Prediction of the influence of the heat generated by viscous friction on the efficiency of chromatography columns

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Journal of Chromatography A, 1177 (2008) 92–104

Prediction of the influence of the heat generated by viscous friction on the efficiency of chromatography columns Krzysztof Kaczmarski a , Fabrice Gritti b,c , Georges Guiochon b,c,∗ a

Department of Chemical and Process Engineering, Rzesz´ow University of Technology, 35-959 Rzesz´ow, Poland b Department of Chemistry, The University of Tennessee, Knoxville, TN 37996-1600, United States c Division of Chemical Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831, United States Received 30 August 2007; received in revised form 30 October 2007; accepted 5 November 2007 Available online 12 November 2007

Abstract The combination of the heat balance in a chromatographic column percolated by a stream of mobile phase and of the model of band migration under linear conditions along such a column permits the calculation of the axial and radial temperature distributions in the column, of the elution band profiles, and of the column efficiency under different sets of experimental conditions. The calculated results are always consistent with the experimental results published by different groups and often in good quantitative agreement. Minor discrepancies arise from difficulties in deriving precise estimates of the heat transfers from the column due to the massive endfittings of the column and to uncontrolled heat transfer from the column tube to ambient air. © 2007 Elsevier B.V. All rights reserved. Keywords: Axial temperature profiles; Column efficiency; Heat generation; Heat transfer; Radial temperature profiles; Viscous friction

1. Introduction Chromatographic columns are packed with fine particles in order to achieve fast mass transfer, hence high column efficiency. The current trend in the field is toward the use of still finer particles in order to achieve faster analyses and higher throughputs in analytical laboratories. During the last few years, new packing materials have been developed and the average size of the particles of new packing materials has decreased from ca. 5 to 1.7 ␮m. There is now a number of intermediate sizes. The choice depends on the nature of the analytical problem and on the maximum pressure at which the equipment available may safely operate. This last condition is important because the viscous friction of the mobile phase forced to percolate through a chromatographic column causes intense resistance to the flow and requires high values of the head pressure. Operating columns in the optimum range of mobile phase velocities requires pressures

∗ Corresponding author at: Department of Chemistry, The University of Tennessee, 552 Buehler Hall, Knoxville, TN 37996-1600, United States. Tel.: +1 8659740733; fax: +1 8659742667. E-mail address: [email protected] (G. Guiochon).

0021-9673/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2007.11.009

between 100 and 400 bar if they are packed with 5 ␮m particles, in excess of 1000 bar if with 1.7 ␮m particles, depending on the column length, the mobile phase viscosity and the molecular mass range of the analytes. While the friction forces cause hydraulic resistance, they also generate heat everywhere in the column. This heat must be evacuated under steady-state conditions. The evacuation of this heat outside the column causes the formation of both a radial and an axial temperature gradient. Due to the radial distribution of the temperature, there is a radial viscosity gradient of the mobile phase, the latter being more viscous in the colder region against the wall than in the warmer central region. Due to this radial viscosity distribution, there is a radial profile of the mobile phase velocity, which is faster in the central region of the column than close to its wall. Then, the chromatographic bands cannot follow piston flow migration but become warped, which causes an apparent decrease of the column efficiency since the signal recorded is proportional to the average concentration of the bulk eluent exiting the column. The radial distribution of the temperature inside the column causes also a radial gradient of the retention parameters, since they depend on the temperature. Generally, adsorption equilibrium constants decrease with increasing temperature, so retention factors will be smaller in

K. Kaczmarski et al. / J. Chromatogr. A 1177 (2008) 92–104

the central region of the column than near its wall. This radial variation of the retention factors across the column will cause a loss of column efficiency. The consequences of this heat effect on the efficiency of analytical columns were generally negligible under the experimental conditions prevailing until a few years ago, although numerous authors reported such losses at high flow velocities, in analytical [1–4] and preparative [5,6] applications. With advanced columns using particles smaller than 2–3 ␮m and operating them with inlet pressures in excess of 400–500 bar, this is no longer so. It becomes important to understand the phenomenon described above in detail, quantitatively. It is, thus, necessary to model this phenomenon. The numerical calculation of the influence of the combination of an axial and a radial gradient of temperature on the column efficiency is an extremely difficult problem. It requires the systematic calculation of the eluted band profiles for a series of flow rates. This calculation is exact only if the band profiles are obtained as numerical solutions of the coupled system of the two partial differential equations that express the heat balance and the mass balance equations. Approximate solutions have been suggested [7–11]. Although useful under such experimental conditions that the heat effect is moderate, their results become incorrect and misleading for high flow velocities under high-pressure differentials. The aim of this paper is to report on the results of our calculations of the axial and radial temperature distributions inside a chromatographic column as functions of the experimental conditions (physico-chemical characteristics of the mobile phase, the solid phase, and the column tube, pressure gradient, mobile phase flow rate, and thermal boundary conditions) and to discuss the influence of these distributions on the column efficiency. We considered the temperature distributions in three particular cases of heat transfer from the column: (1) when the temperature of the column wall remains equal to the ambient temperature of the laboratory (very fast heat transfer from the external surface of the column to the surrounding air); (2) under typical laboratory conditions, when the column is not insulated but the surrounding air is still; and (3) when the column is carefully insulated. 2. Mathematical models Our approach to this problem consists in modeling it and calculating numerical solutions of the model. Two different models are needed and they must be coupled. The first model deals with the heat balance. It expresses how the heat that is generated by the viscous friction of the mobile phase during its percolation along the column is evacuated from the column under steady-state conditions. Part of this heat is carried away from the column by a mobile phase hotter at its exit from the column than at its entrance into it. The rest escapes through heat losses across the column wall and the end connecting devices. The second model accounts for the propagation of a band of solute along a column that is no longer isothermal. These two models are coupled because the parameters that affect both the thermodynamics and the kinetics of the band propagation depend on the local temperature. However, we consider here only analytical chromatography, the concentration of analytes is small and

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the effects of the variations of the analyte concentration during the band migration on the temperature distributions can be neglected. Finally, note that the column being packed in a cylindrical tube and one boundary condition being set on the wall of this tube, the problem is axially symmetrical and the models should be formulated in cylindrical coordinates. 2.1. The heat balance In formulating the heat balance, it was assumed that heat is generated inside the column due to viscous friction and that it is conducted through the packed bed, the column wall and dissipated into the air surrounding the column. The model assumptions are the following: 1. For packed beds, the axial heat dispersion and axial heat conductivity can be neglected. 2. The radial heat dispersion is negligible in comparison with the radial conductive heat transfer. 3. The mobile phase is incompressible. 4. The conductive heat transfer in the packed bed can be expressed by its effective conductivity. 5. The mobile phase flow velocity is a function of the radial coordinate but it remains constant in the axial direction. 6. The heat conductivity of the bed and its specific heat capacity are independent of the local temperature (the thermal variations of these parameters are too small within the temperature range experienced under realistic conditions). 7. Heat is conducted in both axial and radial directions of the column tube. 2.1.1. Equations The validity of the first three assumptions was discussed earlier [7,8]. Under this set of seven assumptions, the heat balance for an infinitesimal volume element of a packed bed can be given in cylindrical coordinates as [7,8,12]:   1 ∂T ∂2 T ∂T m m ∂T (εt cp + (1 − εt )cs ) + cp u = λr,ef + 2 + hv ∂t ∂z r ∂r ∂r (1) where εt is the total column porosity, cpm the mobile phase heat capacity (J/(m3 K)), cs the solid phase heat capacity (J/(m3 K)), T the local temperature (K), u the superficial velocity of the mobile phase (m/s), λr,ef the effective bed conductivity (W/(m K)) and hν is the amount of heat generated per unit volume of bed due to the work performed in the column by the stream of the viscous fluid percolating through the bed, expressed in W/m3 . The heat generated is by the product of the superficial velocity and the pressure gradient [7,8,12]: hv = −u

∂P ∂z

(2)

The heat balance for the column wall can be formulated as follows:   1 ∂T ∂T ∂2 T ∂2 T (3) cw = λw + 2 + λw 2 ∂t r ∂r ∂r ∂z

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where cw is the wall heat capacity (J/(m3 K)) and λw is the wall heat conductivity (W/(m K)). 2.1.2. Initial and boundary conditions The solution of Eqs. (1) and (3) require proper initial and boundary conditions. These conditions were formulated as follows: • Initial condition for t = 0,

we have : T (r, z) = Text

(4)

• Boundary conditions for Eq. (1): for t > 0, we have: At z = 0,

T (r, z) = T0

2.2.1. Equation In a cylindrical system of coordinates, the mass balance equation of the ED model is written as follows:   1 ∂C ∂2 C ∂q ∂2 C ∂C ∂C (7) +F +w = Dz,a 2 + Dr,a + 2 ∂t ∂t ∂z ∂z r ∂r ∂r where C and q are the analyte concentrations in the mobile and the stationary phases (g/L), respectively, Dz,a and Dr,a the axial and the radial apparent dispersion coefficients (m2 /s), w = u/εt the interstitial velocity and F = (1 − εt )/εt is the phase ratio. 2.2.2. Initial and boundary conditions • Initial condition

(5a)

for t = 0,

∂T =0 ∂z

(5b)

At r = 0,

∂T =0 ∂r

(5c)

t > tinjection ,

(5d)

For t > 0,

λr,ef

∂T ∂T = λw ∂r ∂r

For 0 < t < tinjection ,

where Text is the ambient temperature in the laboratory, T0 the inlet mobile phase temperature, L the column length (m) and Ri is the internal column wall diameter (m). • Boundary conditions for Eq. (3): for t > 0, ∂T (r, z) = −he (Text − Tw (r, z)) ∂z

(6a)

∂T (r, z) = he (Text − Tw (r, z)) ∂z

(6b)

At z = 0,

λw

At z = L,

λw

At r = Ri ,

λr,ef

At r = Re ,

λw

∂T (r, z) ∂T (r, z) = λw ∂r ∂r

∂T (r, z) = he (Text − T (r = Re , z)) ∂r

(6c) (6d)

where Re is the external column wall diameter (m) and he is the heat transfer coefficient between the external surface of the column tube and ambient temperature in the laboratory (W/(m2 K)). Under steady-state conditions, these equations can be simplified by neglecting the time dependent terms and initial conditions. 2.2. The mass balance In the formulation of the mass balance for an analyte in the column, it is assumed that the contributions to band broadening due to the finite mass transfer resistances and to axial dispersion can be lumped into an apparent dispersion coefficient. Under this assumption the mass balance equation is a simple extension of the equilibrium–dispersive model [13].

(8)

• Boundary conditions

At z = L,

At r = Ri ,

C=0 z = 0, C(r, z) = C0 ,

z = 0, C(r, z) = 0

at z = L,

∂C =0 ∂z

(9a)

(9b)

∂C =0 (9c) ∂r ∂C at r = Ri , =0 (9d) ∂r Eq. (7) must be combined with an appropriate isotherm equation: at r = 0,

q = f (C, T )

(10)

It should be noticed that the concentration in the solid phase, q, in equilibrium with the mobile phase concentration, C, depends on the temperature. The isotherm equation accounts for the temperature dependence of the retention time. 2.3. Method of calculation model solutions We need to investigate the influence of the heat generated by the stream of viscous mobile phase during its percolation through the chromatography column on the efficiency of this column. For this purpose, the mass balance model must be solved under steady-state conditions. This means that, in principle, it is sufficient to solve the stationary or steady-state version of the model expressed in Eqs. (1) and (3), neglecting the time dependent terms. However, the steady-state solution can also be derived from the limit of the solution of the time-dependent version of the model when time increases indefinitely, which is the method that was used in this work. The heat balance model was solved using a computer program that was written using the orthogonal collocation on finite elements method (OCFE) [14]. For this purpose, the column length was divided into five subdomains, the column radius into two subdomains and only one subdomain was used for the column wall. In each subdomain, three collocation nodal

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points were applied. The spatial derivatives were discretized, following the OCFE method. The set of ordinal differential equations obtained through this process was solved using the VODE solver [15]. The calculation time required to achieve the steady-state solution using a PC Athlon 2.5 GHz was less than 30 s. The mass balance model, expressed by Eq. (7), was solved using the same technique and assuming the steady-state temperature distributions calculated as explained above.

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3.4. Measurement of the temperature profiles at different axial and radial positions

The C18-Vydac 218TP column used in this work was packed and given by Grace (Columbia, MD, USA). The main characteristics the column are: length 25 cm, internal and external diameters of the column tube: 0.46 and 0.65 cm, respectively, total porosity εt = 0.66, external porosity εe = 0.402. The median particle size is equal to 4.98 ␮m. Six snap-on strain relief thermocouples of type T (copperConstantan junction, 1 mm junction size), one mini-hypodermic thermocouple model HYP0 of type T (needle diameter 0.2 mm), a handheld thermometer with two inputs and 100 adhesive pads purchased from Omega Engineering, Inc. (Stamford, CT, USA) were used to measure the temperature profiles. The ambient air temperature of the laboratory was Text = 295 K.

Seven thermocouples were used to measure simultaneously the longitudinal and the radial temperature profiles. One thermocouple was dipped into a beaker filled with water and placed by the instrument, to provide the external temperature, Text , at the moment when the longitudinal and radial temperature profiles were measured. The five snap-on strain relief thermocouples are specially designed for surface temperature measurements. They were placed at regular 5-cm intervals along the column and taped with adhesive. The HYPO needle thermocouple (1 in. long needle and 0.2 mm O.D.) was used to measure the radial temperature profile of the mobile phase exiting from the outlet frit of the column, immediately against the surface of this frit. To reach this surface, the outlet endfitting of the column was drilled to a diameter equal to the inner diameter of the column. This special custom-made endfitting is called later in this text the special endfitting. The temperature was measured successively at 11 equally spaced radial positions. The measurements of the radial and axial temperature profiles were carried out at four flow rates (0.9, 1.8, 2.8 and 3.6 mL/min) with the water-rich mobile phase and at three flow rates (1.5, 3.0 and 4.5 mL/min) with pure methanol. To assess the pressure gradient along the column, its inlet was directly connected to the instrument while the column outlet was kept unconnected. The total inlet pressure required to achieve these flow rates were approximately 102, 193, 285 and 350 bar, respectively, with the water-rich eluent and 100, 200 and 290 bar, respectively, with pure methanol. The pressure drops inside the chromatographic system that were required to achieve these flow rates in the absence of a column were equal to 15, 26, 39 and 48 bar, respectively, with the water-rich eluent and 17, 29 and 41 bar with pure methanol, respectively. The net pressure drops along the column were equal to 87, 167, 246 and 302 bar, respectively, with the water-rich eluent and 87, 171 and 249 bar with pure methanol, respectively. Between two consecutive changes in the flow rate, the eluent was let to percolate through the column for one full hour so that the column could reach a steady-state thermal equilibrium. The temperatures of the seven thermocouples were then recorded.

3.3. Apparatus

4. Results and discussion

The two solvents were pumped into the chromatographic column using the pump of a HP 1090 liquid chromatograph (Agilent Technologies, Palo Alto, CA, USA). This instrument includes a multi-solvent delivery system (with three 1 L tanks), an auto-sampler with a 250 ␮L sample loop, a column thermostat (i.e., an insulated air bath), a diode-array UV-detector and a data station. Compressed nitrogen and helium bottles (National Welders, Charlotte, NC, USA) are connected to the instrument to allow continuous operations of the pump, the auto-sampler and solvent sparging. The daily variation of the ambient temperature in the laboratory never exceeded ±1 ◦ C. The inlet column pressure was measured by the pressure gauge installed in the liquid chromatograph.

4.1. Column temperature distributions when the wall temperature is controlled

3. Experimental The experimental conditions were previously described in detail [16]. We indicate below only the most important of the experimental conditions. 3.1. Chemicals Two different mobile phases were used to measure the temperature profile along the column. The first one was pure methanol. Because pure water does not wet chemically bonded silica-packing materials well, the second one was a 97.5/2.5 (v/v) mixture of water and methanol (water-rich eluent). Both water and methanol were HPLC grade, purchased from Fisher Scientific (Fair Lawn, NJ, USA). 3.2. Materials

4.1.1. Calculations assuming a constant mobile phase viscosity and a homogeneous column The results of the numerical method of calculation of solutions of the system of Eqs. (1) and (3) were verified by comparing them with the results of the analytical solution of the steady-state equivalent of Eq. (1) derived by Poppe et al. [7]. These authors performed experiments on a 20 cm × 0.46 cm stainless column, packed with silica LiChrosorp Si60 (Merck, Darmstadt) with an particle size of 5 ␮m. The mobile phase was iso-octane. The column was placed in a plastic jacket. Its temperature was kept

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Fig. 1. Experimental data (symbols) [7] and calculated (solid lines) axial temperature profiles. Plots of T = T − Text along the column. The pressures drops were 200, 150, 100 and 50 bar, from top to bottom.

constant by placing it in a vigorous air stream at constant temperature. The thermal records showed that the tube temperature remained constant and was equal to the air temperature. The axial column temperature profile was measured in eight points by thermocouples. For our calculation purpose, we assumed, like Poppe et al., that the heat conductivities of stainless steel and of the mobile phase were equal to 90 and 0.146 W/(m K), respectively. Poppe et al. [7] had assumed that the wall temperature was the same as the external air temperature; however he did not give the value of the external temperature nor the exact value of the heat capacity of iso-octane, for which he performed the calculations. In our work we assumed Text = Twall = 295 K and that the heat capacity of the mobile phase is equal to 1.56 × 106 J/(m3 K). The axial temperature profiles obtained are shown in Fig. 1, which compares the results of the measurements of T = T − Text (symbols), already reported in Fig. 3 of Ref. [7], and the profiles calculated with our program, as explained above (solid lines), for the four mobile phase flow rates, 3.9, 2.9, 1.85 and 0.95 cm3 /min, used by Poppe et al. [7]. The T in Fig. 1 denotes temperature in column center. The corresponding pressure drops were approximately 200, 150, 100 and 50 bar, respectively. The calculations assumed piston-flow of the mobile phase. The line corresponding to the experimental data acquired at 150 bar is almost overlaid with the analytical solution presented by Poppe et al. [7], which corresponds also to this pressure (no other solution was given by these authors). The small discrepancy between their and our numerical solutions can arise from the unknown value of the eluent heat capacity assumed by Poppe et al. for their calculations. In our calculations we took into account the heat transfer through column stainless wall of the column. However, as it was expected, the gradient of temperature across the tube wall was practically negligible. The comparison of the experimental and theoretical axial temperature distributions shows clearly, especially at the high values of the pressure drop, that the temperature near the column inlet is overestimated and that the temperature at the column

outlet is underestimated. Poppe et al. [7] suggested that the disagreement between the temperatures calculated and observed is mainly due to the changes of the mobile phase viscosity along the column, caused by the temperature increase. It may also be due to possible changes of the column permeability along the column that would be correlated with an axial variation of the external column porosity. The heat capacity and the heat conductivity of the mobile phase have a rather small impact on the temperature profile because these parameters change only little, even with marked variations of the temperature. It should also be noted that the true radial profile of the mobile phase velocity in an isothermal column is not that of piston flow, as it was assumed in our calculations. Among other authors, Farkas and co-workers [17,18] investigated the radial profiles of the velocity distribution in actual columns. It was proven that the radial distribution of the flow velocity in wide columns is parabolic. In narrower columns, such as those used in this work, the flow distribution is rather proportional to rn with n > 2. The differences observed between the velocities along the column axis and near its wall can easily exceed 10%, meaning that the heat produced by viscous friction should be 10% greater in the column center than in the wall area. This suggests that the temperature in the column center should be higher than expected for the results of calculations based on the theoretical assumption of piston flow conditions. On the other hand, however, it should be remembered that heat is rapidly distributed in the radial direction, due to the temperature gradient what tends to flatten the radial temperature distribution. To check if the radial velocity distribution of the mobile phase can affect the temperature distribution, we performed further calculations for the following two possible radial flow patterns    r 2 u = uaverage −0.09524 + 1.047619 (11) R and



u = uaverage −0.09278

 r 7 R

 + 1.020619

(12)

and for the high back pressure of 200 bar. In both cases, the maximum difference between the temperatures calculated for piston flow and for the mobile phase flow distribution assumed above (Eqs. (12) and (13)) were less than about 0.09 K at the column outlet. This result means that neglecting the actual radial flow distribution cannot explain the observed discrepancy between the experimental results and those of the theoretical calculations. In the rest of this work, all the calculations were performed assuming piston flow. 4.1.2. Influence of the temperature dependence of the mobile phase viscosity Because the radial and axial distributions of the temperature are not flat, neither are the radial and axial distributions of the mobile phase viscosity. This has for a consequence that the heat power generated by the friction forces is not evenly distributed throughout the bed. The viscosity and the external porosity of the bed are related with the pressure drop, P, or the pressure

K. Kaczmarski et al. / J. Chromatogr. A 1177 (2008) 92–104

gradient, P/L, through the Blake–Kozeny equation [12]: dp2 P ε3e L (1 − εe )2 150η

(13a)

P (1 − εe )2 150η =u L ε3e dp2

(13b)

u= or

where η is the mobile phase viscosity and dp is the average diameter of the particles of adsorbent used to pack the column bed. The viscosity of liquids is related to the temperature. For iso-octane, this numerical relationship is given by the formula [19]:   888.09 −5 2 Log10 (η) = −5.9245+ +0.012955T −1.36×10 T T (14) As for all liquids, the viscosity decreases with increasing temperature. To evaluate the maximum possible influence of the axial distribution of the mobile phase viscosity on the axial temperature distribution, we assumed that the column is perfectly insulated (i.e., the column is operated under adiabatic conditions). Nearly adiabatic conditions are most important for chromatographers because the loss of column efficiency is smallest under such conditions, due to the negligible temperature gradient in radial direction—see Section 4.3. The heat transfer model was solved with a combination of Eqs. (14), (13b) and (2), assuming a constant value of the external porosity equal to 0.37, a typical value encountered for HPLC columns. Fig. 2 compares the temperature profiles at column center, T, to the ambient temperature, Text , obtained with a pressure drop P = 200 bar, for the viscosity given by Eq. (14) (solid line) and for a constant value of the viscosity equal to the viscosity at the average temperature, 301.45 K (symbols). As was expected, the

Fig. 2. Comparison of the axial temperature profiles calculated for the viscosity at 301.45 K (symbols) and for the viscosity given by Eq. (14) (solid line), at P = 200 bar, under adiabatic conditions.

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use of the average value of the viscosity gives a temperature that is lower in the initial part of the column than that calculated using the viscosity given by Eq. (14) because the pressure gradient and the heat generated decrease with decreasing viscosity. However, the differences are practically negligible. From this discussion, it follows that neither the influence of the temperature on the viscosity nor the radial velocity distribution are responsible for the increase of T that is observed experimentally in the second half of the column, near its outlet (see Fig. 1). 4.1.3. Influence of the axial heterogeneity of the column bed Another possible explanation of the observed difference between experimental and calculated axial temperature profiles is the assumption made that the column packing is not homogenous in the axial direction. To investigate the influence of an axial variation of the total column porosity on the axial temperature distribution, we have assumed that the column porosity decreases linearly from the column inlet to its outlet according to the relationship:   2x 2x εe (x) = aεe,ave 1 − (15) + εe,ave L L where εe,ave is the average total porosity and a is an adjustable parameter. Eq. (15) was introduced into the Blake–Kozeny equation, Eq. (13b), from which the pressure gradient was calculated. The viscosity was assumed to be constant and equal to that of iso-octane at the average temperature of 297 K. Fig. 3A compares the axial temperature profiles calculated with a = 1.025 (dashed line) and 1.05 (solid line). In the first case, the external porosity increases from 0.36075 to 0.37925, in the second from 0.3515 to 0.3885. The qualitative agreement between theoretical and experimental profiles increases when the porosity increases in the axial direction. However, the temperatures calculated are lower than the experimental ones, as if the heat flux from the column to the surrounding air was lower than when the wall temperature is equal to the temperature of the air. The column wall temperature might have been higher than that of the air. This might explain the observed differences between experimental and calculation results. Unfortunately, Poppe et al. did not measure the external temperature of the tube. Fig. 3B compares the experimental data of Poppe et al. and the results of numerical calculations made assuming the axial column porosity profile found by Wong et al. [20]. These authors prepared a column by connecting a series of four standard 5 cm × 0.46 cm stainless steel tubes and packed it with Nucleosil C18. Then, the column was divided into 5-cm sections and each section was tested for efficiency and porosity. The masses of adsorbent in these four sections were: 0.3904, 0.4043, 0.4087 and 0.4210 g, respectively, in the order in which these sections were filled. However, the authors did not give the porosity distribution. Assuming typical values for the matrix silica density (2.12 g/cm3 ), the density of the C18 ligands (0.78 g/cm3 ), the volume fraction of the silica matrix (0.24) and of C18 ligands (0.1) and the internal porosity (0.39), the external column porosities of the four columns are 0.3376, 0.3279, 0.3249 and 0.3163,

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formed in an air-conditioned laboratory (temperature Text ), with the HPLC instrument operating under typical conditions. This means that we did not use any special means to avoid the influence of the possible air turbulence on the heat transfer between the ambient air and the column wall. The temperature of the column wall (Tw ) and the radial temperature distribution (Tr ) were calculated by solving the set of Eqs. (1) and (3), using the effective thermal conductivity in the radial direction and assuming that heat transfer takes place by natural convection from the column to the surrounding space. In still air, heat is transferred from the column to the surrounding atmosphere by natural convection. It was suggested [21] to calculate the external heat transfer coefficient for a horizontal tube by using the correlation:  3 2 m he dp d ρ gβT cp η =A (16) λ η2 λ

Fig. 3. Comparison between the experimental temperatures and those calculated. (A) Calculations made with a = 1.025 (dashed line) and a = 1.05 (solid line); see Eq. (15) for definition of a and (B) calculations made with the experimental axial porosity distribution found in Ref. [20].

respectively. These porosities decrease nonlinearly from the inlet to the outlet of the column. The temperature profile calculated in this case is very similar to the one calculated for a linear decrease of the porosity. The local pressure drop is a strongly non-linear function of the column porosity (see Eqs. (13)). It is difficult to model the porosity distribution in such way that the pressure drop remains equal to 200 bar. However, the comparison presented in Fig. 3 remains valid if the total pressure drop in the different columns considered remains the same. In our calculations, we adjusted slightly the constant of the Blake–Kozeny equation, from 150 to 156.2 for a = 1.05 and to 157.3 for a = 1.025. This allowed the fulfillment of this requirement with an error smaller than 0.02%. The constant in Blake–Kozeny equation was changed to 94.8 in the case of the calculation presented in Fig. 3B. 4.2. Distribution of the temperature in a column the wall temperature of which is not controlled In this section, we interpret the experimental results reported earlier and described in Section 3. The experiments were per-

where β = 1/T, g the gravity acceleration, η the air viscosity (kg/(m s)), λ is the thermal conductivity of air (W/m/K) and d is the column diameter (m) and ρ is the air density (kg/m3 ). Values of the coefficients A and m suitable for our experimental conditions are 1.09 and 0.2, respectively [21]. External heat transfer decreases with decreasing temperature difference between the tube wall and the ambient air. Under our experimental conditions, the heat transfer coefficient obtained is about 11 W/(m2 K) at T = 5 K and 8 W/(m2 K) at T = 1 K. The thermal conductivity of a porous medium impregnated with a liquid depends on the geometry of the solid bed and on the properties of the medium [22]. The limiting cases are discussed below. If radial conduction takes place in parallel in the solid and the fluid phases, the effective conductivity is the weighted arithmetic mean of the heat conductivities: λr,ef = εt λelu + εlig λlig + εs λs

(17)

On the other hand, if the radial conduction takes place in series, as if all the media layers were parallel to the column axis, the effective conductivity is the weighted harmonic mean: 1 λr,ef

=

εlig εt εs + + λelu λlig λs

(18)

In the case studied, the total porosity is εt = 0.66 and the volume fraction occupied by the organic ligands and by the solid silica matrix are εlig = 0.1 and εs = 0.24, respectively [16]. The heat conductivity of the materials involved are: pure methanol λelu = 0.21, water λelu = 0.60; C18 ligands (treated as a solid phase) λlig = 0.35 and solid silica λs = 1.40 W/(m K).The effective conductivity calculated from Eq. (17) are: • methanol phase λr,ef = 0.51 W/(m K) and • water-rich phase λr,ef = 0.77 W/(m K). The effective conductivity calculated from Eq. (18) are: • methanol phase λr,ef = 0.28 W/(m K) and • water-rich phase λr,ef = 0.64 W/(m K).

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Actually, heat transfer is more complicated than expressed in either Eqs. (17) or (18). The first equation overestimates the effective conductivity and the second underestimates it. However, the second method seems more suitable for the packed beds used in chromatography due to the relatively small contact area between adsorbent particles, giving the bed a layer-like structure composed of the solid matrix, the ligands covering the surface of the pores and the fluid impregnating these pores. In the calculations giving the results reported here, the heat conductivity of the tube was taken as 15 W/(m K) [16]. In the interpretation of the experimental data, we encountered two major problems: (1) the chromatographic instrument itself generates heat and (2) significant heat losses are due to the massive column endfittings. As stated earlier, in Section 3, narrow connecting tubes must be used to reduce the extra-column band broadening contributions. The permeability of these tubes is low, causing important pressure drops along them, up to 50 bars at the highest mobile phase velocity. Such large pressure drops cause important increases of the mobile phase temperature, which, fortunately, are much alleviated by the narrow diameter and the small thickness of the connecting tubes used. So, the main source of temperature increase inside the chromatograph is the heat generated by the pump. We measured the difference between the eluent temperature at the column inlet and the ambient temperature and found the values of: 1.2, 1.2, 1.4 and 1.6 K for the water-rich eluent at flow rates of 0.9, 1.8, 2.8 and 3.6 mL/min, respectively. With methanol, at flow rates of 1.5, 3.0 and 4.5 mL/min, the same temperature difference was 1.2, 1.3 and 1.4 K, respectively. The inlet mobile phase temperature was assumed to be equal to the temperature measured for the wall of the connecting steel tube bringing the mobile phase to the column. Obviously, the temperature of the incoming mobile phase depends on possible fluctuations of the ambient temperature. Small changes of this temperature generate small changes of the temperature of the mobile phase entering the column. These temperature changes can influence markedly the values of the external heat transfer coefficient and of the temperature profiles. The second problem in the modeling of the temperature distribution inside the column was to account for the heat losses at the endfittings, which have a large surface area, about 90 times larger than the tube cross-section area. These endfittings are well connected to the column tube hence act as heat radiators. Their external surface is complex and difficult to model. However, due to the large heat conductivity of steel, the cooling process of the column by the endfittings can be approximated by assuming that the effective external heat transfer coefficient of both end sections of the column tube to ambient air is 90 times larger than that from the column side-wall. As an example, the temperature distribution calculated for a pressure drop equal to 304 bar and a column side-wall effective external heat transfer coefficient equal to 47 W/(m2 K) is shown in Fig. 4. It is clearly visible that the column inlet and outlet endfittings influence the temperature distribution only close to the column ends. In the next two sections, the cooling effects due to the column endfittings were taken into account in all the calculations performed.

99

Fig. 4. Axial temperature profiles calculated when taking into account the heat loss through the endfittings. The gray area denotes the column wall.

4.2.1. Results for the water-rich system Fig. 5 compares the best approximations of the calculated temperature profiles and the experimental data of [16]. The adjustable parameter was the apparent external heat transfer coefficient. The estimated, best values of this coefficient were he = 47, 37, 26 and 15 W/(m2 K) from the highest to the lowest temperature. These values are higher than those derived from the correlation in Eq. (16), but the predicted trend of a decrease of the heat transfer coefficient with decreasing temperature difference, T, is fulfilled. Fig. 6 compares the calculated axial profiles of the wall temperature with the data measured using the special outlet endfitting designed to enable systematic measurements of the radial temperature distribution. The experimental wall temperature is slightly higher than in the cases discussed previously, due to a change in the heat transfer conditions. This explains why lower best estimates of the external heat transfer coefficient (equal to 38, 33, 16 and 11 W/(m2 K), from the highest to the lowest pressure drop) are now found. Fig. 7 compares the measured (symbols) and the calculated (solid line) radial temperature profiles of the mobile phase at the column outlet for the water-rich mobile phase. The solid

Fig. 5. Comparison of the experimental data (symbols) [16] and the calculated (solid line) profiles of column wall temperatures in a MeOH–H2 O 2.5/97.5 (v/v) solution. The measurements were performed with a normal endfitting. The pressure drops were 304, 246, 167 and 89 bar (from top to bottom), the mobile phase flow rates were 3.6, 2.8, 1.8 and 0.9 mL/min.

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Fig. 6. Comparison of the experimental data (symbols) and the calculated (solid lines) profiles of the wall temperature with MeOH–H2 O 2.5/97.5 (v/v). The measurements were made with the special outlet endfitting [16]. The pressure drops were 304, 246, 167 and 89 bar (from top to bottom).

lines are extended into the column wall area. As expected, there is practically no temperature drop at the wall. The calculations were performed for external heat transfer coefficients equal to he = 38, 33, 16 and 11 W/(m2 K), from the highest to the lowest pressure drop, despite the fact that the experimental conditions presented in Figs. 6 and 7 were different, meaning that the values of the heat transfer coefficients must be different also. This is why the calculated temperature profiles are shifted downward compared to experimental results. At all inlet pressures, the calculated temperature differences between the column center and its wall are larger than those measured. This means that the cooling effect due to the endfittings is lower that what is predicted by the calculations. Ignoring the heat

Fig. 7. Comparison of experimental data (symbols) [16] and calculated (solid lines) radial profiles of the mobile phase temperature at column outlet, with MeOH–H2 O 2.5/97.5 (v/v). The measurements were made using the special outlet endfitting [16]. Pressure drops: 302, 246, 167 and 87 bar (top to bottom). The dashed line gives the temperature distribution at 2.5 cm before the column end, for an inlet pressure of 302 bar.

Fig. 8. Comparison of the measured (symbols) and calculated (solid line) temperature profiles of the column wall with MeOH. Measurements performed with a normal endfitting. Pressure drops: 249, 171 and 87 bar (up to bottom); mobile phase velocity: 4.5, 3.0 and 1.5 mL/min.

losses due to the endfittings gives a radial temperature profile at the column end that is identical to the one depicted by the dashed line in Fig. 7, which shows the radial temperature distribution at 2.5 cm upstream the column end. This result means that the calculated difference between the temperatures in the column center and at its wall is twice lower than the one measured when the heat losses due to the endfittings are ignored. It means also that the cooling effect due to the endfittings is restricted to the very end of the column. 4.2.2. Results for the methanol system Figs. 8 and 9 show the best approximations of the experimental temperature profiles, first for a normal (Fig. 8) and then for the

Fig. 9. Comparison of the measured (symbols) and calculated (solid line) column wall temperature for MeOH. Measurements were performed for special outlet endfitting. The pressure drops were equal from up to bottom: 249, 171 and 87 bar.

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4.3. Column efficiency Due to the heat generated by viscous friction, the temperature in a chromatographic column increases in both the axial and the radial directions. The adsorption process depends strongly on the local temperature. Typically, the equilibrium constant decreases with increasing temperature according to van’t Hoff equation [13]:   E (19) K = K0 exp RT

Fig. 10. Comparison of the measured (symbols) and calculated (solid line) radial temperature profiles of the eluent at the column outlet with MeOH. Measurements performed with the special outlet endfitting. The pressure drops were: 249, 171 and 87 bar (top to bottom).The dashed line gives the temperature distribution 2.5 cm before the column end, when cooling by the large column endfitting is calculated for an inlet pressure of 249 bar.

special (Fig. 9) outlet endfitting. As in the case of the water-rich system, the adjustable parameter was the external heat transfer coefficient. The best estimates of this coefficient were, in the first case he = 82, 70 and 45 W/(m2 K) (from the highest to the lowest temperature difference) and in the second case he = 130, 82 and 90 W/(m2 K). These values are much higher than those predicted by Eq. (16) and even higher than those obtained for water. However, the trend of the heat transfer coefficient decreasing with decreasing temperature difference is again verified. Fig. 10 compares the measured [16] and calculated radial profiles of the mobile phase temperature at the column outlet, with pure methanol as the mobile phase. The calculations were performed for the following values of the external heat transfer coefficients he = 130, 82 and 90 W/(m2 K) (from the highest to the lowest pressure drop). The measured temperatures are lower than those calculated and, for the lowest pressure drop, they are even lower than the inlet temperature. This unusual result is explained by the cooling effect due to the evaporation of methanol at the column endfitting [16]. Despite this effect, the calculated radial temperature profiles are in good qualitative agreement with the results of the experimental measurements. In the calculations giving the results presented above, the thermal expansion of the mobile phase was ignored. The effect of this expansion on the eluent temperature increase can be calculated by completing the right hand side of the steady-state version of Eq. (1) of the model by the term αTu∇P [8], where α is the thermal expansion coefficient (a = 1.199 × 10−4 and 0.207 × 10−4 for methanol and water, respectively, at 20 ◦ C). The effect of the thermal expansion on the temperature increase (T − Text ) is marginal for water and small for methanol. Neglecting the thermal expansion of the mobile phase gives a maximum error of 0.7% in the first case, 4% in the second.

where E is the activation energy of adsorption and R is the universal gas constant. For proteins, a more complicated relationship can be observed due to the influence of temperature on the tertiary structure of the molecule. At low temperature, the equilibrium constant increases with increasing temperature, but at higher temperatures, it decreases [23]. In this work, it has been assumed that the equilibrium constant always decreases with increasing temperature. For relatively small changes of the temperature, as those that take place in the present case (inlet pressures below ca. 400 bar), Eq. (19) can be approximated with a linear expression: K = K(T0 ) − δ(T − T0 )

(19b)

To investigate the influence of the heat generated by viscous friction on the column efficiency, we assumed that the analyte adsorbs according to a linear isotherm: q=qs (K(T0 )−δ(T − T0 ))C = (H(T0 ) − δ (T − T0 ))C

(20)

For the calculations, the temperature T0 was taken as equal to 295 K, the Henry constant H(T0 ) as equal to 30, and the parameter δ as equal to 0.25. Typically, the decrease of the Henry constant of phenol is about δ = 0.25/K around 295 K [24].The calculations were performed in three different cases: (i) The temperature of the external column wall is kept equal to 295 K, (ii) The column in not thermally insulated but subject to natural convective, (iii) The column is thermally insulated. In all cases it was assumed that the mobile phase flow followed piston flow behavior. The cooling effect due to the endfittings was also ignored and the inlet temperature of the mobile phase was assumed to be equal to the ambient temperature. The model depicted by the set of Eqs. (1)–(9) and (20) was solved, using for the numerical calculations the values of the physicochemical parameters discussed above for the water-rich or the pure methanol mobile phases, but the external heat transfer coefficient. The maximum pressure drop was either 304 or 249 bar. In the case (iii), the heat transfer through the steel tube was neglected. Moreover, it was assumed that the thickness of the polyurethane foam used to insulate the column was 1cm and that the thermal conductivity of this foam was equal to 0.022 W/(m K). The average external heat transfer coefficient

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calculated from Eq. (16) was about 10 and 5 W/(m2 K) in cases (ii) and (iii), respectively. The mass transfer equation was solved assuming column efficiencies of N = 3000 or 15000 theoretical plates, at constant column temperature. For N = 3000 the axial apparent dispersion coefficient is equal to 2.28 × 10−7 m2 /s in the water-rich phase and to 2.85 × 10−7 m2 /s in pure methanol. To calculate the dispersion coefficient in the radial direction, the equation derived by Knox was applied [25,26]. According to Knox and co-workers [25,26], the radial plate height is Hr = 0.06dp

(21)

This corresponds to Da,r =

u × 0.03dp εe

(22)

Values of 1.3 × 10−9 m2 /s for water-rich system and 1.68 × 10−9 m2 /s for pure methanol were calculated for the radial dispersion coefficient. For the calculations of the band profiles, 100 and 500 subdomains were used in the OCFE method for N = 3000 and N = 15,000, respectively. Before analyzing the results obtained, it is convenient to compare the radial profiles of temperature calculated earlier at the end of the column and along the column in the three cases listed above. These results are presented in Figs. 11, 12a and 12b. In the first case, the differences between the axial and the wall column temperatures at the column end were 2.1 K in pure methanol and 0.9 K in the water-rich phase. The smaller value for the water-rich phase is due to it having a heat conductivity twice higher than that of methanol, causing faster heat transfer in the radial direction, which flattens the temperature profile. The temperature of the water-rich phase was smaller than that of pure methanol, due to the higher heat capacity of water. The same rules apply in the other two cases. In the second case, the temperature differences between the column axis and its wall were 0.8 and 0.25 K for methanol and the water-phase, respectively.

Fig. 11. Comparison of radial temperature profiles calculated at the column outlet. M and W denote the temperatures obtained with methanol and water, respectively. (1) Constant wall temperature, T = Text = 295 K; (2) heat transfer by natural convection; (3) column insulated with polyurethane foam.

Fig. 12. (a) Comparison of the axial profiles of the difference between the temperatures calculated at the axis (TC ) and near the wall of the column (Twall ), for pure methanol. (1) Wall temperature 295 K, case (i); (2) external heat transfer coefficient 130 W/(m2 s); (3) natural convection, case (ii); (4) insulated column, case (iii) and (b) comparison of the axial profiles of the difference between the temperatures calculated at the axis and near the wall of the column, for the water-rich phase. (1) Wall temperature 295 K, case (i); (2) natural convection, case (ii).

In the third case, they were 0.25 and 0.08 K. The temperatures of the eluents increase with increasing heat transfer resistance to the surrounding air but at the same time, this causes a decrease of the temperature differences between the center of the column and its wall. The calculated temperature profiles agree with the experimental data presented in [7,16]. The profiles presented in Fig. 12a and b shows the variations along the column of the calculated temperature differences between the column center and its wall, Tcw = Tc − Tw . With pure methanol, the calculations were performed in four cases: (1) a wall temperature of 295 K and case (i); (2) an external heat transfer coefficient of 130 W/(m2 s) and heat transfer by natural convection, case (ii); (3) natural convection and case (ii); and (4) an insulated column, case (iii). With water, the calculations were made for a wall temperature of 295 K in case (i); and natural convection in case (ii). The difference Tcw rapidly reaches its maximum value for a constant wall temperate. The amplitude of Tcw decreases with increasing resistance of heat

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The loss of column efficiency strongly depends on the isothermal efficiency, Nizot . For example, with methanol as the mobile phase and an external tube temperature kept at 295 K, the column efficiency decreases by about 5% for Nizot = 3000 but by 20% for Nizot = 15,000. In Section 4.2 it was shown that heat transfer through the endfittings of the column could considerably change the radial temperature profiles near the column outlet. However, the influence of the cooling due to these endfittings does not spread far from the column end and it practically has no effect on the column efficiency (this would not be true for very short columns). For example for methanol phase the calculated efficiency in case (ii) was N = 2995. 5. Conclusions Fig. 13. Peak profiles calculated with methanol as the mobile phase. (1) Case (i); (2) he = 130 W/(m2 s); (3) case (ii); (4) case (iii). Nizot = 3000.

transfer to the column surrounding fluid and becomes very low for a well-insulated column. For the calculations performed in case (ii), an initial negative value of Tcw is clearly visible at the column inlet, with a rapid increase at the column end. This effect is related to the heating of the entering fluid by the steel wall, which rapidly conducts heat from the middle part of the column. From the same reason the temperature difference increases near the column end. Because the column efficiency depends essentially on the radial temperature profile and also on the manner in which it varies along the column, this discussion suggests that the largest loss of column efficiency should be expected in case (i), when a column temperature is kept equal to the ambient temperature. In this case, the value observed for Tcw is the largest, a constant temperature pattern is established at the very beginning of the column, and the so-called entrance length is short. In this case, one should expect the highest loss of column efficiency. For a low external heat transfer resistance (case 2 in Fig. 12a) the final value of Tcw is similar to that obtained for a constant wall temperature, however the entrance length is longer, so the loss of column efficiency should be smaller than in the previous case. For strictly natural convection conditions (case (ii)) or for the insulated column, Tcw is low and increases slowly along the column, so the efficiency loss should be marginal. The above presumptions are confirmed by the results of the calculation of chromatograms made in cases (i)–(iii), using the combination of models described earlier, in the theory section. The chromatograms are shown in Fig. 13. The influence on the retention is more striking than the efficiency loss. The efficiencies calculated are presented in Table 1.

The systematic investigation of numerical solutions of our models of the heat transfer in chromatographic columns and of the propagation of chromatographic bands in these columns provides estimates of the axial and radial temperature distributions and affords useful new results. First, under steady-state conditions, both the axial and the radial temperature distributions in columns percolated under high pressure drops have significant amplitude. This amplitude increases with increasing inlet pressure but it was shown that the radial velocity profile, which is due to the radial distribution of the viscosity has practically no influence on the radial temperature distribution. Also the change in the mobile phase viscosity due to the axial temperature gradient affects the local pressure gradient but seems to have only a minor influence on the temperature gradient. On the other hand, a possible heterogeneity of the column external porosity in the axial direction would considerably influence the axial distribution of the heat production, hence the temperature gradient in the axial direction. However, there is not much information in the literature on the degree of heterogeneity of this porosity. Second, although near both column ends the large surface area of the steel column endfittings intensifies heat transfer to ambient air, this has no effect on the column efficiency. Third, the lost of column efficiency is greatest when the temperature of the external wall of the column is kept constant, e.g., equal to the ambient temperature. Under such conditions, the temperature gradient in the radial direction is the largest and the entrance length of the column the shortest. Those are most disadvantageous conditions for the preservation the column efficiency. Even under uncontrolled heat transfer conditions from the column to a surrounding fluid, the loss of column efficiency is smaller. The column efficiency is nearly unaffected when the heat transfer from the column is due only to natural heat convec-

Table 1 Column efficiency for different heat transfer resistances Mobil phase

Methanol N = 3000 for isothermal conditions Methanol N = 15,000 for isothermal conditions water-rich N = 3000 for isothermal conditions

Column efficiency Case (i)

he = 130 W/(m2 s)

Case (ii)

Case (iii)

2,858 11,942 2,973

2,918

2,997

–

3,000

3,000 14,945 –

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tion or if the column is well insulated. However, this means that the insulating material must be protected from the condensation of humidity. Condensed moisture dramatically affects the properties of insulating materials and considerably increases their heat conductivity. Finally, the loss of column efficiency mainly depends on the amplitude of the temperature gradient in the radial direction, the smaller this gradient the smaller the loss of efficiency. The temperature gradient is minimized if the external heat transfer resistance of the column tube dominates over the inside heat transfer resistance of the column. The efficiency loss is also a function of the column length, its diameter and the influence of the temperature on the adsorption constant of the analyte considered. In conclusion, the exact impact of the heat generated by viscous friction on the column efficiency must be analyzed individually, for any specific chromatographic system. Nomenclature

a cpm cs cw C dp Dz,a Dr,a E F g he hv H Hr P q R Re T Text Tcw u w

parameter mobile phase heat capacity solid heat capacity wall heat capacity concentration in mobile phase adsorbent diameter axial apparent dispersion coefficient radial apparent dispersion coefficient activation energy phase ratio gravity acceleration external heat transfer coefficient heat generation per unit volume due to deformation of viscous fluid Henry constant radial plate high pressure concentration in stationary phase gas constant external column wall diameter temperature external temperature = Tc − Tw temperature differences between column center and column wall superficial velocity interstitial velocity

Greek letters parameter δ average bulk porosity εe,ave external porosity εe εt total column porosity η viscosity λr,ef effective bed conductivity λw wall conductivity ρ density Subscripts 0 inlet value w wall References [1] H. Poppe, J.C. Kraak, J. Chromatogr. 282 (1983) 399. [2] I. Halasz, R. Endele, J. Asshauer, J. Chromatogr. 112 (1975) 37. [3] A. De Villiers, H. Lauer, R. Szucs, S. Goodal, P. Sandra, J. Chromatogr. A. 1113 (2006) 84. [4] D.T.T. Nguyen, D. Guillarme, S. Heinisch, M.P. Barrioulet, J.L. Rocca, S. Rudaz, J.L. Veuthey, J. Chromatogr. A. 1113 (2006) 84. [5] A. Brandt, G. Mann, W. Arlt, J. Chromatogr. A. 769 (1997) 109. [6] O. Dapremont, G.B. Cox, M. Martin, P. Hilaireau, H. Colin, J. Chromatogr. A. 796 (1998) 81. [7] H. Poppe, J.C. Kraak, J.F. Huber, Chromatographia 14 (1981) 515. [8] H.-J. Lin, C. Horvath, Chem. Eng. Sci. 56 (1981) 47. [9] G. Desmet, J. Chromatogr. A. 1116 (2006) 89. [10] U.D. Neue, M. Kele, J. Chromatogr. A. 1149 (2007) 236. [11] F. Gritti, G. Guiochon, J. Chromatogr. A. 1166 (2007) 47. [12] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, New York, 2002. [13] G. Guiochon, A. Felinger, A.M. Katti, D. Shirazi, Fundamentals of Preparative and Nonlinear Chromatography, Second ed., Elsevier, Amsterdam, 2006. [14] K. Kaczmarski, G. Storti, M. Mazzotti, M. Morbidelli, Comput. Chem. Eng. 21 (1997) 641. [15] P.N. Brown, A.C. Hindmarsh, G.D. Byrne, available at http://www.netlib.org. [16] F. Gritti, G. Guiochon, J. Chromatogr. A 1138 (2007) 141. [17] T. Farkas, M.J. Sepaniak, G. Guiochon, AIChE J. 43 (1997) 1964. [18] T. Farkas, G. Guiochon, Anal. Chem. 69 (1997) 4592. [19] B.E. Poling, J.M. Prausnitz, J.P. O’Connell, Chemical Properties Handbook, Fifth ed., McGraw-Hill, New York, 2001. [20] V. Wong, R.A. Shalliker, G. Guiochon, Anal. Chem. 76 (2004) 2601. [21] D.W. Green (Ed.), Perry’s Chemical Engineers’ Handbook, Seventh ed., McGraw-Hill, 1997. [22] A. Bejan, A.D. Kraus, Heat Transfer Handbook, John Wiley & Sons, 2003. [23] P. Szabelski, A. Cavazzini, K. Kaczmarski, J. Van Horn, G. Guiochon, Biotechnol. Prog. 18 (2002) 1306. [24] K. Kaczmarski, F. Gritti, G. Guiochon, Chem. Eng. Sci. 61 (2006) 5895. [25] D. Horne, J.H. Knox, L. McLaren, Sep. Sci. 1 (1966) 531. [26] J.H. Knox, G.R. Laird, P.A. Raven, J. Chromatogr. 122 (1976) 129.