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Research Papers

Assessment of Convective Heat Transfer Correlations Against an Expanded Database for Different Fluids at Supercritical Pressures PUBLIC ACCESS

[+] Author and Article Information
Hussam A. M. Zahlan

Nuclear Safety Experiments Branch,
Canadian Nuclear Laboratories,
286 Plant Road,
Chalk River, ON K0J 1J0, Canada
e-mail: Hussam.zahlan@cnl.ca

Laurence K. H. Leung

R&D Facilities & Operations Research Branch,
Canadian Nuclear Laboratories,
286 Plant Road,
Chalk River, ON K0J 1J0, Canada
e-mail: Laurence.leung@cnl.ca

Yan-Ping Huang

Reactor Engineering Research Division,
Nuclear Power Institute of China,
P.O. Box 436-72,
Chengdu 610213, Sichuan, China
e-mail: hyanping007@163.com

Guang-Xu Liu

Reactor Engineering Research Division,
Nuclear Power Institute of China,
P.O. Box 436-72,
Chengdu 610213, Sichuan, China
e-mail: Liugx0711@163.com

1Corresponding author.

Manuscript received May 5, 2017; final manuscript received August 14, 2017; published online December 4, 2017. Assoc. Editor: Thomas Schulenberg.

ASME J of Nuclear Rad Sci 4(1), 011004 (Dec 04, 2017) Paper No: NERS-17-1050; doi: 10.1115/1.4037720 History: Received May 05, 2017; Revised August 14, 2017

Canadian Nuclear Laboratories (CNL) has recently expanded the supercritical heat transfer (SCHT) databank with additional data provided by the Nuclear Power Institute of China (NPIC). These additional data cover flow conditions beyond the current databank, and are applicable for improving or validating existing correlations. The expanded databank comprises more than 41,000 points of heat-transfer measurements with different fluids flowing vertically upward in tubes, annuli, and bundles at supercritical (SC) pressures. It has been applied in assessing the prediction accuracy of 24 heat-transfer correlations, which were derived from experimental data obtained with water or nonaqueous fluids (such as carbon dioxide) flowing in tubes. For the correlation assessment, a sensitivity analysis has been performed by applying the measured wall temperature as an independent parameter. The assessment against the bundle data was based on cross-sectional-averaged flow conditions and the hydraulic diameter. The iterative approach (i.e., without prior knowledge of the wall temperature) overpredicted the wall temperature, which is conservative in safety analyses.

FIGURES IN THIS ARTICLE
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Development and safety analysis of supercritical water-cooled reactors (SCWRs) require predictions of heat transfer during normal operation and postulated accident scenarios. Experimental data are the basis for development of a theoretical model or a correlation. However, reliable water data are still scarce and cover only a limited range of flow conditions, particularly for the rod-bundle geometry. Recently, a number of experiments have been performed using surrogate fluids, such as carbon dioxide (CO2) and refrigerants (Refrigerant–12 and Refrigerant–22). With the low critical pressures and temperatures of these fluids compared to those of water, these experiments were able to cover a wide range of flow conditions.

Unlike the change of thermo-physical properties at low subcritical pressures, properties at the vicinity of the critical (or pseudo-critical) point are highly dependent on temperature (T) and to a much less extent on pressure (P). Although the maximum change occurs at the critical point, these properties generally decrease with increasing pressure. However, the change remains obvious at pseudo-critical point.2 The implications on momentum and energy transport include significant differences in velocity and temperature fields in the radial direction from those for uniform properties at low-subcritical pressures. And, under certain flow conditions, buoyancy, and acceleration effects, attributed mainly to large changes in density, start influencing heat transfer. Besides the nonlinear dependence of fluid's thermophysical properties with temperature, buoyancy in the momentum equation implicitly incorporates wall temperature (Tw) dependence. Therefore, momentum and energy equations become strongly coupled. The description and solution of the turbulent mixed-convection flow are further complicated. An example of this intricacy would be the case of application of correlations to data. Churkin and Deev [1] discussed the unavoidable issues in convergence of iterative solutions, which leads some studies to directly applying correlations to data (especially, for example, for a case of application of a large number of correlations to a large size database) without the iteration for Tw. This complication calls for the need for development of accurate correlations with a different formulation of the convective heat transfer at near critical and supercritical (SC) pressure. Several reviews of convective heat transfer correlations at SC pressure have been published. For example, the studies [27] presented overviews and assessments of supercritical heat transfer (SCHT) correlations against both SC water and SC CO2 data for tube. References [810] and recently [11], based on their rod-bundle water data, presented a more up-to-date assessment of correlations. The present study focuses on evaluation of correlations independently against the expanded databank, regardless of heat transfer mode.

Canadian Nuclear Laboratories' (CNL) heat transfer database consists of measurements obtained with water, CO2, refrigerants, and helium in different flow channels including round tubes, annuli, and bundle subassemblies at SC pressures. Table 1 lists the range of flow conditions, heat flux (q), and geometric parameters of the recently compiled rod-bundle data. The listed flow conditions correspond to the cross-sectional average values. CNL and Nuclear Power Institute of China (NPIC) collaborated in the framework of thermal-hydraulics and safety assessment in support of SCWR concept development. NPIC has performed a number of heat transfer experiments with tubes, annuli, and bundles in water or carbon dioxide flow at SC pressures. An exchange of SCHT databases for tubes was made between CNL and NPIC to consolidate data [16]. Table 2 lists the number of the data contributed by NPIC and the total number of data in the updated CNL databases for water and CO2 flows in tubes. Figure 1 shows the overall ranges of the reduced pressure, P/Pc, and Reynolds number, Re, for tube databases for water and CO2 flows.

Data screening: The compiled data were subjected to quality assurance tests and screening. Those not meeting the following criteria have been excluded from the databank for the present study:

  • Fluid: Data for fluids other than water, CO2, R–12, and R–22 were excluded from this study.

  • Flow direction: Data for flows other than vertical upward were also excluded.

  • Flow geometry: This study is interested in round tube, concentric circular annulus, and rod-bundle subassemblies; other flow geometries were removed from the correlation assessment database.

  • Thermal development region: Data collected at z/d < 30 were left out.

  • Tube diameter: Round tubes with sizes d < 2 mm were also excluded.

  • Duplicates, outliers, and heat balance inconsistencies.

Tube Data-Based Correlations for Water Flow.

Single-phase and SCHT correlations were compiled and described in Zahlan et al. [17]. Recently developed SCHT correlations have also been included in the assessment. The Appendix lists the form of all assessed correlations. Wang et al. [18] at CNL compiled a round tube database for SCHT to water and CO2. The CNL water database included more than 5000 data points for vertical flow. Wang et al. [18] modified the Jackson's [19] correlation using a subset of the CNL water database. They proposed a correlation for upward flow and a correlation for downward flow. Five correlations including the proposed ones were assessed against the CNL database. The results showed that the proposed two correlations provided the closest agreement with the data. Wang and Li [9] applied the heat transfer deterioration (HTD) criterion suggested by Yamagata et al. [20] to screen datasets of Yamagata et al. [20], Kirillov et al. [21], and Zhu et al. [22]. They compiled 1916 data points for normal heat transfer (NHT) in tubes with vertical-upward flow of water. These data were applied in deriving a new correlation (based on a modification of the correlation of Hu [23]) and assessing 15 heat-transfer correlations. The assessment was based on the Nusselt number rather than the heat-transfer coefficient or the Tw. Chen and Fang [10] compiled 5366 data points from 13 different sources for water flowing vertically upward in tubes. They categorized each data point into one of three heat transfer modes (i.e., normal, enhanced, or deteriorated heat transfer) and assessed selected correlations against their database. Identification of the heat transfer mode of the data was based on the ratio of experimental to predicted heat transfer coefficient (HTC, h) using the Dittus–Boelter correlation [24]. Chen and Fang [10] proposed a general form of a SCHT correlation with eight dimensionless groups. Using regression analysis software, the number of dimensionless groups was reduced, and numerical coefficients and exponents of the general form of the correlation were found and optimized based on the experimental data using the least squares method. Similar to the correlation of Kuang et al. [25], the correlation includes both q and Tw as independent parameters.

Tube Data-Based Correlations for Carbon Dioxide Flow.

Krasnoschekov and Protopopov [26] modified their previously derived correlation [27] based on CO2 data. The modified correlation was recommended for the following ranges:

8×104<Reb<5×105
0.85<Prb<65,0.9<μbμw<3.6,0.09<ρwρb<1.0
0.02<cp¯cpb<4

and

46<q<2600kW/m2

Jackson proposed many correlations for SCHT, e.g., see Refs. [19,28]; the original version was based on the Krasnoschekov and Protopopov [26] correlation. Later, Wang et al. [18] modified Jackson's correlation [19] based on CNL data compilation for CO2. Gupta et al. [29] proposed three correlations for NHT based on the experimental data collected at Chalk River Laboratories by Pioro and Khartabil [30] with the CO2 loop. In each of these three correlations, Nusselt, Reynolds, and Prandtl numbers are evaluated at one of the three fluid temperatures, bulk fluid (Tb), fluid at the wall (Tw), or film temperature (Tfilm), which is defined as the arithmetic average between Tb and Tw. Gupta et al. [29] showed that the correlation evaluated at Tw had the best agreement with the data. Yang [31] modified the Petukhov et al. correlation [32] based on the CO2 data collected at CNL by Pioro and Khartabil [30] and proposed two correlations, one for NHT and the other one for HTD. The Appendix lists the CO2 data-based SCHT correlations.

Rod-Bundle Data-Based Correlations.

Dyadyakin and Popov [33] reported the first correlation for a seven-rod bundle in water flow. Spacing between rods was maintained with a helical spacer. Richards [15] analyzed an experimental dataset of SCHT measurements in a seven-rod bundle cooled with Refrigerant–12 (R–12) flowing vertically upward. Richards [15] applied selected correlations for tube and rod bundle to the seven-rod bundle database and proposed a correlation. The correlation was restricted to NHT of R–12 in a seven-rod bundle flow channel. The correlation is applicable to G < 1200 kg/m2 s and bulk fluid temperature (Tb) 70–140  °C. Wang et al. [11] conducted an experimental study of SCHT to water flowing vertically upward through a 2 × 2 rod bundle, which was inserted into a square channel with rounded corners. They assessed eight correlations and modified the Jackson's correlation [19] with their collected rod bundle data. The modified exponent in the density ratio, however, is much higher than the corresponding one in the original correlation, which can be sensitive at the pseudocritical point and create scatter in the correlation trend. These correlations and their database-characteristics are listed in the Appendix.

Twenty-four correlations were applied to the round tube, annulus, and rod bundle databases. Thermophysical properties were estimated from the NIST Software [34]. The HTC was calculated from the Nusselt number of the correlations. Based on the calculated HTC, q and Tb, Tw was estimated for each correlation. For correlations requiring Tw as an independent parameter, the experimental Tw was applied. This approach is reasonable to establish the closeness of the correlation to the measurement. However, it is not appropriate in design and safety analyses where the Tw is not known a priori. A sensitivity analysis has been performed to understand the implication of the iterative approach in establishing the wall-temperature prediction.

The correlation providing the best agreement with wall-temperature measurements, through the direct use of the Tw, was applied iteratively to the water database for NHT in tubes [35]. Based on the independent flow parameters and q (i.e., P, G, q and Tb), the calculation started with two assumed values for Tw (>Tb) following the Secant method. Using NIST Software [34], the assumed Tw values with the pressure were used to calculate the values of the Tw-based thermophysical properties in the correlation. HTC value was determined from the Nusselt number of the correlation. The values of HTC, q, and Tb are used to determine new values of Tw. The algorithm stops when the assumed and determined values of Tw nearly coincide. Otherwise, this procedure is repeated using the updated values of Tw. Under certain conditions, the procedure resulted in convergence issues and in more than one solution. These issues are attributed to the strong nonlinear dependence of thermophysical properties in the correlation on local temperature (as indicated in Churkin and Deev [1]). The challenge at this moment is to identify the most relevant solution and to reduce the prediction uncertainty.

Correlations' Application to the Extended Databank.

Round tube and annulus databases: As discussed earlier, the present evaluation of correlations used the entire databases. The round tube database covered the two fluids water and CO2 while the annulus database was for water only.

Rod bundle database: Rod bundle flow is very different from a simple tube flow. Estimation of bulk fluid enthalpy (Hb) and mass flux (G), for each subchannel, is usually performed with a subchannel code. A different method from subchannel analysis, for rod bundle subassemblies, is based on parameters averaged over the flow cross section and is called cross section average-parameter method [36]. Hb and G are reduced to rod bundle cross section average parameters, which are calculated based on total power and mass flow rate through all subchannels without the consideration of imbalance in flow properties between the subchannels. Thus, heat transfer in a rod bundle is simplified to an equivalent circular tube case rather than an individual subchannel. Tw of interest is the mean temperature and HTC is the average in a cross section based on equivalent hydraulic diameter (dhy). Cross-section average calculation of heat transfer enables the use of tube-based correlations, for instance for preliminary thermal-hydraulic analysis of flows in rod bundles.

Uncertainty Assessment.

Heat-transfer correlations are applied in predicting the cladding temperature in SCWR fuel assemblies. Therefore, the assessment examines the prediction accuracy of Tw through the expression of prediction error, e, in terms of the difference in predicted and measured Tw, respectively, Tw,cor and Tw,exp, i.e., Display Formula

(1)e=100Tw,corTw,expTw,  exp%

where Tw is measured in  °C. The average eavg and standard deviation σ values of e were calculated for all cases. Percentages of data predicted within the error ranges of ±5% (e5),±10% (e10),±15% (e15),±20% (e20),  and>|±20%| (e>20) were also calculated for each correlation.

In total, eight tables of uncertainty numbers are presented, Tables 310. The tables show the assessment results for all tabulated correlations against all databases—regardless of applicable fluid or heat transfer mode.

Against Round Tube and Annulus Databases.

Results for the round tube database: The results of the assessment of all correlations against the water database for tube, with a total number of 20,825 data points of screened data, are presented in Table 3. The table shows that the Chen and Fang [10] correlation has by far the lowest average error and standard deviation and the highest percentage of data predicted within the specified error ranges, discussed in Sec. 4.2. This correlation showed also a similar performance for the CO2 database (Table 4) with 16,796 data points. The standard deviation for the Chen and Fang [10] correlation for the tube database for water and CO2 was less than 6%.

Iteration results: Chen and Fang [10] correlation was applied to the water database for NHT for tube with 8373 data points. From this database, 110 data points showed issues in convergence. Results showed that average error and standard deviation in terms of Tw were 5.0% and 10.0%, respectively. However, the direct application of the correlation to the same database showed that the average error and standard deviation were 0.0% and 1.0%, respectively. The correlation dependency on Tw is highly nonlinear close to pseudo-critical temperature (Tpc), which is believed to be the cause of the convergence issues in iteration. The overprediction of the wall temperature using the iterative approach is considered conservative for design and safety analyses. However, further analyses are needed to establish the general trend over the entire database.

Results for the annulus database: Table 5 presents uncertainty numbers of all correlations applied to the concentric circular annulus database for water with 1078 data points, collected by Licht et al. [47]. Results of the assessment of correlations for this database show that the Jackson's [19] correlation had the best agreement with the data.

Against Rod Bundle Database.

Error analysis of the correlations against the bundle data was based on average wall temperature, because most of the data (for different rod bundles) were reported in terms of average Tw and not maximum Tw. Also, for general temperature representation and correlation assessment based on different subchannels and spacers, the average wall temperature was deemed more meaningful than the maximum wall temperature. The latter was reported to be affected by spacer design and measurement location. However, if studying a specific spacer design, the correlations' assessment would be more useful based on the maximum wall temperature.

Water database: Correlations were also assessed against the rod bundle databases for water. The single-phase correlation of Dittus–Boelter [24], twenty SC heat transfer correlations for tube, and three correlations for rod bundle were applied to the plain (no spacers in the heated length) 2 × 2 rod bundle database (639 data points). The best estimation of the experimental data (Table 6) was also achieved by the Chen and Fang [10] correlation, followed by the Bishop et al. [37] and Griem [41] correlations. Similarly, the correlations were applied to the wire-wrapped rod bundle database with 372 data points. Table 7 shows the results of this assessment. The Chen and Fang [10] correlation had the best agreement with the database for wire-wrapped 2 × 2 rod bundle followed by the Wang et al. [18] correlation, originally developed based on CO2 data.

CO2 database: The results for the CO2 database (three-rod bundle with spacers) with 1155 data points are presented in Table 8. Here, the uncertainty numbers are larger than the ones reported earlier for the plain and spacer-equipped rod bundle water data. The best agreement with the data was obtained by the Chen and Fang [10] correlation followed by the Wang et al. [18] correlation for water.

R–12 and R–22 databases: These databases are for the seven- and three-rod bundle subassemblies, respectively. The results for the seven-rod bundle are presented in Table 9, and Table 10 presents the results for the three-rod bundle. For these two databases, Chen and Fang [10] showed the closest agreement with the experimental data.

Graphical Comparison of Best-Estimate Correlations Against Data.

Best-estimate correlations are those correlations which approximated experimental data closer than others in terms of the different uncertainty numbers, described in Sec. 4.1. The comparison between correlations and experiment is presented in Figs. 25. Representative tests were selected for these figures. Figures 2, 3, and 5 are composed of two plots showing the HTC and the Tb. The HTC plot is positioned on the top of the Tw plot in a vertical configuration. Where applicable, Tpc was presented on the plots with a straight dash-dotted line. Figure 4 shows standard deviation variation with reduced pressure and reduced temperature for the best-estimate correlations, assessed against the water database for tube.

Plots of correlations against tube and annulus databases: Figure 2 shows the results of the application of 4 different correlations to the water database for tube geometry. This figure compares heat transfer as predicted by the best-estimate correlations against two different experiments. One dataset was collected with the SC water experimental facility at NPIC, while the second dataset was reported by Jackson [48].3 Chen and Fang [10] correlation almost followed the experimental trend. Figure 3 presents a similar comparison for the carbon dioxide database for tube geometries. The left top plot of this figure shows some scatter in HTC as predicted by the Chen and Fang [10], Wang et al. [18], and Watts and Chou [40] correlations. The correlation scatter resembles the scatter of the experimental data by Zahlan et al. [35,49]. The right-hand side plots of Fig. 3 show a maximum in HTC at the pseudo-critical temperature as demonstrated by the NPIC data. Figure 4 shows variation of σ with reduced pressure and reduced Tb for the best-estimate correlations applied to the water database for round tube. Performance of the correlations varied along the reduced pressure. Unlike the other two correlations presented in Fig. 4, the Chen and Fang [10] correlation showed the lowest values of σ with P/Pc and Tb/Tc. In the lowest Tb/Tc (liquid-like regions), the correlations showed high σ, which almost decreased with increasing the reduced Tb.

Plots of correlations against rod bundle databases: Turbulent flow and convective heat transfer in rod bundle geometries differ from those in circular tubes. Nevertheless, a preliminary estimation of heat transfer in subchannels can be done with tube-based correlations. Figure 5 shows the variation of HTC and Tw versus Tb for the best-estimate correlations against two sets of data collected at similar flow conditions for 2 × 2 rod bundles. In Fig. 5, a comparison of the experimental heat transfer in a plain 2 × 2 rod bundle, from the data by Wang et al. [11], and the corresponding one in a wire-wrapped 2 × 2 rod bundle, from the data by Wang et al. [14] is presented. Although more information is needed about the location of the wire wrap along the heated length, one observation can be reported here about the wire wrap effect on SCHT in the 2 × 2 rod bundle, which is the local increase in HTC just upstream of Tpc. The best-estimate correlations based on the plain rod bundle data are the Chen and Fang [10], the Griem [41] and the Bishop et al. [37] correlations. Another observation is that the correlations approximated closer experimental heat transfer in the plain rod bundle than that in the wire-wrapped rod bundle.

CNL and NPIC have shared their databases of SCHT for tube geometries. The CNL databases were expanded by adding compiled and original experimental data obtained in the data exchange.

A diversified databank of experimental heat transfer and correlations for the SC pressure region has been compiled at CNL. The databank covers different flow geometries including tube, annulus, and rod bundle, and different fluids including water, CO2, R–12, and other fluids.

In this paper, a correlation assessment was performed against the CNL expanded databank. In total, 24 correlations were applied to the expanded CNL databases for tube, annulus, and rod bundle. This application was independent of the correlations' limitation in terms of range of flow conditions, applicable heat transfer mode, and type of fluid for a correlation database.

Uncertainty analysis of the assessed correlations revealed best-estimate correlations, which were presented in tables and graphs. Chen and Fang [10] correlation showed the best approximation of the experimental Tw; however, it has a strong functional dependence on both Tw and q, and the estimated HTC/Tw by the correlation followed the scatter of the data. The correlation is sensitive to small changes in Tw especially near critical and pseudocritical temperatures; this dependency might lead to convergence issues when applied in system codes.

Chen and Fang [10] correlation was applied iteratively to the water database of NHT for tubes. The calculated average error and standard deviation of the correlation were 5.0% and 10.0%, respectively.

Correlations for circular tube geometry were also applied to the data for noncircular geometry. The application was based on the cross section average parameter concept and dhy. The agreement between correlations and plain rod bundle data was reasonable, and better than that between the correlations and the data for rod bundle with spacers.

Although the Chen and Fang [10] correlation shed light on the correlations' parameters best-describing SCHT, improvements are still required. To avoid divergence issues, a correlation trend is desired to be free from scatter, similar to the corresponding physical phenomena. The challenge at this moment is to optimize the solution, which will also reduce prediction uncertainty.

Reliable new data for convective heat transfer to water are still required for SCWR. Finally, it was found that most of the existing data are for tubes, and rod bundle data are particularly scarce.

The work performed in Canada was supported by the Atomic Energy of Canada Limited.

The work performed in China was supported by the National Science Fund for Distinguished Young Scholars (No. 11325526) and the International Science and Technology Cooperation of China (No. 2012DFG61030).

  • cp =

    specific heat at constant pressure, J/kg K

  • cp¯ =

    averaged specific heat within the range of (Tw − Tb);  ((HwHb)/(Tw  Tb)), J/kg K

  • d, D =

    tube inner diameter, m

  • e =

    a measure of deviation of predicted wall temperature from corresponding measurement (=100((Tw,corTw,exp)/Tw,exp)%)

  • e5, e10, etc. =

    percentage of data within specified error range (±5%, ±10%, etc.)

  • f =

    friction factor; (τwG28 ρ), τw is the wall sear stress (Pa)

  • g =

    gravitational acceleration, m/s2

  • G =

    mass flux (kg/m2 s)

  • h =

    heat transfer coefficient, W/m2 K

  • H =

    specific enthalpy (J/kg)

  • k =

    thermal conductivity, W/m K

  • P =

    pressure, Pa

  • q =

    heat flux, W/m2

  • T =

    temperature,  °C or K

  • z =

    axial distance from the inlet of the heated section, m

 Greek Symbols
  • βb =

    thermal expansion coefficient, βb=(1/ρ)(ρ/T)P, 1/K

  • μ =

    dynamic viscosity, Ns/m2 = kg/ms

  • ν =

    kinematic viscosity, m2/s

  • ρ =

    fluid density, kg/m3

  • σ =

    standard deviation (=100(i=1n(eieavg)2)/n%), n is number of data points

 Subscripts
  • ac =

    acceleration

  • avg =

    average

  • b =

    bulk

  • c =

    critical

  • cor =

    correlation

  • exp =

    experimental

  • h =

    heated

  • hy =

    hydraulic

  • pc =

    pseudo-critical

  • q =

    heat flux

  • sel =

    selected

  • w =

    wall

 Dimensionless Numbers
  • Gr* =

    modified Grashof number based on q(=(gβbqd4/kbνb2))

  • Nu =

    Nusselt number (=hd/k)

  • Pr =

    Prandtl number (=μcp/k)

  • Pr¯ =

    averaged Prandtl number (=(Hw − Hb)μb/(kb × (Tw − Tb)))

  • Qb* =

    thermal loading group (=βbqd/kb)

  • Re =

    Reynolds number (=GD/μ)

  • πac =

    nondimensional acceleration number (=(q/G)(βb/cp))

  • πq =

    nondimensional heat flux number (=(q/G) (β/cp))

 Abbreviations
  • HTC =

    heat transfer coefficient

  • HTD =

    heat transfer deterioration

  • NHT =

    normal heat transfer

  • NPIC =

    Nuclear Power Institute of China

  • SC =

    supercritical

  • SCHT =

    supercritical heat transfer

  • SCWR =

    supercritical water-cooled reactor

Appendix: Assessed Correlations
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Cheng, X. , and Schulenberg, T. , 2001, “ Heat Transfer at Supercritical Pressures—Literature Review and Application to an HPLWR,” Forschungszentrum Karlsruhe, Karlsruhe, Germany, Technical Report No. FZKA 6609.
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Kirillov, P. , Pomet'ko, R. , Smirnov, A. , Grabezhnaia, V. , Pioro, I. , Duffey, R. , and Khartabil, H. , 2005, “ Experimental Study on Heat Transfer to Supercritical Water Flowing in 1- and 4-m-Long Vertical Tubes,” International Conference on Nuclear Energy Systems for Future Generation and Global Sustainability (GLOBAL’05), Tsukuba, Japan, Oct. 9–13, Paper No. GL518DF.
Zhu, X. , Bi, Q. , Yang, D. , and Chen, T. , 2009, “ An Investigation on Heat Transfer Characteristics of Different Pressure Steam–Water in Vertical Upward Tube,” Nucl. Eng. Des., 239(2), pp. 381–388. [CrossRef]
Hu, Z. , 2001, “ Heat Transfer Characteristics of Vertical Up Flow and Inclined Tube in the Supercritical Pressure and Near-Critical Pressure Region,” Ph.D. thesis, Xi'an Jiaotong University, Xi'an, China (in Chinese).
Dittus, F. , and Boelter, L. , 1930, “ Heat Transfer in Automobile Radiators of the Tubular Type,” Univ. Calif. Publ. Eng., 2(13), pp. 443–461.
Kuang, B. , Zhang, Y. , and Cheng, X. , 2008, “ A New, Wide-Ranged Heat Transfer Correlation of Water at Supercritical Pressures in Vertical Upward Ducts,” Seventh International Topical Meeting on Nuclear Reactor Thermal Hydraulics, Operation and Safety (NUTHOS-7), Seoul, South Korea, Oct. 5–9, Paper No. 189.
Krasnoschekov, E. , and Protopopov, V. , 1966, “ Experimental Study of Heat Exchange in Carbon Dioxide in the Supercritical Range at High Temperature Drops,” High Temp., 4(3), pp. 375–382.
Krasnoschekov, E. , and Protopopov, V. , 1959, “ Heat Transfer at Supercritical Region in Flow of Carbon Dioxide and Water in Tubes,” Therm. Energy, 12, pp. 26–30.
Jackson, J. , 2009, “ Validation of an Extended Heat Transfer Equation for Fluids at Supercritical Pressure,” Fourth International Symposium Supercritical Water-Cooled Reactors (ISSCWR-4), Berlin, Mar. 8–11, Paper No. 24.
Gupta, S. , Saltanov, E. , Mokry, S. , Pioro, I. , and Trevani, L. , 2013, “ Developing Empirical Heat-Transfer Correlations for Supercritical CO2 Flowing in Vertical Bare Tubes,” Nucl. Eng. Des., 261, pp. 116–131. [CrossRef]
Pioro, I. , and Khartabil, H. , 2005, “ Experimental Study on Heat Transfer to Supercritical Carbon Dioxide Flowing Upward in a Vertical Tube,” ASME Paper No. ICONE13-50118.
Yang, S.-K. , 2013, “ Heat Transfer Modes in Supercritical Fluids,” 15th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-15), Pisa, Italy, May 12–17, Paper No. NURETH15-547.
Petukhov, B. , Krasnoschekov, E. , and Protopopov, V. , 1961, “ An Investigation of Heat Transfer to Fluids Flowing in Pipes Under Supercritical Conditions,” International Heat Transfer Conference, Boulder, CO, Aug. 28–Sept. 1, pp. 569–578.
Dyadyakin, B. , and Popov, A. , 1977, “ Heat Transfer and Thermal Resistance of Tight Seven Rod Bundle, Cooled With Water Flow at Supercritical Pressures,” Trans. VTI, 11, pp. 244–253 (in Russian).
Lemmon, E. , Mclinden, M. , and Friend, D. , 2002, “ Thermophysical Properties of Fluid Systems, NIST Standard Reference Database Number 23,” NIST Reference Fluid Thermodynamic and Transport Properties Database, Version 7.0, National Institute of Standards and Technology, Gaithersburg, MD.
Zahlan, H. , Groeneveld, D. , and Tavoularis, S. , 2015, “ Measurements of Convective Heat Transfer to Vertical Upward Flows of CO2 in Circular Tubes at Near-Critical and Supercritical Pressures,” Nucl. Eng. Des., 289, pp. 92–107. [CrossRef]
Groeneveld, D. , 1995, “ CANDU Reactor Thermalhydraulics Course,” McMaster University, Hamilton, ON, Canada, Paper No. ARD-TD-567.
Bishop, A. A. , Sandberg, R. O. , and Tong, L. S. , 1965, “ Forced Convection Heat Transfer to Water at Near-Critical Temperatures and Supercritical Pressures,” Symposium on Chemical Engineering Under Extreme Conditions, London, June 14–17, Vol. 2, pp. 7–85.
Swenson, H. , Carver, J. , and Kakarala, C. , 1965, “ Heat Transfer to Supercritical Water in Smooth-Bore Tubes,” ASME J. Heat Transfer, 87(4), pp. 477–484. [CrossRef]
Gupta, S. , Mokry, S. , and Pioro, I. , 2011, “ Developing a Heat-Transfer Correlation for Supercritical-Water Flowing in Vertical Bare Tubes and Its Application in SCWRs,” ASME Paper No. ICONE19-43503.
Watts, M. J. , and Chou, C. T. , 1982, “ Mixed Convection Heat Transfer to Supercritical Pressure Water,” Seventh International Heat Transfer Conference, Munich, Germany, Sept. 6–10, Vol. 3, pp. 495–500.
Griem, H. , 1996, “ A New Procedure for the Prediction of Forced Convection Heat Transfer at Near- and Supercritical Pressure,” Heat Mass Transfer, 31(5), pp. 301–305. [CrossRef]
Koshizuka, S. , and Oka, Y. , 2000, “ Computational Analysis of Deterioration Phenomena and Thermal-Hydraulic Design of SCR,” First International Symposium on Supercritical Water-Cooled Reactor Design and Technology (SCR-2000), Tokyo, Japan, Nov. 6–8, Paper No. 302.
Mokry, S. , Gospodinov, Y. , Pioro, I. , and Kirillov, P. , 2009, “ Supercritical Water Heat-Transfer Correlation for Vertical Bare Tubes,” ASME Paper No. ICONE17-76010.
Cheng, X. , Yang, Y. , and Huang, S. , 2009, “ A Simple Heat Transfer Correlation for SC Fluid Flow in Circular Tubes,” 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13), Ishikawa, Japan, Sept. 27–Oct. 2, Paper No. N13P1047.
Sieder, E. , and Tate, G. , 1936, “ Heat Transfer and Pressure Drop of Liquids in Tubes,” Ind. Eng. Chem., 28(12), pp. 1429–1435. [CrossRef]
Gnielinski, V. , 1976, “ New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow,” Int. Chem. Eng., 16(2), pp. 359–368.
Licht, J. , Anderson, M. , and Corradini, M. , 2008, “ Heat Transfer to Water at Supercritical Pressures in a Circular and Square Annular Flow Geometry,” Int. J. Heat Fluid Flow, 29(1), pp. 156–166. [CrossRef]
Jackson, J. , 2009, “ Heat Transfer Studies at Manchester With Carbon Dioxide at Supercritical and Near-Critical Pressures,” International Atomic Energy Agency, Vienna, Austria, Ref. JDJ/IAEA/CRP/Report No. 1.
Zahlan, H. , and Leung, L. , 2017, “ An Assessment of Round Tube Correlations for Convective Heat Transfer at Supercritical Pressure,” CNL Nucl. Rev. (accepted).
Filonenko, G. , 1954, “ Gidravlicheskoye soprotivleniye v trubakh,” Teploenergetika, 1(4), pp. 40–44 (in Russian).
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References

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Hall, W. , Jackson, J. , and Watson, A. , 1968, “ A Review of Forced Convection Heat Transfer to Fluids at Supercritical Pressures,” Proc. Inst. Mech. Eng., 182(3I), pp. 10–22.
Jackson, J. , and Hall, W. , 1979, “ Forced Convection Heat Transfer to Fluids at Supercritical Pressure,” Turbulent Forced Convection in Channels and Bundles, Vol. 2, S. Kakaç and D. B. Spalding , eds., Hemisphere Publishing, Washington, DC, pp. 563–611.
Jackson, J. , and Hall, W. , 1979, “ Influences of Buoyancy on Heat Transfer to Fluids Flowing in Vertical Tubes Under Turbulent Conditions,” Turbulent Forced Convection in Channels and Bundles, Vol. 2, S. Kakaç and D. B. Spalding , eds., Hemisphere Publishing, Washington, DC, pp. 613–641.
Cheng, X. , and Schulenberg, T. , 2001, “ Heat Transfer at Supercritical Pressures—Literature Review and Application to an HPLWR,” Forschungszentrum Karlsruhe, Karlsruhe, Germany, Technical Report No. FZKA 6609.
Pioro, I. , Khartabil, H. , and Duffey, R. , 2004, “ Heat Transfer to Supercritical Fluids Flowing in Channels—Empirical Correlations Survey,” Nucl. Eng. Des., 230(1–3), pp. 69–91. [CrossRef]
Petukhov, B. , and Polyakov, A. , 1988, “ Heat Transfer at Supercritical Pressures,” Heat Transfer in Turbulent Mixed Convection, Hemisphere Publishing, New York, pp. 183–201.
Zahlan, H. , Groeneveld, D. , Tavoularis, S. , Mokry, S. , and Pioro, I. , 2011, “ Assessment of Supercritical Heat Transfer Prediction Methods,” Fifth International Symposium on Super-Critical-Water-Cooled Reactors (ISSCWR-5), Vancouver, BC, Canada, Mar. 13–16, Paper No. P008.
Wang, C. , and Li, H. , 2014, “ Evaluation of the Heat Transfer Correlations for Supercritical Pressure Water in Vertical Tubes,” Heat Transfer Eng., 35(6–8), pp. 685–692. [CrossRef]
Chen, W. , and Fang, X. , 2014, “ A New Heat Transfer Correlation for Supercritical Water Flowing in Vertical Tubes,” Int. J. Heat Mass Transfer, 78, pp. 156–160. [CrossRef]
Wang, H. , Bi, Q. , Wang, L. , Lv, H. , and Leung, L. , 2014, “ Experimental Investigation of Heat Transfer From a 2 × 2 Rod Bundle to Supercritical Pressure Water,” Nucl. Eng. Des., 275, pp. 205–218. [CrossRef]
Eter, A. , Groeneveld, D. , and Tavoularis, S. , 2016, “ An Experimental Investigation of Supercritical Heat Transfer in a Three-Rod Bundle Equipped With Wire-Wrap and Grid Spacers and Cooled by Carbon Dioxide,” Nucl. Eng. Des., 303, pp. 173–191. [CrossRef]
Gu, H. , Li, H. , Hu, Z. , Liu, D. , and Zhao, M. , 2015, “ Heat Transfer to Supercritical Water in a 2 × 2 Rod Bundle,” Ann. Nucl. Energy, 83, pp. 114–124. [CrossRef]
Wang, H. , Bi, Q. , and Leung, L. , 2016, “ Heat Transfer From a 2 × 2 Wire Wrapped Rod Bundle to Supercritical Pressure Water,” Int. J. Heat Mass Transfer, 97, pp. 486–501. [CrossRef]
Richards, G. , 2012, “ Study of Heat Transfer in a 7-Element Bundle Cooled With the Upward Flow of Supercritical Freon-12,” Master's thesis, University of Ontario Institute of Technology, Oshawa, ON, Canada.
2015, “ Supercritical Heat Transfer Database Exchange Between CNL and NPIC,” private communications.
Zahlan, H. , Leung, L. , Huang, Y. , and Liu, G. , 2017, “ General Assessment of Convection Heat Transfer Correlations for Multiple Geometries and Fluids at Supercritical Pressure,” CNL Nucl. Rev., epub.
Wang, S. , Yuan, L. , and Leung, L. , 2010, “ Assessment of Supercritical Heat-Transfer Correlations Against AECL Database for Tubes,” Second Canada–China Joint Workshop on Supercritical Water-Cooled Reactors (CCSC-2010), Toronto, ON, Canada, Apr. 25–28.
Jackson, J. , 2002, “ Consideration of the Heat Transfer Properties of Supercritical Pressure Water in Connection With the Cooling of Advanced Nuclear Reactors,” 13th Pacific Basin Nuclear Conference (PBNC 2002), Shenzhen City, China, Oct. 21–25, p. 240.
Yamagata, K. , Nishikawa, K. , Hasegawa, S. , Fugii, T. , and Yoshida, S. , 1972, “ Forced Convective Heat Transfer to Supercritical Water Flowing in Tube,” Int. J. Heat Mass Transfer, 15(12), pp. 2575–2593. [CrossRef]
Kirillov, P. , Pomet'ko, R. , Smirnov, A. , Grabezhnaia, V. , Pioro, I. , Duffey, R. , and Khartabil, H. , 2005, “ Experimental Study on Heat Transfer to Supercritical Water Flowing in 1- and 4-m-Long Vertical Tubes,” International Conference on Nuclear Energy Systems for Future Generation and Global Sustainability (GLOBAL’05), Tsukuba, Japan, Oct. 9–13, Paper No. GL518DF.
Zhu, X. , Bi, Q. , Yang, D. , and Chen, T. , 2009, “ An Investigation on Heat Transfer Characteristics of Different Pressure Steam–Water in Vertical Upward Tube,” Nucl. Eng. Des., 239(2), pp. 381–388. [CrossRef]
Hu, Z. , 2001, “ Heat Transfer Characteristics of Vertical Up Flow and Inclined Tube in the Supercritical Pressure and Near-Critical Pressure Region,” Ph.D. thesis, Xi'an Jiaotong University, Xi'an, China (in Chinese).
Dittus, F. , and Boelter, L. , 1930, “ Heat Transfer in Automobile Radiators of the Tubular Type,” Univ. Calif. Publ. Eng., 2(13), pp. 443–461.
Kuang, B. , Zhang, Y. , and Cheng, X. , 2008, “ A New, Wide-Ranged Heat Transfer Correlation of Water at Supercritical Pressures in Vertical Upward Ducts,” Seventh International Topical Meeting on Nuclear Reactor Thermal Hydraulics, Operation and Safety (NUTHOS-7), Seoul, South Korea, Oct. 5–9, Paper No. 189.
Krasnoschekov, E. , and Protopopov, V. , 1966, “ Experimental Study of Heat Exchange in Carbon Dioxide in the Supercritical Range at High Temperature Drops,” High Temp., 4(3), pp. 375–382.
Krasnoschekov, E. , and Protopopov, V. , 1959, “ Heat Transfer at Supercritical Region in Flow of Carbon Dioxide and Water in Tubes,” Therm. Energy, 12, pp. 26–30.
Jackson, J. , 2009, “ Validation of an Extended Heat Transfer Equation for Fluids at Supercritical Pressure,” Fourth International Symposium Supercritical Water-Cooled Reactors (ISSCWR-4), Berlin, Mar. 8–11, Paper No. 24.
Gupta, S. , Saltanov, E. , Mokry, S. , Pioro, I. , and Trevani, L. , 2013, “ Developing Empirical Heat-Transfer Correlations for Supercritical CO2 Flowing in Vertical Bare Tubes,” Nucl. Eng. Des., 261, pp. 116–131. [CrossRef]
Pioro, I. , and Khartabil, H. , 2005, “ Experimental Study on Heat Transfer to Supercritical Carbon Dioxide Flowing Upward in a Vertical Tube,” ASME Paper No. ICONE13-50118.
Yang, S.-K. , 2013, “ Heat Transfer Modes in Supercritical Fluids,” 15th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-15), Pisa, Italy, May 12–17, Paper No. NURETH15-547.
Petukhov, B. , Krasnoschekov, E. , and Protopopov, V. , 1961, “ An Investigation of Heat Transfer to Fluids Flowing in Pipes Under Supercritical Conditions,” International Heat Transfer Conference, Boulder, CO, Aug. 28–Sept. 1, pp. 569–578.
Dyadyakin, B. , and Popov, A. , 1977, “ Heat Transfer and Thermal Resistance of Tight Seven Rod Bundle, Cooled With Water Flow at Supercritical Pressures,” Trans. VTI, 11, pp. 244–253 (in Russian).
Lemmon, E. , Mclinden, M. , and Friend, D. , 2002, “ Thermophysical Properties of Fluid Systems, NIST Standard Reference Database Number 23,” NIST Reference Fluid Thermodynamic and Transport Properties Database, Version 7.0, National Institute of Standards and Technology, Gaithersburg, MD.
Zahlan, H. , Groeneveld, D. , and Tavoularis, S. , 2015, “ Measurements of Convective Heat Transfer to Vertical Upward Flows of CO2 in Circular Tubes at Near-Critical and Supercritical Pressures,” Nucl. Eng. Des., 289, pp. 92–107. [CrossRef]
Groeneveld, D. , 1995, “ CANDU Reactor Thermalhydraulics Course,” McMaster University, Hamilton, ON, Canada, Paper No. ARD-TD-567.
Bishop, A. A. , Sandberg, R. O. , and Tong, L. S. , 1965, “ Forced Convection Heat Transfer to Water at Near-Critical Temperatures and Supercritical Pressures,” Symposium on Chemical Engineering Under Extreme Conditions, London, June 14–17, Vol. 2, pp. 7–85.
Swenson, H. , Carver, J. , and Kakarala, C. , 1965, “ Heat Transfer to Supercritical Water in Smooth-Bore Tubes,” ASME J. Heat Transfer, 87(4), pp. 477–484. [CrossRef]
Gupta, S. , Mokry, S. , and Pioro, I. , 2011, “ Developing a Heat-Transfer Correlation for Supercritical-Water Flowing in Vertical Bare Tubes and Its Application in SCWRs,” ASME Paper No. ICONE19-43503.
Watts, M. J. , and Chou, C. T. , 1982, “ Mixed Convection Heat Transfer to Supercritical Pressure Water,” Seventh International Heat Transfer Conference, Munich, Germany, Sept. 6–10, Vol. 3, pp. 495–500.
Griem, H. , 1996, “ A New Procedure for the Prediction of Forced Convection Heat Transfer at Near- and Supercritical Pressure,” Heat Mass Transfer, 31(5), pp. 301–305. [CrossRef]
Koshizuka, S. , and Oka, Y. , 2000, “ Computational Analysis of Deterioration Phenomena and Thermal-Hydraulic Design of SCR,” First International Symposium on Supercritical Water-Cooled Reactor Design and Technology (SCR-2000), Tokyo, Japan, Nov. 6–8, Paper No. 302.
Mokry, S. , Gospodinov, Y. , Pioro, I. , and Kirillov, P. , 2009, “ Supercritical Water Heat-Transfer Correlation for Vertical Bare Tubes,” ASME Paper No. ICONE17-76010.
Cheng, X. , Yang, Y. , and Huang, S. , 2009, “ A Simple Heat Transfer Correlation for SC Fluid Flow in Circular Tubes,” 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13), Ishikawa, Japan, Sept. 27–Oct. 2, Paper No. N13P1047.
Sieder, E. , and Tate, G. , 1936, “ Heat Transfer and Pressure Drop of Liquids in Tubes,” Ind. Eng. Chem., 28(12), pp. 1429–1435. [CrossRef]
Gnielinski, V. , 1976, “ New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow,” Int. Chem. Eng., 16(2), pp. 359–368.
Licht, J. , Anderson, M. , and Corradini, M. , 2008, “ Heat Transfer to Water at Supercritical Pressures in a Circular and Square Annular Flow Geometry,” Int. J. Heat Fluid Flow, 29(1), pp. 156–166. [CrossRef]
Jackson, J. , 2009, “ Heat Transfer Studies at Manchester With Carbon Dioxide at Supercritical and Near-Critical Pressures,” International Atomic Energy Agency, Vienna, Austria, Ref. JDJ/IAEA/CRP/Report No. 1.
Zahlan, H. , and Leung, L. , 2017, “ An Assessment of Round Tube Correlations for Convective Heat Transfer at Supercritical Pressure,” CNL Nucl. Rev. (accepted).
Filonenko, G. , 1954, “ Gidravlicheskoye soprotivleniye v trubakh,” Teploenergetika, 1(4), pp. 40–44 (in Russian).

Figures

Grahic Jump Location
Fig. 1

Spread of the water and CO2 databases for tube over reduced pressure and Reynolds number

Grahic Jump Location
Fig. 4

Variation of σ with reduced pressure and reduced Tb for the best-estimate correlations, applied to the water database for round tube

Grahic Jump Location
Fig. 2

Comparison of experimental h and Tw versus Tb and best-estimate circular tube correlations for water

Grahic Jump Location
Fig. 3

Comparison of experimental h and Tw versus Tb and best-estimate circular tube correlations for CO2

Grahic Jump Location
Fig. 5

Comparison of experimental h and Tw versus Tb and best-estimate correlations for water in 2 × 2 rod bundle

Tables

Table Grahic Jump Location
Table 3 Correlation uncertainty against water database for tube (20,825)
Table Grahic Jump Location
Table 2 Number of data in the CNL tube database
Table Grahic Jump Location
Table 1 Flow conditions and numbers of the recently compiled bundle data
Table Grahic Jump Location
Table 6 Correlation uncertainty against 2 × 2 rod bundle database for water (639)
Table Grahic Jump Location
Table 7 Correlation uncertainty against wire-wrapped rod bundle database for water (2 × 2-/3-rod; 353/19 data points, respectively)
Table Grahic Jump Location
Table 8 Correlation uncertainty against three-rod bundle database for CO2 (1155)
Table Grahic Jump Location
Table 9 Correlation uncertainty against seven-rod bundle database for R–12 (129)
Table Grahic Jump Location
Table 10 Correlation uncertainty against three-rod bundle database for R–22 (192)
Table Grahic Jump Location
Table 4 Correlation uncertainty against CO2 database for tube (16,796)
Table Grahic Jump Location
Table 11 Single-phase correlations—based on water data
Table Grahic Jump Location
Table 12 Correlations for SCHT to water
Table Grahic Jump Location
Table 13 Rod-bundle correlations and main characteristics of their databases
Table Grahic Jump Location
Table 14 Correlations for SCHT to CO2
Table Grahic Jump Location
Table 5 Correlation uncertainty against water database for annulus (1078)

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