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Validation of Pore-Scale Simulations of Hydrodynamic Dispersion in Random Sphere Packings

Published online by Cambridge University Press:  03 June 2015

Siarhei Khirevich ,
Alexandra Höltzel  and
Ulrich Tallarek
Siarhei Khirevich*
Affiliation:
Department of Chemistry, Philipps-Universität Marburg, Hans-Meerwein-Strasse, 35032 Marburg, Germany
Alexandra Höltzel*
Affiliation:
Department of Chemistry, Philipps-Universität Marburg, Hans-Meerwein-Strasse, 35032 Marburg, Germany
Ulrich Tallarek*
Affiliation:
Department of Chemistry, Philipps-Universität Marburg, Hans-Meerwein-Strasse, 35032 Marburg, Germany
*
Email:khirevic@staff.uni-marburg.de
Email:hoeltzel@staff.uni-marburg.de
Corresponding author.Email:tallarek@staff.uni-marburg.de
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Abstract

We employ the lattice Boltzmann method and random walk particle tracking to simulate the time evolution of hydrodynamic dispersion in bulk, random, monodisperse, hard-sphere packings with bed porosities (interparticle void volume fractions) between the random-close and the random-loose packing limit. Using Jodrey-Tory and Monte Carlo-based algorithms and a systematic variation of the packing protocols we generate a portfolio of packings, whose microstructures differ in their degree of heterogeneity (DoH). Because the DoH quantifies the heterogeneity of the void space distribution in a packing, the asymptotic longitudinal dispersion coefficient calculated for the packings increases with the packings’ DoH. We investigate the influence of packing length (up to 150 dp, where dp is the sphere diameter) and grid resolution (up to 90 nodes per dp) on the simulated hydrodynamic dispersion coefficient, and demonstrate that the chosen packing dimensions of 10 dpx 10 dpx 70 dp and the employed grid resolution of 60 nodes per dp are sufficient to observe asymptotic behavior of the dispersion coefficient and to minimize finite size effects. Asymptotic values of the dispersion coefficients calculated for the generated packings are compared with simulated as well as experimental data from the literature and yield good to excellent agreement.

Keywords

47.56.+r81.05.Rm05.40.-a61.43.Bn02.60.CbPacking microstructuredegree of heterogeneitypacking algorithmhydrodynamic dispersionrandom sphere packings
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 3 , March 2013 , pp. 801 - 822
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Bear, J., Dynamics of fluids in porous media, Dover Publications, 1988.Google Scholar
[2]Dullien, F. A. L., Porous media: fluid transport and pore structure, 2 ed., Academic Press, 1992.Google Scholar
[3]Aste, T., Saadatfar, M., and Senden, T. J., Geometrical structure of disordered sphere packings, Phys. Rev. E, 71 (2005), 061302.CrossRefGoogle ScholarPubMed
[4]Mizutani, R., Takeuchi, A., Osamura, R. Y., Takekoshi, S., Uesugi, K. and Suzuki, Y., Submi-crometer tomographic resolution examined using a micro-fabricated test object, Micron, 41 (2010), 90–95.Google Scholar
[5]Piller, M., Schena, G., Nolich, M., Favretto, S., Radaelli, F. and Rossi, E., Analysis of hydraulic permeability in porous media: from high resolution X-ray tomography to direct numerical simulation, Transp. Porous Media, 80 (2009), 57–78.Google Scholar
[6]Manz, B., Gladden, L. F. and Warren, P. B., Flow and dispersion in porous media: lattice-Boltzmann and NMR studies, AIChE J., 45 (1999), 1845–1854.Google Scholar
[7]Bruns, S., Mullner, T., Kollmann, M., Schachtner, J., Holtzel, A. and Tallarek, U., Confocal laser scanning microscopy method for quantitative characterization of silica monolith morphology, Anal. Chem., 82 (2010), 6569–6575.CrossRefGoogle ScholarPubMed
[8]Bruns, S. and Tallarek, U., Physical reconstruction of packed beds and their morphological analysis: core-shell packings as an example, J. Chromatogr. A, 1218 (2011), 1849–1860.Google Scholar
[9]Khirevich, S., Daneyko, A., Höltzel, A., Seidel-Morgenstern, A. and Tallarek, U., Statistical anal-ysis of packed beds, the origin of short-range disorder, and its impact on eddy dispersion, J. Chromatogr. A, 1217 (2010), 4713–4722.CrossRefGoogle Scholar
[10]Khirevich, S., Höltzel, A., Daneyko, A., Seidel-Morgenstern, A. and Tallarek, U., Structure-transport correlation for the diffusive tortuosity of bulk, monodisperse, random sphere packings, J. Chromatogr. A, 1218 (2011), 6489–6497.Google Scholar
[11]Daneyko, A., Höltzel, A., Khirevich, S. and Tallarek, U., Influence of the particle size distribution on hydraulic permeability and eddy dispersion in bulk packings, Anal. Chem., 83 (2011), 3903–3910.CrossRefGoogle ScholarPubMed
[12]Giddings, J. C., ‘Eddy’ diffusion in chromatography, Nature, 184 (1959), 357–358.CrossRefGoogle Scholar
[13]Schenker, I., Filser, F. T., Gauckler, L.J., Aste, T. and Herrmann, H. J., Quantification of the heterogeneity of particle packings, Phys. Rev. E, 80 (2009), 021302.Google Scholar
[14]Jodrey, W. S. and Tory, E. M., E. M., Computer simulation of close random packing of equal spheres, Phys. Rev. A, 32 (1985), 2347–2351.CrossRefGoogle Scholar
[15]Allen, M. P. and Tildesley, D. J., Computer simulation of liquids, Oxford University Press, 1989.Google Scholar
[16]Zinchenko, A. Z., Algorithm for random close packing of spheres with periodic boundary conditions, J. Comput. Phys., 114 (1994), 298–307.Google Scholar
[17]Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N., Spatial tessellations: concepts and applications of Voronoi diagrams, 2 ed., John Wiley & Sons, 2000.CrossRefGoogle Scholar
[18]Succi, S., The lattice Boltzmann equation for fluid dynamics and beyond, Oxford University Press, 2001.Google Scholar
[19]Pan, C., Luo, L.-S., and Miller, C. T., An evaluation of lattice Boltzmann schemes for porous medium flow simulation, Comput. Fluids, 35 (2006), 898–909.Google Scholar
[20]Ginzbourg, I. and Adler, P. M., Boundary flow condition analysis for the three-dimensional lattice Boltzmann model, J. Phys. II, 4 (1994), 191–214.Google Scholar
[21]Guo, Z., Zheng, C. and Shi, B., Discrete lattice effects on the forcing term in the lattice Boltz-mann method, Phys. Rev. E, 65 (2002), 046308.CrossRefGoogle ScholarPubMed
[22]Tikhonov, A. N. and Samarskii, A. A., Equations of mathematical physics, Dover Publications, 1990.Google Scholar
[23]Zienkiewicz, O. C., Nithiarasu, P. and Taylor, R. L., The finite element method for fluid dynamics, Elsevier Butterworth-Heinemann, 2005.Google Scholar
[24]Devkota, B. H. and Imberger, J., Lagrangian modeling of advection-diffusion transport in open channel flow, Water Resour. Res., 45 (2009), W12406.CrossRefGoogle Scholar
[25]Lin, B. and Falconer, R. A., Tidal flow and transport modeling using ULTIMATE QUICKEST scheme, J. Hydraul. Eng., 123 (1997), 303–314.CrossRefGoogle Scholar
[26]Leonard, B. P., The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection, Comput. Methods Appl. Mech. Eng., 88 (1991), 17–74.CrossRefGoogle Scholar
[27]Hassan, A. E. and Mohamed, M. M., On using particle tracking methods to simulate transport in single-continuum and dual continua porous media, J. Hydrol., 275 (2003), 242–260.Google Scholar
[28]Khirevich, S., Holtzel, A., Seidel-Morgenstern, A. and Tallarek, U., Time and length scales of eddy dispersion in chromatographic beds, Anal. Chem., 81 (2009), 7057–7066.Google Scholar
[29]Salamon, P., Fernandez-Garcia, D. and Gomez-Hernandez, J. J., Modeling tracer transport at the MADE site: the importance of heterogeneity, Water Resour. Res., 43 (2007), W08404.CrossRefGoogle Scholar
[30]Rudnick, J. A. and Gaspari, G. D., Elements of the random walk: an introduction for advanced students and researchers, Cambridge University Press, 2004.Google Scholar
[31]Delay, F., Ackerer, P. and Danquigny, C., Simulating solute transport in porous or fractured formations using random walk particle tracking: a review, Vadose Zone J., 4 (2005), 360–379.CrossRefGoogle Scholar
[32]Salamon, P., Fernàndez-Garcia, D. and Gomez-Hernandez, J. J., J., J., A review and numerical assessment of the random walk particle tracking method, J. Contam. Hydrol., 87 (2006), 277–305.CrossRefGoogle ScholarPubMed
[33]Maier, R. S., Kroll, D. M., Bernard, R. S., Howington, S. E., Peters, J. F. and Davis, H. T., Pore-scale simulation of dispersion, Phys. Fluids, 12 (2000), 2065–2079.CrossRefGoogle Scholar
[34]Freund, H., Bauer, J., Zeiser, T. and Emig, G., Detailed simulation of transport processes in fixed-beds, Ind. Eng. Chem. Res., 44 (2005), 6423–6434.CrossRefGoogle Scholar
[35]Kloeden, P. E. and Platen, E., Numerical solution of stochastic differential equations, Springer-Verlag, 1995.Google Scholar
[36]Szymczak, P. and Ladd, A. J. C., Boundary conditions for stochastic solutions of the convection-diffusion equation, Phys. Rev. E, 68 (2003), 036704.CrossRefGoogle ScholarPubMed
[37]Israelsson, P. H., Kim, Y. D. and Adams, E. E., A comparison of three Lagrangian approaches for extending near field mixing calculations, Environ. Modell. Software, 21 (2006), 16311649.CrossRefGoogle Scholar
[38]Khirevich, S., High-performance computing of flow, diffusion, and hydrodynamic dispersion in random sphere packings, PhD thesis, Philipps University of Marburg, Germany, 2010.Google Scholar
[39]Khirevich, S., Daneyko, A. and Tallarek, U., Simulation of fluid flow and mass transport at extreme scale, Technical Report FZJ-JSC-IB-2010-03, Forschungszentrum Julich, Julich Supercomputing Centre, 2010.Google Scholar
[40]Brenner, H., Dispersion resulting from flow through spatially periodic porous media, Philos. Trans. R. Soc. A, 297 (1980), 81–133.Google Scholar
[41]Hlushkou, D. and Tallarek, U., Transition from creeping via viscous-inertial to turbulent flow in fixed beds, J. Chromatogr. A, 1126 (2006), 70–85.Google Scholar
[42]Giddings, J. C., Dynamics of chromatography: principles and theory, Marcel Dekker, 1965.Google Scholar
[43]Schure, M. R., Maier, R. S., Kroll, D. M. and Davis, H. T., Simulation of packed-bed chromatog-raphy utilizing high-resolution flow fields: comparison with models, Anal. Chem., 74 (2002), 6006–6016.CrossRefGoogle Scholar
[44]Taylor, G., Dispersion of soluble matter in solvent flowing slowly through a tube, Philos. Trans. R. Soc. A, 219 (1953), 186–203.Google Scholar
[45]Aris, R., On the dispersion of a solute in a fluid flowing through a tube, Philos. Trans. R. Soc. A, 235 (1956), 67–77.Google Scholar
[46]Dutta, D., Ramachandran, A. and Leighton, D. T., Effect of channel geometry on solute dispersion in pressure-driven microfluidic systems, Microfluid. Nanofluid., 2 (2006), 275–290.Google Scholar
[47]Lowe, C. P. and Frenkel, D., Do hydrodynamic dispersion coefficients exist?, Phys. Rev. Lett., 77 (1996), 4552–4555.Google Scholar
[48]Maier, R. S. and Bernard, R. S., Lattice-Boltzmann accuracy in pore-scale flow simulation, J. Comput. Phys., 229 (2010), 233–255.CrossRefGoogle Scholar
[49]Koch, D. L., Hill, R. J. and Sangani, A. S., Brinkman screening and the covariance of the fluid velocity in fixed beds, Phys. Fluids, 10 (1998), 3035–3037.Google Scholar
[50]Frenkel, D. and Ernst, M. H., Simulation of diffusion in a two-dimensional lattice-gas cellular automaton: a test of mode-coupling theory, Phys. Rev. Lett., 63 (1989), 2165–2168.CrossRefGoogle Scholar
[51]Merks, R. H. M., Hoekstra, A. G. and Sloot, P. M. A., The moment propagation method for advection-diffusion in the lattice Boltzmann method: validation and Peclet number limits, J. Comput. Phys., 183 (2002), 563–576.Google Scholar
[52]Maier, R. S., Schure, M. R., Gage, J. P. and Seymour, J. D., Sensitivity of pore-scale dispersion to the construction of random bead packs, Water Resour. Res., 44 (2008), W06S03.Google Scholar
[53]Seymour, J. D. and Callaghan, P. T., Generalized approach to NMR analysis of flow and dis-persion in porous media, AIChE J., 43 (1997), 2096–2111.Google Scholar
[54]Augier, F., Idoux, F. and Delenne, J. Y., Numerical simulations of transfer and transport properties inside packed beds of spherical particles, Chem. Eng. Sci., 65 (2010), 1055–1064.Google Scholar
[55]Scheven, U. M., Harris, R. and Johns, M. L., Intrinsic dispersivity of randomly packed mono-disperse spheres, Phys. Rev. Lett., 99 (2007), 054502.Google Scholar
[56]Sahimi, M., Flow and transport in porous media and fractured rock: from classical methods to modern approaches, Wiley-VCH, 1995.Google Scholar
[57]Maier, R. S., Kroll, D. M. and Davis, H. T., Diameter-dependent dispersion in packed cylinders, AIChE J., 53 (2007), 527–530.Google Scholar
[58]Cundall, P. A. and Strack, O. D. L., A discrete numerical model for granular assemblies, Geotechnique, 29 (1979), 47–65.Google Scholar
[59]Scheven, U. M., Dispersion in non-ideal packed beds, AIChE J., 56 (2010), 289–297.Google Scholar
[60]Saffman, P. G., A theory of dispersion in a porous medium, J. Fluid Mech., 6 (1959), 321–349.CrossRefGoogle Scholar
[61]Saffman, P. G., Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries, J. Fluid Mech., 7 (1960), 194–208.Google Scholar
[62]Kuttanikkad, S. P., Pore-scale direct numerical simulation of flow and transport in porous media, PhD thesis, Ruprecht Karls University of Heidelberg, Germany, 2009.Google Scholar