.. _sphx_glr_api_gallery_fluid_engineering_pipe_flow:

Pipe flow
---------

Laminar and turbulent flow in pipes, head losses and friction factors.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Gotthilf Hagen in 1839, timing water through brass tubes, and Jean Léonard Marie Poiseuille in 1840, studying blood flow in glass capillaries, found independently that slow flow through a tube grows with the pressure drop and with the fourth power of the radius. The law Q = \pi R^4\Delta p/(8\mu L) (~engineeringkit.fluid_engineering.poiseuille_flow) follows from a parabolic velocity profile (~engineeringkit.fluid_engineering.poiseuille_velocity), and in Darcy&#x27;s form it is the laminar friction factor f = 64/Re (~engineeringkit.fluid_engineering.laminar_friction_factor).">

.. only:: html

  .. image:: /api/gallery/fluid_engineering/pipe_flow/images/thumb/sphx_glr_plot_01_hagen_poiseuille_thumb.png
    :alt:

  :doc:`/api/gallery/fluid_engineering/pipe_flow/plot_01_hagen_poiseuille`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Hagen and Poiseuille: laminar pipe flow (1839-1840)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Julius Weisbach wrote the friction loss in a pipe in 1845 as h_f = f\,(L/D)\,v^2/(2g), with a coefficient that he fitted to experiments. Henry Darcy&#x27;s tests on new and incrusted cast-iron mains in Paris, published in 1857, showed that the coefficient depends on the diameter and the state of the wall. The Darcy-Weisbach equation (~engineeringkit.fluid_engineering.darcy_head_loss), with loss coefficients for the fittings (~engineeringkit.fluid_engineering.minor_head_loss), is still how pipelines are sized. This example follows the energy and hydraulic grade lines of water flowing from a reservoir through two pipes to a nozzle.">

.. only:: html

  .. image:: /api/gallery/fluid_engineering/pipe_flow/images/thumb/sphx_glr_plot_02_darcy_weisbach_thumb.png
    :alt:

  :doc:`/api/gallery/fluid_engineering/pipe_flow/plot_02_darcy_weisbach`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Weisbach and Darcy: head loss in pipes (1845-1857)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="In 1883 Osborne Reynolds ran water from a tank through glass tubes and fed a thin streak of dye into the entrance. At low speed the streak stayed a straight thread; above a critical speed it broke up and coloured the whole tube. The critical speed fell as the tube widened and as the water warmed, and Reynolds showed that every case met at one value of \rho v D/\mu (~engineeringkit.fluid_engineering.reynolds_number). This example computes the critical speed against diameter and temperature, and shows the jump in friction factor across the transition.">

.. only:: html

  .. image:: /api/gallery/fluid_engineering/pipe_flow/images/thumb/sphx_glr_plot_03_reynolds_transition_thumb.png
    :alt:

  :doc:`/api/gallery/fluid_engineering/pipe_flow/plot_03_reynolds_transition`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Reynolds's dye experiment (1883)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Johann Nikuradse, in Prandtl&#x27;s laboratory at Göttingen, glued graded sand to the inside of brass pipes and measured friction over six roughnesses, from a radius 15 to 507 times the grain size, published in 1933. The curves share the smooth-pipe law at low Reynolds number (~engineeringkit.fluid_engineering.smooth_pipe_friction_factor), leave it once the grains poke through the viscous sublayer, and settle on a constant that depends only on the roughness, 1/\sqrt f = 1.74 + 2\log_{10}(R/k_s) (~engineeringkit.fluid_engineering.nikuradse_friction_factor). This example draws the two limits and finds where each roughness becomes fully rough.">

.. only:: html

  .. image:: /api/gallery/fluid_engineering/pipe_flow/images/thumb/sphx_glr_plot_04_nikuradse_roughness_thumb.png
    :alt:

  :doc:`/api/gallery/fluid_engineering/pipe_flow/plot_04_nikuradse_roughness`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Nikuradse's sand-roughened pipes (1933)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Commercial pipes do not behave like Nikuradse&#x27;s sand: their roughness is uneven, and friction leaves the smooth law gradually rather than with Nikuradse&#x27;s dip. Cyril Colebrook, after experiments with Cedric White on pipes coated with mixed grains, added the smooth and rough laws inside one logarithm in 1939,">

.. only:: html

  .. image:: /api/gallery/fluid_engineering/pipe_flow/images/thumb/sphx_glr_plot_05_colebrook_equation_thumb.png
    :alt:

  :doc:`/api/gallery/fluid_engineering/pipe_flow/plot_05_colebrook_equation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Colebrook's turbulent friction factor (1939)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Lewis Moody, a Princeton professor of hydraulics, plotted Colebrook&#x27;s equation for the roughness of commercial pipes in 1944, together with the laminar line, on one log-log chart (~engineeringkit.fluid_engineering.visualizers.plot_moody_diagram). Engineers read friction factors from it for the next half-century. Moody also showed how to solve the three classic pipe problems with it: find the head loss, the flow, or the diameter. This example draws the chart and solves the third, which needs iteration even with colebrook_friction_factor in hand.">

.. only:: html

  .. image:: /api/gallery/fluid_engineering/pipe_flow/images/thumb/sphx_glr_plot_06_moody_diagram_thumb.png
    :alt:

  :doc:`/api/gallery/fluid_engineering/pipe_flow/plot_06_moody_diagram`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Moody's diagram (1944)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Colebrook&#x27;s equation needs iteration, which was a nuisance on the first programmable calculators. In 1976 Prabhata Swamee and Akalank Jain fitted explicit formulas for all three of Moody&#x27;s pipe problems. Their friction factor,">

.. only:: html

  .. image:: /api/gallery/fluid_engineering/pipe_flow/images/thumb/sphx_glr_plot_07_swamee_jain_thumb.png
    :alt:

  :doc:`/api/gallery/fluid_engineering/pipe_flow/plot_07_swamee_jain`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Swamee and Jain's explicit friction factor (1976)</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/fluid_engineering/pipe_flow/plot_01_hagen_poiseuille
   /api/gallery/fluid_engineering/pipe_flow/plot_02_darcy_weisbach
   /api/gallery/fluid_engineering/pipe_flow/plot_03_reynolds_transition
   /api/gallery/fluid_engineering/pipe_flow/plot_04_nikuradse_roughness
   /api/gallery/fluid_engineering/pipe_flow/plot_05_colebrook_equation
   /api/gallery/fluid_engineering/pipe_flow/plot_06_moody_diagram
   /api/gallery/fluid_engineering/pipe_flow/plot_07_swamee_jain