Diameter effects on heat transfer enhancement mechanism to supercritical pressure of methane in horizontal tubes

ABSTRACT

The flow and heat transfer characteristics of methane under supercritical pressure are crucial for storing and transporting liquefied natural gas. This study employs a renormalization group k-ε model with enhanced wall functions to investigate methane’s mixed convective heat transfer in pipes with different diameters, revealing two heat transfer enhancement characteristics. When the channel diameter changes, the mechanisms influencing heat transfer performance differ, resulting in peaks of varying nature in the heat transfer curve. The enhanced heat transfer mechanisms are explained in detail by comparing the effects of buoyancy and specific heat capacity at various temperatures. Further analysis reveals that increasing heat flux leads to the superposition of buoyancy and specific heat effects, producing a single peak in the heat transfer coefficient. The relative variations of the Nusselt number and synergy angle explain the phenomenon of heat transfer enhancement superposition. A new criterion for enhanced heat transfer due to buoyancy is proposed, Bo* = 4.21 × 10⁻⁸, which can be used to evaluate the impact of buoyancy on the enhanced heat transfer.

Keywords:
Supercritical fluid; Methane; Heat transfer enhanced; Diameter effect; Synergy principle

1. INTRODUCTION

As an efficient and clean fossil energy, natural gas (NG) is widely used in fuel, power generation, transportation, aerospace, and other fields [¹]. Methane, the main component of natural gas, has different physical properties from fluids in a normal state at supercritical pressure. Near the pseudo-critical region, its specific heat, density, thermal conductivity, and viscosity parameters change rapidly, resulting in complex heat transfer characteristics. In recent years, the flow heat transfer experiment and numerical simulation with water and CO₂ as a working medium have been extensively studied [², ³]. The flow and heat transfer of methane under supercritical pressure has also been carried out and some achievements have been made.

Heat transfer of supercritical pressure fluid is affected by operating pressure, physical properties, boundary conditions, and flow direction [⁴,⁵,⁶,⁷], which leads to its complex heat transfer characteristics. Most of the studies on the flow heat transfer of supercritical pressure fluids focus on water [⁸,⁹,¹⁰,¹¹], CO₂ [¹²,¹³,¹⁴,¹⁵,¹⁶,¹⁷], different types of coolant [¹⁸,¹⁹,²⁰,²¹,²²], and some mixed working medium [²³,²⁴,²⁵]. KHOSHVAGHT-ALIABADI et al. [¹²] studied the heat transfer process of CO₂ in a conical horizontal pipe equipped with twisted inserts of non-uniform structures. They discussed the effects of gravity, operating conditions, and pipe shape on heat transfer. The conclusion is that the buoyancy effect in small-diameter channels is weak, resulting in less enhanced heat transfer performance. However, introducing twisted inserts in gradually expanding channels can improve heat transfer effectively. SHI et al. [¹³] concluded that dual wells can alleviate the increase in reservoir pressure. Compared to the minimum well spacing, the maximum well spacing can increase the reservoir capacity by 2.41 × 10⁹ kg. SHAO et al. [¹⁴] selected coal-fired and natural gas combined cycle power plants as representatives and proposed a system model for integration with CCS. The results showed that CCS technology reduced CO₂ emissions, leading to a significant increase in LCOE. Therefore, it is essential to promote technological advancements that reduce the specific heat consumption of CO₂ capture. However, few studies have used methane as a working medium [²⁶]. SHOKRI [²⁷] studied the methane’s flow in the rectangular tube, obtained the relationship between pseudo-critical temperature and methane density with pressure, and deduced a new Nusselt number relationship equation under the Numerical condition. LI et al. [²⁸] studied methane flow in a vertical tube and explored the influence of heat flux, mass flux, and flow direction on heat transfer characteristics. They concluded that the increase in heat flux reduced the heat transfer coefficient (HTC), and high mass flux led to the phenomenon of double peaks in wall temperature. When methane flowed upward and downward, the heat transfer deteriorated and enhanced respectively, the discriminant equation of critical heat flux (q) was also proposed. These conclusions provide a new reference for a study of heat transfer of supercritical methane fluid in tubes.

PIORO and DUFFEY [²⁹] cited experimental data from 440 publications and classified the heat transfer characteristics of supercritical fluids into three categories, namely normal heat transfer (NHT), heat transfer deterioration (HTD), and heat transfer enhancement (HTE). KURGANOV et al. [³⁰] summarized the specific characteristics of three heat transfer modes. Normal heat transfer generally occurred at low heat flux (or low q/G) and high mass flow rate, and deteriorating heat transfer generally occurred at high heat flux (or high q/G) and low mass flow rate, the typical phenomenon was the wall temperature occurred a peak. SHITSMAN [³¹] pointed out that the reason for the deterioration was that high temperature in the near wall area could not be timely reduced under high heat flux, resulting in ineffective heat exchange and a significant decrease in the heat transfer coefficient (HTC). For the heat transfer enhancement phenomenon, a typical characteristic is the wall temperature occurs in a valley. There are relatively few studies on heat transfer enhancement. The tube wall is in a high-temperature state due to continuous heating, which causes the heat transfer to show a trend of deterioration, and the heat transfer enhancement phenomenon is difficult to appear under high heat flux. The enhancement phenomenon still exists in the flow process. For example, under the conditions of low mass flow rate, low heat flux, or small tube diameter, the deterioration degree of the fluid is reduced and the HTC is increased. When the turbulence intensity increases, the mixing between fluids in different areas is promoted, the heat transfer is enhanced, and the high temperature of the wall surface is effectively reduced [³²]. The heat transfer situation is significantly improved at this time, and the HTE phenomenon will occur. In addition, when the physical properties of the fluid are changed, the phenomenon of heat transfer enhancement will also occur, which is also one of the significant of studying fluid heat transfer under supercritical pressure. ACKERMAN [³³] observed a phenomenon under supercritical pressure (similar to subcritical nuclear boiling under subcritical pressure), compared with normal forced convection, where boiling greatly improved heat transfer and the heat transfer coefficient increased rapidly. LEE and HALLER [³⁴] also observed obvious heat transfer enhancement of supercritical water when the bulk temperature was close to the pseudo-critical region.

Reviewing the literature in recent years, the enhancing phenomenon has been mentioned, but no in-depth research has been done, and no consistent conclusions on the mechanism of enhancing. It is found that the enhancement phenomenon is mostly related to the buoyancy effect. ADEBIYI and HALL [³⁵] conducted a supercritical carbon dioxide experiment and found that buoyancy promoted heat transfer. When the buoyancy effect was considered, the heat transfer at the bottom of the tube was enhanced while the heat transfer at the top deteriorated, because the buoyancy changed the flow state of the fluid and the turbulence effect was strengthened. WEN et al. [³⁶] found that the turbulent kinetic energy in the bottom wall region was stronger than that in the top region. In addition, most of the physical models are smooth circular tubes, with a few spiral tubes and finned tubes. In circular tubes, the researches on vertical tubes occupy the majority, there are few on horizontal flow [³⁷]. Compared with vertical tubes, the buoyancy effect in horizontal tubes can be observed more obviously. The effect of buoyancy on flow and heat transfer is different in vertical and horizontal tubes. In vertical tubes, flow direction, and gravity direction are vertical, it can be divided into two situations: in the same direction and the opposite direction. In horizontal tubes, the flow direction is perpendicular to the direction of gravity, resulting in different effects of buoyancy on flow, the result may be enhanced heat transfer, deteriorated heat transfer, or normal heat transfer. LIAO and ZHAO [³⁸] experimentally studied the flow of CO₂ in a vertical tube, pointing out that the Nu number increases with the increase of tube diameter, and the buoyant force has an enhancing effect on up flowing. SHIRALKAR and GRIFFITH [³⁹] conducted an experimental study on CO₂ in a vertical tube diameter of 3.175−6.35 mm and observed that the wall temperature peak gradually increased with the tube diameter, the curve gradually became steeper, heat transfer enhancement phenomenon was significant. The above conclusions are based on the study in vertical tubes, while in practical heat exchanger equipment, the heat exchange tube is mostly horizontal.

The influences of heat flux, mass flow, flow direction, q/G, buoyancy, and other factors on heat transfer have been fully explored in experiments and simulations. By contrast, mature and consistent conclusions have not been drawn on the diameter of heat transfer. KURGANOV et al. [³⁰] discussed the specific characteristics of three typical heat transfer channels and pointed out that future supercritical fluid research should focus on the effect of geometric parameters on the heat transfer characteristics. GAO and BAI [⁴⁰] studied the influence of buoyancy on the heat transfer characteristics of water and pointed out that the diameter may also have an important influence on the heat transfer characteristics, more attention should be paid to the diameter. Therefore, the research literature on the influence of tube diameter on heat transfer urgently needs to be considered. YILDIZ and GROENEVELD [⁴¹] summarized the experimental data of carbon dioxide, water, R-22, and R-23 in 3.18−38.1 mm diameter. Most of the data showed that the heat transfer coefficient (HTC) decreased with the increase in diameter, and the heat transfer deterioration phenomenon was more likely to occur in large diameters. However, it’s not as if a large diameter means a lower HTC. SONG et al. [⁴²] studied the vertical upward flow of carbon dioxide in two tubes with different diameters of 4.4 mm and 9.0 mm, there appeared a phenomenon of the heat transfer coefficient of big diameter tube is higher than small diameter tube when bulk temperature closed to pseudo-critical temperature. KIM et al. [⁴³] adopted the same equipment and experimental conditions as Song, and an additional 8 mm and 10 mm diameter were tested, and the same HTE phenomenon was observed. BAE et al. [⁴⁴] found in the DHT region (deteriorating heat transfer region) that the HTC of large diameter was slightly higher than that of small diameter when a certain volume enthalpy value was reached. CHENG et al. [⁴⁵] simulated the heat transfer characteristics of hydrocarbon fuels in five diameters. They pointed out that the effect of buoyancy in large diameters was significantly stronger than that in small diameters, and the heat transfer enhancement was also significant in small diameters, then according to JACKSON and HALL’s [⁴⁶] buoyancy criterion, the condition for heat transfer enhancement was Bo*≥1 × 10⁻⁸. Other research on diameters has also made some theoretical progress, GUO et al. [⁴⁷] compared the flow resistance of circulating channels with different diameters and found that larger diameters had greater pressure drop. The aspect ratio had no significant effect on the heat transfer characteristics [⁴⁸].

Based on the aforementioned literature, current research on supercritical methane mainly focuses on vertical pipes, horizontal rectangular pipes, or pipes with a single diameter. The reasons and mechanisms of enhanced heat transfer in horizontal microchannels with different diameters have not been summarized. Especially near the inlet, heat transfer phenomena do not always deteriorate, and enhanced heat transfer can also occur during the deterioration process. The flow and heat transfer characteristics of supercritical methane in pipes of different diameters are crucial for the storage and transportation of LNG. However, research on this issue is still rare. This paper summarizes the mechanism of enhanced heat transfer in horizontal circular pipes with diameters ranging from 2 mm to 10 mm based on the physical properties of supercritical methane and the buoyancy effect and proposes a new criterion for determining the influence of buoyancy during enhanced heat transfer. This work will help improve the heat transfer conditions of supercritical fluids, prevent heat transfer deterioration, and thus enhance the economic performance and safety of LNG heat exchange equipment.

2. NUMERICAL MODEL

2.1. Physical model

Due to methane’s different flow and heat transfer parameters in the axial and radial directions, it is necessary to establish a three-dimensional cylindrical tube model to solve the corresponding problems. Based on previous simulations and experimental results, and ensuring sufficient methane flow (pipe characteristic length l/d > 60). Therefore, the length will increase in the same proportions as the inner diameter increases. The inner diameter is d = 2–10 mm, wall thickness δ = 1 mm, length l = 0.333−1.666 m, with methane flowing in the positive z-direction and gravity in the negative x-direction.

In microscale tube diameter (d < 2 mm), the heat transfer coefficient has a significantly higher effect than that of large diameter. Still, the small diameter cannot produce enough buoyance to affect the change of fluid flow state, so the enhancing phenomenon often cannot occur in the local scope. Without considering the research field of flow heat transfer in micro-circular tubes, the conventional diameter range of the circular tube model in many experimental studies [⁴¹,⁴²,⁴³,⁴⁴,⁴⁵] is 2 mm−38.1 mm. The flow and heat transfer processes in five different inner diameters were studied with 2 mm as the starting diameter and 2 mm as the gradient diameter. The physical model and dimension parameters of the horizontal tube are shown in Figure 1 and Table 1.

Figure 1
Physical model.

Table 1
Inner diameter and length in the physical model.

NO.	INNER DIAMETER (mm)	LENGTH (m)	L / D-	REFERENCE
1	2	0.333	166.666	ZHU et al. [23]
2	4	0.666		HE et al. [20]
3	6	0.999		LI et al. [28]
4	8	1.333		KIM et al. [43]
5	10	1.666		ZHANG et al. [15]

2.2. Governing equations

The vector form of the three-dimensional steady-state control equation includes the conservation of mass, momentum, and energy [²⁸], as shown below.

Continuity equation:

(1)$$\frac{\partial }{\partial xi}\left(\rho u_i\right)=0$$

Momentum equation:

(2)$$\frac{\partial }{\partial x_j}\left(\rho u_i{u}_j\right)=\frac{\partial }{\partial x_i}+\text{}\rho g_i+\frac{\partial }{\partial x_j}\left[(\mu +{\mu }_t)\left(\frac{\partial u_i}{\partial x_i}\text{}+\frac{\partial u_j}{\partial x_i}\right)-\frac{2}{3}(\mu +{\mu }_t)\frac{\partial u_k}{\partial x_k}\right]$$

Energy equation:

(3)$$\frac{\partial }{\partial x_i}\left(\rho u_i{c}_{p}T\right)=+\frac{\partial }{\partial x_j}\left(\lambda \frac{\partial T}{\partial x_i}\right)+S_T$$

Turbulent kinetic energy (k)-equation:

(4)$$\frac{\partial }{\partial x_i}\left(\rho u_{i}k\right)=+\frac{\partial }{\partial x_i}\left((\eta \text{}+\frac{{\eta }_t}{{\sigma }_k})\frac{{\partial }_k}{\partial x_i}\right)+C_k+C_b-\rho \epsilon$$

Dissipation rate (ɛ)-equation:

(5)$$\frac{\partial }{\partial x_i}\left(\rho u_i\epsilon \right)=\frac{\partial }{\partial x_i}\left((\eta \text{}\text{}+\frac{{\eta }_t}{{\sigma }_k})\frac{{\partial }_{\epsilon }}{\partial x_i}\right)+C_{1\epsilon }\frac{\epsilon }{k}(G_k\text{}+C_{3\epsilon }{G}_b)+C_{2\epsilon }\rho \frac{{\epsilon }^2}{k}$$

Where μ_t is the turbulent viscosity and S_T is the viscous dissipation; C_k and C_b are the turbulent kinetic energy generated by the velocity and buoyancy forces respectively; σ_k and σ_ε are the turbulent kinetic energy Prandtl number and the turbulent dissipation Prandtl number.

The continuity equation, momentum equation, and energy equation are solved by the finite volume method, and the turbulence model is solved by a second-order discrete scheme.

The heat transfer coefficient and the Nusselt number, which represents the convective heat transfer intensity [²¹, ²⁷], are calculated as follows.

(6)$$h=q_w/\left(T_w-T_f\right)$$

(7)$$Nu=\left(h·d\right)/\lambda$$

Where q_w is the constant heat flux applied to the wall, kW/m²; C_k is the wall temperature, which is divided into top wall temperature T_w,t and bottom wall temperature T_w,b, K; C_f is the temperature of methane fluid in the mainstream region, K; d is inner diameter, mm; λ is the thermal conductivity, W/m⸱K; h is the heat transfer coefficient, W/(m²·K); Nu is dimensionless quantity representing the magnitude of convective heat transfer intensity.

2.3. Physical properties and boundary condition

Since the pressure difference between the inlet and outlet of methane flow is very small and almost neglected, the flow is regarded as a constant pressure flow. Thermal-physical properties of methane at 7 MPa were obtained by REFPROP, density (ρ), viscosity (μ), specific heat (c_p), and thermal conductivity (λ) with temperature are shown in Figure 2. Near the pseudo-critical region, specific heat is most significantly affected by temperature, followed by density, thermal conductivity, and viscosity, and the pseudo-critical temperature T_pc where c_p peak occurs is 205.2 K.

Figure 2
Thermo-physical properties of methane at 7 MPa.

Different working conditions on the boundary conditions of the adaptability of different [⁴⁹], methane flow under high temperature and pressure conditions, especially near the wall where the velocity gradient is large, should guarantee under complex conditions of higher calculation accuracy. The boundary conditions are set as follows, inlet mass flow G = 300 kg/(m²·s), outlet was pressure outlet, constant heat flux q_w = 100 kW/m² was adopted on the wall, which increased to 150 kW/m² in high heat flux conditions, methane inlet temperature T_in = 190 K. SIMPLE algorithm was adopted for coupling pressure and velocity, second-order upwind discrete scheme was used for energy and continuity equation, first-order upwind discrete scheme was used for turbulent kinetic energy and turbulent dissipation equation. The residual was 10⁻⁶, calculation is regarded as converged when the monitored methane outlet temperature is unchanged.

2.4. Meshing and mesh independence analysis

To accurately capture the velocity and temperature parameters of methane in the boundary layer, it should be ensured that there are enough dense and regular grid nodes [⁵⁰]. The structured grid of the horizontal tube is divided as shown in Figure 3, encryption is adopted on the near-wall surface. Table 2 lists the number of mesh 1-mesh 3 nodes, the temperature values, and deviations between mesh 1-mesh 3 and mesh 4. Under the condition of ensuring calculation accuracy, mesh 2 with a small number of meshes was selected for calculation. Figure 4 shows the wall temperature calculation results of mesh 1-mesh 4.

Figure 3
Section grid of the tube.

Table 2
The verification of mesh independence.

MESH	NODES	T/K	y +	DEVIATION RELATIVE TO MESH 4 /%
1	2072800	220.8601	0.954	0.1902
2	2186400	220.5074	0.928	0.03026
3	2353000	220.4676	0.902	0.01665
4	3317600	220.4407	0.867	0.00

Figure 4
Comparison of wall temperature with different node numbers.

The dimensionless parameter y⁺ represents the distance from the first layer grid to the wall surface. To accurately solve the near-wall region dominated by viscosity, y⁺ of the first-layer grid at the wall surface should be guaranteed to be less than 1 [⁵¹, ⁵²]. Use the CFD-Online mesh calculation tool to set the height of the first mesh layer such that the y⁺ value near the wall is less than 1. The details are as follows:

2.5. Turbulence model verification

The selection of a turbulence model is very important for supercritical fluid flow. It is necessary to ensure continuous and accurate results can be obtained under complex conditions when simulating the methane flow in different tubes. RNG and SST models have better prediction accuracy for flow in horizontal tubes [⁵³, ⁵⁴]. In this study, Standard, realizable, RNG, omega, and transition turbulence models were tested respectively.

A simulation was carried out based on DU [⁵⁵] supercritical pressure CH₄ experimental condition to verify the accuracy and reliability of turbulence model calculation. Figure 5 shows the comparison curves of the simulated results of different turbulence models and the experimental wall temperature. Due to the differences in pressure, working medium, and boundary conditions, the simulated and experimental values are different to some extent, but the variation rules of the two models are generally consistent. Among many turbulence models, the RNG k-ε turbulence model has a high agreement with the experimental, and the maximum deviation between the HTC and the experimental is 1.08%, which proves that the RNG k-ε turbulence model has a high reliability.

Figure 5
Comparison between simulated and experimental results from DU [⁵⁵].

3. RESULTS AND DISCUSSION

3.1. Heat transfer enhancement characteristics and mechanism at low flux

This section discussed methane’s heat transfer enhancement characteristics in diameters from 2 mm to 10 mm under low heat flux. Sections 3.1.1 and 3.1.2 described the effect of diameters on the heat transfer characteristics and the mechanism of heat transfer enhancement, respectively.

3.1.1. Diameter effects of heat transfer enhanced characteristic at low flux

Wall temperature and heat transfer coefficient distribution with z/d under the inner diameter (ID) from 2 mm to 10 mm at 7 MPa and G = 300 kg/(m²⸱s), q_w = 100 kW/m² are shown in Figure 6 and Figure 7 respectively.

Figure 6
Wall temperature distribution of different tube diameters tube at q_w = 100 kW/m², G = 300 kg/(m²·). (a) Top wall temperature distribution (b) Bottom wall temperature distribution.

Figure 7
Wall heat transfer coefficient distribution of different tube diameters tube at q_w = 100 kW/m², G = 300 kg/(m²·s). (a) Top wall heat transfer coefficient distribution (b) Bottom wall heat transfer coefficient distribution.

In smaller IDs of 2 mm and 4 mm, the top wall temperature T_w,t increases greatly with the ID, and the curve of T_w,t rises continuously in the z/d direction. When the ID increases to 6 mm, the T_w,t appears as a valley, it is more obvious at 8 mm and 10 mm ID. Compared with T_w,t, the valley phenomenon in bottom wall temperature T_w,b is more significant. It appears earlier than T_w,t in Figure 6 (b). It can be concluded from (a) and (b) in Figure 6 that the valley of T_w,t, and T_w,b gradually move towards the inlet of methane flow with the increase of ID. However, there is no valley in both top and bottom wall temperatures at a smaller diameter of 2 mm.

As shown in Figure 7 (a) and (b), the heat transfer coefficient of the top and bottom walls are T_w,t and T_w,b. Combined with Figure 6, it can be found that the valley point of T_w,t at z/d = 40.7 (ID = 6 mm), z/d = 47.5 (ID = 8 mm), and z/d =6 1.8(ID = 10 mm) corresponds to the peak point of. The valley point of T_w,b at z/d = 42.5 (ID = 4 mm), z/d = 41.3 (ID = 6 mm), z/d = 40.2 (ID = 8 mm) and z/d = 39.9 (ID = 10 mm) corresponds to the peak point of h_w,b. The variation of the h_w,t, and h_w,b peak point with the ID is consistent with the trend of valley point of wall temperature. In addition, there is a difference between the h_w,t,and h_w,b, the h_w,b appears at the second peak point, and it moves to the direction of z/d with the increase of ID. However, there is no corresponding valley in the T_w,b in Figure 6 at the same z/d.

When the wall temperature is above pseudo-critical temperature T_pc, the heat transfer will deteriorate. However, the appearance of the wall temperature valley indicates that the heat transfer of methane is enhanced. Figure 6 and Figure 7 can be summarized, the increase in diameter causes the decrease of wall temperature at a certain position, which leads to the occurrence of enhancing heat transfer at this position, and the enhancing phenomenon becomes obvious with the increase of diameter. This is in agreement with the phenomena observed by other scholars [⁴¹,⁴³]. In addition, when the diameter increases, the heat transfer coefficient will be lower. Figure 7 illustrates the heat transfer enhancing phenomenon will not disappear due to the lower heat transfer coefficient.

Figures 8 and 9 show the temperature (T_w,t, T_w,b, mainstream temperature T_f) and heat transfer coefficient (h_w,t, h_w,b) of 2 mm and 6mm ID, respectively. According to the variation in the wall temperature, the heat transfer is divided into stages a to e, where a is the entrance stage.

Figure 8
Heat transfer characteristics of 6 mm inner diameter tube at q_w = 100 kW/m², G = 300 kg/(m²·s). (a) Wall temperature distribution (b) Wall heat transfer coefficient distribution.

Figure 9
Heat transfer characteristics of 2 mm inner diameter tube at qw = 100 kW/m², G = 300 kg/(m²·s). (a) Wall temperature distribution (b) Wall heat transfer coefficient distribution.

As shown in Figure 8, with the constant heat flux on the tube wall, wall temperature under 6 mm ID rises continuously. In stage a (0 ≤ z/d ≤ 18), as the temperature rises in the mainstream area lags behind the wall temperature, the temperature difference ∆T between the wall and mainstream increases gradually, and the heat transfer coefficient decreases continuously in this stage. In stage b (0 ≤ z/d ≤ 18), T_f begins to rise, T_w,b decreases and the first valley point (h_w,t valley point 1) appears. Temperature difference ΔT decreases, h_w,b increases, and the peak point (h_w,b peak 1) appears at z/d = 45, the first HTE occurred. The valley of T_w,t is generated at stage c, and the h_w,t peak point appears at z/d = 64. After stage c, methane fluid in the mainstream begins to enter the pseudo-critical region, T_f rises further, and the second valley point of T_w,b appears in stage d. The second HTE has occurred. Finally, in stage e, methane gradually moves away from the pseudo-critical region, Physical properties of the fluid in the mainstream region decrease and tend to be gentle, ΔT increases, and the enhancing effect is gradually weakened. As shown in Figure 9, the reduction of HTC in stage a of 2 mm ID is the same principle as that of 6 mm ID, both of which are from the principle of lagged temperature rise in the mainstream area. Methane fluid in the mainstream area enters directly the pseudo-critical region from stage b, ΔT decreases first and then increases from a to b. ΔT_min is at z/d = 72, where the T_w,b reaches the peak. As the T_w,b rises rapidly, ΔT_min does not appear, and h_w,t does not appear at the peak. Subsequently, the mainstream fluid in stage c is moved away from the pseudo-critical region, the HTC decreases continuously.

Compared with a large diameter of 6 mm, the buoyancy effect is significant in a large diameter, enhancing of buoyancy effect in 2 mm ID on heat transfer is almost negligible. The HTC in 2 mm ID does not appear as a peak until the main flow enters the pseudo-critical region (z/d = 72). Therefore, heat transfer enhancement characteristics of methane fluid in small and large IDs are not the same. When the wall temperature is higher than the T_pc and the mainstream temperature is lower than the T_pc (methane is in a liquid-like fluid), HTE occurs in the larger ID tubes, but not in smaller ID tubes. In addition, when the methane fluid in the mainstream region entered the pseudo-critical region, physical property variation led to the HTE in both large and small tube diameters.

3.1.2. Mechanism of diameter effects on heat transfer enhanced at low flux

This section describes the mechanism of heat transfer enhancement by the cross-section of temperature and velocity of the first heat transfer enhancement and the physical properties of the second enhancement.

In Figure 10 and Figure 11, (a) to (e) are respectively the temperature and velocity fields of the first HTE section under 2–10 mm ID. From 2 mm to 10mm tube, the first HTE sections are z/d = 60, z/d = 65, z/d = 46, z/d = 42 and z/d = 39, respectively. In a 2 mm tube, the temperature distribution of the enhancing section is symmetrical in the vertical direction, when ID is increased to 4 mm, the temperature field of the symmetry is broken. The reason is the density difference between mainstream and near the wall caused by the methane fluid temperature difference, under the effect of buoyancy lift, leading to mainstream methane fluid moving down, high temperature of methane fluid relative to moving upwards, the temperature of the cross-section presents “M”-shape distribution. “M”-shape temperature distribution becomes more obvious with the increase of ID, the asymmetry reaches the highest at 10mm ID. Under the buoyancy force, mainstream cryogenic methane fluid moves down the wall, the heat transfer rate increases rapidly when the high and low-temperature fluid meets at the bottom wall, and heat transfer is improved. With the increase of ID, high-temperature fluid occupies more and more space in the top wall area, while the T_w,b remains almost constant due to the HTE, it can be observed from Figure 6 that there is no significant difference in T_w,b under different inner diameters. It can be observed from Figure 11, that the increase of methane flow velocity in the bottom wall promotes heat transfer, the flow velocity of the top wall decreases, which deteriorates heat transfer, and the velocity difference between the top and bottom increases continuously with the ID. In addition, the first HTE occurs in the like-liquid region, buoyancy still has a certain enhancing effect on the top wall, and the HTE peak point of the top wall can be observed in Figure 7 (a).

Figure 10
Temperature distribution of HTE cross sections of different inner diameter tubes. (a) d = 2 mm (b) d = 4 mm (c) d = 6 mm (d) d = 8 mm (e) d = 10mm.

Figure 11
Velocity distribution of HTE cross sections of different inner diameter tubes. (a) d = 2 mm (b) d = 4 mm (c) d = 6 mm (d) d = 8 mm (e) d = 10 mm.

Taking 8 mm diameter as an example, Figure 12 shows the second HTE point (z/d = 95) and the c_p of the front and rear sections, where r/R = −1 is the bottom, r/R = 1 is the top, and r/R = 0 is mainstream area. In the z/d = 60, the c_p peak point appears near the wall, leading to the high-temperature fluid being concentrated in the bottom, it is difficult to transfer heat with the fluid in the mainstream area, so there was no significant HTE. In z/d = 75, the fluid in the mainstream area enters the pseudo-critical region, and the c_p peak point shifts towards the direction of r/R increasing. The fluid temperature near the mainstream area rises, the temperature difference ΔT decreases, HTC begins to increase and the heat transfer is enhanced. Enhanced continued until z/d = 93, and the second heat transfer enhancement point appeared. It can be observed from the corresponding temperature distribution in Figure 12 (b) that the temperature of the mainstream area keeps rising, the volume occupied by the low-temperature fluid keeps decreasing, and ΔT is further reduced. In the z/d = 95, the HTE caused by c_p change is the most significant. After that, at z/d = 112.5 and 130, the c_p peak continues to move to the mainstream area, and the peak value keeps decreasing, leading to the bottom wall being again in a high-temperature environment, HTC begins to decline, and the enhancing effect is weakened. Therefore, the second HTE is caused by the regular change of the c_p of the fluid in the mainstream area crossing the pseudo-critical region.

Figure 12
The c_p distribution of the second HTE point section. (a) Distribution of the second HTE point section (b) The c_p distribution of the near bottom wall.

3.2. Heat transfer enhancement characteristics and synergy principle at high flux

This section discusses methane’s enhanced heat transfer characteristics under p = 7 MPa, q_w = 150 kW/m², l/d = 166.666, and G = 300 kg/(m²·s). Sections 3.2.1 and 3.2.2 respectively described the effect of diameters on the heat transfer characteristics and the synergy principle of heat transfer enhancement.

3.2.1. Diameter effects of heat transfer enhancement characteristic at high flux

When q_w increases, the HTC of all IDs decreases (Figure13). The HTE peak point of 2 mm ID moves down and further towards the inlet. When ID increases to 4mm, the HTE peak point is prominent due to the combined action of the buoyancy effect and c_p. When the ID is 6 mm, the first and second HTE peak points decrease and move towards the inlet. The above analysis shows that the enhancing phenomenon of ID on heat transfer still exists at high q_w, and due to the increase in q_w, the second HTE occurs earlier. When ID increases to 8 mm and 10 mm, the specific heat-enhancing peak point coincides with the buoyancy-enhancing peak point, two enhancing effects are superimposed to form the HTE peak Point 3. The double peak of HTC turns into a high single peak after superimposing, and the superposition peak moves towards the inlet with the increase of ID. However, this is not a phenomenon in the case of low q_w. In addition, since the distance between the first and the second enhancing point is relatively close to the 4 mm tube, the superposition of enhancing is easy to occur when q_w increases. It is worth noting that in the larger ID of 8 mm and 10 mm, the HTC of z/d = 55 to 86 (ID = 8 mm tube) and z/d = 48 to 78 (ID = 10 mm tube) is much higher than that at the low q_w. It can be seen that superposition enhancement is crucial to the improvement of heat transfer.

Figure 13
Comparison of heat transfer coefficient with different diameters between q = 100 kW/m² and q = 150 kW/m². (a) d = 2 mm (b) d = 4 mm (c) d = 6 mm (d) d = 8 mm (e) d = 10 mm.

Table 3 presents the total pressure drop in the pipeline at heat flux of 100 and 150 kW/m². Due to radial shear stress, the total pressure drop increases with higher heat flux. At the same heat flux, the total pressure drop for a 2 mm diameter pipe is more than twice that of a 10 mm diameter pipe. The variation in total pressure drop is primarily influenced by the pipe diameter rather than by the heat flux density, as smaller channels increase the radial velocity gradient. However, when observing the entire pipeline, the heat transfer coefficient is only minimally affected by the pressure drop, so it is not considered.

Table 3
The variation of pressure drop under different q_w.

DIAMETER (mm)	qw(kW/m2)	P (Pa)	qw (kW/m2)	P (Pa)
2	100	627.4	150	834.9
4	100	468.2	150	641.7
6	100	364.8	150	491.5
8	100	298.2	150	414.8
10	100	253.1	150	361.4

3.2.2. Synergy principle in heat transfer enhanced at high flux

GUO et al. [⁵⁶] proposed the field synergy principle (FSP) according to the coordination between the velocity field and the temperature field and pointed out that the higher the synergy number S_t between the velocity field and temperature gradient is, the more flow can promote heat transfer. TAO et al. [⁵⁷] further found that large diameters can improve coordination. TAO et al. [⁵⁸] proposed that field synergy Angle (or included angle) β be used as the discriminant index of coordination. This section provides the mechanism explanation for the enhancing superposition phenomenon in the diameters of 4 mm, 8 mm, and 10 mm based on the field coordination principle. The brief derivation equation involving the FSP is as follows:

(8)$$N{u}_x={\mathrm{Re}}_{x}Pr{\int }_0^l(\overrightarrow{U}\cdot \nabla \overrightarrow{\mathrm{T}})dy$$

(9)$$\overrightarrow{U}\cdot \nabla \overrightarrow{T}=\left|\overrightarrow{U}\right|\cdot \left|\nabla \overrightarrow{T}\right|\cdot \cos \beta$$

(10)$$S_t=\left|\overrightarrow{U}\right|\cdot \left|\nabla \overrightarrow{T}\right|\cdot \cos \beta =\frac{N{u}_x}{R{e}_{x}Pr}$$

Where $\overrightarrow{U}$ is velocity vector; $\nabla \overrightarrow{T}$ is temperature gradient; Pr is dimensionless parameters; Re is related to flowing; β is volume expansivity, (1/K).

According to the above equation, the change of Pr before the pseudo-critical region can be ignored, when G and ID are determined, the Re is determined accordingly. Therefore, the increase of Nu requires the integrand $\overrightarrow{U}\nabla \overrightarrow{T}$ to reach the maximum. Further, it can be determined that when β reaches the minimum, the coordination between the velocity vector and temperature gradient is the highest, and the heat transfer will be enhanced. The field synergy angle β in the circular cross-section is shown in Figure 14, blue arrows represent the velocity vector, red dotted line is the isotherm. When the included angle β between the velocity vector and the direction of the temperature gradient decreases, the coordination will be improved, and heat transfer will be enhanced. Figure 15 shows the included Angle on the section at the superimposed enhancing point of 4 mm, 8 mm, and 10 mm ID. With the increase of ID, the eddy phenomenon and “M”-shaped temperature distribution are gradually obvious, the β decreases continuously under the synergistic effect of the velocity field and temperature field. Figure 16 shows a more accurate β degree and Nu number. The β of 4 mm, 8 mm, and 10 mm ID is 89.79°, 89.77° and 89.76°, respectively. The β is smaller and the Nu number is higher under large ID, indicating that the velocity and temperature have higher synergy and enhancing effect on heat transfer under large ID tubes. In 8 mm and 10 mm diameters, the β reaches the highest at z/d = 76.7 and z/d = 77.1, respectively, and the enhancing phenomenon is the most obvious.

Figure 14
Synergy angle between velocity field and temperature gradient in circle tube. (a) Velocity field contributes nothing to heat transfer (b) Velocity field contributes to heat transfer.

Figure 15
Distribution of velocity field and temperature gradient in large inner diameters tube at q_w = 150 kW/m², G = 300 kg/(m²⸱s). (a) d = 4 mm (b) d = 8 mm (c) d = 10 mm.

Figure 16
Distribution of β and Nu number in large inner diameter tubes at qw = 150 kW/m², G = 300 kg/(m²·s).

The above analysis results show that, under high heat flux, the increase of ID promotes the coordination between the velocity and temperature, and improves the HTC. The superposition of HTE makes the HTC much higher than that of the small ID with the same z/d. At a certain position, the HTC of enhancement of superposition is 1.15 times that of adjacent smaller ID (HTC of 10 mm ID superimposed enhancing point is compared with that of 8 mm ID at low q_w). On the other hand, from the point of view of heat flux, the increase of heat flux will cause the second enhancement to occur in advance, if the interval of the first and second HTE is shorter, the high heat flux can also cause a superposition of HTE. Comprehensively, the superposition is caused by the combination of heat flux and pipe diameter. The field synergy principle represents the coupling degree between the influence of pipe diameter on the flow field and the influence of heat flow on the temperature field.

3.2.3. The effect of flow acceleration on heat transfer

According to K_v proposed by MC ELIGOT et al. [⁵⁹], it is used to characterize the effect of flow acceleration on heat transfer characteristics:

(11)$$K_v=\frac{4q_w{d}_{in}{\beta }_b}{R{e}_b^2\text{}{\mu }_b{c}_p}$$

GU et al. [⁶⁰] discussed the flow acceleration effect of methane in horizontal channels, stating that when K_v is less than 10⁻⁶, the effect of flow acceleration on heat transfer can be neglected.

Under the conditions of P = 7 MPa, q_w = 100~150 kW/m², and G = 300 kg/(m²·s), the axial K_v distribution for different pipe diameters is plotted. From the graph (Figure 17), it can be seen that K_v decreases overall along the flow direction for different pipe diameters, and K_v is smaller for higher mass flow rates. K_v is much smaller than the critical threshold, so the effect of flow acceleration on heat transfer can be neglected in this paper.

Figure 17
Distribution of K_v of different inner diameters. (a) high flux (b) low flux.

3.3. Buoyancy effect on heat transfer characteristics with different diameters

To further study the effect of buoyancy in different diameters on heat transfer characteristics, this section adopts the Bo* equation proposed by JACKSON and HALL [⁴⁶] to represent the buoyancy force in different pipe diameters.

(12)$$Bo*=Gr*/R{e}^{\mathrm{3.425}}\cdot P{r}^{0.8}$$

(13)$$Gr*=\frac{\beta g{d}^{4}qw}{{\lambda }_b{v}_b^2}$$

(14)$$Re=\frac{Gd}{\mu }$$

(15)$$Pr=\frac{{\mu }_b{c}_p}{{\lambda }_b}$$

Where Gr* is dimensionless parameters, μ is velocity, (Pa·s), and v is the kinematic viscosity, (m²/s).

Figure 18 and Figure 19 respectively show the results of buoyancy of five IDs under gravity and no gravity conditions. Ignoring Bo* sudden drop caused by the entrance effect, the z/d > 20 is analyzed. When gravity is not considered, Bo* decreases monotonically for all IDs, and no significant difference. HTC of different IDs decreases continuously without obvious peak and cross phenomenon. For 2 mm ID, the results are almost the same with and without gravity. When gravity is considered, the Bo* of 4 mm ID no longer decreases monotonically but decreases monotonically after a slight peak. Then the peak becomes obvious with the increase of ID, and the trend of the peak point is the same as that of the HTE point. The peak point of Bo* reaches a maximum of 6.94 × 10⁻⁸ under 10 mm ID. The HTC peak becomes obvious with the increase of ID and moves towards the entrance. In some locations, the HTC of large IDs is even higher than that of small IDs. In addition, it should be explained that since the heat transfer occurs after the flow state changes, the corresponding peak of HTC will lag behind the peak of. In other words, the heat transfer can only be carried out between the mainstream flow and near-wall fluid after the eddy current is generated, so the peak of HTC will lag behind the peak of Bo*. Such as in the 4 mm diameter, the peak of Bo* appears at z/d = 42.7, while the corresponding peak of HTC appears at z/d = 60.4. However, with the increase of ID, the hysteresis phenomenon is gradually weakened.

Figure 18
istribution of Bo* of different inner diameters when considering gravity. (a) Bo* (b) heat transfer coefficient distribution.

Figure 19
Distribution of Bo* of different inner diameters when not considering gravity. (a) Bo* (b) heat transfer coefficient distribution.

The above analysis shows that buoyancy is the cause of heat transfer enhancement. In small-diameter tubes, buoyancy has no significant effect on heat transfer, when the diameter increases, buoyancy affects heat transfer. According to the calculation results of Bo* of different diameters, when ≥ 4.21 × 10⁻⁸ is considered, the buoyancy will significantly affect the heat transfer characteristics and enhance the heat transfer. Without considering gravity, Bo* is less than 4.21 × 10⁻⁸ at all diameters and keeps decreasing. Therefore, the existence of buoyancy is the dominant factor for heat transfer enhancement. With the increase of diameter, the critical value of buoyancy on heat transfer enhancement caused by the increase of diameter is 4.21 × 10⁻⁸.

4. CONCLUSIONS

A numerical study was conducted on the flow and heat transfer characteristics of methane within the inner diameter range of 2 to 10 mm under supercritical pressure, and the heat transfer characteristics of methane within different diameter ranges were obtained. The enhancement mechanisms of heat transfer due to diameter were analyzed, and the effects of high heat flux density and buoyancy on the enhanced heat transfer characteristics were studied. The following conclusions were drawn:

• (1)

  T_w and T_f are located on either side of T_pc, causing the occurrence of the first HTE phenomenon. As the pipe diameter increases, the effect of buoyancy causes the “M”-shaped temperature distribution to become more pronounced, with the HTE peak at the inlet gradually shifting toward the entrance and not disappearing.

• (2)

  The appearance of the second HTE peak is due to the fluid temperature in the main flow region being higher than T_pc, and the drastic changes in the properties enhance the heat transfer performance of the channel. In small diameter channels, due to the higher flow velocity, the buoyancy effect is weakened, and heat transfer enhancement is primarily influenced by the variation in c_p. In contrast, in large-diameter channels, heat transfer enhancement is caused by both the buoyancy effect and the variation in c_p.

• (3)

  The methane in the main flow region reaches the near-critical temperature in advance under high heat flux density, causing the second HTE peak to appear earlier. In large-diameter channels, the first and second HTE peaks overlap, forming a high-enhanced peak. According to FPS, as the inner diameter increases, the field synchronization angle decreases, while Nu increases. This indicates that, under high heat flux density, the coordination between the velocity field and the temperature gradient improves as the inner diameter increases.

• (4)

  Buoyancy effect, as the main factor in heat transfer enhancement for large diameter pipes, has a threshold value of Bo* = 4.21 × 10⁻⁸.When Bo* exceeds the threshold value, the buoyancy effect strengthens during the heat transfer enhancement process, causing the HTC to reach a peak. In the absence of gravity, all channels have Bo* values below the threshold, and buoyancy does not affect heat transfer.

5. ACKNOWLEDGMENTS

This work was supported by the National Key Research and Development Program (2023YFB3211005) and the Liaoning Provincial Department of Science and Technology (2023JH1/10400016).

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Publication Dates

• Publication in this collection
  03 Mar 2025
• Date of issue
  2025

History

• Received
  05 Dec 2024
• Accepted
  20 Jan 2025