Heat Transfer Correlations for Supercritical Water in Vertically Upward Tubes

Supercritical pressure water (SCW) has been widely used in many engineering fields and industries, such as fossil fuel-fired power plants, newly developed Gen-IV nuclear power plants and so forth. Heat transfer characteristics of SCW are of great importance for both design and safe operation of the related systems. Many heat transfer correlations have been developed in the history for predicting the heat transfer characteristics of SCW. However, the prediction accuracy of the existing correlations is less than satisfactory, especially in the cases with deteriorated heat transfer (DHT) because of the severe and quick variation in thermal physical properties of SCW in the vicinity of the fluids ’ pseudo critical point. It is very necessary to develop new correlations for the heat transfer of SCW to meet the engineering requirements for satisfactory prediction of the heat transfer behavior of SCW. In this chapter, experimental data on heat transfer of SCW are extensively collected from published literatures, and the performance of the existing heat transfer correlations for SCW are reviewed and quantitatively evaluated against the collected experimental data, and then a new heat transfer correlation for SCW with high prediction accuracy is proposed.


Introduction
Supercritical pressure fluids (SCFs) have been widely used in many engineering fields and industries, such as fossil fuel-fired power plants, newly developed Gen-IV nuclear power plants, chemical processes, and so forth. Heat transfer characteristics of SCFs are of great importance for both design and safe operation of the related systems. As one of the most popular SCFs, supercritical pressure water (SCW) has been selected as the ideal working fluid in nuclear power plants and fossil-fired power stations, and the heat transfer behavior of SCW has attracted more and more attention from both scholars and engineers in these fields [1,2].
As well known, most of the newly designed thermal power plants are operated at supercritical pressures, and the operating parameters of future fossil-fired power plants have been suggested to raise to even higher levels to gain higher thermal efficiency, and at the same time, to obtain effective gains in reduction of the pollution products emission. For example, the U.S. Department of Energy has ever proposed a research program (AD 760), in which the operating parameters of the future fossil-fired power plants are suggested to raise to 37.9 MPa/732°C/760°C with an efficiency of up to 47%. A program similar to AD 760, i.e., the AD 700 thermal program, has also been proposed in Europe, and planned to build the ultrasupercritical steam condition at 720°C and 35 MPa in the future thermal power plants so as to obtain a cycle efficiency up to 50% [2]. There is no doubt that the heat transfer characteristics of SCW is an important issue for the thermal power plants with high level parameters. The higher the operating parameters, the higher the thermal efficiency, and as well, accordingly, the more important is the precise prediction of heat transfer characteristics of SCW. It should be noted that deteriorated heat transfer (DHT) is one kind of special heat transfer phenomena of SCW that might happen in the heat transfer tube and even lead to the burst of heat transfer tube or other failure accidents in the plants. On the other hand, the supercritical water-cooled reactors (SCWRs) has been selected as one of the six most promising reactors concepts presented at the Generation IV International Forum due to its high thermal efficiency and other features [3]. Similar to the situation in supercritical pressure thermal power plants, the heat transfer of SCW is an important issue for SCWRs as well. Furthermore, due to the possible risks in release of radioactive medium under conditions with tube or vessel failures, the heat transfer of SCW is of much more concern for the safe operation of SCWRs than that of thermal power stations, and as a result, precise knowledge about the heat transfer characteristics of SCW is one of the major tasks for the design and operation of SCWRs [3,4]. Besides, heat transfer of SCW is of significance for other applications with SCW as the working fluid, such as SCW gasification system [5].
Two methods have been proposed by different scholars to predict the heat transfer characteristics of SCW: the look-up Table [6,7] and the heat transfer correlations [8]. The look-up table for heat transfer of SCW is based on real experimental data. The prediction accuracy of such look-up table is pretty high; however, the applicable ranges of the operation parameters (e.g., pressures, mass fluxes, and heat fluxes) of the look-up table are limited by the experimental conditions. In the development of ultra-supercritical pressure boilers or other heat exchangers with high operation parameters, the applicability of the look-up table seems to be limited. Unlike the look-up table, heat transfer correlations have been developed on the basis on experimental data, dimensionless analysis, and theoretical analysis of the heat transfer phenomena. The applicability of heat transfer correlations are much flexible than that of look-up tables, and as a result, heat transfer correlations for SCW have found wide applications in related industries and engineering.
In the last few decades, many heat transfer correlations for SCW were proposed (see Table 3 in the later section of this chapter). Generally, the existing heat transfer correlations could be divided into three categories [9], as listed in Table 1. The first type heat transfer correlation for SCW was built on the basis of the Types Forms Remarks represents thermophysical properties correction terms; Gr* represents the buoyancy correction terms; q + represents the thermal acceleration correlation terms Dittus-Boelter et al.'s correlation [10]. Under supercritical pressure conditions, there exists no phase change of the fluid, but the thermophysical properties of the SCW experience dramatic change in the vicinity of pseudocritical point which might have great impact on the heat transfer behavior of the SCW. In view of this fact, a few correction terms consisting of the fluid thermophysical properties, such as ρ w /ρ b , μ w /μ b , λ w /λ b among the others, were introduced to the first type correlations, resulting in the emerge of second type heat transfer correlations for SCW. With the deepening research on heat transfer phenomena of SCW, buoyancy effect and thermal acceleration were considered to be the main reasons for DHT of SCW and then two special correction terms, i.e., Gr* and q + proposed by Jackson [11] and were added to the second type correlations, yielding the third type heat transfer correlations for SCW. Pioro et al. [8,12] have conducted a comprehensive review of the existing heat transfer correlations for SCW and argued that none of the existing correlations could predict the heat transfer of SCW accurately under all heat transfer regimes. Lei et al. [2] ever put emphasis on the deterioration heat transfer of SCW by analyzing the physical mechanism of special heat transfer behavior of the SCW and tried to build new correlation for SCW, and found that most of the existing correlations for heat transfer of SCW focused mainly on the enhanced heat transfer (EHT) regimes of SCW, and these correlations were not capable of providing satisfactory prediction accuracy when it was applied to the DHT regime of SCW, and he thus suggested that excluding DHT data in the development of heat correlations should be responsible for the distinct deviations between the calculated heat transfer values and the experimental data of SCW in the DHT regimes. Another reason for the unsatisfied prediction accuracies of the existing correlations for heat transfer of SCW might be, to some extent, attributed to old thermophysical properties data adopted in the development of some of the existing heat transfer correlations [13]. Therefore, a new and accurate correlation for heat transfer of SCW is expected to be effectively accurate under both the DHT and EHT regimes of SCW, and special emphasis should be directed onto the DHT phenomena, which is the most dangerous situation to not only thermal power plants but also to the Gen-IV SCWRs and other heat transfer equipment with SCW as the working fluid.
In this chapter, experimental data on heat transfer of SCW are collected, including both the EHT regimes and the DHT regimes. Existing heat transfer correlations for SCW are well reviewed. Detailed assessments of prediction accuracy of the existing correlations are also conducted against the collected experimental data. On the basis of the evaluation of the prediction behavior of the existing heat transfer correlations for SCW, a new correlation is proposed for the heat transfer of SCW to cover both the EHT regime and the DHT regime. The prediction capability of the new correlation is also assessed against the collected experimental data.

Experimental data and existing heat transfer correlations
Since 1960s, a series of experimental investigations on heat transfer of SCW have been conducted, and a large amount of experimental data have been accumulated. A comprehensive survey of experimental investigations on heat transfer correlations of SCW has been provided by Pioro et al. [4,8,14] and showed that most of previous experimental data for heat transfer of SCW focused primarily on the flow of SCW in vertical circular tubes [2]. In the present study, approximately 12,704 data points and 250 experimental cases about the flow and heat transfer of SCW in vertical circular tubes are retrieved and collected from the published literatures.

Assessment method
As seen in Table 3, the heat transfer correlations for SCW have been proposed in different years and might be developed on different basis of experimental data. As a result, the applicability of each correlation might be different. As reported by Pioro et al. [8] and Lei et al. [2], there exist distinct discrepancies between the results predicted by different correlations. It is necessary to quantitatively evaluate the prediction performance of the existing correlations.
In this part, the prediction performance of the existing heat transfer correlations are quantitatively estimated by introducing four parameters, i.e., σ 1 (mean relative deviation, MRD), σ 2 (mean absolute deviation, MAD), σ 3 (standard deviation, SD), and ρ xy (correlation coefficient between the predicted values and experimental values), as defined by Eq. (1) through Eq. (5).
where e i is where D(X) refers to the variance of the experimental data X, D(Y) refers to the variance of the calculated results Y, and Cov(X, Y) is the covariation of X and Y [9]. The closer the ρ xy is to 1.0, the better the correlation is [9].  Table 3.
Existing heat transfer correlations for SCW in vertical upward tubes.

Comparison of the predicted values by using existing correlations to the experimental data
As described in the preceding section, the existing heat transfer correlations listed in Table 3 could generally be divided into three categories (see Table 1). For simplicity, two representative correlations are selected from each category of the correlations. Figure 1(a) shows the comparison of the results calculated by the first type of correlations for heat transfer of SCW to the experimental data. This type of correlations is represented by the Dittus-Boelter et al.'s correlation [10] and Gorban et al.'s correlation [8] here. It is seen from Figure 1(a) that most of the Nu values predicted by the correlations are out of AE20% error band, indicating a generally low prediction accuracy of this type of correlations. As explained in the preceding sections, no consideration in these correlations of the dramatic change of thermophysical properties of SCW in the vicinity of its pseudocritical point might be responsible for the low prediction accuracy of the correlations. Figure 1(b) gives the representative comparison of the results calculated by the second type of correlations for SCW to the experimental data. This type of correlations is represented by the Mokry et al. [3]and Xu [32] correlations here. It can be seen from Figure 1(b) that most of predicted results by the correlations concentrate around the line of Nu cal = Nu exp , suggesting a remarkable improvement in the prediction accuracy of the correlations in comparison to that of the first type of the correlations. This improvement in the prediction accuracy might due to introducing thermophysical properties correction terms into the correlations.
Comparison of the results calculated by the third type correlations to the experimental data are illustrated in Figure 1(c). Here, the third type correlations are represented by the Kuang et al.'s correlation [60] and Liu's correlation [66].
It is seen in Figure 1(b), (c) that the prediction accuracy of the Mokry et al.'s correlation [3], Xu's correlation [32], and Liu's correlation [66] are in roughly the same level. It is surprising that although other two more correction factors (i.e., Gr* and q + ) are introduced into the correlations, the prediction accuracy of Kuang et al.'s correlation [60] is unexpectedly worse than that of the Mokry et al.'s correlation [3] and Xu's correlation [32] in the present study. Another two correlations, i.e., Yu et al.'s correlation [62] and Liao's correlation [9], which are of forms similar to that of Kuang et al.'s correlation [60], provide similar prediction performance to that of Kuang et al.'s correlation [60] in this study. This result indicates that adding more correction factors in the correlations does not always produce better accuracy, and the correction terms added to the correlations should be selected carefully. Figures 2, 3 depict the prediction performance of the 34 existing correlations listed in Table 3, under the EHT condition (experimental data from Yamagata et al. [25] is used) and the DHT condition (experimental data from Herkenrath et al. [21] is used), respectively.
It is seen from Figure 2 that under the EHT conditions, most of existing correlations could provide relatively good prediction accuracy in the enthalpy region lower than 1600 kJ/kg and the region higher than 2800 kJ/kg of SCW (i.e., in the regions far away from the pseudocritical point). However, in the enthalpy region of 1600-2300 kJ/kg (a region around the pseudocritical point, named in lots of papers as the large specific heat region, and is hereafter abbreviated as LSHR), the predicated values of many heat transfer correlations, such as Yamagata et al.'s correlation [25], Domin et al. correlation [51] and Swenson et al. correlation [17], are much higher than the corresponding experimental values, implying low prediction capability of these correlations in the LSHR. Careful analysis of the forms of Yamagata et al. correlation [25], Domin's correlation [51], and Swenson et al.'s correlation [17] shows that only one thermophysical correction property factor is employed in the abovementioned three correlations. None of the 34 correlations could give good prediction accuracy in the whole enthalpy region of SCW. It is well known that the thermophysical properties of SCW experience dramatic change in the vicinity of pseudocritical point (i.e., in the LSHR), and with this view in mind, it is suggested that one thermophysical property correction factor might not be sufficient, and, however, proper and enough correction factors should be used in the corrections in order to capture the effect of the dramatic variation in thermophysical properties on the heat transfer of SCW in LSHR.  [56]. It is seen from Figure 3 that the predicted results of Grass et al.'s correlation [53] and Petukhov et al.'s correlation [56] agree pretty well with the corresponding experimental data under DHT conditions through the whole enthalpy region of SCW studied here. Unfortunately, as seen in Figure 2, the prediction results of the above two correlations do not agree satisfactorily with the corresponding experimental data under the EHT conditions, especially in the vicinity of pseudocritical point (in the LSHR). The reason for this result might be attributed to that most of the experimental data adopted in the development Petukhov et al.'s correlation [56] were under DHT regimes. Little  Table 3.
information can be found about the experimental data used for developing Grass et al.'s correlation [53].
It is confusing that under EHT regimes, the prediction results of Miropolskii and Shitsman correlation [50] and Yu et al.'s correlation [62] agree relatively well with the corresponding experimental (see Figure 2) data, however, under DHT regimes, the prediction results of these two correlations are in quite different trend from that of the experimental data, indicating that these two correlations could not capture the variation characteristics of the experimental data under the DHT regimes (see Figure 3). As reported early by Pioro et al. [8] and Lei et al. [2], experiment data under DHT regime were excluded in the development of many of the heat transfer correlations, and this exclusion of the data under DHT regimes results in the low prediction accuracy under DHT regimes. It is seen again from  Table 3 the above results that the applicable scope of each heat transfer correlation is limited by the scope of the experimental database employed. Thus, it is of great importance to develop a new heat transfer correlation with high prediction accuracy Author σ 1 (%) σ 2 (%) σ 3 (%) ρ xy % of data within the error bands AE10% AE20% AE30% over a wide range of experimental parameters covering the NHT regime, the EHT regime, and the DHT regime [4]. In order to gain comprehensive understanding of prediction performance of the existing correlations, quantitative analyses are conducted by employing four parameters, i.e., σ 1 , σ 2 , σ 3 , and ρ xy as defined in the previous section of this chapter, and the results of σ 1 , σ 2 , σ 3 , and ρ xy for each correlation are listed in Table 4.
It is seen from Table 4 that the predicted values of most of the existing correlations falling into the AE10% error band are lower than 50%, and that falling into the AE30% error band are lower than 90%, indicating that no of these heat transfer correlations could give satisfactory predict accuracy under both DHT and EHT regimes. Generally, the prediction performance of Mokry et al.'s correlation [3]and Wang and Li's correlation [67] are compareatively the best among the 34 correlations. More than 90% of the calculated values of these two correlations fall into the AE30% error band. However, it should be noted that only about 50% of the calculated values of these two correlations fall into the AE10% error band, and the prediction accuracy the correlations needs to be improved further.
The Domin's correlation [51] and the Swenson et al.'s correlation [17] exhibit special prediction features, as seen in Figure 2. It is seen from Table 4 that the predicted values of Domin's correlation [51] and Swenson et al.'s correlation [17] falling into AE30% error band are lower than 50%. This result is in accordance with that in Figure 2, that is, under EHT conditions, the prediction accuracy of these correlations are quite low in the LSHR of SCW although relatively good prediction performance is observed in the enthalpy region lower than 1600 kJ/kg and higher than 2800 kJ/kg of SCW. As seen in Figure 3, the Domin's correlation [51] and Swenson et al.'s correlation [17] could not provide prediction accuracy either under DHT conditions.

Development of the new heat transfer correlation
As we discussed earlier, insufficient description of the severe variation of thermophysical properties of SCW in the LSHR is one of the main reasons for the low prediction accuracy of the existing heat transfer correlations. Proper correction terms should be selected carefully to reflect the impact of thermophysical properties of SCW in LSHR on the heat transfer.
Based on the analysis of the existing heat transfer correlations, a general form of heat transfer correlations for SCW is suggested as follows: where F 1 ÁÁÁ F n are correction terms, defined as one of the parameters such as ρ w /ρ b , μ w /μ b , λ w /λ b , C p , and Gr*, and so on, and C 0 , C 1 , C 2 , k 1 , ÁÁÁ k 2 are constant indices for the corresponding correction terms. As mentioned in the previous section, one thermophysical correction factor might not be sufficient to capture the effect of the severe variation of thermophysical properties of SCW on the heat transfer in LSHR. On the other hand, introducing too many correction terms into the correlation does not always mean a high prediction accuracy. It was shown that strong linear correlations existed among the correction terms, and such linear correlation might limit, even reduce, the prediction performance of the correlations [68]. Based on the multicollinearity analysis as conducted in [68], four correction terms are finally adopted in the new heat transfer correlation, i.e., Experimental data including both the DHT regimes and the EHT regimes (as listed in Table 2) are adopted simultaneously in this study to develop the new heat transfer correlation, in hope to get high prediction accuracy under both the DHT and the EHT regimes. Based on the experimental database ( Table 2), the constant indices in Eq. (7) are determined via the Levenberg-Marquardt method [69]. The final new heat transfer correlation for SCW is obtained as Eq.  Figure 4 displays the comparison of the calculated Nu by using the new heat transfer correlation to the corresponding experimental results. It is seen from Figure 4 that the relative errors of most of the calculated values are much lower than AE20%. Table 5 gives the quantitative evaluation of the prediction performance   Table 5.
Quantitative analysis of the new heat transfer correlation.
of the new heat transfer correlation. It is clearly seen from Table 5 that the new correlation has a mean absolute deviation of 9.23%, and 78.57% of the predicted results fall into the 10% error band, and 95.18% of the predicted results fall into the 20% error band, and 99.39% of the predicted results fall into the 30% error band, indicating that the prediction accuracy of the new correlation is better than that of the existing heat transfer correlations.