docs: def Nu_Petukhov(Re, Pr, rho_w=None, rho_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 = \frac{(f/8)Re_b \bar Pr_b}{1+900/Re_b+12.7(f/8)^{1/2}
          (\bar Pr_b)^{2/3}-1)}
          
          f = f_d\left(\frac{\rho_w}{\rho_b}\right)^{0.4}
          \left(\frac{\mu_w}{\mu_b}\right)^{0.2}
  
          f_d = [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 [-]
      rho_w : float, optional
          Density at the wall temperature, [kg/m^3]
      rho_b : float, optional
          Density at the bulk temperature, [kg/m^3]
      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 heat transfer database in [2]_, this correlation was 5th most
      accurate in the enhanced heat transfer category, and second in the normal
      heat transfer category with MADs of 13.8% and 12.0% respectively.
      
      If the extra viscosity and density information is not provided, it will not
      be used.
  
      Examples
      --------
      >>> Nu_Petukhov(1E5, 1.2, 330, 290., 8e-4, 9e-4)
      254.8258598466738
  
      References
      ----------
      .. [1] Petukhov, B.S., V.A. Kurganov, and V.B. Ankudinov. "HEAT TRANSFER 
         AND FLOW RESISTANCE IN THE TURBULENT PIPE FLOW OF A FLUID WITH 
         NEAR-CRITICAL STATE PARAMETERS." High Temperature 21, no. 1 (1983): 
         81-89.
      .. [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.
      '''
      fd = (1.82*log10(Re) - 1.64)**-2
      if rho_w and rho_b:
          fd *= (rho_w/rho_b)**0.4
      if mu_w and mu_b:
          fd *= (mu_w/mu_b)**0.2
      return (fd/8.)*Re*Pr/(1 + 900./Re + 12.7*(fd/8.)**0.5*(Pr**(2/3.)-1))