Supercritical fluids (SCFs) become more and more important in various engineering applications. In nuclear power systems, SCFs are considered as coolant of the reactor core such as the supercritical water-cooled reactor (SCWR), superconducting magnets and blankets in the fusion reactors, or as fluid in the energy conversion systems of the next generation nuclear reactors. Accurate determination of heat transfer and the temperature of the structural material (e.g., fuel rod cladding) is of crucial importance for the system design. Thus, extensive studies on heat transfer to SCFs have been carried out in the past five decades and are still ongoing worldwide. However, no breakthrough is recognized or expected in the near future. In this paper, the status, main challenges, and future R&D needs are briefly reviewed. Three aspects are taken into consideration, i.e., experimental studies, numerical analysis, and model development for the prediction of heat transfer coefficient (HTC). Several key challenges and also the important subjects of the future R&D needs are identified. They are (a) data base for turbulence quantities, (b) multisolution of wall temperature, (c) extensive Reynolds-averaged Navier–Stokes (ERANS) method, and (d) new prediction method for HTC.

## Introduction

Supercritical fluids (SCFs) have been widely applied in energy and chemical engineering. In the frame of the GEN-IV nuclear energy system development, supercritical water and supercritical CO_{2} have been considered as coolant in the nuclear reactor or as working fluids in the energy conversion system. Knowledge of hat transfer to SCFs is required for the design and for the safe operation of nuclear reactors as well as other heat transport systems.

Due to the strong variation of thermal–physical properties in the vicinity of the pseudo-critical point, heat transfer of SFCs shows abnormal behavior compared to that of conventional fluids. One of the main features of heat transfer to SCFs is the strong dependence on heat flux, especially for the bulk temperatures close to the pseudo-critical value.

Heat transfer to SCFs has been extensively investigated since more than five decades, especially in the last decade in the frame of the development of supercritical water-cooled reactors (SCWRs). There are several review papers summarizing the R&D status [1–4]. R&D activities on heat transfer to SCFs have been carried out in all three aspects, i.e., experimental studies, numerical analysis, and model development for the prediction of heat transfer coefficient (HTC). There exist an enormous large number of research works on HT to SCFs, published not only in international journals but also at international conferences such as “International Symposium on SCWR.” Due to the limited content, this paper only cited several studies, which were mainly carried out at the Shanghai Jiao Tong University (SJTU) or at the Karlsruhe Institute of Technology (KIT).

In addition to the R&D activities at individual research institutions, there are also programs of international collaboration. In the frame of the GEN-IV international forum, five countries or regions, i.e., Canada, China, EU, Japan, and Russia, are participating in the joint R&D program of SCWR system. Under the SCWR system, there are three projects. One of them is related to thermal–hydraulics and safety. Research works on heat transfer to SCFs belong to the key tasks of this project. Another international collaboration is the Coordinated Research Program (CRP), organized by the International Atomic Energy Agency with the title “understanding and prediction of thermal–hydraulics phenomena relevant to super critical water-cooled reactors (SCWRs)” [5]. One of the main focuses of the CRP is again the heat transfer to SCFs.

In spite of big efforts made in the last decades worldwide, no breakthrough has been achieved in the understanding of the mechanisms and in the prediction of heat transfer coefficient. Based on the last more than 15 years studies on heat transfer to SCFs, the authors summarize briefly the status, the main challenges, and the future needs related to all the three aspects of heat transfer to SCFs, i.e., experimental study, numerical analysis, and model development for the prediction of HTC.

## General Features of Heat Transfer

Heat transfer at supercritical pressure is mainly characterized by the thermal–physical properties which vary strongly, especially near the pseudo-critical line. Figure 1 shows the specific heat in dependence on pressure and temperature.

It is seen that at each pressure there is a local maximum of the specific heat capacity. In supercritical pressure range, the line connecting the maximum values of the specific heat is called pseudo-critical line. The pseudo-critical temperature increases with increasing pressure. At a pressure of 25 MPa, the pseudo-critical temperature is 384 °C. Figures 2 and 3 show the density and the Prandtl number versus temperature at different pressure values.

Near the pseudo-critical line, the density decreases dramatically. Due to the sharp increase in specific heat, there exists a large peak of the Prandtl number at the pseudo-critical point. It is agreed that the behavior of both parameters has strong effect on the heat transfer to SCFs and is expressed by the dependence of HTC on heat flux. Figure 4 shows schematically the ratio of HTC to the reference value at zero heat flux.

The HTC ratio increases from 1.0 with increasing bulk temperature and reaches its maximum at a bulk temperature still far below the pseudo-critical point. After then, it decreases with the bulk temperature approaching the pseudo-critical point. It shows a minimum value at a bulk temperature around the pseudo-critical point. At bulk temperatures far beyond the pseudo-critical point, the ratio approaches to unity again. All the R&D efforts made in the past, present, and in the future are focused on the understanding and the description of this behavior. In the following status, challenges and future needs in the experimental studies, numerical analysis, and model development are summarized.

## Experimental Studies

As mentioned in Refs. [2–4], there exist a large number of experimental studies carried out in the last five decades. Many new test facilities were constructed for heat transfer studies in supercritical water, CO_{2}, and Freon R134a.

In the frame of a bilateral collaboration between SJTU and KIT, also partially under collaboration with other international partners, two experimental data banks have been established for vertical tubes. One is for supercritical water and the other for supercritical Freon R134a. The parameter ranges of the data banks are illustrated in Table 1. Totally 31,500 data points in supercritical water and 39,000 data points in supercritical R134a were included in the data banks.

Fluid | D (mm) | P (MPa) | G (kg/m^{2} s) | Q (kW/m^{2}) | Data points |
---|---|---|---|---|---|

Water | 1.3–46.0 | 22.3–41.0 | 100–3600 | 100–9400 | 31,500 |

R134a | 7.6–0.0 | 4.22–5.51 | 300–2500 | 10–200 | 39,000 |

Fluid | D (mm) | P (MPa) | G (kg/m^{2} s) | Q (kW/m^{2}) | Data points |
---|---|---|---|---|---|

Water | 1.3–46.0 | 22.3–41.0 | 100–3600 | 100–9400 | 31,500 |

R134a | 7.6–0.0 | 4.22–5.51 | 300–2500 | 10–200 | 39,000 |

The experimental data in supercritical water are from 24 different sources with a large range of tube diameters. The largest number of data points is from EU [6] and China [7,8]. The test data of R134a were obtained at SJTU [9] and KIT [10] in a narrow range of tube diameter. Assessment of the experimental data for SC water shows that more than 50% of the data points fall into the following parameter range, which is important for SCWR applications:

diameter (

*D*): 6–12 mmpressure (

*P*): 23–26 MPamass flux (

*G*): 0–1500 kg/m^{2}swall temperature: >

*T*_{PC}

The above test data are divided into three groups according to the range of mass flux, i.e., (0, 500), (500, 1000), and (1000, 1500), and the number of data points is presented over the ratio of heat flux to mass flux, as shown in Fig. 5. Most test data have the mass flux in the range of 500–1500 and are well uniformly distributed in the *G*/*q* range between 0.3 and 1. According to Styrikovich et al. [11], heat transfer deterioration would occur at the *q*/*G* ratio larger than 0.58. Therefore, it is expected that in more than 50% of the tests, heat transfer deterioration could occur.

Preliminary assessment of the quality of the data bank was made. First, attention was paid to the reproducibility of the test results. In general, it was found that large deviations between experimental data from different sources occur. In addition, significant deviations from the same source are also observed. Figure 6 compares the experimental data of HTC from SJTU in two different tubes, whose diameters differ only slightly from each other. The test data characterized with SJTU1 are from 7.6 mm tubes [8], whereas the test data noted with SJTU2 are obtained in 7.0 mm tubes. For both tests other parameters such as pressure, mass flux, heat flux, and bulk temperature are nearly identical. In all four cases large deviations are obvious. The ratio goes as low as 0.4, in case the fluid temperature increases. Similar large deviations were also observed by comparing test data from other different sources. Figure 7 shows two test examples in Freon R134a, one from SJTU and the other from KIT. Both tests were carried out with the same parameters and in the same diameter tubes. As seen, at some conditions there is an excellent agreement, whereas at some conditions, the deviation is significant. Furthermore, Fig. 8 presents the measured HTC in CO_{2} at the same test facility, in the same tube, and by the same researcher, but on different dates. Again, large deviation occurs.

The difference between the test data are usually explained with the experimental uncertainties and/or quality of heated surface. However, to the opinion of the present author, other two aspects could also be responsible for the large deviation between the test data.

### Multisolutions.

The large deviations between test data forces the authors to assume that this deviation obey a physical phenomenon, the so-called *multisolutions*. That means under the same experimental conditions, i.e., test channel geometry, pressure, mass flux, heat flux, and bulk temperature, the wall temperature has different values. As schematically indicated in Fig. 9, an experiment is carried out under controlled wall temperature, i.e., in a flow channel with fixed pressure, mass flux, and fluid bulk temperature, which is lower than the pseudo-critical value, the wall temperature can be controlled and increased continuously. The measured HTC versus the wall temperature shows possibly the behavior as indicated in Fig. 9. As well accepted, with the increase in the wall temperature close or high than the pseudo-critical value, the near wall layer has the temperature close to the pseudo-critical value and high specific heat, which may lead to high HTC. A further increase in the wall temperature, far away from the pseudo critical value, the high specific heat in the near wall layer disappears and HTC decreases again. This behavior is presented with curve “A” in Fig. 9.

the curve with constant heat flux is also indicated in Fig. 9, characterized with curve “B.” Depending on the thermal–hydraulic parameters (test conditions), both curves may cross each other at only one point or more than one point. If only one cross point is possible, this test case can only give one value of wall temperature, or single solution. In case more than one cross point is possible, several wall temperatures are then possible during this test, i.e., multisolutions. Therefore, from the theoretical points of view, multisolutions may occur, which could be the main reason for the large deviation between the test data.

In the future, efforts have to be made to further clarify the possibility of multisolution and the conditions, where multisolution may occur. The authors have found out that according to several HTC correlations, multisolutions are possible (see Fig. 11). Similar efforts using computational fluid dynamics (CFD) analysis or even with experimental methods would be extremely important. To the opinion of the present authors, this work has the highest priority and has not only impact on future experiments, but also on CFD validation and on the development of HTC prediction methods.

### Effect of Upstream Flow Conditions.

Another possibility leading to the large deviation between the test data could be the effect of the upstream flow condition on HTC. Figure 10 shows the CFD results indicating the local heat flux versus the bulk temperature for supercritical R134a in a 10-mm diameter tube at pressure 4.58 MPa and mass flux 600 kg/m^{2} s. The results are obtained with the boundary conditions of constant wall temperatures 110 °C, 130 °C, and 150 °C, respectively. Two different inlet temperatures of 50 °C and 60 °C are assumed. As recognized that in the entrance region, e.g., *L/D < *50, curves with the same inlet temperature, but different wall temperatures cross each other. This indicates that at the same bulk temperature and the same heat flux, different values or multisolutions of wall temperatures are possible. Moreover, in both the entrance region and the region far away from the inlet (*L/D* > 50), curves with the same wall temperature, but different inlet temperature can also cross each other. That means not only in the entrance region, but also in the region far away from the entrance, the effect of the inlet condition or the upstream flow condition on HTC is not negligible.

## Numerical Analysis

Numerical simulation of heat transfer to SCFs with CFD approach has also a long history. The early studies were reviewed by Cheng and Schulenberg [2]. As well agreed, the main difficulties in the CFD analysis are related to the turbulence modeling at supercritical conditions. Due to a large variation of thermal–physical properties, especially near the pseudo-critical line, there exists a strong buoyancy effect and acceleration effect near the heated wall. The applicability of conventional turbulence models to such conditions is not verified.

Although both Reynolds-averaged Navier–Stokes (RANS) and large eddy simulation (LES) approaches were used, most studies are focused on RANS methods. Due to a sharp variation of properties near the heated wall, a fine numerical mesh structure is necessary. Therefore, low-Reynolds *k–ε* models are preferred than a high-Reynolds *k–ε* model. Another feature in the CFD simulation is the strong buoyancy effect resulted by the large variation of density near the pseudo critical point. Thus, efforts were made to improve the modeling of the buoyancy effect on the turbulence quantities, such as turbulent heat flux. In conventional fluids, Reynolds analogy is widely applied and constant turbulent Prandtl number is assumed. This assumption might be valid in case the molecular Prandtl number is close to 1. However, at supercritical conditions, the molecular Prandtl number varies in a large range, as shown in Fig. 3. Obviously, the validity of the Reynolds analogy is questionable. Therefore, efforts were made in the past, to solve the turbulent heat flux with other methods such as algebraic flux model, instead of turbulent Prandtl number approach [12].

In addition to the RANS methods, CFD simulations were also carried out with LES method [13,14]. The main difficulties identified in the LES simulations consist in the requirement in large computing expenditure and the poor convergence of mesh sensitivity. Furthermore, LES approach did not automatically lead to an improved numerical accuracy.

In general, it can be concluded that the accuracy of the CFD simulation is determined by the reliability of the turbulence modeling. In spite of the large number of CFD simulations, none of the studies gives sufficient accuracy in the description of heat transfer at supercritical conditions [3]. No successful breakthrough with the conventional RANS or LES methods is expected in the near future. Future requirements and challenging tasks consist mainly of two aspects:

- (a)
Data Base for Turbulence Quantities in the Near Wall Region.

Up to now there is still no reliable experimental technique, which enables the experimental measurement of turbulent quantities in the region with strong density variation. More advanced measurement techniques are thus highly requested. In addition, reference data base from direct numerical simulation (DNS) is also extremely valuable. Several institutions are performing DNS and providing detailed turbulence quantities for the assessment or validation of turbulence models used in RANS or LES methods. However, due to the high requirements in the computing effort, DNS is nowadays only possible for flows with low Reynolds number. Future efforts should be focused on the extension of the parameter range of DNS.

- (b)
Extensive Reynolds-averaged Navier–Stokes (ERANS) Approach.

The conventional RANS method is based on the idea, that the thermal–hydraulic parameters such as velocity, temperature, and pressure, are divided into two parts, i.e., time-averaged value and time-dependent fluctuation. In the reality, not only the fluctuations of velocity and temperature, but also of the thermal–physical properties occur, which unfortunately are not considered in the conventional RANS method. Taking the energy conservation equation as example$\u2202(\rho \u22c5h)\u2202t+\u2202(\rho \u22c5ui\u22c5h)\u2202xi=\u2202\u2202xi[\lambda \u2202T\u2202xi]$(2)

The last four terms in the right side of the above equation appear due to the density fluctuation, which are not taken into consideration in the conventional RANS method. Due to the strong variation of density near the pseudo critical region, it is expected that these terms are not negligibly small and should be taken into consideration. Future studies should identify the importance of the effect of the properties fluctuation (also other properties such as specific heat) on the turbulence modeling.

## Prediction Methods

*stands for Nusselt number calculated with the conventional correlation without considering the effect of heat flux on the heat transfer coefficient. The focus is on the derivation of the correction factor*

_{0}*F*, taking the effect of heat flux into consideration. The approaches applied by the most authors are similar and pure empirical. In summary, the correction factor depends on a large number of dimensionless parameters, e.g.,

In the past, more and more parameters were selected to describe the correction factor. However, the complex structure of the correlations does not always lead to an obvious improvement in the prediction accuracy. Assessment of the existing correlations reveals that the deviation between the correlations is large, especially for the case with high heat fluxes [2].

### Requirements on New HT Correlations.

is applied to calculate HTC of supercritical water in 10 mm diameter circular tube with pressure 23.0 MPa, mass flux 1000 kg/m^{2} s, and bulk temperature 350 °C. The dependence of HTC on the wall temperature is illustrated in Fig. 11.

In this figure, HTC according to Eq. (1) at the heat flux 1000 kW/m^{2} is also presented. It is clearly seen that three values of wall temperature are identified. In this case, the phenomenon of multisolutions appears. Therefore, future efforts need to be made in the following aspects:

- (a)
Reliable experimental data base. Any effort to develop prediction models of HTC is based on the experimental data base. The applicability and the accuracy of the HTC depend directly on the data base used.

- (b)
Development of HTC correlations by considering the following aspects:

Selection of key dimensionless parameters based on phenomenological analysis and reduction in the number of the parameters as far as possible;

Elimination of the application of wall temperature, to avoid possible multisolutions. As indicated before, application of the wall temperature may lead to artificial multisolution. Also in case the physical phenomenon “multisolution” occurs, introduction of the wall temperature in the model does not necessarily lead to the correct solution, but may cause numerical instability. Prediction of the multisolution phenomenon requires a completely new approach of modeling.

In this model, only one dimensionless parameter *π _{A}* is applied to predict the correction factor. Moreover, wall temperature is not required. In the future, this correlation needs to be further optimized or improved, if more comprehensive and reliable experimental data base is available. This study is now on going at KIT.

### Look-Up List Approach.

In the past, efforts were also made to develop look-up table (LUT) for the prediction of HTC of SCFs [17]. LUT has been widely and successfully applied in the prediction of critical heat flux (CHF) [18]. However, the main challenges in the development of the LUT for the prediction of HTC in SCFs are

For a given flow channel, there are at least four parameters affecting the HTC, i.e., pressure, mass flux, fluid temperature, and heat flux, one parameter more than for the prediction of CHF. Therefore, significant large experimental data base is required for the development of the LUT.

Near the pseudo critical line, HTC varies strongly and is very sensitive to the variation of the thermodynamic parameters such as pressure and temperature. Therefore, in this region, very small intervals of the thermodynamic parameters are required to achieve sufficient accuracy of LUT.

Instead of LUT, an alternative approach is simply to compile the original experimental data as a “list” (look-up list (LUL)). HTC for a given combination of the thermal–hydraulic parameters would be then determined by an interpolation methodology. The LUL contains purely the original test data without loss of accuracy. At KIT, studies are ongoing to develop an effective and accurate interpolation methodology. Preliminary results indicate that LUL would be a promising approach for the prediction of HTC in SCFs.

## Summary

This paper reviews briefly the status, challenges, and future needs in the studies on heat transfer to SCFs. Three aspects, i.e., experimental studies, numerical analysis, and model development for the prediction of heat transfer coefficient are considered. The main outcomes from this review paper are:

The missing in the breakthrough success in spite of more than five decades intensive investigation is mainly due to the lack in the physical understanding of the phenomenon involved in the heat transfer to SCFs.

Experimental studies are mainly focused on the measurement HTC. The large deviation between the test data emphasizes the necessity to examine the reproducibility and to identify the uncertainty of the test data.

The phenomenon “multisolutions” is theoretically possible and also produced by some heat transfer correlations and CFD simulations. It is strongly recommended, to carry out further investigation to clarify the possibility of the multisolution phenomenon and the parameter ranges where the multisolution phenomenon may occur.

Due to the strong variation of properties, no “fully” development of the hydraulic and thermal boundary near the heated wall could be achieved. The dependence of local heat transfer on the upstream flow condition (or entrance flow condition) requires further investigation.

For flow conditions with strong variation of thermal–physical properties, it is recommended to extend the conventional RANS method by including the fluctuation of the thermal–physical properties. This approach is called ERANS in this study.

Related to the model development the author recommends to exclude the application of wall temperature, to select dimensionless parameters based on phenomenological analysis and to propose the model structure as simple as possible.

The LUL approach shows promising features. Further efforts are required to develop an effective and accurate interpolation methodology.

To solve the above key challenges more effectively, tight international collaboration is highly suggested.

## Acknowledgment

Responsibility for the content of this report lies with the authors. The contribution of the colleagues from KIT and SJTU, especially Florian Feuerstein, Denis Klingel, Zheng Yang, and Meng Zhao is appreciated. The test data in Fig. 8 were provided by Florian Feuerstein.

## Funding Data

German Federal Ministry of Economics and Energy (BMWi) (Contract No. 1501524).

Deutsche Forschungsgemeinschaft (DFG) (Project No. INST 121384/8-1 FUGG).

## Nomenclature

### Greek Symbols

### Subscripts

### Nondimensional Numbers

- Gr =
Grashof number, $\u2009((g\u22c5\beta \u22c5\rho 2\u22c5\Delta T\u22c5D3)/(\mu 2))$

- Nu =
Nusselt number, $\u2009((h\u22c5D)/\lambda )$

- Pr =
Prandtl number, $((CP\u22c5\mu )/\lambda )$

- Re =
Reynolds number, $((G\u22c5D)/\mu )$

*π*=_{A}acceleration number, $((\beta \u22c5q)/(CP\u22c5G))$

*π*=_{B}buoyancy number, $((\beta \u22c5D\u22c5q)/\lambda )$

*π*=_{C}ratio of specific heat, $((CP\xaf)/(CP,B))$

### Abbreviations

- CFD =
computational fluid dynamics

- CHF =
critical heat flux

- CRP =
Coordinated Research Program

- DNS =
direct numerical simulation

- HTC =
heat transfer coefficient

- ERANS =
extensive Reynolds-averaged Navier–Stokes

- KIT =
Karlsruhe Institute of Technology

- LES =
large eddy simulation

- LUL =
look-up list

- LUT =
look-up table

- RANS =
Reynolds-averaged Navier–Stokes

- SCF =
supercritical fluid

- SCWR =
supercritical water-cooled reactor

- SJTU =
Shanghai Jiao Tong University