.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/fluid_engineering/momentum/plot_01_pelton_wheel.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` 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_momentum_plot_01_pelton_wheel.py: Pelton's water wheel (1880) =========================== Lester Pelton, a millwright in the California gold country, patented in 1880 a wheel whose buckets split the jet in two and turn each half almost back on itself. By the momentum equation a vane that turns a jet through :math:`\theta` is pushed with :math:`\rho A (V - u)^2 (1 - \cos\theta)` (:func:`~engineeringkit.fluid_engineering.jet_force`); a wheel, which always has a bucket in the jet, takes :math:`\rho Q u (V - u)(1 - \cos\theta)` (:func:`~engineeringkit.fluid_engineering.pelton_wheel_power`). Turning the jet through nearly 180° and running the buckets at half the jet speed leaves the water with almost no speed at all. This example compares Pelton's buckets with the flat paddles of the hurdy-gurdy wheels they replaced. .. GENERATED FROM PYTHON SOURCE LINES 20-25 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from engineeringkit.fluid_engineering import G, jet_force, pelton_wheel_power .. GENERATED FROM PYTHON SOURCE LINES 26-30 A jet from 300 m of head ------------------------ A 0.1 m nozzle with velocity coefficient 0.98. .. GENERATED FROM PYTHON SOURCE LINES 30-38 .. code-block:: Python H, d_jet, rho = 300.0, 0.1, 1000.0 V = 0.98 * np.sqrt(2 * G * H) A = np.pi * d_jet**2 / 4 Q = A * V jet_power = 0.5 * rho * Q * V**2 print(f"jet speed {V:.1f} m/s, flow {Q:.3f} m³/s, jet power {jet_power / 1e6:.2f} MW") .. rst-class:: sphx-glr-script-out .. code-block:: none jet speed 75.2 m/s, flow 0.590 m³/s, jet power 1.67 MW .. GENERATED FROM PYTHON SOURCE LINES 39-43 Power against bucket speed -------------------------- Flat paddles turn the jet through 90°; Pelton's buckets through 165°. .. GENERATED FROM PYTHON SOURCE LINES 43-54 .. code-block:: Python u = np.linspace(0, V, 200) fig, ax = plt.subplots(figsize=(6, 4)) for label, theta in [("flat paddles, 90°", np.pi / 2), ("Pelton buckets, 165°", np.radians(165))]: eta = pelton_wheel_power(rho, Q, V, u, theta) / jet_power ax.plot(u / V, eta, label=label) print(f"{label}: best efficiency {eta.max():.3f} at u/V = {u[eta.argmax()] / V:.2f}") ax.set(xlabel="bucket speed / jet speed", ylabel="hydraulic efficiency", title="Pelton wheel") ax.legend() fig.tight_layout() .. image-sg:: /api/gallery/fluid_engineering/momentum/images/sphx_glr_plot_01_pelton_wheel_001.png :alt: Pelton wheel :srcset: /api/gallery/fluid_engineering/momentum/images/sphx_glr_plot_01_pelton_wheel_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none flat paddles, 90°: best efficiency 0.500 at u/V = 0.50 Pelton buckets, 165°: best efficiency 0.983 at u/V = 0.50 .. GENERATED FROM PYTHON SOURCE LINES 55-61 The force on a single vane -------------------------- One bucket held still feels the full reversal force; one running away at half the jet speed feels a quarter of it, since it receives water at half the rate and turns it at half the relative speed. .. GENERATED FROM PYTHON SOURCE LINES 61-72 .. code-block:: Python theta = np.radians(165) held, _ = jet_force(rho, A, V, theta) running, _ = jet_force(rho, A, V, theta, vane_speed=V / 2) print(f"bucket held still {held / 1e3:.1f} kN, running at V/2 {running / 1e3:.1f} kN") assert np.isclose(pelton_wheel_power(rho, Q, V, V / 2, theta) / jet_power, (1 - np.cos(theta)) / 2) assert np.isclose(running, held / 4) assert pelton_wheel_power(rho, Q, V, V / 2, theta) > 1.9 * pelton_wheel_power(rho, Q, V, V / 2, np.pi / 2) plt.show() .. rst-class:: sphx-glr-script-out .. code-block:: none bucket held still 87.3 kN, running at V/2 21.8 kN .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.064 seconds) .. _sphx_glr_download_api_gallery_fluid_engineering_momentum_plot_01_pelton_wheel.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/momentum/plot_01_pelton_wheel.ipynb :alt: Launch JupyterLite :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_01_pelton_wheel.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_01_pelton_wheel.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_01_pelton_wheel.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_