.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluid_engineering/pumps/plot_02_euler_turbine_equation.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_pumps_plot_02_euler_turbine_equation.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_pumps_plot_02_euler_turbine_equation.py:


Euler's turbine equation (1754)
===============================

Leonhard Euler, studying Segner's reaction water wheel at the Berlin
academy, wrote the torque of a rotor as the change in the angular momentum
of the water passing through it. Multiplied by the speed, it gives the head
a pump adds or a turbine takes,
:math:`H = (u_2V_{t2} - u_1V_{t1})/g`
(:func:`~engineeringkit.fluid_engineering.euler_turbomachine_head`). This
example draws the ideal head of a centrifugal impeller for backward,
radial and forward blades, and shows that the head scales with the square
of the speed, the affinity law
(:func:`~engineeringkit.fluid_engineering.affinity_laws`).

.. GENERATED FROM PYTHON SOURCE LINES 18-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.fluid_engineering import affinity_laws, euler_turbomachine_head








.. GENERATED FROM PYTHON SOURCE LINES 24-32

The outlet velocity triangle
----------------------------

Water leaves an impeller of radius :math:`r_2` and width :math:`b_2`
with radial speed :math:`V_{r2} = Q/(2\pi r_2 b_2)`. Relative to the
blade it leaves at the blade angle :math:`\beta_2`, so its swirl is
:math:`V_{t2} = u_2 - V_{r2}\cot\beta_2`. With no swirl at the inlet,
the ideal head falls linearly with flow for backward-curved blades.

.. GENERATED FROM PYTHON SOURCE LINES 32-49

.. code-block:: Python


    r2, b2, rpm = 0.15, 0.02, 1750
    omega = rpm * 2 * np.pi / 60
    u2 = omega * r2
    Q = np.linspace(0, 0.08, 50)
    V_r2 = Q / (2 * np.pi * r2 * b2)

    fig, ax = plt.subplots(figsize=(6, 4))
    heads = {}
    for label, beta in [("backward, β₂ = 25°", 25), ("radial, β₂ = 90°", 90), ("forward, β₂ = 140°", 140)]:
        heads[beta] = euler_turbomachine_head(u2, u2 - V_r2 / np.tan(np.radians(beta)))
        ax.plot(Q * 1000, heads[beta], label=label)
    ax.set(xlabel="flow [L/s]", ylabel="ideal head [m]", title=f"Euler head of an impeller at {rpm} rpm")
    ax.legend()
    fig.tight_layout()
    print(f"shut-off head u2^2/g = {heads[25][0]:.1f} m for every blade angle")




.. image-sg:: /api/gallery/fluid_engineering/pumps/images/sphx_glr_plot_02_euler_turbine_equation_001.png
   :alt: Euler head of an impeller at 1750 rpm
   :srcset: /api/gallery/fluid_engineering/pumps/images/sphx_glr_plot_02_euler_turbine_equation_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    shut-off head u2^2/g = 77.1 m for every blade angle




.. GENERATED FROM PYTHON SOURCE LINES 50-58

Backward blades are used in practice: their head falls with flow, so the
pump is stable on its system curve and its power does not run away.

The head at twice the speed
---------------------------

At the same flow coefficient, the velocity triangle keeps its shape and
every speed doubles, so the head goes up four times.

.. GENERATED FROM PYTHON SOURCE LINES 58-70

.. code-block:: Python


    Q_bep, H_bep = 0.04, float(np.interp(0.04, Q, heads[25]))
    Q2, H2, _ = affinity_laws(Q_bep, H_bep, 0.0, 2.0)
    V_r2_fast = Q2 / (2 * np.pi * r2 * b2)
    H2_euler = euler_turbomachine_head(2 * u2, 2 * u2 - V_r2_fast / np.tan(np.radians(25)))
    print(f"at 2x speed: affinity law {H2:.1f} m, Euler {H2_euler:.1f} m")

    assert np.allclose([heads[25][0], heads[90][0], heads[140][0]], u2**2 / 9.80665)
    assert np.allclose(heads[90], heads[90][0]) and np.all(np.diff(heads[25]) < 0) and np.all(np.diff(heads[140]) > 0)
    assert np.isclose(H2, H2_euler)

    plt.show()




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

 .. code-block:: none

    at 2x speed: affinity law 257.2 m, Euler 257.2 m





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

   **Total running time of the script:** (0 minutes 0.087 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_pumps_plot_02_euler_turbine_equation.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/pumps/plot_02_euler_turbine_equation.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_euler_turbine_equation.ipynb <plot_02_euler_turbine_equation.ipynb>`

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

      :download:`Download Python source code: plot_02_euler_turbine_equation.py <plot_02_euler_turbine_equation.py>`

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

      :download:`Download zipped: plot_02_euler_turbine_equation.zip <plot_02_euler_turbine_equation.zip>`


.. only:: html

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

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