Modeling of thermal processes in high pressure liquid chromatography

Journal of Chromatography A, 1216 (2009) 6575–6586

Contents lists available at ScienceDirect

Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma

Modeling of thermal processes in high pressure liquid chromatography II. Thermal heterogeneity at very high pressures Krzysztof Kaczmarski a,∗ , Fabrice Gritti b,c , Joanna Kostka a , Georges Guiochon b,c,∗∗ a b c

Department of Chemical and Process Engineering, Rzeszów University of Technology, Ul. W. Pola 2, 35-959 Rzeszów, Poland Department of Chemistry, The University of Tennessee, Knoxville, TN, 37996-1600, USA Division of Chemical Sciences, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

a r t i c l e

i n f o

Article history: Received 20 March 2009 Received in revised form 15 June 2009 Accepted 27 July 2009 Available online 6 August 2009 Keywords: VHPLC Axial temperature profiles Radial temperature profiles Column efficiency Heat generation Heat transfer Viscous friction Peak profiles

a b s t r a c t Advanced instruments for liquid chromatography enables the operation of columns packed with sub2 ␮m particles at the very high inlet pressures, up to 1000 bar, that are necessary to achieve the high column efficiency and the short analysis times that can be provided by the use of these columns. However, operating rather short columns at high mobile phase velocities, under high pressure gradients causes the production of a large amount of heat due to the viscous friction of the eluent percolating through the column bed. The evacuation of this heat causes the formation of significant axial and radial temperature gradients. Due to these thermal gradients, the retention factors of analytes and the mobile phase velocity are no longer constant throughout the column. The consequence of this heat production is a loss of column efficiency. We previously developed a model combining the heat and mass balance of the column, the equations of flow through porous media, and a linear isotherm model of the analyte. This model was solved and validated for conventional columns operated under moderate pressures. We report here on the results obtained when this model is applied to columns packed with very fine particles, operated under very high pressures. These results prove that our model accounts well for all the experimental results. The same column that elutes symmetrical, nearly Gaussian peaks at low flow rates, under relatively low pressure drops, provides strongly deformed, unsymmetrical peaks when operated at high flow rates, under high pressures, and under different thermal environments. The loss in column efficiency is particularly important when the column wall is kept at constant temperature, by immersing the column in a water bath. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Industrial laboratories are under heavy pressure to increase analytical throughput. This requires the operation of chromatographic columns at high mobile phase velocities but without sacrificing the resolution between the sample components. Both the efficiency of chromatographic columns and the optimum range of velocities at which they should be operated increase in proportion to the inverse of the average size of the particles with which they are packed. Accordingly, there has been a constant historical trend to reduce this size. Doing so permits the achievement of the same column efficiency with shorter columns that can be operated at higher velocities, the two features combining to give faster analyses. However, operating a column packed with very fine particles at high

∗ Corresponding author. Tel.: +48 17 854 36 55. ∗∗ Corresponding author. Tel.: +1 865 974 0733; fax: +1 865 974 2667. E-mail addresses: [email protected] (K. Kaczmarski), [email protected] (G. Guiochon). 0021-9673/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2009.07.049

mobile phase velocities requires the use of very high inlet pressures, up to 1000 bar with sub-2 ␮m particles. Such high pressures gradients along a column cause the production of large amounts of heat due to the work of the viscous friction of the percolating mobile phase against the column bed. The evacuation of this heat results in an important thermal heterogeneity of the column and in losses of column efficiency. Numerous papers [1–19] deal with theoretical and experimental investigations of the consequences of heat generation by viscous friction in chromatographic columns, on the heterogeneity of the distributions of the temperature, the mobile phase velocity, its viscosity and its density throughout the column, and especially on the consequences of this thermal heterogeneity on the column efficiency. Recently, the profiles of peaks eluted from a column operated under natural convection conditions or immersed in a water bath were illustrated in a wide range of mobile phase flow rates [20]. It was shown that in the case of the thermostated columns, the peaks profiles were Gaussian at low flow rates but became trapezoidal at high flow rates (i.e., at very high inlet pressures).

6576

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586

In a previous paper [21], we modeled the behavior of these columns, combining the heat and the mass balance equations of the column, an isotherm model for the analyte, and the equations accounting for flow in porous media. This model takes into account the influence of the axial and radial distributions of the local temperature and pressure on the values of the viscosity and the density of the mobile phase, on its velocity, and on the Henry constants of the analytes. It was validated by proving that it predicts adequately the temperature distribution along the wall of a column packed with conventional 5 ␮m particles and the band profiles of analytes eluted from this column. The main goal of this new work was to investigate the validity of this heat and mass transfer model under the typical experimental conditions used in very high pressure liquid chromatography (VHPLC). For this purpose, we first compare the temperature profiles recorded along the external surface of several columns of various lengths, operated at very high inlet pressures and those calculated with our model. We report on the distributions of axial and radial temperature and of mobile phase velocity calculated for different thermal boundary conditions. Finally, we compare the calculated and measured concentration profiles of bands eluted from a column immersed in a water bath, under experimental conditions identical to those used earlier [20]. Although Gaussian at low flow rates, these profiles deform progressively with increasing flow rate and are strongly deteriorated at high flow rates. The good agreement between experimental and calculated profiles validates the model proposed earlier [21]. 2. Mathematical models The mathematical model used in this work to account for the consequences of the heat generated in a column by the friction of the mobile phase percolating through its bed is exactly the same as applied in previous paper [21]. However, in its application, we no longer assume that the mobile phase heat capacity is constant. This model is briefly described in this section. It combines three separate models: (1) a model of heat transfer; (2) a model of mass transfer; and (3) a model for the mobile phase velocity distribution. These three models are coupled, so the problem is solved by handling them simultaneously. The first model expresses how the heat generated by viscous friction is evacuated from the column under steady-state conditions. The second model accounts for the propagation of a compound band along a column that is no longer isothermal. The equilibrium constant depends on the local temperature and pressure; so does the migration rate of a concentration. The third model accounts for the distribution across the column of the mobile phase velocity, which depends on the local temperature and pressure and is given by the equations of hydrodynamics in porous media.

4. The dependency of the heat conductivity of the bed on the local temperature can be neglected. 5. Heat is conducted in both the axial and the radial directions of the column tube. 2.1.1. Equations Under this set of assumptions, the heat balance for an infinitesimal volume element of a packed bed can be given in cylindrical coordinates as [2,3,21–23]: (εt cpm + (1 − εt )cs )

 = r,ef

∂T ∂P ∂T ∂T − εt T˛ + cpm uz + cpm ur ∂t ∂t ∂z ∂r

1 ∂T ∂2 T + 2 r ∂r ∂r



− uz (1 − ˛T )

∂P ∂z

where εt is the total column porosity, cpm is the mobile phase heat capacity (J/m3 /K), cs is the solid phase heat capacity (J/m3 /K), T is the local temperature (K), uz is the superficial velocity of the mobile phase in the axial direction (m/s), ur is the superficial velocity of the mobile phase in the radial direction (m/s), r,ef is the effective bed conductivity (W/m/K). The coefficient ˛ (1/K) is the coefficient of thermal expansion of the mobile phase. The heat power generated locally, inside the column, due to the viscous friction is the product of the superficial velocity and the pressure gradient [2,3,22,23]: hv = −uz

∂P ∂z

(2)

The heat balance for the column wall can be formulated as follows: ∂T cw = w ∂t



1 ∂T ∂2 T + 2 r ∂r ∂r



+ w

∂2 T ∂z 2

In formulating the heat balance equation, we assumed that heat is generated inside the column due to the viscous friction of the mobile phase against the bed and that it is conducted away through the packed bed and the column wall, to be dissipated into the air surrounding the column. The model assumptions are the following: 1. For packed and monolithic columns, the axial heat dispersion and the axial heat conductivity of the bed can be neglected (but not those of the tube). 2. The radial heat transfer can be expressed by the effective radial conductivity. 3. The mobile phase flow velocity is a function of the radial and the axial coordinates but the mobile phase mass flux is constant in the axial direction.

(3)

where cw is the wall heat capacity (J/m3 /K), and w is the wall heat conductivity (W/m/K). The system of Eqs. (1) and (2) was coupled with a typical set of initial and boundary conditions [21]. It was assumed that the initial bed and wall temperatures are equal to the ambient temperature in the laboratory, Text . The temperature at the column inlet, T0 , is equal to the inlet mobile phase temperature. The temperature gradients in the column center and at its outlet are equal to zero. More over it was assumed that the heat fluxes at the boundary between the bed and the column wall are equal, and finally that the heat is transferred from the column tube to the surrounding air by convection and radiation for column operated under natural convection conditions or that the wall temperature is equal to Tther when the column is thermostated. In the case of a column operated under natural convection conditions the effective heat transfer coefficient, he is the sum of the convective hcon and the radiation hrad heat transfer coefficients. he = hcon + hrad

2.1. The heat balance equation

(1)

(4)

The heat transfer coefficient by radiation was derived from the following equation: 2 2 hrad = C0 ε(Tw + Text )(Tw + Text )

(5)

Which directly results from Stefan–Boltzman law [22]. The constant C0 is equal to 5.669 × 10−8 W/m2 K4 and ε is the emissivity of the steel surface. The heat transfer by convection was calculated from the following correlation [24]: hcon DC =K 



DC3 2 g(Tw − Text ) cp  (Tw + Text )/22 

0.2

(6)

where DC is the external diameter of the column wall, , ,  and cp are the air density, its viscosity, thermal conductivity and heat capacity, respectively. A value of K = 1.09 is recommended.

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586

Under steady-state conditions, the heat balance model can be simplified by removing the time-dependent terms, i.e., the first two terms in Eq. (1), the first term in Eq. (3)) and the initial conditions.

The above system of equations has to be closed by the following relationship:

 L

P =

2.2. The mass balance equation

0

In the formulation of the mass balance of an analyte in the column, we assume that the contributions to band broadening due to the finite mass transfer resistances and to axial dispersion can be lumped into an apparent dispersion coefficient. Under this assumption the mass balance equation is a simple extension of the equilibrium-dispersive model [25].

6577

∂P − ∂z



dz

(12)

where P is the actual pressure drop along the column, measured from the instrument gauges. 2.4. Method of calculation of solutions of the models

where: q is the concentration in the solid phase, C the concentration in the mobile phase, E the activation energy of adsorption, R the universal gas constant and Vm the difference between the partial molar volumes of the solute in the adsorbed layer and in the liquid phase.

To solve the time-dependent mass balance equation, only the steady-state temperature profile is required. To spare computing time it is convenient to solve separately first the heat balance equation and later the mass balance equation. The heat and pressure profiles were derived from the limit of the solution of the timedependent version of the model (Eqs. (1) and (3)) coupled with Eqs. (9)–(12) after a sufficiently long time. The mass balance model was solved last using the steady-state distributions of the temperature and the pressure obtained earlier. The heat (Eqs. (1) and (3)) and the mass (Eq. (7)) differential balance equations were solved using the method of orthogonal collocation on finite elements (OCFE) in the analogical version previously described [26]. The spatial derivatives were discretized, following the OCFE method. The set of ordinary differential equations obtained through this process was solved using the VODE solver [27]. The calculation of the steady-state temperature and pressure gradient distributions requires some additional explanations. As stated earlier, the local values of the viscosity and the density are functions of the local temperatures, which depend themselves on the pressure gradient and on the local velocity. But the velocity and the pressure gradient are functions of the local viscosity and the local density. The solution of this circular relationship requires that the temperature and the pressure gradient should be calculated using a trial and error method. The principle of this method in this case consists in choosing the  parameter in the Blake, Kozeny and Carman correlation in such way that the value calculated for the pressure drop using Eq. (12) agrees with the one measured. The details of calculations are given in [21]. As was stated earlier, the axial mobile velocity uz (r, z) was calculated from Eq. (9). According to the continuity equation, the local radial interstitial velocity, wr = ur /εt , can be calculated with a simple numerical approximation [21]:

2.3. Mobile phase velocity distribution and pressure calculation

(r2 wr (r2 , z2 ) − r1 wr (r1 , z2 ))

2.2.1. Equation In a cylindrical system of coordinates, the mass balance equation of the ED model is written as follows: ∂C ∂C ∂2 C ∂q ∂C +F + wz + wr = Dz,a 2 + Dr,a ∂t ∂t ∂z ∂r ∂z



1 ∂C ∂2 C + r ∂r ∂r 2



(7)

where C and q are the analyte concentrations in the mobile and the stationary phases (g/l), respectively, Dz,a and Dr,a are the axial and the radial apparent dispersion coefficients (m2 /s), respectively, wz = uz /εt and wr = ur /εt are the axial and radial interstitial velocities, respectively, and F = (1 − εt )/εt is the phase ratio. Eq. (7) was solved assuming that initially there is not solute in the column. The solute is injected into the column during the time tinjection . The gradient of concentration at the column outlet, between the column center and its wall region is equal to zero. The details of the initial and boundary conditions are presented in [21]. 2.2.2. Isotherm equation Eq. (7) must be combined with an appropriate isotherm equation. In this work, we consider a linear isotherm. However the Henry constant is a function of the temperature and the pressure [25]. Taking this into account, the following isotherm model was used:



q = CHo exp(E/RT ) exp

−Vm

P − Pref



RT

(8)

The local value of the mobile phase velocity was calculated from the following equation [21]: uz (r, z) =

uo o (r, z)(/)z

where

 

= z

(9)

2 Ri2

 0

R

(r, z)r dr (r, z)

(10)

and uo , o are the mobile phase superficial velocity and density at the column inlet. The local pressure gradient was calculated according to the correlation developed by Blake, Kozeny and Carman [22], using the following equation [21]:



∂P − ∂z



=

uo  o (ε3e /(1 − εe )2 )(dp2 /)(/)z

(11)

where  is an empirical parameter generally considered as equal to 150 [22].

=−

r22 − r12

 w (r , z ) − w (r , z ) z 2 2 z 2 1 z2 − z1

2

+ wz (r2 , z2 )

1 (r2 , z2 ) − (r2 , z1 ) z2 − z1 (r2 , z2 )

 (13)

Eq. (13) represents a simple finite difference scheme for the calculation of the radial component of the velocity in the nodal point (r2 , z2 ), knowing the radial velocity in the previous nodal point, the known axial velocity and the mobile phase density. 2.5. Methods of calculation of the physico-chemical parameters 2.5.1. Density The density of a the eluent was obtained from the classical Tait equation for isothermal compressibility [14,28], assuming a first order variation of the expansion coefficient with the temperature:



(P, T ) = (P o , Tref )

P + b + b1 T

c

105 + b + b1 T

2 exp(˛1 [T − Tref ] + ˇ1 [T 2 − Tref ])

(14)

6578

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586

The best values of the coefficients b, b1 , ˛1 , ˇ1 were derived from data given in ref. [20] 2.5.2. Viscosity The temperature and pressure dependencies of the eluent viscosity can be expressed as (P, T ) = 10(A+B/T ) (1 + [P − 105 ])

(15)

The best values of the coefficients A, B and were derived from data given in ref. [29]. 2.5.3. Thermal expansion coefficient The coefficient of thermal expansion of the mobile phase ˛ was calculated from the well known dependency ˛=−

1 ∂  ∂T

(16)

after combination with Eq. (14). 2.5.4. Heat capacity The relationship between the mobile phase heat capacity cp (J/kg/K), the temperature and the pressure can be derived from the following thermodynamic condition:



∂cp ∂P



 = −T T

∂2

∂T 2



 =T p

1 ∂2  2 − 3 2 ∂T 2 



∂ ∂T

2

cpm (P, T ) = cpm (P o , Tref ) + a1 (T − Tref )



+ a2 (T

2 − Tref ) + (P, T )

P

Po

∂cp dP ∂P

(18)

The value of the heat capacity was calculated from Eq. (18), after combination with Eqs. (17) and (14) and calculation of the integral. 2.5.5. The thermal conductivity The thermal conductivity of a porous medium impregnated with a liquid depends on the geometry of that solid, on its porosity, and on the thermal properties of the medium and of its components [16]. In the case of a two-component heterogeneous system that has a chaotic structure, Zarichnyak and Novikov [30] proposed the following equation for the calculation of the effective conductivity: R,ef = ε2t elu + ε2s s + 4εt εs

elu s elu + s

3s − 2εt (s − elu ) 3elu + εt (s − elu )

3.2.1. Columns The four columns used in this study were all from Waters (Mildford, MA, USA). They were packed with 1.7 ␮m particles of the bridged ethylsiloxane/silica-C18 packing material (BEH). The characteristics of the adsorbent particles are: pore diameter – 130[Å], surface area – 185 [m2 /g], bonded phase – endcapped BEH-C18, total carbon – 18 [%], surface coverage – 3.1 [␮mol/m2 ]. The main characteristics of the packed columns used are summarized in Table 1. These columns were made of type 316 stainless steel, for which the emissivity is ε = 0.28 and the heat conductivity w = 16 (W/m/K).

All the columns were operated with an Acquity UPLC liquid chromatograph (Waters, Milford, MA, USA). This instrument includes a quaternary solvent delivery system, an autosampler with a 10 ␮l 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 extracolumn volume of the instrument is 13.5 ␮l, as measured with a zero-volume union connector in place of the column. The flow rate delivered by the high pressure pumps of 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 is set at 2.0 ml/min. All the measurements were carried out with the column either operated under natural convection conditions or immersed in a thermostated water bath. The laboratory temperature was constant and equal about 22 ◦ C. The daily variation of the ambient temperature never exceeded ±1 ◦ C. 3.4. Measurement of the temperature

(19)

Another equation was proposed by Abbott et al. [31] R,ef = elu

3.2. Materials

3.3. Apparatus (17)

After integration of Eq. (17) the final relationship is obtained [14]:

2

The solvents were filtered before use on an SFCA filter membrane, 0.2 ␮m pore size (Suwannee, GA, USA). Eleven polystyrene standards were used to acquire the ISEC data needed to estimate the column porosities (MW ) 590, 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, USA). Naphtho[2,3a]pyrene was used as the solute and was purchased from Fisher Scientific (Fair Lawn, NJ, USA).

(20)

In the above equations, the porosity εs is the ratio of the volume of the solid phase in the bed to the geometrical volume of the column and s is the solid phase conductivity. 3. Experimental 3.1. Chemical Two different mobile phases were used in this work. First, pure acetonitrile, HPLC grade, purchased from Fisher Scientific (Fair Lawn, NJ, USA). Second, a 85/15 (v/v) aqueous solution of acetonitrile. Dichloromethane and tetrahydrofuran, both HPLC grade, were used to estimate the hold-up volumes of the columns using the pycnometric method. They were also purchased from Fisher Scientific.

The columns wall temperature was measured at several different equidistant axial positions, with thermocouples. The temperature of the eluent exiting the column was recorded with another thermocouple. The column was kept horizontally in the oven, in direct contact with the laboratory atmosphere but Table 1 The main characteristics of the packed columns [17]. Dimension i.d. (mm) × L (mm)

2.1 × 30

2.1 × 50

2.1 × 100

2.1 × 150

Total porositya External porosityb Particle porosity Blake–Kozeny–Carman constantc effective bed conductivity (W/m/K) for acetonitrile

0.635 0.372 0.419 138

0.642 0.373 0.429 152

0.641 0.377 0.424 130

0.639 0.380 0.418 142

0.39

0.39

0.39

0.38

a b c

Measured by pycnometry (THF-CH2 Cl2 ). Measured by inverse size exclusion chromatography (polystyrene standards). Calculating with constant K = 1.09 in Eq. (6).

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586

protected from forced air convection. All information on the measurements of the temperature of the external column wall was given elsewhere [13]. 4. Results and discussion 4.1. Calculations of the column temperature and the mobile phase velocity distributions For column inlet pressures below ca. 100 bar, the heat generated by viscous friction is small, so the temperature and all the column physico-chemical parameters can be regarded as constant throughout the whole column. In contrast, for an inlet pressure of 1000 bar, the difference between the column inlet and outlet temperatures may exceed 20 ◦ C and the difference between the temperatures close to the wall of the column and in its center may reach 5 ◦ C. The temperature and pressure gradients cause important changes of the eluent properties, especially its viscosity and density, with a smaller change expected for the heat capacity. The variations of the density, viscosity and heat capacity with temperature and pressure were calculated from the correlations in Eqs. (14), (15) and (18). The values of the parameters of the mobile phase in these equations are listed in Table 2 [20]. To solve the heat balance model, we need the effective heat conductivity. For a this purpose, Eq. (19) was used to calculate the effective conductivities of silica and of the C18 ligands, and to obtain s,ef . Afterwards, the effective conductivity was obtained from s,ef and elu using Eqs. (19) or (20). These two equations give very similar values. For example, for acetonitrile–water, the values calculated for R,ef were: 0.378 (Eq. (19)) and 0.375 W/m/K (Eq. (20)). These values were obtained by taking for the eluent a conductivity elu = 0.219 W/m/K, for that of the C18 ligands (considered as solid octadecane) lig = 0.35, and for solid silica s = 1.40 W/m/K. In the calculation of the temperature distribution the value R,ef = 0.378 was used for T = 295 K. The R,ef does not change much with the temperature. For example, neglecting the changes in the heat conductivity of silica and C18 with the temperature, the value of the effective conductivity is R,ef = 0.370 at T = 329 K. The effective conductivity of pure acetonitrile is about 0.37 at T = 295 K. Gritti and Guiochon [14] analyzed the impact of the radial heat dispersion on the effective heat conductivity and found it rather low, less than about 5%.

6579

The calculation of the wall temperature profile for a column operated under natural convection conditions is affected by the cooling effect of the massive endfittings, which have a complicated shape, similar to that of a cylinder and a cone, each 1 cm long, connected. The cylindrical part of the endfitting is part of the column wall and covers 1 cm of this tube. To account approximately for the cooling effect of the endfitting, we assumed that its additional contribution to heat transfer from the column to the surrounding air is proportional to the ratio of the external surface of the endfitting to the surface of the column in contact with the endfitting, a ratio of about 3.5. This means that the effective heat transfer coefficient, he , was calculated from Eq. (4) coupled with Eqs. (5) and (6). The convective heat transfer was obtained from Eq. (6), using the external column diameter Dc in the region not occupied by the endfitting and the external diameter of the endfitting for the part of the column that is covered by the endfitting. The value calculated for he in the latter case was multiplied by 3.5. 4.2. Temperature distributions for columns operated under natural convection conditions Figs. 1–3 compare the experimental (symbols) and the theoretical (lines) temperature distributions along the external wall of four

Table 2 Complete list of parameters of the eluents used in the calculation. Parameter

CH3 CN

CH3 CN/H2 O, 85/15, v/v

Tref P0

298.15 [K] 105 [Pa]

298.15 [K] 105 [Pa]

Density (P0 ,Tref ) ˛1 ˇ1 b b1 c Viscosity A B Heat capacity cp,m a1 a2

776.6 [kg/m3 ] −3.304 × 10−4 [K−1 ] −1.756 × 10−6 [K−1 ] 3.403 × 108 [Pa] −7.53 × 105 [Pa K−1 ] 0.125 −1.757 386 [K] 6.263 × 10−9 [Pa−1 ] 1.762 × 106 [J m−3 K−1 ] −2.116 × 103 [J m−3 K−2 ] 0.528 [J m−3 K−3 ]

Heat conductivity  0.188 [W m−1 K−1 ] (298 K)

817.9 [kg/m3 ] 1.404 × 10−4 [K−1 ] −2.343 × 10−6 [K−1 ] 4.319 × 108 [Pa] −9.50 × 105 [Pa K−1 ] 0.132 −2.500 648 [K] 4.534 × 10−9 [Pa−1 ] 2.121 × 106 [J m−3 K−1 ] 0.162 [J m−3 K−2 ] −3.636 [J m−3 K−3 ] 0.219 [W m−1 K−1 ] (298 K)

Fig. 1. Comparison of the measured (symbols) and calculated (lines) temperature profiles along the wall of a column eluted with pure acetonitrile. Columns length, corresponding mobile phase velocities at column inlet and inlet pressures are equal: (1)—3 cm, 2 ml/min, 438 bar, (2)—5 cm, 2 ml/min, 775 bar, (3)—10 cm, 1.45 ml/min, 973 bar, (4)—15 cm, 0.95 ml/min, 979 bar, respectively. Natural convection conditions: (a) calculation performed for heat transfer coefficient calculated from Eq. (6) with constant K = 1.09, (b) calculation performed for constant K in Eq. (6) equal 2.7, 2.8, 1.5 and 1.9 (from shortest to longest column).

6580

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586

Fig. 2. Comparison of the measured (symbols) and calculated (lines) temperature profiles along the wall of a column eluted with pure acetonitrile. Inlet pressures: 973, 765, 559 and 357 bar (from top to bottom); corresponding mobile phase velocities: 1.45, 1.15, 0.85 and 0.55 ml/min at column inlet, respectively. Natural convection conditions. Column L = 10 cm. Calculation performed for constant K in Eq. (6) equal 1.5, 1.5, 1.3 and 1.4 (from top to bottom).

columns operated under natural convection conditions and eluted with pure acetonitrile. In Fig. 1a and b, the columns (lengths 3 and 5 cm) are eluted at the maximum possible mobile phase velocity or (lengths 10 and 15 cm) at the maximum possible pressure drop allowed by the system pump. On the other hand, Figs. 2 and 3 compare the temperature profiles obtained for the 10 and 5 cm long columns at different mobile phase flow rates. Note that the heat generated by the friction of the mobile phase flowing through the column bed is evacuated from the column wall to the surrounding air by both convection and radiation. However, the proportion of heat transferred by radiation to the overall heat loss is relatively low, less than 10%.

The calculations of the temperature profiles were done first with the convective heat transfer coefficient derived from Eq. (6), using the recommended value K = 1.09. However, the wall temperature distributions calculated were always larger than those measured, with differences as large as 5 ◦ C. There are at least two reasons for these discrepancies: (1) the method used to account for the cooling of the column by its massive endfitting is too simple and underestimates the cooling; (2) the wall temperatures measured are probably incorrect and underestimated. This second possibility was checked by repeating several times the wall temperature measurements for the 10 cm long column, at a distance of 7 cm from the column inlet, and at Fv = 1.45 ml/min, each time changing the exact location of the thermocouples. The differences between the different temperatures measured were 3 ◦ C. Such large differences are due to the difficulty in achieving a sufficiently good contact between the thermocouple and the steel wall and in avoiding the heat loss from the thermocouple to the surrounding air. The validity of the second explanation was confirmed by measurements of the temperature of the column eluent. These measurements are more precise than those of the wall temperature because the thermocouple is placed in the mobile phase stream (see Fig. 4 in [13]) but some additional cooling effect is possible, due now to evaporation of the mobile phase. The measured and calculated average differences, L T, between the eluent temperatures at the column outlet and inlet are listed in Table 3. The calculated temperature is still higher than the experimental one, but the maximum difference is less than about 2 ◦ C. The values of the wall temperature measured for all the columns, at all the different flow rates used are in good agreement with those calculated when the constant K in Eq. (6) is estimated on the basis of the experimental temperature profiles—see Figs. 1b, 2 and 3. However, in this case, the differences L T are underestimated by up to 3 degrees (see Table 3). This result also confirms the second assumption made above. In Table 3 are also reported the maximum temperature differences in the radial direction, R T, for all the investigated columns. These differences are important, up to about 5 ◦ C over a distance of only 1 mm. In summary, we showed that the proposed heat transfer model predicts adequately the temperature distributions measured along the column wall. However, further improvements must be made in the experimental procedures of measurement of the wall temperature, in the method of calculation of the effective heat transfer coefficient, and in the estimation of the cooling effect of the column endfittings. Any increase of either the formed temperature gradient or the pressure gradient increases the magnitude of the radial and axial gradients of mobile phase viscosity and density. The decrease of the viscosity along the column can be considerable and reach 50%. In the radial direction, the viscosity difference rarely exceeds about 5%. Much smaller, but still significant changes of the mobile phase density may take place in the axial direction. The density may decrease

Table 3 Comparison of the measured and calculated temperature difference, L T of eluent between column outlet and inlet and maximum difference in radial direction, R T. L [cm]

Fig. 3. Comparison of the measured (symbols) and calculated (lines) temperature profiles along the wall of a column eluted with pure acetonitrile. Inlet pressures: 642, 505, 370 and 237 bar (from top to bottom); corresponding mobile phase velocities: 1.65, 1.3, 0.95 and 0.6 ml/min at column inlet, respectively. Natural convection conditions. Column L = 5 cm. Calculation performed for constant K in Eq. (6) equal 4.1, 3.6, 3.6 and 3.1 (from top to bottom).

3 5 10 15

Fv,inlet [ml/min]

2.000 2.000 1.450 0.950

L T [◦ C]

R T [◦ C] a

b

exptl.

calcd.

calcd.

calcd.a

calcd.b

10.0 18.9 20.0 13.2

11.9 21.1 21.0 13.7

10.2 16.9 18.6 9.7

3.6 4.7 3.1 1.5

4.4 6.0 3.3 1.6

a Calculation performed for heat transfer coefficient calculated from Eq. (6) with constant K = 1.09. b Calculation performed for constant K in Eq. (6) equal 2.7, 2.8, 1.5 and 1.9 (from shortest to longest column).

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586

by 10%. In the radial direction, the density remains practically constant, with differences being less than 0.5%. 4.3. Efficiency of columns operated under natural convection conditions The column efficiency strongly depends on the distributions of the temperature and the mobile phase velocity across the column. Examples of profiles of temperature and velocity calculated for a 5 cm long column are shown in Fig. 4. These calculations were performed with the same parameters as those used for Fig. 1a. For the two shortest columns, 3 or 5 cm long, the temperature increases from the column inlet to its outlet. On the other hand, for the two longest columns, 10 or 15 cm long, the temperature profile goes through a maximum that is reached before the column outlet, at distances of 8 and 11 cm from the inlet, respectively. Beyond these points, the temperature decreases. This is partially due to the cooling effect of the endfitting and partially to the lower heat power generated near the column end. The power of heat generation is

6581

proportional to the product uP. For the 15 cm long column, this product is 1.5 times larger at column inlet than at column outlet. The very high rate of heat transfer along the column steel wall influences considerably the temperature profiles along the beds. The warmer part of the metal tube (the region near the column outlet) sends heat back toward the column entrance region and warms up the incoming mobile phase. On the other hand, the outlet part of the tube is cooled by the cooler middle part of the tube. As a consequence, a positive radial temperature gradient forms in the column entrance region and a negative radial temperature gradient in the outlet region. The absolute value of the magnitude of the positive gradient can be greater than that of the negative gradient. However, the region in which the mobile phase temperature is lower than the wall temperature is relatively short, about 1 cm for the 3 cm long column and 1.5 cm for the 15 cm long column. These temperature gradients cause the formation of similar positive and negative radial gradient of the axial component of the mobile phase velocity. The maximum difference between the mobile phase velocities in the column central and wall regions is less than about 2–3%. The effect of the inverse velocity profiles at column inlet and outlet under natural convection conditions has been mentioned by Rozing and coworkers [32]. The changes in the signs of the radial temperature and velocity gradients have a positive impact on the column efficiency. The positive gradient causes the analyte band to move more slowly in the column center than in the wall region. The converse is true for a negative gradient: the analyte band moves faster in the central than in the wall region. Finally, the loss of column efficiency, which is due to the negative gradient being predominant along the column, is compensated to some degree by the positive gradient. Finally, we need to compare the magnitudes of the axial and radial components of the mobile phase velocity. The radial velocity was calculated, using Eq. (13). The maximum absolute value of the radial velocity component was found to be slightly higher than it is under standard HPLC conditions [21] but it is still about 1000 times less than the axial velocity. So, in our proposed model, the radial component of the mobile phase velocity has no practical influence on the column efficiency. 4.4. Temperature distributions for the thermostated column

Fig. 4. Evolution along the column of the calculated radial temperature profiles (a), T(r, z), and axial mobile phase velocity uz (r, z) (b). Column: 5 cm × 0.21 cm. Other parameters as in Fig. 1a.

Serious problems were encountered in comparisons of experimental temperature distributions with the temperature profiles calculated for columns working under natural convection conditions, due to the lack of precision of the measurements. To obviate these problems, we compared the calculated and measured concentration distribution profiles measured at the column outlet under well-defined thermal conditions, those of a column thermostated in a water bath. The heat transfer from the column wall to a turbulent stream of cooling water is several hundred times larger than that achieved under natural convection conditions. With such a large rate of heat transfer, the temperature of the external surface of the metal tube is practically the same as that of the water. Due to the high thermal conductivity of metal, the radial temperature gradient through the tube wall is practically negligible. The experiments were performed with a water stream kept at a temperature of 299 K, and at the highest possible experimental inlet pressure, for a 50 mm × 2.1 mm column, using 85/15 (v/v) solution of acetonitrile in water. Fig. 5 shows the calculated distributions of the temperature and the mobile phase velocity for this column, operated under these conditions, assuming that the inlet mobile phase temperature is equal 299 K. As the figure shows, the temperature increases rapidly along the column axis, to reach its maximum value at a distance of about 30% of the column length. Beyond that distance, the temperature decreases slowly. This decrease of temperature is explained by the

6582

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586

Fig. 6. Temperature difference between column center, Tc , and the wall, Tw : asterisks denote maximum temperature difference and squares the difference at column outlet. Calculation for column length equal 5 cm.

later, when the pressure drop becomes larger than 300 bar (hence the radial temperature difference larger than 1 K), the peak shapes are more and more deformed and different from the Gaussian band profile. 4.5. Peak profiles eluted from the thermostated column

Fig. 5. Evolution along the column of the calculated radial temperature profiles (a), T(r, z), and axial mobile phase velocity uz (r, z) (b). Column: 5 cm × 0.21 cm thermostated in water bath in temperature equal 299 K. Acetonitrile–water used as mobile phase. Inlet pressures: 808 bar and corresponding mobile phase velocities: 1.5 ml/min.

decrease of the pressure gradient along the column [21], to which the amount of heat generated is proportional. It should be noted that the radial temperature difference between the column wall and its center could reach 6 K, over a distance of only 1 mm. Moreover, this difference is no less than 5 K along about 70% of the column length. This strong radial temperature gradient causes the development of a significant radial variation of the axial eluent velocity, which reaches 10% along about 70% of the column length. These radial temperature, viscosity and velocity gradients cause dramatic changes in the elution band peak profiles, as discussed in the next section. Fig. 6 shows the maximum temperature difference between the column center, Tc , and its wall, Tw (*) and the same difference but at the column outlet (squares) as a function of the mobile phase velocity and the pressure drop along the column. For pressure drops less than about 300 bar, the radial temperature difference is less than 1 K and probably inconsequential; for a pressure drop of 500 bar, it increases to 2 K; and for 800 bar it reaches 5–6 K. As can be seen

The profiles of the elution bands of naphtha[2,3-a]pyrene were recorded to investigate the influence of the heat generated by viscous friction on their shapes and on the column efficiency. The measurements were performed under analytical conditions, for mobile flow rates ranging from 0.03 to 1.8 ml/min, using the 50 mm × 2.1 mm column immerged in a water bath thermostated at 299K eluted with a 85/15 (v/v) mixture of acetonitrile and water. The mass balance of naphtho[2,3-a]pyrene (see Eq. (7)) was solved using its isotherm model (Eq. (8)). The stationary phase temperature and the velocity profiles were provided independently by the solution of the heat balance model. The isotherm model parameters: Ho = 2.936 × 10−4 and E/R = 3245 were derived from measurements of the retention times of naphtho[2,3-a]pyrene at 299, 310 and 329 K at the mobile phase flow rate of 0.12 ml/min, at which the heat effects are negligible. The difference of partial molar volumes of the solute in the adsorbed and the liquid phases is Vm = −1.19 × 10−5 m3 /mol in pure acetonitrile, under a reference pressure Pref of 25 bar (data not published). In the 85/15 (v/v) acetonitrile–water eluent, this value was corrected to Vm = −1.1 × 10−5 m3 /mol, on the basis of the measurement of the peak retention time at a mobile phase flow rate of 1.5 ml/min, with the column in a water bath at 299 K. To solve the mass balance equation, the radial and axial dispersion coefficients are needed. The radial dispersion coefficient, Da,r , was calculated from the following plate height equation derived by Knox et al. [33,34]. Hr = 0.06dp +

1.4 dp 

(21)

where  is the reduced velocity. This corresponds to: Da,r =

0.03dp u + 0.7Dm εt

(22)

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586 Table 4 Radial and axial dispersion coefficients and HETPs. Fv [ml/min]

Da,r ×1010 [m2 /s]

Da,z ×1010 [m2 /s]

hz = Hz /dp

hexp

(Tc − Tw )a [K]

0.12 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

9.47 9.71 10.00 10.31 10.62 10.93 11.25 11.57 11.91 12.24 12.59 12.93 13.31 13.66 14.01

31.7 41.0 54.0 73.0 100 160 180 200 200 200 250 350 500 600 850

4.14 3.22 2.82 2.86 3.14 4.18 4.03 3.92 3.48 3.14 3.56 4.57 6.03 6.72 8.89

7.67 6.98 4.42 3.89 6.21 9.08 11.5 14.2 22.8 31.2 39.5 53.5 72.3 92.2 114.9

0.03 0.08 0.18 0.32 0.51 0.74 1.02 1.35 1.71 2.13 2.63 3.15 3.74 4.39 5.08

a Temperature difference between column center, Tc , and column wall, Tw , at column outlet.

The molecular diffusion coefficient Dm was estimated from the Scheibel equation, often recommended in the literature [25,35]. This coefficient was calculated for the physico-chemical conditions at the column inlet and outlet and the average values of Dm were taken for the calculations. The axial dispersion coefficient, Da,z was estimated by a trial and error method, in order to obtain the best possible agreement between the calculated and the experimental

6583

peak profiles, for each mobile phase velocity. The values of the axial and radial dispersion coefficients are given in Table 4. The quotient Da,z to Da,r increases from 3, at the smallest mobile phase velocity, to 60 at the highest velocity. Fig. 7a–d compare the calculated (solid line) and the experimental (dotted lines) band profiles for the column thermostated at 299 K. The relative error between the predicted and the measured retentions time is always less than 1%. The agreement between the experimental and the theoretical peak profiles is very good and sometimes even excellent. The peak profiles are almost Gaussian for pressure drops less that about 300–350 bar. At higher pressures, the band shapes slowly become a trapezoidal profile. The large degradation of the elution peak profiles at high mobile phase flow rates is due to the large difference between the retention coefficients and the mobile phase velocities in the core region and the wall area of the column. As illustrated by the results of the calculations illustrated in Fig. 8, the elution profiles of the analyte in the column central region can be 1 cm ahead of the profiles that are moving near the column wall. All the local profiles remain Gaussian, but the overall or bulk elution profile is the integral of all these contributions and is not Gaussian. Note that the heights of the different profiles in Fig. 8 depend on their radial position. The values of the reduced experimental plate heights, hexp , and of the reduced axial plate heights, hz = Hz /dp = (2εt Da,z /u)/dp , derived from the estimated values of the axial dispersion coefficients are compared in Table 4. The difference between the temperatures of the column in its center, Tc , and near its wall, Tw , at the column

Fig. 7. Comparison of experimental (symbols) and theoretical (solid lines) concentration profiles of the naphtho[2,3-a]pyrene. Column: 5 cm × 0.21 cm thermostated in water bath in temperature equal 299 K. Acetonitrile–water used as mobile phase. (a) Fv = 0.4, 0.3, 0.2, 0.12 ml/min (from left to right) and P = 210, 161, 114, 75 bar respectively. (b) Fv = 0.8, 0.7, 0.6, 0.5 ml/min (from left to right) and P = 419, 365, 313, 261 bar respectively. (c) Fv = 1.5, 1.1, 0.9 ml/min (from left to right) and P = 808, 580, 470 bar respectively. (d) Fv = 1.2, 1.0 ml/min (from left to right) and P = 664, 525 bar, respectively.

6584

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586

suppose that the plate height increase due to the radial temperature difference is proportional to the power 1.6 of this difference. It is interesting to compare the experimental variation of hT with the temperature difference between the center and the wall of the column (∼ T1.6 ) to that theoretically predicted by the Aris’s model for small T (∼ T2 ). Both approaches agree well but the empirical equation illustrates the degree of approximation of the Aris’s model [18]. 4.6. Influence of tube thermal conductivity on column efficiency

Fig. 8. Calculated dimensionless concentration profile of naphtho[2,3-a]pyrene inside a thermostated column for T = 299 K. Calculation made for column 5 cm × 0.21 cm and mobile phase velocity of acetonitrile–water equal 1.5 ml/min at column inlet.

outlet is also reported. The value of hexp was estimated from the equation hexp = L/(N × dp ), where the number of theoretical plates was calculated using the method of moments. At small velocities, the temperature gradient is negligible and these plate height values should be similar but they are not. The observed differences between hz and hexp are due to the impossibility to match accurately the idealized shapes of the calculated peak profiles to the small tailing that is clearly visible in the experimental bands. The value of hexp is almost equal to that of hz at the smallest mobile phase velocity, when the number of theoretical plates is calculated 2 from the equation N = 5.54(tr /w12 ) where tr is the retention time and w12 is the peak width at half peak height. The dependency of the reduced plate height hz on the mobile phase velocity Fv resembles the Van Deemter relationship – at low mobile phase velocities, hz decreases with increasing velocity while it increases at high velocities. For medium values of Fv , however, a small local maximum is observed (between 0.5 and 1 ml/min). This maximum is most probably due to a lack of accuracy of the Da,z estimate. For mobile phase velocities larger than about Fv = 0.8 ml/min, an increase or decrease of the apparent dispersion coefficient Da,z by about 20–30% slightly affects the calculated band profiles. More over the error made in estimating Da,z may be due to the impossibility to match exactly, point by point, the experimental and calculated bands. Despite this problem, it is clear from Table 4 that the difference between hexp and hz increases strongly with increasing mobile phase flow rate. This difference is indirectly related to the increase of the radial temperature gradient and is denoted hT . For mobile phase velocities larger than about 0.005 m/s (corresponding to a pressure drop larger than 550 bar), the column efficiency is mainly determined by hT the contribution of which to hexp exceeds 90%. For the chromatographic system discussed here, the experimental value of the plate height is correlated to the radial temperature difference at the column outlet and to the axial plate height by the following empirical equation (valid for u > 0.002 m/s). hexp = hz + hT ;

hT = 7.81(Tc − Tw )1.61

As stated in Section 4.3, inside a column operated under natural convection conditions, there are a positive axial temperature gradient and a positive radial mobile phase velocity gradient at the column inlet. There are also negative gradients of these two parameters at the column outlet. A loss of column efficiency is due to the positive radial gradient of the mobile phase velocity being predominant along the first part of the column length. This loss is compensated to a degree by the negative radial gradient that predominates along the last part of the column length. The magnitudes of these two gradients depend on the thermal conductivity of the column tube. Accordingly, the efficiency of columns operated under natural convection conditions should also depend on the thermal conductivity of their tubes. It becomes interesting to find out whether the influence of the two gradients on the column efficiency could compensate each other totally for some value of this thermal conductivity. To answer this question, the temperature distributions along and across the column bed and the column efficiency were calculated for the following conditions: the column length is 5 cm, the inlet pressure is 808 bar, a 85/15 (v/v) aqueous solution of acetonitrile is used as the mobile phase, the inlet mobile phase velocity is 1.5 ml/min, the Blake–Kozeny–Carman constant is 150, the solute used is—naphtho[2,3-a]pyrene, the axial and radial dispersion coefficients were: 8.5 × 10−8 m2 /s and 1.4 × 10−9 m2 /s. Fig. 9 illustrates the dependency of the calculated number of theoretical plates on the heat conductivity of the column tube, from w = 0.5 W/m/K to w = 402 W/m/K (the conductivity of copper). The upper line depicts the results of the calculations made when ignoring the endfittings contribution to heat losses and the lower line those taking them into account. The column efficiency is highest for w = 0 and reaches 3122 theoretical plates. In both cases, when the tube heat conductivity increases, the column efficiency quickly

(23)

Because along about 80% of the column length (see Fig. 5a), the radial temperature gradient across each cross-section of the column is similar to the one observed at the column outlet, we can

Fig. 9. Dependency of the calculated number of theoretical plates on the thermal conductivity of the column tube. Upper line – endfittings were ignored, lower line – endfittings were taken into account.

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586

6585

The results of our experiments and calculations show that, when columns are operated under natural convection conditions, the difference between the temperatures at the inlet and the outlet of the column can reach up to 20 K and the difference between the temperatures in the center of the column and near its wall up to 6 K. The radial temperature gradient forms slowly along column. When the column is immersed in a stream of water at a constant temperature equal to that of the incoming mobile phase the radial temperature gradient forms rapidly at the beginning of the column and remains nearly constant afterwards. The calculated concentration profiles of bands eluted from a column immersed in a water bath at the temperature of the mobile phase entering the column agree very well with those recorded experimentally, in the whole range of inlet pressures and mobile phase velocities that could be used. These results prove the validity of the proposed heat and mass transfer model. Fig. 10. Comparison of the temperature distribution along the column wall (Tw ) and along the column center (Tc ). Solid line—w = 0 W/m/K, dashed line—w = 20 W/m/K, dotted line—w = 402 W/m/K.

decreases at first, then it increases to local maximum reached for w = 10–20 and finally it decreases. It should be noticed that, in the cases studied, the local maximum of the column efficiency is reached for the conductivity of stainless steel. With stainless steel (w = 16 W/m/K), the loss of column efficiency when the endfittings are ignored is about 4% whereas for copper (w = 402 W/m/K), it is 20%. Fig. 10 explains the reason of this behavior. When w = 0, there is no radial temperature gradient anywhere in the column, so there is no loss of efficiency. With increasing wall heat conductivity, the magnitudes of the positive and the negative radial temperature gradients increase, the average value of the negative gradient being larger than that of the positive gradient. The influences of both gradients on the column efficiency compensate to a different degree with increasing w . Finally, when w is greater than about 20 the column efficiency decreases with increasing w . It is worth noting that with increasing tube thermal conductivity, the axial gradient of the wall temperature decreases and that, for w = 402 W/m/K, the wall temperature is nearly constant all along the column. 5. Conclusions The original model of the behavior of VHPLC columns that we developed combines the numerical solutions of the heat and the mass balance equations with the pressure and temperature dependence of the equilibrium constants of analytes, and the equations of flow through porous media. This model provides the distributions of the local temperature and pressure and uses them to derive their influence on the local values of the physico-chemical parameters of the phase system of the chromatographic column, i.e., the viscosity, density and heat capacity of the mobile phase, its velocity and the Henry constant. The model was validated by comparing the calculated and the experimental temperature profiles along four columns and the calculated and recorded concentration profiles of peaks eluted from these columns. These 3, 5, 10 and 15 cm long columns were packed with 1.7 ␮m particles, eluted with pure acetonitrile or with an acetonitrile–water mixture at inlet pressures up to 1000 bar, and operated under two sets of experimental conditions, the columns being either immersed in a thermostated water bath or left in a closed box under natural convection conditions. The elution profiles provided by the calculations were compared with those provided by the experiments under the same sets of conditions.

Nomenclature

cpm cs cw C dp Dz,a DC Dr,a Dm E F Fv g h hcon he hrad hv H Hr K L N P q R tr T Text Tc T0 Tw Tther u Vm w

mobile phase heat capacity solid heat capacity wall heat capacity concentration in the mobile phase adsorbent diameter axial apparent dispersion coefficient external diameter of the column wall radial apparent dispersion coefficient molecular diffusion coefficient activation energy phase ratio volumetric mobile phase flow [ml/min] gravity acceleration reduced plate height convective heat transfer coefficient external heat transfer coefficient radiation heat transfer heat generated per unit volume due to viscous friction Henry constant radial plate height equilibrium constant column length number of theoretical plates pressure concentration in the stationary phase gas constant retention time temperature external temperature temperature in the column center inlet eluent temperature column wall temperature thermostat temperature superficial velocity partial molar volume interstitial velocity

Greek symbol ˛ coefficient of thermal expansion εe external porosity εt total column porosity  viscosity r,ef effective bed conductivity wall conductivity density w

specific volume  parameter in the Blake–Kozeny–Carman correlation

6586

K. Kaczmarski et al. / J. Chromatogr. A 1216 (2009) 6575–6586

Subscripts w wall ext external Acknowledgment This work was partially supported by grant N N204 002036 of the Polish Ministry of Science and Higher Education. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

I. Halasz, R. Endele, J. Asshauer, J. Chromatogr. 112 (1975) 37. H. Poppe, J.C. Kraak, J.F. Huber, Chromatographia 14 (1981) 515. H.-J. Lin, C. Horvath, Chem. Eng. Sci. 56 (1981) 47. H. Poppe, J.C. Kraak, J. Chromatogr. 282 (1983) 399. T. Welsch, M. Schmid, J. Kutter, A. Kálman, J. Chromatogr. A 728 (1996) 299. A. Brandt, G. Mann, W. Arlt, J. Chromatogr. A 769 (1997) 109. O. Dapremont, G.B. Cox, M. Martin, P. Hilaireau, H. Colin, J. Chromatogr. A 796 (1998) 81. A. De Villiers, H. Lauer, R. Szucs, S. Goodal, P. Sandra, J. Chromatogr. A 1113 (2006) 84. D.T.T. Nguyen, D. Guillarme, S. Heinisch, M.P. Barrioulet, J.L. Rocca, S. Rudaz, J.L. Veuthey, J. Chromatogr. A 1113 (2006) 84. F. Gritti, G. Guiochon, J. Chromatogr. A 1131 (2006) 151. G. Desmet, J. Chromatogr. A 1116 (2006) 89. U.D. Neue, M. Kele, J. Chromatogr. A 1149 (2007) 236. F. Gritti, G. Guiochon, J. Chromatogr. A 1138 (2007) 141.

[14] [15] [16] [17] [18] [19] [20] [21]

[22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

[33] [34] [35]

F. Gritti, G. Guiochon, Anal. Chem. 80 (2008) 5009. F. Gritti, G. Guiochon, J. Chromatogr. A 1187 (2008) 165. K. Kaczmarski, F. Gritti, G. Guiochon, J. Chromatogr. A 1177 (2008) 92. F. Gritti, G. Guiochon, Anal. Chem. 80 (2008) 6488. F. Gritti, M. Martin, G. Guiochon, Anal. Chem. 81 (2009) 3365. M.M. Fallas, M.R. Hadley, D.V. McCalley, J. Chromatogr. A 1216 (2009) 3961. F. Gritti, G. Guiochon, J. Chromatogr. A 1216 (2009) 1353. K. Kaczmarski, J. Kostka, W. Zapała, G. Guiochon, Modeling of thermal processes in high pressure liquid chromatography: I. Low pressure onset of thermal heterogeneity, J. Chromatogr. A 1216 (2009) 6560. R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, 2002. A. Bejan, A.D. Kraus, Heat Transfer Handbook, John Wiley & Sons, 2003. GreenF D.W. (Ed.), Perry’s Chemical Engineers’ Handbook, Seventh ed., McGraw-Hill, 1997. G. Guiochon, A. Felinger, A.M. Katti, D. Shirazi, Fundamentals of Preparative and Nonlinear Chromatography, Second ed., Elsevier, Amsterdam, 2006. K. Kaczmarski, G. Storti, M. Mazzotti, M. Morbidelli, Comput. Chem. Eng. Sci. 21 (1997) 641. P.N. Brown, A.C. Hindmarsh, G.D. Byrne, available at http://www.netlib.org. M. Martin, G. Guiochon, J. Chromatogr. A 1090 (2005) 16. J. Thompson, T. Kaiser, J. Jorgenson, J. Chromatogr. A 1134 (2006) 201. Y.P. Zarichnyak, V.V. Novikov, Inzhenerno-Fizicheskii Zhurnal 34 (1978) 648. S. Abbott, P. Achener, R. Simpson, F. Klink, J. Chromatogr. 218 (1981) 123. K. Broeckhoven, J. Billen, G. Desmet, K. Choikhet, G. Rozing, About the effects of ultra-high pressure on performance in HPLC, in: HPLC 2007, Ghent, Belgium, 2007. D. Horne, J.H. Knox, L. McLaren, Sep. Sci. 1 (1966) 531. J.H. Knox, G.R. Laird, P.A. Raven, J. Chromatogr. 122 (1976) 129. K. Miyabe, N. Ando, G. Guiochon, J. Chromatogr. A 1216 (2009) 4377.