Predictions of overloaded concentration profiles in supercritical fluid chromatography

Journal of Chromatography A 1639 (2021) 461926

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Journal of Chromatography A
journal homepage: www.elsevier.com/locate/chroma

Predictions of overloaded concentration profiles in supercritical fluid
chromatography
Marek Les´ ko a, Jörgen Samuelsson a, Emelie Glenne a, Krzysztof Kaczmarski b,∗,
Torgny Fornstedt a,∗
a
b

Department of Engineering and Chemical Sciences, Karlstad University, SE-651 88 Karlstad, Sweden
Department of Chemical Engineering, Rzeszów University of Technology, PL-35 959 Rzeszów, Poland

a r t i c l e

i n f o

Article history:
Received 19 November 2020
Revised 17 January 2021
Accepted 18 January 2021
Available online 22 January 2021
Keywords:
Supercritical fluid chromatography
Overloaded concentration profiles
Heat balance
Mass balance

Temperature gradients
Density gradients

a b s t r a c t
Here, overloaded concentration profiles were predicted in supercritical fluid chromatography using a combined two-dimensional heat and mass transfer model. The heat balance equation provided the temperature and pressure profiles inside the column. From this the density, viscosity, and mobile phase velocity
profiles in the column were calculated. The adsorption model is here expressed as a function of the density and temperature of the mobile phase. The model system consisted of a Kromasil Diol column packed
with 2.2-μm particles (i.e., a UHPSFC column) and the solute was phenol eluted with neat carbon dioxide
at three different outlet pressures and five different mobile phase flow rates. The proposed model successfully predicted the eluted concentration profiles in all experimental runs with good agreement even
with high-density drops along the column. It could be concluded that the radial temperature and density
gradients did not significantly influence the overloaded concentration elution profiles.
© 2021 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY license ( />
1. Introduction
Interest in supercritical fluid chromatography (SFC) is still growing, mainly due to the method’s potential for environmental
friendliness as well as fast and efficient separations. The diffusion coefficients in SFC can in some cases be several times greater
than those observed in HPLC. Generally, the viscosity of the mobile phase in SFC is much lower than that in LC and, as a result,
we observe a much lower pressure drop along the column [1].
However, SFC is more complex than liquid chromatography due
to the high compressibility of the supercritical fluid [1,2]. The isenthalpic expansion of the mobile phase leads to column cooling,
and significant temperature gradients can form in the axial and
radial directions. These gradients influence the density, viscosity,
and velocity of the mobile phase as well as the diffusion and dispersion coefficients, which cannot be assumed to be constant as
is the case in HPLC. With the very high expansion of the mobile
phase, phenomena similar to those in ultra-high-pressure liquid
chromatography (UHPLC) can be observed, but their complexity is
much greater. It is therefore difficult to fully understand all the
∗

Corresponding authors.
E-mail

addresses:

(T. Fornstedt).

(K.

Kaczmarski),

different aspects that influence the supercritical fluid chromatographic process. Moreover, all the involved relationships are interdependent, and without numerical calculations, it is hard to explain the impact of the mobile phase expansion on the separation
and peak shape. Mathematical modeling of SFC is also a great challenge because the above issues, at least in the axial direction of the
column, should be taken into account.
The current most advanced numerical model is a twodimensional equilibrium-dispersive (ED) model coupled with heat
transfer, velocity distribution, and pressure distribution and has
successfully been applied to model analytical separation in SFC [3].
This model has also been used to explain sources of peak splitting
using neat CO2 as eluent [4], and has enabled theoretical analysis
of pressure, temperature, and density drops along SFC columns [5].
Moreover, it was successfully used for modeling retention in analytical supercritical fluid chromatography for CO2 -methanol mobile
phase [3].
Research into SFC has focused mainly on the analytical condition, with few papers addressing the prediction of overloaded concentration profiles. Vajda and Guiochon [6] modeled the elution
bands of methanol, which was eluted with neat carbon dioxide
on silica columns. They determined the isotherm parameters using
frontal analysis for different average pressures in the column. The
sets of isotherm parameters were then used to model the over-

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M. Le´sko, J. Samuelsson, E. Glenne et al.

Journal of Chromatography A 1639 (2021) 461926

the x and r directions, respectively (m s–1 ), α is the mobile phase
thermal expansion coefficient (K–1 ), λeff is the effective bed conductivity (W m–1 K–1 ), λw is the wall conductivity (W m–1 K–1 ),
P is the local pressure (Pa), and T is the local temperature (K).
Using the above heat transport model, temperature profiles have
been calculated for every applied flow rate and outlet pressure.
The coefficient of thermal expansion of the mobile phase was
calculated using the following equation:

loaded concentration profiles by assigning them to the section of
the column with the same pressure as the average pressure applied during their determination. A linear change in pressure along
the column was assumed. The velocity of the mobile phase in each
section of the column was calculated, additionally assuming a linear change in the temperature of the mobile phase.
Wenda and Rajendran [7] considered the separation of the
enantiomers of flurbiprofen on the chiral stationary phase in SFC
under both linear and nonlinear conditions. They determined the
competitive Langmuir and Bi-Langmuir isotherm models under different experimental conditions using the inverse method (only the
mass transfer model was used). The experiments were conducted
in such a way that the pressure drop did not exceed 35 bar while
the maximum density drop was 15 g L–1 . A procedure similar to
that of Wenda and Rajendran has been used by others [8–12]. In
this procedure, the overloaded bands are modeled but the variations in the isotherm parameters and the mobile phase velocity
with temperature and pressure are neglected or only the velocity
changes in the axial direction are included. As far as we know, no
prediction has been made of the overloaded profiles in SFC with
large temperature and density changes included in the calculation
of the heat transport model.
The overarching aim of this study is to predict overloaded concentration profiles under non- isopycnic conditions in SFC and,
when doing so, to account for all changes in the mobile phase
properties in the axial and radial directions. The overloaded concentration profiles will be calculated using the two-dimensional

equilibrium dispersive model. Here an adsorption model, which is
a function of the temperature and density of the mobile phase,
must be applied. The mathematical model, as well as the means
of estimating the parameters and calculating the temperature distribution and overloaded concentration profiles, will be validated
using basic experimental data obtained at different mobile phase
flow rates and outlet pressures, with the significant changes in
temperature and especially density being observed. A further aim
is to examine the thermal effects occurring in the SFC column and
their possible impact on the studied model cases.

α=−

λeff = εt2 λelu + εs2 λs + 4εt εs

The equilibrium dispersive (ED) model is used to describe the
mass balance over the column. The ED model in two dimensions
(i.e., the axial and radial directions) can be written as:

∂ c ( 1 − εt ) ∂ q 1
∂
∂c
1 ∂
∂c
+
+ div(uc ) =
D
+
r Dar
∂t
εt ∂ t εt

∂ x ax ∂ x
r ∂r
∂r
(5)
where c and q are the analyte concentrations in the mobile and
stationary phases, respectively (g L–1 ), Dax and Dar are the apparent dispersive coefficients in the axial and radial directions, respectively (m2 s–1 ), and u is a vector of the mobile phase velocity (see
Eq. (13)).
The axial apparent dispersive coefficient, Dax , was calculated
from the following equation [13,16]:

Dax =

εt

k1 = Fe
Fe =
Deff

k1
1 + k1

+

2

u2x dp
dp
1
+
εt εe Fe 6 10Deff kext

(6)

εp + ( 1 − εp )

1 − εe

εe
Dm εp
=
τ

∂q
,
∂c

and
(7)

where DL is the local dispersion coefficient (m2 s–1 ), Dm is the
molecular diffusion coefficient (m2 s–1 ), kext is the external mass
transfer coefficient (m s–1 ), dp is the particle diameter (m), and τ
is the tortuosity coefficient.
The apparent radial dispersion coefficient, Dar , was calculated
based on the plate height equation derived by Knox [17]:

∂T
∂T
∂T
∂P

− εt T α
+ CP,m ux
+ CP,m ur
∂t
∂t
∂x
∂r
1 ∂T
∂ 2T
∂P
+
− ux ( 1 − α T )
(1)
r ∂r
∂x
∂ r2

(εtCP,m + (1 − εt )CP,s )

∂ 2T
∂ x2

DL εe

where

The heat transport models for the column bed (1) and the column wall (2) are as follows:

+ λw

(4)

2.2. Mass balance equation

2.1. The heat balance equation

∂T
1 ∂T
∂ 2T
= λw
+
∂t
r ∂r
∂ r2

λelu λs
λelu + λs

where the porosity, εs , is the ratio of the volume of the solid phase
in the bed to the geometric volume of the column, and λs and λelu
are the solid phase and eluent conductivities, respectively (W m–1
K–1 ).

Proper modeling of the concentration profiles in SFC requires
using a model consisting of three sub-models: (i) a heat transfer
model, (ii) a mass transfer model, and (iii) a model of the distribution of the mobile phase velocity. The model used here was
successfully applied in modeling the temperature distribution inside the column and the elution of analytical concentration profiles
during SFC [3,4,13,14]. Here, the description is limited to the most
important equations. A more detailed derivation of the isotherm
model in the overloaded case is presented below.

cP ,w

(3)

where ρ is the mobile phase density (kg m–3 ). Here, the effective
thermal conductivity coefficient was calculated using the Zarichnyak correlations [15]:

2. Mathematical model

= λeff

1 ∂ρ
ρ ∂x

Da,r =

0.03dp ux

εt

+ 0.7 Dm

(8)

The external mass transfer coefficient, kext , was calculated from
the Wilson and Geankoplis correlation [18]:

(2)

Sh =

1.09

εe

Re0.33 Sc0.33

(9)

where

where CP,m , CP,s , and CP,w are the mobile phase, solid phase, and
wall heat capacity, respectively (J m–3 K–1 ). The coefficient ε t is total porosity, ux and ur are the superficial mobile phase velocities in

Sh =
2

kext dp
ux ρ dp
, Re =
and Sc =
Dm
η

η
ρ Dm

(10)

M. Le´sko, J. Samuelsson, E. Glenne et al.

Journal of Chromatography A 1639 (2021) 461926

where Sh, Re, and Sc are the Sherwood, Reynolds, and Schmidt
numbers, respectively. The dispersion coefficient, DL , was calculated from the following relationship [19]:

density on the retention factor, the following equation for nonlinear chromatography with Langmuir adsorption can be written as:

DL = γ1 Dm + γ2 ux dp

q = qs

εt

(11)

where γ 1 and γ 2 are the geometrical constants. It was assumed
that γ 1 = 0.7 whereas γ 2 was estimated. The molecular diffusion coefficient, Dm , was estimated from the Wilke–Chang equation [20] for the physicochemical conditions at the column inlet
and outlet, and the average values of Dm were taken for the calculations. The density and viscosity of carbon dioxide were taken
from REFPROP 10 [21].
The tortuosity parameter, τ , was calculated from the following
correlation [22]:

τ = εp + 1.5 ( 1 − εp )

2 R
∫
R2i 0

ρ (r, x )
rdr
η (r, x )

(13)

∂P
uoρ o (1 − εe )2
=ξ
∂x
εe3 dp2 ρ /η

(14)

(15)

where ξ is an empirical parameter considered to equal 150,
whereas the pressure drop was adjusted by estimating the external
porosity.
The radial mobile phase velocity was calculated from the continuity equation:

(16)

2.4. Adsorption isotherm model
In this work, we applied the Langmuir model based on an equation proposed by Martire and Boehm [24]. According to these authors, the dependence of the retention factor on the reduced density and reduced temperature can be written as follows:

ln(k ) = c0 +

ρ2

c1
ρR
+ c 2 ρR + c 3
+ c4 R
TR
TR
TR

(17)

where ρ R and TR are the reduced density and temperature, respectively. Parameters c0 to c4 are adjustable parameters to be estimated based on experimental data. Taking into account the definition of the retention factor, the linear isotherm model can be
written as:

q=c

εt

1 − εt

exp c0 +

ρ2
c1
ρR
+ c 2 ρR + c 3
+ c4 R
TR
TR
TR

+ c 2 ρR c

(19)

The SFC system was an ACQUITY UPC2 system (Waters Corporation, Milford, MA, USA) equipped with a PDA detector, an autosampler with 100-μL loop was used for all injections. The Waters
system utilizes an injection method called the “Partial Loop with
Needle Overfill” which also inject a small amount of weak needle
wash, used to transfer the sample liquid from a vial to the injector
loop. In this case, MeOH was used as weak needle wash. The mass
flow of the eluent was measured using a CORI-FLOW M12 Coriolis
mass flow meter (Bronkhorst High-Tech B.V., Ruurlo, Netherlands)
after the mixer. The pressures at the inlet and outlet of the column were measured using two EJX530A absolute pressure transmitters (Yokogawa Electric Corporation, Tokyo, Japan) connected to
the column using a tee. The temperature of the column wall was
measured at 20, 50, and 80% of the column length using three PT100 four-wire resistance temperature detectors with an accuracy
of ±0.2 °C (Pentronic AB, Gunnebo, Sweden). The temperature and
pressure signals were logged using a PT-104 data logger from Pico
Technology (St. Neots, UK).
All experiments were conducted at a set column temperature of
25 °C (298.15 K). Three set instrumental outlet pressures were considered: 90, 110, and 150 bar. The flow rate of the mobile phase at
all used outlet pressures ranged from 0.5 to 2.5 mL min–1 with
0.5 mL min–1 interval increments. The highest system pressure did
not exceed 400 bar. In this condition, the chromatographic system
was working in a subcritical state of carbon dioxide. The column
with small-diameter particles and a variable mobile phase flow
rate allowed us to obtain the experimental data with both small
and large significant changes in the properties of the mobile phase
inside the column. In this context it should be mentioned that the
mobile phase used in this study as neat carbon dioxide with low
temperature as well the use of the column with small-diameter
particles are not typical for preparative SFC. However, it fully enabled the achievement of the main aim of this work which is the

numerical modeling of overloaded elution concentration profiles by
considering the changes in mobile phase properties in the axial
and radial directions.
The solute was dissolved in toluene at concentrations of 0.75 g
L–1 and 20 g L–1 for analytical and preparative injections, respectively. The injection volume was 2 μL in the analytical mode. Overloaded elution profiles were investigated by injecting 5, 10, and
20 μL of phenol. Competition for the active site between the diluent and the solute is limited and it is not included in the simula-

x

1 ∂
(∂ρ ux )
=0
( ρ ur ) +
r ∂r
∂r

c1
TR

3.2. Instrumentation and procedure

and uo and ρ o are the mobile phase superficial velocity and the
density, respectively, calculated at the column inlet.
The pressure along the column was calculated using the correlation derived from the Blake, Kozeny, and Carman correlation:

−

exp c0 +

The mobile phase consisted of neat carbon dioxide (CO2 ,

99.99%) obtained from AGA Gas AB (Lidingö, Sweden). As solute,
phenol (puriss; Sigma-Aldrich, St. Louis, MO, USA) was used dissolved in toluene (AnalaR NORMAPUR; VWR International, Radnor, PA, USA). All separations were conducted on a Kromasil Diol
column (100 × 3.0 mm, 2.2-μm particle diameter; Nouryon, Amsterdam, Netherlands). Nitrous oxide (99.998%; Sigma-Aldrich) was
used to determine the dead volume of the columns, according to
˚
Asberg
et al. [25].

where (ρ /ηx ) denotes the average value of the ρ /η ratio at a given
axial position:

ρ / ηx =

1−tt

+ c 2 ρR c

3.1. Chemicals and column

The local value of the mobile phase velocity was calculated according to the following equation [23]:

η (r, x )(ρ /ηx )

εt

c1
TR

3. Experimental

(12)

uo ρ o

qs +

exp c0 +

where qs is the saturation capacity.

2.3. Mobile phase velocity distribution and pressure distribution

ux (r, x ) =

1−tt

(18)

The above equation was successfully used in predicting retention and concentration profiles at analytical scale [3,4,14]. Including the first-order dependency of the reduced temperature and
3


M. Le´sko, J. Samuelsson, E. Glenne et al.

Journal of Chromatography A 1639 (2021) 461926
Table 1
Physicochemical properties of the column included in the study.

tions. Two replicates of the injections were always made. The column was equilibrated for at least 40 min after changing the mobile
phase flow rate to achieve steady-state conditions. The analytical

and overloaded profiles were detected at 210 and 230 nm, respectively. The signal of the eluted overloaded profiles was converted
to concentration using calibration curves determined separately for
all considered mobile phase flow rates. The calibration curves were
derived based on three peaks and for the conditions prevailing in
the autosampler and at the outlet of the column. Additionally, the
concentration profiles were adjusted to maintain the mass balance.

Property

Value

Particle size [μm]
Total porosity
External porosity
Particle porosity
Thermal conductivity [W m–1 K–1 ]
External heat transfer coefficient [W m–2 K–1 ]

2.2
0.7886
0.3462
0.6767
0.7998
22.85

4.1. Temperature, pressure, density, viscosity, and velocity
distributions

3.3. Method of estimating model parameters and simulations
To accurately calculate concentration profiles given the isenthalpic expansion of the mobile phase, first, the heat and pressure

profiles inside the column must be calculated (see section 2.1). The
unknown parameters of the model (i.e., external heat transfer coefficient, external porosity, and solid phase conductivity) were estimated using the column inlet pressure and the column wall temperature measured 8 cm from the inlet at a flow rate of 2.5 mL
min–1 and set outlet pressure of 110 bar. Under these conditions,
we expect the greatest changes in the mobile phase properties.
The adsorption isotherm model parameters were estimated in
two steps. First, the analytical version of the isotherm model
(Eq. 19) was used. The model parameters were estimated from the
phenol retention times under three different conditions, as follows:
(i) a set outlet pressure of 90 bar and a flow rate of 0.5 mL min–1 ,
(ii) a set outlet pressure of 110 bar and a flow rate of 2.5 mL min–1 ,
and (iii) a set outlet pressure of 150 bar and a flow rate of 1.5 mL
min–1 . Also, the true value of the geometric constant in the correlation of the dispersion (see Eq. 11) had to be determined. This was
done based on the analytical peak recorded at a set outlet pressure
of 150 bar and a mobile phase flow rate of 1 mL min–1 . Second,
the saturation capacity was estimated from one overloaded elution
concentration profile obtained at a set outlet pressure of 110 bar
and a mobile phase flow rate of 2.5 mL min–1 . When estimating
the saturation capacity, the parameters determined in the first step
were kept constant.
When calculating the concentration profiles, first the axial and
radial temperature gradients in steady state were calculated as a
function of the physicochemical properties of the mobile phase and
its flow rate. Then the mass transfer model was solved using the
temperature and pressure obtained from solving the heat transfer model. The physicochemical properties of the mobile phase, at
each stage of integrating the model equations, were calculated using REFPROP v. 10 [21]. To discretize the spatial derivatives of the
model, the orthogonal collocation on finite elements method was
used [26]. To solve the system of ordinary differential equations,
the Adams–Moulton method implemented in the CVODE procedure
was used [27]. In all estimations of the unknown parameters, the
inverse method with the Levenberg–Marquardt algorithm was applied [28].

To accurately model elution profiles in SFC under non-isopycnic
conditions, the expansion of the mobile phase also needs to be
considered. The viscous friction will heat the column, whereas the
expansion of the mobile phase will cool the column. Therefore, the
heat and pressure profiles inside the column have to be calculated.
For this purpose, the two-dimensional heat transfer model was
used (see section 2.1). Several unknown parameters must be determined: the external porosity, the effective thermal conductivity
of the solid phase, and the external heat transfer coefficient (see
section 3.3 for more details). In Table 1, these unknown parameters are presented using typical values found in chromatographic
systems. The external heat transfer coefficient is used in boundary conditions to solve Eq. (1) and capture the heat transfer between the column oven and the column wall. Here, the external
heat transfer coefficient is 22.85 W m–2 K–1 for the air-controlled
temperature mode. The external porosity is 0.3462 based on the
Blake, Kozeny, and Carman correlation and the permeability coefficient is 150. The effective conductivity of the solid phase was
estimated to 0.7998 W m–1 K–1 , see section 3.3 for more details.
The matrix of the particle is inorganic silica with a conductivity of
1.4 W m–1 K–1 . However, as observed here the ligands of the stationary phase are 2-(3-propoxy) ethane-1,2-diol silanes, have significantly reducing the heat conductivity of the stationary phase.
The heat balance and estimated velocity distribution were validated by comparing the calculated pressure at the column inlet
and the calculated wall temperature in three positions (i.e., 2, 5,
and 8 cm from the column inlet) with the corresponding experimental values. The calculated and measured temperatures were in
good agreement in all runs. The maximum relative error was less
than 0.13%, but the error did not exceed 0.05% in most cases. This
indicates that the maximum difference was less than 0.41 K, but
the difference did not exceed 0.2 K in many cases. The maximum
relative error of the calculated inlet pressure was 1.84%. The maximum difference in the calculated pressure was 2.76 bar, whereas
the difference in over 50% of runs was about 1 bar or even significantly less (for more details, see Tables S.1 and S.2 in Supplementary material). This good agreement confirms the validity of using
the heat transfer model and the correctness of the pressure and
temperature distribution calculations in the studied case.
As expected, the greatest changes in the properties of the mobile phase occur at the lowest considered backpressure and the
highest mobile phase flow rate. Fig. 1a presents the calculated temperature of the column bed and wall at a mobile phase flow rate

of 2.5 mL min–1 and a set outlet pressure of 90 bar. The temperature drop along the column exceeds 5 K while the temperature
difference between the column center and the inner column wall
is higher than 1 K close to the column outlet and reaches the highest value equals 1.70 K at the end of the column. At a set outlet
pressure of 150 bar, the temperature drop is smaller: we observe
2.54 K and 0.91 K for the temperature drop along the column and
the temperature difference in the radial direction, respectively (see
Fig. 1b).

4. Results and discussion
The aim of this study is to numerically model overloaded elution concentration profiles in SFC by considering the changes in
mobile phase properties in the axial and radial directions. This was
done in two steps, which are presented below. First, in section 4.1,
the heat transfer model is calibrated and validated. Moreover, the
influence of the temperature and pressure distribution on the
properties of the mobile phase is discussed. In section 4.2, the
results of calculating the overloaded concentration profiles using
the temperature and pressure distributions are presented and discussed.
4


M. Le´sko, J. Samuelsson, E. Glenne et al.

Journal of Chromatography A 1639 (2021) 461926

Fig. 1. Temperature distribution (in Kelvin) calculated at set pressures at the column outlet: (a) 90 bar, (b) 150 bar. The mobile phase flow rate is 2.5 mL min–1 . Radius 0 is
the center of the column. The dotted line represents the inner column wall.

Table 2
Estimated parameters of the isotherm model.

One may be surprised about the small observed radial temperature gradient in these cases. However, it can be explained by
comparing the column’s geometry and heat transport. Here, the
ratio of column length (L) to its radius (R) is equal to about 60.
Let us assume that the heat Q transported in the axial (Qx ) and
radial direction (Qr ) are equal. We know that Qx is proportional
to R2 · gradx (T ) and Qr is proportional to 2RL · gradr (T ). Because
Qx = Qr we get that gradr (T ) = R/2L · gradx (T ). In our case, 2L /
R is equal to about 120, so in other words, the changes in temperature in the radial direction as compared to axial direction would
be 120 times smaller.
Fig. 2a shows the density of the mobile phase along the column at a mobile phase flow rate of 2.5 mL min–1 and a set outlet
pressure of 90 bar (note that the column wall is not plotted as
in Fig. 1). Here, we observe a density drop of 116 kg m–3 along
the column and of 10 kg m–3 in the radial direction at the column outlet. Like as the temperature changes, at a higher set outlet pressure of 150 bar, the density changes inside the column are
smaller, being 98 kg m–3 and 4 kg m–3 in the axial and radial directions, respectively (see Fig. 2b). Note that the scale differs between Fig. 2a and 2b due to the large difference in the density at
the 90 and 150 bar set outlet pressures. It should also be noted
that the calculated pressure drop (not presented in the figure) at
the 2.5 mL min–1 volumetric flow rate exceeds 200 bar and increases with increasing outlet pressure, i.e., 203 and 221 bar at
the 90 and 150 bar set outlet pressures, respectively. In this region, CO2 is a compressible liquid (i.e., subcritical condition). Here,
the viscosity increases as the pressure increases and decreases as
the temperature increases. Compare Fig. 2 in which the viscosity
is plotted at set outlet pressures of (c) 90 and (d) 150 bar, respectively. The mobile phase properties in this region are thus similar
to those of typical liquids.
From Eq. (13) follows that the properties of the mobile phase
influence the mobile phase velocity, which is no longer constant
across and along the column and can have a significant influence
on the eluted profiles. Fig. 3 shows the calculated superficial velocity inside the column at a 2.5 mL min–1 volumetric flow rate
and set outlet pressures of (a) 90 and (b) 150 bar, respectively. The
thermal expansion of the mobile phase results in the increase of
the mobile phase velocity and thus the increase of the concentration profile movement. The increase in velocity is closely related
to the density drop, see Fig. 2a and 2b. In the cases studied, the

superficial velocity increases from the inlet to the outlet of the
column by 12 and 10% at set outlet pressures of 90 and 150 bar,

Parameter

Value

Standard deviation

95% confidence level of
Student’s t-test

co
c1
c2
qs

0.1663
6.781
-2.230
252.5

1.248E-03
2.224E-02
3.260E-03
1.653E-01

2.499E-03
4.452E-02
6.527E-03

3.248E-01

respectively. The radial gradient of the velocity due to the temperature gradient is also observed, see Fig. 3a and 3b. The velocity
increases from the colder center of the column to the warmer wall
region, except in the region near the column inlet where the opposite tendency is observed.
4.2. Overloaded concentration profiles
Above, the temperature and pressure distributions, as well as
the velocity, viscosity, and density of the mobile phase, were calculated; here they are used to calculate overloaded concentration
profiles for solving the two-dimensional mass transport model.
One of the most important things in simulating concentration profiles is to use an appropriate adsorption isotherm model – in this
case, the modified Langmuir isotherm model, which also considers
the reduced density and reduced temperature (see Eq. [19]). The
model parameters were estimated in two steps (see section 3.3)
and are presented in Table 2. It is worth noting that the first three
parameters, i.e., c1 to c3 in Eq. (17), are enough to forecast the retention times in the considered experimental condition. It follows
from the calculated confidence level that the most important term
in the isotherm model is parameter c2 , which is connected with
the reduced density (see Table 2). In other words, the density of
the mobile phase is the major factor that controls solute retention. Moreover, the values of parameters c1 and c2 are positive and
negative, respectively. This indicates that the solute retention decreases at a constant temperature with increasing density and that
the retention increases with decreasing temperature at constant
density; both these observations are typical of SFC. In Fig. 4, the
retention factor for phenol is plotted as a function of the density
and temperature of the mobile phase.
The γ 2 parameter in Eq. (11) had to be determined. This parameter was estimated based on the analytical peak obtained at a
mobile phase flow rate of 1 mL min–1 and a set outlet pressure
5


M. Le´sko, J. Samuelsson, E. Glenne et al.

Journal of Chromatography A 1639 (2021) 461926

Fig. 2. Density (a, b) and viscosity (c, d) distributions calculated at the column outlet: (a, c) 90 bar, (b, d) 150 bar. The mobile phase flow rate is 2.5 mL min−1 .

Fig. 3. As in Fig. 1, but the velocity (m s−1 ) profiles were calculated.

of 150 bar. This peak was chosen because the smallest temperature and density gradients are observed in this condition. The estimated value of the parameter was validated based on other analytical peaks and good agreement was indicated. The final value
of γ 2 was 17.8, which would seem to be very high. However, the
analytical peaks in all considered runs are not sharp. It should also
be emphasized that this column is not a common column available for sale. It was specially ordered and the particles are smaller
in diameter than those used in common SFC columns. The increased dispersion could be because of imperfectly spherical particles and/or heterogeneous packing of the column. Now only the

saturation capacity in the adsorption model needs to be determined (see section 3.3 for details). The saturation capacity was determined to be 252.5 g L–1 (see Table 2).
Finally, the estimated adsorption isotherm model was used to
predict band profiles for all other experimental conditions than
those used during model calibration. Fig. 5 shows the experimental
and simulated elution profiles for 5-, 10-, and 20-μL injections of
the sample with a concentration of 20 g L–1 and at mobile phase
flow rates of (a) 0.5 mL min–1 ,
(b) 1 mL min–1 , (c) 2 mL min–1 , and (d) 2.5 mL min–1 . The set
outlet pressure is 110 bar (for comparison with 90 and 150 bar see
6


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Journal of Chromatography A 1639 (2021) 461926

Supplementary Material Figs. S1 and S2, respectively). The agreement between the calculated and experimental concentration profiles is good. In all considered experiments, the proposed model

can forecast the overloaded concentration profiles well. It can be
observed that the shock layers of the peaks are not as sharp as
predicted. The front dispersion increases with decreasing injection
volume and increases slightly with increasing mobile phase flow
rate. To investigate whether this effect is due to dispersion, elution
concentration profiles were calculated for different dispersion coefficients, see Fig. 6. The calculations clearly show that the distortion
was caused by the dispersion.
As stated above, good agreement between the calculated and
experimental concentration profiles was obtained. Only a small
shift in retention times was observed, caused by imperfect prediction of the retention times using a linear isotherm model. However, including the two rejected parts of linear isotherm model parameters did not improve the model’s ability to predict the solute
retention. However, the predicted heights and widths of the calculated elution profiles are in very good agreement with corresponding experimental profiles, even though the saturation capacity was
estimated based on one peak. This indicates the correct calculation of the saturation capacity and, especially, of the association
equilibrium constant, which changes along the column. As can be
seen, the density drops along the column exceed 100 kg m–3 at

Fig. 4. Calculated retention factor of the solute as a function of density at four different mobile phase temperatures (19, 21, 23, and 25°C). The retention factor was
calculated based on the estimated isotherm model parameters.

Fig. 5. Comparison of experimental (dotted line) and simulated (solid line) concentration profiles at mobile phase flow rates of (a) 0.5 mL min–1 , (b) 1 mL min–1 , (c) 2 mL
min–1 , and (d) 2.5 mL min–1 . The set pressure at the outlet of the column is 110 bar. Phenol was used as a solute with a sample concentration of 20 g L–1 . The injection
volumes were 20, 10, and 5 μL.
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M. Le´sko, J. Samuelsson, E. Glenne et al.

Journal of Chromatography A 1639 (2021) 461926

pressure, at which the largest density drops occur. The underlying
reason for this is that the effects of radial temperature and density

gradients and mobile phase velocity gradients cancel each other
out. However, it is worth emphasizing the phenomena occurring
in SFC even though they do not have significant effects in this particular case. Let us discuss this in more detail for the second half
of the column – that is, the part from the middle of the column
to the outlet in the axial direction. The temperature increases from
the center of the column to its wall (see Fig. 1), while the density
of the mobile phase is more or less inversely proportional to the
temperature (see Fig. 2). Thus, the density decreases from the column center to the column wall. The velocity of the mobile phase
is higher close to the wall and not in the center of the column,
as is generally observed in LC systems (see Fig. 3), so the velocity gradient will make the solute move faster in the wall region.
However, the speed of the solute migration also depends on the
retention factor, which depends on both the temperature and density of the mobile phase (see Fig. 4). From Fig. 4 we can observe
that, in this particular case, the density was the most important
factor, and that temperature generally only slightly affected the retention. As stated above, the density of the mobile phase is highest
in the center of the column, so the retention factor is the lowest
in the center of the column. In the first half of the column, from
the inlet to the middle, the radial gradients discussed above are
smaller. For the temperature, the gradient is opposite to that at the
end of the column, because the temperature of the column wall is
lower than the temperature of the mobile phase entering the column (see Figs. 1a and 1b).

Fig. 6. The simulated concentration profile calculated for different values of the
parameter gamma 2 (γ 2 , in Eq. 11). The experimental profile is plotted with a blue
dotted line while the calculated profiles are plotted with a green dash dot line (for
γ 2 = 1), with a red dash line (γ 2 = 10) and a black solid line (γ 2 = 17.8), respectively. The set outlet pressure is 90 bar while the mobile flow rate is 2.5 mL min−1 .
The injection volume is 5 μL. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)

4.3. Comparison of one and two-dimensional models
In this section, we compare the solution of the proposed twodimensional column model (2D) with its one-dimensional version

(1D). The advantage of a 1D model is that the calculations could
be performed several dozen times faster as compared with the 2Dmodel, which is of great importance e.g. if model parameters are
estimated with the inverse method or process is optimized.
In the 1D model we make the following assumptions:
1) The column temperature is equal to oven temperature, – we
neglect the drop of temperature in the axial direction, which
seems to be justified in this case because temperature drop was
generally less than 5 °C
2) Gradients of physicochemical parameters do not exist in the radial direction
3) The pressure inside the column is decreasing linearly and only
in the axial direction, from column inlet to the column outlet
– this assumption is justified by the observation made among
others in [29]
4) Mass flow in any cross-section of the column is constant and
equal mass flow at the inlet to the column.

Fig. 7. Calculated solute concentration profile inside the column just before the
zone elutes at a mobile phase flow rate of 2.5 mL min–1 and a set outlet pressure
of 90 bar. Otherwise, the conditions are as in Fig. 6d.

We can now write the mass balance equation in 1D-model as:

the highest mobile phase flow rate, causing a drastic change in the
retention factor along the column. With such a density drop, the
retention factor can increase by approximately 80%. Without taking into account the adsorption isotherm calculations at each point
along the column, the peak retention and its shape could not be
correctly simulated.
As shown in this study, two-dimensional column models allow
a more detailed investigation of the movement of elution profiles,
since they account for both the axial and radial directions. This is

essential because the temperature, viscosity, and mobile phase velocity vary in both the axial and radial directions. Fig. 7 shows the
calculated elution concentration profile near the column outlet at
the highest mobile phase flow rate and a set outlet pressure of
90 bar. In Fig. 7 one does not observe any difference in solute migration velocity in the radial direction, even at the lowest outlet

∂ c ( 1 − εt ) ∂ q 1 ∂
∂
∂c
+
+
D
(u c ) =
∂t
εt ∂ t εt ∂ x x
∂ x ax ∂ x

(20)

Having pressure distribution we calculate density as a function
of position along a column on the base of REFPROP 10.
According to assumption, 4 flow rate is calculated from the
equation:

ux ( x ) =

uo ρ o
ρ (x )

(21)

The axial dispersion coefficient, Dax , is calculated as before.
Fig. 8 presented the comparison of peaks calculated at different flow rates (0.5 mL min−1 and 2.5 mL min−1 ) and back pressures (90 bar and 110 bar) with both models. The red dash line
8


M. Le´sko, J. Samuelsson, E. Glenne et al.

Journal of Chromatography A 1639 (2021) 461926

Fig. 8. Comparison of the solutions of models 1D (dash line) and 2D (solidline) at different mobile flow rate and the pressure at the outlet of the column: (a) F = 0.5 mL
min−1 , Pout = 90 bar, (b) F = 2.5 mL min−1 , Pout = 90 bar, (c) F = 0.5 mL min−1 , Pout = 110 bar, (d) F = 2.5 mL min−1 , Pout = 110 bar.

represents the solution with the 1D model and the black line with
the 2D model. As can be seen, the peaks are shifted slightly - the
difference does not exceed 2%. The difference in the position of
calculated concentration distribution depends on changes in flow
rate and temperature along the column. Larges deviations were observed at the highest flow rate, see Fig. 8b and 8d as compared
to the lower flow rate, see Fig. 8a and 8c. Because the flow rate
is increasing and the temperature is decreasing their influence on
retention is reciprocal. However, in the 1D model, the temperature changes are ignored, which can result in observed changes in
peaks positions. It should be also noticed that axial velocity, calculated with Eq. (21) is not the same as calculated in the 2D-model,
see Eq. (13).
The differences between elution zones calculated with 1D and
2D models will depend on the temperature and pressure dependency of adsorption. The aim of this study is not to formulate a
rule of thumb under which the 2D model could be replaced with
a simple 1D model, without serious loss of accuracy. Therefore, the
simpler 1D model should be used with the great caution.

5. Conclusion
This study shows that it is difficult, even impossible, to obtain

a complete census of all interactions and thus to fully understand
how temperature, density, and velocity gradients affect separation
in SFC, without the aid of numerical calculations.
The overloaded concentration profiles in SFC were predicted
using combined two-dimensional heat and mass transfer models
connected with the model that allowed the calculation of the superficial velocity of the mobile phase. The isotherm model was
derived based on a relation used for predicting the retention factor in SFC. The equilibrium constant in the isotherm model was
a function of the temperature and density of the mobile phase.
Thus, the influences of mobile phase properties on retention and
of the movement of the elution profiles due to velocity distribution on the shape of the concentration profiles and elution times
were taken precisely into account.
The prediction was made using the experimental overloaded
band profiles of phenol eluted with liquid carbon dioxide on a col-

9


M. Le´sko, J. Samuelsson, E. Glenne et al.

Journal of Chromatography A 1639 (2021) 461926

umn packed with 2.2-μm-diameter silica gel particles with diol ligands. The experimental data consisted of elution profiles recorded
at three injection volumes, five mobile phase flow rates, and three
set outlet pressures. Moreover, the mass flow, the pressure at the
column inlet and outlet, and column wall temperature were controlled. The parameters of the heat transfer model, as well as the
isotherm model parameters were estimated based on experimental
data using the inverse method. The calculated temperature distribution and pressure along the column agreed very well with the
corresponding experimental values.
Depending on the conditions of the chromatography process,
insignificant to large changes in the mobile phase temperature

and density were observed. At the highest mobile phase flow rate
(2.5 mL min–1 ) and lowest set outlet pressure (90 bar), the density drop along the column was 116 kg m–3 while the density
changes in the radial direction at a distance of 1.5 mm reached
11 kg m–3 ; these correspond to the temperature drops, which were
5 and 1.7 °C in the axial and radial directions, respectively. The mobile phase temperature and density drop resulted in other changes
in mobile phase properties. As a result of these changes, the superficial velocity increased from the column inlet to outlet by 12%
at the highest considered volumetric flow rate. The retention factor
in such conditions increased by up to 80%. Increasing the set outlet pressure slightly decreased the radial and axial gradients, while
reducing the mobile phase flow rate to 0.5 mL min–1 caused the
column to operate under more or less flat profiles of temperature,
density, and mobile phase velocity distribution.
Concentration profiles calculated using the two-dimensional ED
model taking into account the changes in mobile phase properties
and velocity distribution agreed very well with the corresponding experimental concentration profiles. A small shift in the elution times was observed. However, the shape of the concentration
profiles was perfectly maintained in all considered runs. Moreover,
distortion of the shock concentration was observed as a result of
the increased dispersion. The radial gradient was not observed to
influence the concentration profiles.
We showed that the applied methodology and mathematical model, including the isotherm model derived for overloaded
SFC, lead to good agreement between theoretical and experimental results. The model allowed us to predict overloaded concentration profiles with good accuracy over a wide range of mobile
phase changes. In the end, we also formulated a simpler onedimensional column model which with caution in some cases can
be used for modeling overloaded SFC. We must stress that the twodimensional column model is general and will work in all cases
and is therefore used in this study.

Acknowledgments and funding information
This work was supported by the Swedish Knowledge Foundation via the project “BIO-QC: Quality Control and Purification
for New Biological Drugs” (grant number 20170059) and by the
Swedish Research Council (VR) via the project “Fundamental Studies on Molecular Interactions aimed at Preparative Separations and
Biospecific Measurements” (grant number 2015–04627).

Supplementary materials
Supplementary material associated with this article can be
found, in the online version, at doi:10.1016/j.chroma.2021.461926.

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Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.

CRediT authorship contribution statement
Marek Les´ ko: Conceptualization, Methodology, Investigation,
Validation, Resources, Writing - original draft, Visualization. Jörgen
Samuelsson: Conceptualization, Methodology, Validation, Formal
analysis, Investigation, Writing - review & editing, Supervision. Emelie Glenne: Methodology, Formal analysis, Investigation.
Krzysztof Kaczmarski: Conceptualization, Methodology, Software,
Formal analysis, Data curation, Writing - review & editing, Supervision. Torgny Fornstedt: Conceptualization, Data curation, Writing
- review & editing, Project administration, Funding acquisition.
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