This section presents the variation in the HTCs in straight tubes with inclination of 7.16° due to the effects of a finite-thickness wall under Con. III.

Figure 4 shows the axial dimentionless heat flux (q/q _av, ratio of local heat flux to the average heat flux on the inner wall) of three regions along the straight tube with wall thickness of 2 mm under Con. III. Unlike a uniformly heated wall, the heat capacity along three regions (top, side, and bottom) in the tube with a finite-thickness wall changes with non-uniform turbulent kinetic energy, which results in differences in the circumferential dimentionless heat flux. At the lowest density, the dimentionless heat flux along the top region are the minimum, followed by those on the side.

Figure 5 shows the axial heat transfer coefficients h of SCW along straight tubes under Con. III. The values of h in tubes with and without the finite-thickness wall are compared. The heat transfer coefficients at the bottom are the highest, followed by those at the sides and top. Unlike the differences in the heat flux factor along the three regions in the straight tube with wall thickness of 2 mm, the heat transfer coefficients in straight tubes appear less sensitive to the wall.

This section presents the variation in the HTCs in helical tubes with helix diameter of 105.5 mm, caused by the effects of finite-thickness wall.

Figure 6 shows the axial dimentionless heat flux, q/q _av, of four regions along the helical tube with wall thickness of 2 mm under Con. III. In the helical tube with zero-thickness wall, a uniform heat flux was imposed on the tube wall, so the effects of the wall have been ignored. While in the helical tube with wall thickness of 2 mm given volumetric heat source, the heat flux along four regions is significantly different. With the outlet enthalpy much lower than i _pc (thermophysical properties change dramatically near i _pc in Figure 1), the dimentionless heat flux of the four regions change gently along the helical tube. The values of q/q _av represent the heat transfer capacity. The average values of q/q _av at the top and bottom are the same (1.06), whereas those on the inside and outside are 0.80 and 1.09, respectively. It can be concluded that the heat transfer capacity on the outside is the best, followed by those at the top and bottom, whereas that on the inside is the worst. The relative difference in q/q _av between the top and bottom, caused by buoyancy, is lower than 1%, whereas that between the inside and bottom, caused by the centrifugal force, is 36% in Figure 6. Under a large inlet mass flux, the relative value of the centrifugal force is much higher than that of buoyancy in the helical tube.

Figure 7 shows the axial wall temperatures, T _w, along the four regions in helical tubes with helix diameter of 105.5 mm under Con. III; the values of T _w in tubes with and without finite-thickness walls are compared. In Figure 7A, the difference in the wall temperature between the inside and outside is 6.2 K in the tube without the finite-thickness wall. In Figure 7B, the wall temperature difference between the inside and outside is 4.4 K in the tube with wall thickness of 2 mm. The finite-thickness wall contributes to the circumferential heat exchange, significantly reducing the wall temperature difference between the inner and outer regions by 29%.

Figure 8 shows the axial heat transfer coefficients, h, of SCW along four regions in the helical tube with 105.5 mm helix diameter under Con. I and Con. II. When the fluid enthalpy is close to i _pc, the heat transfer capacity increases with increase in the fluid velocity. However, as shown in Figure 8, (ρ·C _p), which affects the ability to accommodate and carry heat, reaches the maximum before i _pc. Correspondingly, under Con. I and Con. II, the maximum heat transfer coefficients at the four regions appear before i _pc. In Figure 8A without the finite-thickness wall, the difference in the average heat transfer coefficient between the outer and inner regions is 6.5 kW/(m²·K) (relative difference 131%) under Con. I and 11.7 kW/(m²·K) (relative difference 142%) under Con. II. In Figure 8B with wall thickness of 2 mm, the difference in the average heat transfer coefficient between the outer and inner regions is 6.2 kW/(m²·K) (relative difference 128%) under Con. I and 11.4 kW/(m²·K) (relative difference 137%) under Con. II. The finite-thickness wall contributes to reducing the difference in h between the outer and inner regions.

According to the development of the circumferential maximum wall temperature along the helical tubes, six typical temperature regions were selected: (i) the maximum wall temperature is lower than T _pc (T _{w, max} < T _pc), (ii) the maximum wall temperature is higher but the average wall temperature is lower than T _pc (T _{w, av} < T _pc < T _{w, max}), (iii) the average wall temperature is higher but the minimum wall temperature is lower than T _pc (T _{w, min} < T _pc < T _{w, av}), (iv) the minimum wall temperature is higher but the fluid temperature is lower than T _pc (T _b < T _pc < T _{w, min}), (v) the fluid temperature reaches T _pc (T _b = T _pc), (vi) and the fluid temperature exceeds T _pc (T _pc < T _b).

Figure 9 shows the development of the dimensionless circumferential wall temperature (T _w/T _w,max) in the helical tube with 105.5 mm helix diameter under Con. I. Figure 9A depicts the case of the tube without the finite-thickness wall; Figure 9B depicts the case of the tube with wall thickness of 2 mm. The maximum wall temperature appears when θ increases from 0° to 360°, with the preceding wall temperature changing drastically before reaching T _w,max. The non-uniformities in the wall temperature shown in Figure 9 result from the combined effects of buoyancy and centrifugal forces. θ of T _w,max aids the comparison of the relative scale between the centrifugal force and buoyancy. In Figure 9A, for the temperature regions from T _{w, max} < T _pc to T _b < T _pc < T _w,min, the maximum wall temperature moves from the inside to the top (from 64° to 54°), which proves that the buoyancy effect is enhanced. For the temperature region of T _w,max = T _pc, the maximum wall temperature moves from the top to the inside (from 54° to 72°), which proves that with increase in the fluid temperature, the density decreases whereas the fluid velocity increases, resulting in increased centrifugal force. For the temperature region of T _pc < T _b, the maximum wall temperature appears at θ _{Tw, max} = 90° (inner region). The circumferential (T _w/T _w,max) differences in helical tubes with and without a finite-thickness wall are depicted in Figure 10, where the value is 37 K in Figure 9A and 13 K in Figure 9B. The finite-thickness wall contributes to circumferential heat diffusion, which homogenizes the circumferential wall temperature distribution. The intervals of θ _{Tw, max} in helical tubes with and without the finite-thickness wall are 36° in Figure 9A and 27° in Figure 9B. The finite-thickness wall reduces the interval of θ _{Tw, max}.

Figure 10 shows the development of the circumferential heat transfer coefficient h of SCW in the helical tube with 105.5 mm helix diameter under Con. I. Figure 10A depicts the results for the tube without a finite-thickness wall, and Figure 10B depicts the results for the tube with wall thickness of 2 mm. In Figure 10A, for the temperature regions from T _{w, max} < T _pc to T _b = T _pc, circumferential h _max increases from 7.7 kW/(m²·K) to 21.7 kW/(m²·K). For the temperature region of T _pc = T _b, the average circumferential h is the maximum (17.4 kW/(m²·K)), with the maximum relative difference of 216%. For the temperature region of T _pc < T _b, circumferential h drops sharply in the range of 1.9–5.9 kW/(m²·K), which deteriorates the heat transfer. The maximum circumferential h in helical tubes without and with finite-thickness walls are 14.8 kW/(m²·K) (Figure 10A) and 11.1 kW/(m²·K) (Figure 10B), respectively. The finite-thickness wall contributes to the circumferential heat diffusion, which homogenizes the circumferential h distribution.

As shown in Figure 1, the thermophysical properties of SCW change dramatically with increase in temperature. These changes induce buoyancy in the cross section and axial thermal acceleration. To quantitatively analyze the effect of buoyancy and thermal acceleration on the HTCs in the tubes, Hall and Jackson (Hall and Jackson, 1969; Hall, 1971) proposed the buoyancy factor, Bo*. McEligot et al. (1970); (Jackson et al., 1989) proposed the acceleration factor, Kv. Bo * = Gr * R e 3.425 Pr 0.8 (14) where, Gr * = g β d 4 q λ ν 2 (15) R e = G d μ (16) Pr ⁡ = μ C p λ (17) where, Gr^* is the Grashof number, Re is the Reynolds number, Pr is the Prandtl number, g denotes the acceleration due to gravity, d denotes the inner diameter, q denotes the heat flux, β denotes the isothermal compression coefficient, λ denotes the thermal conductivity, υ denotes the kinematic viscosity, G denotes the mass flux, μ denotes the dynamic viscosity, and C _p denotes the specific heat.

Jackson et al. (1989); (Koshizuka et al., 1995) proposed a theoretical model for the relationship between Bo* and heat transfer coefficients under heating. When Bo* <6 × 10⁻⁷, the buoyancy effect on the HTCs is relatively weak; when 6 × 10⁻⁷ < Bo* <1.2 × 10⁻⁶, the buoyancy degrades the heat transfer; when 1.2 × 10⁻⁶ < Bo* <8 × 10⁻⁶, the buoyancy helps to recover the heat transfer; when Bo* >6 × 10⁻⁷, the buoyancy enhances the heat transfer. Kv = 4 q w d α p R e 2 μ C p (18) where, α _p refers to volume expansivity.

McEligot et al. (1970) proposed that when Kv < 3 × 10⁻⁶, the thermal acceleration effect on the HTCs is relatively weak, when Kv ≥ 3 × 10⁻⁶, the thermal acceleration deteriorates the heat transfer.

Figure 11 shows the maximum Bo* and Kv calculated using Equation 14−(18) for the nine cases with various inlet mass flux and helix diameter. The effects of the helix diameter on the maximum Bo* and Kv are relatively weak, whereas obvious differences in the maximum Bo* and Kv are observed under various inlet mass fluxes. In Figure 11A, for the range of 1.2 × 10⁻⁶ < Bo*_max < 8 × 10⁻⁶ under Con. I and Con. II, the buoyancy affects the heat transfer and helps in recovering it. Under Con. III, for Bo* <6 × 10⁻⁷, the buoyancy effect on the HTCs is relatively weak. In Figure 11B, Kv < 3 × 10⁻⁶, which indicates that the thermal acceleration effect on the HTCs is relatively weak.

In Section 3.1.2, it can be concluded that the finite-thickness wall contributes to the circumferential heat exchange, significantly reducing the wall temperature and h difference between the inner and outer regions. This section mainly focuses on analyzing the effects of mass flux on heat transfer in the helical tubes with wall thickness of 2 mm.

Figure 12 shows the axial heat transfer coefficients, h, of SCW along four regions in the helical tube with 105.5 mm helix diameter and wall thickness of 2 mm under Con. III. Comparing Figure 12 with Figures 8B, D, the relative differences in the average heat transfer coefficient in the top-to-bottom direction are 5%, 3%, and much less than 1% under Con. I, Con. II, and Con. III, respectively, which indicates that the buoyancy effect is weakened with increasing mass flux. In the inner-to-outer direction, the relative differences in the average heat transfer coefficient were 128%, 137%, and 211% under Con. I, Con. II, and Con. III, respectively, which indicates that with increasing mass flux, the heat transfer capacity in the outer region is enhanced and the difference in the heat transfer capacity between the inner and outer regions widens.

Figure 13 shows the development of the dimensionless circumferential wall temperature (T _w/T _w,max) in the helical tube with 105.5 mm helix diameter and wall thickness of 2 mm under various inlet mass fluxes. The maximum differences in the wall temperature under Con. I, Con. II, and Con. III are 11 K, 9 K, and 4 K, respectively. With increase in the mass flux, Re increases and the heat transfer capacity is enhanced, which reduces the difference in the wall temperature. In addition, θ _{Tw, max} shifts from 70° to 90°, and the effect of buoyancy is weakened whereas that of the centrifugal force is strengthened.

Figure 14 shows the development of the circumferential dimensionless wall temperature (T _w/T _w,max) in helical tubes with various helix diameters and wall thickness of 2 mm under Con. I. Figures 14A, B depict the results for the cases of helix diameters of 70.5 mm and 140.5 mm, respectively. With θ increasing from 0° to 360°, T _{w, max} occurs between the top region (0°) and inner region (90°); the wall temperature before the maximum value varies more dramatically than that after the maximum value. Comparing Figure 14 with Figure 9B, with the helix diameter increasing from 70.5 mm to 140.5 mm, θ _{Tw, max} decreases from 79° to 70°–66° in the T _{w, max} < T_pc section, shifting from the inner region to the top region; in the section T _pc < T _b, the maximum difference in T _w/T _{w, max} decreases from 0° to 360° (0.019, 0.018, 0.017), and T _w gradually moves to the high-temperature zone.

Figure 15 shows the development of the circumferential heat flux factor q/q _av along the helical tube with wall thickness of 2 mm under Con. I. Figures 15A, B, C depict the results for helix diameters of 70.5 mm, 105.5 mm, and 140.5 mm, respectively. Under the effect of centrifugal force, the mass flux in the inner region is lower than that in the outer region, resulting in poor heat transfer capacity. In Figure 15B, for the case of T _{w, max} < T _pc to T _b < T _pc < T _{w, min}, the buoyancy drains the fluid flow close to the wall from bottom to top, which shifts θ _{Tw, max} from 70° to 54°. With increasing fluid temperature, the fluid velocity increases and density decreases, which enhances the centrifugal force, resulting in lower heat transfer capacity in the inner region. In the case of T _pc < T _b, (q/q _av)_min appears at θ _{Tw, max} = 90° (inner region). q/q _av is greatly affected by the helix diameter before the fluid temperature reaches T _pc. In the region of T _{w, max} < T _pc, the maximum difference in q/q _av for helix diameters of 70.5 mm, 105.5 mm, and 140.5 mm are 0.40 (Figure 15A), 0.34 (Figure 15B), and 0.33 (Figure 15C), respectively. With increase in the helix diameter, the difference in q/q _av decreases.