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ARTICLES The Change of Pressure Drop during Large-Scale Chromatography of Viscous Samples Attila Felingert and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1501, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6120
The variation of the inlet pressure during the injection and the elution of a band of a viscous compound, or pressure-drop profile, was calculated in overloaded elution chromatography, assuming a significant difference between the viscosities of the solvent and the pure solute. The viscosity of the mixture was calculated using the GrunbergNissan equation. By means of the numerical integration of the mass balance equation, the concentration profile along the length of the column was calculated. Then, using the instantaneous concentration profile to derive the local viscosity, hence the local pressure gradient, the pressure drop was determined by integrating the local pressure gradient along the column. The change of pressure drop calculated is significant only when the concentration of the injected sample is very high. If a dilute sample is injected, even large sample volumes will not really change the pressure drop. The maximum pressure drop observed in each case is practically independent of the absorption isotherm. The only determining factors are the solute concentration in the sample, the amount of sample injected, and the solute diffusivity.
Introduction It is now generally recognized that in order to achieve reasonable values of production rate in preparative chromatography, one has to operate the columns under conditions of serious concentration overload. To achieve a sufficient degree of concentration overload, however, the injection of highly concentrated sample solution is required. In order to achieve reproducible results and properly control the operation of a preparative chromatographic column, including the switching of the fractionation valves at the correct times, a constant mobile-phase flow rate should be maintainedthroughoutthe cycle. When the dynamic viscosities of the main components of the sample and of the solvent differ markedly, as is usually the case, the pressure drop raises sharply during the injection of the sample and then decreases more or less slowly during sample migration along the column and returns to ita initial value during the exit of the band. This 1990) phenomenon has been observed and reported (Mann, but not yet studied, to the best of our knowledge. The behavior of a viscous sample plug migrating along a porous packed channel through which it is pushed by a less viscous eluent is unstable (Saffman and Taylor, 1958; Greenkorn, 1983). The deformations of the boundary between the more viscous sample solution and the less viscous mobile phase caused by local inhomogeneities of the packing material are self-amplified. They may eventually lead to fingering or even to the breakup of the sample plug into several smaller ones (Saffman and Taylor, 1958;
* Author to whom correspondence should be addressed at the
University of Tennessee. + On leave from the Department of Analytical Chemistry, University of VeszprBm, VeszprBm H-8201, Hungary. 87567938/93/3009-0450$04.00/0
Greenkorn, 1983; Czok et al., 1991). Although this phenomenon has been reported and amply documented in size exclusion chromatography (Czok et al., 1991), it does not seem to be as detrimental in adsorption chromatography, probably because in most cases the equilibrium isotherm is convex upward. Then, under overloaded conditions, the front boundary is a shock layer but is hydrodynamically stable, with the more viscous fluid pushing the less viscous one. The rear boundary, which is hydrodynamically unstable, is diffuse and flow instabilities do not arise. Nevertheless, the possibility that such a phenomenon takes place should be kept in mind, and as far as possible, one should avoid the use eluents which are much less viscous than the sample (Czoket al., 1991). One way to study and monitor the behavior of viscous bands in liquid chromatographyis to follow the variations of the inlet pressure during the course of a cycle. In this work, we report on the resulta of calculationsof the pressure drop of the column, under different experimental conditions, during a separation cycle in elution chromatography. For the sake of simplicity, we consider a pure component as sample and assume that the flow rate of the mobile phase remains unchanged. In practice, if the initial pressure rise at the end of the injection is too high, the flow rate will be lowered during the first part of the cycle and raised to another value at a set time.
Theory Flow of Liquids through Porous Materials. Darcy’s law is the basic equation to describe the flow of fluids. It states that the local flow velocity is directly proportional to the pressure gradient (Greenkorn,1983;Carman, 1956).
0 1993 American Chemical Society and American Institute of Chemlcal Engineers
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It can be written as
where u is the flow velocity, B is a numerical coefficient depending on the column permeability and the mobilephase viscosity,p is the pressure, and z is the length. Given that the flow rate is inversely proportional to the fluid viscosity and the permeability is proportional to the square of the average size of the stationary-phase particles, we can write eq 1 as
where ko is the specific column permeability (ca. 1X d, is the average particle diameter of the stationary phase, and q is the dynamic viscosity of the mobile phase. In liquid chromatography, the compressibility of the mobile phase is very small, and under steady-state conditions the equation of continuity of the mobile phase can be integrated to give u = uo = constant, where uo is the outlet velocity. When the viscosity of the sample and that of the mobile phase do not differ significantly, eq 2 can be integrated and the total pressure drop can be calculated as (3)
When the solute and mobile-phase viscositiesare different, the viscosityof the mixture will depend on the composition of the mobile phase and the viscosities of the pure components. The mobile-phase viscosity will be a function of time and location, and the pressure drop at constant flow rate will vary during the elution of the sample band. The Viscosity of Liquid Mixtures. Many models have been proposed to calculate the viscosity of liquid mixtures from the viscosities of the pure components. These models rely on interpolation. The most recommended method for calculating mixture viscosities is the Grunberg-Nissan equation (Grunberg and Nissan, 1949; Reid et al., 1987). This one-constant equation is reasonably accurate, except for aqueous solutions. It gives the viscosity of low-temperature liquid mixtures as
where t i and qm are the viscosities of pure compound i and of the mixture, respectively, x is the mole fraction, and Gij is an interaction parameter, a function of temperature which depends on the components i and j (Gii = 0). The constant Gij can be determined from experimental data, or for binary mixtures it can be calculated using a group contribution method (Reid et al., 1987). Teja and Rice (1981)proposed an analogous relationship to calculate the viscosity of liquid mixtures, based on a corresponding states treatment for the mixture compressibility factors. This method is more accurate than the previous one for polar-polar mixtures, particularly for aqueous solutions. However, in this work-since no specifications were made for particular compounds-we used eq 4 and ignored the excess term. In this manner, eq 4 becomes identical to the one proposed by Arrhenius in 1887 for binary mixtures: In qm = x1 In q1 + x z In qz (5) Although the results obtained with this equation might
not account for the behavior of any particular mixture, they will give an excellentapproximation of the phenomena which actually take place in a column upon the injection of a viscous feed. Band Profile Calculations. Numerical integration of the system of mass-balance equations of chromatography permits the calculation of the band profiles at the column outlet, as well as the concentration profiles of each component along the column during elution (Rouchon et al., 1987; Guiochon et al., 1988). We have used the equilibrium-dispersive model of chromatography, assuming near equilibrium between the two phases and using an apparent dispersion term to account for the finite rate of mass transfer (Guiochon et al., 1988). The mass-balance equations for each component of the sample are written as
aci
aqi
aci
a%,
-+ F -+ u -= D,at at az az2 where Ci and qi are the concentrations of component i in the mobile and stationary phases, respectively, t is the time, F is the phase ratio, (1- €)/e, e is the total porosity, and D, is the apparent dispersion coefficient. This system of partial differential equations is solved using a finite difference scheme written for eq 6 with D, = 0 (ideal model) (Guiochon et al., 1988; Czok and Guiochon, 1990). The values of the time and length increments of the integration are chosen in order to provide a numerical dispersion exactly equal to the desired apparent dispersion (Czok and Guiochon, 1990):
At =
W ( k ' + 1) U
(7b)
Although it is easy to calculate the individual band profiles of the components of a complex mixture and the pressuredrop profile during the elution of this mixture, we have performed the calculations with only a single component. For a given total sample size, the effect calculated is most important if the sample contains a pure component, and the elution band is less widely spread out. Calculation of the Pressure Profile. Since the viscosity of a mixture is a nonlinear function of the pure component viscosities, the pressure drop in the column depends both on the amount of viscous solutes in the column and on their concentration profiles. If it is assumed that the flow rate is kept constant through the column, the numerical integration of the mass-balance equation (eq 6) gives the composition profile of the mobile phase along the column length at any time. From this profile, the viscosity profile is easily derived. Knowing the local viscosities of the mobile phase, we can calculate the local pressure drop:
The s u m of the local pressure drops gives the total pressure drop through the column: UAz
Ap = -C q m , i
kodE r=l
where N = L1Az = LIH is the number of theoretical plates in the column. The profile of the pressure drop versus time during the elution can be obtained as we repeat the above calculations with At time increments.
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x 6
5 N
4 0 Q
Q
\
Q
3
Q
Ir! c
m
I 0
200
400
C (mg/mL) Figure 1. Viscosity of the mixture. Plot of the dynamicviscosity of the mixture against the concentration of the sample.
Column Characteristics. Through all of the calculations, we used a typical 25 cm long column paqked with particles of average diameter d , = 20 pm. The total column porosity is e = 0.8, and the molecular diffusivity of each component in the mobile phase is Dm = 1 X 105 cmz/s. The column efficiencywas calculated using the Knox plate height equation
where h = H/d,is the reduced plate height, H is the actual height equivalent to a theoretical plate, and Y = ud,/D, is the reduced mobile-phase velocity. Adsorption Isotherms. In all of the calculations, we assumed that the component of the sample behaved according to the Langmuir isotherm model
where a and b are numerical coefficients. In some calculations,the numericalvalues of these coefficientswere varied in order to study the influence of the column saturation capacity. However, unless noted otherwise, the parameters were a = 24 and b = 0.1 mL/mg. Other Specifications. The viscosities of the solute and the solvent were chosen as 91 = 50.0 CPand vz = 0.345 cP, respectively. The molecular weights were A 4 1 = 500 and M Z= 41, and the density of the solution was p = 0.8 g/mL. The mobile-phase flow rate was 1.0 mL/min for a 4.6 mm i.d. column, corresponding to a velocity of 0.125 cm/s.
Results and Discussion As the viscous sample is pumped into the column under a constant flow rate, the back pressure increases nearly linearly. A maximum in the column pressure drop is reached at the end of the injection. When the entire sample
0
100
200
300
c,
400
500
600
("L)
Figure 2. Maximum pressure at the end of an injection. Plot of the maximum pressure drop, achieved at the end of the injection, against the concentration of the injected sample, for different sample volumes. Values calculated by eq 13 (-) and by numerical integration (0). Sample volume: 1,0.5 mL; 2,l.O mL; 3, 1.5 mL; 4,2.0 mL; 5, 2.5 mL; 6,3.0 mL. has been pumped into the column, the injected zone of viscous sample broadens during its migration along the column, due to the axial dispersion, and becomes less concentrated. Since the viscosity-concentration relationship is strongly nonlinear, as shown in Figure 1, one has to know the concentration distribution of the viscous sample along the length of the column to calculate the pressure drop. On the other hand, due to this strong nonlinearity, when the solute concentration is modest, the viscosity of the mixture is only slightly increased. However, the maximum pressure drop reached at the end of the injection can be estimated simply by neglecting the amount of axial dispersion which takes place during the injection. If the volume of injected sample is V,, it will occupy a plug whose length is lp = VdSt in the column. The rest of the column, (L - lp), is filled with the pure solvent. The viscosity of the sample can be calculated from its concentration (12) vo = exp(xo In vl + (1- xo) In vz) where xo is the mole fraction of the sample injected. The maximum pressure drop is flm,
= ~ U(
~
0
+2 vz(L - 1,)) ,
(13)
Because axial dispersion proceeds at a finite rate, the pressure profile calculated while axial dispersion is neglected should be slightly higher than the one actually recorded or the one predicted when the axial dispersion is taken into account. In the derivation of eq 13, the adsorption of the sample component on the stationary phase during the injection
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453
0
5
time (min)
time (min) Figure 3. Pressure-drop profiie during a cycle. Pressure-drop profile during a cycle (top) and concentration profiie at elution (bottom)obtained upon injection of 1.0-mLsamples of solutions and 600 (-) mglmL. with concentrations of 200 (- - -), 400 ("a),
was neglected. This simplificationis reasonable,however, because in the present calculationswe assume a high degree of column overload. Thus, the retention time of the front is quite short As a matter of fact, in the examplediscussed, it is barely longer than the dead time of the column, to = 3.3 min, while the retention time of the same compound at infinite dilution would be t~ = 23.3 min. In addition, as we will see later, the adsorption isotherm, and especially the column saturation capacity, has practically no effect on the maximum value of the inlet pressure. In Figure 2, we plot the maximum pressure drop versus the sample concentration for different sample volumes. The maximum pressure drop was calculated as explained above, using either the eq 13 (solid lines) or a numerical integration of the mass-balance equation followed by the summation of the local pressure drops (symbols), as explained in the previous section. As the axial dispersion was neglected in the calculation leading to eq 13, the estimated pressure drop is always higher than that obtained by numerical integration. Nevertheless,the data in Figure 2 show that the difference is minimal. In Figure 2 and all of the following figures, we do not use the true inlet pressure, but instead use the pressure relative to the initial column back pressure (i.e., the inlet pressure when only pure solvent is pumped through the column). As eq 9 shows clearly, the linear mobile-phase velocity is directly proportional to the column back pressure and inversely proportional to the square of the particle diameter. Thus, the results can be conveniently expressed in nondimensional units. A change in the mobile-phaseflow rate of the particle size would not change the shape of the inlet pressure profiles. In this manner,
Figure 4. Influence ofthe sample concentration on the pressuredrop profiie. Pressure-drop profile during a cycle (top) and concentra$ionprofile at elution (bottom)obtained upon injection of 3.0-mL'samples of solutions with concentrations of 67 (- - -), 133 and 200 (-) mg/mL. the plots of the relative increases of the inlet pressure are independent of these characteristics. We see in Figure 2 that one has to overload the column considerably to observe a significant back pressure increase. Because of the nonlinear relationship between viscosity and concentration, the volume overload itself does not lead to a high increase in the pressure drop; only whena strong volume overload is accompanied with a high concentration overload can a large increase in the pressure drop be observed. For instance, in the case illustrated in Figure 2,the pressure drop increases by only 7 % when a 1.0-mL spnple of a 300 mg/mL solution is injected, but it increases by 42% when the same volume of a 600 mg/ mL solution is injected. However, in most cases, the solubility of the sample does not allow this amount of concentration overload, and consequentlyonly a moderate pressure increase can be noticed. In Figure 3, some typical pressure profiles (top) are compared with the corresponding chromatograms (bottom). The volume of the sample remained the same in all three cases, 1.0 mL, but three different concentrations were used 200,400,and 600 mg/mL, respectively. The maximum pressure drop when the sample is pumped into the column increases almost linearly with increasing concentration. The rate of this increase is determined by the sample Concentration. When the sample zone migrates along the column, the pressure drop decreases as a consequence of the increased axial dispersion and the broadened zone. Finally, there is a rapid decrease in the pressure drop when the sample starts to elute from the column. Note that the injection of 1.0 mL of a 200 mg/mL sample results in an already significant degree of overload (the loading factor is loo%),but in that case the increase (-e),
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m
I
7
0 0 (0
n o
$3
\
E
-0
U S
I
5
time (min) Figure 6. Influence of the sample volume on the pressure-drop
profile. Pressure-drop profile during a cycle (top) and concentration profile at elution (bottom) obtained upon injection of 1.0-(- -), 2.0and 3.0-mL (-1 samples of a solution with a concentration of 200 mg/mL.
-
(m-),
in the pressure drop is only 3 5%. However, as we will show, the adsorption isotherm itself has no real influence on the maximum pressure drop. It is the volume and the concentration of the sample that determine the change of pressure drop. In Figure 4, we show the results of the same calculations as those performed for Figure 3 using the same values of the loading factor, but this time 3 times larger samples which are 3 times less concentrated are used. The volume injected was 3.0 mL, and the concentrations were 67,133, and 200 mg/mL, respectively. In this instance the sample occupies a much larger fraction of the column. Thus, the maximum pressure drop is lower than in Figure 3, but the pressure drop decreases more slowly during the migration of the sample plug. We also note that, when the band which exits from the column still exhibits a flat concentration plateau due to a considerable degree of volume overload, a linear decrease in the pressure drop during the elution of this plateau is observed; the absolute value of the slope of the pressure profile is then similar to the one observed during the injection. When we compare the curves in Figures 3 and 4 corresponding to the same values of the loading factor, we see that, while the pressure drop increases by only 10% when a 3.0-mL sample of 200 mg/ mL is injected, the pressure drop increases by 41% if a 1.0-mLsample of 600 mg/mL is injected. This is obviously a consequence of the strongly nonlinear relationship between viscosity and concentration. The effect of a strong degree of volume overload is illustrated in Figure 5, which shows the concentration profiles at column exit and the pressure-drop profiles calculated upon injection of LO-,2.0-, and 3.0-mL samples of a 200 mg/mL solution. We observe that the maximum
0
0
5
10
time (min) Figure 6. Influence of the isotherm on the pressure-drop profile. Pressure-drop profile during a cycle (top) and concentration profile at elution (bottom) obtained upon injection of a 1.0-mL sample of a solution with a concentration of 600 mg/mL on stationary phases with saturation capacitiesof 120 (-), 240 (-), and 480 (- - -) mg/mL, respectively.
pressure drop is proportional to the volume injected. The sample is too dilute in this case, however, to generate large pressure-drop changes (the relative value of the maximum pressure drop calculated is 10%). The adsorption isotherm has no significant influence on the maximum value of the pressure drop, but affects the pressure-drop profile during sample migration. In Figure 6, we show the results of similar calculations performed with different values of the saturation capacity of the stationary phase, qs = 120, 240, and 480 mg/mL, while keeping constant the retention factor (k’= 6), the volume of the sample (1.0 mL), and its concentration (600 mg/mL). The slope of the rising pressure-drop profile during the injection is practically independent of the isotherm. Increasing or decreasing by a factor 2 the saturation capacity barely changes the maximum pressure drop observed (no more than f2 7%). For this reason, since the maximum value of the inlet pressure is almost independent of the adsorption isotherm, it can be estimated fairly well be eq 13. On the contrary, the pressure drop falls markedly faster during the migration and elution of the zones when the column saturation capacity is higher. Finally, the molecular diffusion coefficient of the compound injected in the mobile phase has a marked influence on the pressure-drop profile observed during the experiment. For example, increasing this diffusion coefficient 5-fold, to D, = 5 X 10-5 cm2/s, significantly changes the pressure-drop profile, which becomes less steep, with a maximum pressure drop reduced by 7 76. The assumption of a 5 times smaller diffusivity ( D , = 2
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However, as the adsorption isotherm itself has no significant influence on the maximum value of the pressure drop, the increase of the pressure drop observed at the end of the injection cannot be characterized by the loading factor. Furthermore, the loading factor does not distinguish between volume and mass overload, whereas the maximum pressure drop depends strongly on the amount and the concentration of the compound injected. The diffusion coefficient has a marked impact on the pressure-drop profile. If the diffusivity is high, the sample plug disperses significantly faster, and band spreading during the injection can result in a wider band and a much smaller pressure-drop increase.
Acknowledgment This work has been supported in part by Grant CHE9201662 from the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. We acknowledge support of our computational effort by the University of Tennessee Computing Center.
Literature Cited
5
0
10
time (min) Figure 7. Influence of the solute diffusivity on the pressuredrop profile. Pressure-drop profile during a cycle (top) and concentration profile at elution (bottom) obtained upon injection of a 1.0-mL sample of a solution with a concentration of 600 mg/mL. Diffusion coefficient of the solute in the mobile phase: 2 x 1V (-); 1 X 106 5 X 1od (- - -1 cm%. (e..);
X l W cm2/s) results in a steeper pressure-drop profile having a 2% higher maximum value.
Conclusion To observe a significant change in the pressure drop during the separation and elution of viscous samples, there must be a large difference between the viscosity of the pure components and that of the solvent. Since the dependence of the mixture viscosity on the concentration is strongly nonlinear, a significant change in the inlet pressure can be observed only under experimental conditions corresponding to a serious degree of concentration overload. In fact, such a high degree of overload is required that in most cases it cannot be achieved because of the solubility limits of the sample.
Carman, P. C. Flow of Gases Through Porous Media; Academic Press: New York, 1956. Czok,M.; Guiochon, G. The Physical Sense of Simulation Models of Liquid Chromatography: Propagation through a Grid or Solution of the Mass Balance Equation? Anal. Chem. 1990, 62,189-200.
Czok, M.; Katti, A. M.; Guiochon, G. Effect of Sample Viscosity in High-Performance Size-ExclusionChromatography and ita Control. J. Chromatogr. 1991,550,705-719. Greenkorn, R. A. Flow Phenomena in Porous Media; Marcel Dekker: New York, 1983; pp 220-225. Grunberg, L.; Nissan, A. H. Mixture Law for Viscosities. Nature 1949,164,799-800.
Guiochon, G.; Golshan-Shirazi, S.; Jaulmes, A. Computer Simulation of the Propagation of a Large-Concentration Band in Liquid Chromatography. Anal. Chem. 1988,60,1866-1866. Mann, G. Communication to PREPSO, Ghent, Belgium, April 1990.
Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hik New York, 1987. Rouchon,P.; Schonauer,M.; Valentin,P.; Guiochon,G. Numerical Simulation of Band Propagation in Nonlinear Chromatography. Sep. Sci. Technol. 1987,22,1793-1833. Saffman, P. G.; Taylor, G. The Penetration of a Fluid into a porous Medium or Hele-ShawCell Containing a More Viscous Liquid. R o c . R. SOC.London, Ser. A 1958,245, 312-329. Teja, A. S.; Rice, P. Generalized Corresponding states Method for the Viscosities of Liquid Mixtures. Znd. Eng. Chem.Fund. 1981,20,77-81.
Accepted March 5, 1993.
