r'''Calculates Nusselt number for natural convection around a vertical
    isothermal cylinder according to the results of [1]_  correlated by
    [2]_, also as presented in [3]_, [4]_, and [5]_.

    .. math::
        Nu_H = 0.555 Ra_H^{0.25},\; 10^{5} < Ra < 10^{9}

        Nu_H = 0.021 Ra_H^{0.4},\; 10^{9} < Ra < 10^{12}

    Parameters
    ----------
    Pr : float
        Prandtl number [-]
    Gr : float
        Grashof number [-]
    turbulent : bool or None, optional
        Whether or not to force the correlation to return the turbulent
		 result; will return the laminar regime if False; leave as None for
        automatic selection

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

    Notes
    -----
    Transition between ranges is not smooth. If outside of range, no warning is
    given.

    Examples
    --------
    >>> Nu_vertical_cylinder_Kreith_Eckert(.7, 2E10)
    240.25393473033196

    References
    ----------
    .. [1] Eckert, E. R. G., Thomas W. Jackson, and United States. Analysis of
       Turbulent Free-Convection Boundary Layer on Flat Plate. National
       Advisory Committee for Aeronautics, no. 2207. Washington, D.C.: National
       Advisoty Committee for Aeronautics, 1950.
    .. [2] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat
       Transfer. Cengage, 2010.
    .. [3] Morgan, V.T., The Overall Convective Heat Transfer from Smooth
       Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and
       J.P. Hartnett, V 11, 199-264, 1975.
    .. [4] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical
       Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6
       (June 1, 2008): 521-36. doi:10.1080/01457630801891557.
    .. [5] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From
       Vertical Cylinders." In Natural Convection from Circular Cylinders,
       23-42. Springer, 2014.

def Nu_vertical_cylinder_Kreith_Eckert(Pr, Gr, turbulent=None):
    r'''Calculates Nusselt number for natural convection around a vertical
    isothermal cylinder according to the results of [1]_  correlated by
    [2]_, also as presented in [3]_, [4]_, and [5]_.

    .. math::
        Nu_H = 0.555 Ra_H^{0.25},\; 10^{5} < Ra < 10^{9}

        Nu_H = 0.021 Ra_H^{0.4},\; 10^{9} < Ra < 10^{12}

    Parameters
    ----------
    Pr : float
        Prandtl number [-]
    Gr : float
        Grashof number [-]
    turbulent : bool or None, optional
        Whether or not to force the correlation to return the turbulent
		 result; will return the laminar regime if False; leave as None for
        automatic selection

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

    Notes
    -----
    Transition between ranges is not smooth. If outside of range, no warning is
    given.

    Examples
    --------
    >>> Nu_vertical_cylinder_Kreith_Eckert(.7, 2E10)
    240.25393473033196

    References
    ----------
    .. [1] Eckert, E. R. G., Thomas W. Jackson, and United States. Analysis of
       Turbulent Free-Convection Boundary Layer on Flat Plate. National
       Advisory Committee for Aeronautics, no. 2207. Washington, D.C.: National
       Advisoty Committee for Aeronautics, 1950.
    .. [2] Kreith, Frank, Raj Manglik, and Mark Bohn. Principles of Heat
       Transfer. Cengage, 2010.
    .. [3] Morgan, V.T., The Overall Convective Heat Transfer from Smooth
       Circular Cylinders, in Advances in Heat Transfer, eds. T.F. Irvin and
       J.P. Hartnett, V 11, 199-264, 1975.
    .. [4] Popiel, Czeslaw O. "Free Convection Heat Transfer from Vertical
       Slender Cylinders: A Review." Heat Transfer Engineering 29, no. 6
       (June 1, 2008): 521-36. doi:10.1080/01457630801891557.
    .. [5] Boetcher, Sandra K. S. "Natural Convection Heat Transfer From
       Vertical Cylinders." In Natural Convection from Circular Cylinders,
       23-42. Springer, 2014.
    '''
    Ra = Pr*Gr
    if turbulent or (Ra > 1E9 and turbulent is None):
        return 0.021*Ra**0.4
    else:
        return 0.555*Ra**0.25