.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/performance/plot_01_breguet_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_01_breguet_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_01_breguet_range.py:


The Breguet range equation
==========================

Louis Breguet's range formula (1923) follows from three facts: thrust equals
drag, drag equals weight divided by :math:`L/D`, and fuel flow is the
thrust-specific fuel consumption times thrust. Integrating the weight loss
gives :math:`R = \frac{V}{g c_T}\frac{L}{D}\ln\frac{W_i}{W_f}`
(:func:`~engineeringkit.aerodynamics.breguet_range`): range is the product of
engine efficiency, aerodynamic efficiency and a logarithmic fuel fraction.
This example checks it against a time-stepped cruise and shows why airliners
fly near the best-glide lift coefficient.

.. GENERATED FROM PYTHON SOURCE LINES 16-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

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

    g = 9.80665
    V, tsfc = 230.0, 1.6e-5
    W0, Wf = 7.0e5, 5.0e5
    LD = 18.0








.. GENERATED FROM PYTHON SOURCE LINES 27-29

Step the cruise in time
-----------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-39

.. code-block:: Python


    W, x, dt = W0, 0.0, 10.0
    while W > Wf:
        fuel_flow = tsfc * W / LD
        W -= fuel_flow * g * dt
        x += V * dt
    print(f"time-stepped range: {x / 1e3:7.0f} km")
    print(f"Breguet range:      {float(breguet_range(V, tsfc, LD, W0, Wf)) / 1e3:7.0f} km")
    assert np.isclose(x, breguet_range(V, tsfc, LD, W0, Wf), rtol=1e-3)





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

 .. code-block:: none

    time-stepped range:    8878 km
    Breguet range:         8878 km




.. GENERATED FROM PYTHON SOURCE LINES 40-45

Range is proportional to L/D
----------------------------

Flying at a lift coefficient away from the best-glide value lowers
:math:`L/D` and therefore range in direct proportion.

.. GENERATED FROM PYTHON SOURCE LINES 45-61

.. code-block:: Python


    cd0, AR, e = 0.018, 9.5, 0.82
    cl = np.linspace(0.2, 1.0, 100)
    ld = glide_ratio(cl, drag_polar(cl, cd0, AR, e))
    cl_star = float(best_glide_cl(cd0, AR, e))
    fuel_fraction = np.linspace(0.05, 0.45, 5)
    fig, axes = plt.subplots(1, 2, figsize=(9, 3.6))
    axes[0].plot(cl, breguet_range(V, tsfc, ld, W0, Wf) / 1e3)
    axes[0].axvline(cl_star, color="gray", ls="--", label=f"best-glide $C_L$ = {cl_star:.2f}")
    axes[0].set(xlabel="cruise $C_L$", ylabel="range [km]", title="Fly at maximum L/D")
    axes[0].legend()
    axes[1].plot(fuel_fraction, breguet_range(V, tsfc, LD, W0, W0 * (1 - fuel_fraction)) / 1e3, "o-")
    axes[1].set(xlabel="fuel weight fraction", ylabel="range [km]", title="Logarithmic in fuel fraction")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/performance/images/sphx_glr_plot_01_breguet_range_001.png
   :alt: Fly at maximum L/D, Logarithmic in fuel fraction
   :srcset: /api/gallery/aerodynamics/performance/images/sphx_glr_plot_01_breguet_range_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.203 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_performance_plot_01_breguet_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_01_breguet_range.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_breguet_range.ipynb <plot_01_breguet_range.ipynb>`

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

      :download:`Download Python source code: plot_01_breguet_range.py <plot_01_breguet_range.py>`

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

      :download:`Download zipped: plot_01_breguet_range.zip <plot_01_breguet_range.zip>`


.. only:: html

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

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