A Modified Turbulent Model for the Supercritical Water Flows in the Vertical Upward Channels
Graphical Abstract
Introduction
The supercritical water-cooled reactor (SCWR) is one of the proposed six Generation IV reactors [1]. The working fluid used as the coolant in the fuel bundle of the Canadian SCWR core is the supercritical water. The thermal physical properties of the supercritical water change dramatically near the pseudo-critical point (as shown in Fig. 1) [2]. The abnormal heat transfer phenomenon, either the heat transfer enhancement or the heat transfer deterioration could appear in the upward channels at the supercritical condition [3], [4], [5], [6]. When the heat transfer deterioration happens, the heat transfer coefficient is lower, which may lead to higher wall temperatures that might be above the maximum allowable temperature for the cladding surface of the fuel rods. Therefore, an accurate prediction of the wall temperature is very important before the fuel assembly is put into use in the reactor. Thus far, several researchers have made efforts on both the experimental and numerical studies on the fluid flow and heat transfer of the supercritical water in the circular tube [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. The works showed the effects of the buoyancy and the thermal acceleration due to the sharp variations of thermal physical properties of the supercritical water near the pseudo-critical point might be the main reasons of the abnormal heat transfer phenomenon.
Extensive experimental studies using the supercritical water in the vertical upward tube have been made by several researchers, such as the works by Shen et al. and Pioro [10], [12], [13]. Most of the studies focused on the investigation of the heat transfer characteristics or developing the heat transfer correlations at the supercritical conditions. However, there were just a few experimental works for the supercritical water in the upward fuel bundle of the reactor [14], [18], [19], [20], [21] until now. In addition to the experimental studies, researchers have performed many numerical studies by the CFD simulations of flow and thermal field in the supercritical water channels [7], [8], [9], [10], [22], [23], [24]. Most of the CFD studies applied the Reynolds-averaged Navier – Stokes approach. Different turbulent models have been assessed against the available experimental data. The performance of the turbulent models varied case-by-case. It mainly depends on the operating conditions, such as the heat to mass flux ratio, the geometry of the flow channel, and the flow direction [14], [25]. Among them, the heat to mass flux ratio plays a dominant role. However, there is no general consensus on the criterion of the heat to mass flux ratio for the onset of the heat transfer deterioration. Due to the limited experimental data under the supercritical conditions, the turbulent Prandtl number was assumed to be a constant in the previous CFD simulations. The dramatic variations of the thermal physical properties near the pseudo-critical point makes the predictive assessment more difficult. The strong buoyancy and thermal acceleration effect caused by the strong variations of the thermophysical properties of the supercritical water near the pseudo-critical point should be considered in the simulations. Most turbulent models used in CFD simulations were developed for incompressible and constant-property flows. For conventional fluids without large variations of thermal physical properties, can be treated as a constant based on the Reynolds analogy assumption, ranging from 0.8 to 0.9. However, the changes sharply at supercritical conditions [26], [27], [28], [29], [30], [31], [32], [33]. Therefore, an appropriate treatment of the turbulent Prandtl number at the supercritical condition is needed. There are no available experimental data of for the supercritical fluid now. Several models have been proposed in literatures. Myong et al. [27] proposed a variable model for the heat transfer in a fully developed turbulent pipe flow which was heated by a constant heat flux. The fluid used in the numerical simulations were not mentioned. Two models were put developed by Kays and Crawford [28], [29]. The parameters used in the models were derived from the experimental studies on the heat transfer of transformer oil, water, air, but, not under a supercritical condition. Tang et al. [30] introduced a variable model for the heat transfer of the supercritical carbon dioxide in the upward tube. Kong et al. [31] then assessed the accuracy of Tang et al.’s model for the heat transfer of the supercritical water in the upward tube. The results showed the prediction is not satisfactory. This seems to signify that the difference of the critical parameters may lead to the variations of the turbulent Prandtl number although the heat transfer characteristics of supercritical fluids are similar. Jiang et al. [32] and Bae [33] developed similar models for supercritical carbon dioxide and supercritical water in circular tubes. The models were both functions of non-dimensional distance from the wall (y+). After investigating the accuracy of the previous turbulent models, Kong et al. [31] introduced a new variable model considering the effects of the pressure, turbulent viscosity, and molecular Prandtl number for the heat transfer of the supercritical water in an upward tube. The predictions of the wall temperatures by this model are satisfactory except for deteriorated heat transfer operating conditions where there are still large discrepancies.
The objectives of the present study are (1) to propose a new variable model for supercritical fluid flows, (2) to find the best existing turbulent models for the prediction of the wall temperature in the supercritical flow channels, and (3) to modify the existing turbulent models using the proposed variable model.
Section snippets
Configuration of the flow channels
In this work, the experimental data used for the assessment of the simulations of the supercritical water in the upward circular tube and the multiple fuel rods channels are from Mokry et al. [34] and Li [14], respectively. The experimental uncertainties of the wall temperatures for Cases 1–3 are ± 3.0% and Cases 4 − 5 are ± 1.5˚Ϲ. The configurations of these two types of channels are shown in Fig. 2 and the geometrical and operating parameters used in all simulations are listed in Table 1.
Numerical model and governing equations
The
Evaluation of standard turbulent models
Five experimental cases are selected in the study to assess the performance of different turbulent models, among them Cases 1–3 are for the flows in an upward circular tube and Cases 4 − 5 are for flows in upward multiple fuel rods channels. Since an accurate wall temperature prediction of the fuel rod in the reactor is significant for the design and safety of the SCWR, the wall temperatures are compared with the experimental data in the present work.
Turbulent Prandtl number
The discrepancies between the numerical results and the experimental data for supercritical fluid flow and heat transfer can be due to the improper treatment of the momentum and heat eddy diffusivity at the supercritical conditions. The turbulent Prandtl number used in the governing equations is a non-dimensional parameter which measures the relationship of the momentum eddy diffusivity and the heat transfer eddy diffusivity. It can be defined as:
In the energy
Conclusions
In this study, Aa new variable model is developed, and modified Realisable k – model, the k – ω SST model and the RSM are proposed using the new variable to improve the performance of the numerical models for supercritical fluid flow and heat transfer. The assessment of the proposed turbulent models was carried out for the supercritical water flows in both the upward circular tube and the upward channel with multiple fuel rods. The wall temperatures predicted by the modified
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery grant [grant number #04757].
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