docs: def Nu_Yamagata(Re, Pr, Pr_pc=None, Cp_avg=None, Cp_b=None, T_b=None,
                 T_w=None, T_pc=None):
      r'''Calculates internal convection Nusselt number for turbulent vertical
      upward flow in a pipe under supercritical conditions according to [1]_.
          
      .. math::
          Nu_b = 0.0138 Re_b^{0.85}Pr_b^{0.8}F
          
          F = \left(\frac{\bar C_p}{C_{p,b}}\right)^{n_2} \text{ if } 
          \frac{T_{pc}-T_b}{T_w-T_b} < 0
  
          F = 0.67Pr_{pc}^{-0.05} \left(\frac{\bar C_p}{C_{p,b}}\right)^{n_1} 
          \text{ if } 0 < \frac{T_{pc}-T_b}{T_w-T_b}  < 1
  
          F = 1\text{ if } \frac{T_{pc}-T_b}{T_w-T_b} > 1
  
          n_1 = -0.77(1 + 1/Pr_{pc}) + 1.49
          
          n_2 = 1.44(1 + 1/Pr_{pc}) - 0.53
      
          \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, [-]
      Pr_pc : float, optional
          Prandtl number at the pseudocritical temperature, [-]
      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]
      T_b : float
          Bulk temperature, [K]
      T_w : float
          Wall temperature, [K]
      T_pc : float
          Pseudocritical temperature, i.e. temperature at P where Cp is at a 
          maximum, [K]
  
      Returns
      -------
      Nu : float
          Nusselt number with bulk fluid properties, [-]
  
      Notes
      -----
      For the data used to develop the correlation, P varied from 22.6 to 29.4 
      MPa, and D was 7.5 and 10 mm. G varied from 310-1830 kg/m^2/s, q varied 
      from 116 to 930 kW/m^2, and bulk temperature varied from 230 to 540 decrees 
      Celsius. 
      
      In the database in [3]_, the correlation was considered but not tested.
      In [2]_, the correlation was considered but no results were reported.
      
      For enhanced heat transfer database in [2]_, this correlation was the
      second best with a MAD of 11.5%. In the database in [3]_, the correlation
      was the second best as well.
      
      If the extra information is not provided, the correlation will be used
      without the corrections.
  
      Examples
      --------
  
      References
      ----------
      .. [1] Yamagata, K, K Nishikawa, S Hasegawa, T Fujii, and S Yoshida. 
         "Forced Convective Heat Transfer to Supercritical Water Flowing in 
         Tubes." International Journal of Heat and Mass Transfer 15, no. 12 
         (December 1, 1972): 2575-93. doi:10.1016/0017-9310(72)90148-2.
      .. [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. 
      '''
      F = 1
      if all([T_b, T_w, T_pc, Pr_pc, Cp_avg, Cp_b]):
          E = (T_pc - T_b)/(T_w - T_b)
          if E < 0:
              n2 = 1.44*(1 + 1/Pr_pc) - 0.53
              F = (Cp_avg/Cp_b)**n2
          elif 0 < E < 1:
              n1 = -0.77*(1 + 1/Pr_pc) + 1.49
              F = 0.67*Pr_pc**-0.05*(Cp_avg/Cp_b)**n1
      return 0.0138*Re**0.85*Pr**0.8*F