.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/heat_transfer/transient/plot_02_kelvin_age_of_earth.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_heat_transfer_transient_plot_02_kelvin_age_of_earth.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_heat_transfer_transient_plot_02_kelvin_age_of_earth.py:


Kelvin's age of the Earth (1862)
================================

In 1862 William Thomson, later Lord Kelvin, treated the Earth as a
semi-infinite solid that began at a uniform molten temperature and has
cooled from its surface ever since. The temperature below the surface is
then :math:`T_s + (T_i - T_s)\operatorname{erf}(x/2\sqrt{\alpha t})`
(:func:`~engineeringkit.heat_transfer.semi_infinite_temperature`), and the
surface gradient :math:`(T_i - T_s)/\sqrt{\pi\alpha t}` falls with time. From
the gradient measured in mines, about 1 °F per 50 ft, he found an age of
about 98 million years. This example redoes his arithmetic in SI units and
checks the gradient against the profile itself.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.heat_transfer import semi_infinite_temperature








.. GENERATED FROM PYTHON SOURCE LINES 23-28

Kelvin's numbers
----------------

An initial excess of 7000 °F, a gradient of 1 °F per 50 ft and a
diffusivity of 400 ft² per year, converted to SI.

.. GENERATED FROM PYTHON SOURCE LINES 28-38

.. code-block:: Python


    year = 365.25 * 86400
    dT = 7000 * 5 / 9  # K
    gradient = (5 / 9) / (50 * 0.3048)  # K/m
    alpha = 400 * 0.3048**2 / year  # m^2/s
    age = dT**2 / (np.pi * alpha * gradient**2)
    print(f"diffusivity {alpha:.3e} m²/s, gradient {gradient * 1e3:.1f} mK/m")
    print(f"age of the Earth: {age / year / 1e6:.0f} million years")
    assert 95e6 < age / year < 100e6





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

 .. code-block:: none

    diffusivity 1.178e-06 m²/s, gradient 36.5 mK/m
    age of the Earth: 97 million years




.. GENERATED FROM PYTHON SOURCE LINES 39-44

The gradient from the profile
-----------------------------

Differentiate the error-function profile numerically at the surface and
compare with the closed form.

.. GENERATED FROM PYTHON SOURCE LINES 44-60

.. code-block:: Python


    Ts, Ti = 273.0, 273.0 + dT
    dx = 1.0
    numeric = (semi_infinite_temperature(Ti, Ts, dx, alpha, age) - semi_infinite_temperature(Ti, Ts, 0.0, alpha, age)) / dx
    print(f"surface gradient from the profile {numeric * 1e3:.3f} mK/m")
    assert np.isclose(numeric, gradient, rtol=1e-4)

    depth = np.linspace(0, 400e3, 400)
    fig, ax = plt.subplots(figsize=(6, 4))
    for myr in [1, 10, 98, 1000]:
        ax.plot(semi_infinite_temperature(Ti, Ts, depth, alpha, myr * 1e6 * year) - 273.0, depth / 1e3, label=f"{myr} Myr")
    ax.invert_yaxis()
    ax.set(xlabel="temperature [°C]", ylabel="depth [km]", title="Kelvin's cooling Earth")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/heat_transfer/transient/images/sphx_glr_plot_02_kelvin_age_of_earth_001.png
   :alt: Kelvin's cooling Earth
   :srcset: /api/gallery/heat_transfer/transient/images/sphx_glr_plot_02_kelvin_age_of_earth_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    surface gradient from the profile 36.454 mK/m




.. GENERATED FROM PYTHON SOURCE LINES 61-68

Why he was wrong
----------------

Radioactive heating (discovered in 1896) and, above all, convection in the
mantle, argued by John Perry in 1895, keep the interior hot. The radiometric
age, 4.54 billion years, would need a surface gradient about
:math:`\sqrt{4540/98}` times smaller than the one measured.

.. GENERATED FROM PYTHON SOURCE LINES 68-72

.. code-block:: Python


    print(f"the gradient for 4.54 Gyr would be {gradient * np.sqrt(age / (4.54e9 * year)) * 1e3:.1f} mK/m")

    plt.show()




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

 .. code-block:: none

    the gradient for 4.54 Gyr would be 5.3 mK/m





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

   **Total running time of the script:** (0 minutes 0.037 seconds)


.. _sphx_glr_download_api_gallery_heat_transfer_transient_plot_02_kelvin_age_of_earth.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/heat_transfer/transient/plot_02_kelvin_age_of_earth.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_kelvin_age_of_earth.ipynb <plot_02_kelvin_age_of_earth.ipynb>`

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

      :download:`Download Python source code: plot_02_kelvin_age_of_earth.py <plot_02_kelvin_age_of_earth.py>`

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

      :download:`Download zipped: plot_02_kelvin_age_of_earth.zip <plot_02_kelvin_age_of_earth.zip>`


.. only:: html

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

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