r'''Calculates internal convection Nusselt number for turbulent vertical
    upward flow in a pipe under supercritical conditions according to [1]_.
    Correlation includes an adjustment for the thermal entry length.
    One of the most common correlations for supercritical convection.
        
    .. math::
        Nu_b = 0.0069 Re_b^{0.9} \bar Pr_b^{0.66}
        \left(\frac{\rho_w}{\rho_b}\right)^{0.43}(1+2.4D/x)

        \bar{Cp} = \frac{H_w-H_b}{T_w-T_b}
        
    Parameters
    ----------
    Re : float
        Reynolds number with bulk fluid properties, [-]
    Pr : float
        Prandtl number with bulk fluid properties and an average heat capacity
        between the wall and bulk temperatures [-]
    rho_w : float, optional
        Density at the wall temperature, [kg/m^3]
    rho_b : float, optional
        Density at the bulk temperature, [kg/m^3]
    D : float, optional
        Diameter of tube, [m]
    x : float, optional
        Axial distance along the tube, [m]

    Returns
    -------
    Nu : float
        Nusselt number with wall fluid properties, [-]

    Notes
    -----
    For the data used to develop the correlation, P varied from 22.8 to 27.6 
    MPa, and D was x/D varied from 30-365. G varied from 651-3662 kg/m^2/s and 
    q varied from 310 to 3460 kW/m^2. T_b varied from 282 to 527 degrees 
    Celsius.
    
    Cp used in the calculation of Prandtl number should be the average value
    of those at the wall and the bulk temperatures.

    For enhanced heat transfer, this was the 11th most accurate correlation in 
    [2]_ with a MAD of 19.0%. On the overall database in [3]_, it was the 
    most accurate correlation however.
    
    If the extra density information is not provided, it will not be used.
    If both diameter and axial distance are not provided, the entrance 
    correction is not used.

    Examples
    --------
    >>> Nu_Bishop(1E5, 1.2, 330, 290., .01, 1.2)
    265.3620050072533

    References
    ----------
    .. [1] Bishop A.A., Sandberg R.O., Tong L.S. (1965) Forced convection heat 
       transfer to water at near-critical temperature and supercritical 
       pressures. In: AIChE J. Chemical engineering symposium series, no. 2.
       Institute of Chemical Engineers, London
    .. [2] Chen, Weiwei, Xiande Fang, Yu Xu, and Xianghui Su. "An Assessment of
       Correlations of Forced Convection Heat Transfer to Water at 
       Supercritical Pressure." Annals of Nuclear Energy 76 (February 2015): 
       451-60. doi:10.1016/j.anucene.2014.10.027.
    .. [3] Yu, Jiyang, Baoshan Jia, Dan Wu, and Daling Wang. "Optimization of 
       Heat Transfer Coefficient Correlation at Supercritical Pressure Using 
       Genetic Algorithms." Heat and Mass Transfer 45, no. 6 (January 8, 2009): 
       757-66. doi:10.1007/s00231-008-0475-4.
    .. [4] Jäger, Wadim, Victor Hugo Sánchez Espinoza, and Antonio Hurtado. 
       "Review and Proposal for Heat Transfer Predictions at Supercritical 
       Water Conditions Using Existing Correlations and Experiments." Nuclear 
       Engineering and Design, (W3MDM) University of Leeds International 
       Symposium: What Where When? Multi-dimensional Advances for Industrial 
       Process Monitoring, 241, no. 6 (June 2011): 2184-2203. 
       doi:10.1016/j.nucengdes.2011.03.022.

def Nu_Bishop(Re, Pr, rho_w=None, rho_b=None, D=None, x=None):
    r'''Calculates internal convection Nusselt number for turbulent vertical
    upward flow in a pipe under supercritical conditions according to [1]_.
    Correlation includes an adjustment for the thermal entry length.
    One of the most common correlations for supercritical convection.
        
    .. math::
        Nu_b = 0.0069 Re_b^{0.9} \bar Pr_b^{0.66}
        \left(\frac{\rho_w}{\rho_b}\right)^{0.43}(1+2.4D/x)

        \bar{Cp} = \frac{H_w-H_b}{T_w-T_b}
        
    Parameters
    ----------
    Re : float
        Reynolds number with bulk fluid properties, [-]
    Pr : float
        Prandtl number with bulk fluid properties and an average heat capacity
        between the wall and bulk temperatures [-]
    rho_w : float, optional
        Density at the wall temperature, [kg/m^3]
    rho_b : float, optional
        Density at the bulk temperature, [kg/m^3]
    D : float, optional
        Diameter of tube, [m]
    x : float, optional
        Axial distance along the tube, [m]

    Returns
    -------
    Nu : float
        Nusselt number with wall fluid properties, [-]

    Notes
    -----
    For the data used to develop the correlation, P varied from 22.8 to 27.6 
    MPa, and D was x/D varied from 30-365. G varied from 651-3662 kg/m^2/s and 
    q varied from 310 to 3460 kW/m^2. T_b varied from 282 to 527 degrees 
    Celsius.
    
    Cp used in the calculation of Prandtl number should be the average value
    of those at the wall and the bulk temperatures.

    For enhanced heat transfer, this was the 11th most accurate correlation in 
    [2]_ with a MAD of 19.0%. On the overall database in [3]_, it was the 
    most accurate correlation however.
    
    If the extra density information is not provided, it will not be used.
    If both diameter and axial distance are not provided, the entrance 
    correction is not used.

    Examples
    --------
    >>> Nu_Bishop(1E5, 1.2, 330, 290., .01, 1.2)
    265.3620050072533

    References
    ----------
    .. [1] Bishop A.A., Sandberg R.O., Tong L.S. (1965) Forced convection heat 
       transfer to water at near-critical temperature and supercritical 
       pressures. In: AIChE J. Chemical engineering symposium series, no. 2.
       Institute of Chemical Engineers, London
    .. [2] Chen, Weiwei, Xiande Fang, Yu Xu, and Xianghui Su. "An Assessment of
       Correlations of Forced Convection Heat Transfer to Water at 
       Supercritical Pressure." Annals of Nuclear Energy 76 (February 2015): 
       451-60. doi:10.1016/j.anucene.2014.10.027.
    .. [3] Yu, Jiyang, Baoshan Jia, Dan Wu, and Daling Wang. "Optimization of 
       Heat Transfer Coefficient Correlation at Supercritical Pressure Using 
       Genetic Algorithms." Heat and Mass Transfer 45, no. 6 (January 8, 2009): 
       757-66. doi:10.1007/s00231-008-0475-4.
    .. [4] Jäger, Wadim, Victor Hugo Sánchez Espinoza, and Antonio Hurtado. 
       "Review and Proposal for Heat Transfer Predictions at Supercritical 
       Water Conditions Using Existing Correlations and Experiments." Nuclear 
       Engineering and Design, (W3MDM) University of Leeds International 
       Symposium: What Where When? Multi-dimensional Advances for Industrial 
       Process Monitoring, 241, no. 6 (June 2011): 2184-2203. 
       doi:10.1016/j.nucengdes.2011.03.022. 
    '''
    Nu = 0.0069*Re**0.9*Pr**0.66
    if rho_w and rho_b:
        Nu *= (rho_w/rho_b)**0.43
    if D and x:
        Nu *= (1 + 2.4*D/x)
    return Nu