Anal. Chem. 2005, 77, 3986-3992
Experimental Validation of the Tetrahedral Skeleton Model Pressure Drop Correlation for Silica Monoliths and the Influence of Column Heterogeneity Nico Vervoort,*,† Haruko Saito,‡ Kazuki Nakanishi,‡ and Gert Desmet†
Departement of Chemical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium, and Department of Material Chemistry, Graduate School of Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto 606-8501, Japan
This paper describes the use of computational fluid dynamics for the calculation of the flow resistance through computer-generated models resembling silica monoliths. This study was undertaken to determine the effect of skeleton heterogeneity on the flow resistance and, more precisely, to test the hypothesis that increased skeleton heterogeneity decreases the flow resistance. To evaluate the proposed model, 24 real silica monoliths have been prepared using the same method, covering a wide range of skeleton sizes (2.2 µm < ds < 8 µm) and porosities (0.47 < E < 0.66). The permeability of these monoliths was determined by pressure drop measurements, and structural information was obtained by image analysis of laser scanning confocal microscopy-generated 3D images of the skeleton structure. The results indicate that the presence of preferential flow paths due to an increased heterogeneity of the flow through pore space reduces the flow resistance of monolithic media. It is also shown that the pore size is hence a much better suited scaling dimension than the skeleton size to reduce the permeability of monolithic columns. The field of liquid chromatography has recently seen the successful introduction of a novel silica monolithic stationary phase material. This monolithic material was initially developed by Nakanishi and co-workers1-4 and is now commercialized by Merck (Darmstadt, Germany) as Chromolith columns and by Phenomenex (Torrance, CA) as Onyx column. Another type of monolithic column is the polymer-based columns, mostly either acrylate or styrene-divinyl based, which are, among others, commercialized by BIA Separations (Ljubljana, Slovenia) and LC Packings/Dionex (Sunnyvale, CA). This paper focuses entirely on silica monoliths * Corresponding author tel.: +32 (0)2/629.33.27, fax: +32 (0)2/629.32.48, e-mail:
[email protected]. † Vrije Universiteit Brussel. ‡ Kyoto University. (1) Nakanishi, K.; Takahashi, R.; Soga, N. J. Non-Cryst. Solids 1992, 147, 291295. (2) Minakuchi, H.; Nakanishi, K.; Soga, N.; Ishizuka, N.; Tanaka, N. Anal. Chem. 1996, 68, 3498-3501. (3) Minakuchi, H.; Nakanishi, K.; Soga, N.; Ishizuka, N.; Tanaka, N. J. Chromatogr. A 1997, 762, 135-146. (4) Minakuchi, H.; Nakanishi, K.; Soga, N.; Ishizuka, N.; Tanaka, N. J. Chromatogr. A 1998, 797, 121-131.
3986 Analytical Chemistry, Vol. 77, No. 13, July 1, 2005
since they are the most widely used monolithic-type column and have a fixed geometry, while polymeric columns can exhibit very different shapes, making it very difficult to establish a universal correlation applicable to every type of polymeric column. Silica monolithic columns differentiate themselves from particulate-packed columns by their increased flow permeability combined with excellent mass transfer properties. The interesting flow properties are a direct result of their unique, sponge-like geometry and the fact that the skeleton size and the size of the macropores (flow through pores) can be tailored independently of each other, allowing for the combination of large flow through pores and small skeleton sizes. This is in contrast to particulate columns where the size of the flow through pore is linked to the size of the particle. As a consequence, a reduction of the particle size inevitably leads to a reduction of the flow through pore size, increasing the flow resistance. The correct determination of the flow resistance and the establishment of a mathematical correlation linking the expected flow resistance to the column parameters such as the external porosity and the skeleton size are important because this piece of information is indispensable5 in calculating and predicting the optimal bed porosity and skeleton size of silica monoliths. Another reason for obtaining such a relation is that it would allow the determination of the geometrical column parameters, such as skeleton size and porosity, from the observed flow rate and pressure drop of a column with unknown geometry. For packed bed columns the equation describing the relation between the pressure drop, flow rate, and columns parameters is known as the Kozeny-Carman equation, which is given by:
usf )
∆Pdp 3 1 Kc (1 - )2 ηL
(1)
with Kc ) 180 for a packed bed of spheres. Unfortunately this relation does not accurately describe the flow properties of silica monoliths since the Kozeny-Carman relation is derived for a packed bed of spherical particles having a porosity of about 40% whereas silica monoliths have a markedly different geometry and have much higher porosities, up to 85%. (5) Gzil, P.; Vervoort, N.; Baron, G. V.; Desmet, G. Anal. Chem. 2004, 76, 67076718. 10.1021/ac0502798 CCC: $30.25
© 2005 American Chemical Society Published on Web 05/14/2005
This shortcoming of the Kozeny-Carman relation was already discussed in more detail in a previous study6 where a more suitable correlation between the external porosity and the flow resistance of silica monoliths was presented. The correlation was established with the aid of computational fluid dynamics (CFD) simulation software, capable of calculating the flow and pressure drop through a representative unit cell, termed tetrahedral skeleton model (TSM). This representative unit cell was designed in such a way that it mimics the geometry, such as the topology and pore connectivity, of silica monoliths. The TSM was also used to model the band broadening of silica monoliths.7,8 Using the TSM structures to model the flow through silica monoliths, the following equation, relating the flow resistance to the external porosity, was proposed:
φx ) 55
(1 - )
1.55
(2)
Comparison of the calculated flow resistances from the TSM with experimental values for real silica monoliths, taken from various sources in the literature, revealed that the flow resistance of the TSM model was markedly higher than the flow resistance of real silica monoliths.6 The observed deviation of the experimental flow resistance from the proposed TSM flow resistances was explained by the fact that the simulated TSM geometry is perfectly homogeneous whereas real monoliths have a more heterogeneous skeleton, and thus a more heterogeneous pore size distribution. Studies of the flow resistance of porous polymer monoliths9-11 showed that the pressure drop of these materials was also much lower than expected. Calculations made by these authors also showed that the low flow resistance indeed can be caused by pore size heterogeneity. The present study will further evaluate this hypothesis by simulating the flow and pressure drop through more heterogeneous TSM models. In contrast to the previous study,6 where the experimental data was collected from several different sources scattered through various literature sources, all data was now collected from self-synthesized and characterized monoliths. This ensures that all the experimental data are determined in a consistent way and that no differences in measuring methods or data interpretation can bias the results. The experimental data were gathered from a set of 24 silica monoliths, having skeleton sizes ranging from 2.2 to 8 µm and an external porosity varying between 0.47 and 0.66. All relevant geometrical data of the 24 silica monoliths have been summarized in Table 1. Also, the skeleton heterogeneity of these monoliths will be determined to see if monoliths having higher skeleton size variance also have lower flow resistance. Since it is inferred that the flow resistance is not only related to the average skeleton diameter but also related to the variance (6) Vervoort, N.; Gzil, P.; Baron, G. V.; Desmet, G. Anal. Chem. 2003, 75, 843850. (7) Vervoort, N.; Gzil, P.; Baron, G. V.; Desmet, G. J. Chromatogr. A 2004, 1030, 177-186. (8) Gzil, P.; Vervoort, N.; Baron, G. V.; Desmet, G. J. Sep. Sci. 2004, 27, 887896. (9) Hahn, R.; Jungbauer, A. Anal. Chem.2000, 72, 4858. (10) Zochling, A.; Hahn, R.; Ahrer, K.; Urthaler, J.; Jungbauer, A. J. Sep. Sci. 2004, 27, 819. (11) Mihelic, I.; Nemec, D.; Podgornik, A.; Koloini, T. J. Chromatogr. A 2005, 1065, 59-67.
Table 1. Geometrical Data of the 24 Silica Monoliths
dp_LSCM (µm)
ds_LSCM (µm)
0.467 0.470 0.514 0.515 0.516 0.534 0.534 0.534 0.547 0.549 0.551 0.558 0.559 0.566 0.576 0.579 0.584 0.587 0.588 0.614 0.627 0.635 0.648 0.664
7.05 6.96 2.90 3.58 4.38 3.31 3.40 4.16 7.51 8.65 3.73 3.28 3.79 4.34 4.38 6.24 9.35 8.07 5.37 6.74 4.53 3.71 6.25 4.35
8.06 7.86 2.92 2.87 3.79 3.04 3.24 3.88 6.21 7.11 3.09 2.80 3.13 3.36 3.20 4.53 6.65 5.68 3.58 4.24 3.02 2.20 3.39 2.23
dp_Hg (µm)
ds_SEM (µm)
3.20
3.81
2.90
4.15
3.65
4.63
8.72 7.36 4.70
9.95 6.22 4.21
3.57
3.09
of the skeleton thickness, a method was needed for the accurate determination of both the average value of the skeleton thickness and the variance. While determination of the average skeleton diameter is possible with mercury porosimetry, this technique does not allow for the determination of the variance of the skeleton diameter. A more suitable method for the determination of both the average value and the variance is by image analysis of the skeleton. As described in refs 12 and 13, 3D images of the skeleton can be generated by laser scanning confocal microscopy (LCMS). EXPERIMENTAL SECTION Monolithic Sample Preparation. First, poly(ethylene oxide) (PEO) (MW 100 000) and D-sorbitol as polyhydric acid were homogeneously dissolved in 1 M nitric acid. Then, 6.5 g of tetraethoxysilane (TEOS) was added under vigorous stirring in an ice-cooled condition. The weight ratio of the starting composition is TEOS:1 M NHO3:PEO10:D-sorbitol ) 1:1.23-1.54:0.12-0.16: 0-0.08. After 30 min of stirring, the resultant homogeneous solution was transferred into a polypropylene tube (i.d. 6 mm) and allowed to gel at 40 °C in a closed condition. After aging at the same temperature for 10 h, the resultant gel was immersed in water for 3 h in order to remove residual acids and solvent exchanged with 1.5 M aqueous urea for 3 h. The following hydrothermal treatment took 5 h at 110 °C in 1.5 M aqueous urea, and after 2 h of solvent exchange with water the obtained gels were dried and heat treated at 600 °C for 5 h in order to remove residual organics. The dried monoliths were then wrapped inside a Teflon shrink wrap tube, placed inside a 15 mm diameter polypropylene tube, and capped with Teflon connection fittings on both sides to allow (12) Jinnai, H.; Koga, T.; Nishikawa, Y.; Hashimoto, T.; Hyde, S. T. Phys. Rev. Lett. 1997, 78, 2248. (13) Jinnai, H.; Nishikawa, Y.; Koga, T.; Hashimoto, T. Macromolecules 1995, 28, 4782-4784.
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coupling of the columns to conventional HPLC equipment. The space between the shrink wrap tube and the polypropylene tube was then filled with epoxy and cured at 120 °C, after which the columns are ready for use. Pressure Drop Measurement. The pressure drop measurements were done at room temperature (20 °C), using pure water as the permeating fluid and using flow rates ranging from 0.1 to 2.0 mL/min. Since the pressure drop value on the read-out display of the pump contains system pressure in addition to column pressure, each of the reported pressure drop values is obtained by subtracting the pressure drop of empty column from the measured value for the packed column. The relationship between flow rate and pressure drop followed Darcy’s law, and the Darcy permeability (K) was calculated from the slope of the plot of the pressure drop versus the linear velocity. Determination of Geometrical Properties of the Real Monoliths. The structures and geometrical properties of the prepared gels were observed by scanning electron microscopy (SEM), Hg intrusion, nitrogen adsorption, and laser scanning confocal microscopy (LSCM). The skeleton thickness and pore size of the monoliths were determined by drawing a number of lines across the 3D image of the skeleton at different locations and under different angles. From the obtained line length distributions, the average value of the skeleton diameter could be calculated as well as the standard deviation or variance. The advantage of the line method over mercury porosimetry is that it allows for the determination of both the average skeleton diameter and the variance. The structures of the obtained gel skeleton surface were studied using LSCM (Carl Zeiss, LSM 510 Pa). For this purpose, the prepared gels were cut into 2.5 mm thick slices and immersed in a mixture of formamide, benzyl alcohol, and fluorescein with the same refractive index as that of the gel skeleton. This process is needed to allow the laser light to transmit the sample and to obtain a better contrast. A laser with a 488 nm wavelength was used as the excitation source for the fluorescein. A long pass filter (LP505) was installed in front of the photomultiplier in order to detect the fluorescent light (having a wavelength of approximately 519 nm), and an oil-immersed 63×/NA ) 1.40 (Plan-Apochromat, Carl Zeiss) objective was employed. The increment along the optical axis of the microscope perpendicular to the focal plane (∆z) used for observation was 0.35 µm. The lateral and axial resolutions (Rlateral and Raxial) were 0.178 and 0.461 µm, respectively. The laser was scanned in the lateral plane, measuring fluorescent intensity in a two-dimensional optically sliced image composed of N2 (N ) 512) pixels, where N is the number of pixels along the edge of the two-dimensional image. The LSCM images were digitized, and three-dimensional pictures were created by stacking the digitized 2D images. The porosity was measured by calculating the area of the pores in the series of digitized images. The characteristic length, the most probable value of the summation of pore diameter and skeleton thickness, was determined by the structure factor (S(q)), which was obtained from the fast-Fourier transformation of the series of digitized images averaged in the qx-qy plane. CFD Setup and Calculation Procedures. The homogeneous TSMs were created, as described in a previous paper,6 by intersecting a number of cylinders, having a length (ls) and diameter (ds) in a tetrahedral manner (Figure 1a). To estimate 3988 Analytical Chemistry, Vol. 77, No. 13, July 1, 2005
Figure 1. Example of a 3D computer generated model of the simulated homogeneous TSM1 (a) and the heterogeneous TSM2 model (b).
the effect of introducing a certain pore heterogeneity (or skeleton heterogeneity), two sets of artificial monoliths with a different degree of heterogeneity were created, and the flow resistance was calculated in a similar manner as that for the homogeneous TSM used in ref 6. The heterogeneous models were made by increasing the size of the model and by varying the diameter of several cylinders in a random manner (Figure 1b). In contrast to the homogeneous TSM (TSM1) simulated flow domain, consisting of only 7 tetrapods and 32 cylinders, which represent the smallest entity needed to reconstruct an infinite perfectly ordered monolith, the heterogeneous models (further referred to as TSM2 and TSM3) are composed of 78 tetrapods and a total of 255 cylinders, generating a computational cell approximately 12 times the size of the computational domain of the homogeneous model. The CFD simulations were carried out with a commercial CFD package, FLUENT (v.6). The software was installed on a PC with an Intel Pentium IV processor running at 2800 MHz and equipped with 2 Gb RAM. The grids were generated with GAMBIT (v.2) software, run on the same hardware configuration. The sidewall
of the model were set as “symmetry wall” (implying the use of a slip boundary condition for the calculation of the velocity field), the inlet plane of the fluid was set as velocity (specifying a fixed velocity of 1 cm/s at the inlet), and the outlet plane was set as pressure outlet (specifying a pressure of 1 atm at the outlet). Water, with a density of 1000 kg/m3 and a viscosity of 10-3 kg/ ms, was chosen as the working fluid. The corresponding Reynolds numbers (based upon the average skeleton diameter as the characteristic length) were typically of the order 0.001-0.1, such that it can safely be concluded that the flow conditions were always strictly laminar. The pore volume space was discretized with an unstructured tetrahedral grid having a cell count between 40 000 and 100 000 cells for the homogeneous models and a cell count of 300 000-1 000 000 cells for the heterogeneous model. All simulations were carried out using a second-order discretization scheme, the residual drop was at least 10-6, and it was checked that all numerical results were grid independent. From the average fluid x-velocity (ux) and pressure drop, the Darcy permeability can be calculated using Darcy’s law stating that for laminar flows through porous media:
Kx )
uxηL ∆P
(3)
The flow resistance can then be calculated as:
φx,s )
ds2 Kx
(4)
wherein ds is the diameter of the skeleton. In eq 4 the permeability is calculated based upon the average x-velocity of the fluid in the interstitial space, which can be calculated from
ux )
Q πR2ext
(5)
Determination of Pore Heterogeneity of the TSM Models. The degree of pore and skeleton heterogeneity of the TSM models and real monoliths were both determined by image analysis using the same line-based method as described in the Experimental Section. For the real silica monoliths, LSCM was used to create 3D images of the gel skeleton, and from these 3D images the skeleton and pore size were determined by drawing a number of random oriented lines across the image (Figure 2a). From the measured distribution of line lengths, the average value of the skeleton and pore diameter could be calculated as well as the standard deviation or variance. In the case of the TSM structures a similar technique was adopted, using the existing 3D computergenerated drawing files to prepare a series of 2D slices (Figure 2b). In these slices, a number of random oriented lines was subsequently drawn. The average pore and skeleton size and the standard deviation of the obtained line length distributions were calculated using IMAQ Vision Builder image analysis software. RESULTS AND DISCUSSION Determination of the Flow Resistance for the TSM and Real Silica Monoliths. The plots of the pressure drop versus
Figure 2. Illustration of the line drawing method applied to 2D slices generated from 3D computer drawings of a heterogeneous TSM model (a) and applied onto 3D image generated by LSCM method (b).
the linear x-velocity of the fluid (plot not shown) showed a linear relationship between these two parameters. This observed linear relationship is in full agreement with Darcy’s law for laminar flow through porous media, and thus the use of eqs 3 and 4 for the calculation of the permeability is valid. Since the calculation of the flow resistance is based upon Darcy’s law, the accuracy with which it is calculated is linked to the accurate determination of the linear flow velocity. Determination of the linear x-velocity (ux) for the TSM cases is very straightforward since this value is extracted from the CFD software itself, but for real silica monoliths, this is not the case. For real silica monoliths, and in chromatography in general, the linear velocity is usually determined by the common practice of measuring the residence time of a non-retained tracer compound. These measurements assume that the tracer plug is not retained by the column packing material but does diffuse into the porous skeleton of the stationary phase material, thus having a zone retention factor k0′′ * 0. Since the zone retention factor of the t0 marker is not equal to 0, the average x-velocity of the tracer is lower than the average interstitial x-velocity. This also means that the theoretical correlation (eq 2), which is based upon the use of the linear interstitial velocity (ux), cannot be used directly for Analytical Chemistry, Vol. 77, No. 13, July 1, 2005
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Figure 3. Flow resistance of homogeneous TSM model scaled upon u0 (s) velocity and ux velocity (- -) and experimental data obtained from real silica monoliths (O).
Figure 4. SEM picture of real silica monolith.
comparison with experimental data based upon u0. To allow direct use of eq 2 for comparison with experimental data, based upon u0, the obtained flow resistance correlation should be rescaled so that it is also based upon u0. Since the u0 value can be written as
u0 ) ux (1 + k0)
(6)
and k0′′ can be calculated as
k0′′ )
1- int
(7)
The flow resistance should hence be written as
(
φ0 ) φx 1 +
1- int
)
Figure 5. Ratio of pore size (dp) over skeleton size (ds) for real silica monoliths plotted versus external porosity.
(8)
Assuming the internal porosity (int) of the monoliths is about 0.5 and rescaling the calculated TSM flow resistance values according to eq 8, we see that the flow resistance curves shift to somewhat higher values, especially in the low porosity region (Figure 3). Since the value of the flow resistance depends on whether the ux velocity or the u0 velocity are used, care should be taken when evaluating flow resistance data using theoretical correlations. Determination of Geometrical Parameters. The skeleton and pore diameter for the real silica monoliths, determined through image analysis, yielded values that seem to be consistent with the dimensions observed from the SEM images taken from a selection of real monoliths (Figure 4). Furthermore, a plot of the ratio of the pore diameter (dp) over the skeleton diameter (ds) versus the external porosity (Figure 5) shows that the values for the real silica monoliths correlate quite well with the following relationship between dp/ds and the external porosity determined for the TSM in a previous paper7:
dp ) ds
x (
32 1 55 1 -
1.55
)
(9)
The agreement of the experimental relationship between dp/ds and with eq 9 suggests that the TSM model has a good topological resemblance to the real monoliths. Furthermore, the 3990 Analytical Chemistry, Vol. 77, No. 13, July 1, 2005
fact that it is clearly independent of the size of the monolith and also to a large extent of the degree of heterogeneity (TSM data and real monolith data coincide quite well), this geometrical relationship can be used with confidence to accurately estimate, for example, the skeleton size from a measurement of the other two characteristic geometrical features of a monolith (i.e., the external porosity and the pore size). The latter two can for example be determined using mercury intrusion. The availability of eq 9 then circumvents the need for an additional SEM measurement to determine the skeleton size. Table 1 shows that the values for the pore size determined through mercury porosimetry seem to be slightly underestimated when compared to the dimensions observed in SEM pictures. Comparing the pore size data obtained with LSCM with those obtained by Hg porosimetry, we see that the dp_Hg values are consistently smaller than those obtained using LSCM in the low porosity ( < 0.6) region but that for higher porosities the ratio of dp_Hg over dp_LSCM approaches unity (Table 1). For the simulated TSM structures, the dimensions of the skeleton and pore are of course exactly known, since these dimensions were a priori fed into the geometry generator. Other properties such as porosity, pressure drop, and fluid velocity were extracted from the CFD software using its standard post-processing capabilities and calculated in the same manner as in ref 6. Image analysis of both the LSCM-generated pictures of real silica monoliths and the computer-generated 3D pictures of the
Figure 6. Standard deviation of the pore size scaled according to the average pore size plotted vs external porosity.
Figure 7. Flow resistance data based upon u0 velocity and the skeleton diameter of experimental data (O), homogeneous TSM model TSM1 (s), and the two heterogeneous TSM models TSM2 (- -) and TSM3 (‚‚‚).
TSM model allowed for the calculation of the standard deviation of the pore and skeleton size value. Scaling these values according to the average skeleton size showed that for the real monoliths all the columns possessed a significant degree of heterogeneity. Plotting this ratio versus the porosity of the column (Figure 6) did not reveal any strong correlation, but from the few values in the high porosity ( > 0.6) region, we do see that these values are somewhat higher than the average value in the low porosity region ( < 0.6). This could suggest that the heterogeneity of the skeleton in the high porosity region increases or that for such high porosities the geometry of the skeleton has a different topology. To assess the amount or degree of heterogeneity that was built into the different TSM structures a similar analysis was performed which showed that the standard deviation/average skeleton size value was different for all three employed models: the homogeneous TSM1 model having the lowest value (0.16), the TSM2 model having a value of 0.38, and the TSM3 model having the highest value (0.47). Comparison of the Flow Resistance Data from Real Silica Monoliths and TSM. The flow resistances for both the simulated TSM and the real silica monoliths are plotted versus the column porosity in Figure 7. The flow resistance is in this case based upon the average u0 velocity and scaled according to the skeleton diameter (ds). From this figure it can be seen that the flow resistance data do follow the general trend predicted by the TSM,
Figure 8. Flow resistance data based upon u0 velocity and the pore diameter of experimental data (O), homogeneous TSM1 model (s), and the two heterogeneous TSM2 (- -) and TSM3 (‚‚‚).
but the values for the real silica monoliths are lower than those predicted by the homogeneous TSM model. Looking at the ratio of experimental flow resistance over the TSM flow resistance, we see that this ratio is more or less constant at an average value of about 0.5, except for porosities higher than 0.6 were the ratio decreases to a lower value. An important factor in the analysis of the flow resistance is the scaling dimension that is used. For the analysis of performance or flow resistance of packed beds the obvious scaling factor to use is the particle diameter, since this is the only relevant dimension present in the packed bed since all other dimensions, such as the pore size, can be directly related to the particle size. In monolithic media this is however not the case since there is no direct relation between the pore size and the skeleton size. The choice of a suitable scaling parameter is still the subject of discussion and various investigations.14-17 For the purpose of describing the flow resistance of monoliths, it is logical to use the pore size as the relevant scaling parameter, since this is the most important dimension governing the pressure drop and flow rate. Using the pore size as the scaling parameter, it is noted that the curve relating the flow resistance to the porosity becomes much flatter (Figure 8), suggesting that the pore size is indeed much more suitable as a scaling parameter than the skeleton size. Another advantage of using the pore size is that this parameter can be determined much easier than the skeleton size (e.g., by Hg porosimetry). Influence of the Pore Heterogeneity on the Flow Resistance of the TSM. The influence of the pore or skeleton heterogeneity on the flow resistance of the TSM was determined by simulating the flow and pressure drop through three different structures having different degree of heterogeneity. The first structure is a perfectly homogeneous structure (TSM1) whereas the second (TSM2) and third structures (TSM3) have been made more heterogeneous by varying the skeleton diameter at different locations in the model, the third model being more heterogeneous than the second. (14) Meyers, J. J.; Liapis, A. J. Chromatogr. A 1998, 827, 197. (15) Meyers, J. J.; Liapis, A. J. Chromatogr. A 1999, 852, 3. (16) Tallarek, U.; Leinweber, F. C.; Seidel-Morgenstern, A. Chem. Eng. Technol. 2002, 25, 1177. (17) Leinweber, F. C.; Tallarek, U. J. Chromatogr. A 2003, 1006, 207.
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The resulting flow resistances, based upon the average u0 velocity and scaled by the average skeleton diameter are plotted as a function of the model porosity in Figure 7, showing that the flow resistance of the perfectly homogeneous TSM is higher than the flow resistance of the more heterogeneous models and that the flow resistance decreases with increasing heterogeneity of the model (i.e., if not all flow paths have the same size or shape). The same trend is still observed when scaling the flow resistance upon the average pore size (Figure 8), and again the flow resistance value is more constant. Fitting the obtained flow resistance data, based upon average x-velocity (ux) yields the following generally applicable equation:
φx,p ) C
(1 - )
n
(10)
a gradual decrease of the flow resistance with increasing heterogeneity). These CFD calculations thus confirm the hypothesis that increased heterogeneity of the flow path reduces the flow resistance of porous media below that of the perfectly ordered TSM case. Scaling analysis also showed that reducing the flow resistance according to the pore size of the monolith results in more flatter curves in the plot of flow resistance versus porosity. The pore size is hence a much better suited scaling dimension than the skeleton size to reduce the permeability of monolithic columns. ACKNOWLEDGMENT The authors would like to acknowlegde Dr. Kris Pappaert from the CHIS department at the Vrije Universiteit Brussel for aiding with the image analysis of the structures. SYMBOLS
Taking n ) 0.21 as the value for the exponent (when fitting with both C and n as free fitting parameters, n ) 0.21 ( 0.01 for all cases), it is found that for C ) 51 TSM1, C ) 48 for TSM2, and C ) 46 for TSM3. CONCLUSIONS To evaluate the quality of the TSM model proposed in a previous paper,7 experimental flow resistance data were obtained from 24 uniformly prepared real silica monoliths and were compared to the theoretical flow resistance predicted by the TSM model. From the structural information, obtained from image analysis of the TSM structures and the LSCM generated images, the relationship between the pore and skeleton size and the porosity, previously established7 for the idealized TSM structure, could also be validated for real monoliths. This relationship thus allows one to estimate one unknown structural parameter from the measurement of the two others. Analyzing the standard deviation of the skeleton size showed a relative high degree of heterogeneity of the monolith skeleton for all monoliths over the entire porosity range and for all skeleton sizes. Comparison of the experimental and theoretical flow resistance data showed that the experimental data do follow the general trend predicted by the TSM model but are about a factor of 2 smaller. Since it was inferred that this deviation could be the result of pore (or skeleton) heterogeneity, additional heterogeneous TSM models were created. The heterogeneity of these models was again determined by 3D image analysis, showing a different degree of heterogeneity for the three models. Calculation of the flow rate and pressure drop in these models shows that there is a clear correlation between the flow resistance and the heterogeneity (i.e.,
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Analytical Chemistry, Vol. 77, No. 13, July 1, 2005
d
skeleton or pore diameter (m)
k′′
zone retention coefficient (-)
Kc
Kozeny-Carman constant (-)
L
column length (m)
R
column radius (m)
∆P
pressure drop (Pa)
Q
volumetric flow rate (m3/s)
u
fluid velocity (m/s)
Greek Letters
porosity (-)
φ
flow resistance (-)
η
fluid viscosity (Pa‚s)
σ
standard deviation
Subscripts x
based upon x-velocity
0
based upon unretained tracer velocity
p
pore
s
skeleton
int
internal
ext
external
Received for review February 15, 2005. Accepted April 15, 2005. AC0502798
