r'''Calculates Nusselt number for natural convection around a horizontal
    cylinder according to the Kuehn-Goldstein [1]_ correlation, also shown in
    [2]_. Cylinder must be isothermal.

    .. math::
        \frac{2}{Nu_D} = \ln\left[1 + \frac{2}{\left[\left\{0.518Ra_D^{0.25}
        \left[1 + \left(\frac{0.559}{Pr}\right)^{3/5}\right]^{-5/12}
        \right\}^{15} + (0.1Ra_D^{1/3})^{15}\right]^{1/15}}\right]

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

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

    Notes
    -----
    [1]_ suggests this expression is valid for all cases except low-Pr fluids.
    [2]_ suggests no restrictions.

    Examples
    --------
    >>> Nu_horizontal_cylinder_Kuehn_Goldstein(0.69, 2.63E9)
    122.99323525628186

    References
    ----------
    .. [1] Kuehn, T. H., and R. J. Goldstein. "Correlating Equations for
       Natural Convection Heat Transfer between Horizontal Circular Cylinders."
       International Journal of Heat and Mass Transfer 19, no. 10
       (October 1976): 1127-34. doi:10.1016/0017-9310(76)90145-9
    .. [2] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From
       Vertical Cylinders." In Natural Convection from Circular Cylinders,
       23-42. Springer, 2014.

def Nu_horizontal_cylinder_Kuehn_Goldstein(Pr, Gr):
    r'''Calculates Nusselt number for natural convection around a horizontal
    cylinder according to the Kuehn-Goldstein [1]_ correlation, also shown in
    [2]_. Cylinder must be isothermal.

    .. math::
        \frac{2}{Nu_D} = \ln\left[1 + \frac{2}{\left[\left\{0.518Ra_D^{0.25}
        \left[1 + \left(\frac{0.559}{Pr}\right)^{3/5}\right]^{-5/12}
        \right\}^{15} + (0.1Ra_D^{1/3})^{15}\right]^{1/15}}\right]

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

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

    Notes
    -----
    [1]_ suggests this expression is valid for all cases except low-Pr fluids.
    [2]_ suggests no restrictions.

    Examples
    --------
    >>> Nu_horizontal_cylinder_Kuehn_Goldstein(0.69, 2.63E9)
    122.99323525628186

    References
    ----------
    .. [1] Kuehn, T. H., and R. J. Goldstein. "Correlating Equations for
       Natural Convection Heat Transfer between Horizontal Circular Cylinders."
       International Journal of Heat and Mass Transfer 19, no. 10
       (October 1976): 1127-34. doi:10.1016/0017-9310(76)90145-9
    .. [2] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From
       Vertical Cylinders." In Natural Convection from Circular Cylinders,
       23-42. Springer, 2014.
    '''
    Ra = Pr*Gr
    return 2./log(1 + 2./((0.518*Ra**0.25*(1. + (0.559/Pr)**0.6)**(-5/12.))**15
                  + (0.1*Ra**(1/3.))**15)**(1/15.))