r'''Calculates Nusselt number for natural convection around a sphere
    according to the Churchill [1]_ correlation. Sphere must be isothermal.

    .. math::
        Nu_D=2+\frac{0.589Ra_D^{1/4}} {\left[1+(0.469/Pr)^{9/16}\right]^{4/9}}
        \cdot\left\{1 + \frac{7.44\times 10^{-8}Ra}
        {[1+(0.469/Pr)^{9/16}]^{16/9}}\right\}^{1/12}

    Parameters
    ----------
    Pr : float
        Prandtl number [-]
    Gr : float
        Grashof number [-]

    Returns
    -------
    Nu : float
        Nusselt number, [-]

    Notes
    -----
    Although transition from laminar to turbulent is discrete in reality, this
    equation provides a smooth transition in value from laminar to turbulent.
    Checked with the original source.

    Good for Ra < 1E13. Limit of Nu is 2 at low Grashof numbers.

    Examples
    --------
    >>> Nu_sphere_Churchill(.7, 1E7)
    25.670869440317578

    References
    ----------
    .. [1] Schlunder, Ernst U, and International Center for Heat and Mass
       Transfer. Heat Exchanger Design Handbook. Washington:
       Hemisphere Pub. Corp., 1987.

def Nu_sphere_Churchill(Pr, Gr):
    r'''Calculates Nusselt number for natural convection around a sphere
    according to the Churchill [1]_ correlation. Sphere must be isothermal.

    .. math::
        Nu_D=2+\frac{0.589Ra_D^{1/4}} {\left[1+(0.469/Pr)^{9/16}\right]^{4/9}}
        \cdot\left\{1 + \frac{7.44\times 10^{-8}Ra}
        {[1+(0.469/Pr)^{9/16}]^{16/9}}\right\}^{1/12}

    Parameters
    ----------
    Pr : float
        Prandtl number [-]
    Gr : float
        Grashof number [-]

    Returns
    -------
    Nu : float
        Nusselt number, [-]

    Notes
    -----
    Although transition from laminar to turbulent is discrete in reality, this
    equation provides a smooth transition in value from laminar to turbulent.
    Checked with the original source.

    Good for Ra < 1E13. Limit of Nu is 2 at low Grashof numbers.

    Examples
    --------
    >>> Nu_sphere_Churchill(.7, 1E7)
    25.670869440317578

    References
    ----------
    .. [1] Schlunder, Ernst U, and International Center for Heat and Mass
       Transfer. Heat Exchanger Design Handbook. Washington:
       Hemisphere Pub. Corp., 1987.
    '''
    Ra = Pr*Gr
    Nu = 2 + (0.589*Ra**0.25/(1 + (0.469/Pr)**(9/16.))**(4/9.)*(
         1 + 7.44E-8*Ra/(1 + (0.469/Pr)**(9/16.))**(16/9.))**(1/12.))
    return Nu