data points ofhe compound studied for a given back pressure at different volu-

etric flow rates. In each case a set of isotherm parameters can beetermined by modeling the set of adsorption isotherm data points.ach of these sets of isotherm parameters can be assigned to anverage pressure value calculated based on the column inlet andutlet pressure. This method provides estimates of the isothermarameters in sections of the column for a given experimental setp. The parameters can be assigned to the column section with theame pressure value as the average pressure applied during theetermination of the given set of isotherm parameters.

The use of the average volumetric flow rate to calculate thesotherm parameters provides an easy modeling method and offershe opportunity to extend to supercritical fluid chromatographyhe numerical methods developed earlier for this purpose in liquidhromatography.

The mapping of the variations of the isotherm parameters along chromatographic column needs more experimental work, includ-

ng systematic measurements made for several test compoundsnder different sets of experimental conditions. A rigorous compar-

son between band profiles calculated using the average volumetricow rate or considering a continuous variation of the isotherm

[

[[[

gr. A 1333 (2014) 116–123 123

parameters along the column is needed to validate or falsify thefinding reported here of a good agreement between band pro-files calculated with these two different methods. The optimizationprocedures of preparative scale separations are simpler when theaverage system and isotherm parameters related to the averagedensity of the mobile phase are introduced. To take into accountthe continuous change along the column of almost all parame-ters would make the optimization process very complicated. Thestudy of the validity and the applicability of these average valuesof the isotherm parameters and the system properties for differ-ent experimental systems is necessary because the consequenceswould have great importance.

Acknowledgments

This work was supported in part by grant CHE-1108681 of theNational Science Foundation, by financial and technical supportof the Waters Technologies Corporation, and by the cooperativeagreement between the University of Tennessee and the Oak RidgeNational Laboratory. We thank Martin Gilar (Waters TechnologyCorporation) for fruitful discussions and for his support.

References

[1] G. Guiochon, A. Felinger, D.G. Shirazi, A. Katti, Fundamentals of Preparative andNonlinear Chromatography, 2nd edition, Elsevier Academic Press, Amsterdam,2006.

[2] S. Golshan-Shirazi, G. Guiochon, Anal. Chem. 60 (1988) 2634.[3] D. Zhou, K. Kaczmarski, A. Cavazzini, X. Liu, G. Guiochon, J. Chromatogr. A 1020

(2003) 199.[4] A.M. Katti, M. Diack, M.Z. El Fallah, S. Golshan-Shirazi, S.C. Jacobson, A. Seidel-

Morgenstern, G. Guiochon, Acc. Chem. Res. 25 (1992) 366.[5] M. Enmark, J. Samuelsson, P. Forssén, T. Fornstedt, J. Chromatogr. A 1240 (2012)

123.[6] P. Forssén, L. Edström, M. Lämmerhofer, J. Samuelsson, A. Karlsson, W. Lindner,

T. Fornstedt, J. Chromatogr. A 1269 (2012) 279.[7] F. Gritti, G. Guiochon, J. Chromatogr. A 1097 (2005) 98.[8] F. Gritti, G. Guiochon, J. Chromatogr. A 1043 (2004) 159.[9] A. Felinger, F. Gritti, G. Guiochon, J. Chromatogr. A 1024 (2004) 21.10] G. Guiochon, A. Tarafder, J. Chromatogr. A 1218 (8) (2011) 1037.11] R. Scott, P. Kucera, J. Chromatogr. 149 (1978) 93.12] C. Lochmüller, L. Mink, J. Chromatogr. 409 (1987) 55.13] F. Kamarei, A. Tarafder, F. Gritti, P. Vajda, G. Guiochon, J. Chromatogr. A (2013),

http://dx.doi.org/10.1016/j.chroma.2013.08.063.14] M. Enmark, P. Forssén, J. Samuelsson, T. Fornstedt, J. Chromatogr. A 1312 (2013)

124.15] A. Rajendran, O. Kräuchi, M. Mazzotti, M. Morbidelli, J. Chromatogr. A 1092

(2005) 149.16] M.D. LeVan, C.A. Costa, A.E. Rodrigues, A. Bossy, D. Tondeur, AIChE J. 34 (1988)

996.17] I. Langmuir, J. Am. Chem. Soc. 38 (11) (1916) 2221.18] K. Horváth, J.N. Fairchild, K. Kaczmarski, G. Guiochon, J. Chromatogr. A 1217

(2010) 8127.19] R.M. McCormick, B.L. Karger, Anal. Chem. 52 (1980) 2249.20] J. Samuelsson, L. Edström, P. Forssén, T. Fornstedt, J. Chromatogr. A 1217 (2010)

4306.

21] P. Forssén, L. Edström, J. Samuelsson, T. Fornstedt, J. Chromatogr. A 1218 (2011)

5794.22] P. Vajda, G. Guiochon, J. Chromatogr. A 1305 (2013) 293.23] P. Vajda, G. Guiochon, J. Chromatogr. A 1309 (2013) 41.24] P. Vajda, G. Guiochon, J. Chromatogr. A 1308 (0) (2013) 139.