Measurement of the axial and radial temperature profiles of a chromatographic column

Journal of Chromatography A, 1138 (2007) 141–157

Measurement of the axial and radial temperature profiles of a chromatographic column Influence of thermal insulation on column efficiency Fabrice Gritti a,b , Georges Guiochon a,b,∗ b

a 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 28 August 2006; received in revised form 18 October 2006; accepted 20 October 2006 Available online 4 December 2006

Abstract The temperatures of the metal wall along a chromatographic column (longitudinal temperature gradients) and of the liquid phase across the outlet section of the column (radial temperature gradients) were measured at different flow rates with the same chromatographic column (250 mm ×4.6 mm). The column was packed with 5 ␮m C18 -bonded silica particles. The measurements were carried out with surface and immersion thermocouples (all junction Type T, ±0.1 K) that measure the local temperature. The column was either left in a still-air bath (ambient temperature, Text = 295–296 K) or insulated in a packing foam to avoid air convection around its surface. The temperature profiles were measured at several values of the inlet pressure ( 100, 200, 300 and 350 bar) and with two mobile phases, pure methanol and a 2.5:97.5 (v/v, %) methanol:water solution. The experimental results show that the longitudinal temperature gradients never exceeded 8 K for a pressure drop of 350 bars. In the presence of the insulating foam, the longitudinal temperature gradients become quasi-linear and the column temperature increases by +1 and +3 K with a water-rich (heat conductivity  0.6 W/m/K) and pure methanol (heat conductivity  0.2 W/m/K), respectively. The radial temperature gradients are maximum with methanol (+1.5 K at 290 bar inlet pressure) and minimum with water (+0.8 K at 290 bar), as predicted by the solution of the heat transfer balance in a chromatographic column. The profile remains parabolic all along the column. Combining the results of these measurements (determination of the boundary conditions on the wall, at column inlet and at column outlet) with calculations using a realistic model of heat dispersion in a porous medium, the temperature inside the column could be assessed for any radial and axial position. © 2006 Published by Elsevier B.V. Keywords: Chromatographic column; Temperature profiles; Longitudinal and radial temperature gradients; Mobile phase heat conductivity; C18 -bonded silica; Mobile phase; Methanol; Water

1. Introduction Since the inception of HPLC, chromatographic columns have been packed with smaller and smaller particles, leading to dramatic reductions in analysis times. After a period in the 1980s and 1990s, when most analytical columns were packed with 4 or 5 ␮m silica particles, the race toward the use of smaller particles has started again. Packing shorter columns with finer particles (now dp < 2 ␮m) allows a further important decrease of analysis times, the reason for these shorter analysis times being that the optimum mobile phase velocity increases with ∗ Corresponding author at: Department of Chemistry, University of Tennessee, Knoxville, TN 37996-1600, USA. Fax: +1 865 974 2667. E-mail address: [email protected] (G. Guiochon).

0021-9673/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.chroma.2006.10.095

decreasing particle diameter [1]. As a matter of fact, the optimum reduced velocity and the minimum column HETP are such that νopt ∝ dp−1 and Hmin ∝ dp . Unfortunately, high flow rates require the use of ultra-high column inlet pressure (> 1 kbar) [2,3] because the column permeability, k0 , decreases as the square of the particle diameter (k0 ∝ dp2 ). Thus, the race toward using smaller particles can be pursued only as long as: (1) the instrument contributions to band broadening are drastically controlled and (2) pumps delivering mobile phases at very high pressures are available. Conventional equipment cannot operate at inlet pressures exceeding 400 bar. To withstand higher pressures, specially designed and built pumps, valves and connecting tubes are needed. Packing shorter columns with smaller diameter particles alleviates the problem to some extent. Much faster analyses are possible but at the cost of a reduced efficiency.

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While classical columns are packed with 4–5 ␮m particles and have lengths ranging between 15 and 30 cm, modern columns are packed with 1.5 ␮m particles and have lengths of 3 cm. The former provide 15–30,000 plates; the latter barely 10,000 but the analyses are about 30 times faster. To achieve both fast and highly efficient separations, it becomes necessary to operate columns at pressures in the range of 1–2 kbar. Then, however, the compression of the mobile phase stores in the liquid an important amount of energy. This energy is released in the column during the decompression of the mobile phase. It is degraded into heat by the viscous friction. The amount of energy stored increases with increasing pressure. Its release generates a power that increases with decreasing column length and particle size, and with increasing inlet pressure and flow rates (all parameters that are closely related). The release of this energy increases the local temperature and causes a temperature gradient along the chromatographic column and, unless the column is well insulated thermally, a temperature gradient across the column, due to heat exchanges with the outside atmosphere. Both gradients may affect drastically the column HETP and limit the efficiency of chromatographic columns [4,5]. Theoretical calculations of temperature profiles were made in the 1980s [6,7]. They predict the intensities of both longitudinal and radial temperature gradients. The consequences of these gradients are that the adsorption properties of the packing material are uniform neither along the column nor over its cross-section. So, the values measured for the retention factors and for all the thermodynamic functions related to them are average and may be most difficult to account for since their temperature dependence is exponential, not linear. More specifically, the viscosity of the mobile phase changes across the column section, solvents being less viscuous in the central zone, where they are warmer, than in the region near the wall, where they are colder. The mobile phase being isobaric over the column cross-section under steady state, the local mobile phase velocity will be higher in the column center than close to its wall. So, sample bands propagate faster in the central region of the column than in the region close to its wall. This causes a severe loss of efficiency as any radial heterogeneity [4–10]. The radial temperature heterogeneity causes also the components that are in the center of the column to be less strongly adsorbed, hence less retained, and to move faster not only because the mobile phase moves faster but mostly for this reason [5]. Finally, when applying a concentration gradient (such as a solvent gradient), one might expect to observe also a gradient of concentration along the column radius, not simply along the column length since the solvent moves more slowly along the column wall. This effect adds up to the different consequences of the friction heat generation. In spite of considerable theoretical efforts, the practical consequences of the temperature heterogeneity of chromatographic columns on their HETPs and their velocity dependencies have not yet been clarified. In the 1970–1980s, Lin and Horv´ath [6], Poppe and Kraak [7] and Halasz et al. [9] proposed models for the viscous dissipation of energy in packed beds, studied the influence of the thermal conditions on the efficiency of HPLC columns, and demonstrated the effect. Recent experimen-

tal results have confirmed it [8]. However, further theoretical and experimental studies need to be done because the available models do not take properly into account the simultaneous effects of the longitudinal and the radial temperature gradients. No general HETP equation can predict accurately HETP data curves under non-isothermal steady-state conditions. Friction effects occuring in the chromatographic column ends up to be a serious limitation to the increase of the speed and efficiency of HPLC separations. In this work, we aimed at measuring the temperature profiles of fluids percolating through a given chromatographic column under various sets of steady-state conditions (constant pressure drop or constant temperature gradients). The temperatures were measured along the external surface of the column wall. The radial temperatures were measured immediately downstream the frit placed at the column outlet. A complete temperature map inside the chromatographic column will be derived from a dynamic model of heat dispersion in a chromatographic column [6], when the temperature of the column wall is not uniform. 2. Theory 2.1. Propagation of heat in a chromatographic bed without temperature wall control Without loss of generality, the heat balance equation is written in the assumption of cylindrical symmetry for heat dispersion through the chromatographic column and for heat exchanges across the wall. Accordingly, the temperature profile T = T (r, z) of the column is a function of the radial coordinate r and the column length z. Let consider a volume element of column between the longitudinal coordinates z and z + dz and between the radial coordinates r and r + dr (Fig. 1). In this treatment, we assume, as a first approximation, that the mobile phase is not compressible, which is a reasonable first approximation for

Fig. 1. Longitudinal section of the column between the axial positions z and z + dz. The symmetry of the heat dispersion problem imposes the elementary volume of the column included between the radial coordinates r and r + dr. Three distinct heat fluxes must be considered into the heat balance equation: one radial (diffusion Jrad ), one longitudinal (diffusion Jlong ) and one by convection in the direction of the flow.

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liquids. So, the heterogeneous column bed (made of the solvent, the silica support and the C18 -bonded organic layer) is an apparent homogeneous and continuous medium that has effective or apparent heat conductivities equal to λz and λr for its longitudinal and radial heat diffusivities, respectively. The longitudinal and radial heat densities that enter and leave the elementary volume dV = dS × dz = 2πr × dr × dz are: For the longitudinal gradients jlong (z) = −λz

∂T |z ∂z

and jlong (z + dz) = −λz

∂T |z+dz ∂z (1)

and for the radial gradients, jrad (r) = −λr

∂T |r ∂r

and jrad (r + dr) = −λr

∂T |r+dr ∂r

(2)

During a short period of time, dt, certain amounts of heat enter and leave the elementary volume, due to the convection of the mobile phase through the packed bed. The convective heat densities entering and leaving the elementary volume are: jconv (z) = e uS × cpm × T (z) jconv (z + dz) =

e uS × cpm

and

× T (z + dz)

(3)

where e is the external porosity of the column, uS the superficial velocity and cpm is the heat capacity of the mobile phase per unit of volume of mobile phase. During the same time, dt, heat is generated in the elementary volume due to the dissipation of viscous forces. The corresponding heat can be calculated from the pumping power. It is equal to the product of the superficial velocity and the pressure drop dP < 0, according to the flow convention given in Fig. 1. Hence, the dissipation power released per unit volume is directly proportional to the pressure gradient steepness: qviscous = −uS ×

dP dz

(4)

Finally, the accumulation of heat in the elementary volume dV that contributes to elevate its temperature by dT is equal to: cpp × dS × dz × dT = [jlong (z) − jlong (z + dz)] × dS × dt + [2πrjrad (r) − 2π(r + dr)jrad (r + dr)] × dz × dt + [jconv (z) − jconv (z + dz)] × dS × dt + qviscous × dS × dz × dt

(5)

p cp

where is the heat capacity of the packing material (solvent, silica support and layer of C18 chains bonded to silica) per unit of volume of packing material. The previous equation writes:   ∂(∂T /∂z) p × dz × dS × dt cp × dS × dz × dT = λz ∂z   1 ∂(r(∂T/∂r)) + λr × dz × dS × dt − e uS × cpm r ∂r ×

∂T dP × dz × dS × dt − uS × dz × dS × dt ∂z dz

(6)

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Finally, under steady-state conditions, the temperature profile, T (r, z), becomes independent of the time and the general heat balance equation is written:  2  ∂ T ∂2 T ∂T dP 1 ∂T + λ − e uS cpm + − uS (7) 0 = λr z 2 2 ∂r r ∂r ∂z ∂z dz In the problem studied, the boundary conditions are fixed by the temperature of the mobile phase entering the column and by the external temperature in the air surrounding the column, T = Text . The driving force are the viscuous forces, only, and we have to face a problem of forced convection similar to this describe by Bird et al. [11] through a circular tube. The general boundary conditions of the elliptic partial differential equation are known for T (z = 0) = Text , ∂T /∂r(r = 0) = 0. Two are missing. Indeed, ∂T /∂r(r = Ri ) and ∂T /∂z(z = L) are a priori unknown. They depend on the heat exchange between the external surface of the column and the atmosphere surrounding it (hc , r = Ri ) and on the heat exchange between the liquid inside and outside the column tube (he , z = L), respectively. They are not trivial and depends on the experimental conditions. The four boundary conditions for the resolution of the elliptic partial differential equation can be assumed as: BC1 : at z = 0, T = Text ∂T BC2 : at r = 0, =0 ∂r ∂T BC3 : at r = Ri , −λr = hc [T − Tc ] ∂r ∂T ∂Te BC4 : at z = L, −λz + e uS cpm T = −λm + uS cpm Te ∂z ∂z where Tc is the temperature of the column stainless steel tube and Te is the temperature of the liquid just after the position z = L. The value of ∂Te /∂z is not known a priori. In the first series of experiments, the column is let free in a still-air bath. This means that the temperature of the column wall is not fixed. The temperature profile along the column wall depends on the kinetics of heat transfer between the packing material and the column wall (at r = Ri , with Ri the internal radius of the column tube), of the heat conduction through the stainless steel tube of thickness e = Re − Ri , with Re the external radius of the column tube, and of the heat transfer between the external surface of the tube and the surrounding air at temperature Text at r = Re . Consider a thin slice of column tube of width dz. The heat dQc transferred between the whole packing material and the inner surface of the column tube during dt is written (Fig. 2): dQc = hc × 2πRi × [T (Ri , z) − Tc (Ri , z)] × dz × dt

(8)

where Tc (Ri , z) is the temperature of the inner surface of the column tube at abscissa z and hc is the heat transfer coefficient between the packing material and stainless steel. An equation similar to Eq. (7) applies to account for heat conduction through the stainless steel tube, except that there is no heat convection and no heat generated in the column tube. Under steady-state conditions, a stationary regime is reached

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The eight boundary conditions written above are represented schematically in Fig. 3. Similar to Eq. (8), the elementary heat dQe transferred from the surface of the stainless steel tube to the surrounding air is written: dQe = he × 2πRe × [Tc (Re , z) − Text ] × dz × dt

Fig. 2. Description of the different radial interfaces that the radial heat fluxes crosses. Packing material/stainless steel, stainless steel/air or Stainless steel/foam jacket.

and: ∂ 2 Tc 1 ∂Tc ∂ 2 Tc + (9) + ∂r 2 r ∂r ∂z2 The four boundary conditions necessary to solve the partial differential equation Eq. (9) are:

0=

BC5 : at z = 0, Tc = Text BC6 : at r = Ri , −λc

∂Tc = hc [T − Tc ] ∂r

BC7 : at r = Re , −λc

∂Tc = he [Tc − Text ] ∂r

BC8 : at z = L, −λc

∂Tc = he [Tc − Text ] ∂z

(10)

where he is the heat transfer coefficient between the stainless steel surface and the surrounding air. In the case of the second series of experiments in which the column is strapped inside a foam jacket, dQe can be considered as equal to 0, as a first approximation, because foam is a very good insulating material. The new boundary conditions are easily derived assuming hc = 0. Then, the only heat exchanges that take place with the outside are heat losses at both column ends, due to longitudinal heat convection in the mobile phase and longitudinal heat conduction along the tube wall. p ¯ Note that the Peclet number for heat transfer (Pe = dcp uS /λ) is well smaller than 1 ( 0.005) with respect to the length p of the column (d = 25 cm, cp  3 Jm−3 K−1 , uS  0.003 ms−1 −1 −1 and λ¯  0.5 W m s for water rich eluents). Since the flow rates applied in chromatography are associated with very low Reynolds number and laminar flow, the heat exchanged through diffusion is the governing heat transfer force in a chromatographic column. 2.2. Longitudinal temperature gradient in packed beds Hal´asz et al. [9] demonstrated that viscous heat dissipation leads to the formation of a longitudinal temperature gradient under adiabatic conditions. Experimental results discussed later show that a longitudinal temperature gradient takes place also under non-adiabatic conditions. A simple estimate of the longitudinal temperature gradient under adiabatic conditions, T = TL − T0 , or differ-

Fig. 3. The eight boundary conditions (noted BCi , i = 1 . . . 8) required to solve the elliptic partial differential Eqs. (7) and (9).

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ence between the inlet and outlet temperatures, is [9]:

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where P is the pressure drop along the column (Pa) and cpm is the specific heat capacity of the solvent (J m−3 K−1 )

This model assumes that the effective axial heat conduction is negligible with effective radial heat conduction. The calculation of the radial temperature profile θ(ρ, l) can be computed with six significant digits by considering the first 1500 Bessel functions of the first kind in the sum. The calculations were performed using the Mapple 7.0 software for Windows.

2.3. Radial temperature profile in packed beds with non-uniform wall temperature

2.4. Radial temperature profile in packed beds with uniform wall temperature

Consider a column, the wall of which is not kept at a constant temperature but depends on its radial heat loss. In this case, Lin and Horv´ath [6] demonstrated that, once the longitudinal temperature profile, θw = T (Ri , z)/Text , is known along the column wall (i.e., at ρ = r/Ri = 1), the radial temperature profile can be written as follows:   l xf lf θw (x) − 1 − θ(ρ, l) = 1 + + M M 0    ∞ 2  αn J0 (αn ρ) −(l − x)α2n × exp dx M J1 (αn ) M

If the temperature of the wall, T (Ri , z) = T (Ri ) = Text , is kept constant and when the radial temperature profile is fully developed (i.e., if the axial position considered is beyond the “entrance” length), Poppe et al. [7] have showed that the radial temperature profile is parabolic and given by:

T =

P cpm

(11)

n=1

T (ρ) = Text +

uS (dP/dz)R2i (1 − ρ2 ) 4λ

(16)

3. Experimental 3.1. Chemicals

(12) where x is the integration variable (0 < x < l). J0 and J1 are Bessel functions of the first kind and the αn ’s are the positive roots of J0 (αn ) = 0. The parameters l, f and M are defined as follows: - l is the dimensionless axial position normalized to the column tube length L. z l= (13) L - f is a dimensionless parameter given by: f =

3.2. Materials

uS PR2i λm Text e (1 + k0 )(1 + β)L

(14)

where λm is the heat conductivity of the mobile phase, k0 the volume ratio of the intraparticulate void space to the interstitial void space (i.e., k0 = (1 − e )p /e ) and the value of β can be assumed equal to zero when the solid packing has a much smaller heat conductivity than the liquid percolating through the bed. - M is given by: M=

Two mobile phases were used for the measurement of the temperature profile of the column. The first is pure methanol, the second a 97.5/2.5 (v/v) mixture of water and methanol (water rich eluent). Two other mixtures (15/85 and 80/20) were prepared to record the analytical chromatograms of two analytes within a reasonable time of elution (nor too short nor too large). Both water and methanol were HPLC grade, purchased from Fisher Scientific (Fair Lawn, NJ, USA). The analytes, caffeine and benzo[a]pyrene, were purchased from the same manufacturer.

uS ρm cpm R2i

(15)

λm (1 + k0 )(1 + β)L

where ρm is the mobile phase density.

The C18 -Vydac 218TP column used in this work was packed and given by Grace (Columbia, MD, USA). The main characteristics of the column tube, the bare porous silica and the bonding material used are summarized in Table 1. Six snap-on strain relief thermocouples of type T (copperConstantan junction, 1 mm junction size), one mini-hypodermic thermocouple model HYP0 of type T (needle diameter 0.2 mm), a handheld thermometer with two inputs and 100 adhesive pads were purchased from Omega Engineering, Inc. (Stamford, CT, USA). A foam jacket was used to isolate thermally the chromatographic column from the external air (the ambient air temperature of the laboratory was Text = 295 K).

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

Column dimension (mm × mm)

Particle size (␮m)

Mesopore ˚ size (A)

Specific surface 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

5.0

280

70

Polymeric

7.7

89.4

5.0

2.76

a

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

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sure gauge installed in the liquid chromatograph. The precision of the gauge is ±2 bar. Its accuracy was checked by inserting a manometer between the tube connecting the pump to the column and the column inlet. The error measured was +7%. In other words, the pressure given by the instrument was systematically 7% higher than the true column inlet pressure. 3.4. Measurement of the temperature profiles at different axial and radial positions

Fig. 4. Experimental set-up used for the acquisition of the temperatures along the column wall (thermocouples T1–T5) and in the liquid over the column crosssection just after the outlet frit (Hypo thermocouple T6). All temperatures were measured relatively to the external temperature Text (thermocouple T7).

3.3. Apparatus The two solvents were pumped into the chromatographic column using the pump of a HP 1090 liquid chromatograph (Agilent Technologies, Palo Alto, CA, USA). This instrument includes a multi-solvent delivery system (with three 1 L tanks), an auto-sampler with a 250 ␮L sample loop, a column thermostat (i.e., an insulated air bath), a diode-array UV-detector and a data station. Compressed nitrogen and helium bottles (National Welders, Charlotte, NC, USA) are connected to the instrument to allow continuous operations of the pump, the auto-sampler and solvent sparging. The flow-rate accuracy was controlled by pumping the pure mobile phase at 25 ◦ C and 1 mL/min for 50 min, directly into a volumetric glass of 50 mL. The relative error was less than 0.4%, so we can estimate the long-term accuracy of the flow-rate at 4 ␮L/min at flow rates around 1 mL/min. The daily variation of the ambient temperature never exceeded ±1 ◦ C. The inlet column pressures were measured by the pres-

Seven thermocouples were used to measure simultaneously the longitudinal and the radial temperature profiles. One thermocouple was dipped into a beaker filled with water to provide the external temperature, Text , at the moment when the longitudinal and radial temperature profiles were measured. The five snapon strain relief thermocouples are specially designed for surface temperature measurements. They were placed and taped with the adhesive pads on the column wall as shown schematically in Fig. 4. The same 5 cm interval separated two successive thermocouples in the raw. The HYPO needle thermocouple (1 in. long needle, 0.2 mm o.d.) was used to measured the radial temperature profile of the mobile phase exiting from the outlet frit of the column, immediately against the surface of this frit. To reach this surface, the outlet endfitting of the column was drilled to a diameter equal to the inner diameter of the column (see Fig. 4). As shown in Fig. 5, the temperature was measured successively at 11 equally spaced radial positions. Each position was set with the digital caliber supporting the HYPO thermocouple (one fourth of a full rotation corresponds to a vertical translation of 0.40 mm). The precision of these thermocouples is ±0.1 K. The longitudinal temperature gradient was measured under four different conditions: (1) the column placed in a still-air bath and fitted with the drilled outlet endfitting, (2) the column placed in the same still-air bath but fitted with a standard endfitting and a short outlet connecting tube (0.025 in. i.d.), (3)

Fig. 5. Zoom on the experimental set-up used to measure the radial temperature profile. The Hypo thermocouple is made of a terminal needle whose diameter is 0.2 mm. This needle is placed horizontally, parallel to the column axis and was displaced vertically by means of a digital caliper. The minimum distance between two consecutive radial measurements was 0.44 mm.

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Table 2 Physico-chemical properties of water and methanol according to reference [7] at ambient temperature (298 K) Solvent

Thermal conductivity (W/m K)

Heat capacity (×106 ) (J/m3 K)

Viscosity (×10−4 ) (Pa s)

Water Methanol

0.60 0.21

4.19 1.99

8.9 4.6

the column placed in a foam jacket and fitted with the drilled outlet endfitting and (4) the column placed in a foam jacket and fitted with the normal endfitting. In cases (1) and (2), heat transfer could take place between the surface of the column tube and the external air by convection. In cases (3) and (4), heat exchanges between the column wall and the atmosphere was considered to be negligible. In cases (1) and (3), the eluent evaporates at the outlet of the column, cooling it at the level of the meniscus (Fig. 4). This cannot be avoided. In cases (2) and (4), there is no such evaporation nor any cooling of the outlet endfitting. Obviously, the measurement of the radial temperature gradients could only be made with the drilled endfitting. It was either measured with or without the foam jacket around the column. A third series of measurements were made in which the endfitting

was insulated with a 2-mm thick layer of parafilm to reduce the amount of heat lost through eluent evaporation (especially at low flow rates). Four (0.9, 1.8, 2.8 and 3.6 mL/min) and three (1.5, 3.0 and 4.5 mL/min) different flow rates were used with the water-rich and the pure methanol, respectively. The inlet pressures required to achieve these flow rates were approximately 100, 185, 290 and 350 bar with the water-rich eluent and 100, 200 and 290 bar with pure methanol. Between two consecutive changes in the flow rate, the eluent was let to percolate for one full hour so that the column could reach a steady-state thermal equilibrium. The temperatures of the seven thermocouples were then recorded. Measurements were also carried out at extremely low flow rates of methanol to check on the measurement errors originating from methanol evaporation at the column outlet.

Fig. 6. Measurement of the longitudinal temperature profiles. Top: Still-air conditions. Bottom: Foam jacket around the column tube. Left: Special outlet endfitting to allow the measurement of the radial temperature profile (see Figs. 4 and 5). Right: Normal outlet endfitting. (A) Eluent: Water–methanol (97.5, 2.5, v/v). (B) Eluent: Methanol. Note the quasi-linear profile when the foam jacket and the normal outlet endfitting is used.

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Fig. 6. (Continued ).

4. Results and discussion The two mobile phases used were chosen for their very different thermal properties. Pure water might have been more different still but a mixture of methanol and water was chosen instead because pure water has a contact angle smaller than 90 ◦ C with C18 -bonded silica. It may not enter all the mesopores of the adsorbent. Adding a small amount of methanol (2.5%, v/v) to pure water allows the wetting of the surface and avoids that tiny air bubbles could remain in some small mesopores. This is important for our measurements because the heat conductivity of air is much less than that of all liquids and the presence of air in the column would considerably reduce the apparent thermal conductivity of the packed bed. The thermal conductivity, heat capacity and viscosity of water and methanol are summarized in Table 2. Note that water conducts heat three times as fast, stores twice as much thermal energy, and is almost twice as viscous as methanol.

temperature, Text , are plotted versus the column length in the four cases selected, for the water-rich eluent (Fig. 6A) and for methanol (Fig. 6B). The precision of the thermocouples being ±0.1 K, the size of the symbols is approximately equal to an error bar of ±0.2 K. Obviously, we cannot measure directly the actual eluent temperature inside a chromatographic column. This temperature is necessarily higher than the temperature of the external wall. Hence, all the data given in Fig. 6A and B are underestimates of the true temperature of the packing material. The thermal conductivities of stainless steel (λc ) and fused silica at ambient temperature are approximately 15 and 1.4 W/m K, respectively. Assume the longitudinal temperature gradients to be smoother than the radial temperature gradients and the heat transfer resistance between the liquid and the metal to be negligible, it is possible to derive an approximate estimate of the temperature difference between the internal and external surfaces of the stainless steel tube. The equality of the radial fluxes through the packing and through the tube writes:

4.1. Longitudinal temperature gradients The temperature differences between the column wall at five different positions (see Fig. 4) and the ambient laboratory

λRi

T (r = 0) − T (r = Ri ) T (r = Ri ) − T (r = Re ) = λc R e Ri Re − R i (17)

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Fig. 7. Plot of the amplitude of the longitudinal temperature profile (T5–T1) vs. the column drop pressure (P). Note the linearity of the plots but the difference between the slopes given by Eq. (12) and the measurement (see explanations in the text). (A) Eluent: Water–methanol (97.5, 2.5, v/v). (B) Eluent: Methanol.

As a first approximation, one can assume that the heat conductivity of the column bed (i.e., the packing material and the eluent, λ) is 10 times less than that of stainless steel. For the column used, Ri = 0.23 cm and Re = 0.30 cm. Hence, T (r = Ri ) − T (r = Re ) 1 Re − R i < 0.02  T (r = 0) − T (r = Ri ) 10 Re

(18)

In other words, the temperature of the column tube across its thickness is quasi uniform at any given axial position along the tube. The temperature measured on the external wall is very close to that of the liquid phase percolating through the bed against the wall. As the flow rate and the inlet column pressure increase, the wall temperature at any axial position and the longitudinal gradient increase, as expected since the heat dissipated in the column is proportional to the column pressure drop. None of the eight longitudinal temperature profiles appear to be linear, whether with the column in the still-air bath or in an insulating jacket, with or without the normal endfitting, or with methanol or the water rich mobile phase. As expected, the amplitudes of the axial gradients are larger with the column in an insulating jacket than in still-air (Fig. 7A and B). For the same pressure drop (ca. 300 bar),

149

the differences T = T5 − T1 measured for the column fitted with standard endfittings and placed in still-air condition (e.g., the actual temperature differences prevailing during a chromatographic analysis) are +2 and +3 K with water and methanol as eluents, respectively. These values are +3.3 and +4.7 K when the column is in the insulating jacket. This result is expected since much less heat is lost by the external surface of the column tube. Heat can be lost only at both ends of the column which explains the larger temperature gradients measured. It is observed, however, that Eq. (11) does no predict these results accurately. Although the temperature difference between the column outlet and inlet increases linearly with increasing inlet pressure. the proportionality constant measured differs from the reciprocal of the heat capacity of the eluent (cpm ). In deriving Eq. (11), besides assuming adiabatic conditions, which can never be satisfied experimentally, Halasz et al. [9] omitted that a fraction of the heat dissipated in the column serves also to heat the solid packing material, which is initially at room temperature. The slope of the plot P versus T gives apparent heat capacities of 7.4 × 106 and 5.7 × 106 J m−3 K−1 with water and methanol as eluents, respectively while the actual heat capacities of water and methanol (Table 2) are only 4.19 × 106 and 1.99 × 106 J m−3 K−1 , respectively. The heat absorbed by the packing material causes the axial temperature to be lower than the value of T predicted by Eq. (11) and the experimental result gives necessarily an overestimate of the heat capacity of the solvent, if not corrected. Nevertheless, the measurements give a reasonable order of magnitude and the correct ranking of the heat capacities (cp,H2 O > cp,CH3 OH ). Since water stores more energy than methanol when heated, the amplitude and the steepness of the longitudinal temperature gradients are necessary larger with methanol than with water. The heat capacity of the whole column bed can be derived from the data in Table 1 and the calculation of the volume fractions of the solvent, the solid silica and the octadecyl bonded chains. Assuming densities of 0.78 and 2.12 g cm−3 for the C18 bonded phase and silica, respectively, the volume fractions of eluent, bonded phase and silica in the chromatographic column are 0.66, 0.10 and 0.24, respectively. The specific heats of water, methanol, octadecane (i.e., solid C18 [12]) and silica are 4.19, 1.99, 1.67 and 1.49 J m−3 K−1 , respectively. Hence, cp ,H2 O = 0.66 × 4.19 + 0.10 × 1.67 + 0.24 × 1.49 = 3.29 Jm−3 K−1

(19)

cp ,CH3 OH = 0.66 × 1.99 + 0.10 × 1.67 + 0.24 × 1.49 = 1.84 Jm−3 K−1

(20)

Energy conservation (e.g., viscous heat produced by eluent friction equals heat used to increase the bed temperature) writes, using the volume fractions φi determined above [9]: P =

φMP cpm + φC18 cp,C18 + φSiO2 Cp,SiO2 φMP

T

(21)

This equation demonstrates that the proportionality constant between the column inlet pressure and the temperature increase,

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T , between the column inlet and outlet is not directly equal to the specific heat of the single mobile phase, cpm . The proportionality constants calculated for water and methanol as eluents are 4.98 × 106 and 2.78 × 106 J m−3 K−1 , respectively. These values are still inferior to the apparent values measured, 7.4 × 106 and 5.7 × 106 J m−3 K−1 , respectively, because the condition of adiabaticity cannot be reached during our experiments. As expected when there are heat losses, the theoretical temperature increase predicted for a given pressure drop is higher than the one measured experimentally, hence the proportionality constant measured is higher than the value calculated. The use of the insulating foam jacket increases the amplitude and the steepness of the longitudinal temperature gradient. It also affects the shape of the axial temperature profile (see Fig. 6A and B). The same behavior takes place with both eluents. Under still-air conditions, the temperature profile is convex downward between the axial positions 1 and 3 (e.g., over the first half of the column). A sharp temperature increase is observed between the axial positions 2 and 3. In contrast, the temperature profile is linear between the axial positions 1 and 3 when the jacket prevents radial heat losses from the stainless steel surface to ambient

air. Between positions 3 and 4, the column wall temperature tends toward a limit under still-air conditions while it increases more slowly than upstream this position with the jacket. Finally, between axial positions 4 and 5, the temperature increases significantly under still-air conditions. Using the foam jacket, this end part of the temperature profile depends much on whether the standard or the open endfitting, the use of which is necessary to perform the measurements of the radial temperature profile, is fitted to the column outlet. With methanol as eluent, the temperature increase along this last section is strongly affected by the use of the open endfitting. As discussed later, evaporation of methanol from the open endfitting cools the column end, an effect that increases with decreasing flow rate, as seen in Fig. 6A and B and disappears when the standard endfitting is used. 4.2. Radial temperature gradients A special stainless steel endfitting was fastened to the column to allow the measurement of the radial temperature profile of the eluent over the column cross-section (see Figs. 4 and 5). It lets the HYPO thermocouple to come into contact with the outlet frit,

Fig. 8. Plot of the radial temperature profiles measured at different flow rates (or column drop pressures). (A) Eluent: Water–methanol (97.5, 2.5, v/v). (B) Eluent: Methanol. (C) Eluent: Methanol with paraffin around the special outlet endfitting. Note that due to the rapid vaporization of methanol at the column outlet, a cold source is created around the special endfitting and temperature smaller than the external temperature Text are measured. The effect is considerably attenuated by using some paraffin (P = 104 bar).

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measuring the temperature of the eluent just downstream the frit, at any point of the cross-section. The position of the HYPO thermocouple was guided vertically with a digital caliper (Fig. 5). Eleven equally spaced measurements were made, from the bottom to the top of the column cross-section. The temperature measurement is direct and its precision is ±0.1 K. Fig. 8A–C show plots of the differences between the eluent temperature at the column outlet (z = L) and at radial coordinates r and the eluent temperature in the reservoir (Text ). First, we observe that the temperature are symmetric with the maximum temperature in center region of the column. The difference between the average temperature of the eluent at the column outlet and the exterior temperature Text (at which the eluent is pumped) versus the drop pressure is shown in Fig. 9A–C. Finally, Fig. 10A–C show the amplitude of the radial temperature gradient versus the column drop pressure. With water used as the eluent, two different experimental conditions were studied, the column being either left in stillair or wrapped in a foam jacket. Figs. 8A, 9A and 10A show that there is almost no “jacketing” effect on the temperature profile measured at the column outlet (z = L). The average eluent temperature, calculated as the average of the eleven local measurements, increases linearly with increasing column inlet pressure, up to +6 K at about 300 bar. The maximum amplitude

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of the radial gradient or difference between the temperatures at the center of the column and along its inner wall is 0.8 K, at the same drop pressure. In contrast, it was difficult to measure accurately the radial temperature profiles when methanol was used as the eluent. Figs. 8B, 9B and 10B show plots of the radial temperature profile and of the dependencies of the average cross-section temperature and of the radial temperature amplitude at the column outlet, respectively, versus the inlet pressure. Important differences are observed with methanol between the measurements made with the column in still-air or in the insulating jacket. The temperatures measured with the jacket are significantly higher than those in still-air, by +3 to +4 K. At very low flow rates, the temperature measured for the eluent is lower than the ambient temperature (Figs. 8B and 9B). The converse is observed for the amplitude of the radial temperature difference (Fig. 10B). This is due to the measurement process. Methanol has a high vapor pressure at room temperature, much higher than that of water and a somewhat lower latent heat of vaporization (35.3 kJ/mol versus 40.7 kJ/mol). Because the end of the column is not placed into an atmosphere saturated in methanol or water, vaporization takes place. Methanol vaporizes faster than water. Heat is directly transferred from the stainless steel endfitting to the liquid methanol in contact. At low flow rates, the

Fig. 9. Plot of the temperature difference between the average eluent temperature at the column outlet (T 6) and the external temperature (Text ) vs. the column drop pressure (P). (A) Eluent: Water–methanol (97.5, 2.5, v/v). (B) Eluent: Methanol. (C) Eluent: Methanol with paraffin around the special outlet endfitting. Note the linearity of the plots.

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Fig. 10. Same as in Fig. 9, except the radial temperature amplitude.

effects on both the radial and the axial temperature profiles are important. In order to alleviate this problem, the endfitting was wrapped in a thick paraffin layer, to hinder heat transfers from the stainless steel to methanol or air (see Fig. 4). The new results are shown in Figs. 8C, 9C and 10C. The effect is striking at high values of the inlet pressure. The average eluent temperature (Fig. 9B) and the amplitude of the radial temperature gradient (Fig. 10B) are now the same for the column in still-air or in the foam jacket. The thermal behaviors of the column in still-air and in the foam jacket are now almost equivalent. At low flow rates, however, the temperature measured for the eluent is still below the ambient temperature. Heat conduction still takes place through the eluent, from the frit to the free meniscus where evaporation takes place and it is faster than heat convection caused by the eluent flow. Improved experimental set-ups could consist in placing the column outlet in an atmosphere saturated with methanol vapor or in replacing methanol with another solvent having a low heat conductivity and a high boiling point, e.g., xylene, which has the same viscosity as methanol (η = 0.61 cP, λxylene = 0.13 W m−1 K−1 , boiling point = 139 ◦ C). The modified endfitting makes it possible to compare the thermal behaviors of the column in water and methanol at high flow rates, using the same inlet pressure. With a pressure of 300 bar, the flow rates are 2.8 and 4.5 mL/min with water and

methanol, respectively. The average outlet temperatures are ca. +6 and +8 K above ambient temperature and the amplitudes of the radial temperature gradient are 0.8 and 1.3 K with water and methanol, respectively. Under steady-state conditions with a constant wall temperature, the radial temperature profile is fully developed and the maximum amplitude is given from Eq. (16) as T (ρ = 0) − Text =

uS (dP/dz)R2i 4λ

(22)

To estimate λ, the volumetric average of the heat conductivities were calculated, based on the volume fractions of the neat eluent (0.66), the alkyl bonded chains (0.10) and silica (0.24) in the chromatographic column (see earlier). λ,H2 O = 0.66 × 0.60 + 0.10 × 0.35 + 0.24 × 1.40 = 0.77 W/m K

(23)

λ,CH3 OH = 0.66 × 0.21 + 0.10 × 0.35 + 0.24 × 1.40 = 0.51 W/m K

(24)

The superficial linear velocities of the eluent are uS = 0.28 and 0.45 cm/s for water and methanol, respectively; the pressure gradient along the column (L = 0.25 m) is 1.2 × 108 Pa/m and the inner radius of the column tube is Ri = 0.23 × 10−2 m.

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Table 3 Dimensionless parameters M (Eq. (15)) and f (Eq. (14)) [6] (298 K) obtained for a column drop pressure P = 300 bar and with the parameters β given in the text and k0 = 0.65 Solvent

λm (W/m K)

cp (J/kg K)

η (×10−4 ) (Pa s)

ρ (kg/m3 )

uS (×10−2 ) (m/s)

M

f (×10−3 )

Water Methanol

0.60 0.21

4190 2513

8.9 5.6

1000 792

0.28 0.45

0.1254 0.2735

7.61 34.93

The inner radius and the length of the column tube are Ri = 0.23 cm and L = 25 cm.

Accordingly, the theoretical model predicts amplitudes of 0.6 and 0.9 K, in satisfactory agreement with the experimental values. The 50% underestimation should not be surprising since the prediction is based on the assumptions made in Eqs. (23) and (24). Also, the column wall temperature may not be exactly equal to Text , as assumed in reference [7]. Finally, it is useful to calculate the radial temperature profile at different axial positions along the column. This requires knowledge of the inner wall temperature, T (Ri , z). As explained previously, this temperature can be approximated satisfactorily by the temperature, T (Re , z), of the external surface of the column wall, due to the high heat conductivity of stainless steel and the small thickness of the wall (0.07 cm). In order to simplify the numerical integration of Eq. (12), we consider only cases in which the axial temperature gradient is nearly linear, e.g., in which both the foam jacket and the normal outlet endfitting are used. We also assume that the wall temperature at the column inlet is equal to Text and that the wall temperature at the column outlet is the temperature measured by the HYPO thermocouple at the inner surface of the tube. The data necessary for the calculation of the radial profiles at different axial positions are given in Table 3 for both solvents. In this calculation, the value of β must be determined first. According to Horv´ath et al. [6], β is a positive factor the magnitude of which depends on the relative conductivities of the solid to the fluid in the column. It is negligible when the packing material has a much smaller conductivity than the mobile phase but it also depends on the flow conditions. The best values of β were

Fig. 11. Comparison between the experimental (measured at the column outlet, e.g. z = L) and calculated (Eq. (13)) radial temperature profiles with methanol (β = 2.22) and the water-rich (β = 0.49) as eluents. Foam jacket placed around the column and paraffin covering the special outlet endfitting. Note the larger temperature amplitude with methanol, the solvent with the smallest heat conductivity.

calculated so the amplitudes of the radial temperature gradients at the column outlet are equal to those given by Eqs. (14) and (15). For instance, the amplitudes were 0.75 K with water (Fig. 10A) and 1.60 K with methanol (Fig. 10C) when the column was wrapped in the foam jacket. The corresponding values of β were 2.22 and 0.49, respectively (Fig. 11). Fig. 12A and B show the radial temperature profiles calculated at different axial positions (10, 30, 50, 70, 90 and 100% of the column length L). Interestingly, when the inlet temperature of the eluent is equal to the external temperature, the profiles are nearly parallel until the axial position considered is very close to the column entrance (see Fig. 12B, where is shown the result of the calculation made for an abscissa equal to 1% the column

Fig. 12. Calculation of the radial temperature profiles (Eq. (13)) for different axial positions (z = 0.01, 0.1, 0.3, 0.5, 0.7 and 0.9 L) assuming a linear longitudinal temperature gradient. (A) Water-rich eluent. (B) Methanol eluent. Note the quasi-parallel radial profiles from the column inlet to the column outlet.

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length). This shows that the temperature gradient remains constant all along the column, except very near the entrance. Heat dissipation begins to take place immediately inside the column and the same temperature gradient propagates from the entrance to the column outlet, the temperatures in all points of the cross-section increasing by the same amount proportional to the migration distance. 4.3. Influence of the temperature gradients on the column efficiency Table 4 summarizes the efficiency data measured on a C18 bonded silica column eluted with two different solvents. In the

first case, the mobile phase was water-rich (85%, v/v) and the analyte was caffeine. In the second case, the mobile phase was pure methanol and a heavier compound, benzo[a]pyrene, was chosen in order to achieve a sufficiently large retention. In each case, the measurements were carried out at two different flow rates, corresponding to back pressures of 155 bar and 350 bar, respectively, 1.0 and 2.25 mL/min with the water-rich solution and 2.2 and 4.9 mL/min with the pure methanol solution. Three consecutive injections were made to assess the reproducibility of the results. The data were also acquired in the absence and presence of the foam jacket. Consistent with the previous section, a significant loss in the column efficiency ( 20%) is observed when the column is operated without the foam jacket with a

Fig. 13. Comparison between the chromatographic band profiles observed in the presence and absence of the isulating foam jacket. The experimental conditions are listed in Table 4.

27,790 – – 27,790 –

P = 350 bar (Fv = 4.8 mL/min) Still-air Foam

353 351 352 352 0.28

Three consecutive measurements were made (N1 –N3 ) in each experimental condition, except with the highest retention factor (k ≈ 30).

N1 N2 N3 Average efficiency RSD (%) Efficiency increase (%)

10,958 11,151 11,225 11,111 1.24 +1.22

3160 3102 3123 3128 0.94

3896 3845 3814 3852 1.07 +23.1

1246 1244 1242 1244 0.16 −0.43 1249 1246 1253 1249 0.28 10,919 10,988 11,024 10,977 0.49

421 415 411 416 1.21 +18.1

28,660 – – 28,660 – +3.1

6209 – – 6209 – +38.9

8628 – – 8628 –

Benzo[a]pyrene, MeOH/H2 O, 80/20 (v/v) k ≈ 30 P = 150 bar P = 360 bar (Fv = 1 mL/min) (Fv = 2.5 mL/min) Still-air Foam Still-air Foam Benzo[a]pyrene, MeOH k ≈ 3 P = 155 bar (Fv = 2.2 mL/min) Still-air Foam Caffeine, MeOH/H2 O, 15/85 (v/v) k ≈ 15 P = 155 bar P = 350 bar (Fv = 1 mL/min) (Fv = 2.5 mL/min) Still-air Foam Still-air Foam

Table 4 Impact of the foam jacket (smaller radial temperature gradients and higher longitudinal temperature gradient) on the column efficiency by comparison to still-air experimental condition

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pressure drop P = 350 bar. In contrast, there is only a negligible loss of efficiency with a pressure drop of 155 bar. Fig. 13 compares the band profiles recorded with and without the foam jacket wrapped around the column tube. Obviously, the column efficiency increases when the amplitude of the radial temperature gradient is reduced. A very significant efficiency gain, more than 20%, can be obtained with a conventional column operated at high inlet pressures if it is thermally insulated. A still larger effect should be expected when using ultra high pressure systems with column inlet pressures in the range of 1 kbar.  As shown in Table 4, the higher the retention factor k , the more the column efficiency increases when the foam jacket is wrapped around the column. The specific heat capacity of the mobile phase becomes a secondary factor. For instance, when benzo[a]pyrene is eluted with a 20:80 (v/v) mixture of methanol  and water (k = 30) the increase in column efficiency due to the foam jacket is close to 40% while it is barely 20% when the same  analyte is eluted with pure methanol (k = 3) in spite of the lower heat capacity of this second mobile phase. This experimental result demonstrates that the radial temperature gradient is nefarious for the efficiency of chromatographic columns when the analyte band can effectively probe the temperature differences inside the column. 5. Conclusion Using modern temperature sensors that were not available when earlier work on the friction heating of HPLC columns was first carried out [6,7,9], we measured the axial and the radial temperature profiles at the external boundaries of a chromatographic column (i.e., along the external column wall and across the eluent exiting the outlet frit of the column). We compared these results to those calculated using theoretical models available [4–10]. The experimental results demonstrate that the longitudinal gradients are quasi-linear at any radial position inside the column when cooling of the column wall by air convection is eliminated by wrapping the column in a foam jacket. Typically, operating a 25 cm long column packed with 5 ␮m particles at the flow rate generated by an inlet pressure of 300 bar generates longitudinal temperature gradients between 24 and 30 K/m, depending on the thermal properties of the eluent. The average radial temperature gradients are negligible for a column wrapped in a foam jacket. When cooling of the column wall by air convection is allowed, the axial temperature profiles are curved. The radial profiles are nearly parabolic and steep, up to 300–700 K/m, depending also on the thermal properties of the eluent. These experimental results validate the use of Eq. (12), which assumes that longitudinal heat conduction is negligible in comparison to radial heat conduction. From a practical point of view, these results confirm that a chromatographic column packed with fine particles (average size, 5 ␮m or smaller) and operated at high flow rates is not isothermal. The lack of thermal homogeneity has serious consequences on the column efficiency. Since the optimum velocity of a column increases with decreasing particle size, the smaller

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the particles, the more drastic these consequences. Under isocratic conditions, two main thermal effects contribute to cause an increase in peak width, hence, a loss in column efficiency. First, the solvent being warmer in the central zone of the column than along its walls, it is less viscous there. This causes a radial gradient of the eluent velocity, hence a difference between the elution retention times of all compounds. Assuming a radially homogeneous bed permeability, a one K temperature difference between the column center and its wall causes a relative difference of 2.4% in the linear velocity for a mixture of methanol and water  (25/75, v/v). Furthermore, the retention factors, k , decrease with increasing temperature. So, they depend on the radial coordinate. Typically, a one K difference between the central and the wall zones of the column causes differences of 0.7, 1.4, 2.1 and 2.7% in the elution times at T = 295 K of compounds having average adsorption energies of 5, 10, 15 and 20 kJ/mol, respectively. The combination of these two effects causes an important increase of the elution band width, hence a loss of efficiency [5]. To counteract this effect, it was suggested to pump into the column a mobile phase cooler than the external temperature, so that the bands would move more slowly in the central region of the column than along the wall at the beginning of the column, faster at the end [7]. The compensation of the effects is difficult to manage and could be effective only in a narrow range of retention factors. Further experiments are needed to assess the loss of column efficiency due to the thermal effects as described here, especially with columns packed with small particles. Longitudinal and radial temperature gradients coexist unless the column wall can be efficiently thermostated. An axial gradient is far less detrimental for the column efficiency than a radial one but, if its amplitude is too high, it will result in the end of the column not retaining the sample components, hence not contributing to retention. Practically, these temperature gradients certainly limit the efficiency of columns packed with small particles. On the theoretical front, they have to be taken into account in attempts at validating models of mass transfer kinetics (film mass transfer, particle diffusivity) which contribute significantly to the overall HETP at high flow rates. Nomenclature cpm p cp cp ,

mobile phase heat capacity (J/m3 /K) column packing heat capacity (J/m3 /K) H2 O specific heat of the packing using water-rich mobile phase (J/m3 /K) cp , CH3 OH specific heat of the packing using methanol mobile phase (J/m3 /K) cp,H2 O specific heat of water (J/m3 /K) cp,CH3 OH specific heat of methanol (J/m3 /K) cp,C18 specific heat of solid octadecane (J/m3 /K) cp,SiO2 specific heat of neat silica (J/m3 /K) e column tube thickness (m) f parameter defined in Eq. (14) hc heat transfer coefficient between the packing and the inner surface of the column tube (J/m2 /s/K)

he

jlong jrad jconv k0 L l M P qviscous Qc Qe r Re Ri S t T Te Text Tc uS V x z

heat transfer coefficient between the external surface of the column tube and the lab atmosphere (J/m2 / s/K) axial heat flux density defined in Eq. (1) (W/m/K) radial heat flux density defined in Eq. (2) (W/m/K) convective heat flux density defined in Eq. (3) (W/m/K) ratio of the intraparticulate to the interstitial void space or (1 − e )p /e column length (m) axial coordinate normalized to the column length parameter defined in Eq. (15) pressure (Pa) friction power released (W/m3 ) heat exchanged between the packing and the inner surface of the column tube (J) heat exchanged between the external surface of the column tube and the lab atmosphere (J) radial column coordinate (m) external column radius (m) internal column radius (m) column cross-section (m2 ) time variable (s) temperature (K) temperature of the mobile phase after the column outlet (K) laboratory temperature (K) column tube temperature (K) superficial linear velocity (m/s) volume (m3 ) integration variable in Eq. (12) longitudinal column coordinate (m)

Greek letters β positive factor in Eq. (14) φC18 volume fraction of bonded octadecane chains in the packing φMP volume fraction of mobile phase in the packing φSiO2 volume fraction of neat silica in the packing λc heat conductivity of the column tube (W/m/K) λm heat conductivity of the mobile phase (W/m/K) λz apparent longitudinal heat conductivity of the column packing (W/m/K) λr apparent radial heat conductivity of the column packing (W/m/K) λ,H2 O apparent heat conductivity of the column packing using water-rich mobile phase (W/m/K) λ,CH3 OH apparent heat conductivity of the column packing using methanol mobile phase (W/m/K) θ temperature of the packing normalized to the inlet temperature θw axial temperature of the column wall normalized to the lab temperature ρ radial coordinate normalized to the internal radius of the column ρm mobile phase density (kg/m3 ) e external column porosity p particle porosity

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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] J.C. Giddings, Dynamics of Chromatography, Marcel Dekker, New York, 1965. [2] A.D. Jerkovich, J.S. Mellors, J.W. Jorgenson, LC–GC Eur. 1106 (2006) 112. [3] A.D. Jerkovich, J.S. Mellors, J.W. Thompson, J.W. Jorgenson, LC–GC Eur. 77 (2005) 6292.

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