MODELING OF THE HEAT TRANSFER IN A SUPERCRITICAL CO2/DME MIXTURE FLOWING IN COOLED HELICALLY COILED TUBES

The heat transfer of supercritical CO2/DME mixtures was modeled in this study for a mass ratio of 95/5 for cooling in horizontal helically coiled tubes. The CO2/DME heat transfer coefficient was higher in the high-temperature zone than with pure CO2. The heat transfer of CO2/DME (95/5) was predicted for various mass fluxes, heat fluxes and pressures. The CO2/DME heat transfer coefficient increased with the mass flux due to the increased turbulent diffusion, and first increased but then decreased with the heat flux. The peak heat transfer coefficient of CO2/DME shifted toward the high-temperature region as the operating pressure increased. The effects of buoyancy and the centrifugal force were also analyzed to better understand the heat transfer mechanisms in helically coiled tubes. The gravitational buoyancy effect on the heat transfer decreased with mass flux while increased with heat flux. Higher heat fluxes strengthened the centrifugal buoyancy effect on the heat transfer at the beginning of the cooling process but weakened the centrifugal buoyancy effect later in the cooling process. The present study gives insight into the flow and heat transfer processes in helically coiled tubes which is useful for heat exchanger designs and refrigerant selection.


INTRODUCTION
The use of traditional chlorofluorocarbon (CFC), hydrochlorofluorocarbon (HCFC) and hydrofluorocarbon (HFC) refrigerants should be reduced because of global warming and ozone depletion (Groll, 2020). Natural working fluids such as ammonia (NH3), hydrocarbons and carbon dioxide (CO2) are attractive options for next generation refrigerants since they have zero ozone depletion potentials (ODP) and low global warming potentials (GWP). Among the natural working fluids, CO2 is a promising refrigerant due to its favourable characteristics (non-toxic, non-flammable, high volumetric cooling capacity) and eco-friendly properties (ODP=0, GWP=1) so CO2 has been used in heat pumps (Byrne et al., 2009), air conditioners and other industrial processes (Kouta et al., 2016;Milani et al., 2017;Hou et al., 2017;Leng et al., 2016).
Due to the low critical temperature of CO2 (31.1℃), CO2 heat pumps use trans-critical cycles. Thus, the heat absorption occurs in the evaporator at subcritical conditions while the heat rejection takes place in the gas cooler at supercritical condition. The physical properties of supercritical fluids vary rapidly near the pseudo-critical point (Huang et al., 2020). The heat transfer in supercritical CO2 has been investigated for various temperatures, pressures, tube diameters, heat fluxes, mass fluxes and buoyancy forces. One of the most important heat transfer characteristics of supercritical CO2 is the peak heat transfer coefficient near the pseudo-critical point which decreases with increasing pressure (Pitla et al., 2001a;Pitla et al., 2001b;Liao and Zhao, 2002a;Yoon et al., 2003). The heat transfer rate increases with decreasing tube diameter for 1-7.75 mm tubes (Oh and Son, 2010;Dang and Hihara, 2004). The significant influence of buoyancy due to density variations causes the heat transfer to deteriorate for upward cooling flows, whereas the heat transfer rate increases for downward and horizontal flow during cooling (Jiang et al., 2013;Li et al., 2010;Wang et al., 2017a). However, in trans-critical heat pumps, the gas cooler operating pressure exceeds the CO2 critical pressure (7.38 MPa) and usually exceeds 10 MPa (Ehsan et al., 2018), which requires high strength equipment and leads to large throttling losses (Kauf, 1999;Liao et al., 2000;Austin and Sumathy, 2011).
Dimethyl ether (DME) also has outstanding thermophysical properties as a refrigerant such as high thermal conductivity, large latent heat, zero ODP and low GWP. However, pure DME cannot be directly used as a refrigerant due to its flammability (Liu et al., 2018). However, mixtures of CO2 and DME (CO2/DME) use lower operating pressures than CO2 trans-critical heat pumps while also suppressing the flammability of DME (Koyama et al., 2006;Onaka et al., 2008). The condensation (Afroz et al., 2008) and evaporation (Onaka et al., 2010) heat transfers of CO2/DME mixtures flowing in horizontal smooth tubes have been experimentally studied for various mass fluxes. The results showed that the heat transfer coefficient increased with increasing DME mass fraction and that the pressure drops of CO2/DME mixtures were lower than that of pure DME (Afroz et al., 2008;Onaka et al., 2010). However, as with pure CO2, the heat rejection with CO2/DME takes place at supercritical conditions. Previous studies of the heat transfer of CO2/DME have been limited to condensation and evaporation processes for subcritical pressures. Therefore, more studies of the heat transfer characteristics of supercritical CO2/DME are needed for designing gas coolers in heat pumps.

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org 2 In addition, the secondary flow induced by the centrifugal force in helically coiled tubes leads to better heat transfer rates as seen in heat exchangers for various applications (Naphon and Wongwises, 2006;Gu et al., 2020). Many studies have investigated the heat transfer in helically coiled tubes to study the effects of the geometric and flow parameters on the heat transfer (Ghorbani et al., 2010;Piazza and Ciofalo, 2010;Chingulpitak and Wongwises, 2010;Ferng et al., 2012). The supercritical CO2 heat transfer coefficient in a helically coiled tube is higher than in a straight tube due to the coupled effects of the centrifugal and buoyancy forces (Zhang et al., 2015;Yang, 2016;Xu et al., 2016;Liu et al., 2017).
In this study, the heat transfer was modeled for cooling of a supercritical CO2/DME mixture in various horizontal helically coiled tubes. The heat transfer coefficients of a supercritical CO2/DME mixture with a 5% DME mass fraction were calculated and compared with those of pure CO2. Then, the heat transfer rates were calculated for various mass fluxes, heat fluxes and pressures. Finally, the effects of buoyancy and centrifugal forces were analyzed to provide a better understanding of the heat transfer mechanisms in a helically coiled tube.

Geometry and Grid
A three-dimensional geometry was created to model the horizontal helically coiled tube as shown in Fig. 1. The helical radius (R) was 18 mm, the tube radius (r) was 2 mm, the pitch (P) was 34 mm, and the total length of the helically coiled tube was 560 mm. The computational domain was discretized into hexahedral elements using ANSYS ICEM as shown in Fig. 2. The wall y+ was less than one to ensure the calculation accuracy near the wall. Four cases were calculated with four meshes of 315070, 447528, 630630 and 798720 cells for p=8.0 MPa, Tin=320 K, G=239.2 kg·m -2 ·s -1 and q=13.3 kW·m -2 . The number of grid cells in the helically coiled tube section is described by a, b and c, as shown in Fig. 2, and the number of the elements along the tube is denoted by x. The detailed information of the four meshes is listed in Table 1. The calculated mean fluid temperatures are shown in Fig. 3. Similar results were obtained for Cases 3 and 4. The relative error in the fluid temperature was 0.002%. The calculations in this paper then used the mesh with 630630 elements.
The momentum conservation equation is where p is the static pressure,  is the stress tensor, g   is the gravitational body force and F  is the external body forces. F  also contains other model-dependent source terms such as porous-media and user-defined sources.
The energy conservation equation is where Sh is the contribution of the viscous dissipation. The Shear-Stress Transport (SST) k-ω model has been often used to accurately reproduce experimental data for heat transfer of supercritical CO2 in helically coiled tubes in the literature Wang et al., 2017b). The transport equations for the SST k-ω model are where Γ is the effective diffusivity, G is the generation term, Y is the turbulent dissipation, and D represents the cross-diffusion term. The subscript k represents the turbulent kinetic and  denotes the turbulent frequency.
The supercritical fluid was cooled in the horizontal helically coiled tube as shown in Fig. 4. The inlet boundary condition was a constant mass flux with a fixed pressure at the outlet. The wall boundary conditions were a no-slip boundary with a constant heat flux. The steady-state simulations were performed using Fluent which uses a finite volume method to discretize the governing equations. The pressure-based solver was used. Velocity-pressure coupling used the Semi-Implicit Pressure-Linked Equation (SIMPLE) algorithm. The second-order upwind scheme was used to solve the turbulent kinetic energy equations and the QUICK scheme was used for the specific dissipation rate, momentum and energy equations. The thermophysical properties of CO2 and CO2/DME were derived from the NIST Standard Reference Database, including the specific heat, thermal conductivity, density and viscosity. Then, fitting formulae for the thermophysical properties of CO2 and CO2/DME were imported into the model through user defined functions.

Fig. 4 Schematic of the computational model and boundary conditions
The calculations assumed that the tube wall was smooth and ignored the loss due to frictional resistance. The heat flux was assumed to be uniform and constant along the tube. The thermal effect of viscous dissipation was ignored.
The model was used to simulate CO2/DME (95/5) cooling in the horizontal helically coiled tube for inlet temperatures of 300 to 345 K and pressures of 7.3 to 9.0 MPa. The heat fluxes ranged from 13.3 to 40 kW·m -2 and the mass fluxes ranged from 239 to 430 kg·m -2 ·s -1 . The inlet Re varied from 1.33×10 4 to 8.43×10 4 as calculated by Re ud where u is the fluid velocity, d is the tube diameter, G is the mass flux, ρ is the density and μ is the dynamic viscosity. The critical pressure of CO2 is 7.38 MPa and the critical pressure of DME is 5.37 MPa. The critical pressure of the CO2/DME (95/5) mixture is 7.28 MPa (Chen and Du, 2003) 2 2 2 , , , where pc is the critical pressure and w is the mass fraction.

Model Validation
Experimental results for supercritical CO2 (Wang et al., 2017b) were used to validate the numerical model. The experimental system included a low-temperature thermostat bath, a refrigerant pump, a mass flow meter, a pre-heater, a test section and a syringe pump, as shown in The experimental and calculated results for the average wall temperature and heat transfer coefficient are compared in Fig. 6. The inlet temperature ranged from 300 K to 330 K with a pressure of 8.0 MPa, mass flux of 239.2 kg·m -2 ·s -1 and heat flux of 13.3 kW·m -2 , which were consistent with the experimental conditions. As shown in Fig. 6, the predictions agree well with the experimental data. The cooling of CO2 and CO2/DME (95/5) in horizontal helically coiled tubes was predicted for p=8.0 MPa, q=13.3 kW·m -2 , G=239.2 kg·m -2 ·s -1 and Tin=305-345 K. The heat transfer coefficients for the pure CO2 and the CO2/DME (95/5) mixture reached the peak values near their corresponding pseudo-critical temperatures as shown in Fig. 7. The peaks of the heat transfer coefficients are mainly due to the drastic changes in the thermophysical properties crossing the pseudo-critical point, especially the peak specific heat. The CO2/DME (95/5) peak heat transfer coefficient is less than that for pure CO2 and is shifted to higher temperatures because of the smaller specific heat and larger pseudo-critical temperature in the mixture. The variation of the specific heat, regarded as the most influential thermophysical property for heat transfer [36], is shown in Fig. 8. The peak CO2/DME (95/5) specific heat is less than that of pure CO2. At the same time, the temperature where the specific heat reaches the maximum is higher for CO2/DME (95/5), which indicates a higher pseudo-critical temperature. As shown in Fig. 7, the heat transfer coefficient of CO2/DME is higher than that of pure CO2 in the hightemperature region above 313 K. Therefore, CO2/DME is more suitable for heat pumps with high heat rejection temperatures.  The effects of the mass flux on the heat transfer of the CO2/DME mixture during cooling were modeled for CO2/DME (95/5) at p=8.0 MPa, q=13.3 kW·m -2 , and Tin=320 K with mass fluxes of 239.2 to 430.0 kg·m -2 ·s -1 . The local heat transfer coefficients along the helically coiled tube and the mean heat transfer coefficients are shown in Fig. 9 for the various mass fluxes. The CO2/DME heat transfer coefficient increases at higher mass fluxes due to the increased velocities and turbulent diffusion.  The effects of the heat flux on the heat transfer of CO2/DME (95/5) is shown in Fig. 10. The heat flux varies from 13.3 to 40 kW·m -2 at p=8.0 MPa, G=239.2 kg·m -2 ·s -1 and Tin=320 K. The local heat transfer coefficients along the tube are shown in Fig. 10(a). As the cooling begins, the heat transfer coefficient increases with the heat flux. A possible explanation is that the density variation caused by the temperature difference increases with the heat flux and the increased buoyancy then enhances the heat transfer. As the cooling continues, the high heat flux rapidly cools the fluid to below the pseudo-critical temperature which reduces the heat transfer coefficient in Fig. 10(a). Therefore, near the outlet, the heat transfer coefficient is lowest at the highest heat flux. The mean heat transfer coefficients are shown in Fig.  10(b) for heat fluxes ranging from 13.3 to 40 kW·m -2 . The heat transfer coefficient initially increases with the heat flux because the fluid temperature is close to the pseudo-critical temperature. As the heat flux further increases at q>25 kW·m -2 , the bulk fluid temperature drops below the pseudo-critical temperature and the heat transfer coefficient decreases. The optimum heat flux is 25 kW·m -2 in the present case at p=8.0 MPa, G=239.2 kg·m -2 ·s -1 and Tin=320 K. The heat transfer coefficient of CO2/DME (95/5) is shown in Fig.  11 as a function of the bulk temperature for various pressures for G=239.2 kg·m -2 ·s -1 and q=13.3 kW·m -2 . In the low temperature region of 300-312 K, the heat transfer coefficient decreases as the pressure increases. The heat transfer coefficient then increases with pressure for T>318 K. The heat transfer coefficients for the various pressures all increase gradually to a peak and then decrease with increasing fluid temperature. When the pressure increases, the peak heat transfer coefficient is nearly constant and shifts toward the high-temperature region. This is related to the pseudo-critical temperature of CO2/DME (95/5) increasing with increasing pressure. The ratio of gravitational and centrifugal Richardson numbers characterizes the relative effects of the gravitational buoyancy force and the centrifugal buoyancy force as

Gravitational Buoyancy Force
The variation of gravitational Richardson number of CO2/DME (95/5) with temperature is shown in Fig. 12 for p=8.0 MPa, G=239.2 kg·m -2 ·s -1 , and q=13.3 kW·m -2 . Rig increases to a maximum near the pseudocritical temperature and then drops as the bulk temperature increases further. Thus, the gravitational buoyancy effect is greatest around the pseudo-critical point. The local gravitational Richardson numbers along the helically coiled tube are shown in Fig. 13 for various mass fluxes and heat fluxes at p=8 MPa and Tin=320 K. The gravitational buoyancy decreases as the mass flux increases as shown in Fig. 13(a) and increases with the heat flux as shown in Fig. 13(b).
The effect of gravitational buoyancy on the heat transfer can be identified by comparing the cases with and without gravity. The heat transfer characteristics of supercritical CO2/DME (95/5) without gravity were calculated for various mass fluxes and heat fluxes for p=8 MPa and Tin=320 K. The local heat transfer coefficients are compared to those with gravity in Fig. 14. The heat transfer coefficients without gravity are lower than those with gravity, which indicates that the gravitational buoyancy force enhances the heat transfer for all cases. Previous studies have indicated that the gravitational buoyancy effect is negligible for Rig<0.001 (Liao and Zhao, 2002b). Others have suggested that the natural convection effect caused by gravitational buoyancy should be considered for Rig>0.01 (Yang, 2016;Xu et al., 2016;Du et al., 2010). In this study, the gravitational Richardson number Rig ranges from 0.024 to 0.007 for G=430.0 kg·m -2 ·s -1 (Fig. 13), which is less than 0.01. Therefore, the threshold Rig=0.01 is unreasonable for supercritical CO2/DME because the gravitational buoyancy force significantly influences the heat transfer in all cases. The centrifugal Richardson number is shown in Fig. 15 as a function of temperature for p=8.0 MPa, G=239.2 kg·m -2 ·s -1 and q=13.3 kW·m -2 . As with the gravitational buoyancy effect, the centrifugal buoyancy effect also reaches a maximum around the pseudo-critical temperature. The variation of the Richardson number ratio with temperature is shown in Fig. 16. The Richardson number ratio, ϕ, decreases with increasing temperature from 14.2 to 0.93. Thus, the gravitational buoyancy has a large proportion in the liquid-like region at low temperatures, while the centrifugal buoyancy and the gravitational buoyancy both influence the heat transfer in the gas-like region at high temperatures. The local centrifugal Richardson numbers along the helically coiled tube are shown in Fig. 17 for various mass fluxes and heat fluxes for p=8.0 MPa and Tin=320 K. The centrifugal buoyancy force decreases with mass flux due to the smaller density variations as shown in Fig. 17(a). The effect of the heat flux on the centrifugal buoyancy force is more complex as shown in Fig. 17(b). Ric initially increases with the heat flux and then tends to a constant. Near the tube outlet, Ric at q=35 kW·m -2 becomes slightly lower than for the other cases. This can be explained by considering the definition of Ric in Eq. (7) and Eq. (9) and the density distributions in Fig. 18. At the beginning of the cooling, the higher heat flux results in larger density variations between the tube wall and the fluid, which strengthens the centrifugal buoyancy force. Later, the fluid is rapidly cooled by the heat fluxes, resulting in a significant increase in the fluid density. The density difference between the tube wall and the fluid decreases for q=35 kW·m -2 since the density increase slows as the wall temperature cools from 312.6 K to 298 K as shown in Fig. 19. The rapid increase in the fluid density and the reduced density difference lead to the sharp decrease of Ric at q=35 kW·m -2 .  The local gravitational and centrifugal Richardson number ratio along the tube is plotted in Fig. 20 for various mass fluxes and heat fluxes for p=8 MPa and Tin=320 K. The Richardson number ratio decreases with increasing mass flux as shown in Fig. 20(a). The gravitational buoyancy plays a major role at the low mass flux (G≤239.2 kg·m -2 ·s -1 ) while the centrifugal buoyancy plays a major role at the high mass flux (G≥430.0 kg·m -2 ·s -1 ). Higher heat fluxes result in higher ϕ as shown in Fig. 20(b) which indicates that the effect of the gravitational buoyancy becomes increasing more important as the heat flux increases. The heat transfer characteristics of supercritical CO2/DME during cooling were modeled for flow in a 4 mm inner diameter horizontal helically coiled tube. The CO2/DME (95/5) heat transfer coefficients were compared with those for pure CO2 flows. The results showed the effects of mass flux, heat flux and pressure on the heat transfer coefficients. The effects of the gravitational and centrifugal buoyancy forces were also analyzed to give a better understanding of the heat transfer mechanisms. The CO2/DME mixture was found to be more suitable than pure CO2 for heat pumps with high heat rejection temperatures (above 313 K) due to the higher heat transfer coefficients. The peak heat transfer coefficient decreased and shifted to high temperatures for the CO2/DME mixture. The peak heat transfer coefficient was 6931.5 kW·m -2 ·K -1 at 309 K for the pure CO2 and 3670.3 kW·m -2 ·K -1 at 317 K for the CO2/DME (95/5). The CO2/DME heat transfer coefficient increased with the mass flux from 239.3 to 430.0 kg·m -2 ·s -1 due to the increased turbulent diffusion. The heat transfer coefficient of CO2/DME first increased but then decreased with the heat flux from 13.3 to 35.0 kW·m -2 . The optimum heat flux was 25 kW·m -2 in the present study. The peak heat transfer coefficient of CO2/DME shifted toward the high-temperature region as the operating pressure increases, with temperature of 311 K for 7.3 MPa, 317 K for 8.0 MPa and 323 K for 9.0 MPa. The buoyancy effects caused by gravitational and centrifugal forces enhanced the heat transfer in horizontal helically coiled tubes. The gravitational buoyancy and centrifugal buoyancy effects reached their maximums near the pseudo-critical point. In the liquid-like region of CO2/DME (300~317 K), the gravitational buoyancy more greatly influenced the heat transfer than the centrifugal buoyancy; while in the gas-like region (317 K), both the gravitational buoyancy and the centrifugal buoyancy influenced the heat transfer. The gravitational buoyancy increased with the mass flux but decreased with the heat flux. The centrifugal buoyancy decreased with the mass flux and initially increased with the heat flux while then tends to a constant.
This study provides further understanding of the heat transfer characteristics of supercritical CO2/DME for refrigerant selection, operating parameters optimization and gas coolers design in transcritical heat pumps. NOMENCLATURE cp specific heat (J·kg -1 ·K -1 ) d tube diameter (mm) g acceleration due to gravity (m·s -2 ) G mass flux (kg·m -2 s -1 ) Gr Grashof number h heat transfer coefficient (W·m -2 ·K -1 ) k turbulence kinetic energy (J·kg -1 ) p pressure (Pa) P coil pitch (