.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluid_engineering/measurement/plot_03_torricelli_efflux.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_fluid_engineering_measurement_plot_03_torricelli_efflux.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_fluid_engineering_measurement_plot_03_torricelli_efflux.py:


Torricelli's law of efflux (1644)
=================================

Evangelista Torricelli, Galileo's successor at Florence, found that water
spouts from a small hole in a vessel at the speed a body gains falling
from the free surface, :math:`v = \sqrt{2gh}`, and published it in the
*Opera geometrica* of 1644. The jet narrows just past a sharp-edged hole
(the vena contracta), so the flow is a discharge coefficient of about
0.61 times the ideal one
(:func:`~engineeringkit.fluid_engineering.orifice_flow`). This example
drains a tank through an orifice
(:func:`~engineeringkit.fluid_engineering.tank_drain_time`) and checks
the square-root law of the level against a step-by-step integration.

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import integrate

    from engineeringkit.fluid_engineering import G, orifice_flow, tank_drain_time








.. GENERATED FROM PYTHON SOURCE LINES 25-30

Speed of the jet
----------------

Torricelli's speed is the free-fall speed from the surface, so a jet
pointed upward would just reach the surface again.

.. GENERATED FROM PYTHON SOURCE LINES 30-35

.. code-block:: Python


    h = 2.0
    v = np.sqrt(2 * G * h)
    print(f"jet speed from {h} m of head: {v:.2f} m/s, rising {v**2 / (2 * G):.2f} m")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    jet speed from 2.0 m of head: 6.26 m/s, rising 2.00 m




.. GENERATED FROM PYTHON SOURCE LINES 36-42

Draining a tank
---------------

A water butt 0.6 m across with a 20 mm sharp-edged hole at the bottom.
The level obeys :math:`A_t\,dh/dt = -C_d A_o\sqrt{2gh}`, so
:math:`\sqrt h` falls linearly in time.

.. GENERATED FROM PYTHON SOURCE LINES 42-59

.. code-block:: Python


    A_tank, A_hole, h0, cd = np.pi * 0.3**2, np.pi * 0.01**2, 1.0, 0.61
    T = tank_drain_time(A_tank, A_hole, h0, discharge_coefficient=cd)
    t = np.linspace(0, T, 200)
    h_exact = (np.sqrt(h0) - t / T * np.sqrt(h0)) ** 2

    sol = integrate.solve_ivp(
        lambda t, h: [-orifice_flow(A_hole, max(h[0], 0.0), cd) / A_tank], (0, T), [h0], t_eval=t[:-1], rtol=1e-10, atol=1e-12
    )
    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot(t / 60, h_exact, label="Torricelli, closed form")
    ax.plot(sol.t[::20] / 60, sol.y[0][::20], "o", label="step-by-step integration")
    ax.set(xlabel="time [min]", ylabel="water level [m]", title="Draining a tank through an orifice")
    ax.legend()
    fig.tight_layout()
    print(f"drain time {T / 60:.1f} min; the top half goes in {tank_drain_time(A_tank, A_hole, h0, h0 / 2, cd) / 60:.1f} min")




.. image-sg:: /api/gallery/fluid_engineering/measurement/images/sphx_glr_plot_03_torricelli_efflux_001.png
   :alt: Draining a tank through an orifice
   :srcset: /api/gallery/fluid_engineering/measurement/images/sphx_glr_plot_03_torricelli_efflux_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    drain time 11.1 min; the top half goes in 3.3 min




.. GENERATED FROM PYTHON SOURCE LINES 60-64

The vena contracta
------------------

An ideal jet (:math:`C_d = 1`) would empty the tank 1/0.61 times faster.

.. GENERATED FROM PYTHON SOURCE LINES 64-73

.. code-block:: Python


    ideal = tank_drain_time(A_tank, A_hole, h0, discharge_coefficient=1.0)
    print(f"ideal drain time {ideal / 60:.1f} min")

    assert np.allclose(sol.y[0], h_exact[:-1], atol=1e-6)
    assert np.isclose(T / ideal, 1 / cd)
    assert np.isclose(orifice_flow(A_hole, h, 1.0), A_hole * v)

    plt.show()




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ideal drain time 6.8 min





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.169 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_measurement_plot_03_torricelli_efflux.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/fluid_engineering/measurement/plot_03_torricelli_efflux.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_03_torricelli_efflux.ipynb <plot_03_torricelli_efflux.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_03_torricelli_efflux.py <plot_03_torricelli_efflux.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_03_torricelli_efflux.zip <plot_03_torricelli_efflux.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_