Predictions of overloaded concentration profiles in supercritical fluid chromatography

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.

Journal of Chromatography A 1639 (2021) 461926 Contents lists available at ScienceDirect 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 (http://creativecommons.org/licenses/by/4.0/) 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: kkaczmarski@prz.edu.pl Torgny.Fornstedt@kau.se (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- https://doi.org/10.1016/j.chroma.2021.461926 0021-9673/© 2021 The Authors Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) 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 ( − εt ) ∂ q ∂ ∂c ∂ ∂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 + k1 + u2x dp dp + εt εe Fe 10Deff kext (6) εp + ( − εp ) − 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 ∂T ∂ 2T ∂P + − ux ( − 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 ∂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]: Mathematical model = λeff ∂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 = 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 and γ are the geometrical constants It was assumed that γ = 0.7 whereas γ 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 ( − εp ) R ∫ R2i 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 ρR + c + 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 − εt exp c0 + ρ2 c1 ρR + c ρR + c + c4 R TR TR TR + c 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 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 ∂ (∂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 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 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 M Le´sko, J Samuelsson, E Glenne et al Journal of Chromatography A 1639 (2021) 461926 Table 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 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 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 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 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 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 K while the temperature difference between the column center and the inner column wall is higher than 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) 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 M Le´sko, J Samuelsson, E Glenne et al Journal of Chromatography A 1639 (2021) 461926 Fig 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 is the center of the column The dotted line represents the inner column wall Table 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 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 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 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 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 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 mL min–1 and a set outlet pressure M Le´sko, J Samuelsson, E Glenne et al Journal of Chromatography A 1639 (2021) 461926 Fig 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 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 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 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) mL min–1 , (c) 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 M Le´sko, J Samuelsson, E Glenne et al 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 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 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 Comparison of experimental (dotted line) and simulated (solid line) concentration profiles at mobile phase flow rates of (a) 0.5 mL min–1 , (b) mL min–1 , (c) 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 μL 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 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 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 The simulated concentration profile calculated for different values of the parameter gamma (γ , 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 = 1), with a red dash line (γ = 10) and a black solid line (γ = 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 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 °C 2) Gradients of physicochemical parameters 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 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 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 one does not observe any difference in solute migration velocity in the radial direction, even at the lowest outlet ∂ c ( − εt ) ∂ q ∂ ∂ ∂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, 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 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 M Le´sko, J Samuelsson, E Glenne et al Journal of Chromatography A 1639 (2021) 461926 Fig 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 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- 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 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 References [1] G Guiochon, A Tarafder, Fundamental challenges and opportunities for preparative supercritical fluid chromatography, J Chromatogr A 1218 (2011) 1037– 1114, doi:10.1016/j.chroma.2010.12.047 [2] E Lesellier, C West, The many faces of packed column supercritical fluid chromatography – A critical review, Ed Choice IX 1382 (2015) 2–46, doi:10.1016/j chroma.2014.12.083 [3] M Les´ ko, D.P Poe, K Kaczmarski, Modelling of retention in analytical supercritical fluid chromatography for CO2—Methanol mobile phase, J Chromatogr A 1305 (2013) 285–292, doi:10.1016/j.chroma.2013.06.069 [4] K Kaczmarski, D.P Poe, G Guiochon, Numerical modeling of the elution peak profiles of retained solutes in supercritical fluid chromatography, J Chromatogr A 1218 (2011) 6531–6539, doi:10.1016/j.chroma.2011.07.022 [5] K Kaczmarski, D.P Poe, A Tarafder, G Guiochon, Pressure, temperature and density drops along supercritical fluid chromatography columns II Theoretical simulation for neat carbon dioxide and columns packed with 3-μm particles, J Chromatogr A 1250 (2012) 115–123, doi:10.1016/j.chroma.2012.05.071 [6] P Vajda, G Guiochon, The modeling of overloaded elution band profiles in supercritical fluid chromatography, J Chromatogr A 1333 (2014) 116–123, doi:10.1016/j.chroma.2014.01.034 [7] C Wenda, A Rajendran, Enantioseparation of flurbiprofen on amylose-derived chiral stationary phase by supercritical fluid chromatography, J Chromatogr A 1216 (2009) 8750–8758 [8] P Vajda, G Guiochon, Modifier adsorption in supercritical fluid chromatography onto silica surface, J Chromatogr A 1305 (2013) 293–299, doi:10.1016/j chroma.2013.06.075 [9] F Kamarei, P Vajda, G Guiochon, Comparison of large scale purification processes of naproxen enantiomers by chromatography using methanol–water and methanol–supercritical carbon dioxide mobile phases, J Chromatogr A 1308 (2013) 132–138, doi:10.1016/j.chroma.2013.07.112 [10] F Kamarei, A Tarafder, F Gritti, P Vajda, G Guiochon, Determination of the adsorption isotherm of the naproxen enantiomers on (S,S)-Whelk-O1 in supercritical fluid chromatography, J Chromatogr A 1314 (2013) 276–287 [11] F Kamarei, P Vajda, F Gritti, G Guiochon, The adsorption of naproxen enantiomers on the chiral stationary phase (R,R)-whelk-O1 under supercritical fluid conditions, J Chromatogr A 1345 (2014) 200–206, doi:10.1016/j.chroma.2014 04.012 [12] C Rédei, A Felinger, Modeling the competitive adsorption of sample solvent and solute in supercritical fluid chromatography, J Chromatogr A 1603 (2019) 348–354, doi:10.1016/j.chroma.2019.05.045 [13] K Kaczmarski, D.P Poe, G Guiochon, Numerical modeling of elution peak profiles in supercritical fluid chromatography Part I-Elution of an unretained tracer, J Chromatogr A 1217 (2010) 6578–6587, doi:10.1016/j.chroma.2010.08 035 [14] K Kaczmarski, D.P Poe, A Tarafder, G Guiochon, Efficiency of supercritical fluid chromatography columns in different thermal environments, J Chromatogr A 1291 (2013) 155–173, doi:10.1016/j.chroma.2013.03.024 [15] Y.P Zarichnyak, V.V Novikov, Effective conductivity of heterogeneous systems with disordered structure, J Eng Phys 34 (1978) 435–441, doi:10.1007/ BF00860269 [16] J Kostka, F Gritti, G Guiochon, K Kaczmarski, Modeling of thermal processes in very high pressure liquid chromatography for column immersed in a water bath: Application of the selected models, J Chromatogr A 1217 (2010) 4704– 4712, doi:10.1016/j.chroma.2010.05.018 [17] J.H Knox, G.R Laird, P.A Raven, Interaction of radial and axial dispersion in liquid chromatography in relation to the “infinite diameter effect, J Chromatogr A 122 (1976) 129–145, doi:10.1016/S0 021-9673(0 0)82240-2 [18] E.J Wilson, C.J Geankoplis, Liquid mass transfer at very low reynolds numbers in packed beds, Ind Eng Chem Fundam (1966) 9–14, doi:10.1021/ i160 017a0 02 [19] G Guiochon, D.G Shirazi, A Felinger, A.M Katti, Fundamentals of preparative and nonlinear chromatography, 2nd ed., Academic Press, Boston, MA, 2006 [20] B.E Poling, J.M Prausnitz, J.P O’Connell, The properties of gases and liquids, 5th ed., McGraw-Hill Professional, 2001, doi:10.1036/0070116822 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 10 M Le´sko, J Samuelsson, E Glenne et al Journal of Chromatography A 1639 (2021) 461926 [21] E.W Lemmon, I.H Bell, M.L Huber, M.O McLinden, NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 10.0, National Institute of Standards and Technology, 2018 Standard Reference Data Program [22] M Suzuki, Adsorption engineering, Kodansha, Elsevier, Tokyo : Amsterdam ; New York, 1990 [23] K Kaczmarski, J Kostka, W Zapała, G Guiochon, Modeling of thermal processes in high pressure liquid chromatography: I Low pressure onset of thermal heterogeneity, J Chromatogr A 1216 (2009) 6560–6574, doi:10.1016/j chroma.2009.07.020 [24] D.E Martire, R.E Boehm, Unified molecular theory of chromatography and its application to supercritical fluid mobile phases Fluid-liquid (absorption) chromatography, J Phys Chem 91 (1987) 2433–2446, doi:10.1021/ j100293a045 ˚ [25] D Asberg, M Enmark, J Samuelsson, T Fornstedt, Evaluation of co-solvent fraction, pressure and temperature effects in analytical and preparative super- [26] [27] [28] [29] 11 critical fluid chromatography, J Chromatogr A 1374 (2014) 254–260, doi:10 1016/j.chroma.2014.11.045 K Kaczmarski, M Mazzotti, G Storti, M Morbidelli, Modeling fixed-bed adsorption columns through orthogonal collocations on moving finite elements, Comput Chem Eng 21 (1997) 641–660, doi:10.1016/S0098-1354(96) 030 0-6 P Brown, G Byrne, A Hindmarsh, VODE: A variable-coefficient ODE solver, SIAM J Sci Stat Comput 10 (1989) 1038–1051, doi:10.1137/0910062 R Fletcher, A modified marquardt sub-routine for non-linear least squares, Atomic Energy Research Establishment, Harwell, UK, 1971 D.P Poe, S Helmueller, S Kobany, H Feldhacker, K Kaczmarski, The JouleThomson coefficient as a criterion for efficient operating conditions in supercritical fluid chromatography, J Chromatogr A 1482 (2017) 76–96, doi:10.1016/ j.chroma.2016.12.057  ... 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... 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... Comparison of experimental (dotted line) and simulated (solid line) concentration profiles at mobile phase flow rates of (a) 0.5 mL min–1 , (b) mL min–1 , (c) mL min–1 , and (d) 2.5 mL min–1 The