Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime

In this brief paper, a novel set of correlations for the drag, lift and torque coefficients for non-spherical particles of irregular shape is proposed. Such correlations are developed for a sphericity range between 0.7 and 0.95 and for intermediate Reynolds numbers in the range [1, 200], common in i...

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Autores:
Laín Beatove, Santiago
Castang, C.
García, D.
Sommerfeld. M.
Tipo de recurso:
Article of investigation
Fecha de publicación:
2023
Institución:
Universidad Autónoma de Occidente
Repositorio:
RED: Repositorio Educativo Digital UAO
Idioma:
eng
OAI Identifier:
oai:red.uao.edu.co:10614/15865
Acceso en línea:
https://hdl.handle.net/10614/15865
https://doi.org/10.1016/j.ces.2023.119288
https://red.uao.edu.co/
Palabra clave:
Non-spherical particles
Irregular shape
Particle resolved simulations
Flow resistance coefficients
Dependence on sphericity and Reynolds number
Rights
openAccess
License
Derechos reservados - Elsevier, 2023
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dc.title.eng.fl_str_mv Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime
title Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime
spellingShingle Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime
Non-spherical particles
Irregular shape
Particle resolved simulations
Flow resistance coefficients
Dependence on sphericity and Reynolds number
title_short Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime
title_full Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime
title_fullStr Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime
title_full_unstemmed Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime
title_sort Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime
dc.creator.fl_str_mv Laín Beatove, Santiago
Castang, C.
García, D.
Sommerfeld. M.
dc.contributor.author.none.fl_str_mv Laín Beatove, Santiago
Castang, C.
García, D.
Sommerfeld. M.
dc.subject.proposal.eng.fl_str_mv Non-spherical particles
Irregular shape
Particle resolved simulations
Flow resistance coefficients
Dependence on sphericity and Reynolds number
topic Non-spherical particles
Irregular shape
Particle resolved simulations
Flow resistance coefficients
Dependence on sphericity and Reynolds number
description In this brief paper, a novel set of correlations for the drag, lift and torque coefficients for non-spherical particles of irregular shape is proposed. Such correlations are developed for a sphericity range between 0.7 and 0.95 and for intermediate Reynolds numbers in the range [1, 200], common in industrial and environmental processes. The proposed expressions are derived by fitting the results obtained by means of Particle-Resolved Direct Numerical Simulations (PR-DNS) for a uniform flow around different sets of irregular particles and considering a large number of random orientations. The resulting flow resistance coefficients can be approximated by normal distributions whose first and second order statistical moments (i.e., mean and standard deviation) are described in this work by fitting functions depending of particle Reynolds number and sphericity. The derived correlations are simple, can be easily implemented and allow the tracking of irregular particles in a Lagrangian stochastic framework
publishDate 2023
dc.date.issued.none.fl_str_mv 2023
dc.date.accessioned.none.fl_str_mv 2024-10-15T16:51:43Z
dc.date.available.none.fl_str_mv 2024-10-15T16:51:43Z
dc.type.spa.fl_str_mv Artículo de revista
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dc.identifier.citation.spa.fl_str_mv Laín Beatove, S.; Castang, C. ; García, D. y Sommerfeld, M. (2023). Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime. Chemical Engineering Science. Volumen 282. 7 p. https://doi.org/10.1016/j.ces.2023.119288
dc.identifier.issn.spa.fl_str_mv 25901400
dc.identifier.uri.none.fl_str_mv https://hdl.handle.net/10614/15865
dc.identifier.doi.spa.fl_str_mv https://doi.org/10.1016/j.ces.2023.119288
dc.identifier.instname.spa.fl_str_mv Universidad Autónoma de Occidente
dc.identifier.reponame.spa.fl_str_mv Respositorio Educativo Digital UAO
dc.identifier.repourl.none.fl_str_mv https://red.uao.edu.co/
identifier_str_mv Laín Beatove, S.; Castang, C. ; García, D. y Sommerfeld, M. (2023). Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime. Chemical Engineering Science. Volumen 282. 7 p. https://doi.org/10.1016/j.ces.2023.119288
25901400
Universidad Autónoma de Occidente
Respositorio Educativo Digital UAO
url https://hdl.handle.net/10614/15865
https://doi.org/10.1016/j.ces.2023.119288
https://red.uao.edu.co/
dc.language.iso.eng.fl_str_mv eng
language eng
dc.relation.citationendpage.spa.fl_str_mv 7
dc.relation.citationstartpage.spa.fl_str_mv 1
dc.relation.citationvolume.spa.fl_str_mv 282
dc.relation.ispartofjournal.eng.fl_str_mv Chemical Engineering Science
dc.relation.references.none.fl_str_mv Arcen, B., Ouchene, R., Khalij, M., Tani`ere, A., 2017. Prolate spheroidal particles’ behavior in a vertical wall-bounded turbulent flow. Phys. Fluids 29, 093301.
Bagheri, G., Bonadonna, C., Manzella, I., Vonlanthen, P., 2015. On the characterization of size and shape of irregular particles. Powder Technol. 270, 141–153.
Bagheri, G., Bonadonna, C., 2016. On the drag of freely falling non-spherical particles. Powder Technol. 301, 526–544.
Castang, C., Laín, S., Sommerfeld, M., 2021. Pressure center determination for regularly shaped non-spherical particles at intermediate Reynolds number range. Int. J. Multiph. Flow 137, 103565.
Castang, C., Laín, S., García, D., Sommerfeld, M., 2022. Aerodynamic coefficients of irregular non-spherical particles at intermediate Reynolds numbers. Powder Technol. 402, 117341.
Chhabra, R., Agarwal, L., Sinha, N.K., 1999. Drag on non-spherical particles: an evaluation of available methods. Powder Technol. 101, 288–295.
Connolly, B.J., Loth, E., Smith, C.F., 2020. Shape and drag of irregular angular particles and test dust. Powder Technol. 363, 275–285.
Frohlich, ¨ K., Meinkel, M., Schroder, ¨ W., 2020. Correlations for inclined prolates based on highly resolved simulations. J. Fluid Mech. 901, A5. https://doi.org/10.1017/ jfm.2020.482.
Ganser, G.H., 1993. A rational approach to drag prediction of spherical and nonspherical particles. Powder Technol. 77, 143–152.
Haider, A., Levenspiel, O., 1989. Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Technol. 58, 63–70.
Holzer, ¨ A., Sommerfeld, M., 2008. New and simple correlation formula for the drag coefficient of non-spherical particles. Powder Technol. 184, 361–365.
Holzer, ¨ A., Sommerfeld, M., 2009. Lattice Boltzmann simulations to determine drag, lift and torque acting on non-spherical particles. Comput. Fluids 38, 572–589.
Laín, S., Sommerfeld, M., 2007. A study of the pneumatic conveying of non-spherical particles in a turbulent horizontal channel flow. Braz. J. Chem. Eng. 24, 545–546.
Loth, E., 2008. Drag of non-spherical solid particles of regular and irregular shape. Powder Technol. 182, 342–353.
Michaelides, E.E., Feng, Z., 2023. Review—Drag coefficients of non-spherical and irregularly shaped particles. ASME J. Fluids Eng. 145, 060801.
Ouchene, R., Khalij, M., Arcen, B., Tani`ere, A., 2016. A new set of correlations of drag, lift and torque coefficients for non-spherical particles and large Reynolds numbers. Powder Technol. 303, 33–43.
Sanjeevi, S.K.P., Kuipers, J.A.M., Padding, J.T., 2018. Drag, lift and torque correlations for non-spherical particles from Stokes limit to high Reynolds numbers. Int. J. Multiph. Flow 106, 325–337.
Sanjeevi, S.K.P., Dietiker, J.F., Padding, J.T., 2022. Accurate hydrodynamic force and torque correlations for prolate spheroids from Stokes regime to high Reynolds numbers. Chem. Eng. J. 444, 136325.
Schiller, L., Naumann, A., 1933. Über die grundlegende Berechnung bei des Schwerkraftaufbereitung. Verein Deutsche Ingenieure 44, 318–320.
Sommerfeld, M., Laín, S., Euler-Lagrange Methods. Multiphase Flow Handbook 2nd Ed., Computational Methods, Chapter 2.6, pp. 202-242. CRC Press, Boca Raton FL (USA). Eds. E.E. Michaelides, C.T. Crowe, J.D. Schwarzkopf. ISBN 9781498701006 (2016).
Sommerfeld, M., van Wachem, B., Oliemans, R., 2008. Best Practice Guidelines for Computational Fluid Dynamics of Dispersed Multiphase Flows. ERCOFTAC (European Research Community on Flow, Turbulence and Combustion).
Sommerfeld, M., Laín, S., 2018. Stochastic modelling for capturing the behaviour of irregular-shaped non-spherical particles in confined turbulent flows. Powder Technol. 332, 253–264.
Sommerfeld, M., Qadir, Z., 2018. Fluid dynamic forces acting on irregular shaped particles: Simulations by the Lattice-Boltzmann method. Int. J. Multiph. Flow 101, 212–222.
Tenneti, S., Subramanian, S., 2014. Particle-resolved direct numerical simulations for gas-solid flow model development. Ann. Rev. Fluid Mech. 46, 199–230.
Zastawny, M., Mallouppas, G., Zhao, F., van Wachem, B., 2012. Derivation of drag and lift force and torque coefficients for non-spherical particles in flows. Int. J. Multiph. Flow 39, 227–239.
Zhang, F., He, Y., Xie, W., Wei, N., Li, J., Wang, S., Wang, J., 2023. Drag coefficients for elongated/flattened irregular particles based on particle-resolved direct numerical simulations. Powder Technol. 418, 118290.
dc.rights.eng.fl_str_mv Derechos reservados - Elsevier, 2023
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spelling Laín Beatove, Santiagovirtual::5736-1Castang, C.García, D.Sommerfeld. M.2024-10-15T16:51:43Z2024-10-15T16:51:43Z2023Laín Beatove, S.; Castang, C. ; García, D. y Sommerfeld, M. (2023). Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime. Chemical Engineering Science. Volumen 282. 7 p. https://doi.org/10.1016/j.ces.2023.11928825901400https://hdl.handle.net/10614/15865https://doi.org/10.1016/j.ces.2023.119288Universidad Autónoma de OccidenteRespositorio Educativo Digital UAOhttps://red.uao.edu.co/In this brief paper, a novel set of correlations for the drag, lift and torque coefficients for non-spherical particles of irregular shape is proposed. Such correlations are developed for a sphericity range between 0.7 and 0.95 and for intermediate Reynolds numbers in the range [1, 200], common in industrial and environmental processes. The proposed expressions are derived by fitting the results obtained by means of Particle-Resolved Direct Numerical Simulations (PR-DNS) for a uniform flow around different sets of irregular particles and considering a large number of random orientations. The resulting flow resistance coefficients can be approximated by normal distributions whose first and second order statistical moments (i.e., mean and standard deviation) are described in this work by fitting functions depending of particle Reynolds number and sphericity. The derived correlations are simple, can be easily implemented and allow the tracking of irregular particles in a Lagrangian stochastic framework7 páginasapplication/pdfengElsevierDerechos reservados - Elsevier, 2023https://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessAtribución-NoComercial-SinDerivadas 4.0 Internacional (CC BY-NC-ND 4.0)http://purl.org/coar/access_right/c_abf2Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regimeArtículo de revistahttp://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARTinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/version/c_970fb48d4fbd8a8571282Chemical Engineering ScienceArcen, B., Ouchene, R., Khalij, M., Tani`ere, A., 2017. Prolate spheroidal particles’ behavior in a vertical wall-bounded turbulent flow. Phys. Fluids 29, 093301.Bagheri, G., Bonadonna, C., Manzella, I., Vonlanthen, P., 2015. On the characterization of size and shape of irregular particles. Powder Technol. 270, 141–153.Bagheri, G., Bonadonna, C., 2016. On the drag of freely falling non-spherical particles. Powder Technol. 301, 526–544.Castang, C., Laín, S., Sommerfeld, M., 2021. Pressure center determination for regularly shaped non-spherical particles at intermediate Reynolds number range. Int. J. Multiph. Flow 137, 103565.Castang, C., Laín, S., García, D., Sommerfeld, M., 2022. Aerodynamic coefficients of irregular non-spherical particles at intermediate Reynolds numbers. Powder Technol. 402, 117341.Chhabra, R., Agarwal, L., Sinha, N.K., 1999. Drag on non-spherical particles: an evaluation of available methods. Powder Technol. 101, 288–295.Connolly, B.J., Loth, E., Smith, C.F., 2020. Shape and drag of irregular angular particles and test dust. Powder Technol. 363, 275–285.Frohlich, ¨ K., Meinkel, M., Schroder, ¨ W., 2020. Correlations for inclined prolates based on highly resolved simulations. J. Fluid Mech. 901, A5. https://doi.org/10.1017/ jfm.2020.482.Ganser, G.H., 1993. A rational approach to drag prediction of spherical and nonspherical particles. Powder Technol. 77, 143–152.Haider, A., Levenspiel, O., 1989. Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Technol. 58, 63–70.Holzer, ¨ A., Sommerfeld, M., 2008. New and simple correlation formula for the drag coefficient of non-spherical particles. Powder Technol. 184, 361–365.Holzer, ¨ A., Sommerfeld, M., 2009. Lattice Boltzmann simulations to determine drag, lift and torque acting on non-spherical particles. Comput. Fluids 38, 572–589.Laín, S., Sommerfeld, M., 2007. A study of the pneumatic conveying of non-spherical particles in a turbulent horizontal channel flow. Braz. J. Chem. Eng. 24, 545–546.Loth, E., 2008. Drag of non-spherical solid particles of regular and irregular shape. Powder Technol. 182, 342–353.Michaelides, E.E., Feng, Z., 2023. Review—Drag coefficients of non-spherical and irregularly shaped particles. ASME J. Fluids Eng. 145, 060801.Ouchene, R., Khalij, M., Arcen, B., Tani`ere, A., 2016. A new set of correlations of drag, lift and torque coefficients for non-spherical particles and large Reynolds numbers. Powder Technol. 303, 33–43.Sanjeevi, S.K.P., Kuipers, J.A.M., Padding, J.T., 2018. Drag, lift and torque correlations for non-spherical particles from Stokes limit to high Reynolds numbers. Int. J. Multiph. Flow 106, 325–337.Sanjeevi, S.K.P., Dietiker, J.F., Padding, J.T., 2022. Accurate hydrodynamic force and torque correlations for prolate spheroids from Stokes regime to high Reynolds numbers. Chem. Eng. J. 444, 136325.Schiller, L., Naumann, A., 1933. Über die grundlegende Berechnung bei des Schwerkraftaufbereitung. Verein Deutsche Ingenieure 44, 318–320.Sommerfeld, M., Laín, S., Euler-Lagrange Methods. Multiphase Flow Handbook 2nd Ed., Computational Methods, Chapter 2.6, pp. 202-242. CRC Press, Boca Raton FL (USA). Eds. E.E. Michaelides, C.T. Crowe, J.D. Schwarzkopf. ISBN 9781498701006 (2016).Sommerfeld, M., van Wachem, B., Oliemans, R., 2008. Best Practice Guidelines for Computational Fluid Dynamics of Dispersed Multiphase Flows. ERCOFTAC (European Research Community on Flow, Turbulence and Combustion).Sommerfeld, M., Laín, S., 2018. Stochastic modelling for capturing the behaviour of irregular-shaped non-spherical particles in confined turbulent flows. Powder Technol. 332, 253–264.Sommerfeld, M., Qadir, Z., 2018. Fluid dynamic forces acting on irregular shaped particles: Simulations by the Lattice-Boltzmann method. Int. J. Multiph. Flow 101, 212–222.Tenneti, S., Subramanian, S., 2014. Particle-resolved direct numerical simulations for gas-solid flow model development. Ann. Rev. Fluid Mech. 46, 199–230.Zastawny, M., Mallouppas, G., Zhao, F., van Wachem, B., 2012. Derivation of drag and lift force and torque coefficients for non-spherical particles in flows. Int. J. Multiph. Flow 39, 227–239.Zhang, F., He, Y., Xie, W., Wei, N., Li, J., Wang, S., Wang, J., 2023. Drag coefficients for elongated/flattened irregular particles based on particle-resolved direct numerical simulations. Powder Technol. 418, 118290.Non-spherical particlesIrregular shapeParticle resolved simulationsFlow resistance coefficientsDependence on sphericity and Reynolds numberComunidad generalPublication082b0926-3385-4188-9c6a-bbbed7484a95virtual::5736-1082b0926-3385-4188-9c6a-bbbed7484a95virtual::5736-1https://scholar.google.com/citations?user=g-iBdUkAAAAJ&hl=esvirtual::5736-10000-0002-0269-2608virtual::5736-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0000262129virtual::5736-1ORIGINALSphericity_based_correlations_for_flow_resistance_coefficients_of_non-spherical_particles_of_irregular_shape_beyond_the_Stokes_regime.pdfSphericity_based_correlations_for_flow_resistance_coefficients_of_non-spherical_particles_of_irregular_shape_beyond_the_Stokes_regime.pdfArchivo texto completo del artículo de revista, PDFapplication/pdf3798394https://red.uao.edu.co/bitstreams/193385c8-f850-4b5b-857a-f79b59f992d8/download1eeaf97f13b3b15297094c8e1a13108eMD51LICENSElicense.txtlicense.txttext/plain; charset=utf-81672https://red.uao.edu.co/bitstreams/42a7b275-eac6-4beb-83a2-dbfa2e8d9a88/download6987b791264a2b5525252450f99b10d1MD52TEXTSphericity_based_correlations_for_flow_resistance_coefficients_of_non-spherical_particles_of_irregular_shape_beyond_the_Stokes_regime.pdf.txtSphericity_based_correlations_for_flow_resistance_coefficients_of_non-spherical_particles_of_irregular_shape_beyond_the_Stokes_regime.pdf.txtExtracted texttext/plain38471https://red.uao.edu.co/bitstreams/cf14df99-17a2-49a3-927e-bff5ccb43460/download58d2550c8e3003a22c85e0d720420671MD53THUMBNAILSphericity_based_correlations_for_flow_resistance_coefficients_of_non-spherical_particles_of_irregular_shape_beyond_the_Stokes_regime.pdf.jpgSphericity_based_correlations_for_flow_resistance_coefficients_of_non-spherical_particles_of_irregular_shape_beyond_the_Stokes_regime.pdf.jpgGenerated Thumbnailimage/jpeg15005https://red.uao.edu.co/bitstreams/5f128e86-3698-4507-a9e4-e372e4bd36f3/download70e28adbd428dbe2c61d7710e40c91b5MD5410614/15865oai:red.uao.edu.co:10614/158652024-10-16 03:01:50.489https://creativecommons.org/licenses/by-nc-nd/4.0/Derechos reservados - Elsevier, 2023open.accesshttps://red.uao.edu.coRepositorio Digital Universidad Autonoma de Occidenterepositorio@uao.edu.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