International Journal of Heat and Mass Transfer

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1 International Journal of Heat and Mass Transfer 56 (2013) Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: Convection heat transfer of supercritical pressure carbon dioxide in a vertical micro tube from transition to turbulent flow regime Pei-Xue Jiang a,b,, Bo Liu a,b, Chen-Ru Zhao a,b,c, Feng Luo a,b a Key Laboratory for Thermal Science and Power Engineering of the Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing , China b Beijing Key Laboratory of CO 2 Utilization and Reduction Technology, Department of Thermal Engineering, Tsinghua University, Beijing , China c Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing , China article info abstract Article history: Received 20 April 2012 Received in revised form 17 August 2012 Accepted 18 August 2012 Keywords: Supercritical pressure carbon dioxide Micro tube Convection heat transfer Transient and turbulent flow regime Flow acceleration This paper presents experimental investigations of the convection heat transfer of carbon dioxide at supercritical pressures in a vertical tube with inner diameter of 99.2 lm for various Reynolds numbers, heat fluxes and flow directions. The effects of buoyancy and flow acceleration due to heating and pressure drop are evaluated and analysed. The results show that the effects of flow acceleration are significant and the local wall temperature varies non-linearly for both upward and downward flows at the pressures in the vicinity of critical point and low inlet Reynolds numbers when the heat fluxes are relatively high. The buoyancy effect on the heat transfer is negligible in micron scale tubes at inlet Reynolds (from 2600 to 6700) and various heat fluxes (from 85 kw/m 2 to 748 kw/m 2 ). The flow acceleration due to heating and pressure drop can strongly influence the turbulence and reduce the heat transfer for high heat fluxes and low inlet Reynolds. Comparison of numerical predictions with the experimental data showed that the AKN low Reynolds number turbulence model gave better agreement than the k e realizable turbulence model with the enhanced wall treatment. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Supercritical fluids are used as working fluids in many industrial applications, such as supercritical pressure water cooled reactors (SPWR), nuclear reactors using the supercritical CO 2 indirect cycle, transpiration cooling of high heat flux surfaces, enhanced geothermal systems, trans-critical CO 2 heat pump and refrigeration systems. The platelet transpiration cooling method is one of the most efficient methods for protecting high heat flux surfaces such as rocket thruster walls using hydrogen or methane at supercritical pressures as the coolant flowing through micron scale channels in the platelets. The platelets are formed by bonding together thin metal sheets containing chemically etched coolant micro channels. The coolant flow rate and the flow distributions can be precisely regulated by proper design of the coolant passages. This provides efficient thermal management and can be used for thermal protection of the next generation of liquid rocket engines [1]. Enhanced geothermal systems (EGS) aim to extract geothermal energy from rocks that lack fractures and have low permeability Corresponding author at: Key Laboratory for Thermal Science and Power Engineering of the Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing , China. Tel.: ; fax: addresses: jiangpxtsinghua.edu.cn, jiangpxmail.tsinghua.edu.cn (P.-X. Jiang). with fluid circulation made possible by increasing the permeability through hydraulic fracturing, such as by injecting fluid through deep boreholes to activate existing rock fractures, with the working fluid flow through these fracture networks controlled by a system of injection and production boreholes [2]. CO 2 has been proposed as the working fluid in EGS in response to CO 2 emissions reduction needs, with the CO 2 at supercritical pressures for the conditions of interest in EGS [2]. When fluids are at supercritical pressures, small fluid temperature and pressure variations result in significant changes in the thermophysical properties. The specific heat, c p, reaches a peak at a temperature defined as the pseudo critical temperature, T pc. Convection heat transfer of fluids at supercritical pressures exhibits many special characteristics resulting from the sharp variations of the thermophysical properties, the buoyancy force induced by the radial non-uniform density distribution and flow acceleration due to the fluid expansion as a result of axial density variations resulting from the axial temperature and pressure variations during heating. Eextensive research has been conducted on forced and mixed convection heat transfer of supercritical fluids in normal size tubes in the past 50 years by Shitsman [3], Petukhov [4], Hall [5], Jackson [6], Krasnoshchekov and Protopopov [7], Protopopov [8], Popov and Valueva [9], Kurganov and Kaptilnyi [10], Jiang et al. [11,12], and He et al. [13]. The special features of the convection heat /$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.

2 742 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) Nomenclature Bo non-dimensional buoyancy parameter c p specific heat at constant pressure [J/(kg K)] d tube diameter [m] g gravitational acceleration [m 2 /s] G mass flow rate [kg/s] Gr Grashof number h bulk specific enthalpy [J/kg] h x local heat transfer coefficient [W/(m 2 K)] Kv non-dimensional flow acceleration parameter p pressure [MPa] Pr Prandtl number R tube inner radius [m] Re Reynolds number T temperature [K] u velocity [m/s 2 ] x axial coordinate [m] Greek symbols a p thermal expansion coefficient [1/K] b T isothermal compression coefficient [1/Pa] d tube wall thickness [m] k thermal conductivity [W/(m K)] l dynamic viscosity [Pa s] q fluid density [kg/m 3 ] or electrical resistivity [X m] Subscripts CO 2 carbon dioxide f fluid i inner surface in inlet o outer surface out outlet pc pseudo critical p induced by pressure variation T induced by temperature variation w wall transfer of supercritical fluids due to the sharp variations of the thermophysical properties and the buoyancy in normal size tubes have been widely investigated. Shitsman [3] pointed out that, for relatively large tubes (d in = 8 mm), the local wall temperatures varied in a complex and nonlinear form with heat transfer deterioration observed in buoyancy-aided flows (upward flow in a heated passage) resulting from the buoyancy effect whereas in buoyancyopposed flow cases (downward flow in a heated passage) the local wall temperatures varied smoothly. Jackson and Hall [14,15] explained the buoyancy affected convection heat transfer behaviour for supercritical fluids in channels using a semi-empirical theory and proposed a non-dimensional buoyancy parameter, Bo,to evaluate the significance of buoyancy. They suggested that the modification of the mean flow field by the buoyancy and the variation of the turbulence production were the main factors affecting the heat transfer and local temperature variation along the tube for buoyancy-aided cases and buoyancy opposed cases. For upward flows with <Bo < , the buoyancy will reduce the heat transfer while for <Bo <8 10 6, the heat transfer reduction will be gradually reduced as Bo increases. For Bo >8 10 6, the heat transfer will be enhanced by the buoyancy according to McEligot and Jackson [16]. When the channel size is reduced and the heat flux is increased, the heat transfer is reduced leading to non-monotonically distributions of the wall temperature for both flow directions, with such effects found in a number of investigations, for example those of Domin [17] with water at 227 bar in a 2 mm tube and Shiralkar and Griffith [18] with CO 2 at 75.8 bar in a 3.17 mm tube. These results were explained as being due to the reduced turbulence due to the flow acceleration as the fluid expands due to the increased bulk temperature. Generally, the buoyancy and flow acceleration effects on the convection heat transfer of fluids at supercritical pressures are related to the channel size, with smaller channels size having more significant flow acceleration but less buoyancy effect. The buoyancy and flow acceleration effects on the heat transfer of supercritical fluids in tubes with various inner diameters have also been investigated. Liao and Zhao [19,20] found that the buoyancy effects were significant for all flow orientations even for Reynolds numbers as high as 10 5 according to their experimental investigation of the average convection heat transfer for supercritical CO 2 in horizontal and vertical miniature tubes having diameters of 0.70, 1.40, and 2.16 mm. Their experimental results indicated that for all flow orientations, the Nusselt number decreased substantially for tube diameters less than 1.0 mm, which differs from results in normal size tubes [14,15,21,22]. He et al. s [23] numerical simulations of turbulent convection heat transfer of supercritical pressure CO 2 in a mm inner diameter vertical tube using the low Reynolds number eddy viscosity turbulence model indicated that for a mm diameter vertical mini tube and a large Reynolds number of 10 5, the buoyancy effect was insignificant. Jiang et al. [24,25] investigated the convection heat transfer of CO 2 at supercritical pressures in a 0.27 mm diameter vertical mini tube to show that for inlet Reynolds numbers exceeding , the buoyancy and flow acceleration have little influence on the local wall temperature with no reduction of the convection heat transfer coefficient observed in either flow direction. For relatively low Reynolds numbers (< ) and high heat fluxes (e.g. 113 kw/m 2 ), the local wall temperatures varied non-linearly along the tube for both upward and downward flows and the convection heat transfer coefficients for downward flow were greater than for upward flow. Their experimental results indicated that for mini tubes (e.g mm inner diameter), the flow acceleration due to heating for the studied conditions strongly influenced the turbulence and reduced the heat transfer for high heat fluxes. The buoyancy effect could not be neglected even though relatively small even when the heating was strong. These few studies on the heat transfer of supercritical fluids in mini scale channels need to be further supplemented with more data since such flow are expected to be significantly affected by flow acceleration and less affected by buoyancy. McEligot et al. [26] proposed a non-dimensional flow acceleration parameter, Kv T =4q w b/(qu b c p Re), to evaluate the influence of the flow acceleration on the heat transfer. They suggested that for turbulent flow, the turbulence may be significantly reduced for Kv T P and the flow may even re-laminarise, which would reduce the overall heat transfer. Murphy et al. [27] found that when Kv T , the fluid flow remained turbulent. For convection heat transfer of fluids at supercritical pressures in micro scale tubes, both the temperature increase and the pressure drop along the tube will induce significant density decreases resulting in flow acceleration. Generally, in normal size tubes, the pressure drop is negligible compared with the inlet pressure; thus, the flow acceleration induced by the pressure drop is usually neglected, even in a mini tube with 0.27 mm inner diameter, the flow

3 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) acceleration was almost exclusively due to the thermal expansion [24,25]. When the channel size reduced to micron size, the axial pressure gradient is expected to increase drastically; thus, the influence of flow acceleration induced by the pressure drop on the heat transfer needs to be carefully reconsidered. This paper presents experimental and numerical investigations of the convection heat transfer of CO 2 at supercritical pressures in a vertical tube with inner diameter of 99.2 lm for various Reynolds numbers, heat fluxes and flow directions. The effects of buoyancy and flow acceleration due to heating and pressure drop are evaluated. The results provide a better understanding of the heat transfer characteristics of supercritical fluids in micro channels with the effect of flow acceleration due to both thermal expansion and the pressure drop. 2. Experimental system and data reduction method The experimental system for measuring the heat transfer in a micro tube and an SEM photograph of a cross section of the micro tube are shown in Fig. 1 and details can be found in Ref. [24]. The test section was a 50 mm long vertical smooth stainless steel 1Cr18N9T tube with a 40 mm (400d) heating section and 5 mm (50d) adiabatic sections before and after the heating section. The test section was thermally insulated from the environment. The test section was soldered to a stainless steel tubes at the inlet and outlet with inner diameters of 2 mm and outer diameters of 3 mm which were connected with the test loop by flanges and high-pressure fittings. The test section inner tube diameter was 99.2 lm and the outer diameter was 217 lm. The test section was insulated thermally and electrically from the test loop by a layer of polytetrafluoethylene (PTFE) placed between the flanges and between the screws and the flanges. The flow direction (upward or downward) of CO 2 flowing through the test section was adjusted by a set of valves. Mixers were installed before where the inlet and outlet CO 2 temperatures were measured by accurate RTDs (Pt-100). The CO 2 inlet pressure was measured by a pressure transducer (EJA430A) with the pressure drop through the tube measured by a differential pressure transducer (Model EJA110A), respectively. The local outer wall temperatures of the small tube were measured using micro T- type thermocouples welded onto the outer tube surface. The constant heat flux produced by the current stabilized power source was calculated from the heating current (measured by a digital multimeter) and the electrical resistance of the test section. The system was assumed to be at steady state when the variations of the wall temperatures and the inlet and outlet fluid temperatures were all within ±0.2 C and the flow rate and the inlet pressure variations were within ±0.2% for at least 10 min. The local heat transfer coefficient, h x, at each axial location was calculated as q h x ¼ w ðxþ T w;i ðxþ T f ;b ðxþ where the local heat flux on the inner surface, q w (x), is calculated as: q w ðxþ ¼ I2 R x ðtþ Q loss;dx pd i Dx ¼ I2 qðtþdx=½pðd 2 o d2 i Þ=4Š Q loss;dx pd i Dx The electrical resistivity of the micro tube, q(t), was calculated using a correlation of the measured electrical resistivity at various temperatures. The heat loss, Q loss,dx, of the test section was measured by evacuating the tube before the convection heat transfer tests without CO 2 in the micro tube. The electrical power input to the tube and the wall temperature were then measured with this assumed to be equal to the heat loss in the experiments conducted as a function of the temperature difference between the tube wall and the surroundings. The measured results showed that the heat loss was very small compared with the heat input to the test section (within 4%), which indicates that the test section was well insulated. The local bulk fluid temperature, T f (x), was obtained using the NIST software REFPROP 7.0 referenced from the local bulk fluid enthalpy, h f (x), which was in turn calculated by: h f ðxþ ¼h f ;in þ q wðxþpd i x G The Reynolds number based on the mean bulk temperature was defined as: Re ¼ qud i l ¼ 4G pd i l The inner wall temperature, T w,i (x), was calculated using the measured outer wall temperature, T w,o (x), and the internal heat source, q v, as: T wi ðxþ ¼T wo ðxþþ q mðxþ 16k ½D2 d 2 Šþ q vðxþ 8k D2 ln d D where q v was calculated as: q v ðxþ ¼ I2 R x ðtþ Q s;dx ½pðD 2 d 2 Þ=4ŠDx A numerical model of the two-dimensional heat conduction [28] showed that the influence of two-dimensional heat conduction in the tube wall had little influence except near the inlet and outlet. The non-dimensional buoyancy parameter, Bo, introduced by Jackson and Hall [15], used to evaluate the buoyancy effect was defined as: ð1þ ð2þ ð3þ ð4þ ð5þ ð6þ Fig. 1. Schematic of experimental system and SEM photograph of the test section.

4 744 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) Bo Gr ¼ Re 3:425 Pr 0:8 ð7þ where, Gr ¼ ga pd 4 i q w km 2 ð8þ The non-dimensional flow acceleration parameter, Kv, was calculated as: Kv ¼ m b du b u 2 dx ¼ d Re b dp T dx þ 4q wda p ¼ Kv b Re 2 p þ Kv T l b c p ð9þ where Kv T is the same as the non-dimensional thermal expansion acceleration parameter proposed by McEligot et al. [26] which describes the effect of flow acceleration due to the heating defined as: Kv T ¼ 4q wda p Re 2 l b c p ð10þ Fig. 2. Local wall temperatures for various wall heat fluxes for downward flow p in = MPa, p out = 7.9 MPa, T in = 296 K, G = kg/(m 2 s), Re in = Solid symbols: wall temperatures; lines: fluid temperatures. Kv p is the non-dimensional flow acceleration parameter describing the effect of flow acceleration due to the pressure drop through the tube as: KV p ¼ d Re b dp T dx ð11þ The experimental uncertainty in the local heat transfer coefficient mainly resulted from the heating and temperature measurement uncertainties. Prior to installation, the thermocouples and the RTDs were calibrated by the National Institute of Metrology, PR China. The accuracies of the RTDs which measure the inlet and outlet bulk temperatures were ±0.1 C while the accuracies of the thermocouples measuring the outer wall temperatures were ±0.15 C in the temperature range used in the present study. The accuracy of the pressure transducer (Model EJA430A) was 0.075% of the full range of 14 MPa and the accuracy of the differential pressure transducer (Model EJA110A) was 0.075% of the full range of 5 MPa. A detailed uncertainty analysis showed that the uncertainty of the heat transfer coefficient was ±12.7%. The uncertainty in the inlet pressure was estimated to be ±0.13%. The uncertainty of the mass flow rate was 0.1%. 3. Experimental results and discussion The heat transfer of supercritical CO 2 flowing through a 99.2 lm inner diameter vertical tube was experimentally investigated for inlet pressures of MPa which is close to the ciritical pressure 7.38 MPa, and inlet Reynolds numbers of The distributions of the local wall temperature, local heat transfer coefficient, buoyancy parameter, Bo, and the flow acceleration parameter, Kv, and its two components, Kv T and Kv p, along the test section are presented for various inlet Reynolds numbers, pressures and wall heat fluxes. The results illustrate the effects of buoyancy and flow acceleration due to the thermal expansion and pressure drop on the heat transfer. T pc ¼ 150:55 þ 6: p 1: p 2 þ 5: p 2:5 5: p 3 ð12þ where T pc is in K and the pressure p is in MPa [20].The inlet bulk temperature is lower than T pc for all cases in Fig. 2. The bulk temperature increases along the test section as the CO 2 absorbs the heat generated in the tube wall. Also, the local wall temperature increases with increasing wall heat flux, q w. Within the heat flux range in this study, the local wall temperature increases along the test section, with no abnormal local wall temperature variations observed. The variations of the local non-dimensional buoyancy effect parameter, Bo, are shown in Fig. 3 for the various heat fluxes corresponding to the cases in Fig. 2. For these heat fluxes, Bo has a magnitude of 10 10, much lower than the threshold of given by McEligot and Jackson [16], with buoyancy effects only important for Bo above this value. Thus, the low Bo in Fig. 3 indicate that the buoyancy effect is insignificant. The local wall temperature variations for the upward and downward flow when the other test conditions are held constant are consistent and the differences are quite small, which also indicates that buoyancy is insignificant for the heat and mass flux range in Fig. 2. As mentioned in the introduction, when the tube size is reduced to the micro size, the buoyancy effect is reduced, however, the flow acceleration effect, including both the flow acceleration due to 3.1. Relatively high inlet pressure and Reynolds number The local wall temperature and bulk temperature variations along the test section are shown in Fig. 2 for various heat fluxes at inlet test conditions of p in = MPa, T in = 296 K, and Re in = for downward flow cases. The local pseudo critical temperature, T pc, decreases along the tube because of the significant pressure drop along the test section in the micro tube; T pc was calculated as a function of pressure as: Fig. 3. Local Bo for various wall heat fluxes p in = MPa, p out = 7.9 MPa, T in = 296 K, G = kg/(m 2 s), Re in =

5 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) Fig. 4a. Local Kv T for various wall heat fluxes p in = MPa, p out = 7.9 MPa, T in = 296 K, G = kg/(m 2 s), Re in = Fig. 5. Thermophysical property variations of CO 2 at p = 8.8 MPa. thermal expansion and the flow acceleration due to the pressure drop in the tube is more significant than in large tubes. The local non-dimensional heating flow acceleration parameter, Kv T, is presented in Fig. 4a for various heat fluxes. When the heat fluxes are relatively low, the local Kv T increases along the test section with the bulk temperature increasing and approaching T pc.kv T also increases with increasing heat flux as shown in Fig. 4a. As the heat flux increases further, Kv T increases with the heat flux, reaches a peak, and then decreases rapidly when the wall and fluid temperatures are larger than T pc.kv T has a magnitude of 10 7, with the local maximum reaching for the maximum heat flux in the present test conditions at q w = 748 kw/m 2. The local nondimensional compressible flow acceleration parameter, Kv p, at various heat fluxes is shown in Fig. 4b. Since the tube is very small, the pressure drop along the tube can exceed 1 MPa, with pressure gradients of 20 MPa/m, which causes the fluid density to decrease greatly along the tube and intensifies the flow acceleration. Kv p has a similar magnitude of 10 7 of Kv T as shown in Fig. 4b, which indicates that the flow acceleration induced by the pressure drop cannot be neglected when the tube size is reduced to microns. This differs from the study conducted by McEligot et al. [26], in which the pressure drop induced flow acceleration was negligible compared with the thermal expansion flow acceleration and the streamwise flow acceleration was mainly caused by the thermal expansion due to the heat transfer in larger tubes. As shown in Fig. 4b, for heat fluxes of q w = kw/m 2,Kv p increases along the test section as the bulk temperature increases and approaches T pc. This is mainly related to the variation of the isothermal compressibility, b T, and gradually decrease of q with the fluid temperature as shown in Fig. 5. Higher heat fluxes generate larger Kv p, especially near the outlet where the bulk temperature approaches T pc ; thus, a small fluid temperature increase results in a large increase of the isothermal compressibility and a large decrease of q, so the pressure drop more strongly affects the fluid expansion. When the heat flux increases further to 586 kw/m 2 and 748 kw/m 2,Kv p first increases along the test section, reaches a maximum when the bulk temperature is close to T pc, and then decreases along the test section with a sharp peak in Kv p. The drastic increase and decrease of Kv p near T pc is mainly induced by the peak of the isothermal compressibility, b T, and the variation of q. The heat flux does not always intensify the isothermal compressible flow acceleration. When the heat flux exceeds 586 kw/ m 2, the maximum value of Kv p decreases with increasing heat flux as shown in Fig. 4b due to the combined influence of the Reynolds number and the isothermal compressibility on Kv p, as defined in Eq. (11). For higher heat fluxes, the local bulk temperature increases and exceeds T pc downstream, with the local Re increasing drastically as a result, while the isothermal compressibility, b T, decreases. Thus, the tradeoffs between Re and b T cause the decrease in maximum value of Kv p as the heat flux exceeds the threshold value. Fig. 4b. Local Kv p for various wall heat fluxes p in = MPa, p out = 7.9 MPa, T in = 296 K, G = kg/(m 2 s), Re in = Fig. 6. Local Kv for various wall heat fluxes p in = MPa, p out = 7.9 MPa, T in = 296 K, G = kg/(m 2 s), Re in =

6 746 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) Relatively low inlet pressures and Reynolds numbers Fig. 7. Local Nu variations along the test section for various heat fluxes p in = MPa, p out = 7.9 MPa, T in = 296 K, G = kg/(m 2 s), Re in = The variation of the local non-dimensional flow acceleration parameter, Kv, which is the sum of Kv T and Kv p, along the test section is shown as Fig. 6 for various heat fluxes. A peak is observed for high heat flux cases, with the peak position moving closer to the entrance with increasing heat flux. The Nu variations are presented in Fig. 7. For the heat fluxes q w = 496 kw/m 2 and q w = kw/m 2, the bulk temperature increases along the test section and the wall temperature increase above T pc near outlet; thus, the specific heat, c p, increases which enhances the heat transfer between the fluid and the wall and the Nu increases significantly as shown in Fig. 7. As the heat flux increases further to 684 kw/m 2 and 748 kw/m 2, the wall temperature increases to a value high than T pc and the bulk temperature also increase to above T pc downstream, which indicates that the fluid is in the gas-like where the density, specific heat and thermal conductivity are small and the heat transfer is reduced significantly as a result. The maximum local Kv is below , less than the threshold, , proposed by Murphy et al. [27], as shown in Fig. 6. No local heat transfer deterioration and no abnormal wall temperature variations occur in the experiments as shown in Figs. 2 and 7, which is consistent with the conclusion of Murphy et al. [27] that the local turbulence suppression induced by the flow acceleration is insignificant when Kv The local wall temperature and bulk temperature variations along the test section for various heat fluxes at inlet test conditions of p in = MPa, T in = 297 K, and Re in = 2600 are shown in Fig. 8 for downward flow. The corresponding local Nu variations are presented in Fig. 9. For the relatively low heat flows (q w =85kW/m 2 ), the local wall temperature increases along the test section. The Nu significantly increases and then decrease near the outlet. This is mainly due to the fluid bulk temperature increasing and approaching T pc, so the specific heat increases, which enhances the heat transfer between the fluid and the wall, as shown in Fig. 9. For higher heat fluxes, the local wall temperature exhibits a non-linear variation and sharply increases near the entrance, then decreases near x/d = and then increases downstream. The corresponding local Nu significantly drops to the minimum near x/ d = 100, which indicates the heat transfer is impaired near the entrance and then increase downward, which indicate the heat transfer is recovered as shown in Fig. 9. Fig. 10 presents the corresponding local Bo variations for various heat fluxes. For higher heat fluxes, when the bulk temperature is below T pc,bo increases with increasing heat flux, reaches a maximum when the bulk temperature approaches T pc, and then decreases sharply. This is mainly because when the bulk temperature is lower than T pc, the fluid in the centre is in a liquid-like state, while when the wall temperature is higher than T pc, the fluid near the wall is in a gas-like state, so the radial density difference is quite large and the buoyancy more significantly affects the flow and heat transfer. However, as the heat flux increases further, the bulk temperature exceeds T pc and all the fluid across the crosssection is in a gas-like state, so the density gradient decreases and the buoyancy effect is reduced. The Bo variation with the fluid temperature is quite similar to that of Kv T, which is presented in Fig. 11a because both Bo and Kv T are related to the thermal expansion coefficient, a p, which represents the density variation with temperature at constant pressure. A larger a p mean sharper density variations; thus, the buoyancy and thermal flow acceleration effects are stronger. The density decreases very rapidly around T pc with the thermal expansion coefficient reaching a maximum, so the buoyancy and thermal flow acceleration effects are strong. Bo increases more here as the pressure and inlet Reynolds number decrease compared with the cases in Section 3.1. The local Bo reaches 10 9 when the heat flux is high; however, it is still far below the threshold of ,so the buoyancy effect on the heat transfer is still negligible even if Fig. 8. Local wall temperatures for various wall heat fluxes for downward flow p in = MPa, p out = 7.6 MPa, T in = 297 K, G = 1823 kg/(m 2 s), Re in = 26. Solid symbols: wall temperatures; lines: fluid temperatures. Fig. 9. Local Nu variations along the test section at various heat fluxes p in = MPa, p out = 7.6 MPa, T in = 297 K, G = 1823 kg/(m 2 s), Re in = 2600.

7 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) Fig. 10. Local Bo for various wall heat fluxes p in = MPa, p out = 7.6 MPa, T in = 297 K, G = 1823 kg/(m 2 s), Re in = Fig. 12. Local Kv for various wall heat fluxes p in = MPa, p out = 7.6 MPa, T in = 297 K, G = 1823 kg/(m 2.s), Re in = with decreasing pressure and inlet Reynolds number. Kv p also reaches a peak as the bulk temperature approaches T pc.kv p increases as the heat flux increases for relatively low heat fluxes until the heat flux exceeds a critical value, 173 kw/m 2 in the present tests. Kv p is about 10 7, indicating that the effect of the flow acceleration due to pressure drop on the flow and heat transfer cannot be negligible. The corresponding non-dimensional flow acceleration parameter, Kv, for the various heat fluxes is shown in Fig. 12. As shown in Fig. 9, the Nusselt number decreases for x/d = for higher heat fluxes, where Kv is relatively high, indicating that the flow acceleration begins to suppress the turbulence and reduce the heat transfer when Kv exceeds Numerical simulations and comparison with experiments Fig. 11a. Local Kv T for various wall heat fluxes p in = MPa, p out = 7.6 MPa, T in = 297 K, G = 1823 kg/(m 2.s), Re in = Fig. 11b. Local Kv P for various wall heat fluxes p in = MPa, p out = 7.6 MPa, T in = 297 K, G = 1823 kg/(m 2.s), Re in = the pressure is quite close to p c and the inlet Reynolds number is quite low in the micro tubes. Figs. 11a and 11b show the local Kv T and Kv p variations along the test section. Comparing Figs. 4a and 11a shows that for the same inlet temperature and heat flux conditions, Kv T increases In the numerical simulations, the inner and outer diameters of the vertical stainless steel tube were 99.2 lm and 217 lm asin the experiments. The electrical heating of the test section was modelled as a uniform heat source. The heated section was 40 mm long and the adiabatic sections before and after the heated section were 5 mm long (50d). The flow was assumed to be twodimensional turbulent flow. The conjugate convection heat transfer and heat conduction in the wall with an internal heat source in the vertical tube was numerically simulated using FLUENT 12 with various turbulence models used to model the turbulence. The governing equations for the steady state, two-dimensional turbulent flow of a supercritical pressure fluid in a vertical tube accounting for the temperaturedependent property variations and buoyancy can be written as: Continuity: 1 r x ðqruþþ r ðqrvþ ¼ 0 U-momentum: 1 r x ðqru2 Þþ r ðqrvuþ V-momentum: 1 r x ðqruvþþ r ðqrv 2 Þ ¼ p x þ qg þ 1 2 r x rl e þ r rl U e r þ V x U x ¼ p r þ 1 r x rl U e r þ V x þ 2 r rl V e 2 l e V r r 2 ð13þ ð14þ ð15þ

8 748 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) where l e is the effective viscosity defined by in l e = l + l T in which l T is the turbulent viscosity defined as: l t ¼ qf l c l k 2 =e ð16þ where f l is damping function to account for near-wall effects and C l is constant. Energy equation: 1 r x ðqc prutþþ r ðqc prvtþ ¼ 1 T r x rc p þ x r rc p l Pr þ l T r T l Pr þ l T r T T r ð17þ where Pr is the molecular Prandtl number and r T is the turbulent Prandtl number. The flow of the supercritical fluids can be strongly distorted in comparison with conventional wall shear flows duo to the influences of buoyancy, flow acceleration and non-uniformity of fluid properties, so the use of standard k e turbulence model coupled with a simple wall function is not appropriate. Jiang et al. [29,30] and He et al. [23] performed numerical simulations of convection heat transfer of supercritical fluids and the results showed that the Low-Reynolds number turbulence models were able to reproduce the temperature variations trend influenced by the buoyancy and the flow acceleration. Low-Reynolds number turbulence models, which extend the standard model by including the modeling of the effects of the wall and the molecular diffusion, can be used in the wall region as well as the core, and therefore potentially more suitable for the supercritical fluids. The low Reynolds number eddy viscosity turbulence models, Abe, Kondoh and Nagano (AKN) [31] and the k e realizable turbulence model with the enhanced wall treatment were used in the present study. The constitutive and transport equations can be written as: Turbulent kinetic energy: ðqukþ þ 1 x r ðrqvkþ r ¼ x r r l þ l t qe þ qd r k k þ 1 x r k r l þ l t r k þ P k þ G k where, the shear production term P k is: " ( 2 U P k ¼ l t 2 þ V 2 þ V ) 2 þ U x r r r þ V # 2 x ð18þ ð19þ Fig. 13. Comparisons of experimental and numerical results using the AKN model for downward flow p in = MPa, p out = 7.6 MPa, T in = 297 K, G = 1823 kg/(m 2.s), Re in = of grids, 3600 nodes in the axial direction and ( ) nodes in the radial direction (fluid region + tube wall). Calculations with a more refined mesh showed that the results were grid independent. Figs. 13 and 14 compare the measured and calculated wall temperatures for downward supercritical pressure CO 2 flowing in the vertical heated micro tube for a relatively low Reynolds number (Re in = 2600) at various wall heat fluxes using the AKN low Reynolds number turbulence model and the k e realizable turbulence model with the enhanced wall treatment. When the heat flux is relatively low (85 kw/m 2 ), the local wall temperature increases linearly along the test section, with the wall temperatures predicted using the AKN and k e realizable turbulence models both corresponding well with the measured data, as shown in Figs. 13 and 14. For higher wall fluxes (173 kw/m 2, 221 kw/m 2, and 244 kw/m 2 ), where the wall temperature is higher than T pc, the bulk temperature crosses T pc inside the test section and the thermophysical properties vary dramatically, so the local wall temperature varies non-linearly along the test section and the heat transfer is significantly affected by the flow acceleration. Near the entrance where the heat transfer is affected by the flow acceleration, the AKN low Reynolds number turbulence model over-predicts the effect of the flow acceleration and the calculated temperatures are higher than the measured values. The k e The gravitational production term G k is: k U e r þ V T x r G k ¼ q 0 u 0 g x ¼ bl t C 1t g x ð20þ Turbulence dissipation rate: ðqueþ þ 1 ðrqveþ ¼ x r r x l þ l t r e e x þ 1 r r r l þ l t r k þ C e1 f 1 e k ðp k þ G k Þ C e2 f 2 qe 2 k þ qe e r ð21þ The NIST Standard Reference Database 23 (REFPROP) Version 7 was used to calculate the temperature and pressure dependent properties of CO 2. The SIMPLEC algorithm was used to couple the pressure and the velocities. The QUICK scheme was used for the momentum and energy equations. The convergence criteria required a decrease of at least four orders of magnitude for the residuals. The mesh was refined in the radial direction towards the wall to ensure the y+ value at the first node of the mesh near the wall was less than 1. The whole domain was discretized into a mesh Fig. 14. Comparisons of experimental and numerical results using the k e model for downward flow p in = MPa, p out = 7.6 MPa, T in = 297 K, G = 1823 kg/(m 2.s), Re in = 2600.

9 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) realizable model with the enhanced wall treatment fails to reproduce the heat transfer reduction. For the heat transfer recovery region (x/d > 140), the AKN and k e models are both able to well predict the local wall temperature variation with the results using the AKN model better than that with k e realizable model. Thus, the turbulence model for the heat transfer deterioration region due to flow acceleration should be modified in further studies. 5. Conclusions The convection heat transfer of CO 2 at supercritical pressures in a vertical micro tube with inner diameter of 99.2 lm was experimentally and numerically investigated for various heat fluxes, pressures and inlet Reynolds numbers. The buoyancy effect and flow acceleration due to heating and the pressure drop effect were analysed. This study shows that: (1) When the inlet pressure and Reynolds number are relatively high, p in = MPa and Re in = in this study, the local wall temperature increases along the test section and the heat transfer coefficient is high. When the bulk temperature is lower than T pc and the wall temperature is higher than T pc the heat transfer coefficient decreases as the bulk and wall temperatures increase further. The influences of buoyancy and flow acceleration are very weak. (2) The effects of the flow acceleration are more significant at relatively lower pressures close to p c and lower inlet Reynolds numbers. At pressures of MPa and inlet Reynolds number of 2600, the local wall temperature exhibits non-linear variations near the entrance due to the flow acceleration. The buoyancy effect on the heat transfer is negligible in micro tubes such as used in the present study. The flow acceleration effect due to heating and the pressure drop is significant for low inlet Reynolds numbers (Re in = 2600) and high heat fluxes and the heat transfer is reduced when the non-dimensional flow acceleration parameter, Kv, exceeds about for Re in = (3) The results show that the AKN turbulence model and the k e realizable turbulence model with the enhanced wall treatment are both able to predict the temperature variations for flows which are not significantly affected by the buoyancy and flow acceleration. For flows which are affected by the flow acceleration, the AKN low Reynolds number turbulent model responds to the flow acceleration but overpredicts the wall temperature, while the k e realizable turbulence fails to respond to the flow acceleration. Acknowledgments This project was supported by the Key Project Fund from the National Natural Science Foundation of China (No ). We thank Professor J.D. Jackson in the School of Mechanical, Aerospace and Civil Engineering, the University of Manchester, UK for his many suggestions for this research. We also thank Prof. David Christopher for editing the English. References [1] H.H. Mueggenburg, J.W. Hidahl, E.L. Kessler, Platelet Actively Cooled Thermal Management Devices, AIAA [2] K. Pruess, Enhanced geothermal systems (EGS) using CO 2 as working fluid a novel approach for generating renewable energy with simultaneous sequestration of carbon, Geothermics 35 (2006) [3] M.E. Shitsman, Impairment of the heat transmission at supercritical pressures, Teplofiz. Vys. Temp. 1 (2) (1963) (in Russian). [4] B.S. Petukhov, Heat transfer and friction in turbulent pipe flow with variable physical properties, Adv. Heat Transfer 6 (1970) [5] W.B. Hall, Heat transfer near the critical point, Adv. Heat Transfer 7 (1971) [6] J.D. 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