docs: def Nu_Krasnoshchekov_Protopopov(Re, Pr, Cp_avg=None, Cp_b=None, k_w=None, 
                                   k_b=None, mu_w=None, mu_b=None):
      r'''Calculates internal convection Nusselt number for turbulent vertical
      upward flow in a pipe under supercritical conditions according to [1]_.
          
      .. math::
          Nu_b = Nu_0\left(\frac{\mu_w}{\mu_b}\right)^{0.11}\left(\frac{k_b}{k_w}
          \right)^{-0.33}\left(\frac{\bar C_p}{C_{p,b}}\right)^{0.35}
          
          Nu_0 = \frac{(f/8)Re_b \bar Pr_b}{1.07+12.7(f/8)^{1/2}
          (\bar Pr_b)^{2/3}-1)}
  
          fd = [1.82\log_{10}(Re_b) - 1.64]^{-2}
          
      Parameters
      ----------
      Re : float
          Reynolds number with bulk fluid properties, [-]
      Pr : float
          Prandtl number with bulk fluid properties [-]
      Cp_avg : float, optional
          Average heat capacity between the wall and bulk temperatures, [J/kg/K]
      Cp_b : float, optional
          Heat capacity at the bulk temperature, [J/kg/K]
      k_w : float, optional
          Thermal conductivity at the wall temperature, [W/m/K]
      k_b : float, optional
          Thermal conductivity at the bulk temperature, [W/m/K]
      mu_w : float, optional
          Viscosity at the wall temperature, [Pa*s]
      mu_b : float, optional
          Viscosity at the bulk temperature, [Pa*s]
  
      Returns
      -------
      Nu : float
          Nusselt number with bulk fluid properties, [-]
  
      Notes
      -----
      For the data used to develop the correlation, P varied from 22.3 to 32 MPa,
      Re varied from 2E4 to 8.6E6, Pr from 0.86-86, viscosity ration from 0.9 to 
      3.6, thermal conductivity ratio from 1 to 6, and heat capacity ratio from
      0.07 to 4.5.
      
      For the heat transfer database in [3]_, this correlation was 14th most
      accurate.
      
      If the extra heat capacity, viscosity, and thermal conductivity 
      information is not provided, it will not be used.
  
      Examples
      --------
      >>> Nu_Krasnoshchekov_Protopopov(1E5, 1.2, 330, 290., 0.62, 0.52, 8e-4, 9e-4)
      228.85296737400222
  
      References
      ----------
      .. [1] Krasnoshchekov EA, Protopopov VS (1959) Heat transfer at 
         supercritical region in flow of carbon dioxide and water in tubes. 
         Therm Eng 12:26-30
      .. [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. 
      '''
      fd = (1.82*log10(Re) - 1.64)**-2
      Nu = (fd/8.)*Re*Pr/(1.07 + 12.7*(fd/8.)**0.5*(Pr**(2/3.)-1))
      if mu_w and mu_b:
          Nu *= (mu_w/mu_b)**0.11
      if k_w and k_b:
          Nu *= (k_w/k_b)**-0.33
      if Cp_avg and Cp_b:
          Nu *= (Cp_avg/Cp_b)**0.35
      return Nu