.. 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_01_biot_number.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_01_biot_number.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_01_biot_number.py:


Biot's heated bar and the Biot number (1804)
============================================

In 1804 Jean-Baptiste Biot heated one end of iron bars and read thermometers
set along them. He saw that the temperature profile is set by a contest
between conduction along the bar and loss from its surface. He could not
write a consistent differential equation for it; Fourier did in 1807. The
ratio of the two effects, now the Biot number
:math:`Bi = hL/k` (:func:`~engineeringkit.heat_transfer.biot_number`),
decides whether a body can be treated as a single temperature. This example
cools a plane wall at three Biot numbers and compares the lumped model
:math:`e^{-Bi\,Fo}` with the exact series solution.

.. GENERATED FROM PYTHON SOURCE LINES 17-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import integrate

    from engineeringkit.heat_transfer import biot_number, fourier_number, plane_wall_transient








.. GENERATED FROM PYTHON SOURCE LINES 24-30

The same wall, three coolants
-----------------------------

A 40 mm steel plate (:math:`k = 40` W/(m K), half-thickness
:math:`L = 20` mm) cooled on both faces by still air, forced air and a
water spray.

.. GENERATED FROM PYTHON SOURCE LINES 30-37

.. code-block:: Python


    k, L = 40.0, 0.02
    coolants = {"still air": 20.0, "air jet": 200.0, "water spray": 2000.0}
    Bis = {name: biot_number(h, L, k) for name, h in coolants.items()}
    for name, Bi in Bis.items():
        print(f"{name:12s}: Bi = {Bi:.2f}")





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

 .. code-block:: none

    still air   : Bi = 0.01
    air jet     : Bi = 0.10
    water spray : Bi = 1.00




.. GENERATED FROM PYTHON SOURCE LINES 38-43

Lumped versus distributed
-------------------------

A lumped body follows :math:`\theta = e^{-Bi\,Fo}`. The exact mean
temperature of the wall averages the series over the thickness.

.. GENERATED FROM PYTHON SOURCE LINES 43-65

.. code-block:: Python


    x_star = np.linspace(0, 1, 201)
    Fo = np.linspace(0, 3, 61)[1:]
    fig, axes = plt.subplots(1, 2, figsize=(10, 3.8))
    for name, Bi in Bis.items():
        mean = np.array([integrate.trapezoid(plane_wall_transient(F, Bi, x_star), x_star) for F in Fo])
        lumped = np.exp(-Bi * Fo)
        error = np.max(np.abs(lumped - mean))
        print(f"{name:12s}: worst lumped error {error:.4f} (fraction of the initial excess)")
        line = axes[0].plot(Fo, mean, label=f"exact, Bi = {Bi:.2f}")[0]
        axes[0].plot(Fo, lumped, "--", color=line.get_color())
        axes[1].plot(x_star, plane_wall_transient(0.5, Bi, x_star), color=line.get_color(), label=f"Bi = {Bi:.2f}")
        if Bi <= 0.1:
            assert error < 0.01
        else:
            assert error > 0.1
    axes[0].set(xlabel="Fourier number", ylabel="mean excess temperature", title="Solid: exact, dashed: lumped")
    axes[0].legend()
    axes[1].set(xlabel="x / L (centre to surface)", ylabel="excess temperature", title="Profiles at Fo = 0.5")
    axes[1].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/heat_transfer/transient/images/sphx_glr_plot_01_biot_number_001.png
   :alt: Solid: exact, dashed: lumped, Profiles at Fo = 0.5
   :srcset: /api/gallery/heat_transfer/transient/images/sphx_glr_plot_01_biot_number_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    still air   : worst lumped error 0.0001 (fraction of the initial excess)
    air jet     : worst lumped error 0.0071 (fraction of the initial excess)
    water spray : worst lumped error 0.1045 (fraction of the initial excess)




.. GENERATED FROM PYTHON SOURCE LINES 66-71

What the time scale means
-------------------------

In the still-air case the plate is effectively isothermal, and an hour of
cooling is :math:`Fo` of order 100 on the half-thickness.

.. GENERATED FROM PYTHON SOURCE LINES 71-76

.. code-block:: Python


    alpha = k / (7850.0 * 460.0)
    print(f"one hour is Fo = {fourier_number(alpha, 3600.0, L):.0f}")

    plt.show()




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

 .. code-block:: none

    one hour is Fo = 100





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

   **Total running time of the script:** (0 minutes 0.159 seconds)


.. _sphx_glr_download_api_gallery_heat_transfer_transient_plot_01_biot_number.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_01_biot_number.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_biot_number.ipynb <plot_01_biot_number.ipynb>`

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

      :download:`Download Python source code: plot_01_biot_number.py <plot_01_biot_number.py>`

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

      :download:`Download zipped: plot_01_biot_number.zip <plot_01_biot_number.zip>`


.. only:: html

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

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