.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/performance/plot_02_breguet_propeller_range.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_aerodynamics_performance_plot_02_breguet_propeller_range.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_aerodynamics_performance_plot_02_breguet_propeller_range.py:


Breguet's propeller range (c. 1920)
===================================

Breguet's range equation was first written for propeller aircraft, the only
kind there was. Shaft power :math:`P = DV/\eta_{pr}` burns fuel at a rate
set by the engine's power-specific consumption :math:`c_P`. Drag is
:math:`W/(L/D)`, so the speed cancels and
:math:`R = \frac{\eta_{pr}}{g c_P}\frac{L}{D}\ln\frac{W_i}{W_f}`
(:func:`~engineeringkit.aerodynamics.breguet_range_propeller`). A propeller
aircraft goes furthest at the speed of maximum :math:`L/D`, which is slower
than a jet's best-range speed. This example checks the formula against a
time-stepped cruise and plots range against speed for a light aircraft.

.. GENERATED FROM PYTHON SOURCE LINES 17-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import best_glide_cl, breguet_range_propeller, drag_polar, glide_ratio

    g, eta, psfc = 9.80665, 0.8, 7.6e-8
    W0, Wf = 13.1e3, 11.7e3
    rho, S, Cd0, AR, e = 1.0, 16.2, 0.025, 7.4, 0.8








.. GENERATED FROM PYTHON SOURCE LINES 27-29

Step the cruise in time at constant L/D
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-39

.. code-block:: Python


    LD = float(glide_ratio(best_glide_cl(Cd0, AR, e), drag_polar(best_glide_cl(Cd0, AR, e), Cd0, AR, e)))
    W, x, dt, V = W0, 0.0, 1.0, 50.0
    while W > Wf:
        W -= g * psfc * (W / LD) * V / eta * dt
        x += V * dt
    R = float(breguet_range_propeller(eta, psfc, LD, W0, Wf))
    print(f"(L/D)max = {LD:.2f}: time-stepped {x / 1e3:.0f} km, Breguet {R / 1e3:.0f} km")
    assert np.isclose(x, R, rtol=1e-3)





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

 .. code-block:: none

    (L/D)max = 13.64: time-stepped 1654 km, Breguet 1654 km




.. GENERATED FROM PYTHON SOURCE LINES 40-46

Range against cruise speed
--------------------------

Holding speed fixed, the lift coefficient and so :math:`L/D` follow from
:math:`W = \tfrac12\rho V^2 S C_L` at the mean cruise weight. Range peaks
where :math:`L/D` does.

.. GENERATED FROM PYTHON SOURCE LINES 46-61

.. code-block:: Python


    V = np.linspace(30, 90, 121)
    W_mean = np.sqrt(W0 * Wf)
    CL = 2 * W_mean / (rho * V**2 * S)
    ranges = breguet_range_propeller(eta, psfc, glide_ratio(CL, drag_polar(CL, Cd0, AR, e)), W0, Wf)
    V_best = float(np.sqrt(2 * W_mean / (rho * S * best_glide_cl(Cd0, AR, e))))
    fig, ax = plt.subplots(figsize=(7, 4))
    ax.plot(V, ranges / 1e3)
    ax.axvline(V_best, color="gray", ls="--", label=f"(L/D)max speed {V_best:.1f} m/s")
    ax.set(xlabel="cruise speed [m/s]", ylabel="range [km]", title="Propeller range peaks at maximum L/D")
    ax.legend()
    fig.tight_layout()
    assert abs(V[np.argmax(ranges)] - V_best) < 1.0

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/performance/images/sphx_glr_plot_02_breguet_propeller_range_001.png
   :alt: Propeller range peaks at maximum L/D
   :srcset: /api/gallery/aerodynamics/performance/images/sphx_glr_plot_02_breguet_propeller_range_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.070 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_performance_plot_02_breguet_propeller_range.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/aerodynamics/performance/plot_02_breguet_propeller_range.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_breguet_propeller_range.ipynb <plot_02_breguet_propeller_range.ipynb>`

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

      :download:`Download Python source code: plot_02_breguet_propeller_range.py <plot_02_breguet_propeller_range.py>`

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

      :download:`Download zipped: plot_02_breguet_propeller_range.zip <plot_02_breguet_propeller_range.zip>`


.. only:: html

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

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