The Solution Equivalent of the Navier-Stokes Equation in HPLC

Volume 4, Issue 1, February 2023     |     PP. 1-34      |     PDF (2573 K)    |     Pub. Date: September 18, 2023
DOI: 10.54647/mechanics130034    53 Downloads     975 Views  

Author(s)

Hubert M. Quinn, The Wrangler Group LLC, 40 Nottinghill Road, Brighton, Ma. 02135, United States

Abstract
The Navier-Stokes equation is generally considered the ultimate mathematical expression for the dictates of the Laws of Nature which pertain to transport phenomena in the field of fluid dynamics. It is written and typically discussed, however, in the form and jargon of advanced mathematics. This makes it very difficult for any nonmathematician to understand, and this, in part, is why it remains unsolved for most applications. The essence of the equation, however, has nothing to do with mathematics and everything to do with the underlying physics surrounding the fluid transport mechanisms involved in any given fluid flow embodiment. Accordingly, it is the non-mathematical “solution equivalent” of the N-S equation that is important to the practitioner of fluid dynamics. In the case of HPLC (High Pressure Liquid Chromatography), for instance, this means the physics underlying fluid flow through conduits packed with partially porous solid particles. Recently, 2019, a new fluid flow model (QFFM) was published which contains, embedded in its framework, the “solution equivalent” for the N-S equation in chromatographic columns. This novel fluid flow model teaches that an empty HPLC column is a special case of the same column packed with solid particles. In fact, one is the mirror image of the other. The difference between the two is defined by the choice of independent variables. Thus, by setting the value of three independent variables in the QFFM, the complexity of the advanced mathematics in the Navier-Stokes equation can be avoided. If one considers “matter” and “anti-matter” to be the mirror image of one another, however, one can easily rationalize the rules of engagement which underlie this phenomenon in the context of the Navier-Stokes equation. In this paper we will explain how the QFFM rationalizes the fundamental issues of the Navier-Stokes equation, providing the “solution equivalent”, in the jargon of classical mechanics, as opposed to that of advanced mathematics, for fluid flow through HPLC columns.

Keywords
Conduit Porosity; Hypothetical Q particles; Particle porosity: Packed beds; Column Permeability.

Cite this paper
Hubert M. Quinn, The Solution Equivalent of the Navier-Stokes Equation in HPLC , SCIREA Journal of Mechanics. Volume 4, Issue 1, February 2023 | PP. 1-34. 10.54647/mechanics130034

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. Halasz, M. Naefe, Analytical Chemistry, 44 (1972) 76
[ 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 ] Halasz, R. Endele, K. Unger, Journal of Chromatography, 99 (1974) 377-393
[ 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)