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

    .. math::
        Nu_H = 0.726 Ra_H^{0.25},\; 2\times 10^{8} < Ra < 4\times 10^{10}

        Nu_H = 0.0674 (Gr_H Pr^{1.29})^{1/3},\; 4\times 10^{10} < Ra < 9\times 10^{11}

    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
    -----
    Cylinder of diameters 2.75 inch, with heights of 6, 18, and 36.25 inch.
    Temperature was controlled via multiple separately controlled heating
    sections. Fluids were water and ethylene-glycol. Transition between ranges
    is not smooth. If outside of range, no warning is given. [2]_, [3]_, and
    [4]_ are in complete agreement about this formulation.

    Examples
    --------
    >>> Nu_vertical_cylinder_Touloukian_Morgan(.7, 2E10)
    249.72879961097854

    References
    ----------
    .. [1] Touloukian, Y. S, George A Hawkins, and Max Jakob. Heat Transfer by
       Free Convection from Heated Vertical Surfaces to Liquids.
       Trans. ASME 70, 13-18 (1948).
    .. [2] 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.
    .. [3] 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.
    .. [4] 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_Touloukian_Morgan(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]_, as
    presented in [3]_ and [4]_.

    .. math::
        Nu_H = 0.726 Ra_H^{0.25},\; 2\times 10^{8} < Ra < 4\times 10^{10}

        Nu_H = 0.0674 (Gr_H Pr^{1.29})^{1/3},\; 4\times 10^{10} < Ra < 9\times 10^{11}

    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
    -----
    Cylinder of diameters 2.75 inch, with heights of 6, 18, and 36.25 inch.
    Temperature was controlled via multiple separately controlled heating
    sections. Fluids were water and ethylene-glycol. Transition between ranges
    is not smooth. If outside of range, no warning is given. [2]_, [3]_, and
    [4]_ are in complete agreement about this formulation.

    Examples
    --------
    >>> Nu_vertical_cylinder_Touloukian_Morgan(.7, 2E10)
    249.72879961097854

    References
    ----------
    .. [1] Touloukian, Y. S, George A Hawkins, and Max Jakob. Heat Transfer by
       Free Convection from Heated Vertical Surfaces to Liquids.
       Trans. ASME 70, 13-18 (1948).
    .. [2] 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.
    .. [3] 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.
    .. [4] 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 > 4E10 and turbulent is None):
        return 0.0674*(Gr*Pr**1.29)**(1/3.)
    else:
        return 0.726*Ra**0.25