Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering Applications

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.). 1990 by (1) 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.


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 [1].
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. [2] 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 freeconvection 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 [3].
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 with coal 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 with natural gas 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 [8] and many other sources.
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 [9].

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)) [6]. 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] 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)) 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) 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

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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: P in ≈ 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: P in ≈ 24.5 MPa (P cr = 22.064 MPa) and T in /T out = 340/520°C (T cr = 374°C); also, note that powerconversion cycle in a different LFR designs from other countries is based on SCP carbon dioxide Brayton gas-turbine cycle  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 [6,17]. 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.
Nowadays, the most widely used SCFs are water, carbon dioxide, and refrigerants [9]. 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 [6] (for details, see Table 3).
Also, other applications of SCFs will be discussed in the following chapters and are listed in Pioro and Duffey [9].   Rankine power cycle (see Figure 1) 1,2,3 Thermodynamics diagrams: P-T and T-s can be found in Handbook [6]. 3 Thermodynamics diagrams: P-T and T-s are shown in Figure 10.

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   [26]; and Pioro and Duffey [9] (for more details, see Table 4).

Glossary
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 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., "liquidlike") (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 T pc ) is the point at a pressure above the critical pressure and at a temperature (T pc > T cr ) 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. T-s diagram of generic SCP Rankine "steam"-turbine power cycle (modern advanced coal-fired thermal power plants and future SCWR NPPs) [6,7].

Figure 2.
T-s diagram of generic subcritical-pressure Rankine steam-turbine power cycle (older coal-fired thermal power plants and AGR Torness NPP) [6,7]. Superheated steam is the steam at pressures below the critical pressure, but at temperatures above the critical temperature (see Figure 10).
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.

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) [27] initiative (for details, see [6,[9][10][11][12][13][28][29][30]; 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.     [17]) [18].
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 hydraulicequivalent 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 [9].
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 [9].
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, 30-35):  Schematic of 600-MW th GFR concept considered initially by GIF with direct Brayton helium cycle (Courtesy of GIF) (see also [6]). 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. Therefore, the following conditions can be distinguished at critical and SCPs: a. Wall and bulk-fluid temperatures are below a pseudocritical temperature within a part of (see Figure 12) or the entire heated channel (see Figures 14a,  24a, and 30); .
T-s diagrams of 2400-MW th GFR NPP combined power cycle (based on Figure 8) [18]. b. Wall temperature is above, and bulk-fluid temperature is below a pseudocritical temperature within a part of (see Figures 13a, 31, 34, and 35) or the entire heated channel (see Figure 14b); c. Wall temperature and bulk-fluid temperature is above a pseudocritical temperature within a part of or the entire heated channel (see 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. [46] 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.    Figure 11. Uncertainties of primary parameters are listed in Table 5.

Figure 21.
Heat transfer coefficient vs. bulk-fluid enthalpy in vertical tube with upward flow of SCW at various heat fluxes (data from Yamagata et al. [46]).
An annulus or a one-rod (single-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. 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). Figure 23) [35]: heated rods equipped with four helical ribs.

Figure 23.
Radial cross-sections of annular channel (single rod) and three-rod bundle (for other details, see Figure 22) [35]: heated rods equipped with four helical ribs; all dimensions in mm; and Ukrainian stainless steel has been used for heated rods, by content and other parameters, this steel is very close to those of SS-304. 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   5)). Uncertainties of primary parameters are listed in Table 6.

Figures 28 and
regimes plus sometimes quite significant differences in local HTC values and wall temperatures around the central rod circumference.

Supercritical water (SCW)
Unfortunately, satisfactory analytical methods for practical prediction of forcedconvection 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 [10,48]. 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 [9]). However, an analysis of these correlations has shown that they are more or less accurate only within the particular  Table 6.   [41,42]: heated rods equipped with four helical ribs; all dimensions in mm; and Ukrainian stainless steel has been used for heated rods, by content and other parameters this steel is very close to those of  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.

3-D view (a) and cross-sectional view of vertical seven-rod bundle (b) cooled with upward flow of SCW
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 (T w À T b ); H w ÀH b T w ÀT b , J/kg K. Also, additional terms, ; 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 [49] correlation. In 1942, McAdams [50] 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 [9].
The majority of empirical correlations were proposed in the 1960s-1970s [9], 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 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/m 2 s, and heat flux-70-1250 kW/m 2 . All thermophysical properties of SCW were calculated according to NIST REFPROP software [25]. This correlation has accuracy of AE25% for HTCs and AE15 for wall temperatures (Figure 34). Eventually, this nondimensional correlation can be also used for other SCFs. However, its accuracy can be less or even significantly less in these cases.
Eq. (3) has an uncertainty of about AE25% for HTC values and about AE15% 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, Nu w 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 [51]: 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/m 2 s, and bulk-fluid inlet temperature 320-350°C. Uncertainty is about AE15% for the DHT heat flux.
Wang et al. [33] have evaluated 15 q dht correlations for SCW, and they have concluded that Pioro-Mokry correlation (Eq. (5)) "may be used for preliminary estimations." A recent study was conducted by Zahlan et al. [55,56] 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 (Figure 35). Tables 8 and 9 list results from this study in the form of the overallweighted average and root-mean-square (RMS) errors: (a) within three SC subregions; and (b) for subcritical liquid and superheated steam. Many of the correlations listed in these tables can be found in Zahlan et al. [55,56] and Pioro and Duffey [9]. In their conclusions, Zahlan et al. [55,56] determined that within the SC region, the latest correlation by Pioro-Mokry [51] (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. Chen et al. [57] has also concluded that the Pioro-Mokry correlation for SCW HT "performs best" compared to other 14 correlations.

Supercritical carbon dioxide
The following correlation was proposed by S. Gupta (an MASc student of I. Pioro) [21] for SC carbon dioxide flowing inside vertical bare tubes: Uncertainties associated with this correlation are AE30% for HTC values and AE 20% for calculated wall temperatures (see Figures 36 and 37). Ranges of parameters for the dataset used to develop Eq. (6) are listed in Table 10. Table 11 list mean and root-mean square (RMS) errors in HTC and T w for proposed correlations using equations shown below: 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 Heat loss ≤3.4% Table 6.
Overall average and RMS error within subcritical region [55,56].   DHT within the SC carbon dioxide dataset, the following correlation for the minimal heat flux at which deterioration occurs was proposed: In general, the total pressure drop for forced convection inside a channel can be calculated according to expressions listed in Pioro and Duffey [9] and Pioro et al. [71].

Conclusions
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, threeand 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 [49] significantly overestimates experimental HTC values within the pseudocritical range. The Bishop et al. [60] and Jackson [68] correlations tend also to deviate substantially from the experimental data within the pseudocritical range. The Swenson et al. [61] 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. [51] (bulk-fluid-temperature approach) and Gupta et al. [21] (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 AE25% for HTC values and about AE15% for calculated wall temperature. Also, based on an independent study performed by Zahlan et al. [55,56], 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.
Pr cross-sectional average Prandtl number within the range of (T w -T b );