.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/analytical/chromatography/plot_03_van_deemter_equation.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_analytical_chromatography_plot_03_van_deemter_equation.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_analytical_chromatography_plot_03_van_deemter_equation.py:


The van Deemter equation and the optimum flow velocity
=========================================================

:func:`~chemistrykit.analytical.van_deemter_H` evaluates
:math:`H=A+B/u+Cu`, the sum of eddy diffusion (`A`), longitudinal
diffusion (`B/u`, dominant at low flow) and mass-transfer resistance
(`Cu`, dominant at high flow). The competition between the last two
gives a minimum plate height
:math:`H_{min}=A+2\sqrt{BC}` at :math:`u_{opt}=\sqrt{B/C}`
(:func:`~chemistrykit.analytical.minimum_plate_height`,
:func:`~chemistrykit.analytical.optimum_flow_velocity`).

.. GENERATED FROM PYTHON SOURCE LINES 16-30

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.analytical import minimum_plate_height, optimum_flow_velocity, van_deemter_H

    A, B, C = 1.5, 25.0, 0.05  # plate height in units of 1e-3 cm, u in cm/s
    u = np.linspace(0.5, 60.0, 3000)
    H = van_deemter_H(u, A, B, C)
    u_opt = optimum_flow_velocity(B, C)
    H_min = minimum_plate_height(A, B, C)
    print(f"u_opt = sqrt(B/C) = {u_opt:.3f}; numerical minimum at u = {u[np.argmin(H)]:.3f}")
    print(f"H_min = A + 2 sqrt(BC) = {H_min:.4f}; numerical minimum H = {H.min():.4f}")
    print(f"At u_opt the B and C terms are equal: B/u = {B / u_opt:.4f}, Cu = {C * u_opt:.4f}")





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

 .. code-block:: none

    u_opt = sqrt(B/C) = 22.361; numerical minimum at u = 22.364
    H_min = A + 2 sqrt(BC) = 3.7361; numerical minimum H = 3.7361
    At u_opt the B and C terms are equal: B/u = 1.1180, Cu = 1.1180




.. GENERATED FROM PYTHON SOURCE LINES 31-44

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(7, 4.5))
    ax.plot(u, H, color="black", linewidth=2, label="H = A + B/u + Cu")
    ax.plot(u, np.full_like(u, A), "--", label="A (eddy diffusion)")
    ax.plot(u, B / u, "--", label="B/u (longitudinal diffusion)")
    ax.plot(u, C * u, "--", label="Cu (mass transfer)")
    ax.plot([u_opt], [H_min], "o", color="crimson", label=f"optimum: u = {u_opt:.1f}, H = {H_min:.2f}")
    ax.set_ylim(0, 12)
    ax.set_xlabel("linear velocity u")
    ax.set_ylabel("plate height H")
    ax.set_title("van Deemter curve and its three terms")
    ax.legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/analytical/chromatography/images/sphx_glr_plot_03_van_deemter_equation_001.png
   :alt: van Deemter curve and its three terms
   :srcset: /api/gallery/analytical/chromatography/images/sphx_glr_plot_03_van_deemter_equation_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.052 seconds)


.. _sphx_glr_download_api_gallery_analytical_chromatography_plot_03_van_deemter_equation.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

      :download:`Download Jupyter notebook: plot_03_van_deemter_equation.ipynb <plot_03_van_deemter_equation.ipynb>`

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

      :download:`Download Python source code: plot_03_van_deemter_equation.py <plot_03_van_deemter_equation.py>`

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

      :download:`Download zipped: plot_03_van_deemter_equation.zip <plot_03_van_deemter_equation.zip>`


.. only:: html

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

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