Influence of Viscous Friction Heating on the Efficiency of Columns

Anal. Chem. 2009, 81, 3365–3384

Influence of Viscous Friction Heating on the Efficiency of Columns Operated under Very High Pressures Fabrice Gritti,† Michel Martin,‡ and Georges Guiochon*,† Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, Division of Chemical Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6120, and Laboratoire de Physique et Mećanique des Milieux He´te´roge`nes (PMMH, UMR 7636 CNRS, Universite´ Paris 6, Universite´ Paris 7), Ećole Supe´rieure de Physique et de Chimie Industrielles, 10 Rue Vauquelin, F-75231 Paris Cedex 05, France When columns packed with very fine particles are operated at high mobile phase velocities, the friction of the mobile phase percolating through the column bed generates heat. This heat dissipates along and across the column and axial and radial temperature gradients appear. The wall region of the column tends to be cooler than its center, and due to the influence of temperature on the mobile phase viscosity and on the equilibrium constant of analytes, the band velocity is not constant across the column. This radial heterogeneity of the temperature distribution across the column contributes to band broadening. This phenomenon was investigated assuming a cylindrically symmetrical column and using the general dispersion theory of Aris, which relates the height equivalent to the theoretical plate (HETP) contribution due to a radial heterogeneity of the column to the radial distribution of the linear velocities of a compound peak and to the radial distribution of its apparent dispersion coefficients in the column bed. The former is known from the temperature gradient across the column, the temperature dependencies of the mobile phase viscosity, and the retention factor of the compound. The latter is derived from the known expression of the transverse reduced HETP equation for the column. The values of the HETP calculated with the Aris model and a classical HETP equation were compared to those measured on a 2.1 × 50 mm Acquity BEH-C18 column, run at flow rates of 0.6, 0.95, 1.30, and 1.65 mL/min, with pure acetonitrile as the mobile phase and naphtho[2,3a]pyrene as the retained compound. These two sets of data are in generally good agreement, although the experimental values of the HETP tend to increase faster with increasing mobile phase velocity than the calculated values. The viscous friction of the stream of a mobile phase against the bed of a chromatographic column through which it percolates under a significant pressure gradient generates heat.1 The amount * Corresponding author: (e-mail) [email protected]; (fax) 865 974-2667. † University of Tennessee and Oak Ridge National Laboratory. ‡ Laboratoire de Physique et Mećanique des Milieux He´te´roge`nes. (1) Lin, H.-J.; Horváth, Cs. Chem. Eng. Sci. 1981, 36, 47. 10.1021/ac802632x CCC: $40.75  2009 American Chemical Society Published on Web 04/10/2009

of heat locally produced is proportional to the product of the pressure gradient and the superficial linear velocity. So, the effects of this heat flux are negligible with the 30 cm long columns packed with 10 or 20 µm particles and operated under 40-100 atm that were used in the early development of high-pressure liquid chromotography (HPLC). They are still small with the conventional 15-20 cm long columns packed with 5 µm particles and operated under 200-400 atm that are still widely used.2,3 They become significant with the newly available 5-10 cm long columns packed with 1.5-3 µm particles and operated under 400-1000 atm that are becoming popular because they offer shorter analysis times and lower detection limits. Depending on the column characteristics and particularly on its diameter, the consequences of this local heating of the column on its efficiency may be serious. Considerable attention is now paid to the quantitative analysis of this effect.4 This heat produced locally is dissipated across the packing material in both the radial and the longitudinal direction of the column. In principle, a steady-state thermal equilibrium is achieved asymptotically. However, experience shows that this state is practically reached after a time that is short compared to the time usually invested by analysts to equilibrate the mobile and stationary phases. Afterward, the column temperature remains constant everywhere in the bed and no longer depends on time. Under steady-state conditions, the heat generated in each point of the column flows outside the column and this flow is associated with the formation of longitudinal and radial temperature gradients. The amplitude of these gradients depends on the degree of thermal insulation of the column. If the column is kept adiabatic, there is no radial heat loss through its wall, the column temperature is radially uniform, and all heat generated is evacuated through the column ends, by heat conduction to the equipment and by the fluid leaving the column. Accordingly, the longitudinal gradient is maximum. In contrast, if the temperature of the column wall is kept constant (i.e., by placing the column in a liquid bath kept at room temperature), the axial temperature gradient becomes negligible and a parabolic radial temperature profile forms across the packed bed. The radial temperature gradient is maximum, and the heat generated in the bed is lost (2) Gritti, F.; Guiochon, G. J. Chromatogr. A 2006, 1131, 151. (3) Gritti, F.; Guiochon, G. J. Chromatogr. A 2007, 1138, 141. (4) Gritti, F.; Guiochon, G. J. Chromatogr. A 2007, 1166, 30.

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radially, through the column wall. The effect of the thermal environment of the column on band broadening and on the column performance was illustrated in refs 5 and 6. The reduced height equivalent to the theoretical plate (HETP) plots of naphtho[2,3-a]pyrene deviate more strongly from those expected for a hypothetically isothermal column when the column is immersed in a thermostatted liquid bath than when it is simply left in stillair conditions. Actually, most columns are operated under intermediate conditions of thermal insulation, so radial and longitudinal temperature gradients coexist. Whether the longitudinal or the radial temperature gradient dominates depends on the thermal entrance length or distance along the column beyond which the radial temperature gradient is parabolic.1 If the column length is much larger than the thermal entrance length, the radial temperature gradient is maximum, the longitudinal temperature gradient is minimum, and vice-versa. Because the density and viscosity of the mobile phase, and the retention factors, depend on the local temperature and pressure, the existences of these axial and radial temperature gradients and of the longitudinal pressure gradient cause the mobile phase viscosity to depend on the spatial coordinates. The longitudinal temperature gradient has only minor consequences. Mass conservation of the mobile phase requires that its mass flow rate across any column cross-section remains constant. The longitudinal variation of the mobile phase viscosity along the column only affects the local pressure gradient that will decrease along the column, with negligible effects on the column efficiency. Simple corrections could take care of the small increase of the mobile phase velocity due to the decrease of its density with increasing temperature and decreasing local pressure and the small effect of the temperature on the coefficients of the HETP equation. Since there is a longitudinal pressure gradient along the column but only a negligible radial pressure gradient [The radial temperature gradient causes a radial viscosity gradient, hence a radial gradient of the axial mobile phase velocity. This means that the mobile phase flows faster in the central region of the column than along its wall, causing the streamlines slowly to focus toward the column center. Thus there is a small radial pressure gradient that we will neglect.], the mobile phase linear velocity, ue, the molecular diffusivity, Dm, and all the coefficients of the plate height equation are also functions of both the radial and the axial coordinates. The classical theory of band broadening derived by Van Deemter et al.7 and by Giddings8,9 as well as empirical equations such as that of Knox10 are valid only for an isothermal column. Whenever heat effects are significant and the radial thermal gradient is not flat, the relationship between the apparent column HETP derived from the elution bandwidth and the local value of the HETP given by these classical approaches is most complex (as it is in gas chromatography and in thin layer chromatography). More appropriate and accurate models of band broadening are necessary to account for the experimental data obtained in very high-pressure liquid chromatography (VHPLC).5,6 The application (5) Gritti, F.; Guiochon, G. J. Chromatogr. A 2009, 1216, 1353. (6) Gritti, F.; Guiochon, G. J. Chromatogr. A 2008, 1206, 113. (7) van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. Sci. 1956, 5, 271. (8) Giddings, J. C. Dynamics of Chromatography; M. Dekker: New York, 1965. (9) Giddings, J. C. Unified Separation Science; Wiley: New York, 1991. (10) Knox, J. H. J. Chromatogr. Sci. 1977, 15, 352.

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of the classical theory of band broadening to the columns used in VHPLC requires the calculation of the contribution to band spreading that is due to the radial gradient of the band velocity, itself a result of the radial gradient of temperature. In order to perform this numerical calculation, the column is divided into a series of slices of length dz, the contribution of each slice is determined, and these contributions are integrated from the column inlet to its outlet. The goal of this work is the application of the general dispersion theory of Aris11 to derive the contribution to H due to the radial heterogeneity of a column provided that the length, dz, of the column slice is much larger than the particle size, dp. The interest of the Aris approach is that it takes into account the relaxation of the concentration gradient across the column radius, when the flow distribution is not uniform. In a previous treatment,4 we neglected this phenomenon, which resulted in an overestimate of the efficiency loss due to the temperature radial heterogeneity of the column. The aim of this work is to account for the influence on the HETP of the combination of radial dispersion and radial velocity distribution that arises from the radial temperature gradient. The calculation results will be compared to experimental data measured on a 5 cm long column packed with 1.7 µm particles percolated at high flow rates (0.60, 0.95, 1.30, and 1.65 mL/min, respectively) for which a significant increase of the column wall temperature is observed. THEORY When a band migrates along an homogeneous column, it disperses uniformly along and across this column. This ideal model assumes that the mobile phase percolates following piston flow displacement and that the column is isothermal. Piston flow through a porous medium means that the flow velocity averaged over a region large compared to the average particle size but small compared to the column radius is constant across the column cross-section. Then, band dispersion is characterized by an HETP given, e.g., by the classical Van Deemter equation:7 H)A+

B + Cue ue

(1)

where the numerical coefficients A (which may be a function of the linear velocity, ue),1,8,9,12 B, and C account respectively for the effects of axial dispersion, axial molecular diffusion through the tortuous and constricted porous networks of the bed packing and the particles, and the mass transfer resistances while ue is the interstitial linear velocity of the mobile phase. The three terms in eq 1 are accounted for using various models that can be found in the literature and will be elaborated upon later. When the column is operated under a high pressure gradient and the mobile phase flow velocity is large, the assumption made above fails, significant heat being generated by the friction of the mobile phase against the bed. The coefficients A, B, and C depend on the local temperature hence on the position z,r (because we assume that the column bed is cylindrically symmetrical) inside the column. The apparent column HETP derived from the band variance, σ2, should be given by two successive integrations of (11) Aris, R. Proc. R. Soc. 1959, A252, 538. (12) Gritti, F.; Guiochon, G. Anal. Chem. 2006, 78, 5329.

the radial dispersion coefficient, the smaller the increase of the bandwidth because the faster dispersion process tends to render the column more radially homogeneous. The theoretical treatment of Aris permits the calculation of the additional contribution to the HETP. For an open tubular column, this contribution is written:11 HOTC ) Figure 1. Radial velocity gradient that is formed across the column diameter, as a result of the existence of a radial temperature gradient. Application of the general dispersion theory of Aris to packed HPLC columns. The velocity profiles are determined by the linear velocity of the compound (eq 4), for a retention factor equal to zero. The radial dispersion coefficient is given by eq 5.

eq 1, first along the column and second across it, following an approach previously described.4 However, this earlier work assumed that the band does not disperse in the radial direction, even though a radial concentration gradient is formed as a consequence of the radial temperature distribution. It used as the model of the column a series of coaxial cylinders between which the sample molecules are trapped, mass transfer not being allowed between two adjacent coaxial cylinders. The column behaves as an ideal column in the radial direction, the radial dispersion coefficient being zero. Accordingly, the model overestimates band broadening. Radial mass transfer kinetics is finite in actual columns and it tends to relax the concentration gradients that build up across the column. So, the influence of the column radial heterogeneity is actually smaller than predicted by the model. In this work, the contribution of the radial mass transfer kinetics is taken into account, based on the method suggested by Aris.11 Aris Theory and Its Application to the Open Tubular Columns Used in GC. Aris calculated the additional contribution to band spreading due to an annular distribution of the characteristics of the mobile and the stationary phases, as shown in Figure 1. This approach can be directly applied to the calculation of the HETP of open tubular columns, which are widely used in GC and are also useful in HPLC.13,14 In this case, the stationary phase is uniformly coated onto the inner surface of a capillary tube while the radial distribution of the mobile phase flow velocity is given by Hagen-Poiseuille law (in classical chromatography) or by a more complex cylindrical distribution (in turbulent flow chromatography).15 In capillary electrochromatography, another cylindrical distribution of the mobile phase velocity was used.16 In all these cases, the radial dispersion coefficient is given by a different cylindrical distribution. Under such conditions, a radial concentration gradient builds up. Radial solute dispersion takes place, tending to relax the radial concentration gradient. The combination of the radial distribution of the linear mobile phase velocity in the axial direction and of the radial dispersion increases the bandwidth compared to what it would be with piston flow displacement of the mobile phase but decreases it compared to what it would be in the absence of radial dispersion. The larger (13) Yue, G.; Luo, Q.; Zhang, J.; Wu, S.-L.; Karger, B. L. Anal. Chem. 2007, 79, 938. (14) Desmet, G. J. Chromatogr. A 2006, 1116, 89. (15) Martin, M.; Guiochon, G. Anal. Chem. 1982, 54, 1533. (16) Martin, M.; Guiochon, G. Anal. Chem. 1984, 56, 614.

Ri2 B + CuS + Cm uS uS Dm

(2)

where B is the longitudinal diffusion term (B ) 2Dm), Dm is the molecular diffusion coefficient, Ri is the inner radius of the open tubular column, uS is the cross-section averaged linear velocity, and C is the mass transfer resistance for the stationary phase. The term CmuSRi2/Dm is the additional term caused by the radial distribution of the linear velocities over the column crosssection. This contribution is often referred to as the mass transfer resistance in the mobile phase. The term Cm depends on the retention factor of the compound. In chromatography, assuming a Poiseuille flow profile leads to the well-known value of Cm in the Golay equation:17 Cm )

1 + 6k + 11k2 24(1 + k)2

(3)

where k′ is the retention factor of the solute. There is no coupling term in eq 2 (see coupling theory of Giddings)8 between band spreading due to a flow and to a diffusion mechanism. This is because (1) the Aris model assumes that the flow is strictly unidirectional and (2) there is no eddy diffusion term in a cylindrical open tube. As a result, a sample molecule can be transferred from one flow streamline to a neighborly one only by radial diffusion. It cannot do so due to a flow mechanism. In applications of the general theory of dispersion of Aris, the radial distribution of the axial velocities of the mobile phase arises from the formation of a Poiseuille flow pattern inside open tubes. However, the great attractiveness of the Aris theory lies in the fact that it is valid as long as there is a distribution of the mobile phase velocity (whatever its physical origin), that radial mass transfer between different mobile phase streamlines is made possible by a dispersion mechanism, and that the retention factor k′ of the solute is known. Then the Aris theory permits the calculation of the additional contribution, Cm. As we show in the next section, the Aris theory applies to HPLC packed columns that are not isothermal because they are operated under a high pressure gradient. Extension of the Aris Theory to Columns Used in VHPLC. The work of Aris11 provides an algebraic expression of the contribution to the plate height that arises from any radial distribution across the section of a column of either the amplitude of the axial velocity of the mobile phase and/or of that of the radial distribution of the coefficient of dispersion. The application of this work to the calculation of the plate height contribution arising from the viscous friction heating of columns raises two types of problems. It provides the long-time expression of the apparent (17) Golay, M. J. E. In Gas Chromatography; Desty, D. H., Ed.; Butterworths: London, 1958; p 36.


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axial dispersion coefficient due to the radial heterogeneity of the flow stream and of the associated plate height. First, the Aris model assumes that the flow is unidirectional, parallel to the column axis and that the cross-section average mobile phase velocity is constant along the column. These conditions are not strictly fulfilled, and we discuss in this section why these assumptions may be relaxed and the model applied in the case of VHPLC columns. Second, the approach assumes that, during their elution, the sample molecules have enough time to diffuse across the whole section of the column and sample sufficiently well the velocity profile. For short elution times, the variance of the band does not increase in proportion to the elapsed time, as it does in the long-time limit, but they increase as the square of the time. The degree to which this condition is fulfilled in VHPLC columns will be discussed later, in Appendix 1, once the data necessary to estimate the relevant factors will have been calculated. A priori, the Aris theory cannot be directly applied to columns used in VHPLC for the following four important reasons: 1. The cross-section average linear velocity along the column is not constant but it increases with increasing migration distance, z, along the column, due to the compressibility and to the thermal expansion of the mobile phase. 2. The geometrical distributions of the mobile and the stationary phases inside packed or monolithic chromatographic columns is clearly not homogeneous (as it is in open tubular columns). Columns are actually packed with very small porous silica particles or made of domains of porous material (silica or polymers) separated by networks of throughpores. The actual stream of the mobile phase percolates through this bed by flowing through the macropores that are formed between the packed particles or through the throughpores. These channels are randomly distributed. Their radial distribution is more or less heterogeneous. 3. The flow pattern is not strictly unidirectional. Locally, the mobile phase streamlines flow around the particles or along throughpores that are not parallel to the column axis and participate to the eddy dispersion of the band. The local mobile phase velocities have a radial component. Only the crosssection average of these velocities is zero. 4. The Aris model provides an expression of the dispersion coefficient valid in the long-time limit, for which analyte molecules spend enough time in the column for statistically sampling the radial velocity profile. In VHPLC conditions, this long-time limit may not be reached. This limitation is discussed separately in the section Validity of the Aris Contribution to the HETP of VHPLC Columns. Radial Distribution of the Analyte Velocity. Nevertheless, it is possible to cast the problem differently, under such conditions that the basic assumptions of the Aris treatment are satisfied. First, the column can be divided in a series of successive elementary columns of length dz, along which all cross-section averaged properties remain constant. dz should be much larger than dp. Second, in packed columns the mobile phase stream percolates under the stress caused by a high pressure gradient. At the scale considered (the elementary volume used in the integrations discussed later is very large compared to the average particle size), the radial distribution of linear velocities across 3368


the column is not due to the laws of hydrodynamics, as it is inside an open tubular column, but it results from the radial distribution of the local temperature, hence of the local density and viscosity of the mobile phase and of the retention factor. The following velocity profile of the band under linear chromatographic conditions will describe the flow distribution in Aris’ approach: u(z, x) )

u0(z, x)

1 + k (z, x)

)

eue(z, x) t(1 + k(z, x))

(4)

where x is the dimensionless radial coordinate (x ) r/R), z is the distance along the column, k′(z, x) is the local retention factor, u0(z, x) is the local chromatographic linear velocity, and ue(z, x) is the local interstitial linear velocity. t and e are the total and the interparticle porosities of the packed bed, respectively. Equation 4 incorporates the retention factor of the sample into the velocity profile distribution of its band. Finally, the formulation of the problem becomes fully equivalent to that of the flow along an open tubular column if we consider that the retention factor in the Aris treatment is zero because there is no adsorption of the compound at the column wall of packed columns. The four objections to the use of Aris theory described above are alleviated and this model can account for the description of band spreading in nonisothermal packed columns. The general dispersion theory of Aris simplifies considerably when the symmetry of the problem is cylindrical (see the section Parameter Cm). Radial Distribution of the Apparent Dispersion Coefficient. To use the Aris theory, we need besides the radial velocity distribution an estimate of the radial distribution of the molecular dispersion coefficient in the direction perpendicular to the mobile flow velocity. This dispersion is the driving force toward the relaxation of the radial concentration gradients that are caused by the radial gradient of the axial velocity given by eq 4. We know that bands migrating along a column spread in the transverse direction, due to the combination of radial and eddy diffusion.18-20 The contribution of radial diffusion through the packed bed impregnated of mobile phase is accounted for by an apparent diffusion coefficient, D0, related to the classical B term of the van Deemter equation. Accordingly, D0 ) [eγe + (1 - e)Ω]Dm

(5)

where γe (= 0.60) is the obstruction factor of the packed bed and Ω (= 1.0 at room temperature, for naphtho[2,3-a]pyrene on C18 phases)4 measures the relative diffusivity of the sample inside the particles with respect to the bulk diffusivity Dm. The general expression of Ω will be given later. Eddy dispersion takes place both in the radial and in the axial direction. When the Peclet number, Pe ) uedp/Dm, is less than 0.3, the contribution of this second mechanism may be neglected. However, in the experiments the results of which are reported in this work, the internal column diameter being Ri ) 0.105 cm, the average particle diameter dp ) 1.7 µm, the bulk molecular diffusion coefficient Dm are close to 1.0 × 10-5 cm2/s, and the inlet flow rate is varied between 0.60 and 1.65 (18) Knox, J. H.; Laird, G. R.; Raven, P. J. Chromatogr. 1976, 122, 129. (19) Tallarek, U.; Albert, K.; Bayer, E.; Guiochon, G. AIChE J. 1996, 42, 3041. (20) Tallarek, U.; Bayer, E.; Guiochon, G. J. Am. Chem. Soc. 1998, 120, 1494.

cm3/min; hence, the Peclet number ranges between 12 and 37. In this range of linear velocities, the two dispersion mechanisms coexist, eddy dispersion being the more important.21 The general expression of the radial dispersion coefficient Dr(z, x) is the sum of two contributions, a molecular diffusion one, D0, and an eddy dispersion one, Deddy,18 so Dr(z, x) ) D0 + Deddy ) [eγe + (1 - e)Ω]Dm(z, x) + 1 γr uS(z, x)dp (6) 2 where γr is an empirical constant depending essentially on the size of the packed particles. By definition, γr is smaller than 1. A value of 1 for γr would correspond to the general expression of the dispersion coefficient in the axial direction.22 Knox et al.18 and Eon23 found values of 0.06 and 0.075 for large particle sizes for =70 µm particles packed in a 1 cm i.d. column. For 30 and 5 µm particles, Tallarek et al.19,20 measured values of 0.14 and 0.30, respectively. A quasi-linear correlation exists between the decimal logarithm of γr and the particle size dp. Although no value of γr is available in the literature for columns packed with 1.7 µm particles, we can reasonably expect values of the order of 0.32. Basically, these data suggest that the smaller the particle size, the more efficient the sample mixing in the radial direction. It is possible that this dependence of γr on the average particle size be related to the width of the relative distribution of the particle sizes. Accordingly, we will assume that γr ) 0.32 in further calculations. Equation 6 can be rewritten in reduced units by expressing the reduced plate height (hr ) 2Dr/uSdp) as a function of the reduced average interstitial linear velocity ν ) uedp/Dm over the column diameter.

[

2 γe + hr )

]

1 - e Ω e + γr ν

(7)

Equation 7 shows that γr can be experimentally measured as the limit of hr at high reduced linear velocity. Since γe + [(1 - e)/e]Ω is of the order of 2, the diffusion term varies between 0.12 and 0.33 when the reduced average linear velocity varies between 12 and 37. This term is comparable to the contribution of eddy dispersion to the transverse reduced plate height, γr. It cannot be neglected a priori. As shown in this section, the Aris model can be used to calculate the general plate height equation for columns heated by viscous friction, provided that the long-time limit of validity of the Aris model is reached. It allows to calculate the coefficient Cm in eq 3, which gives the additional HETP contribution due to the radial profile of linear velocities caused by the frictional heating of the bed. Plate Height Equation for Friction Heated Columns. To write the plate height contribution of a column slice of length dz, we need to take into account the radial variations of the local values of the coefficients of eq 1. Coefficient A in eq 1 accounts for eddy dispersion. It results from phenomena that take place at (21) Hlushkou, D.; Tallarek, U. J. Chromatogr. A 2006, 1126, 70. (22) Guiochon, G.; Shirazi, D.; Felinger, A.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography, 2nd ed.; Academic Press: Boston, MA, 2006.

different scales in the column and are due to the anastomosis of the streamlets of the mobile phase percolating through the bed. These phenomena are not accounted for in the Aris treatment. Giddings8 showed that eddy diffusion is due to the combination of transchannel, short-range interchannel, long-range interchannel, and transcolumn exchanges that take place over distances of the order of magnitude of dp/6, dp, 5dp, and Ri (Ri, column radius), respectively. The Aris theory accounts only for the transcolumn effects, which are related to the change of linear velocity between the center and the wall region of the column. Accordingly, the transcolumn effect was not included in the eddy dispersion term, A. The other parameters in eq 1 depend on the position z along the column, due to the influences of the axial temperature and pressure gradients. For the slice of column between the abscissas z and z + dz (with dz . dp), the local plate height, H(z), can be written, based on the cross-section average of the parameters A, B/ue, Dr, and Cue: d2 ¯ ¯ (z) + B (z) + Cue(z) + Cm(z) c u(z) H(z) ) A ue ¯ r(z) D

(8)

The essential contribution of this work is the expression of the Aris term, the fourth term in eq 8. It is important to understand why this term depends on dc, the column diameter, and not on dp, the average particle diameter. Because this term characterizes the kinetics of relaxation of the concentration gradients across the column, the reduced linear velocity used in Aris’ formalism refers to the column diameter dc and to the overall ¯ r, across this diameter. Note also radial dispersion coefficient, D that the cross-section average of the local velocity u(z,x), u(z), in the Aris’ term differs from the local interstitial linear velocity, ue(z,x), which figures in the other terms because, in the Aris’ treatment, u(z) represents the cross-section averaged propagation velocity of the retained compound (or of the band, as defined in eq 4) which is quite different from the mobile phase linear velocity. ¯, Most of the parameters used to account for the coefficients A ¯ B/ue, Cm, Dr, and Cue depend on the local conditions, i.e., on the cross-section profiles of the temperature and the pressure. The calculated overall apparent column HETP, H, is finally given by

H)

1 L

∫

L

0

H(z) dz

(9)

The great advantage of Aris dispersion theory is that it allows a simplified, yet correct treatment of a complex two-dimensional ¯ r, and Cue(z) ¯ (z), [B/ue](z), D problem. The classical parameters A in eq 8 were derived by Aris as functions of the abscissa z, as the cross-section averages of the local parameters A, B/ue, Dr, and Cue, respectively.16 They are (23) Eon, C. H. J. Chromatogr. 1978, 149, 29.


3369

∫ A(z, x)x dx

(10)

¯ B (z) ) 2 ue

∫

B(z, x) x dx ue

(11)

¯ r(z) ) 2 D

∫ D (z, x)x dx

(12)

∫ C(z, x)u (z, x)x dx

(13)

¯ (z) ) 2 A

Cue(z) ) 2

1

0

1

0 1

r

0

1

e

0

In these equations, x ) r/R is the dimensionless radial coordinate. The following four sections detail the calculations of the four parameters A(z,x), B(z,x), C(z,x), and Dr(z,x) for the packed columns used in our work. The local linear velocity ue(z,x) will be derived in the section Mobile Phase Linear Velocity. Parameter A. This parameter accounts for eddy dispersion. It is derived from the coupling theory of Giddings by considering the three main sources of velocity heterogeneity in the packed bed.8 This term was recently revisited.12 It was shown that it should be written: A(ue) )

a3ue3 + a2ue2 + a1ue b3ue3 + b2ue2 + b1ue + 1

(14)

The relationships between the coefficients {a1,a2,a3,b1,b2,b3}, the experimental conditions, and the parameters characterizing the column bed are given elsewhere.12 They depend on the local temperature and pressure, through the molecular diffusion coefficient Dm. Their fundamental expressions were suggested by Giddings. They are related to the diffusion length, the flow persistence length, and the velocity biases for each source of flow heterogeneity. Table 1 summarizes these three parameters that are called ωR, ωλ, and ωβ, respectively. The molecular diffusion coefficient Dm is derived from the classical Wilke and Chang correlation:24 Dm(T, P) ) 7.4 × 10-8

√φSMST ηS(T, P)VA0.6

cm2 /s

(15)

where φS is the association factor of the solvent S, MS is its molecular weight (in grams per mole), VA is the molar volume of the liquid solute at its normal boiling point (in cubic centimeters per mole), and ηS(T,P) is the viscosity of the solvent at pressure P and temperature T (in centipoise). An excellent and simple expression of the simultaneous effects of temperature and pressure on the viscosity of the eluent, ηS(T,P), is given by the following correlation:25 ηS(P, T) ) 10

(A1+B1/T)

(1 + ζ[P - 1])

(16)

where A1, B1, and ζ are the empirical parameters that best fit the experimental data. Examples are given in ref 25 regarding pure acetonitrile as the mobile phase, for pressures and temperatures between 1 and 1500 bar and 293 and 333 K, respectively. (24) Wilke, C. R.; Chang, P. AIChE J. 1955, 1, 264. (25) Gritti, F.; Guiochon, G. J. Chromatogr. A 2008, 1187, 165. (26) Gritti, F.; Guiochon, G. Chem. Eng. Sci. 2006, 61, 7636.

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Table 1. Three Different Origins of Velocity Inequalities Listed by Giddings6 in a Packed Bed short-range long-range transchannel interchannel interchannel flow ωλ mechanism ∆u/u ) ωβ λi diffusion ωR mechanism ∆u/u ) ωβ ωi

1

1.5

5

1 0.5 1/6

0.8 0.48 5/4

0.2 0.1 10

1 0.014

0.8 0.5

0.2 2

Parameter B. This parameter accounts for the static axial diffusion of the solute inside the column, in the interparticle volume and in the pores of the particles:12

(

B ) 2 γe +

)

1 - e Ω Dm e

(17)

In eq 17, Ω is essentially a function of the temperature T for low-molecular-weight compounds and is written as12

[

( )]

βQst DS,0 δ exp Ω(T) ) pγp F(λm) + 2 K(T ) rp Dm,0 RT

(18)

where p = 0.4 is the particle porosity, γp = 0.6 is the internal particle obstructive factor,8 F(λm) ) 0.63 is the pore steric hindrance parameter,12 δ is a structural parameter (dimension of a length) that 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,12 rp is the average pore radius (δ/rp ) 1.25 for regular C18 stationary phases with a surface coverage of the order of 2 µmol/m2),26 K(T) is the Henry’s equilibrium constant, DS,0/ Dm,0 is the ratio of the frequency factor of surface diffusion to the frequency factor of bulk diffusion (=15027 for a retained compound), β is an empirical parameter (=0.7),28 and Qst is the isosteric heat of adsorption. Overall, the factor Ω should be of the order of 1 and this will be confirmed by the experimental values measured for the reduced HETP of the compound on the Acquity BEH-C18 column, at low flow rates. Note that this expression of the B term is valid when the HETP is expressed as a function of the interstitial linear velocity ue. Parameter C. This parameter is the solid-phase mass transfer coefficient. If the film mass transfer resistance is neglected, the C term simply is written as12 C)

(

)

δ0(T ) 2 dp2 1 e 30 1 - e 1 + δ0(T ) Ω(T )Dm(T, P)

(19)

where e is the local external porosity and dp is the average particle size. These parameters can be considered as independent of the temperature and the pressure. In contrast, δ0, which is related to the adsorption strength of the analyte28 is a function of the temperature, with (27) Gritti, F.; Guiochon, G. J. Chromatogr. A 2006, 1128, 45. (28) Miyabe, K.; Guiochon, G. J. Phys. Chem. B 1999, 103, 11086.

δ0(T) )

( )

(20)

1 - t e t - e K ) δ0 t t t

(21)

1 - e [p + (1 - p)K(T )] e

with k′ )

Although p = 0.40 for many packing materials, it can vary rather broadly with the nature of the packing material. In this work, we neglect the effect of the pressure on the distribution constant K(T ) of naphtho[2,3-a]pyrene between the bulk and the C18-bonded stationary phases. This effect is usually limited for neutral polycyclic aromatic hydrocarbons but it can be significant for ionizable analytes29 or for analytes having a large molecular weight. Parameter Cm. This parameter is the coefficient introduced by Aris to account for the curvature of the radial distribution of axial linear velocities. The cylindrical symmetry of the radial distributions of both the propagation velocity of the analyte and its radial dispersion coefficient simplifies considerably the results of the general dispersion theory of Aris. Aris showed that, for an unretained compound (k′ ) 0), this coefficient is given by11 Cm(z)

D0 (z) ) A0(z) ¯r D

(22)

with A0(z) )

I1(z) - 2I2(z) + I3(z) 2

(23)

and Φ2(z, x) dx 2xΨ(z, x)

(24)

0

xΦ(z, x) dx 2Ψ(z, x)

(25)

∫

x3 dx 2Ψ(z, x)

(26)

I1(z) )

∫

I2(z) )

∫

1

I3(z) )

1

1

0

0

with

In eq 28, u(z) is the linear propagation velocity of the compound (and not the chromatographic linear velocity u0), averaged over the column cross-section at the abscissa z. This velocity is u(z) ) u0/(1 + k′), where k′ is the retention factor of the compound studied. Note that, by construction, the function Φ(z,x) is equal to 1 for x ) 1 and all z. The functions Ψ(z,x) is strictly larger than 1 (because of the additional eddy dispersion term in Dr) and the local function φ(z,x) is expected to oscillate around 1 when x varies from 0 to 1. It is important to check that these physical conditions are always verified before beginning the full numerical calculations. The physical origin of the radial heterogeneity of the linear velocity distribution is the heat generated by the viscous friction of the mobile phase percolating through the column bed during its decompression. The center of the column becomes warmer than the wall region. For two reasons (see eq 4), the solute propagates faster in the center region of the column than near its wall: 1. Because the solvent viscosity and density decrease with increasing temperature, the interstitial linear velocity ue is larger in the center of the column. 2. Because the retention factor k′ decreases with increasing temperature, the solute is less retained and, hence, elutes faster in the region close to the center of the column than near its wall. Axial and Radial Temperature Profiles. The heat released is transported by axial convection to the column exit and by radial conduction to the column wall. Due to the finite thermal conductivity, λp, of the column bed made of the eluent and the C18-bonded silica particles, there is a resistance to this heat transfer, which generates a radial temperature gradient. Under steady-state conditions, the radial temperature profile should become parabolic beyond a certain entrance length.1,30 This is true for long columns and small flow rates, in which case the heat produced in the packed column is entirely evacuated through the column wall by conduction. In this case, the radial temperature gradient is maximum:30 ∆RT ) (1 + RT )

Φ(z, x) )

∫ 2x φ(z, x ) dx x

0

u(z, x) ) u(z)

u(z, x) 2

∫ u(z, x)x dx 1

(28)

0

and Ψ(z, x) )

Dr(z, x) D0

(30)

(27)

In these equations, x′ is a dummy variable. The functions φ(x) and Ψ(x) are the dimensionless radial profiles of the mobile phase velocity u and the radial diffusion coefficient Dr, respectively.11 In the particular case discussed here φ(z, x) )

Fv∆P 4πλpL

(29)

(29) Fallas, M. M.; Hadley, M. R.; McCalley, D. V. Influence of Pressure on Retention in Ultrahigh Pressure Liquid Chromatography. 32nd International Symposium on High Performance Liquid Phase Separations and Related Techniques (HPLC-2008), Baltimore, MD, May 10-16, 2008.

where ∆RT is the radial gradient amplitude, Fv the mobile phase flow rate, ∆P the column pressure drop, R the expansion coefficient of the liquid, and T the temperature. The coefficient R is negative because the eluent density decreases with increasing temperature. Conversely, when the column length is much smaller than the thermal entrance length, the radial heat conducted through the packed bed is too small to evacuate most of the heat produced. As a result, the eluent temperature at the column outlet is larger than at the column inlet. The axial temperature gradient is maximum. This takes place with short columns run at high flow rates. This takes place also when the column is strictly adiabatic, independently of its length, in which case no heat is lost through the column wall. The longitudinal temperature gradient is4 (30) Poppe, H.; Kraak, J. C.; Huber, J. F. K.; Van Den Berg, J. H. M. Chromatographia 1981, 14, 515.


3371

∆LT ) (1 + RT )

∆P cpm

1+

1 heReηSL2Kc(1 - e)2

(31)

∆Pcpme3dp2Ri2

In eq 31, cpm is the specific heat of the mobile phase, he is the heat transfer coefficient between the external surface of the column tube and the surrounding medium (e.g., the static atmosphere in the column oven, the insulating material around the column, or the temperature of the water in the water bath), Re is the external column radius, L is the column length, and Kc is the Kozeny-Carman constant that characterizes the permeability of the packed bed. The other parameters were previously defined. The profile of temperature given by eqs 30 and 31 are only extreme solutions of the thermal problem. In most actual systems, both radial and axial temperature gradients coexist. The temperature profiles are far more complex that those predicted by eqs 30 and 31 because the entrance length is usually of the same order of magnitude as the column length, the column is not adiabatic, and the heat losses through the column endfittings are usually much larger than those through the column wall in its middle. The entrance length le is given by1 le )

ueRi2 cpm (1 + k1) λ

with k1 )

p(1 - e) e

(33)

3372


(∫

P(z)

P0

(

)FS(298.25, P0 )

)

χ(P, T) dP ×

exp

(∫

T(z,x)

298.25

)

-R(P0, T) dT (34)

P(z) + b + b0T(z, x) P0 + b + b0T(z, x)

)

c

(35)

× exp(R1[T(z, x) - 298.25] + R2[T(z, x)2 - 298.252]) (36) In this equation, the parameters b, b0, c, R1, and R2 depend on the nature of the eluent. For acetonitrile, b ) 3403 bar, b0 ) -7.53 bar/min, c ) 0.125, R1 ) -3.304 × 10-4 1/K, and R2 ) -1.756 × 10-6 1/K2.31 The first term in the right-hand-side of eq 34 is the reference mobile phase density at 298.25 K and atmospheric pressure P0, the second term accounts for the solvent compression (P(z)), and the third term for the temperature change (T(z,x)). Finally, the local interstitial linear velocity ue(z,x) is written as ue(z, x) )

where FS(z,x) is the local density of the mobile phase and FS(Text,Pinlet) is its density at the column inlet pressure Pinlet and at ambient temperature Text, the temperature at which the eluent enters the column. Equation 33 holds true provided the radial pressure gradients inside the column are negligible. (31) Gritti, F.; Guiochon, G. Anal. Chem. 2008, 80, 5009.

FS(z, x) ) FS(298.25, P0) exp

(32)

where Ri is the column internal radius. If we need to pump a stream of pure acetonitrile through a 1.05 × 50 mm column at a maximum flow rate of 1.65 mL/min, an inlet column pressure of 775 bar is required. Given the values of the external and internal porosities, e ) 0.40 and p ) 0.40, the thermal conductivity of pure acetonitrile λ ) 0.19 W/m K, and its specific heat cpm ) 1.78 106 J/m3 K, the entrance length (eq 32) is about 13 cm which is nearly three times larger than the length of the column. Accordingly, in this case,3 the column is too short for the column temperature to be independent of z. The axial temperature gradient is not flat toward the end of the column, the radial temperature gradients are not completely developed into a parabolic profile, and eqs 30 and 31 give only an approximation of the actual temperature profiles in the column. The temperature profiles, T(r,z), were calculated from the mathematical model described earlier31 and from the experimental profile of the axial temperature along the external surface area of the column. The column was let free in still-air conditions. Mobile Phase Linear Velocity. The local mobile phase linear velocity can be calculated everywhere in the column if we know the local value of the mobile phase density. At any position (z,x) inside the column, the mass conservation through the crosssection area of an annular coaxial tube is written as ue(z, x)FS(z, x) ) ue(Text, Pinlet)FS(Text, Pinlet)

Knowing the compressibility factor χ(P,T) at any temperature and any pressure and the isobaric thermal expansion coefficient R(P0,T) of pure acetonitrile, we obtain the following:25,31

Fv eπRi2

(

Pinlet + b + b0T(z, x) P(z) + b + b0T(z, x)

)

c

×

exp(-R1[T(z, x) - Text] - R2[T(z, x)2 - Text2]) (37) In eq 37, the flow rate Fv is the flow rate set at the column inlet, where the temperature and the pressure of the eluent are Text and Pinlet, respectively. The retention factor k′(z, x) can be written as k′(z, x) )

[

1 - t -Qst K0 exp t RT(z, x)

]

(38)

where K0 is the Henry constant at infinite temperature and Qst is the isosteric heat of adsorption. These parameters will be determined experimentally for the compound studied (here naphtho[2,3-a]pyrene). The function φ(z,x) can now be calculated according to eqs 4, 37, 38, and 28. The function Φ(x) is then calculated with eq 27. Similarly, the function Ψ(z,x) is derived from eqs 5, 37, and 29. The integrals I1, I2, and I3 (eqs 24, 25, and 26) can be calculated, and the term Cm(z) is given by eqs 23 and 22. Pressure Profile along the Column. Many parameters are affected by the local pressure inside the column, e.g., the mobile phase viscosity (ηS), the mobile phase density (FS), and the molecular diffusion coefficient (Dm). In this part, we assume that the pressure is constant across the column at any abscissa, although the temperature is not. Accordingly, we neglect the radial linear velocity component ur compared with the axial linear velocity uz.31,32 (32) Kaczmarski, K.; Gritti, F.; Kostka, J.; Guiochon, G. Influence of the Heat Generated by Viscous Friction on the Efficiency of HPLC and UPLC Separation. 32nd International Symposium on High Performance Liquid Phase Separations and Related Techniques (HPLC-2008), Baltimore, MD, May 1016, 2008.

Table 2. Column Characteristics neat silica

bridged ethylsiloxane/silica hybrid (BEH)

particle size [µm] pore diameter [Å] surface area [m2/g]

1.7 130 185

bonded phase analysis

BEH-C18

total carbon [%] surface coverage [µmol/m2] endcapping

18 3.10 proprietary

Packed Column Analysis serial number dimension i.d. (mm) × L (mm) total porositya external porosityb particle porosity Kozeny-Carman constantc constantc k0 [m-2] × 10-14

014737108255 48 2.1 × 50 0.642 0.373 0.429 146.5 3.841

015137227155 72 2.1 × 100 0.641 0.377 0.424 138.5 3.493

014837220155 23 2.1 × 150 0.639 0.380 0.418 141.7 3.434

a Measured by pycnometry (THF-CH2Cl2). b Measured by inverse size exclusion chromatography (polystyrene standards). c Measured at a flow rate of 0.2 mL or less.29

The cross-section average pressure P(z) must then be calculated in order to determine A, B, Dr, and Cin eqs 10-13 at the axial coordinate z. These parameters depend on the local values of ue(z,x) (eq 37) or Dm(z,x) (eq 15), which are both functions of the local pressure P(z). P(z) will be determined by calculating the local pressure gradient, dP/dz, and by integrating the local permeability equation along the column length at any radial coordinate x. Accordingly, 1 dP (x, z) ) - ηS(x, z)uS(x, z) dz P0

(39)

where P0 is the bed permeability given by the Kozeny-Carman equation. P0 )

e3 Kc(1 - e)2

dp2

(40)

where Kc is the Kozeny-Carman constant that characterizes the column bed (=140).25 Knowing the inlet column pressure and temperature, previously measured, eq 39 is integrated step-by-step for each coaxial tube at the reduced radius coordinate x, in order to determine the local pressure P(x,z). The average pressure P(z) is then calculated according to P(z) ) 2

∫ P(x, z)x dx 1

0

(41)

It is important to recall here that the calculation of the column cross-section averaged pressure, P(z), is based on the physical constraint that there cannot be any radial flow, hence any radial pressure distribution, between adjacent annular elementary cylinders (The flow circulation in each elementary cylinder is independent of that in the other cylinders). In spite of this constraint, the calculated radial pressure gradient is not rigorously zero but it can be considered as negligible, as shown by Kaczmarski et al.32

EXPERIMENTAL Chemicals. The mobile phase used in this work was pure acetonitrile, HPLC grade, purchased from Fisher Scientific (Fair Lawn, NJ). Dichloromethane and tetrahydrofuran, both HPLC grade, were used to estimate the hold-up volumes of the columns with the pycnometry method. The column was successively filled with these two solvents and weighed. The hold-up volume was derived from the difference between these masses and from the known densities of dichloromethane and tetrahydrofuran33 also purchased from Fisher Scientific. The solvents were filtered before use on an SFCA filter membrane, 0.2 µm pore size (Suwannee, GA). Eleven polystyrene standards were used to acquire the ISEC data needed to estimate the column porosities (MW ) 590, 1100, 3680, 6400, 13 200, 31 600, 90 000, 171 000, 560 900, 900 000, and 1 860 000). They were purchased from Phenomenex (Torrance, CA). Naphtho[2,3-a]pyrene was used as the solute and was purchased from Fisher Scientific (Fair Lawn, NJ). Columns. The columns used in this study were generously offered from Waters (Mildford, MA). They were packed with particles of the bridged ethylsiloxane/silica-C18 packing material (BEH). The main characteristics of the packing material used and of the bare porous silica matrix and the characteristics of the packed columns are summarized in Table 2. Apparatus. All the columns were operated with an Acquity UPLC liquid chromatograph (Waters, Milford, MA). This instrument includes a quaternary solvent delivery system, an autosampler with a 10 µL sample loop, a monochromatic UV detector, a column thermostat, and a data station running the Empower data software from Waters. From the exit of the Rheodyne injection valve to the column inlet and from the column outlet to the detector cell, the total extra-column volume of the instrument is 13.5 µL, as measured with a zero-volume union connector in place of the column. We measured a time offset of 0.59 s after the zero injection time was recorded. The flow-rate delivered by the high pressure pumps of (33) Gritti, F.; Kazakevich, Y.; Guiochon, G. J. Chromatogr. A 2007, 1161, 157.


3373

the instrument is true at the column inlet. The flow rate eventually measured at the column outlet depends on the inlet pressure (an effect due to the eluent compressibility). The maximum pressure that the pumps can deliver is 1034 bar. The maximum flow rate was set at 1.70 mL/min during the record of the chromatograms. All the measurements were carried out at a constant laboratory temperature of 22 °C, fixed by the laboratory temperature control system. The daily variation of the ambient temperature never exceeded ±1 °C. Temperature Measurement. The temperature of the external column wall was measured with surface thermocouples as described in a previous report.3 The precision of these measurements was ±0.1 °C. The column was kept horizontally in direct contact with the laboratory atmosphere, but it was protected from forced air convection. Then, the column tube dissipates heat by a natural convection mechanism and radiation. Figure 2A shows the positions of the thermocouples used to measure the temperature of the external surface area of the column tube. RESULTS AND DISCUSSION The main goal of this work was to assess the impact of the radial temperature heterogeneity of the column bed on the apparent column HETP. The previous section provides the methods needed to calculate the local coefficients A, B, C, Dr,

Figure 2. (A) Experimental setup used to measure the temperature profile along the external surface of the column tube. (B) Experimental wall temperature of the column let free under still air conditions as a function of the axial coordinate z. Note the increase of the temperature with increasing flow rate. 3374


and ue of the HETP equation (eqs 10-13 and 22) at any position (x,z) inside the column. The axial and radial temperature profiles in the 2.1 × 50 mm column used in this work were previously determined,31 providing temperature maps calculated based on the energy balance equation and on boundary conditions derived from the records of the bulk temperature of the eluent at the inlet and outlet of the column and from the axial temperature profile of the external stainless steel tube wall. In this work, we measured these same temperatures at four different flow rates (0.60, 0.95, 1.30, and 1.65 mL/min), at which a significant amount of friction energy is liberated inside the column and calculated the column HETP at these flow rates. We compared these results to the experimental HETP of naphtho[2,3-a]pyrene measured for flow rates increasing from 0.03 to 1.70 mL/min. In this section, we first explain briefly how the thermal data required are measured. Then, we discuss the variations of the ¯ r along the column as a ¯ , B/ue, Cue, and D HETP parameters A function of the axial coordinate (0 < z < L, L column length). This discussion provides clear evidence for why it is impossible to define a column HETP in VHPLC experiments. The axial temperature and pressure gradients are too strong, as previously reported in ref 34. Finally, the apparent column HETP is calculated for any axial position and integrated for the whole column. The calculated HETP data are compared to the experimental curve and to the HETP that would be observed if the column were radially homogeneous. This approach provides an accurate estimate of the heat effects on the column efficiency and will be helpful to predict the column efficiency losses observed in VHPLC experiments. Experimental Data Used in the Calculations. The experimental data required for the calculation of the HETP in high pressure gradient, high flow rate liquid chromatography are the temperature and the pressure profiles inside the column, at the flow rate considered. These experimental data are critical for the accurate calculation of the kinetic coefficients intervening in the HETP equation. We also need the isosteric heat of adsorption, Qst of naphtho[2,3-a]pyrene on the BEH-C18 column. These data will then be used to calculate the local values of the kinetic parameters, B, A, Cm, C, Dr (eqs 10-13 and 22), and of u and ue (eqs 37, 38, and 4) that are necessary to determine the overall column HETP given by eqs 8 and 9. Temperature Profile along the Column. The temperature profile along the column wall was acquired by measuring for four different flow rates (0.60, 0.95, 1.30, and 1.65 mL/min) and the wall temperature at z ) 0, 1, 2, 5, 4, and 5 cm. These flow rates are the real flow rate imposed at the column inlet, no matter what the inlet pressure is. We also measured the inlet temperature of the eluent (Tinlet ) 295 K) and its outlet temperature, the pressures of the eluent at the column inlet and outlet (Pinlet ) 237, 370, 505, and 642 bar, respectively), and its outlet pressure (Poutlet ) 15, 16, 19, and 22 bar, respectively). The surface temperature of the external wall was measured using five thermocouples, as shown in Figure 2A. The results are plotted in Figure 2B. As expected, the temperature of the column wall strongly depends on the axial position. Close to the column inlet, the temperature increases steeply with increasing distance from the inlet and the slope of this variation increases rapidly with (34) Neue, U.; Kele, M. J. Chromatogr. A 2007, 1149, 236.

Figure 3. Calculated temperature profile along and across the column for four different inlet flow rates. (top left) 0.60 mL/min. (top right) 0.95 mL/min. (bottom left) 1.30 mL/min. (bottom right) 1.65 mL/min. The detail of the numerical calculations are given in ref 31.

increasing flow rate. Figure 3A-D shows the temperature profiles, T(r,z) inside the bed, across, and along it, that were calculated by setting the experimental wall temperatures in Figure 2B as boundary condition of the energy balance equation, as explained in an earlier publication31 that provides the details of the calculations. The coexistence of both an axial and a radial temperature gradients is noteworthy. Figure 3A-D also shows that the thermal entrance length is of the same order of magnitude as the length of the column, which explains why radial and longitudinal temperature gradients can be observed in the column studied. Pressure Profile along the Column. The pressure profiles P(z) along the column are shown in Figure 4A for the four applied flow rates on the 5 cm long column. These profiles are not linear at high flow rates, due to the large decrease of the eluent viscosity from the column inlet (low temperatures and high pressures) to the column outlet (high temperatures and low pressures). Figure 4B shows the variation of the pressure gradient along the column. It is steeper near the column inlet than close to the column outlet. Isosteric Heat of Adsorption. At high pressure gradients and high velocities, retention factors are no longer constant. They depend on the flow rate and vary along the column. To account for these variations, we need the isosteric heat of adsorption, Qst,

of the studied solute. For each column, the retention factor of naphtho[2,3-a]pyrene was measured at a very low flow rate (

Influence of Viscous Friction Heating on the Efficiency of Columns

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