Due to the involvement of phase change in different cooling mediums, using a three-dimensional model for calculations would consume a significant amount of computational resource and time. Therefore, to quickly obtain the flow and heat transfer characteristics of the two-phase fluid within the loop, a two-dimensional model is adopted for transient calculations. Given the significant temperature gradient observed within the pipe, it is crucial to account for both heat and mass transfer in these regions. The entire phase change process is primarily computed using the Volume of Fluid (VOF) multiphase flow model, which could capture the distinct water/air interface in reservoirs and track the generation, flow, and evolution of bubbles in the circuit. Besides, the self-programmed code serving as auxiliary tools to enable mass and energy exchange between liquid and gaseous coolants during phase transitions. The equations used in the fluid domain are detailed as follows:

Conservation of mass [29]: (1)∂αq∂t+v→∇αq=Sαqρq∑q=1nαq=1where the ρq represents fluid density, αq represents volume traction.

Mass source term: (2)S={−r⋅αlρlT−TsTs,T≥Tsr⋅αgρgTs−TTs,T<Tswhere the T represents temperature, the r represents the local stress tensor caused by fluid viscosity.

Momentum conservation [29]: (3)∂∂t(ρU→)+∇⋅(ρU→U→)=−∇p+∇⋅τ¯¯+∂∂xj(−ρui′uj′¯)where the U represents the velocity, the τ¯¯ represents the local stress tensor caused by fluid viscosity.

SST k-ω turbulence model [30] is used to solve the Reynolds stress term: ∂∂t(ρk)+∂∂xj(ρkuj)=∂∂xj[(μ+μtσk)∂k∂xj]+Gk+Gb−ρε−YM+Sk (4)∂∂t(ρε)+∂∂xj(ρεuj)=∂∂xj[(μ+μtσε)∂ε∂xj]+ρC1Sε−ρC2ε2k+νε+C1εεkC3εGb+Sεthe specific definitions of each term in the Standard k-ω turbulence model are found in ANSYS Fluent documents [31].

Energy conservation [29]: (5)∂∂t(ρE)+∂∂xi[ui(ρE+p)]=∂∂xj[k∂T∂xj+ui(τij)eff]+Shwhere the E represents the internal energy of the fluid, and k represents the heat conductivity of the fluid.

The equation used in the solid domain is mainly the energy equation as follows [32]: (6)∇⋅(ks∇T)=∂∂t(ρE)

The NCS is driven by the density variation of fluid within different sections of the loop. Therefore, the changes in the physical properties of gas-liquid two-phase fluid, especially liquid fluid, cannot be ignored. Throughout the cooling process, the physical properties of liquid fluid do not vary significantly with pressure changes. Thus, the influence of temperature is mainly considered here. Within the temperature variation range of liquid fluid, enough temperature points are selected, and the density ρ, dynamic viscosity μ, thermal conductivity k, and specific heat capacity C_p of the liquid fluid at each temperature are calculated using piecewise linear fitting. The data source of physical properties for four mediums are obtained from the NIST, and their properties are shown in Tables 1–4.

Prior to heating and startup, the system maintains a static condition, featuring a cooling mediums level of h = 13 mm in the tank and an initial system temperature of T_sur. Given that the tank is directly exposed to the atmospheric environment, the upper surface of the tank is designated as a pressure outlet boundary with an outlet pressure of P_out = 1 atm. Upon heating and startup (t > 0 s), a constant heat flux (Q₁, Q₂) is employed for the external wall surface of the U-shaped piping.

The study mainly focuses on numerical simulation, hence, the accuracy of the mathematical model and numerical calculation method need to be validated using experiment. Experimental methods and data from previous studies [33] were used for validation, and the experimental device is shown in the Fig. 4a. In the experiment, the model was heated using a heating plate, and the heating plate was split into two fully symmetrical sections, each individually linked to the control unit to guarantee independent heating capacity and temperature regulation for both plates. The transient temperature changes of the pipe wall were monitored and recorded using an infrared thermal imager. A 5.0 W continuous laser and a high-speed camera were employed to record flow of cooling mediums. With the operating condition of T_sur = 284 K and Q₁ = Q₂ = 10 kW/m², the comparison of the trend of pipe wall temperature and the flow states at selected moments between experimental and numerical results is shown in Fig. 4. It can be observed that under the same operating condition, the calculated numerical results are in good agreement with the experimental results in terms of flow state change and temperature change, with a relative deviation of less than 3%. Therefore, it is concluded that the mathematical model and calculation method adopted in this chapter are effective.

To investigate the change in the flow state of different cooling mediums within the loop under constant heat flux condition on the pipe wall, transient numerical simulations are conducted under operating condition of T_sur = 300 K and Q₁ = Q₂ = 8 kW/m². Based on previous research findings, the evolution of the flow state of liquid water within the loop under uniform heat flux condition could be divided into three stages: expansion stage, localized circulation stage, and phase change stage. Consistent with the flow state of water, when the heating duration is brief (t < 1 s), resulting in a minimal increase in wall temperature (ΔT < 2 K), the fluid inside the pipe absorbs heat from the wall, causing a slight reduction in its density. During this initial phase, the fluid expands outwards along the pipeline. Notably, the fluid velocity close to the pipe wall is notably higher than that at the center. Specifically, across the pipe’s cross-section in the radial direction, fluid velocity increases from the boundary layer and subsequently decreases towards the pipe’s center. This velocity gradient arises due to the rapid temperature rise in the fluid adjacent to the pipe wall during the short heating period. Different cooling media exhibit varying temperature rise rates, which influence the duration of the expansion phase.

Fig. 5 displays the pressure, temperature, velocity, and phase distribution of four cooling mediums at t = 1 s. Among them, the R113 has the highest pressure within the pipe, exceeding 6000 Pa in the horizontal segment pipe, followed by the water, while the ethanol has the lowest pressure within the pipe, about 5000 Pa. This is related to the fluid’s density and heat absorption capacity. During the initial heating stage, the temperature rise of the fluid is small, resulting in insignificant changes in its density. Therefore, the greater density of the fluid, the higher the pressure observed in the horizontal segment of the pipeline. At this point, the liquid water is still in an expansion state. The three additional cooling media, ethanol, fuel oil, and R113 demonstrate increasing axial and radial temperature differences inside the pipe as a result of their rapid temperature increase and significant density variations with rising temperature. Gravity and buoyancy working together create a localized circulatory flow inside the vertical pipe, where fluid descends close to the center and ascends near the wall. This circulation initiates in the pipe’s midsection and gradually spreads towards both the orifice and the elbow over time. As a result, the turbulence intensity in the fluid flow surrounding the orifice is increased. The specific heat capacity of R113 is less than half that of water and thus experiences the fastest temperature rise, combined with low fluid viscosity, resulting in the highest local circulation speed. The fuel oil does not form a circulation in the elbow and the bottom horizontal pipe due to its greater flow resistance, for its viscosity is 5–8 times higher than other cooling mediums at low temperatures. Additionally, the elbow increases the turbulence level of the fluid within the pipe, resulting in irregular flow. During this stage, the flow velocities of the four cooling mediums are very slow, ranging from 1 × 10⁻⁴ to 1 × 10⁻⁶ m/s, with the maximum velocity occurring near the vertical pipe orifice. In summary, due to differences in temperature rise rates and viscosities among different cooling mediums, the time cycles for the evolution of their flow states exhibit slight differences.

As the temperature of the solid wall further increases, Fig. 6 depicts the velocity streamline diagrams and the flow law of four cooling mediums within the pipes at t = 10 s, respectively. Fig. 7 displays the heat flux distribution on the pipe wall at the same time. At t = 10 s, the pressure distributions of the four cooling mediums undergo minimal changes, and the temperature increase is also insignificant. At this point, it can be observed that liquid water has entered a localized circulation state. Liquid water has high thermal conductivity, and its flow speed is accelerated with the occurrence of localized circulation, so it exhibits strong heat exchange capability with the wall. Consequently, the wall heat transfer rate reaches up to 6100 W/m². Ethanol flows rapidly at the pipe orifice and remains in a localized circulation state. This is because as the fluid temperature within the pipe rises, the fluid exits the orifice and exchanges with fluid in the reservoir. However, within the vertical pipe, the flow velocity is lower, and most of the fluid flows in the opposite direction, resulting in an overall wall heat flux that is only 1/3 of that of liquid water. The fuel oil experiences a rapid temperature rise because of the smaller specific heat capacity. Due to its high viscosity and significant flow resistance, the buoyancy force generated is insufficient to overcome the shear force in the boundary layer. Therefore, it exhibits irregular flow within the pipe. Furthermore, the fuel oil has low thermal conductivity and low overall flow velocity, with the overall flow velocity below 0.001 m/s except at the elbow. Consequently, the heat exchange capacity between the fuel oil and the solid wall is relatively weak, showing a wall heat transfer rate of 2800 W/m². The R113 has rapid temperature increase and low viscosity, experiencing low flow resistance and showing an overall flow velocity less than 0.0025 m/s. Moreover, the thermal conductivity of the R113 is only about 1/10 of that of liquid water. Therefore, there exists localized circulation with greater turbulence intensity within the pipe, enhancing its heat exchange capacity. The overall wall heat transfer rate reaches 4300 W/m². In summary, during the initial heating stage, the viscosity and thermal conductivity of the fluid affect the duration of localized circulation, which in turn increases the flow velocity and enhances the heat exchange between the cooling medium and the solid wall.

Fig. 8 presents the pressure, temperature, velocity, and phase distribution of four fluids within pipes at t = 60 s. As the fluid temperature further increases, the pressure of all four cooling mediums decreases, with the maximum drop reaching 1000 Pa. R113 experiences the fastest temperature rise and reaches the highest temperature, approximately 320 K. Liquid water, due to its high specific heat capacity, experiences the slowest temperature rise and remains the lowest in temperature. Additionally, Fig. 8b displays the temperature distribution, all four cooling mediums exhibit lower temperatures in part of the fluid at the elbow, which caused by higher turbulence dissipation at these locations. During this stage, due to its slow temperature rise, liquid water continues to flow in localized circulation within the pipe, and its flow velocity is relatively fast due to the presence of temperature difference. R113 has a relatively low phase transition temperature and a rapid temperature rise and enters the initial phase transition stage, generating tiny bubbles. This further reduces the fluid’s density and viscosity, resulting in the fastest flow velocity, with a center velocity exceeding 0.01 m/s. As the bubbles rise, the fluid flows uniformly towards the pipe orifice. Ethanol, the specific heat is half that of water, and the thermal conductivity is more than twice that of water, resulting in fast temperature rise and small temperature difference. Hence, the density difference formed by the temperature difference is not enough to drive the local circulation. Additionally, the fluid density further decreases upon heating because of its lower pressure within the pipe. At the moment depicted in Fig. 8c, the inflow of the cooling medium from the reservoir results in the fluid within the vertical pipe flowing in the opposite direction. In the next moment, as the fluid within the pipe expands, the overall flow within the vertical pipe shifts towards the pipe orifice, exhibiting irregular overall flow. The fuel oil has high viscosity, and thus remains in an extremely irregular flow state, with high turbulence intensity and relatively fast flow velocity at the elbow pipe. As the heating time progresses, the velocities of all four cooling mediums increase, and the maximum velocity reaches 1 × 10⁻² m/s. In summary, as temperature rises, the thermophysical properties of cooling mediums have a significant impact on the changes in fluid flow states.

Due to the differences in phase change temperatures and temperature rise rates among the four cooling mediums, the time required for each to reach its phase change state is different. The latent heat of phase change and the specific time points of phase change are shown in Table 5. Once the four liquid cooling media reach their respective phase change temperatures, small bubbles initially emerge in the center of the vertical pipe, slowly expanding and coalescing as time passes. These bubbles then swiftly ascend and exit through the pipe orifice due to buoyancy. Fig. 9 illustrates the pressure, temperature, velocity, and phase distribution of the fluids within the pipe 30 s after each of the four cooling media undergoes phase change. The presence of an elbow in the pipe results in complex flow patterns and high turbulence intensity within the horizontal pipe segment. Consequently, despite the upper wall of the horizontal pipe experiencing the quickest temperature increase, bubbles first become apparent in the middle of the vertical pipe because of the relatively lower turbulence levels present. Furthermore, during the liquid-to-gas phase transition, the fluid undergoes rapid expansion, leading to a significant increase in flow velocity, which can exceed 10⁻¹ m/s. Observing the velocity vector diagrams at different locations within the pipes of the four cooling mediums reveals that the velocity changes are relatively drastic at the locations where bubbles form, indicating that the fluid motion intensifies due to the phase change of the liquid. The fuel oil has a higher phase change temperature and undergoes the most intense phase change, resulting in a significant reduction in viscosity, and the local fluid velocity of fuel oil reaches as high as 1 m/s. Ethanol has lower density and viscosity, and hence has the second-fastest flow rate after fuel oil, which local fluid velocity reaches to 0.4 m/s. R113 has lowest phase change temperature and highest fluid density, leading to the slowest flow rate, about 0.2 m/s. In summary, the intensity of the phase change is closely related to the fluid’s phase change temperature. The higher the phase change temperature, the greater the flow velocity and the more intense the heat transfer. Moreover, in the two-phase state, the viscosity of the liquid fluid has a minor impact on the flow velocity.

To investigate the heat transfer between the fluid inside the pipe and the solid wall at this point, Fig. 10 presents the distribution of fluid-solid heat transfer and volume fraction of liquid cooling mediums in the right side of vertical pipe. When the liquid cooling medium volume fraction ranges from 0 to 1, the liquid cooling medium undergoes a phase change during this period, releasing substantial latent heat of phase change and enhancing heat exchange with the solid wall. Hence, the maximum wall heat transfer rate for water, ethanol, fuel oil and R113 can reach approximately 11, 12.5, 30, and 10 kW/m², respectively. The magnitude of wall heat transfer is related to the fluid’s temperature, velocity, and latent heat of phase change. Fuel oil has the highest phase change temperature and the fastest flow velocity. Although the phase change latent heat of fuel oil is only 374 kJ/kg, its wall heat transfer rate is still the highest. Ethanol has the second-fastest flow velocity and the second-highest heat transfer rate about 7500 W/m², but due to the significantly lower phase change temperature, its wall heat transfer rate is far behind fuel oil. This indicates that flow velocity is one of the main factors influencing heat transfer rate. R113 has the slowest flow velocity and extremely low latent heat of phase change, resulting in the lowest wall heat transfer rate. Although the latent heat of phase change of liquid water is 10 times that of fuel oil, its wall heat transfer rate is not prominent among the four cooling mediums due to differences in phase change temperature and flow velocity. When the liquid cooling medium volume fraction drops to 0 and the wall is completely occupied by the vapor region, the fluid-solid heat transfer rate drops sharply. At this point, the external heat absorbed by the solid wall (Q) will be much greater than the heat absorbed and carried away by the internal cooling medium from the solid wall. This results in a rapid increase in the local wall temperature, leading to localized overheating. Moreover, due to the pipe orifice’s connection to the reservoir, the fluid temperature, enhanced by bubble expansion, is greater than the wall temperature at the orifice. Consequently, the heat transfer rate between the fluid and the solid near the outlet (−0.1 < Y < 0 m) drops drastically to a negative value, signifying a shift in heat transfer direction from the fluid to the solid. However, due to the small difference in specific heat capacity between the other three liquid cooling mediums and the quartz glass pipe wall, the temperature difference between the fluid and the solid wall during heating is not significant, and therefore no obvious heat transfer deterioration occurs at the pipe orifice. In summary, the influence of fluid velocity on the heat transfer rate between the fluid and the solid wall is greater than that of the fluid’s latent heat of phase change, and by reasonably regulating the thermophysical properties of the fluid, some instances of heat transfer deterioration can be avoided.

To address the issues of heat transfer deterioration and localized temperature rise caused by bubble blockage under uniform heat flux condition, an investigation is conducted on asymmetric heat flux condition, i.e., enhancing the heat flux on the right-side l₂ pipe wall. Transient calculations are performed under the condition of T_sur = 300 K, Q₁ = 8 kW/m², and Q₂ = 8.01 kW/m². Fig. 11 illustrates the wall heat flux on the right-side pipe wall of four cooling mediums after 10 s of heating. The figure reveals that the relative strength of heat transfer among the four cooling mediums remains unchanged. However, the difference of heat flux between the pipes on both sides causes a density difference in the fluid, which subsequently forms a pressure difference, increasing the fluid flow velocity. Consequently, the heat transfer rate of all four cooling mediums at this point is greater than that when the heat flux is uniform on both sides of the pipe, with a maximum increase of 1100 W/m². Fig. 12 depicts the velocity vector diagrams of the four cooling mediums within the vertical pipe at 30 s of heating. Due to the pressure difference between the pipes on both sides, the heat exchange between hot cooling medium within the pipe and the cold cooling medium in the reservoir is enhanced. Therefore, the temperature of cooling mediums within the pipe rises slowly. Liquid water and R113 exhibit varying degrees of localized circulation. The R113 has low viscosity and significant density change with temperature, and hence experiences violent localized circulation within the pipe, with faster flow velocity and higher wall heat transfer rate up to 3000 W/m². Ethanol has rapid temperature rise rate, low density, and significant density change with the increase of temperature, and thus the pressure difference arising from density difference can overcome its own gravity, forming a counterclockwise overall circulation. The fuel oil has fast temperature rise rate and high viscosity, and thus does not exhibit localized circulation. The fluid within pipes on both sides expands towards pipe orifice, and flow velocity in the right pipe is twice as fast as that in the left. However, due to the small difference in heat flux between the pipes on both sides, the pressure difference is insufficient to overcome the force of gravity, preventing the formation of an overall circulation within the pipe. In summary, different fluid requires different heat flux differences to drive overall circulation due to their distinct densities and magnitude of the density change with the increase of temperature.

When the fluids are heated to the phase change temperature, the distributions of fluid-solid heat transfer and liquid volume fraction on the inner wall of the right vertical pipe of the four cooling mediums are shown in Fig. 13. Due to the presence of a heat flux difference and an increase in heat flux, the times for liquid water, ethanol, fuel oil, and R113 to reach their phase change points advance to 160, 79, 235, and 39 s, respectively. Meanwhile, the existence of the heat flux difference enhances the flow velocities of all four cooling mediums, resulting in higher heat transfer rates during the phase change state compared to uniform heat flux condition. Hence, the maximum wall heat transfer rate for water, ethanol, fuel oil, and R113 can reach 12, 17, 31, and 13 kW/m², respectively. Furthermore, the phase changes of the four cooling mediums become more intense, with a relative decrease in the liquid volume fraction and a narrowing of the heat flux fluctuation range. Additionally, the heat exchange between the fluid in the reservoir and the fluid at the pipe orifice increases due to the pressure difference, mitigating the heat transfer deterioration at the pipe orifice. Therefore, even though the magnitude of the heat flux difference is insufficient to drive an overall circulation of the fluid, it can still enhance fluid-solid heat transfer and reduce the phenomenon of heat transfer deterioration.

To facilitate the directional circulation of the cooling medium solely through the pressure differential on both sides, a heat flux disparity of 100 W/m² arises between the fluids in the pipes on both sides. Throughout the heating process, the evolution of the four cooling mediums can be categorized into three phases: liquid expansion, localized flow, and comprehensive circulation. Before any phase transition occurs in the fluid, the temperature divergence of fluids in the vertical pipes on either side steadily escalates with the duration of heating. Impelled by this pressure differential, the cooling medium establishes a comprehensive counterclockwise flow within the loop, transitioning into the comprehensive circulation phase, as depicted in Fig. 14. During this phase, the temperature and density contrast, axial velocity of the fluids in the vertical pipes on both sides attain their peak values, with the axial flow velocity surging to approximately 1.1 m/s. The density contrast (Δρ) between the fluids on both sides functions as the primary impetus for this comprehensive circulation. The essential role of this driving force is to alter the fluid flow in the opposite direction (clockwise) to the overall flow direction during the localized circulation stage, namely the upward flow near the left vertical pipe wall and the downward flow at the center of the right vertical pipe. Fig. 15 illustrates the fluid flow direction transition within the right vertical pipe as an example. The longer the duration of the localized circulation stage, the greater the maximum Y-direction velocity v of the local flow opposite to the overall flow direction, which means a larger external force is required to overcome this flow direction. Furthermore, the higher the fluid density, the greater the buoyancy force required for overall circulation, and hence the larger density difference needed. Ethanol has a low density and shorter localized circulation time, which facilitates the formation of overall circulation. Fuel oil exhibits earlier overall circulation when the heat flux difference is sufficiently large due to its high viscosity and short localized circulation time. This shortens the duration of the localized circulation stage and correspondingly reduces the local reverse flow velocity v, thus requiring a smaller driving force Δρ for overall circulation. Fig. 16 shows the wall heat flux distribution of the four cooling mediums at this time. After fuel oil enters overall circulation, the flow velocity increases and the wall heat transfer rate rises to 7600 W/m². Ethanol has a long overall circulation time and high fluid velocity, and thus its wall heat transfer rate also increases to a level similar to that of fuel oil, about 7500 W/m². The wall heat transfer rate of water increases to 7000 W/m² for its high thermal conductivity and specific heat capacity. Additionally, the wall heat transfer rate of R113 significantly drops to 5150 W/m² due to the shorter duration of overall circulation. At this time, fluid flow is mainly driven by pressure difference, and viscosity has a minor impact on liquid flow velocity. It can be concluded that overall circulation significantly increases fluid flow velocity and enhances fluid-solid heat transfer in the loop.

To investigate the heat transfer efficiency of the fluid during the phase change circulation phase, additional heat is applied to the system. As the pipe wall’s temperature approaches the fluid’s boiling point, the cooling mediums undergo phase change at high-temperature regions while simultaneously flowing with the circulation, resulting in no significant bubble formation. However, thereafter, T_max does not show a significant increase. After reaching the phase change point 30 s of four cooling mediums, the heat flux on the right pipe wall surface is as illustrated in Fig. 17. Due to the high phase change temperature and long heating duration results in a large pressure difference and rapid flow rate, the wall heat flux of the fuel oil remains the highest, with an average of 15,000 W/m². Furthermore, the emergence of multiple tiny bubbles close to the exit of the right vertical pipe swiftly increases the density disparity (Δρ) of the fluids on both sides, resulting in a notable acceleration of the fluid’s overall circulation rate. The bubbles quickly flow out of the pipe orifice, preventing bubble accumulation, pipe blockage, and deterioration of wall heat transfer. Furthermore, whenever localized high temperatures appear in the loop, the fluid undergoes phase change to absorb heat, ensuring the maximum system temperature (T_max) close to the boiling point of the cooling medium. The calculation results indicate that when the heat flux difference (ΔQ) loaded on the pipe wall on both sides is sufficiently large, all four cooling mediums undergo overall circulation. Moreover, as the duration of overall circulation increases, the fluid flow rate accelerates, increasing overall heat transfer. The influence of the viscosity of the cooling medium on the flow rate and heat transfer becomes insignificant. In addition, although there are differences in heat transfer capacity among the natural circulation systems using different cooling mediums, they all possess self-adaptive cooling capabilities by autonomously adjusting their circulation states based on external heat flux differences.