Researches on specifics of thermophysical properties and heat transfer at supercritical pressures (SCPs) started as early as the 1930s with the study on free-convection heat transfer to fluids at a near-critical point. In the 1950s, the concept of using SC “steam” to increase thermal efficiency of coal-fired thermal power plants became an attractive option. Germany, USA, the former USSR, and some other countries extensively studied heat transfer to SC fluids (SCFs) during the 1950s till the 1980s. This research was primarily focused on bare circular tubes cooled with SC water (SCW). However, some studies were performed with modeling fluids such as SC carbon dioxide and refrigerants instead of SCW. Currently, the use of SC “steam” in coal-fired thermal power plants is the largest industrial application of fluids at SCPs. Near the end of the 1950s and at the beginning of the 1960s, several studies were conducted to investigate a possibility of using SCW as a coolant in nuclear reactors with the objective to increase thermal efficiency of nuclear power plants (NPPs) equipped with water-cooled reactors. However, these research activities were abandoned for some time and regained momentum in the 1990s. In support of the development of SCW-cooled nuclear-power reactor (SCWR) concepts, first experiments have been started in annular and various bundle flow geometries. At the same time, more numerical and CFD studies have been performed in support of our limited knowledge on specifics of heat transfer at SCPs in various flow geometries. As the first step in this process, heat transfer to SCW in vertical bare tubes can be investigated as a conservative approach (in general, heat transfer in fuel bundles will be enhanced with various types of appendages, that is, grids, end plates, spacers, bearing pads, fins, ribs, etc.). New experiments in the 1990–2000s were triggered by several reasons: (1) thermophysical properties of SCW and other SCFs have been updated from the 1950s–1970s, for example, a peak in thermal conductivity in the critical/pseudocritical points was “officially” introduced in 1990s; (2) experimental techniques have been improved; (3) in SCWRs, various bundle flow geometries will be used instead of bare-tube geometry; (4) in SC “steam” generators of thermal power plants, larger diameter tubes/pipes (20–40 mm) are used, however in SCWRs hydraulic-equivalent diameters of proposed bundles will be within 5–12 mm; (5) with Research and Development (R&D) of next-generation or Generation-IV nuclear-power-reactor concepts, new areas of application for SCFs have appeared—for example, SCP helium was proposed to be used as a reactor coolant, SCP Brayton and Rankine cycles with SC carbon dioxide as a working fluid are being developed, etc. A comparison of thermophysical properties of SCFs with those of subcritical-pressure fluids showed that SCFs as single-phase fluids have unique properties, which are close to “liquid-like” behavior below critical or pseudocritical points and are quite similar to the behavior of “gas-like” substances above these points. A comparison of selected SCW heat transfer correlations has shown that their results may differ from one to another by more than 200%. Based on these comparisons, it became evident that there is a need for reliable, accurate, and wide-range SCW heat transfer correlation(s) to be developed and verified. Therefore, the objective of this chapter is to summarize in concise form specifics of supercritical-fluids thermophysical properties and heat transfer in power-engineering applications.
- supercritical water
- carbon dioxide
- forced convective heat transfer
1.1 Historical note on using supercritical fluids (SCFs)
The use of supercritical fluids (SCFs) in various processes is not new and, actually, is not a human invention. Nature has been processing minerals in aqueous solutions at near or above the critical point of water for billions of years. In the late 1800s, scientists started to use this natural process in their labs for creating various crystals. During the last 50–60 years, this process, called hydrothermal processing (operating parameters: water pressure from 20 to 200 MPa and temperatures from 300 to 500°C), has been widely used in the industrial production of high-quality single crystals (mainly gem stones) such as sapphire, tourmaline, quartz, titanium oxide, zircon and others .
Also, compressed water, that is, water at a supercritical pressure (SCP), but at a temperature below T cr ≈ 374°C, exists in oceans at the depth of ∼2.2 km and deeper. If at this depth there is an active underwater volcano with the temperature of a magma above T cr of water, conditions for existence of supercritical water (SCW) can be reached.
The first works devoted to the problem of heat transfer at supercritical pressures (SCPs) started as early as the 1930s. Schmidt et al.  investigated free-convection heat transfer to fluids at a near-critical point with the application to a new effective cooling system for turbine blades in jet engines. They found that the free-convection heat transfer coefficient (HTC) at the near-critical state was quite high, and decided to use this advantage in single-phase thermosyphons with an intermediate working fluid at the near-critical point .
In the 1950s, the idea of using SC “steam” (actually, SCW) appeared to be rather attractive for the Rankine power cycle. The objective was to increase a thermal efficiency of coal-fired thermal power plants (ThPPs) (see Table 1 ). This change, that is, substantially higher operating pressures in the Rankine cycle from subcritical ones, and, correspondingly to that, higher inlet-turbine temperature up to 625°C, has allowed increasing of thermal efficiencies from 40–43% to 50–55% (gross) (in total by 7–15%). Currently, SCP coal-fired thermal power plants (world electricity generation 38%—the largest source for electricity generation; in India—77%; China—65%; Germany—37%; and in USA—30%) are the second ones by thermal efficiencies after gas-fired combined-cycle ThPPs (world electricity generation 23%—second largest source for electricity generation; in Russia—59%; UK—44%; Italy—42%; and in USA—34%) [4, 5]. More details on ThPPs can be found in Pioro and Kirillov  and many other sources.
|No.||Power plant||Gross thermal efficiency|
|1||Combined-cycle ThPP (combination of Brayton gas-turbine cycle (fuel—natural gas or LNG); combustion-products parameters at gas turbine: P in ≈ 2.3 MPa and T in ≈ 1650°C) and Rankine cycle steam-turbine parameters: P in ≈ 12.5 MPa and T in ≈ 585°C (T cr = 374°C)||Up to 62%|
|2||SCP coal-fired ThPP (Rankine cycle “steam”-turbine parameters (see Figure 1 ): P in ≈ 23.5–38 MPa (P cr = 22.064 MPa), T in ≈ 540‑625°C (T cr = 374°C) and steam reheat at: P ≈ 0.25·P in and T reheat ≈ 540‑625°C)||Up to 55%|
|3||Subcritical-pressure coal-fired ThPP (older plants; Rankine cycle steam-turbine parameters (see Figure 2 ): P in = 17 MPa (T sat = 352°C), T in = 540°C (T cr = 374°C), and steam reheat at: P ≈ 0.25·P in and T reheat = 540°C)||Up to 43%|
|4||Carbon dioxide-cooled reactor (advanced gas-cooled reactor (AGR)) NPP (Generation-III) (reactor coolant (carbon dioxide): P = 4 MPa and T = 290–650°C; Rankine cycle steam-turbine parameters (see Figure 2 ): P = 17 MPa (T sat = 352°C); T in = 540°C (T cr = 374°C), and steam reheat at: P ≈ 0.25·P in and T in = 540°C)||Up to 42%|
|5||Sodium-cooled fast reactor (SFR) (BN-600; BN-800) NPP (reactor coolant (sodium): P ≈ 0.1 MPa (above sodium level) and T max = 550°C; Rankine cycle steam-turbine parameters (see Figure 3 ): P = 14 MPa (T sat = 337°C); T in = 505°C (T cr = 374°C) and steam reheat at: P ≈ 0.25·P in and T in = 505°C)||Up to 40%|
|6||Pressurized water reactor (PWR) NPP (Generation-III+, new reactors) (reactor coolant (light water): P = 15.5 MPa (T sat = 345°C) and T = 280‑322°C; Rankine cycle steam-turbine parameters (see Figure 4 ): P = 7.8 MPa and T in = T sat = 293°C and steam reheat at P in ≈ 1 MPa and T in ≈ 273°C)||Up to 36‑38%|
|7||Pressurized water reactor (PWR) NPP (Generation-III, current fleet) (reactor coolant: P = 15.5 MPa (T sat = 345°C) and T = 292–329°C; Rankine cycle steam-turbine parameters (see Figure 4 ): P = 6.9 MPa and T in = T sat = 285°C and steam reheat at P in ≈ 1 MPa and T in ≈ 265°C)||Up to 34‑36%|
|8||Boiling-water-reactor (BWR) or advanced BWR NPP (Generation-III and III+, current fleet) (P in = 7.2 MPa and T in = T sat=288°C (direct cycle) and steam reheat at P in ≈ 1 MPa and T in ≈ 268°C (see Figure 4 ))||Up to 34%|
|9||Pressurized heavy water reactor (PHWR) NPP (Generation-III, current fleet) (reactor coolant: P out = 10 MPa (T sat = 311°C) and T = 260–310°C; Rankine cycle steam-turbine parameters: P = 4.6 MPa and T in = T sat = 259°C and steam reheat at l P in ≈ 1 MPa and T in ≈ 240°C)||Up to 32%|
Also, at SCPs there is no liquid-vapor-phase transition; therefore, there is no such phenomenon as critical heat flux (CHF) or dryout. It is only within a certain range of parameters a deteriorated heat transfer (DHT) regime may occur. Work in this area was mainly performed in Germany, USA, former USSR, and some other countries in the 1950–1980s .
1.2 Future applications of SCFs in next-generation nuclear-power reactors and NPPs
At the end of the 1950s and the beginning of the 1960s, early studies were conducted to investigate a possibility of using SCW in nuclear reactors. Several concepts of nuclear reactors using SCW were developed in Great Britain, France, USA, and former USSR. However, this idea was abandoned for almost 30 years with the emergence of light water reactors (LWRs), but regained interest in the 1990s following LWRs maturation ([6, 9, 10, 11, 12, 13]).
This interest was triggered by economical considerations, because nuclear power plants (NPPs) with LWRs (and, especially, with PHWRs) have relatively low thermal efficiencies within the range of 30–36% for Generation-III reactors and up to 37% (38%) for advanced reactors of Generation-III+ (see Table 1 ) compared to those of modern ThPPs (up to 62% for combined-cycle plants and up to 55% for SCP Rankine cycle plants (see Table 1 )) . Therefore, NPPs with various designs of water-cooled reactors at subcritical pressures cannot compete with modern advanced ThPPs. Also, it should be noted that currently, water-cooled reactors are the vast majority of nuclear-power reactors in the world [14, 15]: (1) PWRs—298 units or 68% from the total number of 441 units; (2) BWRs—66 units or 15%; (3) PHWRs—48 units or 11%; (4) light water, graphite-moderated reactors (LGRs)—13 units of 3%.
Therefore, six concepts of nuclear-power reactors/NPPs of next generation, Generation-IV, were proposed (see Table 2 ), which will have thermal efficiencies comparable with those of modern thermal power plants. Supercritical water-cooled reactor (SCWR) is one of these six concepts under development in a number of countries [6, 20]. Analysis of Generation-IV concepts listed in Table 2 shows that SCFs, such as helium and water, will be used as reactor coolants, and SCFs such as helium, nitrogen (or mixture of nitrogen (80%) and helium (20%)), carbon dioxide, and water will be used as working fluids (WFs) in power Brayton and Rankine cycles (critical parameters of selected SCFs are listed in Table 3 ). However, it should be mentioned that helium as the reactor coolant and as the working fluid in Brayton power cycle will be at supercritical conditions, which are far above by pressure and temperature critical parameters, that is, helium will behave as compressed gas.
|No.||Nuclear power plant||Gross eff., %|
|1||Very high-temperature reactor (VHTR) NPP (reactor coolant—helium (SCF): P = 7 MPa and T in/T out = 640/1000°C; primary power cycle—direct SCP Brayton helium-gas-turbine cycle; possible back-up—indirect Brayton or combined cycles (see Figures 5 and 6 ))||≥55|
|2||Gas-cooled fast reactor (GFR) or high-temperature reactor (HTR) NPP (reactor coolant—helium (SCF): P = 9 MPa and T in/T out = 490/850°C; primary power cycle—direct SCP Brayton helium-gas-turbine cycle (see Figure 7 ); possible back-up—indirect SCP Brayton or combined cycles (see Figures 8 and 9 ))||≥50|
|3||Supercritical water-cooled reactor (SCWR) NPP (one of Canadian concepts; reactor coolant—SC light water: P = 25 MPa and T in/T out = 350/625°C (T cr = 374°C); direct cycle; SCP Rankine cycle with high-temperature secondary-steam superheat: T out = 625°C; possible back-up–indirect SCP Rankine “steam”-turbine cycle with high-temperature secondary-steam superheat) (for details of SCP Rankine cycle, see Table 1 Item No. 2 and Figure 1 )||45–50|
|4||Molten salt reactor (MSR) NPP (reactor coolant—sodium-fluoride salt with dissolved uranium fuel: T in/T out = 700/800°C; primary power cycle—indirect SCP carbon dioxide Brayton gas-turbine cycle; possible back-up—indirect Rankine steam-turbine cycle)||∼50|
|5||Lead-cooled fast reactor (LFR) NPP (Russian design BREST-OD-300 * : reactor coolant—liquid lead: P ≈ 0.1 MPa and T in/T out = 420/540°C; primary power cycle—indirect subcritical-pressure Rankine steam cycle: Pin ≈ 17 MPa (P cr = 22.064 MPa) and T in/T out = 340/505°C (T cr = 374°C); high-temperature secondary-steam superheat (in one of the previous designs of BREST-300 NPP primary power cycle was indirect SCP Rankine “steam” cycle: Pin ≈ 24.5 MPa (P cr = 22.064 MPa) and T in/T out = 340/520°C (T cr = 374°C); also, note that power-conversion cycle in a different LFR designs from other countries is based on SCP carbon dioxide Brayton gas-turbine cycle||∼41–43|
|6||Sodium-cooled fast reactor (SFR) NPP (Russian design BN-600: reactor coolant—liquid sodium (primary circuit): P ≈ 0.1 MPa and T in/T out = 380/550°C; liquid sodium (secondary circuit): T in/T out = 320/520°C; primary power cycle—indirect Rankine steam-turbine cycle: Pin ≈ 14.2 MPa (T sat ≈ 337°C) and T in max = 505°C (T cr = 374°C); secondary-steam superheat: P ≈ 2.45 MPa and T in/T out = 246/505°C; possible back-up in some other countries—indirect SCP carbon dioxide Brayton gas-turbine cycle)||∼40|
|No.||Fluid||Molar mass||T cr||P cr||ρ cr||Application in power engineering at SCPs|
|1||Carbon dioxide,1 CO2||44.01||30.978||7.3773||467.6||WF in Brayton and Rankine power cycles (see Figures 5 and 6 )|
|3||Helium,2 He||4.0026||Reactor coolant in VHTR & GFR (see Figure 7 ); WF in Brayton power cycle (see Figure 7 )|
|5||Nitrogen, N2||28.013||‑146.96||3.3958||313.3||WF in Brayton cycle (also, mixture of N2 (80%) & He (20%) is proposed (see Figures 8 and 9 ))|
|6||R-12, CCl2F2||120.91||111.97||4.1361||565.0||Modeling fluid in thermalhydraulic tests|
|7||R-134a, CF3CH2F||102.03||101.06||4.0593||511.9||Modeling fluid in thermalhydraulic tests|
|8||Water3, H2O||18.015||373.95||22.064||322.0||WF in Rankine cycle of coal-fired ThPP; reactor coolant in SCWR; WF in Rankine power cycle (see Figure 1 )|
Nowadays, the most widely used SCFs are water, carbon dioxide, and refrigerants . Quite often, carbon dioxide and refrigerants are considered as modeling fluids and used instead of SCW due to significantly lower critical pressures and temperatures, which decreases the complexity and costs of thermalhydraulic experiments. However, they can be/will be used as working fluids in new SCP power cycles: Brayton and Rankine ones  (for details, see Table 3 ).
Also, other applications of SCFs will be discussed in the following chapters and are listed in Pioro and Duffey .
2. Specifics of thermophysical properties of SCFs
Prior to a general discussion on specifics of forced-convective heat transfer at critical and supercritical pressures, it is important to define special terms and expressions used at these conditions [6, 9]. For a better understanding of these terms and expressions their definitions are listed in Glossary (see below). Specifics of thermophysical properties at SCPs are described in Pioro et al. ; Handbook ; Mann and Pioro ; Gupta et al. ; Pioro and Mokry ; and Pioro and Duffey  (for more details, see Table 4 ).
|No.||Literature source||Fluid||P, MPa||T, °C||Properties|
|1||Piora et al. ||Properties of selected metals, alloys, and diamond|
Properties of selected insulating materials
Radiative properties of selected materials
Properties of selected nuclear fuels
Properties of selected gases at atmospheric pressure
Properties of selected cryogenic gases
Properties of selected fluids on saturation line
Properties of selected supercritical fluids
Properties of selected liquid alkali metals
Thermophysical properties of nuclear-reactor coolants
|2||Handbook ||H2O, CO2, He||‑||‑||T-s diagrams|
|H2O (BWR, PHWR, PWR)||7, 11, 15||50‑375||ρ, k, μ, ν, c p, H, Pr, β|
|H2O (SCW)||P cr, 25, 30, 35, 40||350‑600||ρ, k, μ, ν, c p, H, Pr, β|
|CO2 (SC CO2)||P cr, 8.4, 10.0, 11.7||0‑165||ρ, k, μ, ν, c p, H, Pr, β|
|He||P cr and other pressures||Range of T||k, c p, β|
|Air, Ar, CO2, He, H2, Kr (gases)||0.1||0‑1000||ρ, k, μ, c p, Pr, β|
|CO2 (AGR)||4||250‑1000||ρ, k, μ, c p, H, Pr, β|
|He (VHTR, GFR)||7, 9|
|Na, Pb, Pb-Bi (SFR, LFR)||0.1|
|3||Mann and Piora ||SC R-134a||P cr, 5, 10, 13, 15||‑100‑175||k, c p, β|
|4||Gupta et al. ||SCW|
SC R-134a (three fluids on same graph)
|ρ, k, μ, c p, H, Pr|
|5||Piora and Mokry ||H2O||‑||‑||T-s diagram|
|H2O (SCW)||P cr, 25, 30, 35||350‑600||ρ, k, μ, ν, c p, H, Pr, β|
|R-12 (SC R-12)||P cr, 4.65||0‑350||ρ, k, μ, ν, c p, H, Pr, β|
|6||Piora and Duffey ||R-134a (SC R-134a)||P cr, 4.6||70‑150||ρ, k, μ, ν, c p, H, Pr, β|
Also, profiles of the basic thermophysical properties (density, thermal conductivity, dynamic viscosity, specific heat and specific enthalpy) and Prandtl number for four SCFs: water, ethanol, methanol, and carbon dioxide; at critical and one supercritical pressure, which is 25 MPa for water and the corresponding to that equivalent pressures for all other SCFs vs. reduced temperature (temperature) are shown in Figures 15 – 20 .
3. Specifics of forced-convection heat transfer at supercritical pressures
3.1 Vertical bare tubes
Water is the most widely used coolant or working fluid at SCPs. The largest application of SCW is in SC “steam” generators and turbines, which are widely used in the thermal power industry worldwide. Currently, upper limits of pressures and temperatures used in the thermal-power industry are about 30–38 MPa and 600–625°C, respectively (see Table 1 ). A new direction in SCW application in the power industry has been the development of SCWR concepts (see Table 2 ), as part of the Generation-IV International Forum (GIF)  initiative (for details, see [6, 9, 10, 11, 12, 13, 42, 43, 44]; and Proceedings of the International Symposiums on SCWRs (ISSCWR) (selected augmented and revised papers from ISSCWRs have been published in the ASME Journal of Nuclear Engineering and Radiation Science in 2020, Vol. 6 No. 3; in 2018, Vol. 4, No. 1, and 2016, Vol. 2, No. 1).
Experiments at SCPs are very expensive and require sophisticated equipment and measuring techniques. Therefore, some of these studies (e.g., heat transfer in fuel-bundle simulators) are proprietary and, hence, are not published in open literature.
The majority of studies deal with heat transfer and hydraulic resistance of working fluids, mainly water, carbon dioxide, refrigerants, and helium, in circular bare tubes [9, 28, 45, 46, 47, 48]. A limited number of studies were devoted to heat transfer and pressure drop in annuli and bundles [9, 10, 30, 31, 32, 33, 34, 35, 38, 49].
New experiments in the 1990s–2000s were triggered by several reasons: (1) thermophysical properties of SCW have been updated from the 1950s–1970s, for example, a peak in thermal conductivity in the critical/pseudocritical points was “officially” introduced in the 1990s; (2) experimental techniques have been improved; (3) in SCWRs various bundle flow geometries will be used instead of bare-tube geometry; and (4) in SC steam generators of thermal power plants larger diameter tubes/pipes (20–40 mm) are used, however, in SCWRs hydraulic-equivalent diameters of proposed bundles will be within 5–12 mm.
Accounting that SCW, SC carbon dioxide and SC R-12 are the most widely used fluids, specifics of heat transfer, including generalized correlations, will be discussed in this paper. Specifics of heat transfer and pressure drop at other conditions and/or for other fluids are discussed in the book by Pioro and Duffey .
All primary sources (i.e., all sources found by the authors from a total of 650 references dated mainly from 1950 till beginning of 2006) of heat transfer experimental data for water and carbon dioxide flowing inside circular tubes at supercritical pressures are listed in the book by Pioro and Duffey .
In general, three major heat transfer regimes (for their definitions, see Section 2, Glossary) can be noticed at critical and supercritical pressures (for details, see Figures 12 , 13a , 14 , 21 , 24 , 25 , 27 , 30 , 31 – 35 ):
Normal heat transfer;
Improved heat transfer; and
Deteriorated heat transfer.
Also, two special phenomena (for their definitions, see Section 2, Glossary) may appear along a heated surface: (1) pseudo-boiling; and (2) pseudo-film boiling. These heat transfer regimes and special phenomena appear to be due to significant variations of thermophysical properties near the critical and pseudocritical points and due to operating conditions.
Wall temperature and bulk-fluid temperature is above a pseudocritical temperature within a part of or the entire heated channel;
Upward and downward flows;
Horizontal flows; and
Effect of gravitational forces at lower mass fluxes; etc.
All these conditions can affect SC heat transfer.
Figure 13b shows bulk-fluid-temperature and thermophysical-properties (thermal conductivity, dynamic viscosity, specific heat, and Prandtl number) profiles along the heated length of a vertical bare circular tube (operating conditions in this figure correspond to those in Figure 13a ).
Some researchers have suggested that variations in thermophysical properties near critical and pseudocritical points result in the maximum value of HTC. Thus, Yamagata et al.  found that for SCW flowing in vertical and horizontal tubes, the HTC increases significantly within the pseudocritical region ( Figure 21 ). The magnitude of the peak in HTC decreases with increasing heat flux and pressure. The maximum HTC values correspond to a bulk-fluid enthalpy, which is slightly less than the pseudocritical bulk-fluid enthalpy.
3.2 Vertical annular channel, and three- and seven-rod bundles cooled with SCW
In future SCWRs the main flow geometry will be bundles of various designs [6, 10]. Therefore, a limited number of experiments have been performed in simplified bundle simulators cooled with SCW and heated with an electrical current [30, 31, 32, 33, 34, 35]. An annulus or a one-rod bundle is the simplest bundle geometry (see Figures 22a and 23 ), and Figure 24 shows profiles of bulk-fluid and wall temperatures, and HTC along heated length of vertical annular channel (one-rod bundle). Figures 22b and 23 show three-rod-bundle flow geometry, and Figure 25 shows profiles of bulk-fluid and wall temperatures, and HTC along heated length of vertical three-rod bundle. Figure 26 shows seven-rod-bundle flow geometry, and Figure 27 shows profiles of bulk-fluid and wall temperatures, and HTC along heated length of the vertical seven-rod bundle.
Analysis of data in Figures 25b and 27b shows that all three HT regimes, which were noticed in bare circular tubes, are also possible in annuli and bundle flow geometries ( Table 6 ). Figures 24 and 25 show a comparison between the HTC experimental data obtained in annulus and three-rod bundle with those calculated through the Dittus-Boelter correlation (Eq. (1)). The comparison showed that, in general, there is no significant difference between calculated HTC values and experimental ones. This finding means that in spite of the presence of rod(s) with four helical ribs in SCW flow, which can be considered as an HT enhancement surface(s), there is no significant increase in HTC. However, when q dht values reached in SCW-cooled annulus and 3- and seven-rod bundles were compared to those obtained in bare tubes, it was found that q dht in bare tubes were 1.6–1.8 times lower (see Table 7 ).
|No.||Test section||Operating conditions||q dht, MW/m2||Increase in q dht value compared to that of bare tube|
|1||Bare tube||P = 24.1 MPa and G = 2000 kg/m2s||1.43||1.8|
|2||Annulus||P = 22.6 MPa and G = 2000 kg/m2s||2.55|
|3||Bare tube||P = 24.1 MPa and G = 2700 kg/m2s||1.95||1.6|
|4||Three-rod bundle||P = 22.6 MPa and G = 2700 kg/m2s||3.20|
|5||Bare tube||P = 24.5 MPa and G = 800 kg/m2s||0.54||1.8|
|6||Seven-rod bundle||P = 24.5 MPa and G = 800 kg/m2s||0.96|
3.3 Vertical seven-rod bundle cooled with SC R-12
Figures 28 and 29 show a seven-rod bundle test section, which can be considered as a bare bundle, and Figures 30 and 31 show profiles of bulk-fluid and wall temperatures, and HTC vs. heated length of the central rod at three circumferential locations. Analysis of Figures 30 and 31 shows that we also have here all three HT regimes plus sometimes quite significant differences in local HTC values and wall temperatures around the central rod circumference.
4. Practical prediction methods for forced-convection heat transfer at supercritical pressures
4.1 Supercritical water (SCW)
Unfortunately, satisfactory analytical methods for practical prediction of forced-convection heat transfer at SCPs have not yet been developed due to the difficulty in dealing with steep property variations, especially, in turbulent flows and at high heat fluxes [10, 67]. Therefore, generalized correlations based on experimental data are used for HTC calculations at SCPs.
There are numerous correlations for convective heat transfer in circular tubes at SCPs (for details, see in Pioro and Duffey ). However, an analysis of these correlations has shown that they are more or less accurate only within the particular dataset, which was used to derive the correlation, but show a significant deviation in predicting other experimental data. Therefore, only selected correlations are considered below.
In general, many of these correlations are based on the conventional Dittus-Boelter-type correlation (see Eq. (1)) in which the “regular” specific heat (i.e., based on bulk-fluid temperature) is replaced with the cross-sectional averaged specific heat within the range of (Tw − Tb ); , J/kg K. Also, additional terms, such as: ; etc., can be added into correlations to account for significant variations in thermophysical properties within a cross-section due to a nonuniform temperature profile, that is, due to heat flux.
It should be noted that usually generalized correlations, which contain fluid properties at a wall temperature, require iterations to be solved, because there are two unknowns: (1) HTC and (2) the corresponding wall temperature. Therefore, the initial wall temperature value at which fluid properties will be estimated should be “guessed” to start iterations.
The most widely used heat transfer correlation at subcritical pressures for forced convection is the Dittus-Boelter  correlation. In 1942, McAdams  proposed to use the Dittus-Boelter correlation in the following form, for forced-convective heat transfer in turbulent flows:
However, it was noted that Eq. (1) might produce unrealistic results at SCPs within some flow conditions (see Figure 12 ), especially, near the critical and pseudocritical points, because it is very sensitive to properties variations.
In general, experimental HTC values show just a moderate increase within the pseudocritical region. This increase depends on mass flux and heat flux: higher heat flux—less increase. Thus, the bulk-fluid temperature might not be the best characteristic temperature at which all thermophysical properties should be evaluated. Therefore, the cross-sectional averaged Prandtl number, which accounts for thermophysical-properties variations within a cross-section due to heat flux, was proposed to be used in many SC HT correlations instead of the regular Prandtl number. Nevertheless, this classical correlation (Eq. (1)) was used extensively as a basis for various SC HT correlations .
The majority of empirical correlations were proposed in the 1960s–1970s , when experimental techniques were not at the same level (i.e., advanced level) as they are today. Also, thermophysical properties of SCW have been updated since that time (for example, a peak in thermal conductivity in critical and pseudocritical points within a range of pressures from 22.1 to 25 MPa for water was not officially recognized until the 1990s).
Therefore, new correlations within the SCWRs operating range, were developed and evaluated by I. Pioro with his students (mainly, by S. Mokry et al. (bulk-fluid-temperature approach) and S. Gupta et al. (wall temperature approach)) using the best SCW dataset by P.L. Kirillov and his co-workers and adding smaller datasets by other researchers:
The Pioro-Mokry correlation (Eq. (2)) was verified within the following operating conditions (only for NHT and IHT regimes (see Figures 32 and 33 ), but not for the DHT regime): SCW, upward flow, vertical bare circular tubes with inside diameters of 3–38 mm, pressure—22.8–29.4 MPa, mass flux—200–3000 kg/m2s, and heat flux—70–1250 kW/m2. All thermophysical properties of SCW were calculated according to NIST REFPROP software . This correlation has accuracy of ±25% for HTCs and ±15 for wall temperatures. Eventually, this nondimensional correlation can be also used for other SCFs. However, its accuracy can be less or even significantly less in these cases.
Pioro-Gupta correlation (wall temperature approach) :
Eq. (3) has an uncertainty of about ±25% for HTC values and about ±15% for calculated wall temperatures within the same ranges as those for Eq. (2). Also, it was decided to add an entrance effect to make this correlation even more accurate. This entrance effect was modeled by an exponentially-decreasing term as shown below:
where, is calculated using Eq. (3). It should be noted that this HT correlation is also intended only for NHT and IHT regimes.
The following empirical correlation was proposed by I. Pioro and S. Mokry for calculating the minimum heat flux at which the DHT regime appears in vertical bare circular tubes:
Pioro-Mokry correlation for q dht :
Correlation (Eq. (5)) is valid within the following range of experimental parameters: SCW, upward flow, vertical bare tube with inside diameter 10 mm, pressure 24 MPa, mass flux 200–1500 kg/m2s, and bulk-fluid inlet temperature 320–350°C. Uncertainty is about ±15% for the DHT heat flux.
A recent study was conducted by Zahlan et al.  in order to develop a heat transfer look-up table for the critical/SCPs. An extensive literature review was conducted, which included 28 datasets and 6663 trans-critical heat transfer data. Tables 8 and 9 list results from this study in the form of the overall-weighted average and root-mean-square (RMS) errors: (a) within three SC sub-regions; and (b) for subcritical liquid and superheated steam. Many of the correlations listed in these tables can be found in Zahlan et al.  and Pioro and Duffey . In their conclusions, Zahlan et al.  determined that within the SC region, the latest correlation by Pioro-Mokry  (Eq. (2)) showed the best prediction for the data within all three sub-regions investigated (based on RMS error) (see Table 8 ). Also, the Pioro-Mokry correlation showed quite good predictions for subcritical-pressure water and superheated steam compared to other several correlations (see Table 9 ). Also, it was concluded that Pioro-Gupta correlation (Eq. (3)) was quite close by RMS errors to the Pioro-Mokry correlation.
|Liquid-like||Gas-like||Critical or pseudocritical|
|2||Sieder and Tate ||46||65||97||132||‑||‑|
|3||Bishop et al. ||5||28||5||20||23||31|
|4||Swenson et al. ||1||31||‑16||21||4||23|
|5||Krasnoshchekov et al. ||18||40||‑30||32||24||65|
|6||Hadaller and Banerjee ||34||53||14||24||‑||‑|
|8||Watts and Chou , NHT||6||30||‑6||21||11||28|
|9||Watts and Chou , DHT||2||26||9||24||17||30|
|11||Koshizuka and Oka ||26||47||27||54||39||83|
|13||Mokry et al. [23, 63]||‑5||26||‑9||18||‑1||17|
|14||Kuang et al. ||‑6||27||10||24||‑3||26|
|15||Cheng et al. ||4||30||2||28||21||85|
|16||Gupta et al. ||‑26||33||‑12||20||‑1||18|
Chen et al.  has also concluded that the Pioro-Mokry correlation for SCW HT “performs best” compared to other 14 correlations.
4.2 Supercritical carbon dioxide
The following correlation was proposed by S. Gupta (an MASc student of I. Pioro)  for SC carbon dioxide flowing inside vertical bare tubes:
Uncertainties associated with this correlation are ±30% for HTC values and ± 20% for calculated wall temperatures (see Figures 34 and 35 ). Ranges of parameters for the dataset used to develop Eq. (6) are listed in Table 10 .
|P, MPa||Tin , °C||Tout , °C||Tw , °C||q, kW/m2||G, kg/m2s|
Table 11 list mean and root-mean square (RMS) errors in HTC and Tw for proposed correlations using equations shown below:
|Errors in HTC (for the reference dataset), %|
|Proposed new correlation (T b approach)||0.9%||22.4%|
|Proposed new correlation (T film approach)||0.2%||21.7%|
|Proposed new correlation (T w approach—Eq. (6))||0.8%||20.3%|
|Swenson et al.  correlation||89%||132%|
|Mokry et al.  correlation for SCW||68%||123%|
|Gupta et al.  correlation for SCW||78%||130%|
It was also decided to develop the q dht correlation for SC carbon dioxide based on the dataset obtained by I. Pioro in vertical bare tube with upward flow, which ranges are listed in Table 10 . Therefore, based on the identified 41 cases of DHT within the SC carbon dioxide dataset, the following correlation for the minimal heat flux at which deterioration occurs was proposed:
Supercritical fluids are used quite intensively in various industries. Therefore, understanding specifics of thermophysical properties, heat transfer, and pressure drop in various flow geometries at supercritical pressures is an important task.
In general, three major heat transfer regimes were noticed at critical and supercritical pressures in various flow geometries (vertical bare tubes, annulus, three- and seven-rod bundles) and several SCFs (SCW, SC carbon dioxide, and SC R-12): (1) normal heat transfer; (2) improved heat transfer; and (3) deteriorated heat transfer. Also, two special phenomena may appear along a heated channel: (1) pseudo-boiling; and (2) pseudo-film boiling. These heat transfer regimes and special phenomena appear to be due to significant variations of thermophysical properties near the critical and pseudocritical points and due to operating conditions.
Comparison of heat transfer-coefficient values obtained in bare circular tubes with those obtained in annulus (one-rod bundle)/three-rod bundle (rod(s) equipped with four helical ribs) shows that there are almost no differences between these values. However, the minimal heat flux at which deterioration occurs (q dht) in annulus, and three- and seven-rod bundles are in 1.6–1.8 times higher compared to that recorded in bare tubes.
The current analysis of a number of well-known heat transfer correlations for supercritical fluids showed that the Dittus-Boelter correlation  significantly overestimates experimental HTC values within the pseudocritical range. The Bishop et al.  and Jackson  correlations tend also to deviate substantially from the experimental data within the pseudocritical range. The Swenson et al.  correlation provided a better fit for the experimental data than the previous three correlations within some flow conditions, but does not follow up closely the experimental data within others.
Therefore, new correlations were developed by Pioro with his students Mokry et al.  (bulk-fluid-temperature approach) and Gupta et al.  (wall temperature approach), which showed the best fit for the experimental data within a wide range of operating conditions. These correlations have uncertainties of about ±25% for HTC values and about ±15% for calculated wall temperature. Also, based on an independent study performed by Zahlan et al. [50, 51], Pioro-Mokry correlation (given as Eq. (2)) is the best for superheated steam compared to other well-known correlations. Also, this correlation showed quite good predictions for subcritical-pressure fluids.
The author would like to express his appreciation to his former and current students, S. Clark, A. Dragunov, S. Gupta, M. Mahdi, D. Mann, S. Mokry, R. Popov, G. Richards, Eu. Saltanov, H. Sidawi, E. Tamimi, and A. Zvorykin, for their assistance in the preparation of figures and developing of correlations.
area, m2 specific heat at constant pressure, J/kg K averaged specific heat within the range of (Tw – Tb ); , J/kg K inside diameter, m mass flux, kg/m2s; specific enthalpy, J/kg heat transfer coefficient, W/m2K thermal conductivity, W/m K heated length, m mass-flow rate, kg/s; pressure, Pa heat transfer rate, W heat flux, W/m2; specific entropy, J/kg K temperature, °C film temperature, °C; volume-flow rate, m3/s specific volume, m3/kg axial coordinate, m thermal diffusivity, m2/s; difference dynamic viscosity, Pa·s density, kg/m3 kinematic viscosity, m2/s; Nusselt number; Prandtl number; cross-sectional average Prandtl number within the range of (Tw – Tb ); Reynolds number; average bulk calculated correlation critical deteriorated heat transfer flow heated hydraulic-equivalent inlet maximum minimum outlet pseudocritical saturation thermal wall
specific heat at constant pressure, J/kg K
averaged specific heat within the range of (Tw – Tb ); , J/kg K
inside diameter, m
mass flux, kg/m2s;
specific enthalpy, J/kg
heat transfer coefficient, W/m2K
thermal conductivity, W/m K
heated length, m
mass-flow rate, kg/s;
heat transfer rate, W
heat flux, W/m2;
specific entropy, J/kg K
film temperature, °C;
volume-flow rate, m3/s
specific volume, m3/kg
axial coordinate, m
thermal diffusivity, m2/s;
dynamic viscosity, Pa·s
kinematic viscosity, m2/s;
cross-sectional average Prandtl number within the range of (Tw – Tb );
deteriorated heat transfer
Compressed fluid is the fluid at a pressure above the critical pressure, but at a temperature below the critical temperature (see Figure 10 ).
Critical point (also called a critical state) is the point in which the distinction between the liquid and gas (or vapor) phases disappears (see Figure 10 ), that is, both phases have the same temperature, pressure, and specific volume or density. The critical point is characterized with the phase-state parameters: T cr, P cr and v cr (or ρ cr), which have unique values for each pure substance.
Deteriorated heat transfer (DHT) is characterized with lower values of the HTC compared to those for normal heat transfer (NHT); and hence, has higher values of wall temperature within some part of a heated channel (see Figures 12 , 13a , 24b , 25 , 27 , 31 , and 35 ) or within the entire heated length (see Figure 14b ).
Improved heat transfer (IHT) is characterized with higher values of the HTC compared to those for NHT; and hence, lower values of wall temperature within some part of a heated channel (see Figures 12 , 21 , 25 , 27b , 33 , and 34 ) or within the entire heated length. In our opinion, the IHT regime or mode includes peaks or “humps” in the HTC profile near the critical or pseudocritical points.
Normal heat transfer (NHT) can be characterized in general with HTCs similar to those of subcritical convective heat transfer far from the critical or pseudocritical regions, when they are calculated according to the conventional single-phase Dittus-Boelter-type correlations: Nu = 0.0243 Re 0.8 Pr 0.4 (see Figures 12 , 13a , 14 , 21 , 24 , 25 , 27 , and 30 – 34 ).
Overheated vapor is the vapor at pressures below the critical pressure, and at temperatures above the saturation temperature, but below the critical temperature (see Figure 10 ).
Pseudocritical line is the line, which consists of pseudocritical points (see Figure 10 ).
Pseudo-boiling is a physical phenomenon similar to subcritical-pressure nucleate boiling, which may appear at SCPs. Due to heating of an SCF with a bulk-fluid temperature below the pseudocritical temperature (high-density fluid, i.e., “liquid-like”) (see Figures 10 , 11 , 13b and 15 ), some layers near the heated surface may attain temperatures above the pseudocritical temperature (low-density fluid, i.e., “gas-like”). This low-density “gas-like” fluid leaves the heated surface in a form of variable density volumes (bubbles). During the pseudo-boiling, the HTC usually increases (IHT regime).
Pseudocritical point (characterized with P and Tpc ) is the point at a pressure above the critical pressure and at a temperature (Tpc > Tcr ) corresponding to the maximum value of specific heat at this particular pressure (see Figures 10 , 11 , and 13b ).
Pseudo-film boiling is a physical phenomenon similar to subcritical-pressure film boiling, which may appear at SCPs. At pseudo-film boiling, a low-density fluid (a fluid at temperatures above the pseudocritical temperature, i.e., “gas-like”) prevents a high-density fluid (a fluid at temperatures below the pseudocritical temperature, i.e., “liquid-like”) from contacting (“rewetting”) a heated surface. Pseudo-film boiling leads to the DHT regime.
Supercritical fluid is the fluid at pressures and temperatures that are higher than the critical pressure and critical temperature (see Figure 10 ). However, in the present paper, the term supercritical fluid usually includes both terms—supercritical fluid and compressed fluid.
Supercritical steam is actually supercritical water, because at supercritical pressures fluid is considered as a single-phase substance (see Figure 10 ). However, this term is widely (and incorrectly) used in the literature in relation to supercritical-“steam” generators and turbines.
Superheated steam is the steam at pressures below the critical pressure, but at temperatures above the critical temperature (see Figure 10 ).
Abbreviations and acronyms
|AECL||Atomic Energy of Canada Limited|
|AGR||advanced gas-cooled reactor|
|ASME||American Society of Mechanical Engineers|
|BN||fast sodium (reactor; in Russian abbreviations)|
|BWR||boiling water reactor|
|CHF||critical heat flux|
|CFD||computational fluid dynamics|
|CRL||Chalk River Laboratotries (AECL)|
|DHT||deteriorated heat transfer|
|GFR||Gas-cooled fast reactor|
|HTC||heat transfer coefficient|
|IAEA||International Atomic Energy Agency|
|IHT||improved heat transfer|
|LNG||liquified natural gas|
|LWR||light water reactor|
|NIST||National Institute of Standards and Technology (USA)|
|NHT||normal heat transfer|
|NPP||nuclear power plant|
|PHWR||pressurized heavy water reactor|
|PWR||pressurized water reactor|
|RMS||root-mean square (error)|
|S-CO2 (SC-CO2)||supercritical carbon dioxide|
|SCWR||supercritical water-cooled reactor|
|SFR||sodium-cooled fast reactor|
|ThPP||thermal power plant|
|USA||United States of America|
|USSR||Union of Soviet Socialist Republics|
|VHTR||very high temperature reactor|