Convection with supercritical fluids (ht.conv_supercritical)¶

ht.conv_supercritical.
Nu_McAdams
(Re, Pr)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1].
Found in [2] to fit the enhanced heat transfer regime with a MAD of 10.3% which was better than and of the other reviewed correlations.
\[Nu_b = 0.0243Re_b^{0.8}Pr_b^{0.4}\]Parameters:  Re : float
Reynolds number with bulk fluid properties, []
 Pr : float
Prandtl number with bulk fluid properties, []
Returns:  Nu : float
Nusselt number with bulk fluid properties, []
Notes
This has also been one of the forms of the DittusBoelter correlations. Claimed to fit data for high pressures and low heat fluxes.
References
[1] (1, 2) Mac Adams, William H. Heat Transmission. New York and London, 1942. [2] (1, 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): 45160. doi:10.1016/j.anucene.2014.10.027. Examples
>>> Nu_McAdams(1E5, 1.2) 261.3838629346147

ht.conv_supercritical.
Nu_Shitsman
(Re, Pr_b, Pr_w)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1] and [2] as shown in both [3] and [4].
\[Nu_b = 0.023 Re_b^{0.8}(min(Pr_b, Pr_w))^{0.8}\]Parameters:  Re : float
Reynolds number with bulk fluid properties, []
 Pr_b : float
Prandtl number with bulk fluid properties, []
 Pr_w : float
Prandtl number with wall fluid properties, []
Returns:  Nu : float
Nusselt number with bulk fluid properties, []
Notes
[3] states this correlation was developed with D = 7.8 and 8.2 mm and with a Pr approximately 1. [3] ranked it third in the enhanced heat transfer category, with a MAD as 11.5%
[4] cites a [1] as the source of the correlation. Neither have been reviewed, and both are in Russian. [4] lists this as third most accurate of the 14 correlations reviewed from a database of all regimes.
References
[1] (1, 2, 3) M. E Shitsman, Impairment of the heat transmission at supercritical pressures, High. Temperature, 1963, 1(2): 237244 [2] Miropol’skiy ZL, Shitsman ME (1957). Heat transfer to water and steam at variable specific heat. J Tech Phys XXVII(10): 23592372 [3] (1, 2, 3, 4) 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): 45160. doi:10.1016/j.anucene.2014.10.027. [4] (1, 2, 3, 4) 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): 75766. doi:10.1007/s0023100804754. Examples
>>> Nu_Shitsman(1E5, 1.2, 1.6) 266.1171311047253

ht.conv_supercritical.
Nu_Griem
(Re, Pr, H=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1], also shown in [2], [3] and [4]. Has complicated rules regarding where properties should be evaluated.
\[Nu_m = 0.0169Re_b^{0.8356} Pr_{sel}^{0.432}\omega\]Parameters:  Re : float
Reynolds number as explained below, []
 Pr : float
Prandtl number as explained below, []
 H : float, optional
Enthalpy of water (if the fluid is water), [J/kg]
Returns:  Nu : float
Nusselt number as explained below, []
Notes
w is calculated as follows, for water only, with a reference point from the 1967IFC formulation. It is set to 1 if H is not provided: if Hb < 1.54E6 J/kg, w = 0.82; if Hb > 1.74E6 J/kg, w = 1; otherwise w = 0.82 + 9E7*(Hb1.54E6).
To determine heat capacity to be used, Cp should be calculated at 5 points, and the lowest three points should be averaged. The five points are: Tw, (Tw+Tf)/2, Tf, (Tb+Tf)/2, Tb.
Viscosity should be the bulk viscosity. Thermal conductivity should be the average of the bulk and wall values. Density should be the bulk density.
[2] states this correlation was developed with D = 10, 14, and 20 mm, P from 22 to 27 MPa, G from 300 to 2500 kg/m^2/s, and q from 200 to 700 kW/m^2. It was ranked 6th among the 14 correlations reviewed for enhanced heat transfer, with a MAD of 13.8%, and 6th overall for the three heat transfer conditions with a overall MAD of 14.8%. [3] ranked it 8th of 14 correlations for the three heat transfer conditions.
[2] has an almost complete description of the model; both [3] and [4] simplify it.
References
[1] (1, 2) Griem, H. “A New Procedure for the Prediction of Forced Convection Heat Transfer at near and Supercritical Pressure.” Heat and Mass Transfer 31, no. 5 (1996): 3015. doi:10.1007/BF02184042. [2] (1, 2, 3, 4) 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): 45160. doi:10.1016/j.anucene.2014.10.027. [3] (1, 2, 3, 4) 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): 75766. doi:10.1007/s0023100804754. [4] (1, 2, 3) 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? Multidimensional Advances for Industrial Process Monitoring, 241, no. 6 (June 2011): 21842203. doi:10.1016/j.nucengdes.2011.03.022. Examples
>>> Nu_Griem(1E5, 1.2) 275.4818576600527

ht.conv_supercritical.
Nu_Jackson
(Re, Pr, rho_w=None, rho_b=None, Cp_avg=None, Cp_b=None, T_b=None, T_w=None, T_pc=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1].
\[ \begin{align}\begin{aligned}Nu_b = 0.0183 Re_b^{0.82} Pr^{0.5} \left(\frac{\rho_w}{\rho_b}\right)^{0.3} \left(\frac{\bar C_p}{C_{p,b}}\right)^n\\n = 0.4 \text{ for } T_b < T_w < T_{pc} \text{ or } 1.2T_{pc} < T_b < T_w\\n = 0.4 + 0.2(T_w/T_{pc}  1) \text{ for } T_b < T_{pc} <T_w\\n = 0.4 + 0.2(T_w/T_{pc}  1)[1  5(T_b/T_{pc}1)] \text{ for } T_{pc} < T_b < 1.2T_{pc} \text{ and } T_b < T_w\\\bar{Cp} = \frac{H_wH_b}{T_wT_b}\end{aligned}\end{align} \]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]
 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
The range of examined parameters is as follows: P from 23.4 to 29.3 MPa; G from 7003600 kg/m^2/s; q from 46 to 2600 kW/m^2; Re from 8E4 to 5E5; D from 1.6 to 20 mm.
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.
This is sometimes called the JacksonHall correlation. If the extra information is not provided, the correlation will be used without the corrections.
References
[1] (1, 2) Jackson, J. D. “Consideration of the Heat Transfer Properties of Supercritical Pressure Water in Connection with the Cooling of Advanced Nuclear Reactors”, 2002. [2] (1, 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): 45160. doi:10.1016/j.anucene.2014.10.027. [3] (1, 2) 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): 75766. doi:10.1007/s0023100804754. [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? Multidimensional Advances for Industrial Process Monitoring, 241, no. 6 (June 2011): 21842203. doi:10.1016/j.nucengdes.2011.03.022. Examples
>>> Nu_Jackson(1E5, 1.2) 252.37231572974918

ht.conv_supercritical.
Nu_Gupta
(Re, Pr, rho_w=None, rho_b=None, mu_w=None, mu_b=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1].
\[ \begin{align}\begin{aligned}Nu_w = 0.004 Re_w^{0.923} \bar{Pr}_w^{0.773} \left(\frac{\rho_w}{\rho_b}\right)^{0.186} \left(\frac{\mu_w}{\mu_b}\right)^{0.366}\\\bar{Cp} = \frac{H_wH_b}{T_wT_b}\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number with wall fluid properties, []
 Pr : float
Prandtl number with wall 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]
 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 wall fluid properties, []
Notes
For the data used to develop the correlation, P was set at 24 MPa, and D was 10 mm. G varied from 2001500 kg/m^2/s and q varied from 0 to 1250 kW/m^2.
Cp used in the calculation of Prandtl number should be the average value of those at the wall and the bulk temperatures.
For deteriorated heat transfer, this was the most accurate correlation in [2] with a MAD of 18.1%.
If the extra density and viscosity information is not provided, it will not be used.
References
[1] (1, 2) Gupta, Sahil, Amjad Farah, Krysten King, Sarah Mokry, and Igor Pioro. “Developing New HeatTransfer Correlation for SuperCriticalWater Flow in Vertical Bare Tubes,” January 1, 2010, 80917. doi:10.1115/ICONE1830024. [2] (1, 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): 45160. doi:10.1016/j.anucene.2014.10.027. Examples
>>> Nu_Gupta(1E5, 1.2, 330, 290., 8e4, 9e4) 186.20135477175126

ht.conv_supercritical.
Nu_Swenson
(Re, Pr, rho_w=None, rho_b=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1].
\[ \begin{align}\begin{aligned}Nu_w = 0.00459 Re_w^{0.923} Pr_w^{0.613} \left(\frac{\rho_w}{\rho_b}\right)^{0.231}\\\bar{Cp} = \frac{H_wH_b}{T_wT_b}\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number with wall fluid properties, []
 Pr : float
Prandtl number with wall 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]
Returns:  Nu : float
Nusselt number with wall fluid properties, []
Notes
The range of examined parameters is as follows: P from 22.8 to 27.6 MPa; G from 5422150 kg/m^2/s; Re from 7.5E4 to 3.16E6; T_b from 75 to 576 degrees Celsius and T_w from 93 to 649 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 deteriorated heat transfer, this was the most accurate correlation in [2] with a MAD of 18.4%. On the overall database in [3], it was the 9th most accurate correlation.
If the extra density information is not provided, it will not be used.
References
[1] (1, 2) Swenson, H. S., J. R. Carver, and C. R. Kakarala. “Heat Transfer to Supercritical Water in SmoothBore Tubes.” Journal of Heat Transfer 87, no. 4 (November 1, 1965): 47783. doi:10.1115/1.3689139. [2] (1, 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): 45160. doi:10.1016/j.anucene.2014.10.027. [3] (1, 2) 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): 75766. doi:10.1007/s0023100804754. [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? Multidimensional Advances for Industrial Process Monitoring, 241, no. 6 (June 2011): 21842203. doi:10.1016/j.nucengdes.2011.03.022. Examples
>>> Nu_Swenson(1E5, 1.2, 330, 290.) 217.92827034803668

ht.conv_supercritical.
Nu_Xu
(Re, Pr, rho_w=None, rho_b=None, mu_w=None, mu_b=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1].
\[ \begin{align}\begin{aligned}Nu_b = 0.02269 Re_b^{0.8079} \bar{Pr}_b^{0.9213} \left(\frac{\rho_w}{\rho_b}\right)^{0.6638} \left(\frac{\mu_w}{\mu_b}\right)^{0.8687}\\\bar{Cp} = \frac{H_wH_b}{T_wT_b}\end{aligned}\end{align} \]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]
 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 23 to 30 MPa, and D was 12 mm. G varied from 6001200 kg/m^2/s and q varied from 100 to 600 kW/m^2.
Cp used in the calculation of Prandtl number should be the average value of those at the wall and the bulk temperatures.
For deteriorated heat transfer, this was the third most accurate correlation in [2] with a MAD of 20.5%.
If the extra density and viscosity information is not provided, it will not be used.
References
[1] (1, 2) Xu, F., Guo, L.J., Mao, Y.F., Jiang, X.E., 2005. “Experimental investigation to the heat transfer characteristics of water in vertical pipes under supercritical pressure”. J. Xi’an Jiaotong University 39, 468471. [2] (1, 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): 45160. doi:10.1016/j.anucene.2014.10.027. Examples
>>> Nu_Xu(1E5, 1.2, 330, 290., 8e4, 9e4) 289.133054256742

ht.conv_supercritical.
Nu_Mokry
(Re, Pr, rho_w=None, rho_b=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1], and reviewed in [2].
\[ \begin{align}\begin{aligned}Nu_b = 0.0061 Re_b^{0.904} \bar{Pr}_b^{0.684} \left(\frac{\rho_w}{\rho_b}\right)^{0.564}\\\bar{Cp} = \frac{H_wH_b}{T_wT_b}\end{aligned}\end{align} \]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]
Returns:  Nu : float
Nusselt number with bulk fluid properties, []
Notes
For the data used to develop the correlation, P was set at 20 MPa, and D was 10 mm. G varied from 2001500 kg/m^2/s and q varied from 0 to 1250 kW/m^2.
Cp used in the calculation of Prandtl number should be the average value of those at the wall and the bulk temperatures.
For deteriorated heat transfer, this was the four most accurate correlation in [2] with a MAD of 24.0%. It was also the 7th most accurate against enhanced heat transfer, with a MAD of 14.7%, and the most accurate for the normal heat transfer database as well as the top correlation in all categories combined.
If the extra density information is not provided, it will not be used.
References
[1] (1, 2) Mokry, Sarah, Igor Pioro, Amjad Farah, Krysten King, Sahil Gupta, Wargha Peiman, and Pavel Kirillov. “Development of Supercritical Water HeatTransfer Correlation for Vertical Bare Tubes.” Nuclear Engineering and Design, International Conference on Nuclear Energy for New Europe 2009, 241, no. 4 (April 2011): 112636. doi:10.1016/j.nucengdes.2010.06.012. [2] (1, 2, 3) 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): 45160. doi:10.1016/j.anucene.2014.10.027. Examples
>>> Nu_Mokry(1E5, 1.2, 330, 290.) 246.1156319156992

ht.conv_supercritical.
Nu_Bringer_Smith
(Re, Pr)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under nearsupercritical conditions according to [1] and as shown in [2] and [3].
\[Nu_x = 0.0266Re_x^{0.77}Pr_w^{0.55}\]Parameters:  Re : float
Reynolds number with fluid properties at T_ref, []
 Pr : float
Prandtl number with wall fluid properties, []
Returns:  Nu : float
Nusselt number with fluid properties at T_ref, []
Notes
Fit to data somewhat distant from the critical and pseudocritical regions. Found to fit the data in [3] fourth best; in [2] however, it was ranked so low that no ranking was given.
Tref and the properties therein should be evaluated as follows:
\[ \begin{align}\begin{aligned}T_{ref} = T_b \text{ if } \frac{T_{pc}T_b}{T_wT_b} < 0\\T_{ref} = T_{pc} \text{ if } 0 < \frac{T_{pc}T_b}{T_wT_b} < 1\\T_{ref} = T_w \text{ if } \frac{T_{pc}T_b}{T_wT_b} > 1\end{aligned}\end{align} \]References
[1] (1, 2) Bringer, R. P., and J. M. Smith. “Heat Transfer in the Critical Region.” AIChE Journal 3, no. 1 (March 1, 1957): 4955. doi:10.1002/aic.690030110. [2] (1, 2, 3) 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): 45160. doi:10.1016/j.anucene.2014.10.027. [3] (1, 2, 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): 75766. doi:10.1007/s0023100804754. Examples
>>> Nu_Bringer_Smith(1E5, 1.2) 208.17631753279107

ht.conv_supercritical.
Nu_Ornatsky
(Re, Pr_b, Pr_w, rho_w=None, rho_b=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1] as shown in both [2] and [3].
\[Nu_b = 0.023Re_b^{0.8}(\min(Pr_b, Pr_w))^{0.8} \left(\frac{\rho_w}{\rho_b}\right)^{0.3}\]Parameters:  Re : float
Reynolds number with bulk fluid properties, []
 Pr_b : float
Prandtl number with bulk fluid properties, []
 Pr_w : float
Prandtl number with wall 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]
Returns:  Nu : float
Nusselt number with bulk fluid properties, []
Notes
[2] ranked it thirteenth in the enhanced heat transfer category, with a MAD of 19.8% and 11th in the normal heat transfer with a MAD of 17.6%. [3] ranked it seventh on a combined database.
If the extra density information is not provided, it will not be used.
References
[1] (1, 2) Ornatsky A.P., Glushchenko, L.P., Siomin, E.T. (1970). The research of temperature conditions of small diameter parallel tubes cooled by water under supercritical pressures. In: Proceedings of the 4th international heat transfer conference, ParisVersailles, France. Elsevier, Amsterdam, vol VI, Paper no. B, 8 November 1970 [2] (1, 2, 3) 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): 45160. doi:10.1016/j.anucene.2014.10.027. [3] (1, 2, 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): 75766. doi:10.1007/s0023100804754. Examples
>>> Nu_Ornatsky(1E5, 1.2, 1.5, 330, 290.) 276.63531150832307

ht.conv_supercritical.
Nu_Gorban
(Re, Pr)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1]. Not recommended.
\[Nu_b = 0.0059Re_b^{0.90}Pr_b^{0.12}\]Parameters:  Re : float
Reynolds number with bulk fluid properties, []
 Pr : float
Prandtl number with bulk fluid properties, []
Returns:  Nu : float
Nusselt number with bulk fluid properties, []
Notes
Reviewed in [2] and [3]; [2] did not even rank it, and [3] ranked it 12th of 14.
References
[1] (1, 2) Gorban LM, Pomet’ko RS, Khryaschev OA (1990) Modeling of water heat transfer with Freon of supercritical pressure, 1766, Institute of Physics and Power Engineering, Obninsk [2] (1, 2, 3) 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): 45160. doi:10.1016/j.anucene.2014.10.027. [3] (1, 2, 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): 75766. doi:10.1007/s0023100804754. Examples
>>> Nu_Gorban(1E5, 1.2) 182.5367282733999

ht.conv_supercritical.
Nu_Zhu
(Re, Pr, rho_w=None, rho_b=None, k_w=None, k_b=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1].
\[ \begin{align}\begin{aligned}Nu_b = 0.0068 Re_b^{0.9} \bar{Pr}_b^{0.63} \left(\frac{\rho_w}{\rho_b}\right)^{0.17} \left(\frac{k_w}{k_b}\right)^{0.29}\\\bar{Cp} = \frac{H_wH_b}{T_wT_b}\end{aligned}\end{align} \]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]
 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]
Returns:  Nu : float
Nusselt number with bulk fluid properties, []
Notes
For the data used to develop the correlation, P varied from 22 to 30 MPa, and D was 26 mm. G varied from 6001200 kg/m^2/s and q varied from 200 to 600 kW/m^2.
Cp used in the calculation of Prandtl number should be the average value of those at the wall and the bulk temperatures.
On the overall database in [2], this was the 8th most accurate correlation,and ninth most accurate against normal heat transfer.
If the extra density and thermal conductivity information is not provided, it will not be used.
References
[1] (1, 2) Zhu, Xiaojing, Qincheng Bi, Dong Yang, and Tingkuan Chen. “An Investigation on Heat Transfer Characteristics of Different Pressure SteamWater in Vertical Upward Tube.” Nuclear Engineering and Design 239, no. 2 (February 2009): 38188. doi:10.1016/j.nucengdes.2008.10.026. [2] (1, 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): 45160. doi:10.1016/j.anucene.2014.10.027. Examples
>>> Nu_Zhu(1E5, 1.2, 330, 290., 0.63, 0.69) 240.1459854494706

ht.conv_supercritical.
Nu_Bishop
(Re, Pr, rho_w=None, rho_b=None, D=None, x=None)[source]¶ 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.
\[ \begin{align}\begin{aligned}Nu_b = 0.0069 Re_b^{0.9} \bar Pr_b^{0.66} \left(\frac{\rho_w}{\rho_b}\right)^{0.43}(1+2.4D/x)\\\bar{Cp} = \frac{H_wH_b}{T_wT_b}\end{aligned}\end{align} \]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 30365. G varied from 6513662 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.
References
[1] (1, 2) Bishop A.A., Sandberg R.O., Tong L.S. (1965) Forced convection heat transfer to water at nearcritical temperature and supercritical pressures. In: AIChE J. Chemical engineering symposium series, no. 2. Institute of Chemical Engineers, London [2] (1, 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): 45160. doi:10.1016/j.anucene.2014.10.027. [3] (1, 2) 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): 75766. doi:10.1007/s0023100804754. [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? Multidimensional Advances for Industrial Process Monitoring, 241, no. 6 (June 2011): 21842203. doi:10.1016/j.nucengdes.2011.03.022. Examples
>>> Nu_Bishop(1E5, 1.2, 330, 290., .01, 1.2) 265.3620050072533

ht.conv_supercritical.
Nu_Yamagata
(Re, Pr, Pr_pc=None, Cp_avg=None, Cp_b=None, T_b=None, T_w=None, T_pc=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1].
\[ \begin{align}\begin{aligned}Nu_b = 0.0138 Re_b^{0.85}Pr_b^{0.8}F\\F = \left(\frac{\bar C_p}{C_{p,b}}\right)^{n_2} \text{ if } \frac{T_{pc}T_b}{T_wT_b} < 0\\F = 0.67Pr_{pc}^{0.05} \left(\frac{\bar C_p}{C_{p,b}}\right)^{n_1} \text{ if } 0 < \frac{T_{pc}T_b}{T_wT_b} < 1\\F = 1\text{ if } \frac{T_{pc}T_b}{T_wT_b} > 1\\n_1 = 0.77(1 + 1/Pr_{pc}) + 1.49\\n_2 = 1.44(1 + 1/Pr_{pc})  0.53\\\bar{Cp} = \frac{H_wH_b}{T_wT_b}\end{aligned}\end{align} \]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 3101830 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.
References
[1] (1, 2) 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): 257593. doi:10.1016/00179310(72)901482. [2] (1, 2, 3) 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): 45160. doi:10.1016/j.anucene.2014.10.027. [3] (1, 2, 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): 75766. doi:10.1007/s0023100804754. [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? Multidimensional Advances for Industrial Process Monitoring, 241, no. 6 (June 2011): 21842203. doi:10.1016/j.nucengdes.2011.03.022.

ht.conv_supercritical.
Nu_Kitoh
(Re, Pr, H=None, G=None, q=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1], also shown in [2], [3] and [4]. Depends on fluid enthalpy, mass flux, and heat flux.
\[ \begin{align}\begin{aligned}Nu_b = 0.015Re_b^{0.85} Pr_b^m\\m = 0.69  \frac{81000}{q_{dht}} + f_cq\\q_{dht} = 200 G^{1.2}\\f_c = 2.9\times10^{8} + \frac{0.11}{q_{dht}} \text{ for } H_b < 1500 \text{ kJ/kg}\\f_c = 8.7\times10^{8}  \frac{0.65}{q_{dht}} \text{ for } 1500 \text{ kJ/kg} < H_b < 3300 \text{ kJ/kg}\\f_c = 9.7\times10^{7} + \frac{1.3}{q_{dht}} \text{ for } H_b > 3300 \text{ kJ/kg}\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number with bulk fluid properties, []
 Pr : float
Prandtl number with bulk fluid properties, []
 H : float, optional
Enthalpy of water (if the fluid is water), [J/kg]
 G : float, optional
Mass flux of the fluid, [kg/m^2/s]
 q : float, optional
Heat flux to wall, [W/m^2]
Returns:  Nu : float
Nusselt number as explained below, []
Notes
The reference point for the enthalpy values is not stated in [1]. The upper and lower enthalpy limits for this correlation are 4000 kJ/kg and 0 kJ/kg, but these are not enforced in this function.
If not all of H, G, and q are provided, the correlation is used without the correction.
This correlation was ranked 6th best in [3], and found 4th best for enhanced heat transfer in [2] with a MAD of 12.3%.
For the data used to develop the correlation, G varied from 1001750 kg/m^2/s, q varied from 0 to 1800 kW/m^2, and bulk temperature varied from 20 to 550 decrees Celsius.
This correlation does not have realistic behavior for values outside those used in the study, and should not be used.
References
[1] (1, 2, 3) Kitoh, Kazuaki, Seiichi Koshizuka, and Yoshiaki Oka. “Refinement of Transient Criteria and Safety Analysis for a HighTemperature Reactor Cooled by Supercritical Water.” Nuclear Technology 135, no. 3 (September 1, 2001): 25264. [2] (1, 2, 3) 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): 45160. doi:10.1016/j.anucene.2014.10.027. [3] (1, 2, 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): 75766. doi:10.1007/s0023100804754. [4] (1, 2) 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? Multidimensional Advances for Industrial Process Monitoring, 241, no. 6 (June 2011): 21842203. doi:10.1016/j.nucengdes.2011.03.022. Examples
>>> Nu_Kitoh(1E5, 1.2, 1.3E6, 1500, 5E6) 331.80234139591306

ht.conv_supercritical.
Nu_Krasnoshchekov_Protopopov
(Re, Pr, Cp_avg=None, Cp_b=None, k_w=None, k_b=None, mu_w=None, mu_b=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1].
\[ \begin{align}\begin{aligned}Nu_b = Nu_0\left(\frac{\mu_w}{\mu_b}\right)^{0.11}\left(\frac{k_b}{k_w} \right)^{0.33}\left(\frac{\bar C_p}{C_{p,b}}\right)^{0.35}\\Nu_0 = \frac{(f/8)Re_b \bar Pr_b}{1.07+12.7(f/8)^{1/2} (\bar Pr_b)^{2/3}1)}\\fd = [1.82\log_{10}(Re_b)  1.64]^{2}\end{aligned}\end{align} \]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 0.8686, 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.
References
[1] (1, 2) Krasnoshchekov EA, Protopopov VS (1959) Heat transfer at supercritical region in flow of carbon dioxide and water in tubes. Therm Eng 12:2630 [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): 45160. doi:10.1016/j.anucene.2014.10.027. [3] (1, 2) 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): 75766. doi:10.1007/s0023100804754. [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? Multidimensional Advances for Industrial Process Monitoring, 241, no. 6 (June 2011): 21842203. doi:10.1016/j.nucengdes.2011.03.022. Examples
>>> Nu_Krasnoshchekov_Protopopov(1E5, 1.2, 330, 290., 0.62, 0.52, 8e4, 9e4) 228.85296737400222

ht.conv_supercritical.
Nu_Petukhov
(Re, Pr, rho_w=None, rho_b=None, mu_w=None, mu_b=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1].
\[ \begin{align}\begin{aligned}Nu_b = \frac{(f/8)Re_b \bar Pr_b}{1+900/Re_b+12.7(f/8)^{1/2} (\bar Pr_b)^{2/3}1)}\\f = f_d\left(\frac{\rho_w}{\rho_b}\right)^{0.4} \left(\frac{\mu_w}{\mu_b}\right)^{0.2}\\f_d = [1.82\log_{10}(Re_b)  1.64]^{2}\end{aligned}\end{align} \]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.
References
[1] (1, 2) Petukhov, B.S., V.A. Kurganov, and V.B. Ankudinov. “HEAT TRANSFER AND FLOW RESISTANCE IN THE TURBULENT PIPE FLOW OF A FLUID WITH NEARCRITICAL STATE PARAMETERS.” High Temperature 21, no. 1 (1983): 8189. [2] (1, 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): 45160. doi:10.1016/j.anucene.2014.10.027. Examples
>>> Nu_Petukhov(1E5, 1.2, 330, 290., 8e4, 9e4) 254.8258598466738

ht.conv_supercritical.
Nu_Krasnoshchekov
(Re, Pr, rho_w=None, rho_b=None, Cp_avg=None, Cp_b=None, T_b=None, T_w=None, T_pc=None)[source]¶ Calculates internal convection Nusselt number for turbulent vertical upward flow in a pipe under supercritical conditions according to [1].
\[ \begin{align}\begin{aligned}Nu_b = Nu_0\left(\frac{\rho_w}{\rho_b}\right)^{0.3}\left( \frac{\bar C_p}{C_{p,b}}\right)^{n}\\Nu_0 = \frac{(f/8)Re_b \bar Pr_b}{1.07+12.7(f/8)^{1/2} (\bar Pr_b^{2/3}1)}\\f_d = [1.82\log_{10}(Re_b)  1.64]^{2}\\n = 0.4 \text{ for } T_b < T_w < T_{pc} \text{ or } 1.2T_{pc} < T_b < T_w\\n = n_1 = 0.22 + 0.18T_w/T_{pc} \text{ for } 1 < T_w/T_{pc} < 2.5\\n = n_1 + (5n_1  2)(1  T_b/T_{pc}) \text{ for } T_{pc} < T_b < 1.2T_{pc} \text{ and } T_{b} < T_w\end{aligned}\end{align} \]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]
 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
The range of examined parameters is as follows: P from 23.4 to 29.3 MPa; G from 7003600 kg/m^2/s; q from 46 to 2600 kW/m^2; Re from 8E4 to 5E5; D from 1.6 to 20 mm.
If the extra information is not provided, the correlation will be used without the corrections.
References
[1] (1, 2) Krasnoshchekov, E.A., Protopopov, V.S., Van Fen, Kuraeva, I.V., 1967. Experimental investigation of heat transfer for carbon dioxide in the supercritical region. In Proceedings of the Second AllSoviet Union Conference on Heat and Mass Transfer, Minsk, Belarus, May. [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): 45160. doi:10.1016/j.anucene.2014.10.027. Examples
>>> Nu_Krasnoshchekov(1E5, 1.2) 234.82855185610364