r'''Calculates Nusselt number for a fluid flowing inside a curved 
    pipe such as a helical coil under turbulent conditions, using the method of 
    Schmidt [1]_, also shown in [2]_, [3]_, and [4]_.
            
    For :math:`Re_{crit} < Re < 2.2\times 10 ^4`:
        
    .. math::
        Nu = 0.023\left[1 + 14.8\left(1 + \frac{D_i}{D_c}\right)\left(
        \frac{D_i}{D_c}\right)^{1/3}\right]Re^{0.8-0.22\left(\frac{D_i}{D_c}
        \right)^{0.1}}Pr^{1/3}
            
    For :math:`2.2\times 10^4 < Re < 1.5\times 10^5`:
        
    .. math::
        Nu = 0.023\left[1 + 3.6\left(1 - \frac{D_i}{D_c}\right)\left(\frac{D_i}
        {D_c}\right)^{0.8}\right]Re^{0.8}Pr^{1/3}
        
    Parameters
    ----------
    Re : float
        Reynolds number with `D=Di`, [-]
    Pr : float
        Prandtl number with bulk properties [-]
    Di : float
        Inner diameter of the coil, [m]
    Dc : float
        Diameter of the helix/coil measured from the center of the tube on one
        side to the center of the tube on the other side, [m]

    Returns
    -------
    Nu : float
        Nusselt number with respect to `Di`, [-]
        
    Notes
    -----
    For very low curvatures, reasonable results are returned by both cases
    of Reynolds numbers.

    Examples
    --------
    >>> helical_turbulent_Nu_Schmidt(2E5, 0.7, 0.01, .2)
    466.2569996832083

    References
    ----------
    .. [1] Schmidt, Eckehard F. "Wärmeübergang Und Druckverlust in 
       Rohrschlangen." Chemie Ingenieur Technik 39, no. 13 (July 10, 1967): 
       781-89. doi:10.1002/cite.330391302. 
    .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and 
       Correlations for Convection Heat Transfer and Pressure Losses in 
       Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0
       (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693.
    .. [3] Hardik, B. K., P. K. Baburajan, and S. V. Prabhu. "Local Heat 
       Transfer Coefficient in Helical Coils with Single Phase Flow." 
       International Journal of Heat and Mass Transfer 89 (October 2015): 
       522-38. doi:10.1016/j.ijheatmasstransfer.2015.05.069.
    .. [4] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat
       Transfer, 3E. New York: McGraw-Hill, 1998.

def helical_turbulent_Nu_Schmidt(Re, Pr, Di, Dc):
    r'''Calculates Nusselt number for a fluid flowing inside a curved 
    pipe such as a helical coil under turbulent conditions, using the method of 
    Schmidt [1]_, also shown in [2]_, [3]_, and [4]_.
            
    For :math:`Re_{crit} < Re < 2.2\times 10 ^4`:
        
    .. math::
        Nu = 0.023\left[1 + 14.8\left(1 + \frac{D_i}{D_c}\right)\left(
        \frac{D_i}{D_c}\right)^{1/3}\right]Re^{0.8-0.22\left(\frac{D_i}{D_c}
        \right)^{0.1}}Pr^{1/3}
            
    For :math:`2.2\times 10^4 < Re < 1.5\times 10^5`:
        
    .. math::
        Nu = 0.023\left[1 + 3.6\left(1 - \frac{D_i}{D_c}\right)\left(\frac{D_i}
        {D_c}\right)^{0.8}\right]Re^{0.8}Pr^{1/3}
        
    Parameters
    ----------
    Re : float
        Reynolds number with `D=Di`, [-]
    Pr : float
        Prandtl number with bulk properties [-]
    Di : float
        Inner diameter of the coil, [m]
    Dc : float
        Diameter of the helix/coil measured from the center of the tube on one
        side to the center of the tube on the other side, [m]

    Returns
    -------
    Nu : float
        Nusselt number with respect to `Di`, [-]
        
    Notes
    -----
    For very low curvatures, reasonable results are returned by both cases
    of Reynolds numbers.

    Examples
    --------
    >>> helical_turbulent_Nu_Schmidt(2E5, 0.7, 0.01, .2)
    466.2569996832083

    References
    ----------
    .. [1] Schmidt, Eckehard F. "Wärmeübergang Und Druckverlust in 
       Rohrschlangen." Chemie Ingenieur Technik 39, no. 13 (July 10, 1967): 
       781-89. doi:10.1002/cite.330391302. 
    .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and 
       Correlations for Convection Heat Transfer and Pressure Losses in 
       Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0
       (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693.
    .. [3] Hardik, B. K., P. K. Baburajan, and S. V. Prabhu. "Local Heat 
       Transfer Coefficient in Helical Coils with Single Phase Flow." 
       International Journal of Heat and Mass Transfer 89 (October 2015): 
       522-38. doi:10.1016/j.ijheatmasstransfer.2015.05.069.
    .. [4] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat
       Transfer, 3E. New York: McGraw-Hill, 1998.
    '''
    D_ratio = Di/Dc
    if Re <= 2.2E4:
        term = Re**(0.8 - 0.22*D_ratio**0.1)*Pr**(1/3.)
        return 0.023*(1. + 14.8*(1. + D_ratio)*D_ratio**(1/3.))*term
    else:
        return 0.023*(1. + 3.6*(1. - D_ratio)*D_ratio**0.8)*Re**0.8*Pr**(1/3.)