Consequences of the radial heterogeneity of the column temperature at high mobile phase velocity

Journal of Chromatography A, 1166 (2007) 47–60

Consequences of the radial heterogeneity of the column temperature at high mobile phase velocity Fabrice Gritti a,b , Georges Guiochon a,b,∗ a

b

Department of Chemistry, University of Tennessee, Knoxville, TN 37996-1600, USA Division of Chemical Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6120, USA Received 4 April 2007; received in revised form 27 June 2007; accepted 29 June 2007 Available online 5 July 2007

Abstract When a high velocity stream of mobile phase percolates through a chromatographic column, the bed cannot remain isothermal. Due to the mobile phase decompression, heat is generated along the column. Longitudinal and radial temperature gradients take place along and across its bed. The various consequences of this thermal heterogeneity are calculated and their effects on the column efficiency investigated for a 0.46 cm × 25 cm stainless steel column packed with 5 ␮m particles. The maximum pressure drop applied was varied from 0.1 to 2 kbar. The amplitude of the longitudinal temperature gradient can be estimated on the basis of the integral heat balance equation applied to the whole column and of measurements of the eluent temperature at the column exit. Assuming that the radial gradient is parabolic and the longitudinal gradient linear, the amplitude of the radial gradient can be determined on the basis of the energy balance across the column and of direct measurements of the radial gradient at high inlet pressures. A radial temperature gradient causes a radial distribution of the eluent viscosity, hence of its local velocity. The result is that bands move faster in their center than along the wall, become warped, hence a radial concentration gradient, similar in origin to the one observed in open cylindrical tubes. Diffusion relaxes this gradient. If there is only a longitudinal temperature gradient, the column efficiency would be 30% smaller for a 2 kbar pressure drop than if there is no longitudinal temperature gradient. However, when both a longitudinal and a radial temperature gradient coexist, there is a large loss of efficiency. If the influence of the diffusive relaxation of the radial concentration gradient is neglected, the peak shape would be broad and exhibit a marked shoulder. © 2007 Elsevier B.V. All rights reserved. Keywords: Chromatographic column; Mobile phase friction; Heat effects; Longitudinal temperature gradients; Pressure drop; Radial temperature gradients; Column efficiency; C18 -bonded silica

1. Introduction The axial and radial temperature profiles in a conventional HPLC column (e.g., 150 mm × 4.6 mm) through which the mobile phase percolates at high velocity were recently measured experimentally and discussed [1]. This work was motivated by the high current interest of analysts for performing faster and/or more efficient separations. Due to the compressibility of liquids, an important amount of energy is stored into the mobile phase by the pump and conveyed into the column where it is dissipated into heat as the eluent decompresses. The chromatographic column is heated from its inside, which causes the formation of axial and radial temperature gradients, the axial gradient due to the

∗

Corresponding author. Fax: +1 865 974 2667. E-mail address: [email protected] (G. Guiochon).

0021-9673/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2007.06.067

progressive heating of the eluent, the radial gradient due to radial heat losses. The amplitude of these gradients increases rapidly with increasing inlet pressure. These effects and their consequences are negligible for inlet pressures less than ca. 100 bars. They increase rapidly with increasing inlet pressure. The heat effects have little consequences with conventional column technology. During recent years, however, the industry needs for faster HPLC analysis have lead to important changes in column technology. This evolution is proceeding along three different directions. Two successive generations of monolithic columns which, for the same efficiency than packed columns, have a permeability that is several times higher, have already been developed [2]. New columns packed with fine particles, having diameters between 1 and 3 ␮m, are now available as well as the instruments needed to operate these columns at high mobile phase velocities, hence at pressures in excess of 1000 bar [3]. Finally, operating columns at temperatures higher than

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F. Gritti, G. Guiochon / J. Chromatogr. A 1166 (2007) 47–60

ambient permits the achievement of faster analyses because the viscosity of the mobile phase decreases rapidly with increasing temperature, while the solute diffusivity and the optimum mobile phase velocity for maximum efficiency increase. Whether the mobile phase is pumped into the column under a high inlet pressure or at a high temperature, important stationary longitudinal and radial temperature gradients can be formed inside their beds. The only practical way to reduce the importance of these gradients and their consequences is to reduce the column diameter [4,5], but this introduces new, severe constraints on the extra-column volume of the instrument. A radial temperature gradient causes a radial gradient of the mobile phase viscosity, hence a heterogeneous radial distribution of the mobile phase velocity, which reduces the column efficiency compared to what it would have been under isothermal conditions. On the other hand, a longitudinal temperature gradient is not directly harmful and may even be used to compensate to some extent for this efficiency loss. This would require an instrument design causing an inversion of the direction of the radial gradient somewhere along the column. Furthermore, the longitudinal pressure gradient may cause a significant reduction of the column efficiency for inlet pressures larger than 1 kbar, due to the decrease of the molecular diffusivity of analytes at these pressures. Thus, there are limits to the performance of unidimensional HPLC [6], which explain the new wave of interest for 2D-LC/LC, a combination that may improve the peak capacity achieved in liquid chromatography [7]. It is important to understand the effects of the combination of a high pressure gradient, a radial and an axial thermal gradient in an HPLC column. At high mobile phase velocities, the column efficiency is essentially controlled by the flow pattern (eddy dispersion) and by the mass transfer to and through the stationary phase. The effects of axial diffusion become negligible [8]. Thus, the effect of an axial thermal gradient on the column efficiency is rather small. Diffusion coefficients increase with increasing temperature and the mass transfer kinetics of analytes through the particles accelerate. On the other hand, due to the pressure gradient, the molecular diffusivities of analytes, which decrease with increasing pressure [9], are low at the column inlet and this slows down mass transfer kinetics at the column inlet. This balance between temperature and pressure effects was recently discussed by Neue and Kele [10]. The formation of temperature gradients due to frictional heating in ultra-high-pressure liquid chromatography and its consequences have recently been investigated in the case of 2.1 mm I.D. columns [11]. The goal of this work is to calculate the consequences of the coexistence of a longitudinal pressure gradient, a longitudinal and a radial temperature gradients on the column efficiency under high inlet pressures (2 kbar maximum). We assume a simple, yet realistic model of the stationary profiles of temperature, a linear longitudinal temperature gradient and a parabolic radial temperature gradient. This assumption is based on the results of recent measurements performed with a 4.6 mm × 250 mm column, packed with 5 ␮m particles [1], operated with an inlet pressure of only 350 bar. The experimental temperature profiles constitute the constraints of the thermal problem. Based on the energy balance in the column, the outlet temperature of the elu-

ent can be expressed as a function of the column pressure drop. Knowing the flow rate applied, the variations of the specific volume and the viscosity of the eluent as a function of its temperature and pressure, the integration of Darcy’s law along the column gives the column pressure drop, hence the temperature increase between the column inlet and outlet. Integration of the local HETP over the column length and the column inner diameter permits the calculation of the apparent Van Deemter curve for a strong thermal heterogeneity of the packed bed. 2. Theory 2.1. General expression of the local sample bandwidth under fast elution chromatography Fast elution chromatography is characterized by the use of a high inlet pressure in order to generate a high velocity stream of mobile phase. According to the classical theory of band broadening in chromatographic columns, the contribution of axial diffusion to the band width is negligible because the elution time is short. In contrast, the contribution of eddy dispersion, due to the heterogeneity of the flow pattern in the anastomosed interparticle channels is independent of the linear velocity because the number of diffusion steps is zero and the packed bed structure is stable and not affected by the high-pressure applied. Band broadening at high mobile phase velocity depends essentially on the kinetics of mass transfer between the liquid phase percolating along the bed and the stagnant liquid phase inside the packed particles and of mass transfer across particles. The film mass transfer resistance through the stagnant film of liquid surrounding the particles can be neglected because its influence is consistently weak on the overall mass transfer kinetics [12,13]. According to the general rate model of chromatography, the infinitesimal increment of the band second central moment, dσz2 (in length unit), associated with the migration of the solute by an infinitesimal length dz along the column is given by [14]:  2 dp2 dσz2 δ0 (T, P) 1 e = u(T, P) dz 6 1 − e 1 + δ0 (T, P) (T, P)Dm (T, P) (1) where e (0.40) is the local external porosity and dp is the particle size. These parameters will be considered as independent of the temperature and the pressure. In contrast, δ0 is a parameter related to the adsorption strength of the analyte [14]. It is a function of the temperature and the pressure given by   1 − e [p + (1 − p )K(T, P)] δ0 (T, P) = (2) e where p (although p  0.40 for many packing materials, it can vary rather broadly) is the particle or internal porosity (independent of both the temperature and the pressure) and K(T, P) is the Henry’s constant, which is generally a function of both the temperature and the pressure. In Eq. (1), the parameter  is related to the diffusion rate of the sample inside the particles. It depends on two parallel diffusion processes, pore and surface diffusion. The general expression of

F. Gritti, G. Guiochon / J. Chromatogr. A 1166 (2007) 47–60

this term is:

   α DS,0 βQst (T, P) = p γp F (λm ) + 2 K(T, P) exp rp Dm,0 RT (3) where γp (0.55) is the internal particle obstructive factor [15], F (λm ) (0.70) the pore steric hindrance parameter [8], α a structural parameter (dimension of a length) is defined as the reciprocal of the product of the packing material density, the silica mass percent in the packing material, and the specific surface area of the neat silica used, rp the average pore radius (α/rp  1 for regular C18 stationary phases with a surface coverage of the order of 2 ␮mol/m2 [12]), DS,0 /Dm,0 the ratio of the frequency factor of surface diffusion to the frequency factor of bulk diffusion ( 20 [12] for a retained compound), β an empirical parameter (0.7 [14]), and Qst is the isosteric heat of adsorption (Qst  −15 kJ/mol as a reference value in liquid chromatography for a retained compound). In Eq. (1), Dm (T, P) is the molecular diffusion coefficient. For small molecules, this coefficient can be estimated according to the correlation of Wilke and Chang [16]: √ φS MS T Dm (T, P) = 7.4 × 10−8 (4) ηS (T, P)VA0.6 where φS is the association factor of the solvent S, MS its molecular weight, VA the molar volume of the liquid solute at its normal boiling point, and ηS (T, P) is the viscosity of the solvent at pressure P and temperature T. An excellent expression of the simultaneous effects of temperature and pressure on the viscosity is given by the correlation of Lucas [9] by η(T, P) 1 + D( Pr /2.118)A = η(T, Pvp ) 1 + Cω Pr

(5)

where ω is the acentric factor [17], Pvp the vapor pressure of the liquid at the temperature T, and Pr = (P − Pvp )/Pc with Pc the critical pressure of the solvent. A, C, and D are parameter expressed as a function of the reduced temperature Tr = T/Tc with Tc the critical temperature of the solvent:   4.674 × 10−4 A = 0.9991 − 1.0523Tr−0.03877 − 1.0513 C = −0.07921 + 2.1616Tr − 13.4040Tr2 + 44.1706Tr3 −84.8291Tr4 + 96.1209Tr5 − 59.8127Tr6 + 15.6719Tr7 0.3257 D= − 0.2086 0.2906 [1.0039 − Tr2.573 ] For the conventional solvents used in liquid chromatography, the vapor pressure at 300 K is subatmospheric and the viscosity η(T, Pvp )  η(T, P 0 ). The variation of the solvent viscosity with temperature can be accounted for by the Vogel equation [18]: ln ηS (T, Pvp )  ln ηS (T, P 0 ) = A1 +

B1 T + C1

where A1 , B1 , and C1 are the empirical constants.

(6)

49

At reduced temperatures Tr smaller than 0.75, the solvent viscosity is predicted to be a linear function of the pressure through Eq. (5). Finally, in Eq. (1), u(T, P) is the interstitial velocity of the mobile phase at temperature T and pressure P. It is related to the density of the liquid solvent ρS by [19]: u(T, P) = u(295 K,P 0 )

ρS (295 K,P 0 ) ρS (T, P)

(7)

The density ρS can be calculated from the expansion coefficient αp and the isothermal compressibility factor χT as [19]:  P  0 ρS (T, P) = ρS (295 K,P ) × exp χT (T, P) dP  × exp

P0

T



−αp (T, P ) dT 0

(8)

295

2.2. Maximum amplitude of the temperature gradients for a given pressure drop The goal of this section is the derivation of estimates of the temperature difference that take place along and across the column for a maximum inlet pressure of 2 kbar. These estimates will be based on the temperature measurements made for a pressure drop of 350 bar. 2.2.1. Estimate of the maximum amplitude of the axial temperature gradient We consider a practical upper pressure limit of 2 kbar. The maximum linear velocity of the mobile phase is essentially determined by the bed permeability. The local pressure gradient is dP ηS (T, P) =− u(T, P) dz k0

(9)

where k0 is the permeability of the packed bed, assumed to be constant along the column and independent of the local pressure and liquid velocity. k0 can be estimated from the Kozeny–Carman correlation: k0 =

3e dp2

(10)

Kc (1 − e )2

where Kc is the Kozeny–Carman constant (Kc  180). The energy dissipated inside the chromatographic column per unit of time is the power dissipated by the mobile phase percolating through the column with a linear interstitial velocity u(T, P). The local power per unit volume, Pfriction , is the product of the local superficial velocity u0 (T, P) (e.g., the volumetric flow rate applied divided by the column tube cross-section area) and the local pressure gradient [20]. Accordingly, Pfriction = −u0 (T, P) ×

dP (1 − e )2 Kc = 2 × dz dp e

×ηS (T, P) × u2 (T, P)

(11)

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F. Gritti, G. Guiochon / J. Chromatogr. A 1166 (2007) 47–60

Table 1 Physico-chemical properties of the C18 -Vydac 214TP column provided by the manufacturer (Grace) Column

Column dimension Particle Mesopore Specific surface ˚ (mm × mm) size (␮m) size (A) area (m2 /g)

C18 derivatization

%C C18 bonding and endcapping (%)

% Silica (mass)

C18 surface coverage (␮mol/m2 )

Dead volumea (mL)

C18 -Vydac 214TP

250 × 4.6

Polymeric

7.7

89.4

5.0

2.76

a

5.0

280

70

Measured from thiourea elution with a mixture of methanol and water (30/70, v/v) as the mobile phase.

The overall power dissipated inside the column is obtained by integration of Eq. (11) along the column volume (length L, internal radius R):   Kc (1 − e )2 L R Wfriction = 2π 2 × ηS (T, P) dp e 0 0 ×u2 (T, P) × r dr dz

(12)

This integral can be numerically solved provided that we know the radial and longitudinal profiles of the solvent viscosity and of the interstitial velocity. However, if, as a first approximation, we neglect the effects of the compressibility and the thermal expansion of the solvent, hence assume that the linear velocity u(T, P) is constant all along the column and equal to u(298 K,P 0 ), the frictional heat power delivered in the column tube is Wfriction 

Kc (1 − e )2 × × η × u2 × πR2 L dp2 e

(13)

Wfriction 

3e dp2 P 2 × πR2 × ηL Kc (1 − e )2

(14)

Eq. (14) is derived from Eq. (13) by writing that the superficial velocity is given by u = k0 P/ηL and taking k0 from Eq. (10). The integral heat balance equation allows the calculation of the temperature increase along the column. Under steady state conditions, the power Wfriction dissipated in the column is equal to the power lost through the column external surface, that is to the sum of the difference between the input and output heat convective fluxes (liquid input at temperature Text and output at Tz=L ) and the heat exchanged by the column external surface area (at a temperature Tc (z), surface 2πRe L, column external radius, Re ) and the ambient atmosphere (heat transfer coefficient he between the stainless steel surface of the column and the surrounding air). Accordingly, assuming that the temperature of the column surface, Tc (z), increases linearly with increasing distance along the column Wfriction = Wsurface + Wexit = he × 2πRe L   P2e dp2 Tc,z=0 + Tc,z=L × − Text + 2 ηLKc (1 − e )2 ×e πR2 × cpm × [Tz=L − Text ]

(15)

In a previous experimental study [1], it was found that the average temperature, Tz=L , of a water-rich eluent exiting the column at a flow rate of 3.6 mL/min (inlet pressure 350 bar)

was approximately 7 ◦ C higher than the temperature of that eluent entering the column, when the column was not thermally insulated. The temperatures of the external column wall at its inlet (Tc,z=0 ) and outlet (Tc,z=L ) were +2 and +6 ◦ C higher than the ambient temperature, respectively. This experience demonstrated that the column tube relaxes the axial temperature gradient of the packed bed and the mobile phase due to its large thermal conductivity (15 W/(m K) versus 1.4 W/(m K) for silica and 0.6 W/(m K) for water). Eqs. (14) and (15) permit the calculation of the heat transfer coefficient, he , from the experimental conditions (inlet pressure, P, temperature increase of the mobile phase along the column, TL = Tz=L − Text and temperature increase of the column wall TW = Tc,z=L − Tc,z=0 ):   3e dp2 R2 P − cpm TL P (16) he = ηL2 Kc (1 − e )2 2Re TW Eq. (16) was applied with the following set of experimental parameters [1]: P = 3.5 × 107 Pa (350 bar), TL = 7.0 K, TW = 4 K, Kc = 190, dp = 5 × 10−6 m, e = 0.40, η = 9 × 10−4 Pa s, cpm = 4.19 × 106 J m−3 K−1 , L = 2.5 × 10−1 m, R = 2.3 × 10−3 m, and Re = 3.2 × 10−3 m (see Table 1). The experimental data yield he  15 W/(m2 K) for heat transfer between air at atmospheric pressure and stainless steel. This value is consistent with the range of values found in the literature [21], between 5 and 35 W/(m2 K). The amplitude of the axial thermal gradient can now be estimated from this value of he , assuming that TL = 2 TW . Accordingly, from Eq. (16):   1 P TL = m cp 1 + (he Re ηL2 Kc (1 − e )2 / Pcpm 3e dp2 R2 ) (17) These results lead to two important conclusions. First, under the conditions of the experiment reported [1], Wfriction = 2.23 W, Wsurface = 0.30 W, and Wexit = 1.93 W. This means that less than 15% of the frictional power that is released in the column is lost through radial thermal exchange with ambient air; the rest of the power generated is evacuated with the eluent leaving the column. Obviously this fraction would decrease with increasing column diameter. Second, consider the same column but operate it under a higher pressure P = 2 kbar. Assuming that the average solvent viscosity at 1000 bar is η = 10−3 Pa s, we expect the eluent to leave the column at a temperature of about 45 ◦ C above ambient temperature, if the eluent is water. This temperature would be twice as large with solvents like methanol, ethanol, tetrahy-

F. Gritti, G. Guiochon / J. Chromatogr. A 1166 (2007) 47–60

drofuran, or hexane which have a specific heat capacity that is typically twice lower than that of water or even less [22]. 2.2.2. Estimate of the maximum amplitude of the radial temperature gradient Consider the slice of column between abscissa z and z + dz. To write the heat balance in this slice under steady state conditions, we neglect the axial heat transfer in the packed bed compared to the radial heat loss and assume that, locally, T (z + dz) = T (z) + ( TL /L) dz so that longitudinal heat conduction does not contribute to any gain or loss of heat in the slice. The frictional heat power is dissipated only through radial heat conduction (constant radial flux J) and through convection. Accordingly, two equations can be written, one for the stainless steel tube, the other for the packed bed: J = he × (Tc,Re − Text ) × 2πRe dz = −λc

dTc × 2πr dz, dr (18a)

R < r < Re e u ×

P TL × 2πr dr dz = e u × cpm × L L d(r(dT/dr)) dr 0
×2πr dr dz − λp × ×2π dr dz,

where λc is the thermal conductivity of stainless steel (λc  15 W/(m K)) and Tc,Re is the temperature of the stainless steel tube at r = Re . Integration of Eqs. (18a) and (18b) and combination of these results with Eq. (17) gives: he Re r Tc (r) − Tc (R) = ln Tc (Re ) − Text λc R   e uR2 r2 [ P − cpm TL ] 1 − 2 4λp L R   r2 Phe Re 1 −  4λp cpm R2

(19a)

T (r) − T (R) =

(19b)

The value of he Re /λc is of the order of a few thousands so that one can consider that the radial temperature gradient in the stainless steel tube is negligible compared to that across the packed bed. The maximum amplitude of the radial temperature across the packed bed, TR , is estimated by TR =

Phe Re 4λp cpm

(20)

From previous experimental data performed with water pumped at an inlet pressure of 350 bar, the maximum amplitude of the radial thermal gradient was 0.8 K. This results allows a check of the validity of Eq. (20). Assuming he = 15 W/(m2 K), we find the thermal conductivity of the bed (silica + alkyl + water) to be λp = 0.13 W/(m K), a value close to that of liquids like alkanes. This suggests that the thermal conductivity of the composite bed (silica, λsilica = 1.4 W/(m K), water, λwater =

51

0.6 W/(m K), and the alkane chains λC18 = 0.11 W/(m K)) is controlled by the lowest thermal conductivity, much like the electrical conductivity of a series of conductors. The heat conduction of a series of materials is li 1 = (21) λp λi i

where li is related to length fraction of each material i. The volume fractions of silica, the C18 chains, and water are 0.25, 0.15, and 0.60, respectively. This model predicts a thermal conductivity of 0.39 W/(m K), about three times higher than the values measured, based on a value of he of 15 W/(m2 K). This comparison casts some doubts on the validity of the estimates made of the thermal conductivity of a packed bed filled with a liquid. Eq. (21) may not hold in the case of chromatographic beds and λp should better be estimated from the experimental values of the amplitude of the radial temperature gradient, TR . From the experimental data acquired with a inlet pressure of 350 bar, one can extrapolate that the maximum amplitude of the radial temperature gradient for an inlet pressure of 2 kbar would be of the order of 5 K. This value is derived for water, which has the highest specific heat capacity and thermal conductivity of solvents used in liquid chromatography [22]. For methanol, acetonitrile, tetrahydrofuran, hexane, or ethanol this maximum amplitude is larger. It could be two- to fourfold larger. 2.2.3. Conclusion on these estimates The general heat transfer equations coupled with the results of experimental measurements predict that a column kept in still air and eluted with water with an inlet pressure of 2 kbar should have a longitudinal temperature gradient with an amplitude of 50 K. This gradient might reach 100 K if the column were eluted with a solvent having a lower specific heat capacity, such as methanol, acetonitrile, THF, or a light alkane. This axial gradient can be attenuated according to the following term: he Re ηL2 Kc (1 − e )2 Pcpm 3e dp2 R2 As this term increases, the longitudinal gradient decreases. This term is much lower than 1 in the case reported above (0.03) and the temperature amplitude remains close to P/cpm , within less than 3%. The geometric parameters of the column (length, internal, and external radii) can be adjusted to decrease the gradient amplitude. Long columns having a large external radius, a small internal radius and packed with small particles may experience considerably reduced longitudinal temperature amplitude, which is why columns packed with very small particles and operated under high pressures have narrow bore sizes. Also, solvents with small specific heat capacities should be preferred. If we keep constant the pressure drop (2000 bar), the internal (2.3 mm) and the external radius (3.2 mm), the column length (25 cm), and the nature of the solvent (water, η = 10−3 Pa s and cpm = 4.2 × 106 J m−3 s−1 ), the longitudinal temperature amplitude will be reduced by a factor 2 if the particle size is reduced

52

F. Gritti, G. Guiochon / J. Chromatogr. A 1166 (2007) 47–60

to 0.85 ␮m. This is because the flow rate is markedly reduced (u 1.35 mm/s) and actually becomes too small to achieve fast and efficient separations. Following the same reasoning, keeping all the parameters constant except one, we calculate that reducing the longitudinal temperature amplitude by a factor 2 would require a column length of 1.5 m, or an internal radius of 390 ␮m, or a solvent with η/cpm = 8.2 × 10−9 , or an external column radius of 11.1 cm. Only two approaches are physically acceptable and can be considered further: the use of small inner diameter and/or of long columns, e.g. of capillary columns. The effects of the longitudinal temperature amplitude expected at a 2000 bar inlet pressure can then be reduced by using capillary columns and running them at a linear velocity large enough to approach the minimum of the HETP, which is inversely proportional to the particle size. Otherwise, there is no practical way to reduce the amplitude of the longitudinal temperature gradient below the expected 40–100 K for a 2000 bar inlet pressure with a conventional analytical columns (inner diameter 4.6 mm, length 25 cm). From Eq. (20), one expects the amplitude of the radial temperature gradient to be between 5 and 20 K at a 2000 bar inlet pressure with the column considered in this work. 3. Results and discussion The goal of this work is to predict from a qualitative point of view the possible impact of the use of ultra-high pressure drops, up to 2000 bar, on the HETP of a column. The needed calculations will be based on the use of a conventional chromatographic column (for narrow-bore columns, the temperature is practically homogeneous in the radial direction) with dimensions R = 2.3 mm, Re = 3.2 mm, and L = 25 cm, that is packed with 5 ␮m particles. We assume that the solvent has the same characteristics as pure water (viscosity η = 10−3 Pa s, specific heat capacity cpm = 4.2 × 106 J m−3 s−1 . See column characteristics in Table 1) For an adiabatic column, under these experimental conditions, with an inlet pressure of 2 kbar, the amplitude of the axial temperature gradient is TL = 50 K and that of the radial temperature amplitude is TR = 5 K, as explained in the theory section and derived by extrapolation of our previous experimental measurements [1]. As a first approximation, we assume that the amplitude of the radial temperature gradient is independent of the axial position. Finally, the temperature profile T (r, z)is:   r2 TL T (r, z) = Text + (22) z + TR 1 − 2 L R This model neglects the entrance length of the column, which determines the axial position beyond which the parabolic radial profile is fully developed. It also neglects the role of the stainless steel enclosing the column tube in relaxing the longitudinal temperature gradient close to the wall. Eq. (22) represents the simplest temperature profile that combines both the radial and the axial temperature gradients, based on the experimental knowledge of the maximum amplitudes measured for TR and TL .

3.1. Relationship between the local pressure and the local temperature Although the model of temperature profile described above cannot be exact, it is a first-order approximation that can be used to calculate the pressure profile along the column. The mass flux of solvent through any cross-section of the column is constant (mass conservation), as stated in Eq. (8). The compressibilities χT (T, P) of water and methanol are given by the Tait equation χT (T, P) =

c(295 K) c(T )  P + b(T ) P + b(295 K)

(23)

In the case considered, the coefficients b and c can be considered as independent of the temperature because the range of temperatures used in liquid chromatography are much smaller than the critical temperature, Tc , of the solvent (Tc = 647 and 513 K for water and methanol, respectively [23]). The values of the c and b for methanol and water are 0.1368 and 2996 bar, and 0.148 and 1210 bar, respectively [24]. The thermal expansion coefficient under atmospheric pressure, P 0 , is given by αT (T, P 0 ) = a1 + a2 T

(24)

In the case of water, a1 = −2.08 × 10−3 K−1 and a2 = 7.84 × 10−6 K−2 . For methanol, a1 = 1.14 × 10−3 K−1 and a2 = 0 K−2 [23]. The density of the mobile phase is given by  P  ρS (T, P) c = exp dP ρS (295 K,P 0 ) P0 P + b   T  × exp − (a1 + a2 T ) dT 295 c

=

((P + b)/(P 0 + b)) ea1 (T −295)+a2 ((T

2 −2952 )/2)

(25)

Combining Eqs. (7), (9), and (25) permits the calculation of the local pressure gradient as a function of the local pressure P and temperature T  c ηS (T, P) P 0 + b dP 2 2 × ea1 (T −295)+a2 ((T −295 )/2) =− dz k0 P +b ×u(295 K,P 0 )

(26)

Eq. (26) can be integrated step by step by starting from the column outlet, where the pressure is the atmospheric pressure, to the column inlet, where the pressure has to be determined consistently. First, this imposes the choice of the outlet temperature, hence a fixed arbitrary choice for the longitudinal temperature amplitude, TL . The value of the radial temperature amplitude, TR , is directly dependent on the value of TL by combining Eqs. (17) and (20). Secondly, the integration of Eq. (26) also requires the knowledge of the flow rate Fv , hence the linear velocity u present in this equation. This integration is performed numerically between z = L and z = 0. It gives the overall pressure drop P along the entire column. From Eq. (17), one can

F. Gritti, G. Guiochon / J. Chromatogr. A 1166 (2007) 47–60

53

Fig. 1. Temperature (solid line) and pressure (dashed line) profiles along the column. Note that the pressure profile is not linear while the temperature profile is linear. The local pressure gradient is larger at the column inlet than close to its outlet: (A) eluent water and (B) eluent methanol. Note that the deviation of the pressure profile from linear behavior is larger with methanol, the more compressible solvent.

also calculate independently the expected amplitude of the longitudinal temperature and compare it to the initial arbitrary choice made. This equation is based on heat conservation and writes that the power generated inside the column by the viscous forces is equal to the sum of the heat lost by the external surface of the column to the still air surrounding it, at normal pressure and the heat evacuated from the column by the exiting stream of mobile phase. The eventual difference between the two pressure drops is then minimized by changing and optimizing the variable flow rate Fv . Fig. 1 illustrates the relationship between the local pressure and temperature along the column, assuming a linear temperature profile along the column, a constraint arbitrarily set in our initial assumptions. The radial temperature gradient causes a

radial gradient of mobile phase viscosity, hence a radial gradient of the mobile phase velocity across the column (i.e., along its radius). While the temperature profile along the column remains linear, the pressure profile does not. The variation of the local pressure is steeper near the column inlet than near its outlet because the eluent viscosity increases with increasing pressure and decreases with increasing temperature. Since the column inlet is colder and under a higher pressure than its outlet, the influence of the variation of the local viscosity on the pressure profile is clearly visible in Fig. 1. The effect is more important with methanol (Fig. 1B) than with water (Fig. 1A). Fig. 2 shows plots of the solvent viscosity corresponding to the axial temperature profile (but assuming a constant pressure), for the pressure

Fig. 2. Viscosity profiles of the mobile phase along a column. The solid line shows the actual eluent viscosity at the local temperature and pressure at abscissa z along the column. The dotted line gives the viscosity profile expected under isothermal conditions (295 K) with the actual pressure profile along the column. The dashed line shows the viscosity expected at constant pressure (atmospheric pressure P 0 ) but with the actual temperature profile along the column. Note the significant differences between the three curves: (A) eluent water and (B) eluent methanol.

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Fig. 3. Plot of the linear mobile phase velocity along the column when the compressibility and the thermal expansion of the eluent are taken into account. Pressure drop along the column, 2 kbar. Note that the velocity is about 10% larger at the column outlet than at its inlet: (A) eluent water and (B) eluent methanol.

profile (but assuming a constant temperature), and along the column, after the correlation of Lucas (see Eq. (5)[9]). It is important to note that the viscosity decreases by a factor three (for water) and ten (for methanol) between the column inlet and the column outlet. The difference between the outlet and inlet linear velocities (see Fig. 3) is about 8% (water) and 13% (methanol). This difference results from the simultaneous effects of the pressure and the temperature gradients on the specific volume of the eluent. As a consequence, the local pressure gradient is about 2.5 times (water) and 6.5 times (methanol) larger at the column inlet than at its outlet. Note also that the combination of simultaneous variations of the pressure and the temperature along the column results in a much larger effect than those that would be caused by mere isothermal (at 295 K) or isobaric (under atmospheric pressure) flows (see top and bottom curves in Fig. 2). 3.2. Impact of the axial temperature gradient alone on liquid–solid mass transfer under high pressure In this section, we consider the effect of an axial longitudinal temperature gradient on the high velocity of the HETP curve. Obviously, under the heterogeneous conditions considered here, the HETP is an apparent parameter valid for a specific column and is the result of the convolution of the local values of H over the column length. Eq. (1) is the general equation for the local HETP of totally porous, spherical particles [8]. The numerical integration of Eq. (1) over the column length gives the overall space band variance, σz2 . Divided by the column length, L, this variance gives the efficiency of the column under the pressure and the temperature gradients considered. This efficiency can be compared to that of a hypothetically isothermal, isobaric column, which is postulated when the parameters in Eq. (1) are estimated in conventional chromatography. Obviously, these parameters depend on both the temperature and the pressure, We assume, however, that the compound dis-

tribution between the liquid and the stationary phase does not depend on the local pressure. This is a reasonable assumption in the case of low molecular weight compounds [25]. As a consequence, the parameters δ0 (Eq. (2)) and  (Eq. (3)) depend only on the temperature. We made the calculations assuming a retention factor equal to 5 at 295 K, a moderate value selected to illustrate a typical case encountered in LC. The corresponding isosteric heat of adsorption was −15 kJ/mol, which allows the determination of the Henry’s constant K(T ) and of the temperature dependence of the retention behavior of the analyte, according to the classical Van’t Hoff law:   Qst K(T ) = K0 exp − (27) RT While δ0 , , and K are temperature dependent only, the diffusion coefficient, Dm , depends significantly on both the temperature and the pressure (Eqs. (4) and (5)). Fig. 4 shows plots of Dm versus the axial coordinate, z. Note that the molecular diffusivity is about 3.5 times (water) and 10 times (methanol) larger at the column outlet than it is at the column inlet. This indicates that knowledge of the isothermal, isobaric diffusion coefficient alone is insufficient. The use of this value would lead to an erroneous estimate of the local HETP. The single effect of either temperature at constant pressure (P 0 ) or of pressure at constant temperature (295 K), respectively, is much smaller than the actual effect observed when both temperature and pressure vary simultaneously along the column. As a consequence, it is critical to estimate the viscosity, η, and the diffusion coefficient, Dm , as a function of both the temperature and the pressure in any attempt to estimate the local HETP parameters at a given axial coordinate, z, along the column. From the temperature and the pressure profiles given in Fig. 1 and by combining Eqs. (4) and (5), this determination is straightforward. A plot of the ascending branch of the Van Deemter curve is shown in Fig. 5, as a plot of H versus the linear velocity measured at 295 K, under atmospheric pressure (u = Fv /e S, with S the

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Fig. 4. Same as Fig. 2, except molecular diffusivity profiles of an analyte (molar volume at the boiling point, VA = 100 cm3 /mol). Note how important it is to consider the simultaneous effects of temperature and pressure when estimating the diffusion coefficients: (A) eluent water and (B) eluent methanol.

column cross-section area). Fig. 6 shows a plot of H versus the pressure drop, P. The combined effects of the axial temperature profile and of the pressure gradient on the column efficiency become significant with water and methanol for inlet pressures exceeding about 500 bar. It appears advantageous to work under high inlet pressure since the slope of the high-velocity branch of the Van Deemter curve decreases monotonously with increasing flow rate. However, this apparent efficiency gain should be considered with caution. First, from a practical point of view, it is not useful to operate columns at such high flow rates that the HETP exceeds largely its minimum value. Secondly, from a fundamental point of view, since there is a radial heat loss, there must be a radial temperature gradient (Eq. (20)) across the column. We discuss below the consequences of the existence of this radial temperature gradient on the C-branch of the Van Deemter curve.

3.3. Simultaneous influence of an axial and a radial temperature gradient on mass transfers under high pressure We consider in this section the consequences of the simultaneous variation of the retention factor along the longitudinal and the radial coordinates. It is important to account for the dependence of the axial gradient of temperature on the radial position because this influence of the radial position might cause a significant loss of the overall or apparent column efficiency. Were we able to move a concentration sensor across the exit cross-section of the column, we would observe a radial distribution of the elution time. Sample molecules eluted at the center of the column would come first, those eluted close to the wall would exit last. Giddings calculated this loss of efficiency in his treatment of the apparent plate height in radially nonuniform columns [15]. The consequence of a radial temperature gradient is straightforward. Because we assume that the column bed is radially

Fig. 5. Plots of the contributions of the solid–liquid mass transfer contribution to the column HETP vs. the linear velocity u295 K,P 0 . Comparison between the isothermal (295 K) and isobaric (P 0 ) plot and the plot which considers only the longitudinal temperature gradient ( TL ) and the pressure drop ( P) in the calculation: (A) eluent water and (B) eluent methanol. Note the slight improvement of the column efficiency with increasing pressure drop.

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Fig. 6. Same as Fig. 5, but with the pressure drop in the abscissa of the plots: (A) eluent water and (B) eluent methanol. Note the small temperature effect below P = 400 bar.

homogeneous, the radial distributions of the external porosity and of the permeability of the column bed are constant and the pressure is also constant across the column. Since the viscosity depends on the temperature, however, the local linear velocity is a function of the radial coordinate, r. As a first approximation, we will neglect the effects of the radial relaxation of the concentration gradient that is due to diffusion down the concentration gradient caused by this radial distribution of velocities and will consider the column bed as a set of elementary coaxial cylindrical beds. In other words, we assume that there is not enough time for radial diffusion to reduce significantly the radial concentration gradient caused by the radial velocity gradient. The degree of validity of this assumption can be assessed by estimating the variance, X, of the radial diffusion during the retention time tR , which indicates the distance over which a degree of radial relaxation is achieved. For this purpose, we assume that radial diffusion proceeds at the same rate as axial diffusion, with B = 2γDm , Dm being the molecular diffusivity of the compound considered. According to the general diffusion equation [15], we have: ( X)2 = 2Dm tR

(28a)

The retention time of the sample is tR = t0 (1 + k ) =

L t L (1 + k ) (1 + k ) = u0 e u

(28b)

The relative standard deviation of radial diffusion, X/R, during its elution time is

X 2Dm (1 + k ) t L = (28c) R R2 e u where k is the retention factor and u is the interstitial velocity of the mobile phase. In this work, we have chosen a L = 25 cm long column, with an internal radius R = 2.3 mm. We further assume that k = 5, e = 0.4, and t = 0.6, which are conventional values, and a molecular coefficient of the order of 1 × 10−5 cm2 /s. Numerical calculations show that X/R is smaller than 0.1 if u is larger than 8.5 cm/s. This velocity is in the range of the linear velocities encountered in ultra-high pressure chromatography (see Fig. 3). This means that diffusion will smoothen but only to a moderate extent the profiles that we calculate now. Since we assume that the concentration is not radially relaxed, we must also assume an ideal sample injection distribution over the inlet cross-section area of the column (sample size N0 molecules).

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Once the sample penetrates in a given point of the inlet crosssection area, it will propagate along a straight line parallel to the column axis, at a velocity function of the local temperature and pressure along this path. Both the radial and longitudinal temperature gradients that affect the local flow velocity and the retention factor of the compound must be taken into account. As indicated above, we consider the chromatographic bed as a series of infinitely thin coaxial cylinders of internal radius r and external radius r + dr. The elution time along this cylinder is tR (r) and the time-based band standard deviation σt (r). The amount of sample injected in the cylinder is: dN(r) =

2rdr N0 R2

(29a)

Assuming that the elution peak of this sample out of this cylinder is Gaussian, its profile dC(t, r) is   2r 1 [t − tR (r)]2 dr (29b) dC(t, r) = N0 2 √ exp − R 2σt2 (r) 2π The two quantities tR (r) and σt2 (r) are obtained by integrating the incremental retention time dtR and the incremental space band dispersion dσz2 over the column length for a constant radius r.  L tR (r) = dtR (r) (30) 0

and σt2 (r) =

1 u(r, z = L)2

 0

L

dσz2 (r)

(31)

where u(r, z = L) is the interstitial mobile phase velocity at the radial position r and at the axial position z = L, the column outlet, where the sample is detected. The total chromatogram is obtained by integrating Eq. (29b) from r = 0 to r = R. For this calculation, it is convenient to use the auxiliary variable x = r/R. Accordingly,     2 1 [t − tR (x)]2 C(t) = N0 x exp − dx (32) π 0 2σt2 (x) This calculation requires knowledge of the functions tR (x) and σt (x). To derive these functions, we calculated numerically the values of tR (x) and σt (x) for a series of columns of radii r such that r/R = x = 0.95, 0.85, 0.75, 0.65, 0.55, 0.45, 0.35, 0.25, 0.15, and 0.05 and calculated the values of the coefficients of the third order polynomial function that best fits these values: tR (x) = t0 + t1 x + t2 x2 + t3 x3

(33a)

and σt (x) = s0 + s1 x + s2 x2 + s3 x3

(33b)

An example is shown in Fig. 7, for water pumped through the column at an interstitial velocity of 7.13 cm/s. Combining Eqs. (32), (33a), and (33b), one can numerically calculate the band profile for any inlet pressure. In the following, we show

Fig. 7. Example of the variations of the retention time, tR , and the time peak standard deviation, σt , with the radial coordinate r/R. Note that the elution time and the peak standard deviation are larger close to the column wall (colder region), as expected.

results calculated for inlet pressures of 100 bar (typical case in HPLC), 400 bar (extreme case in HPLC), 800, 1200, 1600, and 2000 bar (typical and extreme cases in UPLC), with either water or methanol as the eluent. The specific volume of the compound at its boiling point was assumed to be 100 cm3 /mol (small molecules). Fig. 8 shows a plot of the overall band variance of the profile given by Eq. (32) divided by the column length L. Since the radial relaxation of the concentration gradient was neglected, this value of the column efficiency is necessarily an underestimate. Yet the figure makes it obvious that, for inlet pressures beyond 400 bar, the column efficiency is strongly affected by the radial thermal heterogeneity. The efficiency gain that was due to the axial temperature gradient vanishes. The calculated chromatograms are shown in Fig. 9. For an inlet pressure of 100 bar, the peak remains nearly Gaussian because the amplitude of the radial temperature gradient is less than 0.2 K. However, for higher inlet pressures, the peak becomes rapidly distorted and exhibits a growing shoulder. As another consequence of the column not being isothermal, the retention volume diminishes, as observed in previous work [19].

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Fig. 8. Same as in Fig. 5, except the addition of the plot regarding the solid–liquid mass transfer term, which takes into account both the linear longitudinal and the parabolic radial temperature gradients of the packing material. Note the nefarious impact of the column radial thermal heterogeneity.

Fig. 9. Calculated chromatograms when both the axial and the radial temperature gradients of the column are taken into account at different pressure drops. Eluent water. Note the apparition of a shoulder and the increasing band width upon increasing the pressure drop.

The rapid decrease of the column efficiency shown in Fig. 8 should be pondered carefully. The model of the column as a series of coaxial cylinders along which the eluent migrates without radial diffusion is certainly approximate. It neglects both radial diffusion and eddy diffusion in the radial direction, which forces the eluent stream to flow in random directions that are on the average parallel to the column axis but, at any given point or time, at an angle with this axis. Homogenization of the concentrations over the column cross-section certainly attenuates the strong efficiency loss. More data collected at very high flow rates, hence with high inlet pressures are required to confirm the conclusions derived from this simple model. Still, de Villiers et al. [11] clearly observed a decrease in the column efficiency when the temperature of the wall was thermostated in a water bath inducing a significant radial temperature gradient. We note also that Dapremont et al. [26] studied experimentally the influence on the elution band profiles of a temperature difference between the column wall (which was kept at a constant temperature) and the column center (which was heated by a stream of warm mobile phase). These authors recorded symmetrical peaks for an isothermal column and strongly distorted profiles with significant radial thermal gradients. When the column center is warmer than its wall, the peak tails. If the column center is colder than the wall, there is a hump on the front of the profile, causing a serious loss of efficiency, as predicted from the calculated profiles in Fig. 10. Any radial temperature heterogeneity of the column reduces seriously its efficiency. These results demonstrates that radial diffusion is not fast enough to average out the contributions of all the coaxial annular cylinders into a single Gaussian peak profile. Fig. 11 shows that the smaller the column diameter, the more symmetrical the elution peak profile. In classical analytical chromatography, the column inner diameter is usually small enough to allow the sample to diffuse across the column and homogenize the radial composition of the band, due to the effective diffusivity of the compound through the column packing (Deff = Dm ) and the radial convective eddies. In classical analytical chromatography also, pressures are mostly

Fig. 10. Experimental chromatograms for three experimental thermal conditions. The temperature of the wall is colder, warmer, or at the same temperature as the feed eluent. Note the strong peak dissymmetry when the temperature of the wall and the eluent differ. Reproduced with permission from [26](Fig. 9).

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Fig. 11. Calculated chromatograms showing the effect of the inner diameter of the column on the peak shape under the condition of radial temperature gradients between the temperature of the wall and the temperature of the feed eluent. Note the disappearing of the peak dissymmetry when the column diameter is small. Reproduced with permission from [26](Fig. 15).

well below 400 bar. However, when the temperature difference between the column wall and the mobile phase entering the column is high enough, significant peak distortion are observed [27]. This situation will be analyzed in depth in a forthcoming communication based on the general dispersion theory of Aris [28]. 4. Conclusion The importance of the impact of the longitudinal and the radial temperature gradient on the efficiency of a chromatographic column operated at high velocities, with a high inlet pressure, can be determined. The calculations were made by extrapolating the profiles measured for an inlet pressure of 350 bar and assuming a realistic linear axial temperature profile and a parabolic radial temperature profile. The results show that the thermal heterogeneity of the column that results from the eluent decompression and its friction against the bed can cause a significant decrease of the column efficiency. The C-term of the classical Van Deemter equation increases with increasing column pressure drop. This effect is particularly important with large diameter columns (our results are based on the use of a 25 cm × 0.46 cm column). The efficiency loss is due to the combination of the radial heterogeneity of the flow profile (faster in the column center, slower along its wall) and of the radial profile of retention factors (lower in the column center, higher close to its wall). The effects of the radial flow and retention factor distribution add up, causing a progressive increase of the slope of the C-branch of the Van Deemter curve. Therefore, it is impossible to carry out fast and efficient separations using conventional columns (15–25 cm × 0.46 cm columns packed with 5 ␮m or smaller particles). The columns would need to be thermally insulated and the separation factor to be independent of temperature. The results reported above are overestimates because they neglect a possible radial relaxation of the radial concentration

59

gradient which is likely to take place at moderate linear velocities. A finite rate of radial diffusion will reduce the nefarious effects of a radial temperature gradient, due to the homogenization of the sample concentration across the column cross-section. A new and more elaborate model should take into account the radial dispersion of the band. Our model neglects also the influence of the stainless steel tube, the thermal conductivity of which is ca. 25 and 75 times larger than those of water and methanol, respectively. The metal tube around the column relaxes the axial temperature gradient in the vicinity of the wall, compared to the same gradient in the column central region. It might even make the column center to be colder than its wall region at the column inlet and the temperatures of the center and the wall to be close half-way along the column. The largest amplitude of the radial temperature gradient should always take place at the column outlet [1]. The energy balance in the whole column states that the temperature gradients will be lower with long, narrow columns packed with small particles. This is why capillary columns packed with sub-2 ␮m particles, which are able to withstand pressures of 2 kbar or more [29] suffer less from the nefarious thermal heterogeneity of the bed at ultra-high pressures [30]. The practicality of such columns as analytical tools remains doubtful, however. From a fundamental point of view, the problem of the thermal heterogeneity of the packed bed should be investigated by integrating the local heat balance equation in the two dimensions, z, r. The actual steady-state temperature profiles obtained across the packed bed will probably differ somewhat from the profiles assumed in this work. The influence of the stainless steel tube could easily be taken into account in this process. Given specific boundary conditions, the calculation results will inform on the true temperature profile and on the impact of the particle size, the length, the inner and outer diameters of the column tube, and the nature of the eluent on the column thermal heterogeneity. This is the current objective of our research. Nomenclature

cpm dp Dm F (λm ) Fv he

mobile phase heat capacity (J/(m3 K)) average particle size (m) molecular diffusion coefficient (m2 /s) pore steric hindrance parameter flow rate (m3 /s) heat transfer coefficient between the external surface of the column tube and the lab atmosphere (W/(m2 K)) J radial heat loss (W) k0 permeability of the packed bed (m2 ) K(P, T ) Henry’s constant at temperature T and pressure P Kc Kozeny–Carman constant L column length (m) MS molecular weight of the solvent (g/mol) P pressure (Pa) P pressure drop (Pa) Pc critical pressure (Pa)

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Pfriction created power due to solvent friction per unit volume of packed bed (W/m3 ) reduced pressure Pr vapor pressure (Pa) Pvp Qst isosteric heat of adsorption (J/mol) r radial column coordinate (m) rp average pore diameter (m) R internal column radius (m) Re external column radius (m) S column internal cross-section (m2 ) tR retention time (s) T temperature (K) Tc critical temperature Tc,z=L temperature of the column surface at the outlet (K) Tc,z=0 temperature of the column surface at the inlet (K) Text laboratory temperature (K) reduced temperature Tr Tz=L temperature of the exiting liquid mobile phase (K) TL longitudinal temperature gradient amplitude of the mobile phase (K) TR radial temperature gradient amplitude (K) longitudinal temperature gradient amplitude of stain TW less steel tube (K) u interstitial linear velocity (m/s) uS superficial linear velocity (m/s) VA molar volume of the solute at its boiling point (m3 /mol) power evacuated by the exiting liquid mobile phase (W) Wexit Wfriction power friction generated in the column (W) Wsurface power evacuated through the lateral surface of the stainless steel column (W) x auxiliary variable in Eq. (32) X standard deviation of radial diffusion (m) z longitudinal column coordinate (m) Greek letters α structural parameter in Eq. (8)(see more detailed in Ref. [8]) αT (T, P) liquid phase isobaric expansion coefficient (K−1 ) β positive factor in Eq. (14) γp particle obstructive factor δ0 (T, P) thermodynamic parameter defined in Eq. (2) external column porosity e p particle porosity ηS (T, P) mobile phase viscosity at temperature T and pressure P (Pa s) λc heat conductivity of the stainless steel column tube (W/(m K)) λp heat conductivity of the packed bed (W/(m K)) λsilica heat conductivity of solid silica (W/(m K)) λwater heat conductivity of pure water (W/(m K)) λC18 heat conductivity of liquid octadecylsilane (W/(m K))

ρS (T, P) mobile phase density at temperature T and pressure P (m3 ) 2 time band second central moment (m2 ) σt 2 space band second central moment (m2 ) σz φS association factor of the solvent χT (T, P) liquid phase isothermal compressibility factor (Pa−1 ) ω acentric factor (T, P) sample particle diffusivity defined in Eq. (3) Acknowledgments This work was supported in part by grant CHE-06-08659 of the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. References [1] F. Gritti, G. Guiochon, J. Chromatogr. A 1138 (2007) 141. [2] G. Guiochon, J. Chromatogr. A, doi:10.1016/j.chroma.2007.05.090, in press. [3] U. Neue, B. Alden, P. Iraneta, M. Savaria, T. Grady, M. Kele, K. Wyndham, Lecture 2102 Presented at HPLC-2006, San Francisco, CA, June 17–22, 2006. [4] J.E. McNair, K.C. Lewis, J.W. Jorgenson, Anal. Chem. 69 (1997) 983. [5] J.E. McNair, K.D. Patel, J.W. Jorgenson, Anal. Chem. 71 (1999) 700. [6] G. Guiochon, J. Chromatogr. A 1126 (2006) 6. [7] G. Guiochon, J. Chromatogr. A, submitted for publication. [8] F. Gritti, G. Guiochon, Anal. Chem. 78 (2006) 5329. [9] K. Lucas, Chem. Ing. Tech. 53 (1981) 959. [10] U. Neue, M. Kele, J. Chromatogr. A 1157 (2007) 236. [11] A. de Villiers, H. Lauer, R. Szucs, S. Goodall, P. Sandra, J. Chromatogr. A 1113 (2006) 84. [12] F. Gritti, G. Guiochon, Chem. Eng. Sci. 61 (2006) 7636. [13] F. Gritti, G. Guiochon, Anal. Chem. 79 (2007) 3188. [14] K. Miyabe, G. Guiochon, J. Phys. Chem. B 103 (1999) 11086. [15] J.C. Giddings, Dynamics of Chromatography, Marcel Dekker, New York, 1965. [16] C.R. Wilke, P. Chang, AIChE J. 1 (1955) 264. [17] B.E. Poling, J.M. Prausnitz, J.P. O’Connell, The Properties of Gases & Liquids, 5th ed., McGraw-Hill, Inc., New York, 2001. [18] H. Vogel, Physik. Z. 22 (1921) 645. [19] F. Gritti, G. Guiochon, J. Chromatogr. A 1131 (2006) 151. [20] H.-J. Lin, S. Horv´ath, Chem. Eng. Sci. 36 (1981) 47. [21] www.cheresources.com [22] H. Poppe, J.C. Kraak, J.F.K. Huber, H.M. van der Berg, Chromatographia 36 (1981) 515. [23] R.C. Weast (Ed.), CRC Handbook of Chemistry and Physics, 68th ed. 1987–1988. [24] M. Martin, G. Guiochon, J. Chromatogr. A 1090 (2005) 16. [25] V.L. McGuffin, C.E. Evans, J. Microcol. Sep. 3 (1991) 513. [26] O. Dapremont, G.B. Cox, M. Martin, P. Hilaireau, G. Guiochon, J. Chromatogr. A 796 (1998) 81. [27] D. Guillaume, S. Heinisch, J.L. Rocca, J. Chromatogr. A 1052 (2004) 39. [28] R. Aris, Proc. Roy. Soc. London, Ser. A 252 (1959) 538. [29] F. Chen, E.C. Drumm, G. Guiochon, J. Chromatogr. A 1083 (2005) 68. [30] K.D. Patel, A.D. Jerkovich, J.C. Link, J.W. Jorgenson, Anal. Chem. 76 (2004) 5777.