{"id":446,"date":"2022-12-11T23:08:59","date_gmt":"2022-12-11T23:08:59","guid":{"rendered":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/chapter\/exercise-11\/"},"modified":"2023-01-14T17:22:23","modified_gmt":"2023-01-14T17:22:23","slug":"exercise-11","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/chapter\/exercise-11\/","title":{"raw":"Exercise 11","rendered":"Exercise 11"},"content":{"raw":"
\r\n

The Hagen-Poiseuille equation, also known as the Hagen-Poiseuille law, Poiseuille Law or Poiseuille equation, is a physical law that describes the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. The assumptions of the equation are that the fluid is incompressible and Newtonian; the flow is laminar through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no acceleration of fluid in the pipe. For velocities and pipe diameters above a threshold, actual fluid flow is not laminar but turbulent, leading to larger pressure drops than calculated by the Hagen-Poiseuille equation as shown here.<\/p>\r\n

[latex]\\displaystyle \\Delta p=\\frac{8\\mu LQ}{\\pi R^{4}}=\\frac{8\\pi \\mu LQ}{A^{2}}[\/latex]<\/p>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u0394p<\/em><\/td>\r\n=<\/td>\r\npressure difference between the two ends (ML\u22121<\/sup>T\u22122<\/sup>)<\/td>\r\n<\/tr>\r\n
L<\/em><\/td>\r\n=<\/td>\r\nlength of pipe (L)<\/td>\r\n<\/tr>\r\n
\u03bc<\/em><\/td>\r\n=<\/td>\r\ndynamic viscosity<\/a> (ML\u22121<\/sup>T\u22121<\/sup>)<\/td>\r\n<\/tr>\r\n
Q<\/em><\/td>\r\n=<\/td>\r\nvolumetric flow rate<\/a> (L3<\/sup>T\u22121<\/sup>)<\/td>\r\n<\/tr>\r\n
R<\/em><\/td>\r\n=<\/td>\r\npipe\u00a0radius<\/a> (L)<\/td>\r\n<\/tr>\r\n
A<\/em><\/td>\r\n=<\/td>\r\ncross section<\/a>\u00a0of pipe (L2<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The equation does not hold close to the pipe entrance. The equation fails in the limit of low viscosity, wide and\/or short pipe. Low viscosity or a wide pipe may result in turbulent flow, making it necessary to use more complex models, such as the Darcy<\/span>-<\/span>Weisbach<\/span> equation<\/span><\/a>. The ratio of length to radius of a pipe should be greater than one forty-eighth [latex]\\left ( \\textrm{i.e.,}> \\frac{1}{48} \\right )[\/latex] of the Reynolds<\/span>\u00a0<\/span>number<\/span><\/a> for the Hagen-Poiseuille law to be valid.<\/p>\r\n\r\n

    \r\n \t
  1. What are the assumptions associated with the Hagen-Poiseulle equation and the Poiseuille law?<\/li>\r\n \t
  2. How does the viscosity of the fluid change the relationship between pressure gradient and flow?<\/li>\r\n \t
  3. How is the equation for laminar flow in a full pipe similar to Darcy\u2019s law?<\/li>\r\n<\/ol>\r\n

    Click here for solution to <\/span>Exercise <\/span>11<\/span><\/a><\/p>\r\n

    Return to where text linked to Exercise <\/span>11<\/span><\/a><\/p>\r\n

    <\/p>\r\n\r\n<\/div>","rendered":"

    \n

    The Hagen-Poiseuille equation, also known as the Hagen-Poiseuille law, Poiseuille Law or Poiseuille equation, is a physical law that describes the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. The assumptions of the equation are that the fluid is incompressible and Newtonian; the flow is laminar through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no acceleration of fluid in the pipe. For velocities and pipe diameters above a threshold, actual fluid flow is not laminar but turbulent, leading to larger pressure drops than calculated by the Hagen-Poiseuille equation as shown here.<\/p>\n

    \"\displaystyle \Delta p=\frac{8\mu LQ}{\pi R^{4}}=\frac{8\pi \mu LQ}{A^{2}}\"<\/p>\n

    where:<\/p>\n\n\n\n\n\n\n\n\n
    \u0394p<\/em><\/td>\n=<\/td>\npressure difference between the two ends (ML\u22121<\/sup>T\u22122<\/sup>)<\/td>\n<\/tr>\n
    L<\/em><\/td>\n=<\/td>\nlength of pipe (L)<\/td>\n<\/tr>\n
    \u03bc<\/em><\/td>\n=<\/td>\ndynamic viscosity<\/a> (ML\u22121<\/sup>T\u22121<\/sup>)<\/td>\n<\/tr>\n
    Q<\/em><\/td>\n=<\/td>\nvolumetric flow rate<\/a> (L3<\/sup>T\u22121<\/sup>)<\/td>\n<\/tr>\n
    R<\/em><\/td>\n=<\/td>\npipe\u00a0radius<\/a> (L)<\/td>\n<\/tr>\n
    A<\/em><\/td>\n=<\/td>\ncross section<\/a>\u00a0of pipe (L2<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    The equation does not hold close to the pipe entrance. The equation fails in the limit of low viscosity, wide and\/or short pipe. Low viscosity or a wide pipe may result in turbulent flow, making it necessary to use more complex models, such as the Darcy<\/span>–<\/span>Weisbach<\/span> equation<\/span><\/a>. The ratio of length to radius of a pipe should be greater than one forty-eighth \"\left ( \textrm{i.e.,}> \frac{1}{48} \right )\" of the Reynolds<\/span>\u00a0<\/span>number<\/span><\/a> for the Hagen-Poiseuille law to be valid.<\/p>\n

      \n
    1. What are the assumptions associated with the Hagen-Poiseulle equation and the Poiseuille law?<\/li>\n
    2. How does the viscosity of the fluid change the relationship between pressure gradient and flow?<\/li>\n
    3. How is the equation for laminar flow in a full pipe similar to Darcy\u2019s law?<\/li>\n<\/ol>\n

      Click here for solution to <\/span>Exercise <\/span>11<\/span><\/a><\/p>\n

      Return to where text linked to Exercise <\/span>11<\/span><\/a><\/p>\n

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