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

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 (T_w) 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 T_w. 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 [2–7] presented overviews and assessments of supercritical heat transfer (SCHT) correlations against both SC water and SC CO₂ data for tube. References [8–10] 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/P_c, 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.

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 T_w. 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 T_w as independent parameters.

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 (T_b), fluid at the wall (T_w), or film temperature (T_film), which is defined as the arithmetic average between T_b and T_w. Gupta et al. [29] showed that the correlation evaluated at T_w 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.

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 (T_b) 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 T_b, T_w was estimated for each correlation. For correlations requiring T_w as an independent parameter, the experimental T_w 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 T_w 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 T_w, 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 T_b), the calculation started with two assumed values for T_w (>T_b) following the Secant method. Using NIST Software [34], the assumed T_w values with the pressure were used to calculate the values of the T_w-based thermophysical properties in the correlation. HTC value was determined from the Nusselt number of the correlation. The values of HTC, q, and T_b are used to determine new values of T_w. The algorithm stops when the assumed and determined values of T_w nearly coincide. Otherwise, this procedure is repeated using the updated values of T_w. 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.

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 (H_b) 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]. H_b 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. T_w of interest is the mean temperature and HTC is the average in a cross section based on equivalent hydraulic diameter (d_hy). 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.

where T_w 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 3–10. The tables show the assessment results for all tabulated correlations against all databases—regardless of applicable fluid or heat transfer mode.

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 T_w 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 T_w is highly nonlinear close to pseudo-critical temperature (T_pc), 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.

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 T_w and not maximum T_w. 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.

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. 2–5. Representative tests were selected for these figures. Figures 2, 3, and 5 are composed of two plots showing the HTC and the T_b. The HTC plot is positioned on the top of the T_w plot in a vertical configuration. Where applicable, T_pc 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 T_b 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/P_c and T_b/T_c. In the lowest T_b/T_c (liquid-like regions), the correlations showed high σ, which almost decreased with increasing the reduced T_b.

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 T_w versus T_b 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 T_pc. 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 T_w; however, it has a strong functional dependence on both T_w and q, and the estimated HTC/T_w by the correlation followed the scatter of the data. The correlation is sensitive to small changes in T_w 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 d_hy. 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 (T_w − T_b); $((Hw−Hb)/(Tw – Tb))$⁠, J/kg K

d, D =

tube inner diameter, m

e =

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

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 $(=100(∑i=1n(ei−eavg)2)/n%)$⁠, $n$ 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 (=μc_p/k)

$Pr¯$ =

averaged Prandtl number (=(H_w − H_b)μ_b/(k_b × (T_w − T_b)))

$Qb*$ =

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/c_p))

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