The analysis of the relations between porosity and tortuosity in granular beds
Wojciech Sobieski
Seweryn Lipiński
Abstrakt
In the paper, functions describing different porosity-tortuosity relations were collected, and then the tortuosity values were calculated for a one granular bed consisting of spherical particles with normal distribution of diameters. Information about the bed porosity and particle sizes was obtained from measurements conducted for an artificial granular bed, consisting of glass marbles. The results of calculations were compared with the results of two other methods of tortuosity determination, performed for the same case (details are not described in this paper): the first of them uses the Path Tracking Method, the second one - information about the velocity components in a creeping flow (the Lattice-Boltzmann Method was applied to obtain the velocity field in the flow). The main aim of our article was to test whether the functions linking tortuosity with porosity, which are available in the literature, give similar results as the methods described above. To achieve this aim, the relative errors between results of calculations for the collected formulas and values from the both previous mentioned methods were calculated.
Słowa kluczowe:
porous media, granular beds, porosity, tortuosityBibliografia
ALLEN R., SUN S. 2014. Investigating the role of tortuosity in the Kozeny-Carman equation. International Conference on Numerical and Mathematical Modeling of Flow and Transport in Porous Media, Dubrovnik, Croatia, 29 September - 3 October.
AHMADI M.M., MOHAMMADI S., HAYATI A.N. 2011. Analytical derivation of tortuosity and permeability of monosized spheres: A volume averaging approach. Physical Review E., (83): 026312.
BEAR J. 1972. Dynamics of Fluids in Porous Media. Courier Dover Publications, New York.
CARMAN P.C. 1997. Fluid Flow through a Granular Bed. Transactions of the Institute of Chemical Engineers, Jubilee Supplement, (75): 32-48.
COOKE A.J., ROWE R.K. 1999. Extension of Porosity and Surface Area Models for Uniform Porous Media. Journal of Environmental Engineering, 125(2): 126-136.
DIAS R.P., TEIXEIRA J.A., MOTA M., YELSHIN A. 2006. Tortuosity variation in a low density binary particulate bed. Separation and Purification Technology, 51(2): 180-184.
DUDDA W., SOBIESKI W. 2014. Modification of the PathFinder algorithm for calculating granular beds with various particle size distributions. Technical Sciences, 17(2): 135-148.
EBNER M., CHUNG D.-W., GARCIA R.E., WOOD V. 2013. Tortuosity Anisotropy in Lithium-Ion Battery Electrodes. Advanced Energy Materials, 4(5): 1301278.
FENG Y.T., HAN K., OWEN D.R.J. 2007. Coupled lattice Boltzmann method and discrete element modelling of particle transport in turbulent fluid flows: Computational issues. International Journal for Numerical Methods in Engineering, 72(9): 1111-1134.
GAO H.-Y., HE Y.-H., ZOU J., XU N.-P., LIU C.T. 2012. Tortuosity factor for porous FeAl intermetallics fabricated by reactive synthesis. Transactions of Nonferrous Metals Society of China, 22(9): 2179-2183.
GOMMES C.J., BONS A.J., BLACHER S., DUNSMUIR J.H., TSOU A.H. 2009. Practical methods for measuring the tortuosity of porous materials from binary or gray-tone tomographic reconstructions. American Institute of Chemical Engineers (AIChE) Journal, 55(8): 2000-2012.
JOHNSON D.L., PLONA T.J., SCALA C., PASIERB F., KOJIMA H. 1982. Tortuosity and Acoustic Slow Waves. Physical Review Letters, 49(25): 1840-1844.
KOCHAŃSKI J., KACZMAREK M., KUBIK J. 2000. Determination of permeability and tortuosity of permeable media by ultrasonic method. Studies for sintered bronze. Journal of Theoretical and Applied Mechanics, 1(39): 923-928.
KONG W., ZHANG Q., GAO X., ZHANG J., CHEN D., SU S. 2015. A Method for Predicting the Tortuosity of Pore Phase in Solid Oxide Fuel Cells Electrode. International Journal of Electrochemical Science, 10(1): 5800-5811.
KOZENY J. 1927. Uber kapillare Leitung des Wassers im Boden. Akademie der Wissenschaften in Wien, Sitzungsberichte, 136(2a): 271-306.
LANFREY P.-Y., KUZELJEVIC Z.V., DUDUKOVIC M.P. 2010. Tortuosity model for fixed beds randomly packed with identical particles. Chemical Engineering Science, 65: 1891-1896.
LE L.H., ZHANG C., TA D., LOU E. 2010. Measurement of tortuosity in aluminum foams using airborne ultrasound. Ultrasonics, 50(1): 1-5.
MATYKA M., KHALILI A., KOZA Z. 2008. Tortuosity-porosity relation in porous media flow. Physical Review E., 78: 026306.
NABOVATI A., SOUSA A.C.M. 2007. Fluid flow simulation in random porous media at pore level using the Lattice Boltzmann Method. Journal of Engineering Science and Technology, 2(3): 226-237.
NAKASHIMA Y., KAMIYA S. 2007. Mathematica Programs for the Analysis of Three-Dimensional Pore Connectivity and Anisotropic Tortuosity of Porous Rocks using X-ray Computed Tomography Image Data. Journal of Nuclear Science and Technology, 44(9): 1233-1247.
NWAIZU C., ZHANG Q. 2012. Characterizing Airflow paths in Grain bulks. Paper NABEC/CSBE. Northeast Agricultural& Biological Engineering Conference Canadian Society for Bioengineering Lakehead University, Orillia, Ontario July 15-18.
Pathfinder Project [on-line]. 2013. University of Warmia and Mazury in Olsztyn http://www.uwm.edu.pl/pathfinder/index.php (access: 7.02.2014).
RIBEIRO A.M., NETO P., PINHO C. 2010. Mean Porosity and Pressure Drop Measurements in Packed Beds of Monosized Spheres: Side Wall Effects. International Review of Chemical Engineering, 2(1): 40-46.
SOBIESKI W. 2009. Calculating tortuosity in a porous bed consisting of spherical particles with known sizes and distribution in space. Research report 1/2009, Winnipeg (Canada).
SOBIESKI W. 2014. The quality of the base knowledge in a research process. Scientific Researches in the Department of Mechanics and Machine Design, University of Warmia and Mazury in Olsztyn, 2: 29-47.
SOBIESKI W., DUDDA W., LIPIŃSKI S. 2016a. A new approach for obtaining the geometric properties of a granular porous bed based on DEM simulations. Technical Sciences, 19(2): 165-187.
SOBIESKI W., LIPIŃSKI S. 2013. PathFinder User’s Guide. University of Warmia and Mazury in Olsztyn. http://www.uwm.edu.pl/pathfinder/index.php (access: 7.02.2014).
SOBIESKI W., LIPIŃSKI S., DUDDA W., TRYKOZKO A., MAREK M., WIĄCEK J., MATYKA M., GOŁĘBIEWSKI J. 2016b. Granular Porous Media. Department of Mechanics and Machine Design, University of Warmia and Mazury, Olsztyn.
SOBIESKI W., ZHANG Q., LIU C. 2012. Predicting Tortuosity for Airflow through Porous Beds Consisting of Randomly Packed Spherical Particles. Transport in Porous Media, 93(3): 431-451.
STARLY B., YILDIRIM E, SUN W. 2007. A tracer metric numerical model for predicting tortuosity factors in three-dimensional porous tissue scaffolds. Computer Methods and Programs in Biomedicine, 87(1): 21-27.
TANG X.-W., SUN Z.-F., CHENG G.C. 2012. Simulation of the relationship between porosity and tortuosity in porous media with cubic particles. Chinese Physics B, 21(10): 100201.
WANG P. 2014. Lattice Boltzmann Simulation of Permeability and Tortuosity for Flow through Dense Porous Media. Mathematical Problems in Engineering, Article ID 694350.
WU Y.S., VAN VLIET L.J., FRIJLINK H.W., VAN DER VOORT MAARSCHALK K. 2006. The determination of relative path length as a measure for tortuosity in compacts using image analysis. European Journal of Pharmaceutical Sciences, 28(5): 433-440.
VALLABH R. 2009. Modeling Tortuosity in Fibrous Porous Media using Computational Fluid Dynamics. PhD Thesis, North Carolina State University, Raleigh, United States.
YU B.M., LI J.H. 2004. A geometry model for tortuosity of flow path in porous media. Chinese Physics Letter, 21(8): 1569-1571.
AHMADI M.M., MOHAMMADI S., HAYATI A.N. 2011. Analytical derivation of tortuosity and permeability of monosized spheres: A volume averaging approach. Physical Review E., (83): 026312.
BEAR J. 1972. Dynamics of Fluids in Porous Media. Courier Dover Publications, New York.
CARMAN P.C. 1997. Fluid Flow through a Granular Bed. Transactions of the Institute of Chemical Engineers, Jubilee Supplement, (75): 32-48.
COOKE A.J., ROWE R.K. 1999. Extension of Porosity and Surface Area Models for Uniform Porous Media. Journal of Environmental Engineering, 125(2): 126-136.
DIAS R.P., TEIXEIRA J.A., MOTA M., YELSHIN A. 2006. Tortuosity variation in a low density binary particulate bed. Separation and Purification Technology, 51(2): 180-184.
DUDDA W., SOBIESKI W. 2014. Modification of the PathFinder algorithm for calculating granular beds with various particle size distributions. Technical Sciences, 17(2): 135-148.
EBNER M., CHUNG D.-W., GARCIA R.E., WOOD V. 2013. Tortuosity Anisotropy in Lithium-Ion Battery Electrodes. Advanced Energy Materials, 4(5): 1301278.
FENG Y.T., HAN K., OWEN D.R.J. 2007. Coupled lattice Boltzmann method and discrete element modelling of particle transport in turbulent fluid flows: Computational issues. International Journal for Numerical Methods in Engineering, 72(9): 1111-1134.
GAO H.-Y., HE Y.-H., ZOU J., XU N.-P., LIU C.T. 2012. Tortuosity factor for porous FeAl intermetallics fabricated by reactive synthesis. Transactions of Nonferrous Metals Society of China, 22(9): 2179-2183.
GOMMES C.J., BONS A.J., BLACHER S., DUNSMUIR J.H., TSOU A.H. 2009. Practical methods for measuring the tortuosity of porous materials from binary or gray-tone tomographic reconstructions. American Institute of Chemical Engineers (AIChE) Journal, 55(8): 2000-2012.
JOHNSON D.L., PLONA T.J., SCALA C., PASIERB F., KOJIMA H. 1982. Tortuosity and Acoustic Slow Waves. Physical Review Letters, 49(25): 1840-1844.
KOCHAŃSKI J., KACZMAREK M., KUBIK J. 2000. Determination of permeability and tortuosity of permeable media by ultrasonic method. Studies for sintered bronze. Journal of Theoretical and Applied Mechanics, 1(39): 923-928.
KONG W., ZHANG Q., GAO X., ZHANG J., CHEN D., SU S. 2015. A Method for Predicting the Tortuosity of Pore Phase in Solid Oxide Fuel Cells Electrode. International Journal of Electrochemical Science, 10(1): 5800-5811.
KOZENY J. 1927. Uber kapillare Leitung des Wassers im Boden. Akademie der Wissenschaften in Wien, Sitzungsberichte, 136(2a): 271-306.
LANFREY P.-Y., KUZELJEVIC Z.V., DUDUKOVIC M.P. 2010. Tortuosity model for fixed beds randomly packed with identical particles. Chemical Engineering Science, 65: 1891-1896.
LE L.H., ZHANG C., TA D., LOU E. 2010. Measurement of tortuosity in aluminum foams using airborne ultrasound. Ultrasonics, 50(1): 1-5.
MATYKA M., KHALILI A., KOZA Z. 2008. Tortuosity-porosity relation in porous media flow. Physical Review E., 78: 026306.
NABOVATI A., SOUSA A.C.M. 2007. Fluid flow simulation in random porous media at pore level using the Lattice Boltzmann Method. Journal of Engineering Science and Technology, 2(3): 226-237.
NAKASHIMA Y., KAMIYA S. 2007. Mathematica Programs for the Analysis of Three-Dimensional Pore Connectivity and Anisotropic Tortuosity of Porous Rocks using X-ray Computed Tomography Image Data. Journal of Nuclear Science and Technology, 44(9): 1233-1247.
NWAIZU C., ZHANG Q. 2012. Characterizing Airflow paths in Grain bulks. Paper NABEC/CSBE. Northeast Agricultural& Biological Engineering Conference Canadian Society for Bioengineering Lakehead University, Orillia, Ontario July 15-18.
Pathfinder Project [on-line]. 2013. University of Warmia and Mazury in Olsztyn http://www.uwm.edu.pl/pathfinder/index.php (access: 7.02.2014).
RIBEIRO A.M., NETO P., PINHO C. 2010. Mean Porosity and Pressure Drop Measurements in Packed Beds of Monosized Spheres: Side Wall Effects. International Review of Chemical Engineering, 2(1): 40-46.
SOBIESKI W. 2009. Calculating tortuosity in a porous bed consisting of spherical particles with known sizes and distribution in space. Research report 1/2009, Winnipeg (Canada).
SOBIESKI W. 2014. The quality of the base knowledge in a research process. Scientific Researches in the Department of Mechanics and Machine Design, University of Warmia and Mazury in Olsztyn, 2: 29-47.
SOBIESKI W., DUDDA W., LIPIŃSKI S. 2016a. A new approach for obtaining the geometric properties of a granular porous bed based on DEM simulations. Technical Sciences, 19(2): 165-187.
SOBIESKI W., LIPIŃSKI S. 2013. PathFinder User’s Guide. University of Warmia and Mazury in Olsztyn. http://www.uwm.edu.pl/pathfinder/index.php (access: 7.02.2014).
SOBIESKI W., LIPIŃSKI S., DUDDA W., TRYKOZKO A., MAREK M., WIĄCEK J., MATYKA M., GOŁĘBIEWSKI J. 2016b. Granular Porous Media. Department of Mechanics and Machine Design, University of Warmia and Mazury, Olsztyn.
SOBIESKI W., ZHANG Q., LIU C. 2012. Predicting Tortuosity for Airflow through Porous Beds Consisting of Randomly Packed Spherical Particles. Transport in Porous Media, 93(3): 431-451.
STARLY B., YILDIRIM E, SUN W. 2007. A tracer metric numerical model for predicting tortuosity factors in three-dimensional porous tissue scaffolds. Computer Methods and Programs in Biomedicine, 87(1): 21-27.
TANG X.-W., SUN Z.-F., CHENG G.C. 2012. Simulation of the relationship between porosity and tortuosity in porous media with cubic particles. Chinese Physics B, 21(10): 100201.
WANG P. 2014. Lattice Boltzmann Simulation of Permeability and Tortuosity for Flow through Dense Porous Media. Mathematical Problems in Engineering, Article ID 694350.
WU Y.S., VAN VLIET L.J., FRIJLINK H.W., VAN DER VOORT MAARSCHALK K. 2006. The determination of relative path length as a measure for tortuosity in compacts using image analysis. European Journal of Pharmaceutical Sciences, 28(5): 433-440.
VALLABH R. 2009. Modeling Tortuosity in Fibrous Porous Media using Computational Fluid Dynamics. PhD Thesis, North Carolina State University, Raleigh, United States.
YU B.M., LI J.H. 2004. A geometry model for tortuosity of flow path in porous media. Chinese Physics Letter, 21(8): 1569-1571.
Sobieski, W., & Lipiński, S. (2017). The analysis of the relations between porosity and tortuosity in granular beds. Technical Sciences, 20(1), 75–85. https://doi.org/10.31648/ts.2912
Wojciech Sobieski
Seweryn Lipiński
