.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/thermodynamics/cycles/plot_04_otto_cycle.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_thermodynamics_cycles_plot_04_otto_cycle.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_thermodynamics_cycles_plot_04_otto_cycle.py:


Otto's four-stroke cycle (1876)
===============================

Nicolaus Otto's 1876 engine compressed its charge before igniting it, a
principle Alphonse Beau de Rochas had described in 1862. Its four strokes
(intake, compression, power, exhaust) are idealized as the air-standard
Otto cycle: isentropic compression through a ratio :math:`r`, heat added at
constant volume, isentropic expansion and heat rejected at constant volume.
The efficiency :math:`1 - r^{1-\gamma}`
(:func:`~engineeringkit.thermodynamics.otto_efficiency`,
:class:`~engineeringkit.thermodynamics.OttoCycle`) depends only on the
compression ratio. Otto's first engine had :math:`r \approx 2.5`; knock
limits petrol engines to about 10-12 today.

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.thermodynamics import OttoCycle, isentropic_pressure_volume, otto_efficiency, thermal_efficiency
    from engineeringkit.thermodynamics.visualizers import plot_pv_diagram








.. GENERATED FROM PYTHON SOURCE LINES 25-30

The cycle on the p-v diagram
----------------------------

One kilogram of air at 100 kPa and 300 K, compression ratio 8, 800 kJ/kg
added by combustion.

.. GENERATED FROM PYTHON SOURCE LINES 30-58

.. code-block:: Python


    R, gamma, r = 287.0, 1.4, 8.0
    cv = R / (gamma - 1)
    p1, T1, q_in = 100e3, 300.0, 800e3
    v1 = R * T1 / p1
    v2 = v1 / r
    p2 = isentropic_pressure_volume(p1, v1, v2, gamma)
    T2 = p2 * v2 / R
    T3 = T2 + q_in / cv
    p3 = R * T3 / v2
    p4 = isentropic_pressure_volume(p3, v2, v1, gamma)
    T4 = p4 * v1 / R
    q_out = cv * (T4 - T1)

    compress = np.linspace(v1, v2, 60)
    expand = np.linspace(v2, v1, 60)
    v_path = np.concatenate([compress, expand, [v1]])
    p_path = np.concatenate([isentropic_pressure_volume(p1, v1, compress, gamma), isentropic_pressure_volume(p3, v2, expand, gamma), [p1]])
    ax = plot_pv_diagram(v_path, p_path, label=f"r = {r:g}")
    ax.set(xlabel="specific volume [m$^3$/kg]", title="Air-standard Otto cycle")

    cycle = OttoCycle(r)
    print(cycle.summary())
    print(f"peak temperature {T3:.0f} K, peak pressure {p3 / 1e6:.1f} MPa")
    print(f"efficiency from the four states {thermal_efficiency(q_in, q_out):.4f}")
    assert np.isclose(thermal_efficiency(q_in, q_out), cycle.efficiency)
    assert np.isclose(T2 / T1, T3 / T4)




.. image-sg:: /api/gallery/thermodynamics/cycles/images/sphx_glr_plot_04_otto_cycle_001.png
   :alt: Air-standard Otto cycle
   :srcset: /api/gallery/thermodynamics/cycles/images/sphx_glr_plot_04_otto_cycle_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    {'compression_ratio': 8.0, 'gamma': 1.4, 'efficiency': np.float64(0.5647247183519378)}
    peak temperature 1804 K, peak pressure 4.8 MPa
    efficiency from the four states 0.5647




.. GENERATED FROM PYTHON SOURCE LINES 59-65

The compression ratio is everything
-----------------------------------

The efficiency does not depend on how much fuel is burned, only on how far
the charge is compressed. Higher octane fuels resist knock and allow higher
compression ratios.

.. GENERATED FROM PYTHON SOURCE LINES 65-76

.. code-block:: Python


    ratios = np.linspace(1, 16, 151)
    fig, ax = plt.subplots(figsize=(6, 3.5))
    ax.plot(ratios, otto_efficiency(ratios))
    for label, rr in (("Otto 1876", 2.5), ("1950s car", 7.0), ("modern petrol", 11.0)):
        ax.plot(rr, otto_efficiency(rr), "o", label=f"{label}: r = {rr:g}, η = {otto_efficiency(rr):.2f}")
    ax.set(xlabel="compression ratio", ylabel="ideal efficiency", ylim=(0, 0.7))
    ax.legend(fontsize=8)
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/thermodynamics/cycles/images/sphx_glr_plot_04_otto_cycle_002.png
   :alt: plot 04 otto cycle
   :srcset: /api/gallery/thermodynamics/cycles/images/sphx_glr_plot_04_otto_cycle_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.049 seconds)


.. _sphx_glr_download_api_gallery_thermodynamics_cycles_plot_04_otto_cycle.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/thermodynamics/cycles/plot_04_otto_cycle.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_otto_cycle.ipynb <plot_04_otto_cycle.ipynb>`

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

      :download:`Download Python source code: plot_04_otto_cycle.py <plot_04_otto_cycle.py>`

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

      :download:`Download zipped: plot_04_otto_cycle.zip <plot_04_otto_cycle.zip>`


.. only:: html

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

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