r'''Calculates Nusselt number for natural convection around a vertical
    plate according to the Churchill-Chu [1]_ correlation, also presented in
    [2]_. Plate must be isothermal; an alternate expression exists for constant
    heat flux.

    .. math::
        Nu_{L}=\left[0.825+\frac{0.387Ra_{L}^{1/6}}
        {[1+(0.492/Pr)^{9/16}]^{8/27}}\right]^2

    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.

    Can be applied to vertical cylinders as well, subject to the criteria below:

    .. math::
        \frac{D}{L}\ge \frac{35}{Gr_L^{1/4}}

    Examples
    --------
    From [2]_, Example 9.2, matches:

    >>> Nu_vertical_plate_Churchill(0.69, 2.63E9)
    147.16185223770603

    References
    ----------
    .. [1] Churchill, Stuart W., and Humbert H. S. Chu. "Correlating Equations
       for Laminar and Turbulent Free Convection from a Vertical Plate."
       International Journal of Heat and Mass Transfer 18, no. 11
       (November 1, 1975): 1323-29. doi:10.1016/0017-9310(75)90243-4.
    .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and
       David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ:
       Wiley, 2011.

def Nu_vertical_plate_Churchill(Pr, Gr):
    r'''Calculates Nusselt number for natural convection around a vertical
    plate according to the Churchill-Chu [1]_ correlation, also presented in
    [2]_. Plate must be isothermal; an alternate expression exists for constant
    heat flux.

    .. math::
        Nu_{L}=\left[0.825+\frac{0.387Ra_{L}^{1/6}}
        {[1+(0.492/Pr)^{9/16}]^{8/27}}\right]^2

    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.

    Can be applied to vertical cylinders as well, subject to the criteria below:

    .. math::
        \frac{D}{L}\ge \frac{35}{Gr_L^{1/4}}

    Examples
    --------
    From [2]_, Example 9.2, matches:

    >>> Nu_vertical_plate_Churchill(0.69, 2.63E9)
    147.16185223770603

    References
    ----------
    .. [1] Churchill, Stuart W., and Humbert H. S. Chu. "Correlating Equations
       for Laminar and Turbulent Free Convection from a Vertical Plate."
       International Journal of Heat and Mass Transfer 18, no. 11
       (November 1, 1975): 1323-29. doi:10.1016/0017-9310(75)90243-4.
    .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and
       David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ:
       Wiley, 2011.
    '''
    Ra = Pr * Gr
    Nu = (0.825 + (0.387*Ra**(1/6.)/(1 + (0.492/Pr)**(9/16.))**(8/27.)))**2
    return Nu