A Review of Flow and Heat Transfer Characteristics of Supercritical Carbon Dioxide under Cooling Conditions in Energy and Power Systems

Abstract

Supercritical carbon dioxide (SCO₂) is widely used in many fields of energy and power engineering, such as nuclear reactors, solar thermal power generation systems, and refrigeration systems. In practical applications, SCO₂ undergoes a cooling process significantly when it is cooled near the pseudo–critical point. Because of the drastic variations in thermo–physical properties, the heat transfer characteristics fluctuate, affecting the heat exchange and overall cycle performance. This paper summarizes extensive experiments and numerical simulations on the cooling process of SCO₂ in various application scenarios. The effects of various working conditions, such as mass flow, working pressure, pipe diameter, flow direction, and channel shapes, are reviewed. The applicability and computational results using different numerical methods under different working conditions are also summarized. Furthermore, empirical correlations obtained in experiments at different conditions are included. The present review can provide a helpful guideline for the design of effective cooling systems or condensers so that the accuracy of the design and efficiency of the system can be improved.

1. Introduction

In recent years, with the growing concern about the environment and climate change, new environmentally friendly and low–carbon technologies or technologies that can utilize renewable energy have been widely developed [1]. Carbon dioxide is a safe, nontoxic, nonflammable, economical, and resource–rich natural working medium widely applied in energy systems. It can be used as a refrigerant for refrigeration systems [2], air conditioners [3], heat pumps [4], and other devices in a trans–critical state [5], as well as an adequate working fluid for systems such as thermal power plants [6], advanced nuclear reactors [7], waste heat recovery [8], and solar thermal power generation [9]. In these systems and cycles, the condensers or coolers undergo a supercritical heat transfer process, which plays an essential role in cycle efficiency. Therefore, the study of supercritical cooling heat transfer process is very important for the design of a supercritical heat exchanger [10].

Reviews of the SCO₂ have been carried out by scholars. Duffey and Pioro [11] performed a comprehensive review of SCO₂ flows in horizontal, vertical, and other geometrical tubes. Three modes of heat transfer were proposed in this study, namely normal heat transfer (NHT), heat transfer deterioration (HTD), and heat transfer enhancement (HTE), and they were defined by the expected heat transfer coefficient values. Cabeza et al. [12], Fang et al. [13], Huang et al. [14], and Cheng et al. [15] performed comprehensive review works on the experimental data of SCO₂ heat transfer and pressure drop characteristics. Rao et al. [16], Li et al. [17], and Ehsan et al. [18] conducted a detailed review of heat transfer characteristics, correlations, and the effects of different operating parameters with SCO₂ under heating and cooling conditions of channels and tubes. Wang et al. [19] reviewed the simulation techniques used in turbulent SCO₂ flow and discussed their advantages, shortcomings, and applicability. In addition, the research on heat transfer deterioration in vertical tubes [20] and the judgment of buoyancy criteria [21] were also reviewed.

As mentioned, early review works mainly focused on the influence of different parameters on heat transfer and pressure drop, as well as the comparison of heat transfer correlations under SCO₂ heating conditions. However, understanding the heat transfer and pressure drop characteristics of SCO₂ under cooling conditions is essential for designing a high–efficiency gas cooler or condenser. The main objective of this review is to assess heat transfer in SCO₂ and understand the unique characteristics under the cooling condition. This paper can provide suggestions for designing high–efficiency heat exchangers in the future to improve the system’s overall efficiency.

2. Thermal–Physical Properties of SCO₂

Supercritical fluid is a type of fluid that reaches or exceeds the critical temperature and pressure. Liquids and gases enter the supercritical stage when heated above the critical temperature T_c and compressed above the critical pressure P_c. Above the critical temperature (T_c = 30.98 °C) and critical pressure (P_c = 7.38 MPa), CO₂ goes into the supercritical state. Working in this state, CO₂ does not undergo a phase transition (Figure 1a). Thermo–physical parameters of SCO₂, e.g., the density and dynamic viscosity, change dramatically when approaching the critical point, as shown in Figure 1b. This is the notable feature of SCO₂ compared to constant–property fluids. The isobaric–specific heat of SCO₂ peaks at the pseudo–critical temperature (T_pc). The rapid and nonlinear changes in the specific heat of SCO₂ against the temperature at several supercritical pressures are shown in Figure 2. The pseudo–critical temperature T_pc of SCO₂ increases when the working pressure increases. Far from the critical point, however, the change has been less pronounced. When SCO₂ is cooling, the initial performance change is small and, when the temperature reaches the T_pc, the performance changes drastically. As the working pressure approaches P_c, the specific heat peak becomes sharper, which makes the heat transfer coefficient increase significantly. Compared with supercritical water, SCO₂ is more suitable as a heat transfer fluid because of its lower critical parameters and lower specific volume values. In a typical SCO₂ recompression Brayton cycle, as shown in Figure 3a, the cooling process 8–1 makes the density and specific heat of CO₂ increase rapidly during cooling near T_c, causing the compressor to deliver a high–density fluid. Therefore, the compression of the high–density fluid by the compressor reduces power consumption and improves the overall thermal performance of the cycle. In the trans–critical CO₂ cycle, process 2–3 in Figure 3b is also the cooling process above the P_c, and the heat transfer process at this stage will affect the performance of the whole cycle. T_pc can be written and calculated as an algebraic function of the working pressure. The unit of working pressure is bar and the unit of result temperature is Celsius [22].

$$T_{pc}=-122.6+6.124p-0.1657p^2+0.01773p^{2.5}-0.0005608p^3$$

(1)

3. Summary of Experimental Studies

By collating the references, we find that many scholars have conducted extensive experimental studies on SCO₂ cooling to study turbulence and heat transfer.

Table 1. Heat transfer experiments under SCO₂ cooling conditions.

References	Pipe Types and Material	Tube Diameter ID (mm)	Tube Length L (mm)	Inlet Temperature T (°C)	Inlet Pressure p (MPa)	Mass Flux G (kg/m2. s)	Flow Form
Liao and Zhao [22]	circular tube, stainless steel (SS)	0.50–2.16	110	20–110	7.4–12	0.02–0.2 kg/min	forced
Pitla et al. [23]	circular tube, SS	4.72	1300/1800	120	8–12	1143–2228.9	–
Yoon et al. [24]	circular tube, copper	7.73	1000	50–80	7.5–8.8	225–450	forced
Dang and Hihara [25]	circular tube, copper	1–6	475	30–70	8–10	200–1200	–
Kuang et al. [26]	circular tube, SS	0.79	635	25–45	8–10	300–1200	forced
Huai et al. [27]	circular tube, aluminum	1.31	500	22–53	7.4–8.5	113.7–418.6	forced
Son and Park [28]	circular tube, SS	7.75	6000	90–100	7.5–10	200–400	–
Bruch et al. [29]	circular tube, copper	6	750	15–70	7.4–12	50–590	mixed
Jiang et al. [30]	circular tube, SS	2	150	55–70	7.8–9.8	Re = 434,069,108,640	mixed
Lv et al. [31]	circular tube, SS	3.8	1600	76.3–93.6	8–10	300–800	–
Oh and Son [32]	circular tube, SS	4.55/7.75	4000/6000	90–100	7.5–10	200–600	forced
Eldik et al. [33]	circular tube, SS	16	–	90–120	8–11	Re = 350,000–680,000	forced
Liu et al. [34]	circular tube, copper	4–10.7	1200	25–67	7.5–8.5	74.1–795.8	–
Ding and Li [35]	circular tube, SS	7.5	3000	29–55	8–10	880.3–1383.3	–
Ma et al. [36]	circular tube, SS	12	1500	50–70	8–10	491–823	mixed
Zhang et al. [37]	circular tube, SS	4.12–9.44	1000	70	8–9	160–320	mixed
Lei et al. [38]	circular tube, SS	1	1200	34.41–42.09	7.5–8.2	467.9–907.9	mixed
Wahl et al. [39]	circular tube, SS	2	1200	10–85	7.7–8.5	400–1300	forced
Li et al. [40,41]	semi–circular channel, SS	1.17 (hydraulic diameter)	500	10–90	7.5–10	326–762	forced
Baik et al. [10]	semi–circular zigzag channel, SS	1.8 (hydraulic diameter)	200	26–43	7.3–8.6	Re = 15,000–100,000	forced
Chu et al. [42]	semi–circular channel, SS	1.4	150	36.85–101.85	8–11	150–650 kg/h	mixed
Chu et al. [43]	semi–circular zigzag channel, SS	2.8	100	43.2–65.9	8–11	200–430 kg/h	mixed
Liu et al. [44]	semi–circular channel, SS	1.14 (hydraulic diameter)	1300	26.4–121.27	7.53–11.97	50.03–780.11	–
Park et al. [45]	semi–circular channel, SS	1.2	640	44–70	7.5–8.5	Re = 10,000–14,400	–
Wang et al. [46]	helically coiled tube, copper	4	500	47.05	8.0042–9.0072	159–318.2	mixed
Xu et al. [47]	helically coiled tube, copper	2–5	560	21.85–56.85	8–9	159.1–954.9	–
Liu et al. [48]	helically coiled tube, copper	2–4	–	20–50	7.5–9	79.6–1273.2	forced
Lee et al. [49]	micro–fin tube, copper	4.6	2400	100	8–10	1200–2000	forced
Yu et al. [50]	spiral circular tube, copper	6.34	13,000	59–94.5	7.6–9.6	395.9–471.9	–
Yang and Liao [51]	square duct, Aluminum	0.5 × 0.5	675	55	7.5	50–450	–
Zhu et al. [52]	fluted tube, copper	8.02–12.57	1200	45–85	8.2–11.2	Re = 13,000–42,000	–

lists primary sources of experimental data on cooling. Their operating conditions are shown in Figure 4. The SCO₂ cooling heat transfer temperature range is mainly concentrated between the T_c and 80 °C, that is, the area marked by the green box in Figure 1a. The experimental working temperature extended below T_c is the background of the trans–critical cycle with SCO₂ as the refrigerant. The typical experiment test loop for SCO₂ cooling is shown in Figure 5. Water cooling is used in all experiments.

3.1. Horizontal Tubes

There have been numerous studies on cooling flows of SCO₂ in horizontal tubes. The turbulent heat transfer and flow features near critical conditions were studied experimentally.

3.1.1. Effect of Mass Flux on Heat Transfer Characteristics

When SCO₂ is cooled, the heat transfer coefficient approaches the peak and then decreases. Under the same conditions, except that the flow rate conditions are changed, the flow rate increases and the heat transfer coefficient increases, which is the same as the case of constant characteristic flow, as shown in Figure 6a. Furthermore, when the mass flux is constant, the heat transfer coefficient increases in the cooling process until SCO₂ cools to the area with large thermo–physical property changes and the heat transfer coefficient reaches the maximum. The heat transfer coefficient drops suddenly when the fluid enters the liquid state. Under different flow rate conditions, the heat transfer coefficient oscillates near the peak. Liao and Zhao [22], Pitla et al. [23], Yoon et al. [24], Dang and Hihara [25], Kuang et al. [26], Huai et al. [27], Son and Park [28], Jing et al. [31], Oh and Son [32], Eldik et al. [33], Liu et al. [34], Ding and Li [35], Zhang et al. [37], Wahl et al. [39], Dong et al. [53], Huai and Koyama [54], and Lv et al. [55] all experimentally investigated heat transfer characteristics influenced by mass flow with various parameter ranges and drew the consistent conclusions.

3.1.2. Effect of Operating Pressure on Heat Transfer Characteristics

The effect of operating pressure on heat transfer characteristics can be seen in Figure 6b. The heat transfer coefficient peak moves towards a higher temperature value with the increase in pressure. This phenomenon is consistent with the transition of the pseudo–critical region to a higher temperature value with increasing pressure. When the pressure is higher than P_c, because the change of thermal physical property decreases with the increase in pressure, the change in heat transfer coefficient with temperature also shows the same trend. The heat transfer coefficient peaks at a pressure close to P_c are more pronounced. Liao and Zhao [22], Pitla et al. [23], Yoon et al. [24], Dang and Hihara [25], Huai et al. [27], Son and Park [28], Jing et al. [31], Oh and Son [32], Eldik et al. [33], Liu et al. [34], Ding and Li [35], Zhang et al. [37], Wahl et al. [39], Dong et al. [53], Huai and Koyama [54], and Lv et al. [55] have given the experimental results of the change in heat transfer coefficient of SCO₂ with temperature in different parameter ranges. Their conclusions on the change in heat transfer coefficient with operating pressure were consistent with the above.

3.1.3. Effect of Tube Diameter on Heat Transfer Characteristics

Liao and Zhao [22] conducted experiments on the cooling flow of SCO₂ on six horizontal stainless steel tubes with different diameters from 0.5 to 2.16 mm. The Nusselt number (Nu) in the measurement temperature range strongly depended on the pipe diameter, and Nu decreased significantly when the pipe diameter was reduced. The authors attributed the cause to the buoyancy effect. The buoyancy effect became less critical with decreased tube diameter, which was still significant, even if Reynolds number (Re) was up to 10⁵. Dang and Hihara [25] experimentally investigated with different diameters. The authors defined an effective heat transfer coefficient. At bulk temperature (T_b) < T_pc, the effective heat transfer coefficient was not affected by tube diameter. At T_b > T_pc, the effective heat transfer coefficient increased slightly with increasing diameter. Similarly, Oh and Son [32] compared the change in heat transfer coefficient in 4.55 mm and 7.75 mm tubes, Liu et al. [34] compared the change in heat transfer coefficient in 6 mm and 10.7 mm tubes, and Zhang et al. [37] compared the change in heat transfer coefficient in four different pipe diameters between 4.12 mm and 9.44 mm, as shown in Figure 7. All the above studies found that the heat transfer coefficient would increase with the increase in tube diameter, regardless of the range of pipe diameter.

3.1.4. Pressure Drop Characteristic

Zhang et al. [37] experimented with different pressure, mass flux, and diameter to investigate pressure drop characteristics under SCO₂ cooling conditions. As the temperature of SCO₂ gradually increased from T_pc, the pressure drop increased accordingly. The dramatic thermo–physical change near T_pc also led to dramatic changes in pressure drop. Under the same operating pressure, the pressure drop increased with the increase in the mass flux. As the working pressure increased, the pressure drop decreased because the change in characteristics away from the critical region became smaller. Pressure drop increased monotonically with the increase in tube diameter. Furthermore, when the temperature of SCO₂ was lower than T_pc, the pressure drop change tended to be flat. The results can be seen in Figure 8. Yoon et al. [24], Dang and Hihara [25], Kuang et al. [26], Huai et al. [27], Son and Park [28], and Liu et al. [34] also conducted experimental studies on the pressure drop characteristics of horizontal tubes with different diameters (from 0.79 mm to 10.7 mm). These scholars all found that the pressure drop increases with the increase in flow rate and decreases with the increase in operating pressure.

3.2. Vertical Tubes

Bruch et al. [29] experimentally studied the influence of pressure, mass flux, and flow direction on heat transfer characteristics in a vertical copper tube with an inner diameter of 6 mm. The effect of pressure was the same as that in the horizontal tube. The heat transfer coefficient of upward flow would increase with the increase in mass flux. Heat transfer increased in a limited mass flux for downward flow due to the mixed convection. The heat transfer coefficient of upward flow was more significant than that of downward flow, as shown in Figure 9. This was caused by buoyancy, and the effect of buoyancy mainly existed in the liquid–like and pseudo–critical regions. Figure 10 shows the relationship between the buoyancy parameter and the dimensionless mixed convection parameter related to Nu. As can be seen from the figure, the mixed convection effects under heating and cooling conditions were comparable.

Other scholars have also studied the vertical flow of SCO₂ with different pipe diameters. Lei et al. [38] studied the heat transfer characteristics in a 1 mm cooling tube. The results indicated that the heat transfer coefficient in the gas–like zone was less affected by the wall heat flux, while that in the liquid–like zone was not affected by the wall heat flux. The buoyancy effect had a crucial effect on the heat transfer process. A decrease in mass flow resulted in increased buoyancy and reduced heat transfer. In forced convection with negligible buoyancy effects, heat transfer enhancement was observed. The heat transfer performance of downward flow was better than that of horizontal flow, as shown in Figure 11. Jiang et al. [30] conducted experiments with a 2 mm cooling tube. It was observed through experiments that, due to the different directions of buoyancy, heat transfer enhancement would occur in the upward flow and the heat transfer deterioration and recovery would occur in the downward flow. Ma et al. [36] experimentally studied the cooling heat transfer in a 12 mm vertical tube. In addition to drawing similar conclusions to other vertical tube experiments, the new findings were that the influence of the mass flux at the water side on the overall heat transfer coefficient was more significant than that at the SCO₂ side. The heat transfer affected by buoyancy increased with the decrease in mass flux at the SCO₂ side. The mass flux variation on the water side and the pressure variation on the SCO₂ side had little effect on buoyancy.

3.3. Enhanced Channels

3.3.1. Semi–Circular Channels

To improve heat transfer efficiency, a printed circuit heat exchanger (PCHE) is designed as a highly effective heat exchanger due to its large UA values [56,57]. It is manufactured by photolithographically etching small channels into steel or other alloy plates and bonding the metal plates together by diffusion welding. The channel cross–sectional shape is usually semi–circular. Li et al. [40,41] conducted experimental investigations on forced convection heat transfer of SCO₂ in PCHE under heating and cooling conditions. The difference in heat transfer between heating and cooling conditions stemmed from the inverse distribution of radial thermo–physical properties.

Moreover, the heat transfer effect under cooling condition was better than that under the heating condition. Baik et al. [10] investigated SCO₂ cooling flow in a semi–circular zigzag channel by experiment and numerical calculation. Round corner channels could reduce pressure drop more than sharp corner channels. Chu et al. [42] experimented with studying the flow and heat transfer of SCO₂ in PCHE with a straight semi–circular channel. The heat transfer ability of SCO₂ was 1.2–1.5 times higher than water and the growth of the heat transfer rate was not linear but had a turning point with the increase in pressure loss. Chu et al. [43] experimentally studied SCO₂ cooling flow in a semi–circular zigzag channel. The results showed that the significant convective thermal resistance was on the SCO₂ side. The buoyancy of SCO₂ in PCHEs during cooling cannot be ignored. Liu et al. [44] and Park et al. [45] also experimented with a PCHE with straight channels in general operating conditions, and the results were the same as above.

3.3.2. Helically Coiled Tubes

The helical gas coolers introduced by Okada are made by twisting straight copper tubes into helical coils. They are widely used in trans–critical CO₂ air conditioning and heat pump systems because of their compact structure, easy fabrication, and high heat transfer performance. Yu et al. [50] experimented with investigating the performance and heat transfer of SCO₂ water–cooled gas cooler with the helically coiled tube. Moreover, a heat exchanger model was developed to predict the heat capacity. The calculated results were consistent with the experimental results. In the 36 experimental conditions tested, 94% of the data error is within ±20%. The heat transfer rate showed local maxima and minima during the trans–critical process due to the sharp rise in specific heat near the pseudo–critical region. Wang et al. [46] investigated the SCO₂ cooled in the 4 mm diameter helically coiled tube. The effects of mass flux and pressure were the same as that of a straight tube. Heat flux mainly affected the radial thermo–physical property distribution of the tube transverse section. When heat flux increased, the heat transfer coefficient increased at T_pc, while remaining almost unchanged in the liquid–like zone. Three existing buoyancy criteria overestimated the effect of buoyancy. Xu et al. [47,58] experimentally investigated the heat transfer of SCO₂ in the cooled helically coiled tubes, and several parameters have analyzed the effects on the exergy and Re. The dimensionless exergy destruction caused by the irreversibility of heat transfer was much greater than that by flow friction. A correlation based on experimental data was proposed to predict the optimal Re and appropriate operating conditions in cooled helically coiled tubes. The proposed correlation was $R{e}_{\mathrm{opt} }/{10}^4=1.258072\times {10}^{-6}{\eta }^{0.235231}{\alpha }^{-1.661865}{\tau }^{1.018487}$, where η stands for the dimensionless passage length of the coil. α means the dimensionless duty parameter. τ represents the dimensionless inlet temperature difference ratio.

3.3.3. Other Types of Channels

Different types of channels have been developed to increase heat transfer capacity. Zhu et al. [52] investigated heat transfer characteristics during the cooling of a fluted tube–in–tube heat exchanger. This tube produced a higher heat transfer coefficient on both sides than the smooth tube, thereby reducing temperature differences in the gas cooler. At any pressure, the smaller the hydraulic diameter of the fluted tube, the higher the heat transfer coefficient. Nu depended mainly on the pitch of the flute. A smaller flute pitch resulted in a more significant temperature gradient near the wall. In the work of Lee et al. [49], experiments were carried out on a 4.6 mm inner diameter micro–fin tube for refrigeration and air conditioning equipment. The heat transfer performance in this channel was the same as that of smooth tubes, but the heat transfer coefficient was 12–39% higher than that of the smooth tubes. Square micro channels (0.5 × 0.5 mm) were studied in the work of Yang and Liao [51]. The results showed that, under near–critical conditions, the cooling range of the test section has great influence on heat transfer and pressure drop.

4. Summary of Numerical Studies

The experiment provides precious data for studying turbulent heat transfer in SCO₂. However, since the experiments are usually performed in small channels under high pressure, the measurement data provided by the existing measurement techniques is minimal. To obtain more detailed information on the heat transfer properties of turbulent SCO₂, detailed information on the velocity field, temperature field, and turbulence distribution is required. In addition, in practical engineering applications, the range of applied operating conditions is far beyond the scope of experimental research. With the development of computer technology, numerical simulations can obtain more abundant flow and heat transfer data than experimental studies, and considerable research has been published.

Table 2. Heat transfer simulations under SCO₂ cooling conditions.

References	Channel Configuration	Tube Diameter D (mm)	Inlet Temperature T (°C)	Inlet Pressure p (MPa)	Inlet Reynolds Number	Turbulence Model	Boundary Conditions
Lei and Chen [62]	Wavy microchannel	1.31 (hydraulic diameter)	36.85–51.85	7–9	127.1–400 kg/m2. s	9 turbulence models	Constant heat flux: −9–−40 kW/m2
Pitla et al. [63]	Circular tube	4.52	121.85	10	3.2 × 105	Nikuradse’s mixing length model and the K equation turbulence model	Constant wall temperature: 29.85 °C
Dang and Hihara [64]	Circular tube	6	65	8	4 × 104–105	BR, JL, LS, MK	Constant heat flux: −6–−33 kW/m2
Han et al. [3]	Square duct	0.5 × 0.5	100	10	105	Elliptic−blending equation	Constant heat flux: −10–−40 kW/m2
Jiang et al. [30]	Circular tube	2	70	8.8	4216/4340	RNG k–ε, YS, AKN, LB	Conjugated heat transfer
Dang and Hihara [65]	Circular tube	0.2	170	8	50–100 kg/m2. s	Laminar flow	Constant heat flux: −1–−2 kW/m2
Du et al. [66]	Circular tube	6	56.85	8	200 kg/m2. s	k–ε, RNG k–ε, RSM, AB, LB, LS, YS, AKN, CHC	Constant heat flux: −6–−33 kW/m2
Cao et al. [67]	Circular/Triangular	0.5	120	8	1866	Laminar flow	Constant wall temperature: 25 °C
Li et al. [40,41]	Semi−circular channel	1.9	10–90	7.5–10	326–762 kg/m2. s	SST k–ω	a piecewise linear temperature distribution
Tanahashi et al. [68]	channel	–	–	8	Reτ = 180	DNS	Constant wall temperature: 36 °C
Du et al. [69]	Circular tube, vertical	6	56.85	8	60,000	LB	Constant heat flux: −6–−33 kW/m2
Yang et al. [70]	Circular tube	0.4	70	8.5	10,000	LB	Conjugated heat transfer
Yang et al. [71]	Circular tube	0.5	120	8	1.6 × 10−5 kg/s	Laminar flow	Constant wall temperature: 23 °C
Rao and Liao [72]	Circular tube, vertical	1.4	120	8	11,650	Standard k–ε	Constant wall temperature: 25 °C
Zhao and Che [73]	Circular tube, vertical	1–5	200–500	7.58	100–600 kg/m2. s	AKN, V2F, YS	Conjugated heat transfer
Yang [74]	helically coiled tube	4	73	8	250 kg/m2. s	RNG k–ε	Constant heat flux: −30 kW/m2
Yang [75]	Circular tube	6–27	56.85	8	200 kg/m2. s	13 turbulence models	Constant heat flux: −33 kW/m2
Pandey [76,77]	Circular tube, vertical	2	58.9	8/8.8	5400	DNS	Constant heat flux: −30.87/−61.74 kW/m2
Baik et al. [10]	semi–circular zigzag channel	1.8	26–43	7.3–8.6	15,000–100,000	Standard k–ε	Conjugated heat transfer
Wang et al. [46]	helically coiled tube	4	320.2	8.0091	239.2 kg/m2. s	SST k–ω	Constant heat flux: −13.3/−18 kW/m2
Xiang et al. [78]	Circular tube	2–6	67	8	30,000	SST k–ω	Constant heat flux: −35/−45 kW/m2
Wang et al. [79]	Circular tube	15.75–24.36	65	8	3.83 × 105	RNG k–ε, YS, AKN, LS	Constant heat flux: −10–−36 kW/m2
Wang et al. [80]	Circular tube	6	56.85–76.85	8–10	200 kg/m2. s	Standard k–ε	Conjugated heat transfer
Zhang et al. [37]	Circular tube	4.12–9.44	70	8–9	160–320 kg/m2. s	Standard k–ε	Constant heat flux: −34.5–−105.4 kW/m2
Chai and Tassou [81]	6 different cross–section geometries	1.22 (hydraulic diameter)	120	7.5/15	2 m/s	Standard k–ε	Constant heat flux: −100–−300 kW/m2
Diao et al. [82]	Circular tube, inclined	6	56.85/66.85	8–10	200 kg/m2. s	LB	Constant heat flux: −33–−42 kW/m2
Guo et al. [83]	Circular tube, vertical	2/6	40–70	8	200–1500 kg/m2. s	SST k–ω with TWL	Constant heat flux: −40/−60 kW/m2
Liu et al. [48]	helically coiled tube	2–4	20–55	7.5–9	0.001–0.004 kg/s	SST k–ω	Constant heat flux: −9/−39.9 kW/m2
Ren et al. [84]	Semi−circular channel	2.8	40–100	7.5/8.1	200–800 kg/m2. s	SST k–ω	Conjugated heat transfer
Wang et al. [85]	Circular tube, inclined	20	65	8	223/382.2 kg/m2. s	AKN	Constant heat flux: −10–−36 kW/m2
Zhang et al. [86]	6 different cross–section geometries	1.16 (hydraulic diameter)	39.85–91.85	8.1–20	3000–18,000	SST k–ω	Constant heat flux: −20/−33 kW/m2
Ren et al. [87]	semi–circular zigzag channel	1.9	40–100	7.5/8.1	400–800 kg/m2. s	SST k–ω	Conjugated heat transfer
Yu et al. [61,88]	spirally fluted tube	11.06	50	8	35,000	RNG k–ε	Constant heat flux: −52–−60 kW/m2
Saeed et al. [59]	semi–circular zigzag channel	1.106	70–110	8	0.5–1.25 g/s	SST k–ω	Conjugated heat transfer
Yang et al. [89]	helically coiled tube	4	73	8	250 kg/m2. s	RNG k–ε with EWT	Constant heat flux: −30 kW/m2

summarizes the numerical simulation work on SCO₂ cooling heat transfer. The main types of pipes summarized in this chapter mainly include circular tubes, semi–circular channels, helically coiled tubes, fluted tubes, square microchannels, and polygon channels. The configuration of some special channels is shown in Figure 12.

Figure 12. Physical model of semi–circular channels [59] (a), helically coiled tubes [60] (b), and fluted tubes [61] (c).

Figure 12. Physical model of semi–circular channels [59] (a), helically coiled tubes [60] (b), and fluted tubes [61] (c).

4.1. In–House Codes

In the early days, the researchers developed in–house code to solve the SCO₂ heat transfer problem in simple geometries. Pitla et al. [63] proposed a mathematical model based on the Favre–averaged, parabolized Navier–Stokes equations in conjunction with Nikuradse’s mixing length model and the k equation turbulence model to simulate the turbulent flow of SCO₂ during in–tube cooling. It was seen that the velocity and temperature laws of the wall for constant property flows were not valid here. Dang and Hihara [64] examined four different turbulence models, namely the Myong and Kasagi (MK) model, the Launder and Sharma (LS) model, the Jones and Launder (JL) model, and the Bellmore and Reid (BR) model, to the applicability of heat transfer coefficient prediction. The JL model had a good prediction of the heat transfer coefficient under most conditions, although it slightly underestimated at large heat flux. Asinari [90] developed a new model based on the BR model, which considered the effects on the turbulence of variable thermo–physical properties. The refinement did not improve the existing results dramatically, and the density fluctuations were smaller than supposed. Dang and Hihara [65] performed the numerical simulations for SCO₂ in a mini tube with Re less than 1000. When heat flux was constant, Nu reached its maximum when T_b > T_pc and its minimum value when T_b < T_pc. f·Re reached its peak at T_b = T_pc. Cao et al. [67] investigated numerically with laminar mixed convective heat transfer of SCO₂ in a horizontal mini tube with a hydraulic diameter of 0.5 mm. The effects of the sharply varied physical properties of SCO₂ and the geometrical characteristics of tubes on the fluid flow and heat transfer were examined. The buoyancy played a major role in this process. As a result, a robust secondary flow occurred within the cross–section, which led to distortions in the velocity and temperature distributions. In addition, heat transfer was enhanced due to buoyancy, while Nu varied unevenly. Rao and Liao [72,91,92] compiled an FORTRAN computer code based on the finite control–volume method to investigate the turbulent convective heat transfer of SCO₂ flowing in a vertical mini tube of 1.4 mm diameter. The results showed that the transport phenomenon and buoyancy influence mechanism in small–diameter tubes were similar to those in large–diameter tubes. In addition, it was consistent with the results of experimental studies. As the pipe diameter increased or the mass flow decreased, heat transfer was more obviously affected by buoyancy. Han et al. [3] studied SCO₂ cooling heat transfer in a pipe using elliptical mixing second–moment turbulent closure. In addition, a heat transfer coefficient correlation for square cross–sectional duct flows was established.

4.2. Commercial Computational Fluid Dynamics Solvers

4.2.1. Horizontal Tubes

Commercial computational fluid dynamics (CFD) software packages have been used to calculate heat transfer in SCO₂, and relatively accurate results have been obtained. The mainstream commercial software currently used are FLUENT and CFX. Since buoyancy is crucial in SCO₂ heat transfer simulations and its behavior varies across channels and flow directions, this part is divided into various geometric arrangements. The typical physical model and boundary conditions used in tubes are shown in Figure 13. Du et al. [66] used FLUENT to numerically investigate SCO₂ cooling heat transfer in a horizontal tube. Nine turbulent models have been compared in the prediction performance in heat transfer, including standard k–ε, RNG k–ε, Reynolds stress model (RSM), and six low Reynolds turbulence models: Abid (AB) model, Lam Bremhorst (LB) model, LS model, Yang–Shih (YS) model, Abe Kondoh Nagano (AKN) model, and Chang Hsieh Chen (CHC) model. The consistency between LB model and experimental data of [25] was the best. Moreover, the effect of buoyancy on the SCO₂ flow enhanced the cooling heat transfer effect, especially near the pseudo–critical point. The mixed convection was the primary heat transfer mechanism in the SCO₂ cooling process. Yang et al. [70] presented a simulation with a 0.4 mm inner diameter microtube. Near the SCO₂ critical point, changes in thermo–physical characteristics led to more significant buoyancy fluctuation, and buoyancy’s impact on the heat transfer coefficient was substantial and complex. Yang [75] investigated the heat transfer in a large horizontal tube with 6–27 mm diameter. Nine turbulence models were compared in this study, the same as Du et al. [66]. The results showed that almost all models could qualitatively exhibit the heat transfer trend. The standard k–ε model with enhanced wall treatment was in the best agreement with the experimental data. When T_b > T_c, the heat transfer coefficient with the wall was more significant than that without wall. Xiang et al. [78] studied the convective heat transfer of SCO₂ in a horizontal tube by using the SST k–ω turbulence model in ANSYS CFX. The secondary flow and vortexes generated on the cross–section were analyzed. The nonuniform variation in heat transfer coefficient was explained by the field synergy principle. Wang et al. [79,93] numerically simulated the cooling heat transfer of SCO₂ in large horizontal tubes (24.36 mm, 20 mm, and 15.75 mm). Results showed that, when T_b > T_pc, the heat transfer coefficient of SCO₂ was increased with the increase in the heat flux and pipe diameter. In addition, at T_b < T_pc, the heat flux and pipe diameter hardly affected the heat transfer coefficient. The buoyancy effect slightly enhanced the turbulent heat transfer of SCO₂ flowing in large–diameter horizontal tubes, which was the opposite result of past studies on small–diameter tubes. Wang et al. [80] simulated SCO₂ flow in a horizontal tube to study the heat transfer mechanism of nonuniform conjugate cooling. From the calculation results, it could be concluded that the actual heat flux loaded on the wall–fluid interface was highly nonuniform. The bottom surface had a minor heat flux and the top had an enormous heat flux. In addition, as the heat flux increased, both the thermally induced flow acceleration and the secondary flow intensity increased. The buoyancy could effectively enhance the cooling heat transfer of SCO₂ in horizontal tubes. Yang et al. [94] obtained the conclusions similar to the above.

4.2.2. Vertical Tubes

Guo et al. [83] used SST k–ω turbulent model with variable turbulent Prandtl number model (TWL model) to investigate the SCO₂ flow and heat transfer in vertical tubes under cooling and heating conditions. The heat transfer coefficient under cooling condition was more significant than that in the heating condition due to more liquid–like fluid near the wall, as shown in Figure 14a. As for cooling conditions, the buoyancy effect enhanced the local heat transfer of the upward flow and deteriorated the local heat transfer of the downward flow, as shown in Figure 14b. Zhang et al. [95] used low Reynolds turbulence model–YS model to investigate heat transfer performance in different flow directions and tube diameters. The results showed that the local heat transfer coefficient reached the maximum when the fluid temperature was near the pseudo–critical point. The heat transfer coefficient was extremely influenced by the tube diameter and flow direction. Jiang et al. [30], Dong et al. [96], Cao et al. [97], and Zhao and Jiang [98] studied the conjugate heat transfer of SCO₂ and water with different tube diameters. The heat transfer coefficient increased with the increase in the mass flow rate of cooling water and significantly with the increase in the mass flow rate of SCO₂.

4.2.3. Inclined Tubes

When the layout space is limited, the inclined layout can increase the heat transfer area of the heat exchange channel as much as possible, thereby improving the heat exchange capacity. Yang et al. [71] numerically simulated laminar mixed convective heat transfer in a 0.5 mm tube at constant wall temperature cooling. The variation in velocity and temperature distributions, secondary flow, friction coefficient, and heat transfer coefficient with inclination angle was studied. Good heat transfer performance was exhibited between inclination angles from −30° to 30°. The impact of the inclination angle on heat transfer decreased as the magnitude of gravity decreased. Wang et al. [85] simulated the turbulent SCO₂ cooling process in 20 mm inclined tubes using the AKN model. The buoyancy force affecting the flow characteristics of SCO₂ could be divided into components parallel to the mainstream and components perpendicular to the mainstream. The former tended to increase the speed of upward flow and enhance turbulence near the wall. Instead, it led to a local “stratification” of downward flow. The main effect of the latter was to form a secondary flow, which makes the velocity peak near the top. In the liquid–like region, heat transfer was significantly affected by buoyancy, and the influence was further enhanced with the increase in heat flux. However, the influence of buoyancy on the heat transfer coefficient of SCO₂ decreased with the increase in pipe diameter in the inclined pipe with small flow rate. Diao et al. [82] came to similar conclusions as above.

4.2.4. Semi–Circular Channels

As mentioned in the previous section, semi–circular channels are mainly used in PCHE. In all the studies of this part, the SST k–ω model is selected for the turbulence model. Kruizenga et al. [40,99,100] numerically studied the cooling flow and heat transfer behavior of SCO₂ in a semi–circular straight channel with a diameter of 1.9 mm through FLUENT and verified it through experiments. Ren et al. [84,87] conducted numerical studies on the local flow and heat transfer properties of SCO₂ in horizontal semi–circular straight and zigzag channels. The effects of mass flux and channel geometry (including pitch length and inclination angle) were discussed. These effects altered the horizontal secondary flow and, thus, affected flow and heat transfer. Saeed et al. [59] designed and analyzed a SCO₂ Brayton recirculating precooler using a zigzag channel under different operating conditions. Keeping the low inlet Re can effectively shorten the length of the precooler. Pinch points may occur inside the precooler when the design value of the channel flow is kept above medium.

4.2.5. Helically Coiled Tubes

Yang [74] used the RNG k–ε turbulence model to numerically simulate the heat transfer of SCO₂ in the helically coiled tube. The effects of tube structure parameters were analyzed, demonstrating that gravity had no significant influence on the calculated heat transfer coefficients under the set conditions. Furthermore, Yang et al. [89] used the RNG k–ε turbulence model with enhanced wall treatment to investigate SCO₂ cooling flow in a noncircular cross–section horizontal helical coil. Heat transfer and pressure drop increased with the increase in polygonal sides, but the increased amplitude decreased gradually. In addition, there was a maximum pressure drop. Wang et al. [46] and Xu et al. [101] used the SST model and drew similar conclusions.

4.2.6. Fluted Tubes

Yu et al. [61] numerically studied the influences of geometric parameters on heat transfer and flow characteristics of SCO₂ cooling in the spirally fluted tube with the RNG k–ε model. The optimal structure was obtained according to the evaluation factors. The buoyancy force benefited the heat transfer coefficient of SCO₂ in a spirally fluted tube. Furthermore, Yu et al. [88,102] analyzed the heat transfer and flow behavior of SCO₂ cooling in helical fluted tubes at different inclination angles. The effect of the component of buoyancy in the mainstream direction was different from that in the radial direction, especially near the wall. The buoyancy effect became more important as the inclination angle increased. The optimum inclination angle that matches the best heat transfer performance was determined through the simulation results. Variations in local heat transfer coefficients were also investigated. Li et al. [103,104] studied the influence of operating parameters on the flow and heat transfer characteristics of SCO₂ in a horizontal spiral groove tube through simulation. The results showed that the overall heat transfer coefficient of the fluid decreased with the increase in cooling pressure but changed little with the inlet temperature. With the increase in mass flow, the influence of the buoyancy effect decreased and the peak value of the local heat transfer coefficient increased. Further, the influence of the groove number on the spiral groove tube was studied. Under the same working conditions, the more grooves the spiral groove tube has, the greater the overall heat transfer coefficient and friction coefficient would be.

4.2.7. Square Microchannels

Square microchannels are widely used in air conditioning, refrigeration, heat pipe, and other systems. Lei and Chen [62] conducted numerical studies on the cooling heat transfer and hydraulic characteristics of SCO₂ in a horizontal wavy microchannel with a square cross–section. The LB model had a better prediction by comparing the results of nine different turbulence models with experimental data. Compared with the straight channel, the heat transfer performance of the wavy microchannel significantly improved, while the pressure drop slightly increased. The change in heat transfer coefficient with operating parameters was consistent with that of a horizontal circular tube. After that, Lei and Chen [105] further studied the cooling heat transfer and pressure loss characteristics of SCO₂ in two different forms of wavy microchannels. The wavelength and wave amplitude had the optimum value for the overall thermal performance.

4.2.8. Other Types of Channels

In recent years, microchannel heat exchangers with different cross–sectional geometries have been developed, so it is necessary to study the effect of cross–sectional geometry. Chai and Tassou [81] numerically investigated the impact of six different cross–sectional geometries on the flow and heat transfer characteristics of SCO₂ in microchannels. Existing heat transfer empirical correlations were compared under the same heating and cooling conditions. Different section geometries under the same boundary conditions led to different heat transfer coefficients and friction factors, as shown in Figure 15. The cross–sections that obtain larger heat transfer coefficients generally have a more significant pressure drop. Zhang et al. [86] conducted a numerical study of SCO₂ with circular, semi–circular, and square cross–sections. A larger heat transfer coefficient, minor fluid friction, and entropy generation can be obtained using a smaller heat flow ratio and working pressure. The peak heat transfer coefficient around T_pc was mainly due to the effective thermal conductivity within the turbulent viscous sublayer.

4.3. Direct Numerical Simulations

The research on turbulent heat transfer of supercritical fluids by the RANS method and self–built model has been summarized above. According to the research, both the traditional turbulence model and the modified turbulence model have different degrees of limitations. Due to its limits and accuracy, the RANS model cannot quantitatively give reliable results for the heat transfer mechanism of supercritical fluids. This is partly due to the treatment of damping functions. In addition, the flow and heat transfer data that the Reynolds average equation can give are also limited. To further explain these physical mechanisms and processes, it is necessary to capture the information of the turbulent flow field accurately to have a deeper comprehension of the heat transfer process, and it is possible to theoretically construct a turbulence model that conforms to the physical process [106,107]. The direct numerical simulations (DNS) method can obtain turbulent flow information in the turbulent flow field, provide a deeper physical understanding of supercritical fluid’s turbulent heat transfer process, and provide reliable data for constructing accurate turbulence models. In recent years, different scholars have conducted some DNS studies on turbulent heat transfer in supercritical fluids, aiming to explain the physical process and mechanism of turbulent heat transfer in supercritical fluids [108,109]. However, there are few DNS studies on SCO₂ cooling conditions at present. Tanahashi et al. [68] used DNS to investigate SCO₂ turbulent channel flow under cooling conditions. The low–speed streaks alter the turbulent transport of fine–scale structure and temperature, leading to high Nu. Pandey et al. [76,77] pioneered the study of SCO₂ cooling conditions in tubes using the DNS method. They found that the combined effects of deceleration and buoyancy enhanced heat transfer in the upward flow and deteriorated heat transfer in downward flow. Quadrant and octant analysis showed that sweep and ejection events decreased when heat transfer deteriorated, leading to turbulence attenuation. Furthermore, the anisotropy of the Reynolds stress tensor showed that the turbulence was significantly modulated in the near–wall zone, whether flow was upward or downward.

5. Summary of Heat Transfer Correlations

Since the 1950s, scholars have been committed to summarizing supercritical heat transfer empirical correlations to guide practical engineering applications [20]. Likewise, many empirical correlations were summarized under the SCO₂ cooling condition, and these correlations are described in detail in

Table 3. Heat transfer correlations under SCO₂ cooling conditions.

References	Correlations	Applied Range	Relative Error
Liao and Zhao [22]	$N{u}_w=0.128{\mathrm{Re}}_w^{0.8}{\mathrm{Pr}}_w^{0.3}{\left(\frac{\mathrm{Gr}}{{\mathrm{Re}}_b^2}\right)}^{0.205}{\left(\frac{{\rho }_b}{{\rho }_w}\right)}^{0.437}{\left(\frac{{\overline{c}}_p}{c_{pw}}\right)}^{0.411}$	$7.4 < \mathrm{p} < 12 \mathrm{MPa}, 20 < \mathrm{T}{\mathrm{b}}_{ }< 110 °\mathrm{C}, 2 < {\mathrm{T}}_{\mathrm{b}}-\mathrm{T}{\mathrm{w}}_{ }<30 °\mathrm{C}, 0.02 < \dot{m}<0$.2 kg/min, 10−5 < Gr/Reb2 < 10−2 0.50 < d < 2.16 mm	±18.9%
Pitla et al. [23]	$Nu=\left(\frac{N{u}_{\mathrm{wall} }+N{u}_{\mathrm{bulk} }}{2}\right)\frac{k_{\mathrm{wall} }}{k_{\mathrm{bulk} }}$	–	±20%
Yoon et al. [24]	$\begin{array}{l}N{u}_{\mathrm{b}}=a{\mathrm{Re}}_{\mathrm{b}}^b{\mathrm{Pr}}_{\mathrm{b}}^c{\left(\frac{{\rho }_{\mathrm{pc}}}{{\rho }_{\mathrm{b}}}\right)}^n\\ a=0.14,b=0.69,c=0.66,n=0,for{T}_{\mathrm{b}}>T_{\mathrm{pc}}\\ a=0.013,b=1.0,c=-0.05,n=1.6,for{T}_{\mathrm{b}}⩽T_{\mathrm{pc}}\end{array}$	–	±12.7%
Dang and Hihara [25]	$$\begin{gathered} Nu=\frac{\left(f_{\mathrm{f}}/8\right)\left(R{e}_{\mathrm{b}}-1000\right)Pr}{1.07+12.7\sqrt{f_{\mathrm{f}}/8}\left(P{r}^{2/3}-1\right)} \\ Pr=\left\{\begin{array}{l}{c}_{{\mathrm{p}}_{\mathrm{b}}}{\mu }_{\mathrm{b}}/k_{\mathrm{b}}, \mathrm{for} c_{{\mathrm{p}}_{\mathrm{b}}}\ge {\overline{c}}_{\mathrm{p}}\hfill \\ {\overline{c}}_{\mathrm{p}}{\mu }_{\mathrm{b}}/k_{\mathrm{b}}, \mathrm{for} c_{{\mathrm{p}}_{\mathrm{b}}}<{\overline{c}}_{\mathrm{p}} \mathrm{and} {\mu }_{\mathrm{b}}/k_{\mathrm{b}}\ge {\mu }_{\mathrm{f}}/k_{\mathrm{f}}\hfill \\ {\overline{c}}_{\mathrm{p}}{\mu }_{\mathrm{f}}/k_{\mathrm{f}}, \mathrm{for} c_{{\mathrm{p}}_{\mathrm{b}}}<{\overline{c}}_{\mathrm{p}} \mathrm{and} {\mu }_{\mathrm{b}}/k_{\mathrm{b}}<{\mu }_{\mathrm{f}}/k_{\mathrm{f}}\hfill \end{array}\right. \\ {\overline{c}}_{\mathrm{p}}=\left(h_{\mathrm{b}}-h_{\mathrm{w}}\right)/\left(T_{\mathrm{b}}-T_{\mathrm{w}}\right) \\ R{e}_{\mathrm{b}}=Gd/{\mu }_{\mathrm{b}} \\ {f}_{\mathrm{f}}={\left[1.82{\log }_{10}\left(R{e}_{\mathrm{f}}\right)-1.64\right]}^{-2} \\ {\mathrm{Re}}_{\mathrm{f}}=Gd/{\mu }_{\mathrm{f}} \end{gathered}$$	–	±20%
Kuang et al. [26,110]	$Nu=0.00158{\mathrm{Re}}^{1.05}{\mathrm{Pr}}^{0.648}{\left(\frac{{\rho }_w}{\rho }\right)}^{0.367}{\left(\frac{\overline{C}p}{Cp}\right)}^{0.4}$	8 ≤ P ≤ 10 MPa, 300 ≤ G ≤ 1200 kg/m2	±20%
Huai et al. [27]	$Nu=2.2186\times {10}^{-2}{\mathrm{Re}}^{0.8}{\mathrm{Pr}}^{0.3}{\left(\frac{{\rho }_w}{\rho }\right)}^{-1.4652}{\left(\frac{\overline{C}p}{Cp}\right)}^{0.0832}$	7.4 ≤ P ≤ 8.5 MPa, 22 ≤ Tr ≤ 53 °C, 113.7 ≤ G ≤ 418.6 kg/m2 s, −0.8 ≤ q ≤ −9 kW/m2	±30%
Son and Park [28]	$\begin{array}{l}N{u}_{\mathrm{b}}={\mathrm{Re}}_{\mathrm{b}}^{0.55}{\mathrm{Pr}}_{\mathrm{b}}^{0.23}{\left(\frac{c_{\mathrm{p},\mathrm{b}}}{c_{\mathrm{p},\mathrm{w}}}\right)}^{0.15}for\frac{T_{\mathrm{b}}}{T_{\mathrm{pc}}}>1\\ N{u}_{\mathrm{b}}={\mathrm{Re}}_{\mathrm{b}}^{0.35}{\mathrm{Pr}}_{\mathrm{b}}^{1.9}{\left(\frac{{\rho }_{\mathrm{b}}}{{\rho }_{\mathrm{w}}}\right)}^{-1.6}{\left(\frac{c_{\mathrm{p},\mathrm{b}}}{c_{\mathrm{p},\mathrm{w}}}\right)}^{-3.4}\hspace{1em}for\frac{T_{\mathrm{b}}}{T_{\mathrm{pc}}}\le 1\end{array}$	–	±17.62%
Han et al. [3]	$Nu=N{u}_b{\left(\frac{{\rho }_w}{{\rho }_b}\right)}^a{\left(\frac{C_{pf}}{C_{pb}}\right)}^b\left(1+C\frac{R{i}^c}{{\mathrm{Pr}}^d}\right)$	–	–
Bruch et al. [29]	$\begin{array}{l}\mathrm{in} \mathrm{turbulent} \mathrm{aiding} \mathrm{mixed} \mathrm{convection}: \mathrm{Gr}/{\mathrm{Re}}^{2.7}<4.2\times {10}^{-5}:\frac{N{u}_b}{N{u}_{FC}}=1-75{\left(\frac{\mathrm{Gr}}{{\mathrm{Re}}^{2.7}}\right)}^{0.46}\\ \hfill \mathrm{Gr}/{\mathrm{Re}}^{2.7}>4.2\times {10}^{-5}:\frac{N{u}_b}{N{u}_{FC}}=13.5{\left(\frac{\mathrm{Gr}}{{\mathrm{Re}}^{2.7}}\right)}^{0.40}\\ \mathrm{in} \mathrm{turbulent} \mathrm{opposing} \mathrm{mixed} \mathrm{convection}: \frac{N{u}_b}{N{u}_{FC}}={\left(1.542+3243{\left(\frac{\mathrm{Gr}}{{\mathrm{Re}}^{2.7}}\right)}^{0.91}\right)}^{1/3}\end{array}$	–	±30%
Dang and Hihara [65]	$Nu=4.364{\left(\frac{c_{p_c}}{c_{p_f}}\right)}^{0.18}{\left(\frac{{\rho }_b}{{\rho }_f}\right)}^{0.47}{\left(\frac{{\mu }_f}{{\mu }_c}\right)}^{0.12}\frac{k_f}{k_b}{\left(\frac{k_c}{k_f}\right)}^{0.69}+1.31{\left(\frac{1}{x^{*}}\right)}^{1/3}\exp \left(-13\sqrt{x^{*}}\right)$	–	–
Oh and Son [32]	$\begin{array}{l}N{u}_b=a^{\prime }{\mathrm{Re}}_b^{b^{\prime }}\cdot {\mathrm{Pr}}_b^{c^{\prime }}\cdot {\left(\frac{{\rho }_b}{{\rho }_w}\right)}^{d^{\prime }}\cdot {\left(\frac{c_{p,b}}{c_{p,w}}\right)}^{e^{\prime }}\\ a^{\prime }=0.023,\hspace{1em}{b}^{\prime }=0.7,\hspace{1em}{c}^{\prime }=2.5,\hspace{1em}{d}^{\prime }=0,\hspace{1em}{e}^{\prime }=-3.5,for\hspace{1em}{T}_b/T_{pc}>1\\ a^{\prime }=0.023,\hspace{1em}{b}^{\prime }=0.6,\hspace{1em}{c}^{\prime }=3.2,\hspace{1em}{d}^{\prime }=3.7,\hspace{1em}{e}^{\prime }=-4.6,for\hspace{1em}{T}_b/T_{pc}\le 1\end{array}$	–	±14.1%
Fang et al. [111]	$\begin{array}{c}Nu=\frac{(f/8)\left({\mathrm{Re}}_b-20\mathrm{Re}{e}_b^{0.5}\right)\overline{\mathrm{Pr}}}{1+12.7{(f/8)}^{1/2}\left({\overline{\mathrm{Pr}}}^{2/3}-1\right)}\left(1+0.001\frac{q}{G}\right)\hfill \\ f=f_{\mathrm{noniso} }-1.36{\left(\frac{{\mu }_w}{{\mu }_b}\right)}^{-1.92}{f}_{ac}\hfill \\ f_{ac}=\frac{D}{\Delta L}\left({\rho }_{b,out}+{\rho }_{b,in}\right)\left(\frac{1}{{\rho }_{b,out}}-\frac{1}{{\rho }_{b, \mathrm{in} }}\right)\hfill \\ f_{\mathrm{noniso} }=f_{iso,b}{\left(\frac{{\mu }_w}{{\mu }_b}\right)}^{0.49{\left({\rho }_f/{\rho }_{pc}\right)}^{1.31}}\hfill \\ f_{iso}=1.613{\left[\ln \left(0.234{\left(\frac{\epsilon }{D}\right)}^{1.1007}-\frac{60.525}{R{e}^{1.1105}}+\frac{56.291}{R{e}^{1.0712}}\right)\right]}^{-2}\hfill \end{array}$	–	±10%
Li et al. [40]	$N{u}_{b,pdf}=0.0781R{e}_{b,pdf}^{0.662}{P}_{b,pdf}^{0.455}{\left(\frac{{\rho }_w}{{\rho }_{b,pf}}\right)}^{0.392}{\left(\frac{{\overline{C}}_{p,pdf}}{C_{p,b,pdf}}\right)}^{0.334}$	–	±25%
Kruizenga et al. [40,99]	$\begin{array}{l}Nu=0.0183{\mathrm{Re}}^{0.82}{\mathrm{Pr}}^{0.5}{\left(\frac{{\rho }_w}{\rho }\right)}^{0.3}{\left(\frac{\overline{C}p}{Cp}\right)}^n{\left(\frac{C_{p,b}}{C_{p-IG}}\right)}^{-0.19}\\ n=0.4 \mathrm{for} T_{\mathrm{b}}<T_{\mathrm{w}}<T_{\mathrm{pc}}\hspace{1em} \mathrm{or} \hspace{1em}1.2T_{\mathrm{pc}}<T_{\mathrm{b}}<T_{\mathrm{w}}\\ n=0.4+0.2\left(\frac{T_{\mathrm{w}}}{T_{\mathrm{pc}}}-1\right) \mathrm{for} T_{\mathrm{b}}<T_{\mathrm{pc}}<T_{\mathrm{w}}\\ n=0.4+0.2\left(\frac{T_{\mathrm{w}}}{T_{\mathrm{pc}}}-1\right)\left(1-5\left[\frac{T_{\mathrm{b}}}{T_{\mathrm{pc}}}-1\right]\right) \mathrm{for} T_{\mathrm{pc}}<T_{\mathrm{b}}<1.2T_{\mathrm{pc}} \mathrm{and} T_{\mathrm{b}}<T_{\mathrm{w}}\end{array}$	–	±19%
Lee et al. [49]	$\begin{array}{l}N{u}_b={\mathrm{Re}}_b^{0.56}{\mathrm{Pr}}_b^{0.27}{\left(\frac{c_{p.b}}{c_{p,w}}\right)}^{0.2}{ \mathrm{for} \mathrm{T}}_{\mathrm{b}}/{\mathrm{T}}_{\mathrm{pc}}>1\\ N{u}_b={\mathrm{Re}}_b^{0.35}{\mathrm{Pr}}_b^{2.0}{\left(\frac{c_{p,b}}{c_{p,w}}\right)}^{-3}{ \mathrm{for} \mathrm{T}}_{\mathrm{b}}/{\mathrm{T}}_{\mathrm{pc}}\le 1\end{array}$		±16.58%
Liu et al. [34]	$N{u}_w=0.01{\mathrm{Re}}_w^{0.9}{\mathrm{Pr}}_w^{0.5}{\left(\frac{{\rho }_w}{{\rho }_b}\right)}^{0.906}{\left(\frac{c_{p,w}}{c_{p,w}}\right)}^{-0.585}$	–	±15%
Ding and Li [35]	$N{u}_b=0.028332{\mathrm{Re}}^{0.837992}{\mathrm{Pr}}^{0.078006}$	1.2 × 105 ≤ Re ≤ 4.97 × 105, 1.80 ≤ Pr ≤ 13.16	±15%
Ma et al. [36]	$\frac{N{u}_{\mathrm{b}}}{N{u}_{\mathrm{Fc}}}=2.61-86.965{\left(\frac{Gr}{R{e}^{2.7}}\right)}^{0.458}$	8 × 104 < Re < 4.9 × 105, 11 < Pr < 130	±30%
Baik et al. [10]	$N{u}_{CO2}=0.8405{\mathrm{Re}}^{0.5704}$	15,000 < Re < 85,000	±7%
Chu et al. [42]	$\frac{N{u}_{\mathrm{b}}}{N{u}_{\mathrm{Fc}}}=\left\{\begin{array}{l}0.58-53{\left(\frac{Gr}{R{e}^{2.7}}\right)}^{0.36},T_w\approx T_{pc}\\ 0.36-22{\left(\frac{Gr}{R{e}^{2.7}}\right)}^{0.42},T_w>T_{pc}\end{array}\right.$	30,000 < Re < 60,000, Tw ≈ Tpc 30,000 < Re < 70,000, Tw > Tpc	±5.2%
Wang et al. [46]	$N{u}_b=0.022986{\mathrm{Re}}_b{}^{0.85665}{\mathrm{Pr}}_b{}^{0.26322}{\left(\frac{{\rho }_b}{{\rho }_w}\right)}^{0.04988}{\left(\frac{{\overline{C}}_p}{C_{pw}}\right)}^{-0.2174}$	8.0091 < P < 9.0026 MPa, 870.0 < Re(a/R)0.5 < 5281, 4.20 < q < 24.3 kW/m2	±15%
Zhang et al. [37]	$N{u}_f=0.000567{\mathrm{Re}}_f^{1.23}{\mathrm{Pr}}_f^{0.83}{\left(\frac{{\rho }_f}{{\rho }_w}\right)}^{0.86}{\left(\frac{{\overline{C}}_{p_f}}{C_{p_f}}\right)}^{0.76}{\left(\frac{G{r}_g}{R{e}^2}\right)}^{0.16}\left[1+{\left(\frac{d}{l}\right)}^{\frac{2}{3}}\right]$	–	±20%
Liu et al. [48]	$Nu=0.02464{\mathrm{Re}}^{0.8275}{\mathrm{Pr}}^{0.1572}{\left(\frac{{\rho }_b}{{\rho }_w}\right)}^{0.0337}{\left(\frac{{\overline{c}}_p}{c_{pw}}\right)}^{-0.0522}{\left(1+3.54d/D\right)}^{1.459}$	7.5 < P < 9 MPa, 1169 < Re(d/D)0.5 < 21,410, 9 < q < 39.9 kW/m2	±20%
Ren et al. [84]	$N{u}_{FC}=0.01882R{e}_b^{0.82}{\overline{Pr}}^{0.5}{\left(\frac{{\rho }_b}{{\rho }_w}\right)}^{-0.3}{\left(\frac{{\mu }_b}{{\mu }_w}\right)}^{0.2887}$ $N{u}_b=\left[1+9.5{\left(\frac{Gr}{R{e}^2}\right)}^{0.958}\right]N{u}_{FC}$	under forced convection conditions: 1.1 × 104 < Reb < 7 × 104, 0.95 < Prb < 48 under mixed convection conditions: 4700 < Re < 7 × 104, 300 K < Tb <370 K, 0.95 < Pr < 49, 4 × 10−5 < Gr/Re2 < 0.07	±15%
Wang et al. [79,93]	$N{u}_f=1.2838\times \frac{\left(f_f/8\right)\left({\mathrm{Re}}_b-1000\right){\mathrm{Pr}}_f}{1.07+12.7\sqrt{f_f/8}\left({\mathrm{Pr}}_f^{2/3}-1.0\right)}{\left(\frac{{\rho }_w}{{\rho }_b}\right)}^{-0.1458}$	25 ≤ Tb ≤ 65 °C, 243.6 ≤ G ≤ 800 kg/m2 s, 5 < q < 36 kW/m2, 7.7 × 104 ≤ Re ≤ 6.3 × 105, 8 ≤ P ≤ 10 MPa, 1.2 ≤ Pr ≤ 13.4, 3.1 × 10−4 ≤ Ri ≤ 0.331	±15%
Zhu et al. [52]	$\begin{array}{l}N{u}_f=a\frac{\left(f_f/8\right)\left({\mathrm{Re}}_b-1000\right){\mathrm{Pr}}_f}{1.07+12.7\sqrt{f_f/8}\left({\mathrm{Pr}}_f^{2/3}-1.0\right)}{\left(\frac{{\mathrm{Pr}}_f}{{\mathrm{Pr}}_w}\right)}^b{\left(\frac{{\rho }_f}{{\rho }_w}\right)}^c{\left(\frac{c_{pf}}{\overline{c_p}}\right)}^d{\left(\frac{D_{out}}{p}\right)}^e\\ a=1.85,\hspace{1em}b=1.5128,\hspace{1em}c=-0.7115,\hspace{1em}d=-2.2726,\hspace{1em}e=0.447,for\hspace{1em}{T}_f/T_{pc}\le 1\\ a=3.7,\hspace{1em}b=0.0479,\hspace{1em}c=0.9577,\hspace{1em}d=-0.2729,\hspace{1em}e=0.447,for\hspace{1em}{T}_f/T_{pc}>1\end{array}$	4000 < Re < 42,000, 2 < Pr < 11	±25%
Liu et al. [44]	$N{u}_b=0.1229{\mathrm{Re}}_b{}^{0.6021}{\mathrm{Pr}}_b{}^{0.3}{\left(\frac{C_{p,w}}{C_{p,b}}\right)}^{0.1310}$	3600 < Re < 36,500, 7.53 ≤ P ≤ 11.97 MPa, 26.4 ≤ Tb ≤ 121.27 °C, 0.9 ≤ Pr ≤ 10.6	±30%
Ren et al. [87]	$N{u}_z=6.3943R{e}^{0.4611}P{r}^{0.4759}{\left(\frac{{\rho }_{\mathrm{w}}}{{\rho }_{\mathrm{b}}}\right)}^{-1.027}{\left(\frac{{\mu }_{\mathrm{w}}}{{\mu }_{\mathrm{b}}}\right)}^{0.3428}{\left(\frac{\overline{c_p}}{c_{p, \mathrm{b}}}\right)}^{0.7601}{\left(\frac{L_{\mathrm{p}}}{D_{\mathrm{h}}}\right)}^{-0.2121}{\left(\frac{\beta \cdot \pi }{180}\right)}^{0.4735}$	2.1 × 104 < Re < 4.8 × 104, 0.9 < Pr < 12, 25° < β < 40°, 6 < Lp < 12 mm, 130.4< ρ < 461.1 kg/m3, 2.022 × 10−5 < μ < 3.276 × 10−5 Pa.s, 1.296 < cp < 29.89 kJ/(kg. K)	±20%
Wahl et al. [39]	$\begin{array}{l}{\mathrm{Nu}}_w={\mathrm{a}}_0^{*}{\mathrm{Re}}_w^{{\mathrm{a}}_1*}{\mathrm{Pr}}_w^{{\mathrm{a}}_2*}{\left(\frac{{\rho }_{\mathrm{b}}}{{\rho }_{\mathrm{w}}}\right)}^{{\mathrm{a}}_3}*{\left(\frac{\overline{cp}}{c{p}_b}\right)}^{{\mathrm{a}}_4}*{\left(\frac{{\lambda }_{\mathrm{b}}}{{\lambda }_{\mathrm{w}}}\right)}^{{\mathrm{a}}_5}*{\left(\frac{{\eta }_{\mathrm{b}}}{{\eta }_{\mathrm{w}}}\right)}^{{\mathrm{a}}_6}\\ a_1=0.771,a_2=0.455,a_3=1.450,a_4=-0.026,a_5=1.604,a_6=-2.623,for\hspace{1em}{T}_w/T_{pc}\ge 1\\ a_1=0.971,a_2=0.388,a_3=1.279,a_4=0.450,a_5=2.158,a_6=-2.923,for\hspace{1em}{T}_w/T_{pc}<1\end{array}$	–	±20%
Yang et al. [89]	$N{u}_f=2.606{\mathrm{Re}}_f^{1.035}{\mathrm{Pr}}_f^{0.954}{\left(\frac{{\rho }_w}{{\rho }_f}\right)}^{-1.678}{\left(\frac{\overline{c_{pf}}}{c_{pf}}\right)}^{0.977}{\delta }^{-0.007}$	δ = a/R = 0.04–0.15, P = 8 MPa, Re = 13,789–80,478, Pr = 1.15–13.17, ρw/ρf = 1.10–1.96	–

. It is worth noting that these correlations are developed under specific experimental data and the error is basically within ±30%. As shown in Figure 16, the correlations show significant bias due to changes in the study parameter ranges. In addition, the early correlations were developed based on the existing forced convection correlations or conventional fluid correlations and did not consider the thermo–physical properties of SCO₂. Recently developed correlations account for changes in the thermo–physical properties of SCO₂, making the results more reliable. Some correlations have begun to take into account buoyancy effects. However, up to now, due to the lack of in–depth research on the heat transfer mechanism of SCO₂ cooling flow, no co–operative relationship can have good prediction performance in the whole range of SCO₂ cooling conditions. It is necessary to conduct further experiments and simulation studies near the regions with the most dramatic changes in thermal properties. The new correlation must consider all the factors that affect the SCO₂ cooling heat transfer and be suitable for all application conditions.

6. Conclusions

SCO₂ has been widely used in different situations. The flow and heat transfer process’ experimental and numerical simulation results under cooling conditions are summarized. The main conclusions of this review are summarized as follows:

(1)

The effects of parameters such as mass flow, pressure, pipe diameter, and buoyancy on flow heat transfer under different channel types have been studied in experiments—the heat transfer coefficient increases as the flow rate increases under cooling conditions. The thermo–physical properties of SCO₂ change drastically near the pseudo–critical point and, the closer to the critical point, the larger the peak value of physical property change. The heat transfer coefficient reaches a more significant peak value in the pseudo–critical region when the operating pressure is close to P_c. Nu decreases with the decrease in tube diameter. The pressure drop exhibits a trend consistent with the heat transfer coefficient.

(2)

In terms of numerical research, most simulation works were based on the commercial software FLUENT or CFX. Detailed velocity, temperature, and turbulence distribution information under different channel types were obtained, and unique phenomena, such as secondary flow and changes in buoyancy along the flow process, were analyzed. However, the RANS model cannot give reliable results quantitatively, and the performance of the same RANS model under different operating conditions varies greatly. Therefore, it is not easy to achieve model generality. Although DNS can only be carried out at low Re at present, it can study the unsteady flow characteristics of SCO₂ turbulent flow with strong buoyancy in the tubes and create a database for establishing new turbulence models.

(3)

A large number of heat transfer correlations have been established. These correlations are fairly predictable within their corresponding parameter ranges but, so far, there is no general correlation that can be used for the entire SCO₂ cooling operating range. Establishing a general correlation requires a clearer understanding of the SCO₂ cooling process.

7. Future Scopes

Many achievements have been made in the cooling heat transfer behavior of SCO₂, contributing to the recent progress. Therefore, several innovative insights can serve as directions for further research in the future:

(1)

There is an urgent need to conduct more detailed mechanism experimental studies to understand the specific causes of irregular heat transfer.

(2)

To improve the simulation results’ accuracy, more abundant numerical information should be obtained. In addition to improving existing RANS models, advanced simulation methods can also be considered under SCO₂ cooling conditions, such as direct simulation Monte Carlo (DSMC), lattice Boltzmann method (LBM), large eddy simulation (LES), and molecular dynamics methods (MDM).

(3)

The structural design and optimization of gas coolers/condensers should be carried out in strict accordance with the heat transfer mechanism, and the state of heat transfer enhancement should be used as much as possible to avoid the form of deterioration. More advanced heat transfer enhancement technology should be used in practical engineering to improve heat transfer efficiency.

(4)

Since the working conditions are constantly changing in the actual process, it is necessary to study the dynamic heat transfer characteristics of the heat exchanger and the thermodynamic characteristics relative to the entire system. Furthermore, not only does the individual heat exchanger need to be optimized, but the whole system needs to be given special attention. Due to the high computational cost and time required for CFD simulations, system optimization studies can be considered using simplified models of gas coolers or condensers (e.g., machine learning techniques) and other components. The coupled model can be used to improve the overall performance of the optimized system.

(5)

After further study of heat transfer and flow mechanisms, all factors that affect heat transfer should be considered in developing empirical correlations and the design of SCO₂ heat exchangers. A correlation that can cover all SCO₂ cooling conditions and show the local heat transfer performance needs to be developed. Accurate heat transfer correlations can improve the accuracy of heat exchanger design, thereby increasing the system’s efficiency.