| Peer-Reviewed

Quinn’s Law of Fluid Dynamics: Supplement #3 A Unique Solution to the Navier-Stokes Equation for Fluid Flow in Closed Conduits

Received: 22 July 2020     Accepted: 5 August 2020     Published: 25 August 2020
Views:       Downloads:
Abstract

The recent publication of Quinn’s Law of Fluid Dynamics brings into focus longstanding contradictions regarding permeability in closed conduits that have littered the fluid dynamics landscape for more than 150 years. In this paper, we will use this new level of understanding to explain these contradictions, in layman’s terms, and resolve them, by introducing for the first time, as far as we know, a unique solution to the Navier-Stokes equation for fluid flow in closed conduits, which is understandable by knowledgeable physicists, engineers, chromatographers and aerospace enthusiasts alike, but who may not necessarily be versed in the abstract jargon of a graduate in advanced mathematics. In addition, we will apply our unique solution to chosen illustrative worked examples, as well as those of third parties from the published literature. In so doing, we will demonstrate the utility of our solution, not only, to packed conduits containing particles having solid skeletons, but also, to empty conduits, which in the context of this new understanding of fluid dynamics in closed conduits, represents a special case of a packed conduit in which the particles are fully porous, i.e., they are made entirely of free space.

Published in Fluid Mechanics (Volume 6, Issue 2)
DOI 10.11648/j.fm.20200602.11
Page(s) 30-50
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Forchheimer Coefficients, Conduit Permeability, Continuity Equation, Porosity, Tortuosity, Packed Conduits

References
[1] Poiseuille, J. L. (1841). "Recherches expérimentales sur le mouvement des liquides dans les tubes de très-petits diamètres." Comptes Rendus, Académie des Sciences, Paris 12, 112 (in French).
[2] H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, Paris, France, 1856.
[3] J. L. M. Poiseuille, Memoires des Savants Etrangers, Vol. IX pp. 435-544, (1846);. Brillouin, Marcel (1930). "Jean Leonard Marie Poiseuille". Journal of Rheology. 1: 345. doi: 10.1122/1.2116329.
[4] H. M. Quinn, “Reconciliation of packed column permeability data, column permeability as a function of particle porosity,” Journal of Materials, vol. 2014, Article ID 636507, 22 pages, 2014.
[5] J. M. Coulson; University of London, Ph. D. thesis, “The Streamline Flow of Liquids through beds comprised of Spherical particles” 1935.
[6] A. O. Oman and K. M. Watson, “Pressure drops in granular beds,” National Petroleum News, vol. 36, pp. R795–R802, 1944.
[7] M. Leva and M. Grummer, “Pressure drop through packed tubes, part I, a general correlation,” vol. 43, pp. 549–554, 1947.
[8] F. A. L. Dullien, Porous Media, Fluid Transport and Pore Structure, Acedemic Press, 2nd edition, 1979.
[9] S. W. Churchill, Viscous Flows: The Practical Use of Theory, Butterworks, 1988.
[10] S. P. Burke and W. B. Plummer, “Gas flow through packed columns,” Industrial and Engineering Chemistry, vol. 20, pp. 1196–1200, 1923.
[11] J. C. Giddings, Dynamics of Chromatography, Part I, Principles and Theory, Marcel Dekker, Inc. New York, 1965.
[12] T. Farkas, G. Zhong, G. Guiochon, Journal of Chromatography A, 849, (1999) 35-43.
[13] M. Rhodes, Introduction to Particle technology, John Wiley & Sons, Inc., p. 83 (1998).
[14] G. O. Brown., 1999-2006, Henry Darcy and His Law, www.biosystems.okstate.edu/Darcy.
[15] I. Halász, R. Endele, and J. Asshauer, “Ultimate limits in high-pressure liquid chromatography,” Journal of Chromatography A, vol. 112, no. C, pp. 37–60, 1975.
[16] F. E. Blake, “The resistance of packing to fluid flow,” Transaction of American Institute of Chemical Engineers, vol. 14, pp. 415–421, 1922.
[17] J. Kozeny, “Uber kapillare Leitung des wassers in Böden,” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, vol. 136, pp. 271–306, 1927.
[18] Carman, P. C., “Fluid flow through granular beds,” Transactions of the Institution of Chemical Engineers, vol. 15, pp. 155–166, 1937.
[19] Bird, R. B., Stewart, W. E., Lightfoot, E. N. Transport Phenomena, John Wiley & Sons, Inc., p. 190.
[20] H. M. Quinn, Reconciliation of Packed Column Permeability Data-Part 1. The Teaching Of Giddings Revisited, Special Topics & Reviews in Porous Media-An International Journal 1 (1), (2010) 79-86.
[21] I. Halasz, M. Naefe, Analytical Chemistry, 44 (1972) 76
[22] G. Guiochon, Chromatographic Review, 8 (1966).
[23] A. E. Scheidegger, The Physics of Flow Through Porous Media, MacMillan Company, New York, NY, USA, 1957.
[24] J. Kozeny, "Ueber kapillare Leitung des Wassers im Boden." Sitzungsber Akad. Wiss., Wien, 136 (2a): 271-306, 1927.
[25] J. C. Giddings, Unified Separation Science, John Wiley & Sons (1991).
[26] Halasz, R. Endele, K. Unger, Journal of Chromatography, 99 (1974) 377-393.
[27] U. Neue, HPLC Columns-Theory, Technology and Practice, Wiley-VCH (1997).
[28] P. C. Carman, Trans. Instn. Chem. Engrs. Vol. 15, (1937) 155-166.
[29] J. M. Godinho, A. E. Reising, U. Tallarek, J. W. Jorgenson; Implementation of high slurry concentration and sonication to pack high-efficiency, meter-long capillary ultrahigh pressure liquid chromatography columns: Journal of Chromatography A, 1462 (2016) 165-169.
[30] L. R. Snyder, J. J. Kirkland, Introduction to Modern Liquid Chromatography, 2nd Edition, John Wiley & Sons, Inc. p. 37 (1979).
[31] G. Guiochon, S. G. Shirazi, A. M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, Boston, Ma, (1994).
[32] S. Ergun and A. A. Orning, “Fluid flow through randomly packed columns and fluidized beds,” Industrial & Engineering Chemistry, vol. 4, no. 6, pp. 1179–1184, 1949.
[33] Ergun, Chem. Eng. Progr. 48 (1952) 89-94.
[34] I. F. Macdonald, M. S. El-Sayed, K. Mow, and F. A. L. Dullien Industrial & Engineering Chemistry Fundamentals 1979 18 (3), 199-208 DOI: 10.1021/i160071a001.
[35] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, 1965.
[36] Reynolds O. 1883. An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels. Philos. Trans. R. Soc. 174: 935–82.
[37] J. Nikuradze, NASA TT F-10, 359, Laws of Turbulent Flow in Smooth Pipes. Translated from “Gesetzmassigkeiten der turbulenten Stromung in glatten Rohren” VDI (Verein Deutsher Ingenieure)-Forschungsheft 356.
[38] J. Nikuradze, NACA TM 1292, Laws of Flow in Rough Pipes, July/August 1933. Translation of “Stromungsgesetze in rauhen Rohren.” VDI-Forschungsheft 361. Beilage zu “Forschung auf dem Gebiete des Ingenieurwesens” Ausgabe B Band 4, July/August 1933.
[39] L. Prandtl, in Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg 1904, A. Krazer, ed., Teubner, Leipzig, Germany (1905), p. 484. English trans. in Early Developments of Modern Aerodynamics, J. A. K. Ackroyd, B. P. Axcell, A. I. Ruban, eds., Butterworth-Heinemann, Oxford, UK (2001), p. 77.
[40] Moody, L. F. (1944). "Friction factors for pipe flow." Trans. ASME, 66: 671-678.
[41] Studies and Research on Friction, Friction Factor and Affecting Factors: A Review Sunil J. Kulkarni *, Ajaygiri K. Goswami; Chemical Engineering Department,, Datta Meghe College of Engineering, Airoli, Navi Mumbai, Maharashtra, India.
[42] Technical Note: Friction Factor Diagrams for Pipe Flow; Jim McGovern Department of Mechanical Engineering and Dublin Energy Lab Dublin Institute of Technology, Bolton Street Dublin 1, Ireland.
[43] B. J. Mckeon, C. J. Swanson, M. V. Zagarola, R. J. Donnelly and A. J. Smits. Friction factors for smooth pipe flow; J. Fluid Mech. (2004), vol. 511, pp. 41-44. Cambridge University Press; DO1; 10.1017/S0022112004009796.
[44] Unified fluid flow model for pressure transient analysis in naturally fractured media; Petro Babak1 and Jalel Azaiez; Journal of Physics A: Mathematical and Theoretical, Volume 48, Number 17.
[45] Quinn, H. M. Quinn’s Law of Fluid Dynamics Pressure-driven Fluid Flow Through Closed Conduits, Fluid Mechanics. Vol. 5, No. 2, 2019, pp. 39-71. doi: 10.11648/j.fm.20190502.12.
[46] Jan H. van Lopik1 Roy Snoeijers1 Teun C. G. W. van Dooren1 Amir Raoof1 Ruud J. Schotting; Transp Porous Med (2017) 120: 37–66 DOI 10.1007/s11242-017-0903-3.
[47] Forchheimer, P.: Wasserbewegung durch boden. Zeit. Ver. Deutsch. Ing 45, 1781–1788 (1901).
[48] Quinn, H. M., Quinn’s Law of Fluid Dynamics; Pressure-driven Fluid Flow through Closed Conduits. Fluid Mechanics. Vol. 5, No. 2, 2019, pp. 39-71. doi: 10.11648/j.fm.20190502.12.
[49] Quinn, H. M., Quinn’s Law of Fluid Dynamics: Supplement #2 Reinventing the Ergun Equation. Fluid Mechanics. Vol. 6, No. 1, 2020, pp. 15-29. doi: 10.11648/j.fm.20200601.12.
[50] Sidiropoulou, M. G., Moutsopoulos, K. N., Tsihrintzis, V. A.: Determination of Forchheimer equation coefficients a and b. Hydrol. Process. 21 (4), 534–554 (2007).
[51] Quinn, H. M., Quinn’s Law of Fluid Dynamics: Supplement #1 Nikuradze’s Inflection Profile Revisited. Fluid Mechanics. Vol. 6, No. 1, 2020, pp. 1-14. doi: 10.11648/j.fm.20200601.11.
[52] Banerjee A., Srinivas P., Mritunjay K. S., Selchar C. D., Kumar G. N. P., Modelling of Flow Through Porous Media Over the Complete Flow Regime., Transport in Porous Media. Doi.org/10.1007/s11242-019-01274-2.
[53] G. N. Pradeep Kumar, Dr. P. Srinivas; A Revisit To Forchheimer Equation Applied In Porous Media Flow; International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 www.ijres.org Volume 2 Issue 6 June. 2014 PP.41-53.
[54] Mohammad-Bagher Salahi; Mohammad Sedghi-Asl; and Mansour Parvizi; Nonlinear Flow through a Packed-Column Experiment. Journal of Hydrologic Engineering December 2014. DOI: 10.1061/(ASCE)HE.1943-5584.0001166.
[55] Li, Z., Wan, J., Zhan, H., Cheng, X., Chang, W., Huang, K., Particle Size Distribution on Forchheimer Flow and Transition of Flow Regimes in Porous Media, Journal of Hydrology (2019), https://doi.org/10.1016/j.jhydrol.2019.04.026.
[56] J. Mazzeo, U. D. Neue, M. Kele, R. S. Plumb; Analytical Chemistry, December 2005, 460-467.
[57] D. Cabooter, J. Billen, H. Terryn, F. Lynen, P. Sandra, G. Desmet; Journal of Chromatography A, 1178 (2008) 108–117.
Cite This Article
  • APA Style

    Hubert Michael Quinn. (2020). Quinn’s Law of Fluid Dynamics: Supplement #3 A Unique Solution to the Navier-Stokes Equation for Fluid Flow in Closed Conduits. Fluid Mechanics, 6(2), 30-50. https://doi.org/10.11648/j.fm.20200602.11

    Copy | Download

    ACS Style

    Hubert Michael Quinn. Quinn’s Law of Fluid Dynamics: Supplement #3 A Unique Solution to the Navier-Stokes Equation for Fluid Flow in Closed Conduits. Fluid Mech. 2020, 6(2), 30-50. doi: 10.11648/j.fm.20200602.11

    Copy | Download

    AMA Style

    Hubert Michael Quinn. Quinn’s Law of Fluid Dynamics: Supplement #3 A Unique Solution to the Navier-Stokes Equation for Fluid Flow in Closed Conduits. Fluid Mech. 2020;6(2):30-50. doi: 10.11648/j.fm.20200602.11

    Copy | Download

  • @article{10.11648/j.fm.20200602.11,
      author = {Hubert Michael Quinn},
      title = {Quinn’s Law of Fluid Dynamics: Supplement #3 A Unique Solution to the Navier-Stokes Equation for Fluid Flow in Closed Conduits},
      journal = {Fluid Mechanics},
      volume = {6},
      number = {2},
      pages = {30-50},
      doi = {10.11648/j.fm.20200602.11},
      url = {https://doi.org/10.11648/j.fm.20200602.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.fm.20200602.11},
      abstract = {The recent publication of Quinn’s Law of Fluid Dynamics brings into focus longstanding contradictions regarding permeability in closed conduits that have littered the fluid dynamics landscape for more than 150 years. In this paper, we will use this new level of understanding to explain these contradictions, in layman’s terms, and resolve them, by introducing for the first time, as far as we know, a unique solution to the Navier-Stokes equation for fluid flow in closed conduits, which is understandable by knowledgeable physicists, engineers, chromatographers and aerospace enthusiasts alike, but who may not necessarily be versed in the abstract jargon of a graduate in advanced mathematics. In addition, we will apply our unique solution to chosen illustrative worked examples, as well as those of third parties from the published literature. In so doing, we will demonstrate the utility of our solution, not only, to packed conduits containing particles having solid skeletons, but also, to empty conduits, which in the context of this new understanding of fluid dynamics in closed conduits, represents a special case of a packed conduit in which the particles are fully porous, i.e., they are made entirely of free space.},
     year = {2020}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Quinn’s Law of Fluid Dynamics: Supplement #3 A Unique Solution to the Navier-Stokes Equation for Fluid Flow in Closed Conduits
    AU  - Hubert Michael Quinn
    Y1  - 2020/08/25
    PY  - 2020
    N1  - https://doi.org/10.11648/j.fm.20200602.11
    DO  - 10.11648/j.fm.20200602.11
    T2  - Fluid Mechanics
    JF  - Fluid Mechanics
    JO  - Fluid Mechanics
    SP  - 30
    EP  - 50
    PB  - Science Publishing Group
    SN  - 2575-1816
    UR  - https://doi.org/10.11648/j.fm.20200602.11
    AB  - The recent publication of Quinn’s Law of Fluid Dynamics brings into focus longstanding contradictions regarding permeability in closed conduits that have littered the fluid dynamics landscape for more than 150 years. In this paper, we will use this new level of understanding to explain these contradictions, in layman’s terms, and resolve them, by introducing for the first time, as far as we know, a unique solution to the Navier-Stokes equation for fluid flow in closed conduits, which is understandable by knowledgeable physicists, engineers, chromatographers and aerospace enthusiasts alike, but who may not necessarily be versed in the abstract jargon of a graduate in advanced mathematics. In addition, we will apply our unique solution to chosen illustrative worked examples, as well as those of third parties from the published literature. In so doing, we will demonstrate the utility of our solution, not only, to packed conduits containing particles having solid skeletons, but also, to empty conduits, which in the context of this new understanding of fluid dynamics in closed conduits, represents a special case of a packed conduit in which the particles are fully porous, i.e., they are made entirely of free space.
    VL  - 6
    IS  - 2
    ER  - 

    Copy | Download

Author Information
  • Department of Research and Development, the Wrangler Group LLC, Brighton, USA

  • Sections