.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/thermodynamics/properties/plot_08_pitzer_acentric_factor.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_properties_plot_08_pitzer_acentric_factor.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_properties_plot_08_pitzer_acentric_factor.py:


Pitzer's acentric factor (1955)
===============================

Van der Waals's law of corresponding states says that all fluids share one
compressibility chart in reduced temperature and pressure. That holds for
argon, krypton and xenon but fails for larger, less spherical molecules.
Kenneth Pitzer and his co-workers at Berkeley added a third parameter in 1955,
the acentric factor
:math:`\omega = -\log_{10}[p_s(0.7\,T_c)/p_c] - 1`
(:func:`~engineeringkit.thermodynamics.pitzer_acentric_factor`). It is zero
for the simple fluids, whose reduced vapour pressure at :math:`T_r = 0.7` is
almost exactly 0.1. The compressibility factor
(:func:`~engineeringkit.thermodynamics.compressibility_factor`) is then
:math:`Z = Z^0 + \omega Z^1`; at moderate pressure, the second-virial form
(:func:`~engineeringkit.thermodynamics.pitzer_virial_compressibility`).

.. GENERATED FROM PYTHON SOURCE LINES 20-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.thermodynamics import (
        peng_robinson_compressibility,
        pitzer_acentric_factor,
        pitzer_virial_compressibility,
        saturation_pressure,
    )
    from engineeringkit.thermodynamics.utils import if97








.. GENERATED FROM PYTHON SOURCE LINES 32-36

The acentric factor of water
----------------------------

From the IF97 saturation line at 0.7 times the critical temperature.

.. GENERATED FROM PYTHON SOURCE LINES 36-42

.. code-block:: Python


    p07 = float(saturation_pressure(0.7 * if97.T_CRIT))
    omega_water = pitzer_acentric_factor(p07, if97.P_CRIT)
    print(f"water: p_s(0.7 Tc) = {p07 / 1e3:.1f} kPa, p_r = {p07 / if97.P_CRIT:.4f}, omega = {omega_water:.3f} (tables: 0.344)")
    assert np.isclose(omega_water, 0.344, atol=0.002)





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

 .. code-block:: none

    water: p_s(0.7 Tc) = 998.4 kPa, p_r = 0.0453, omega = 0.344 (tables: 0.344)




.. GENERATED FROM PYTHON SOURCE LINES 43-50

The compressibility of n-butane
-------------------------------

Smith, Van Ness and Abbott's example: n-butane (Tc = 425.1 K, pc = 3.796 MPa,
omega = 0.200) at 510 K and 25 bar has Z = 0.879 by the generalized virial
correlation. Peng and Robinson's cubic gives a similar value; both fall below
one, as attraction between the molecules dominates.

.. GENERATED FROM PYTHON SOURCE LINES 50-56

.. code-block:: Python


    Tc, pc, omega = 425.1, 3.796e6, 0.200
    z = pitzer_virial_compressibility(510.0, 2.5e6, Tc, pc, omega)
    print(f"n-butane, 510 K, 25 bar: Z = {z:.3f} (virial), {peng_robinson_compressibility(510.0, 2.5e6, Tc, pc, omega):.3f} (Peng-Robinson)")
    assert abs(z - 0.879) < 1e-3





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

 .. code-block:: none

    n-butane, 510 K, 25 bar: Z = 0.879 (virial), 0.858 (Peng-Robinson)




.. GENERATED FROM PYTHON SOURCE LINES 57-63

The third parameter on the chart
--------------------------------

On the critical isotherm, at the same reduced pressure, a fluid with larger
omega deviates more from the ideal gas. (Above about 1.05 Tc the correction
changes sign.)

.. GENERATED FROM PYTHON SOURCE LINES 63-74

.. code-block:: Python


    pr_ = np.linspace(0.01, 0.6, 50)
    fig, ax = plt.subplots(figsize=(6, 3.5))
    for w, name in ((0.0, "argon, ω = 0"), (0.2, "n-butane, ω = 0.20"), (0.344, "water, ω = 0.34")):
        ax.plot(pr_, pitzer_virial_compressibility(Tc, pr_ * pc, Tc, pc, w), label=name)
    ax.set(xlabel="reduced pressure", ylabel="Z", title="Generalized virial correlation at Tr = 1")
    ax.legend()
    fig.tight_layout()
    assert pitzer_virial_compressibility(Tc, 0.5 * pc, Tc, pc, 0.344) < pitzer_virial_compressibility(Tc, 0.5 * pc, Tc, pc, 0.0)

    plt.show()



.. image-sg:: /api/gallery/thermodynamics/properties/images/sphx_glr_plot_08_pitzer_acentric_factor_001.png
   :alt: Generalized virial correlation at Tr = 1
   :srcset: /api/gallery/thermodynamics/properties/images/sphx_glr_plot_08_pitzer_acentric_factor_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.108 seconds)


.. _sphx_glr_download_api_gallery_thermodynamics_properties_plot_08_pitzer_acentric_factor.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/properties/plot_08_pitzer_acentric_factor.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_08_pitzer_acentric_factor.ipynb <plot_08_pitzer_acentric_factor.ipynb>`

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

      :download:`Download Python source code: plot_08_pitzer_acentric_factor.py <plot_08_pitzer_acentric_factor.py>`

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

      :download:`Download zipped: plot_08_pitzer_acentric_factor.zip <plot_08_pitzer_acentric_factor.zip>`


.. only:: html

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

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