International Journal of Heat and Mass Transfer 56 (2013) 741–749

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Convection heat transfer of supercritical pressure carbon dioxide in a verticalmicro 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 100084, Chinab Beijing Key Laboratory of CO2 Utilization and Reduction Technology, Department of Thermal Engineering, Tsinghua University, Beijing 100084, Chinac Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 April 2012Received in revised form 17 August 2012Accepted 18 August 2012

Keywords:Supercritical pressure carbon dioxideMicro tubeConvection heat transferTransient and turbulent flow regimeFlow acceleration

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.08

⇑ Corresponding author at: Key Laboratory for TEngineering of the Ministry of Education, DepartmTsinghua University, Beijing 100084, China. Tel.: +862770209.

E-mail addresses: jiangpx@tsinghua.edu.cn, jiangpxJiang).

This paper presents experimental investigations of the convection heat transfer of carbon dioxide atsupercritical 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 pressuredrop are evaluated and analysed. The results show that the effects of flow acceleration are significant andthe local wall temperature varies non-linearly for both upward and downward flows at the pressures inthe vicinity of critical point and low inlet Reynolds numbers when the heat fluxes are relatively high. Thebuoyancy effect on the heat transfer is negligible in micron scale tubes at inlet Reynolds (from 2600 to6700) and various heat fluxes (from 85 kW/m2 to 748 kW/m2). The flow acceleration due to heatingand pressure drop can strongly influence the turbulence and reduce the heat transfer for high heat fluxesand low inlet Reynolds. Comparison of numerical predictions with the experimental data showed that theAKN low Reynolds number turbulence model gave better agreement than the k–e realizable turbulencemodel with the enhanced wall treatment.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Supercritical fluids are used as working fluids in many indus-trial applications, such as supercritical pressure water cooled reac-tors (SPWR), nuclear reactors using the supercritical CO2 indirectcycle, transpiration cooling of high heat flux surfaces, enhancedgeothermal systems, trans-critical CO2 heat pump and refrigera-tion systems. The platelet transpiration cooling method is one ofthe most efficient methods for protecting high heat flux surfacessuch as rocket thruster walls using hydrogen or methane at super-critical pressures as the coolant flowing through micron scalechannels in the platelets. The platelets are formed by bonding to-gether thin metal sheets containing chemically etched coolant mi-cro channels. The coolant flow rate and the flow distributions canbe precisely regulated by proper design of the coolant passages.This provides efficient thermal management and can be used forthermal protection of the next generation of liquid rocket engines[1]. Enhanced geothermal systems (EGS) aim to extract geothermalenergy from rocks that lack fractures and have low permeability

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hermal Science and Powerent of Thermal Engineering,6 10 62772661; fax: +86 10

@mail.tsinghua.edu.cn (P.-X.

with fluid circulation made possible by increasing the permeabilitythrough hydraulic fracturing, such as by injecting fluid throughdeep boreholes to activate existing rock fractures, with the work-ing fluid flow through these fracture networks controlled by a sys-tem of injection and production boreholes [2]. CO2 has beenproposed as the working fluid in EGS in response to CO2 emissionsreduction needs, with the CO2 at supercritical pressures for theconditions of interest in EGS [2].

When fluids are at supercritical pressures, small fluid tempera-ture and pressure variations result in significant changes in thethermophysical properties. The specific heat, cp, reaches a peak ata temperature defined as the pseudo critical temperature, Tpc. Con-vection heat transfer of fluids at supercritical pressures exhibitsmany special characteristics resulting from the sharp variationsof the thermophysical properties, the buoyancy force induced bythe radial non-uniform density distribution and flow accelerationdue to the fluid expansion as a result of axial density variationsresulting from the axial temperature and pressure variations dur-ing heating.

Eextensive research has been conducted on forced and mixedconvection heat transfer of supercritical fluids in normal size tubesin the past 50 years by Shitsman [3], Petukhov [4], Hall [5], Jackson[6], Krasnoshchekov and Protopopov [7], Protopopov [8], Popovand Valueva [9], Kurganov and Kaptilnyi [10], Jiang et al. [11,12],and He et al. [13]. The special features of the convection heat

Nomenclature

Bo⁄ non-dimensional buoyancy parametercp specific heat at constant pressure [J/(kg K)]d tube diameter [m]g gravitational acceleration [m2/s]G mass flow rate [kg/s]Gr⁄ Grashof numberh bulk specific enthalpy [J/kg]hx local heat transfer coefficient [W/(m2 K)]Kv non-dimensional flow acceleration parameterp pressure [MPa]Pr Prandtl numberR tube inner radius [m]Re Reynolds numberT temperature [K]u velocity [m/s2]x axial coordinate [m]

Greek symbolsap thermal expansion coefficient [1/K]

bT 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/m3] or electrical resistivity [X m]

SubscriptsCO2 carbon dioxidef fluidi inner surfacein inleto outer surfaceout outletpc pseudo criticalp induced by pressure variationT induced by temperature variationw wall

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transfer of supercritical fluids due to the sharp variations of thethermophysical properties and the buoyancy in normal size tubeshave been widely investigated. Shitsman [3] pointed out that, forrelatively large tubes (din = 8 mm), the local wall temperatures var-ied in a complex and nonlinear form with heat transfer deteriora-tion observed in buoyancy-aided flows (upward flow in a heatedpassage) resulting from the buoyancy effect whereas in buoyancy-opposed flow cases (downward flow in a heated passage) the localwall temperatures varied smoothly. Jackson and Hall [14,15]explained the buoyancy affected convection heat transfer behav-iour for supercritical fluids in channels using a semi-empirical the-ory and proposed a non-dimensional buoyancy parameter, Bo⁄, toevaluate the significance of buoyancy. They suggested that themodification of the mean flow field by the buoyancy and the varia-tion of the turbulence production were the main factors affectingthe heat transfer and local temperature variation along the tubefor buoyancy-aided cases and buoyancy opposed cases. For upwardflows with 5.6 � 10�7 < Bo⁄ < 1.2 � 10�6, the buoyancy will reducethe heat transfer while for 1.2 � 10�6 < Bo⁄ < 8 � 10�6, the heattransfer reduction will be gradually reduced as Bo⁄ increases. ForBo⁄ > 8 � 10�6, the heat transfer will be enhanced by the buoyancyaccording to McEligot and Jackson [16]. When the channel size isreduced and the heat flux is increased, the heat transfer is reducedleading to non-monotonically distributions of the wall temperaturefor both flow directions, with such effects found in a number ofinvestigations, for example those of Domin [17] with water at227 bar in a 2 mm tube and Shiralkar and Griffith [18] with CO2

at 75.8 bar in a 3.17 mm tube. These results were explained as beingdue to the reduced turbulence due to the flow acceleration as thefluid expands due to the increased bulk temperature.

Generally, the buoyancy and flow acceleration effects on theconvection heat transfer of fluids at supercritical pressures are re-lated to the channel size, with smaller channels size having moresignificant flow acceleration but less buoyancy effect. The buoy-ancy and flow acceleration effects on the heat transfer of supercrit-ical fluids in tubes with various inner diameters have also beeninvestigated. Liao and Zhao [19,20] found that the buoyancy effectswere significant for all flow orientations even for Reynolds num-bers as high as 105 according to their experimental investigationof the average convection heat transfer for supercritical CO2 in hor-izontal 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 tubediameters less than 1.0 mm, which differs from results in normalsize tubes [14,15,21,22]. He et al.’s [23] numerical simulations ofturbulent convection heat transfer of supercritical pressure CO2

in a 0.948 mm inner diameter vertical tube using the low Reynoldsnumber eddy viscosity turbulence model indicated that for a0.948 mm diameter vertical mini tube and a large Reynolds num-ber of 105, the buoyancy effect was insignificant. Jiang et al.[24,25] investigated the convection heat transfer of CO2 at super-critical pressures in a 0.27 mm diameter vertical mini tube to showthat for inlet Reynolds numbers exceeding 4 � 103, the buoyancyand flow acceleration have little influence on the local walltemperature with no reduction of the convection heat transfercoefficient observed in either flow direction. For relatively low Rey-nolds numbers (<2.9 � 103) and high heat fluxes (e.g. 113 kW/m2),the local wall temperatures varied non-linearly along the tube forboth upward and downward flows and the convection heat trans-fer coefficients for downward flow were greater than for upwardflow. Their experimental results indicated that for mini tubes(e.g. 0.27 mm inner diameter), the flow acceleration due to heatingfor the studied conditions strongly influenced the turbulence andreduced the heat transfer for high heat fluxes. The buoyancy effectcould not be neglected even though relatively small even when theheating was strong. These few studies on the heat transfer ofsupercritical fluids in mini scale channels need to be furthersupplemented with more data since such flow are expected to besignificantly affected by flow acceleration and less affected bybuoyancy.

McEligot et al. [26] proposed a non-dimensional flow accelera-tion parameter, KvT = 4qwb/(qubcpRe), to evaluate the influence ofthe flow acceleration on the heat transfer. They suggested thatfor turbulent flow, the turbulence may be significantly reducedfor KvT P 3 � 10�6 and the flow may even re-laminarise, whichwould reduce the overall heat transfer. Murphy et al. [27] foundthat when KvT 6 9.5 � 10�7, the fluid flow remained turbulent.For convection heat transfer of fluids at supercritical pressures inmicro scale tubes, both the temperature increase and the pressuredrop along the tube will induce significant density decreasesresulting in flow acceleration. Generally, in normal size tubes, thepressure drop is negligible compared with the inlet pressure; thus,the flow acceleration induced by the pressure drop is usually ne-glected, even in a mini tube with 0.27 mm inner diameter, the flow

P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) 741–749 743

acceleration was almost exclusively due to the thermal expansion[24,25]. When the channel size reduced to micron size, the axialpressure gradient is expected to increase drastically; thus, theinfluence of flow acceleration induced by the pressure drop onthe heat transfer needs to be carefully reconsidered.

This paper presents experimental and numerical investigationsof the convection heat transfer of CO2 at supercritical pressures in avertical tube with inner diameter of 99.2 lm for various Reynoldsnumbers, heat fluxes and flow directions. The effects of buoyancyand flow acceleration due to heating and pressure drop are evalu-ated. The results provide a better understanding of the heat trans-fer characteristics of supercritical fluids in micro channels with theeffect of flow acceleration due to both thermal expansion and thepressure drop.

2. Experimental system and data reduction method

The experimental system for measuring the heat transfer in amicro tube and an SEM photograph of a cross section of the microtube are shown in Fig. 1 and details can be found in Ref. [24]. Thetest section was a 50 mm long vertical smooth stainless steel1Cr18N9T tube with a 40 mm (400d) heating section and 5 mm(50d) adiabatic sections before and after the heating section. Thetest section was thermally insulated from the environment. Thetest section was soldered to a stainless steel tubes at the inletand outlet with inner diameters of 2 mm and outer diameters of3 mm which were connected with the test loop by flanges andhigh-pressure fittings. The test section inner tube diameter was99.2 lm and the outer diameter was 217 lm. The test sectionwas insulated thermally and electrically from the test loop by alayer of polytetrafluoethylene (PTFE) placed between the flangesand between the screws and the flanges. The flow direction (up-ward or downward) of CO2 flowing through the test section wasadjusted by a set of valves.

Mixers were installed before where the inlet and outlet CO2

temperatures were measured by accurate RTDs (Pt-100). The CO2

inlet pressure was measured by a pressure transducer (EJA430A)with the pressure drop through the tube measured by a differentialpressure transducer (Model EJA110A), respectively. The local outerwall temperatures of the small tube were measured using micro T-type thermocouples welded onto the outer tube surface. The con-stant heat flux produced by the current stabilized power sourcewas calculated from the heating current (measured by a digitalmultimeter) and the electrical resistance of the test section.

The system was assumed to be at steady state when the varia-tions of the wall temperatures and the inlet and outlet fluid tem-peratures were all within ±0.2 �C and the flow rate and the inletpressure variations were within ±0.2% for at least 10 min.

Fig. 1. Schematic of experimental system a

The local heat transfer coefficient, hx, at each axial location wascalculated as

hx ¼qwðxÞ

Tw;iðxÞ � Tf ;bðxÞð1Þ

where the local heat flux on the inner surface, qw(x), is calculated as:

qwðxÞ ¼I2RxðtÞ � Q loss;Dx

pdiDx¼ I2qðtÞDx=½pðd2

o � d2i Þ=4� � Q loss;Dx

pdiDxð2Þ

The electrical resistivity of the micro tube, q(t), was calculatedusing a correlation of the measured electrical resistivity at varioustemperatures. The heat loss, Qloss,Dx, of the test section was mea-sured by evacuating the tube before the convection heat transfertests without CO2 in the micro tube. The electrical power inputto the tube and the wall temperature were then measured withthis assumed to be equal to the heat loss in the experiments con-ducted as a function of the temperature difference between thetube wall and the surroundings. The measured results showed thatthe heat loss was very small compared with the heat input to thetest section (within 4%), which indicates that the test section waswell insulated.

The local bulk fluid temperature, Tf(x), was obtained using theNIST software REFPROP 7.0 referenced from the local bulk fluid en-thalpy, hf(x), which was in turn calculated by:

hf ðxÞ ¼ hf ;in þqwðxÞpdix

Gð3Þ

The Reynolds number based on the mean bulk temperature was de-fined as:

Re ¼ qudi

l¼ 4G

pdilð4Þ

The inner wall temperature, Tw,i(x), was calculated using the mea-sured outer wall temperature, Tw,o(x), and the internal heat source,qv, as:

TwiðxÞ ¼ TwoðxÞ þqmðxÞ16k

½D2 � d2� þ qvðxÞ8k

D2 lndD

ð5Þ

where qv was calculated as:

qvðxÞ ¼I2RxðtÞ � Qs;Dx

½pðD2 � d2Þ=4�Dxð6Þ

A numerical model of the two-dimensional heat conduction [28]showed that the influence of two-dimensional heat conduction inthe tube wall had little influence except near the inlet and outlet.

The non-dimensional buoyancy parameter, Bo⁄, introduced byJackson and Hall [15], used to evaluate the buoyancy effect was de-fined as:

nd SEM photograph of the test section.

Fig. 2. Local wall temperatures for various wall heat fluxes for downward flowpin = 9.2–9.7 MPa, pout = 7.9 MPa, Tin = 296 K, G = 4625–5043 kg/(m2 s), Rein = 6200–6700. Solid symbols: wall temperatures; lines: fluid temperatures.

Fig. 3. Local Bo⁄ for various wall heat fluxes pin = 9.2–9.7 MPa, pout = 7.9 MPa,Tin = 296 K, G = 4625–5043 kg/(m2 s), Rein = 6200–6700.

744 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) 741–749

Bo� ¼ Gr�

Re3:425Pr0:8 ð7Þ

where,

Gr� ¼ gapd4i qw

km2 ð8Þ

The non-dimensional flow acceleration parameter, Kv, was calcu-lated as:

Kv ¼ mb

u2b

dub

dx¼ � d

Re� bT

dpdxþ 4qwdap

Re2lbcp

¼ Kvp þ KvT ð9Þ

where KvT is the same as the non-dimensional thermal expansionacceleration parameter proposed by McEligot et al. [26] which de-scribes the effect of flow acceleration due to the heating defined as:

KvT ¼4qwdap

Re2lbcp

ð10Þ

Kvp is the non-dimensional flow acceleration parameter describingthe effect of flow acceleration due to the pressure drop through thetube as:

KVp ¼ �d

Re� bT

dpdx

ð11Þ

The experimental uncertainty in the local heat transfer coeffi-cient mainly resulted from the heating and temperature measure-ment uncertainties. Prior to installation, the thermocouples andthe RTDs were calibrated by the National Institute of Metrology,PR China. The accuracies of the RTDs which measure the inletand outlet bulk temperatures were ±0.1 �C while the accuraciesof the thermocouples measuring the outer wall temperatureswere ±0.15 �C in the temperature range used in the present study.The accuracy of the pressure transducer (Model EJA430A) was0.075% of the full range of 14 MPa and the accuracy of the differen-tial pressure transducer (Model EJA110A) was 0.075% of the fullrange of 5 MPa.

A detailed uncertainty analysis showed that the uncertainty ofthe heat transfer coefficient was ±12.7%. The uncertainty in theinlet pressure was estimated to be ±0.13%. The uncertainty of themass flow rate was 0.1%.

3. Experimental results and discussion

The heat transfer of supercritical CO2 flowing through a 99.2 lminner diameter vertical tube was experimentally investigated forinlet pressures of 7.7–9.7 MPa which is close to the ciritical pres-sure 7.38 MPa, and inlet Reynolds numbers of 2600–6700.

The distributions of the local wall temperature, local heat trans-fer coefficient, buoyancy parameter, Bo⁄, and the flow accelerationparameter, Kv, and its two components, KvT and Kvp, along the testsection are presented for various inlet Reynolds numbers, pres-sures and wall heat fluxes. The results illustrate the effects of buoy-ancy and flow acceleration due to the thermal expansion andpressure drop on the heat transfer.

3.1. Relatively high inlet pressure and Reynolds number

The local wall temperature and bulk temperature variationsalong the test section are shown in Fig. 2 for various heat fluxesat inlet test conditions of pin = 9.2–9.7 MPa, Tin = 296 K, andRein = 6200–6700 for downward flow cases. The local pseudo crit-ical temperature, Tpc, decreases along the tube because of the sig-nificant pressure drop along the test section in the micro tube;Tpc was calculated as a function of pressure as:

Tpc ¼ 150:55þ 6:12� 10�1p� 1:66� 10�3p2 þ 5:61

� 10�5p2:5 � 5:61� 10�7p3 ð12Þ

where Tpc is in K and the pressure p is in MPa [20].The inlet bulktemperature is lower than Tpc for all cases in Fig. 2. The bulk tem-perature increases along the test section as the CO2 absorbs the heatgenerated in the tube wall. Also, the local wall temperature in-creases with increasing wall heat flux, qw. Within the heat fluxrange in this study, the local wall temperature increases along thetest section, with no abnormal local wall temperature variations ob-served. The variations of the local non-dimensional buoyancy effectparameter, Bo⁄, are shown in Fig. 3 for the various heat fluxes cor-responding to the cases in Fig. 2. For these heat fluxes, Bo⁄ has amagnitude of 10�10, much lower than the threshold of 5.6 � 10�7-

given by McEligot and Jackson [16], with buoyancy effects onlyimportant for Bo⁄ above this value. Thus, the low Bo⁄ in Fig. 3 indi-cate that the buoyancy effect is insignificant. The local wall temper-ature variations for the upward and downward flow when the othertest conditions are held constant are consistent and the differencesare quite small, which also indicates that buoyancy is insignificantfor the heat and mass flux range in Fig. 2.

As mentioned in the introduction, when the tube size is reducedto the micro size, the buoyancy effect is reduced, however, the flowacceleration effect, including both the flow acceleration due to

Fig. 4a. Local KvT for various wall heat fluxes pin = 9.2–9.7 MPa, pout = 7.9 MPa,Tin = 296 K, G = 4625–5043 kg/(m2 s), Rein = 6200–6700. Fig. 5. Thermophysical property variations of CO2 at p = 8.8 MPa.

P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) 741–749 745

thermal expansion and the flow acceleration due to the pressuredrop in the tube is more significant than in large tubes. The localnon-dimensional heating flow acceleration parameter, KvT, is pre-sented in Fig. 4a for various heat fluxes. When the heat fluxes arerelatively low, the local KvT increases along the test section withthe bulk temperature increasing and approaching Tpc. KvT also in-creases with increasing heat flux as shown in Fig. 4a. As the heatflux increases further, KvT increases with the heat flux, reaches apeak, and then decreases rapidly when the wall and fluid temper-atures are larger than Tpc. KvT has a magnitude of 10�7, with the lo-cal maximum reaching 4.3 � 10�7 for the maximum heat flux inthe present test conditions at qw = 748 kW/m2. The local non-dimensional compressible flow acceleration parameter, Kvp, at var-ious heat fluxes is shown in Fig. 4b. Since the tube is very small, thepressure drop along the tube can exceed 1 MPa, with pressure gra-dients of 20 MPa/m, which causes the fluid density to decreasegreatly along the tube and intensifies the flow acceleration. Kvp

has a similar magnitude of 10�7 of KvT as shown in Fig. 4b, whichindicates that the flow acceleration induced by the pressure dropcannot be neglected when the tube size is reduced to microns. Thisdiffers from the study conducted by McEligot et al. [26], in whichthe pressure drop induced flow acceleration was negligible com-pared with the thermal expansion flow acceleration and thestreamwise flow acceleration was mainly caused by the thermalexpansion due to the heat transfer in larger tubes.

Fig. 4b. Local Kvp for various wall heat fluxes pin = 9.2–9.7 MPa, pout = 7.9 MPa,Tin = 296 K, G = 4625–5043 kg/(m2 s), Rein = 6200–6700.

As shown in Fig. 4b, for heat fluxes of qw = 211–496 kW/m2, Kvp

increases along the test section as the bulk temperature increasesand approaches Tpc. This is mainly related to the variation of theisothermal compressibility, bT, and gradually decrease of q withthe fluid temperature as shown in Fig. 5. Higher heat fluxes gener-ate larger Kvp, especially near the outlet where the bulk tempera-ture approaches Tpc; thus, a small fluid temperature increaseresults in a large increase of the isothermal compressibility and alarge decrease of q, so the pressure drop more strongly affectsthe fluid expansion. When the heat flux increases further to586 kW/m2 and 748 kW/m2, Kvp first increases along the test sec-tion, reaches a maximum when the bulk temperature is close to Tpc,and then decreases along the test section with a sharp peak in Kvp.The drastic increase and decrease of Kvp near Tpc is mainly inducedby the peak of the isothermal compressibility, bT, and the variationof q.

The heat flux does not always intensify the isothermal com-pressible flow acceleration. When the heat flux exceeds 586 kW/m2, the maximum value of Kvp decreases with increasing heat fluxas shown in Fig. 4b due to the combined influence of the Reynoldsnumber and the isothermal compressibility on Kvp, as defined in Eq.(11). For higher heat fluxes, the local bulk temperature increasesand exceeds Tpc downstream, with the local Re increasing drasti-cally as a result, while the isothermal compressibility, bT, decreases.Thus, the tradeoffs between Re and bT cause the decrease in maxi-mum value of Kvp as the heat flux exceeds the threshold value.

Fig. 6. Local Kv for various wall heat fluxes pin = 9.2–9.7 MPa, pout = 7.9 MPa,Tin = 296 K, G = 4625–5043 kg/(m2 s), Rein = 6200–6700.

Fig. 7. Local Nu variations along the test section for various heat fluxes pin = 9.2–9.7 MPa, pout = 7.9 MPa, Tin = 296 K, G = 4625–5043 kg/(m2 s), Rein = 6200–6700.

746 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) 741–749

The variation of the local non-dimensional flow accelerationparameter, Kv, which is the sum of KvT and Kvp, along the test sec-tion is shown as Fig. 6 for various heat fluxes. A peak is observedfor high heat flux cases, with the peak position moving closer tothe entrance with increasing heat flux. The Nu variations are pre-sented in Fig. 7. For the heat fluxes qw = 496 kW/m2 and qw = 586 -kW/m2, the bulk temperature increases along the test section andthe wall temperature increase above Tpc near outlet; thus, the spe-cific heat, cp, increases which enhances the heat transfer betweenthe fluid and the wall and the Nu increases significantly as shownin Fig. 7. As the heat flux increases further to 684 kW/m2 and748 kW/m2, the wall temperature increases to a value high thanTpc and the bulk temperature also increase to above Tpc down-stream, which indicates that the fluid is in the gas-like where thedensity, specific heat and thermal conductivity are small and theheat transfer is reduced significantly as a result. The maximum lo-cal Kv is below 6 � 10�7, less than the threshold, 9.5 � 10�7, pro-posed by Murphy et al. [27], as shown in Fig. 6. No local heattransfer deterioration and no abnormal wall temperature varia-tions occur in the experiments as shown in Figs. 2 and 7, whichis consistent with the conclusion of Murphy et al. [27] that the lo-cal turbulence suppression induced by the flow acceleration isinsignificant when Kv 6 9.5 � 10�7.

Fig. 8. Local wall temperatures for various wall heat fluxes for downward flowpin = 7.7–7.8 MPa, pout = 7.6 MPa, Tin = 297 K, G = 1823 kg/(m2 s), Rein = 26. Solidsymbols: wall temperatures; lines: fluid temperatures.

3.2. Relatively low inlet pressures and Reynolds numbers

The local wall temperature and bulk temperature variationsalong the test section for various heat fluxes at inlet test conditionsof pin = 7.7–7.8 MPa, Tin = 297 K, and Rein = 2600 are shown in Fig. 8for downward flow. The corresponding local Nu variations are pre-sented in Fig. 9. For the relatively low heat flows (qw = 85 kW/m2),the local wall temperature increases along the test section. The Nusignificantly increases and then decrease near the outlet. This ismainly due to the fluid bulk temperature increasing and approach-ing Tpc, so the specific heat increases, which enhances the heattransfer between the fluid and the wall, as shown in Fig. 9. Forhigher heat fluxes, the local wall temperature exhibits a non-linearvariation and sharply increases near the entrance, then decreasesnear x/d = 100–150 and then increases downstream. The corre-sponding local Nu significantly drops to the minimum near x/d = 100, which indicates the heat transfer is impaired near the en-trance and then increase downward, which indicate the heat trans-fer is recovered as shown in Fig. 9.

Fig. 10 presents the corresponding local Bo⁄ variations for vari-ous heat fluxes. For higher heat fluxes, when the bulk temperatureis below Tpc, Bo⁄ increases with increasing heat flux, reaches amaximum when the bulk temperature approaches Tpc, and thendecreases sharply. This is mainly because when the bulk tempera-ture is lower than Tpc, the fluid in the centre is in a liquid-like state,while when the wall temperature is higher than Tpc, the fluid nearthe wall is in a gas-like state, so the radial density difference isquite large and the buoyancy more significantly affects the flowand heat transfer. However, as the heat flux increases further, thebulk temperature exceeds Tpc and all the fluid across the cross-section is in a gas-like state, so the density gradient decreasesand the buoyancy effect is reduced.

The Bo⁄ variation with the fluid temperature is quite similar tothat of KvT, which is presented in Fig. 11a because both Bo⁄ and KvT

are related to the thermal expansion coefficient, ap, which repre-sents the density variation with temperature at constant pressure.A larger ap mean sharper density variations; thus, the buoyancyand thermal flow acceleration effects are stronger. The density de-creases very rapidly around Tpc with the thermal expansion coeffi-cient reaching a maximum, so the buoyancy and thermal flowacceleration effects are strong. Bo⁄ increases more here as the pres-sure and inlet Reynolds number decrease compared with the casesin Section 3.1. The local Bo⁄ reaches 10�9 when the heat flux ishigh; however, it is still far below the threshold of 6 � 10�7, sothe buoyancy effect on the heat transfer is still negligible even if

Fig. 9. Local Nu variations along the test section at various heat fluxes pin = 7.7–7.8 MPa, pout = 7.6 MPa, Tin = 297 K, G = 1823 kg/(m2 s), Rein = 2600.

Fig. 10. Local Bo⁄ for various wall heat fluxes pin = 7.7–7.8 MPa, pout = 7.6 MPa,Tin = 297 K, G = 1823 kg/(m2 s), Rein = 2600.

Fig. 11a. Local KvT for various wall heat fluxes pin = 7.7–7.8 MPa, pout = 7.6 MPa,Tin = 297 K, G = 1823 kg/(m2.s), Rein = 2600.

Fig. 11b. Local KvP for various wall heat fluxes pin = 7.7–7.8 MPa, pout = 7.6 MPa,Tin = 297 K, G = 1823 kg/(m2.s), Rein = 2600.

Fig. 12. Local Kv for various wall heat fluxes pin = 7.7–7.8 MPa, pout = 7.6 MPa,Tin = 297 K, G = 1823 kg/(m2.s), Rein = 2600.

P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) 741–749 747

the pressure is quite close to pc and the inlet Reynolds number isquite low in the micro tubes.

Figs. 11a and 11b show the local KvT and Kvp variations alongthe test section. Comparing Figs. 4a and 11a shows that for thesame inlet temperature and heat flux conditions, KvT increases

with decreasing pressure and inlet Reynolds number. Kvp alsoreaches a peak as the bulk temperature approaches Tpc. Kvp in-creases as the heat flux increases for relatively low heat fluxes untilthe heat flux exceeds a critical value, 173 kW/m2 in the presenttests. Kvp is about 10�7, indicating that the effect of the flow accel-eration due to pressure drop on the flow and heat transfer cannotbe negligible.

The corresponding non-dimensional flow acceleration parame-ter, Kv, for the various heat fluxes is shown in Fig. 12. As shown inFig. 9, the Nusselt number decreases for x/d = 25–200 for higherheat fluxes, where Kv is relatively high, indicating that the flowacceleration begins to suppress the turbulence and reduce the heattransfer when Kv exceeds 6 � 10�7.

4. Numerical simulations and comparison with experiments

In the numerical simulations, the inner and outer diameters ofthe vertical stainless steel tube were 99.2 lm and 217 lm as inthe experiments. The electrical heating of the test section wasmodelled as a uniform heat source. The heated section was40 mm long and the adiabatic sections before and after the heatedsection were 5 mm long (50d). The flow was assumed to be two-dimensional turbulent flow.

The conjugate convection heat transfer and heat conduction inthe wall with an internal heat source in the vertical tube wasnumerically simulated using FLUENT 12 with various turbulencemodels used to model the turbulence. The governing equationsfor the steady state, two-dimensional turbulent flow of a supercrit-ical pressure fluid in a vertical tube accounting for the temperature-dependent property variations and buoyancy can be written as:

Continuity:

1r

@

@xðqrUÞ þ @

@rðqrVÞ

� �¼ 0 ð13Þ

U-momentum:

1r

@

@xðqrU2Þ þ @

@rðqrVUÞ

� �¼ � @p

@xþ qg þ 1

r2@

@xrle

@U@x

� �� ��

þ @

@rrle

@U@rþ @V@x

� �� ��ð14Þ

V-momentum:

1r

@

@xðqrUVÞ þ @

@rðqrV2Þ

� �¼ � @p

@rþ 1

r@

@xrle

@U@rþ @V@x

� �� ��

þ 2@

@rrle

@V@r

� �� ��� 2

leVr2 ð15Þ

Fig. 13. Comparisons of experimental and numerical results using the AKN modelfor downward flow pin = 7.7–7.8 MPa, pout = 7.6 MPa, Tin = 297 K, G = 1823 kg/(m2.s),Rein = 2600.

Fig. 14. Comparisons of experimental and numerical results using the k–e modelfor downward flow pin = 7.7–7.8 MPa, pout = 7.6 MPa, Tin = 297 K, G = 1823 kg/(m2.s),Rein = 2600.

748 P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) 741–749

where le is the effective viscosity defined by in le = l + lT in whichlT is the turbulent viscosity defined as:

lt ¼ qflclk2=e ð16Þ

where fl is damping function to account for near-wall effects and Clis constant.

Energy equation:

1r

@

@xðqCprUTÞ þ @

@rðqCprVTÞ

� �

¼ 1r

@

@xrCp

lPrþ lT

rT

� �@T@x

� �þ @

@rrCp

lPrþ lT

rT

� �@T@r

� �� �ð17Þ

where Pr is the molecular Prandtl number and rT is the turbulentPrandtl number.

The flow of the supercritical fluids can be strongly distorted incomparison with conventional wall shear flows duo to the influ-ences of buoyancy, flow acceleration and non-uniformity of fluidproperties, so the use of standard k–e turbulence model coupledwith a simple wall function is not appropriate. Jiang et al. [29,30]and He et al. [23] performed numerical simulations of convectionheat transfer of supercritical fluids and the results showed thatthe Low-Reynolds number turbulence models were able to repro-duce the temperature variations trend influenced by the buoyancyand the flow acceleration. Low-Reynolds number turbulence mod-els, which extend the standard model by including the modeling ofthe effects of the wall and the molecular diffusion, can be used inthe wall region as well as the core, and therefore potentially moresuitable for the supercritical fluids. The low Reynolds number eddyviscosity turbulence models, Abe, Kondoh and Nagano (AKN) [31]and the k–e realizable turbulence model with the enhanced walltreatment were used in the present study. The constitutive andtransport equations can be written as:

Turbulent kinetic energy:

@ðqUkÞ@x

þ 1r@ðrqVkÞ@r

� �¼ @

@xlþ lt

rk

� �@k@x

� �þ 1

r

� @

@rr lþ lt

rk

� �@k@r

� �þ Pk þ Gk

� qeþ qD ð18Þ

where, the shear production term Pk is:

Pk ¼ lt 2@U@x

� �2

þ @V@r

� �2

þ Vr

� �2( )

þ @U@rþ @V@x

� �2" #

ð19Þ

The gravitational production term Gk is:

Gk ¼ �q0u0gx ¼ �blt

C1t

ke

� �@U@rþ @V@x

� �@T@r

� �gx ð20Þ

Turbulence dissipation rate:

@ðqUeÞ@x

þ 1r@ðrqVeÞ

@r

� �¼ @

@xlþ lt

re

� �@e@x

� �þ 1

r@

@rr lþ lt

rk

� �@e@r

� �

þ Ce1f1ekðPk þ GkÞ � Ce2f2

qe2

kþ qE ð21Þ

The NIST Standard Reference Database 23 (REFPROP) Version 7was used to calculate the temperature and pressure dependentproperties of CO2. The SIMPLEC algorithm was used to couple thepressure and the velocities. The QUICK scheme was used for themomentum and energy equations. The convergence criteria re-quired a decrease of at least four orders of magnitude for the resid-uals. The mesh was refined in the radial direction towards the wallto ensure the y+ value at the first node of the mesh near the wallwas less than 1. The whole domain was discretized into a mesh

of grids, 3600 nodes in the axial direction and (56 + 21) nodes inthe radial direction (fluid region + tube wall). Calculations with amore refined mesh showed that the results were grid independent.

Figs. 13 and 14 compare the measured and calculated wall tem-peratures for downward supercritical pressure CO2 flowing in thevertical heated micro tube for a relatively low Reynolds number(Rein = 2600) at various wall heat fluxes using the AKN low Rey-nolds number turbulence model and the k–e realizable turbulencemodel with the enhanced wall treatment. When the heat flux isrelatively low (85 kW/m2), the local wall temperature increaseslinearly along the test section, with the wall temperatures pre-dicted using the AKN and k–e realizable turbulence models bothcorresponding well with the measured data, as shown in Figs. 13and 14. For higher wall fluxes (173 kW/m2, 221 kW/m2, and244 kW/m2), where the wall temperature is higher than Tpc, thebulk temperature crosses Tpc inside the test section and the ther-mophysical properties vary dramatically, so the local wall temper-ature varies non-linearly along the test section and the heattransfer is significantly affected by the flow acceleration. Nearthe entrance where the heat transfer is affected by the flowacceleration, the AKN low Reynolds number turbulence modelover-predicts the effect of the flow acceleration and the calculatedtemperatures are higher than the measured values. The k–e

P.-X. Jiang et al. / International Journal of Heat and Mass Transfer 56 (2013) 741–749 749

realizable model with the enhanced wall treatment fails to repro-duce the heat transfer reduction. For the heat transfer recoveryregion (x/d > 140), the AKN and k–e models are both able to wellpredict the local wall temperature variation with the results usingthe AKN model better than that with k–e realizable model. Thus,the turbulence model for the heat transfer deterioration regiondue to flow acceleration should be modified in further studies.

5. Conclusions

The convection heat transfer of CO2 at supercritical pressures ina vertical micro tube with inner diameter of 99.2 lm was experi-mentally and numerically investigated for various heat fluxes,pressures and inlet Reynolds numbers. The buoyancy effect andflow acceleration due to heating and the pressure drop effect wereanalysed. This study shows that:

(1) When the inlet pressure and Reynolds number are relativelyhigh, pin = 9.2–9.7 MPa and Rein = 6200–6700 in this study,the local wall temperature increases along the test sectionand the heat transfer coefficient is high. When the bulk tem-perature is lower than Tpc and the wall temperature is higherthan Tpc the heat transfer coefficient decreases as the bulkand wall temperatures increase further. The influences ofbuoyancy and flow acceleration are very weak.

(2) The effects of the flow acceleration are more significant atrelatively lower pressures close to pc and lower inlet Rey-nolds numbers. At pressures of 7.7–7.8 MPa and inlet Rey-nolds number of 2600, the local wall temperature exhibitsnon-linear variations near the entrance due to the flowacceleration. The buoyancy effect on the heat transfer is neg-ligible in micro tubes such as used in the present study. Theflow acceleration effect due to heating and the pressure dropis significant for low inlet Reynolds numbers (Rein = 2600)and high heat fluxes and the heat transfer is reduced whenthe non-dimensional flow acceleration parameter, Kv,exceeds about 6 � 10�7 for Rein = 2600.

(3) The results show that the AKN turbulence model and the k–erealizable turbulence model with the enhanced wall treat-ment are both able to predict the temperature variationsfor flows which are not significantly affected by the buoy-ancy and flow acceleration. For flows which are affected bythe flow acceleration, the AKN low Reynolds number turbu-lent model responds to the flow acceleration but over-predicts the wall temperature, while the k–e realizableturbulence fails to respond to the flow acceleration.

Acknowledgments

This project was supported by the Key Project Fund from theNational Natural Science Foundation of China (No. 50736003).We thank Professor J.D. Jackson in the School of Mechanical, Aero-space and Civil Engineering, the University of Manchester, UK forhis many suggestions for this research. We also thank Prof. DavidChristopher for editing the English.

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