Numerical investigation on heat transfer of supercritical
CO2 in heated helically coiled tubes
Kaizheng Wang1, Xiaoxiao Xu*1,Yangyang Wu1, Chao Liu1, Chaobin Dang2
1.Key Laboratory of Low-grade Energy Utilization Technologies and Systems, Chongqing
University, No. 174 Shazhengjie, Shapingba, Chongqing, 400044, P.R. China
2.Department of Human and Engineered Environmental Studies, The University of Tokyo, 5-1-5
Kashiwanoha, Kashiwa-shi, Chiba 277-8563, Japan

Abstract: Numerical simulation on heat transfer of supercritical CO2 in heatedhelically coiled tubes is performed to evaluate the performance of turbulence models
in predicting heat transfer of supercritical CO2 in the helically coiled tube, and tohelp better understanding the heat transfer mechanism. All turbulence models yield
similar tendencies in heat transfer coefficient in the helically coiled tube. The SST(shear-stress transport) model gives the best prediction to the experimental data due
to accurate predictions of flow separation under adverse pressure gradients. The
parameter of Gr / Re2.7 is incapable of predicting the buoyancy effect onset ofsupercritical CO2 in the helically coiled tube. The turbulent Prandtl number has little
influence on the calculated heat transfer coefficient.
Key words: numerical simulation, helically coiled tube, supercritical CO2, heattransfer* Corresponding author. Tel：+86 023-65112469，Fax：+86 023-65112469

E-mail address: xuxiaoxiao@cqu.edu.cn，(Xu Xiao-xiao).

Nomenclatureainner pipe radius [m]Asurface area [m2]bcoil pitch divided by 2π [m]cpspecific heat at constant pressure [J/(kgK)]

Cu , C , Ck

turbulence models’ constantsdtube diameter [m]Dadditional term in the k-equationEflow energy [W/kg]

f1 , f 2

functions in the dissipation equationf

damping functiongacceleration due to gravity[m/s2]

Gmass flux [kg/(m2 s)]

Gr

Grashof numberhheat transfer coefficient [W/(m2K)]

kturbulence kinetic energy [m2/s2]

ppressure [Pa]Pkturbulent shear production[W/m3]

Pr

Prandtl numberqheat flux [W/m2]

R

curvature radius [m]ReReynolds numberT

temperature [K]uvelocity [m/s]xCartesian coordinates [m]ynormal distance from the inner wall[m]y+, y*

non-dimensional distance from wall

Greek symbols

volume expansion coefficient [K-1]torsion [b/R]

curvature ratio [a /R]

rate of dissipation of k [m2/s3]



global azimuthal angle around the curvature axis [°]thermal conductivity [W/(mK)]

dynamic viscosity [Pas]

density [kg/m3]

 k ,

turbulent Prandtl number for k and shear stress [N/m2]



local polar coordinates in the cross-section [°]Subscriptsbbulk fluidccircumference of cross-sectioncw

node on walligeneral spatial indicespc

pseudo-criticaltturbulent quantity

w

wall

1. Introduction
The heat transfer of supercritical CO2 in helically coiled tubes appears in power
plant technology and cooling systems for heat pump and air-conditioning systems.
Supercritical CO2 is used as an experimental fluid to investigate mechanics ofsupercritical fluids. Supercritical CO2 is use as an experimental fluid for two reasons:first, the properties of fluids have similar trends at supercritical pressures. Second,
CO2 reaches the critical point at much lower temperature and pressure compared tothat of water, which results in the lower cost of performing experiments [1]. The
study of heat transfer of supercritical CO2 in helically coiled tubes is of great
importance in heat exchangers design.
Experimental and numerical studies on the flow and heat transfer ofsupercritical fluids were performed in straight tubes [2-5]. Dang, et al. [6,7]experimentally analyzed the effects of lubricating oil on heat transfer of supercritical
CO2 in horizontal tubes. An experimental investigation on heat transfer ofsupercritical CO2 under cooling condition is conducted by Bruch, et al. [3]. Jiang, etal. [8] carried out experiments on flow and heat transfer of supercritical CO2 in a
porous tube. Experiments on heat transfer of supercritical fluids in micro-tubes wereconducted by Withag, et al. [9], Oh, et al. [10], and Lee, et al. [11].
Due to a relatively small tube-diameter used in practice and a relatively highoperating pressure for supercritical CO2, it is hard to measure the practical flow
pattern of supercritical fluids directly. Numerical calculation may be the only
feasible way to provide such data. Numerical simulation can potentially play a very

important role in improving the understanding of the flow and heat transfer
mechanism.
Jiang, et al. [12] investigated the heat transfer of supercritical CO2 in a verticalsmall tube. The numerical simulation was carried out by using several turbulencemodels. They found that the low Reynolds number turbulence model proposed by
Yang–Shih [13] reproduced the general characteristic occurred in the experiments.
Lots of low Reynolds number turbulence models were employed by He, et al. [14]to simulate heat transfer of supercritical CO2 flowing upwards in a vertical tube.
They concluded that all of models were able to reproduce some generalcharacteristic to some extent. V2F eddy viscosity turbulence model was used in the
numerical computation of heat transfer of supercritical CO2 in inclined pipes [15].
Research was conducted on the heat transfer of supercritical CO2 in a vertical small
tube with inner diameter of 99.2 μm and the AKN low Reynolds number turbulencemodel gave the better prediction of the heat transfer than the k-ε realizableturbulence model [16].
Several modes were applied to simulate the heat transfer behavior in straighttubes for supercritical CO2 and the study showed the JL model [17] gave the best
prediction to the heat transfer than other models [18] based on their experimental
results [19]. They also concluded that turbulent Prandtl did not have a significant
influence on the heat transfer. Cao, et al. [20] numerically investigated the laminarheat transfer of supercritical CO2 in horizontal triangle and circular tubes. The
effects of fluid physical properties and buoyancy were analyzed. The heat transfer ofsupercritical CO2 in the tubes was enhanced due to the effect of buoyancy.
Numerical studies on heat transfer of supercritical CO2 were performed by He, et al.
[21]. They adopted low Reynolds turbulence number of k   model and V2F
types. The effect of buoyancy on heat transfer and turbulence production insupercritical fluids could be very significant.
All the above mentioned studies focused on the heat transfer of supercriticalfluids in straight tubes. Few studies can be found in open literature on the heattransfer of supercritical fluids in helically coiled tubes. A large amount ofexperimental and numerical studies of the flow and heat transfer characteristics in
the helically coiled tube was concentrated in constant-property fluids, withconsideration a variety of situations, such as effects of coiled parameters [22, 23],
phase change flow

[24, 25], Nanofluids flow [26, 27], ice slurries flow [28],Reynolds number effects [29], Dean number effects [22, 23], and so forth.
Jayakumar, et al. [23] numerically investigated the local Nusselt number along thelength and circumference. Numerical investigation on the heat transfer of water inthe helically coiled tube was conducted by Di Liberto, et al. [30]. The symmetries of
the heat transfer and turbulence were founded. The maximum values were located in
the outer side.
Apparently, from the above mentioned researches that focus either on the heattransfer in straight pipes using supercritical fluids, or on the heat transfer in thehelically coiled tubes using constant-property fluids. So, it is greatly significant tostudy the flow and heat transfer characteristics of supercritical CO2 in the helicallycoiled tube.
In this study, the heat transfer of supercritical CO2 is calculated by someturbulence models. The calculated results are compared with experimental results.
The main contents and contributions include the following aspects. First, theperformance of turbulence models in predicting the heat transfer coefficient ofsupercritical CO2 in helically coiled tubes is evaluated. Second, in order tounderstand the heat transfer mechanism, we provide the information of the velocity
and turbulence kinetic energy. Furthermore, we investigate the effect of the
turbulent Prandtl number on the heat transfer of supercritical CO2 in the helicallycoiled tube. The paper provides the mechanism of the heat transfer in the helicallycoiled tube for the design of high-efficiency heat exchanger.

2. Numerical modeling
2.1 Calculation model and Turbulence model
Governing equations for the flow
Continuity: ui   0
xi

(1)

Momentum:
ui 

p
 
 ui u j    gi    t x jxi x j x j 

(2)  T ui (  E  p    k  ui ij xi
xi  xi
(3)

Energy:where t is turbulent viscosity which is based on the turbulence model.
The simulation use a number of turbulence models: standard k   model and
RNG k   model with enhanced wall treatment, SST k   model [31], AB
model [32], LB model [33], LS model [17], YS model [13], AKN model [34] andCHC model [35].
The transport equations for the various models can be expressed in a genericform.
t  k    kui        Pk  Gk     D
xi  k  xi t      ui         C 1 f1Pk  C 1C 3Gk  C 2 f 2     E
xi    xi 
k


t 

C f   k 2

t T
Prt xi

;

 

(4)

(5)



 u u j  ui
Pk  t  i  x j xi  x j

Gk   gi

(3)

(6)

1 
 T

(7)

The damping functions model and constants are presented in Table 1.

2.2 Geometry and mesh model
The mainly geometrical parameters of the helically coiled tube are illustrated inFig. 1. The helically coiled tube can be described by the tube radius (R), the coilradius ( a ) and the coil pitch  2 b  . In this study, the values of R, a , 2πb, are fixed
as 141.5 mm, 9 mm, 32 mm , respectively; and the length of the tube is 5500 mm.
The dimensionless torsion and curvature can be defined as:

a
R
b

R

=

(8)
(9)

Fig. 2 represents the mesh of the simulations which are established in FLUENT
preprocessor GAMBIT [36].

2.3 Boundary condition and numerical scheme
The software FLUENT 6.3 is used in the numerical calculation. The conditions
are specified that supercritical CO2 enters the helically coiled tube from the bottom
at a temperature of 288 K (mass flow inlet boundary condition) and leaves from thetop (outflow boundary condition). The supercritical CO2 is heated by a specified
constant wall heat flux.
The couple of velocity and pressure is solved by The SIMPEC algorithm.
QUICK is used in the discretization of momentum and energy equations. Turbulencekinetic energy and turbulence dissipation rate equations are adopted second order
upwind. When the relative residual of for each governing equation is less than 10-5 ,the numerical calculation is considered converged.
Supplementary Fig.1 shows thermal conductivities, thermal capacities,
densities and dynamic viscosities of supercritical CO2 at a pressure of p = 8.0 MPa.
For the thermodynamic properties, the NIST Standard Reference Database 23 isused. When the pressure is equal to 8.0 MPa, the cp reached a peak value at thepseudo-critical temperature (Tpc) of 307.6 K. The physical properties are used to
calculate in FLUENT software by defining a piecewise-linear function oftemperature for the properties of supercritical CO2.

2.4 Mesh independency
It is important to carry out an independent check of the mesh because mesh
quality plays an important role in numerical calculation. The grid number is set to be7813076 = 2402356, sectionaxial. Here, the sectional number represents the total
number of grids on one cross section of the tube. Generally, y+ should be approachto y+ = 1 to ensure the characteristics near the wall are captured. The dimensionlessdistance y+ in the calculation is smaller than 0.8. The mesh is compressed in theaxial direction in the radial direction towards the helically coiled wall based on thecalculation for each case.
Mesh independency is carried out. Table 2 presents three different numbers ofhexahedral meshes. The SST model is applied at the pressure 8.0 MPa, the mass flux
97.8 kg/(sm2), the heat flux 9.03 kW/m2 and the inlet temperature 288 K. The
comparisons of calculated Tw are showed in Fig. 3. In the Table 2, it can be seenthat the average relative error between Case-2 and Case-3 is 0.0769%. The grid
number which is set to be 2402356 is enough to make the calculated results
independent from grids.

3. Results and discussion
Measured data for supercritical CO2 in helically coiled under heating conditions
are presented by Wang, et al. [37]. The calculation is performed at the pressure 8.0

MPa, the mass flux 97.8 kg/(sm2), the heat flux 9.03 kW/m2 and the inlet
temperature 288 K.
The heat transfer coefficient is calculated by the formulah  qw / Tw  Tb 

(10)where the wall temperature Tw is circumferential averaging wall temperatureand the Tb at a cross section is defined asTb  u C TdA u C dA
p

(11)

p

where dA is an elemental area of the tube across-section.

3.1 Model comparison and selection
Wen, et al. [38] used some turbulence models to simulate the heat transferfeatures of supercritical water in vertical tube, they found that the SST model was
better in predicting heat transfer than k   model. However, Lei, et al. [39]concluded that the RNG k   give better prediction than SST model. So the model
validation work was required. Numerical investigations on heat transfer ofsupercritical fluids were conducted by many researchers. They made conclusions
that different models performed well at different conditions. Some numericalinvestigations on heat transfer of supercritical fluids in straight tubes are shown in
the Table 3
Fig. 4 shows the comparisons of calculated h with the experimental data [37].
Overall, all turbulence models obtain similar tendencies in h along the angle (θ) of
the helically coiled tube.

Fig. 4 (a) shows the calculated results of the standard k   model, the SST
model and the RNG model. The standard k   model and the RNG k   model
overestimate the heat transfer coefficient over the entire angle (θ) range, whereas the
SST model simulates well the experimental h, but slightly overestimated theexperimental h when θ < 720o.
Fig. 4 (b) shows the three turbulence models cannot predict the experimental hwell. The AB model well predicts the measured h over a wild region except in theregion of θ < 1500o. The LB model over-predicts experimental h over a wild rangeof the angle (θ), but underestimates the measured h in the beginning and the end of
the helically coiled tube. The LS model overestimates the measured h in whole
region.
Fig. 4 (c) shows the calculated results of the AKN model, the YS model andthe CHC model. The YS model and the CHC model predict well only in the regionof the maximum experimental h and in the beginning of the helically coiled tube,
respectively. The AKN model predicts well except θ > 1200o.
From the discussion above, among all the turbulence models in this calculation,
the SST model gives the best prediction to the experimental data. The maximum
error between the experimental h and the predictions of the SST model isapproaching to 10%. In an early study [14], because the non-uniformity of fluid
property and buoyancy can lead to extremely distorted flow, the standard k  model with simple wall functions is not appropriate. The literature [44] studies thereasons that the performances of turbulence models are different from one to another.

The different evaluation on turbulence kinetic energy causes different calculation
results.
Fig. 5 shows the trends of wall temperature and bulk temperature predicted by
SST turbulence model. The SST model reproduces the measured temperature very
well whereas slightly underestimates the measured temperature as θ > 1600o. The
maximum error between the calculated temperature and experimental temperature is
within 2%. In conclusion, the SST model gives the best prediction to theexperimental h, wall and bulk temperature, indicating that SST model applied in thispaper are suitable. Thus, hereinafter, CFD results by SST model are used for further
analysis.

3.2 Effects of heat flux
The effects of heat fluxes on the heat transfer performances of supercritical
CO2 in the helically coiled tube are showed in Fig. 6. With increasing heat flux, themaximum value decreases. This can be attributed to the radial distribution ofproperties, especially the influences of thermal conductivity and specific heat. As
the increase in heat flux, the heat transfer capacity (cp) decreases, which results in
reducing the influence of cp on heat transfer . This phenomenon has been discussed
by previous studies. Generally speaking, the increase of the heat flux has asignificant influence on diminishing heat transfer ability.
Compared with the development of h at different mass flux, the profiles of hhave similar tendency, butsome differents on h exist at different mass flux. This ismainly because an increase of mass flux improves the heat transfer ability ofsupercritical CO2 due to an increase of turbulent diffusion.

3.3 Flow fields and circumferential distributions of wall temperature and heattransfer coefficient
To better understand the heat transfer mechanism of supercritical CO2 in thehelically coiled tube, the details of some parameters related to heat transfer and flow,such as velocity, turbulence kinetic energy and temperature, are given at the massflux 97.8 kg/(sm2) and the heat flux 9.03 kW/m2. Fig. 7 shows the calculated
velocity contours at various planes along the length of coil. These contours are
displayed from bottom to top along the length of coil (the inlet plane  =0o ). The
selected across sections are oriented in the same way, as shown in Fig. 1. Outer side,
inner side, bottom and top correspond to  =00 ,  =1800 ,  =2700 and  =900
respectively.
Without buoyancy, fluid flows faster at the outer side of the helically coiled
tube than it does at the inner side due to the effect of centrifugal force (theno-buoyancy pattern in Fig. 7). A secondary flow pattern caused by buoyancy issimilar to that resulted by centrifugal force. It is interested to note that under theinteraction of centrifugal force and buoyancy, the symmetry of flow field is spoiled.
In the Fig. 7, the maximum velocity region is almost located at the bottom of thecross section (  =2800 ) at the beginning. This indicates that the effect of buoyancy
is dominant over the effect of centrifugal force. With the increase of the bulk

temperature of supercritical CO2, the maximum velocity region is move towards the
top of the tube due to the buoyancy which pushes the heated fluid upward.
Fig. 8 shows the profile of velocity at different angle (θ) along the helicallycoiled tube. The velocity increases with the increase of angle along the flow
direction, and this is because of the flow acceleration caused by the change ofdensity. The velocity varies very rapidly near the wall, and this is attributed to the
combined effects of buoyancy force, flow acceleration and centrifugal force.
Fig. 9 and Fig. 10 show the calculated turbulence kinetic energy contours andprofile at different cross sections along the length of the coiled tube. In the Fig. 9,
the changes of turbulence kinetic energy contours are similar to that of velocity
contours. The numerical calculations present the development of the flow field ofsupercritical CO2 heated in helically coiled tube. The buoyancy force changes theorientation of the secondary flow and the distribution of velocity and turbulencekinetic energy in the helically coiled tube.
Buoyancy causes the increases of the velocity gradient near the wall andflattening it at the core region of fluid. The effect of acceleration also leads toturbulence to reduce even further at the core region of fluid. The combined effects
result in more severe flow laminarization in the core region of fluid as shown in Fig.
9 and Fig. 10, but turbulence kinetic energy near the wall retains a fairly high level.
Though the values of the turbulence kinetic energy at θ = 720o where the bulk
temperature approaches Tpc are slightly larger than the values at θ = 270o (Fig. 10),the turbulence kinetic energy enhances the heat transfer since turbulence produced

near the wall play an important role in determining the heat transfer capability. The
combine effects of high turbulence kinetic energy near wall and the maximum
specific heat cp near the pseudo-critical region result in the wall temperature and thebulk temperature becoming flatter in the vicinity of the critical point which can beseen from Fig. 5 and the heat transfer coefficient reaches a maximum at about θ =725o as seen from Fig. 4 (a)The local heat transfer coefficient along the circumference is calculated by theformulahc  qw / Tnw  Tb (12)where the Tnw is a certain node temperature of wall and the Tb is defined as Eq.
(11)
The circumferential distributions of heat transfer coefficient (hc) and wall
temperature (Tnw) at different axial positions are illustrated in Fig. 10. Fig. 11 showsthat the wall temperature close to the top region is higher than other regions;however, the heat transfer coefficients reach a minimum at top region. The effect ofbuoyancy is dominant over the effect of centrifugal force at the beginning of thehelically coiled tube. The buoyancy pushes the heated fluid upward, which results ina maximum wall temperature and a minimum heat transfer coefficient in the top
region. The combine effects of buoyancy and centrifugal force make heavy fluid
local at  =2800 . As a result, the heat transfer coefficient at  =2800 reach a
relatively high level.

3.4 Effects of buoyancy and flow acceleration
Since the buoyancy and flow acceleration significant affect heat transfer,further researches is carried out. The development of heat transfer coefficient andnondimensional value of Gr / Re2.7 calculated by SST model along the angle (θ) isshown in Fig. 12. The Grashof number is defined as
Gr  g b   w  a 3
b b2(13)

where density b and dynamic viscosity b are based on Tb, density  w is basedon Tw, the Tw is peripheral averaging wall temperature.
Full model indicates that the properties in the calculation vary as normal (fullmodel). Fig. 12 also shows the results calculated by SST model using constantdensity while other properties vary as normal (  =const ). In this simulation, thedensity is fixed at constant value based on the fluid temperature of inlet. When  isequal to a constant, the influences of buoyancy and flow acceleration caused by thevariation of density does not exist.
The heat transfer coefficient of no-gravity flow (no-gravity) is lower than thatof the full model, which suggests that the buoyancy force caused by the gravity withdensity variation makes an enhancement in heat transfer. The heat transfer
coefficient calculated by using constant density while other properties vary as
normal is higher over a wild range of the angle (θ). The results showed in Fig. 12
clearly indicate that the change of densities play a significant role in the decrease ofheat transfer due to the flow acceleration.
The term Gr/Re2.7 proposed by Jackson, et al. [45] represents the ratio ofbuoyancy to inertia forces. Where, the Grashof Number represents the naturalconvection of fluid, and the Reynolds Number represents the forced convection of
fluid. The term Gr/Re2.7 is always used to represent the effect of buoyancy. The
criterion for the buoyancy effect is suggested to be greater than 10-5. Buoyancy
effect cannot neglect when Gr/Re2.7 > 10-5. Fig. 12 shows that the effect of buoyancy
increases gradually with the function of the angle (θ), and reaches the maximum
near the pseudo-critical point, and then drops along the helically coiled tube. The
value of Gr/Re2.7 is far below the criterion of 10-5. However, the effect of buoyancy
cannot ignore for the helically coiled tube, because the effect of buoyancy makes adiscrepancy between full model and no-gravity flow. Therefore, it is necessary to
propose a new criterion for supercritical CO2 heated in the helically coiled tube.

3.5 The influence of turbulent Prandtl number Prt
The turbulent Prandtl number in most of the numerical calculations ofsupercritical fluids is fixed at 0.9 or 1.

This is because of the little information

about the effect of property variations. So, to find out what results can be achieved isof great importance by using different Prt.
To study the effect of Prt on heat transfer, the comparisons of h obtained byusing the constant values of 0.85, 0.90, 0.95, and 1.00 for Prt are showed in Fig. 13.
There are no visible differences in heat transfer coefficient calculated by SST modelusing various Prt. Fig. 13 shows that the effect of Prt is ignorable under theconditions based on the experiment. A similar conclusion can be found in [18], they

adopted three different Prandtl numbers and showed that the Prt did not have asignificant influence on the heat transfer.

5. Conclusion
This paper studies the heat transfer of supercritical CO2 in the helically coiled
tube by using numerical calculation method. We can get the following main
conclusion.
1) All turbulence models obtain similar tendencies in h along the angle (θ) of
the helically coiled tube. The SST model gives the best prediction to theexperimental h, wall and bulk temperature. The maximum error between
the experimental data and the predictions of the SST model is within 10%.
2) The results reveal the distribution of flow field. Due to the effects ofbuoyancy and fluid thermal acceleration, the distributions of velocity andturbulencekinetic

energyare

greatlydifferent

from

that

of

constant-property fluids in the helically coiled tube. The effects ofbuoyancy and centrifugal force result in a maximum wall temperature anda minimum heat transfer coefficient in the top region of the cross section.
3) The parameter of Gr / Re2.7 proposed by previous literature is incapable ofrepresenting the influence of buoyancy for the helically coiled tube.
4) The Prt has virtually no effecton the calculated heat transfer coefficientfor the conditions based on the experiment.

Acknowledge
The authors gratefully acknowledge National Natural Science Foundation ofChina (Project No. 51206197), Fundamental Research Funds for the Central
Universities (Project No. cdjzr12140032) and Chongqing Natural Science
Foundation (Project No. CSTC2011BB6094) for providing the financial support forthis study.

Reference[1] S. Gupta, E. Saltanov, S.J. Mokry, I. Pioro, L. Trevani, D. McGillivray,
Developing empirical heat-transfer correlations for supercritical CO2 flowing invertical bare tubes, Nuclear Engineering and Design, 261 (2013) 116-131.
[2] J.H. Song, H.Y. Kim, H. Kim, Y.Y. Bae, Heat transfer characteristics of asupercritical fluid flow in a vertical pipe, J. Supercritical Fluids, 44 (2008) 164-171.
[3] A. Bruch, A. Bontemps, S. Colasson, Experimental investigation of heattransfer of supercritical carbon dioxide flowing in a cooled vertical tube,
International J. Heat and Mass Transfer, 52 (2009) 2589-2598.
[4] M.J. Schuler, T. Rothenfluh, P.R. von Rohr, Simulation of the thermal fieldof submerged supercritical water jets at near-critical pressures, J. Supercritical
Fluids, 75 (2013) 128-137.
[5] B. Niceno, M. Sharabi, Large eddy simulation of turbulent heat transfer at
supercritical pressures, Nuclear Engineering and Design, 261 (2013) 44-55.
[6] C. Dang, K. Hoshika, E. Hihara, Effect of lubricating oil on the flow andheat-transfer characteristics of supercritical carbon dioxide, International J.
Refrigeration, 35 (2012) 1410-1417.
[7] C. Dang, K. Iino, E. Hihara, Effect of PAG-type lubricating oil on heattransfer characteristics of supercritical carbon dioxide cooled inside a smallinternally grooved tube, International J. Refrigeration, 33 (2010) 558-565.
[8] P.X. Jiang, R.F. Shi, Y.J. Xu, S. He, J.D. Jackson, Experimental
investigation of flow resistance and convection heat transfer of CO2 at supercriticalpressures in a vertical porous tube, J. Supercritical Fluids, 38 (2006) 339-346.
[9] J.A.M. Withag, J.L.H.P. Sallevelt, D.W.F. Brilman, E.A. Bramer, G. Brem,Heat transfer characteristics of supercritical water in a tube: Application for 2D and
an experimental validation, J. Supercritical Fluids, 70 (2012) 156-170.
[10] H.K. Oh, C.H. Son, New correlation to predict the heat transfer coefficientin-tube cooling of supercritical CO2 in horizontal macro-tubes, Experimental
Thermal and Fluid Science, 34 (2010) 1230-1241.
[11] H.S. Lee, H.J. Kim, J.I. Yoon, K.H. Choi, C.H. Son, The cooling heattransfer characteristics of the supercritical CO2 in micro-fin tube, Heat and Mass
Transfer, 49 (2013) 173-184.
[12] P.X. Jiang, C.R. Zhao, R.F. Shi, Y. Chen, W. Ambrosini, Experimental
and numerical study of convection heat transfer of CO2 at super-critical pressuresduring cooling in small vertical tube, International J. Heat and Mass Transfer, 52(2009) 4748-4756.
[13] Z. Yang, T. Shih, New time scale based k-epsilon model for near-wallturbulence, AIAA J., 31 (1993) 1191-1198.
[14] S. He, W.S. Kim, P.X. Jiang, J.D. Jackson, Simulation of mixed
convection heat transfer to carbon dioxide at supercritical pressure, Proceedings ofthe Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering
Science, 218 (2004) 1281-1296.
[15] P. Forooghi, K. Hooman, Numerical study of turbulent convection in
inclined pipes with significant buoyancy influence, International J. Heat and Mass
Transfer, 61 (2013) 310-322.
[16] P.X. Jiang, B. Liu, C.R. Zhao, F. Luo, Convection heat transfer ofsupercritical pressure carbon dioxide in a vertical micro tube from transition toturbulent flow regime, International J. Heat and Mass Transfer, 56 (2013) 741-749.
[17] B.E. Launder, B.I. Sharma, Application of the energy-dissipation model of
turbulence to the calculation of flow near a spinning disc, Letters in Heat and Mass
Transfer, 1 (1974) 131-137.
[18] C. Dang, E. Hihara, In-tube cooling heat transfer of supercritical carbon
dioxide. Part 2. Comparison of numerical calculation with different turbulencemodels, International J. Refrigeration, 27 (2004) 748-760.
[19] C. Dang, E. Hihara, In-tube cooling heat transfer of supercritical carbon
dioxide. Part 1. Experimental measurement, International J. Refrigeration, 27 (2004)
736-747.
[20] X.L. Cao, Z.H. Rao, S.M. Liao, Laminar convective heat transfer ofsupercritical CO2 in horizontal miniature circular and triangular tubes, Applied

Thermal Engineering, 31 (2011) 2374-2384.
[21] S. He, W.S. Kim, J.D. Jackson, A computational study of convective heattransfer to carbon dioxide at a pressure just above the critical value, Applied
Thermal Engineering, 28 (2008) 1662-1675.
[22] Y.M. Ferng, W.C. Lin, C.C. Chieng, Numerically investigated effects ofdifferent Dean number and pitch size on flow and heat transfer characteristics in ahelically coil-tube heat exchanger, Applied Thermal Engineering, 36 (2012)378-385.
[23] J.S. Jayakumar, S.M. Mahajani, J.C. Mandal, K.N. Iyer, P.K. Vijayan,
CFD analysis of single-phase flows inside helically coiled tubes, Computers &Chemical Engineering, 34 (2010) 430-446.
[24] C.N. Chen, J.T. Han, T.C. Jen, L. Shao, Thermo-chemical characteristicsof R134a flow boiling in helically coiled tubes at low mass flux and low pressure,
Thermochimica Acta, 512 (2011) 163-169.
[25] A. Owhadi, K.J. Bell, Forced convection boiling inside helically-coiled
tubes, International J. Heat and Mass Transfer, 10 (1967) 397-401.
[26] N. Jamshidi, M. Farhadi, K. Sedighi, D.D. Ganji, Optimization of design
parametersfor

nanofluids

flowing

insidehelicalcoils,

International

Communications in Heat and Mass Transfer, 39 (2012) 311-317.
[27] F. Akbaridoust, M. Rakhsha, A. Abbassi, M. Saffar-Avval, Experimental
and numerical investigation of nanofluid heat transfer in helically coiled tubes atconstant wall temperature using dispersion model, International J. Heat and Mass

Transfer, 58 (2013) 480-491.
[28] N. Haruki, A. Horibe, Flow and heat transfer characteristics of ice slurries
in a helically-coiled pipe, International J. Refrigeration, 36 (2013) 1285-1293.
[29] I. Di Piazza, M. Ciofalo, Numerical prediction of turbulent flow and heattransfer in helically coiled pipes, International J. Thermal Sciences, 49 (2010)653-663.
[30] M. Di Liberto, M. Ciofalo, A study of turbulent heat transfer in curved
pipes by numerical simulation, International J. Heat and Mass Transfer, 59 (2013)112-125.
[31] F.R. Menter, Two-equation eddy-viscosity turbulence models forengineering applications, AIAA J., 32 (1994) 1598-1605.
[32] R. Abid, Evaluation of two-equation turbulence models for predicting
transitional flows, International J. Engineering Science, 31 (1993) 831-840.
[33] C. Lam, K. Bremhorst, A modified form of the k-epsilon model for
predicting wall turbulence, ASME Transactions J. Fluids Engineering, 103 (1981)
456-460.
[34] K. Abe, T. Kondoh, Y. Nagano, A new turbulence model for predicting
fluid flow and heat transfer in separating and reattaching flows—I. Flow fieldcalculations, International J. Heat and Mass Transfer, 37 (1994) 139-151.
[35] K. Chang, W. Hsieh, C. Chen, A modified low-Reynolds-number
turbulence model applicable to recirculating flow in pipe expansion, J. Fluids
Engineering, 117 (1995) 417-423.

[36] GANBIT, User’s Guide, GAMBIT 2.2.30, FLUENT Inc., (2005).
[37] S. Wang, W. Zhang, Z. Niu, J. Xu, Mixed convective heat transfer tosupercritical carbon dioxide in helically coiled tube, CIESC J. , 64 (2013)
3917-3926.
[38] Q.L. Wen, H.Y. Gu, Numerical simulation of heat transfer deterioration
phenomenon in supercritical water through vertical tube, Annals of Nuclear Energy,
37 (2010) 1272-1280.
[39] X. Lei, H. Li, Y. Zhang, W. Zhang, Effect of Buoyancy on the Mechanismof Heat Transfer Deterioration of Supercritical Water in Horizontal Tubes, J. Heat
Transfer, 135 (2013) 071703-071703.
[40] S. He, W.S. Kim, J.H. Bae, Assessment of performance of turbulencemodels in predicting supercritical pressure heat transfer in a vertical tube,
International J. Heat and Mass Transfer, 51 (2008) 4659-4675.
[41] F.Y. Yang, K. Wang, T. Liu, Y.Q. Wang, Z.C. Liu, Convection Heat
Transfer of CO2 at Supercritical Pressures in Microtubes, Chemical Engineering &Technology, 36 (2013) 2051-2056.
[42] S. He, P.X. Jiang, Y.J. Xu, R.F. Shi, W.S. Kim, J.D. Jackson, A
computational study of convection heat transfer to CO2 at supercritical pressures in a
vertical mini tube, International J. Thermal Sciences, 44 (2005) 521-530.
[43] M. Jaromin, H. Anglart, A numerical study of heat transfer to supercriticalwater flowing upward in vertical tubes under normal and deteriorated conditions,
Nuclear Engineering and Design, 264 (2013) 61-70.

[44] S.H. Kim, Y.I. Kim, Y.Y. Bae, B.H. Cho, Numerical simulation of thevertical upward flow of water in a heated tube at supercritical pressure, in:Proceedings of the 2004 International Congress on Advances in Nuclear Power
Plants, ICAPP'04, June 13, 2004 - June 17, 2004, American Nuclear Society,
Pittsburgh, PA, United states, 2004, pp. 1527-1534.
[45] J. Jackson, W. Hall, Turbulent forced convection in channels and bundles,
Hemisphere Publishing Corporation, New York, 1979, pp. 613-640.

Figure captions
Fig. 1. A helically coiled tube with its main geometrical parameters: a , tube radius;R, coil radius; 2 b , coil pitch; The inner (I) and outer (O) sides of the curved duct
are also indicated;  , local polar coordinates in the cross-section; θ, global
azimuthal angle around the curvature axis
Fig. 2 Grid of the helically coiled tube used for the simulations
Fig. 3. Comparisons of calculated Tw at different numbers of hexahedral meshes
Fig. 4. Comparisons of calculated h with the experimental data [37]along the angle(θ) of the helically coiled tube(G = 97.8 kg/(m2s); q = 9.03 kW/m2):（a） k  model ( k   ), RNG- k   model (RNG- k   ), SST model (SST); (b) AB model(AB), LB model (LB), LS model (LS); (c) YS mode (YS), AKN model (AKN),
CHC model (CHC).
Fig. 5. Development of wall temperature and bulk temperature predicted by SST
turbulence model (G = 97.8 kg/(m2s); q = 9.03 kW/m2)
Fig. 6. Calculated heat transfer coefficients using SST turbulence model at different
heat fluxes: (a) G = 100 kg/(m2s); (b) G = 300 kg/(m2s)
Fig. 7. Contours of velocity magnitude and secondary velocity vectors for different
cross sections along the helically coiled tube
Fig. 8. Radial distributions of velocity magnitude predicted by the SST modelFig. 9. Turbulence kinetic energy contours for different cross sections along thehelically coiled tube
Fig. 10. Radial distributions of turbulence kinetic energy predicted by the SST

modelFig. 11. Circumferential distributions of wall temperature and heat transfer
coefficients at different axial positions (Outer side, inner side, bottom and top
correspond to  =00 ,  =1800 ,  =2700 and  =900 respectively): (a) wall temperature;(b) heat transfer coefficients
Fig. 12. Influences of buoyancy and flow acceleration on heat transfer
Fig. 13. Comparison of the heat transfer coefficients predicted by SST model using
various constant Prt

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6.

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Table captions

Table 1 Detail of turbulence number models(a) Constants used in the turbulence models , D and E terms
model

C

C 1

C 2

k



D

E

k 

0.09

1.44

1.92

1.0

1.3

0

0

RNG k  

0.085

 1   4.38
1  0.015 3

1.68

0.71790.7179

0

0

SST

0.09

1.55

1.83

2.0

2.0

0

   k  
2   t ( )2  y y k


AB

0.09

1.45

1.83

1.0

1.4

0

0

LB

0.09

1.44

1.92

1.0

1.3

0

0

LS

0.09

1.44

1.92

1.0

1.3

 k 2  y 

  2u 2 t  2  y 

YS

0.09

1.44

1.92

1.0

1.3

0

AKN

0.09

1.50

1.90

1.4

1.4

0

0

CHC

0.09

1.44

1.44

1.0

1.3

0

0

1.42 

  2u 2  y 

22

 t 

(b) Function used in the turbulence models
model

f

f1

f2

k 

1.0

1.0

1.0

RNG k  

1.0

1.0

1.0

SST

1.0

1.0

1.0

AB


4 
tanh  0.008Re y  1  3 4 
Re
t



1.0

 Re2    Re y  21  exp 1  t   1  exp 
936  
 12  

LB

2 
20.5 
1  exp  0.0165Re y   1 



Ret 


1.0

1  exp   Ret2 

LS



3.4
exp 
2
 1  Ret 50  

1.0

1  0.3exp   Ret2 

YS



11


 1.5 104 Re y  Ret 1  exp  5.0 107 Re3y  105 1.0 10 Re y  




Ret



AKN


  Ret 2   y*  5 1  0.75 exp       1  exp    
 14   Ret   200   


CHC

2 
31.66 
1  exp  0.0215Re y   1  5 4 


Ret 


Note:

  S k  , S   2Sij Sij  , Sij  0.5

1  RetRet
1  Ret1.0  Ret 2   y*   1  0.3exp       1  exp  
 3.1     6.5   
1.0

1  0.01exp   Ret2   1  exp  0.0631Re y 


 

2

k2
yk 0.5y1  ui u j 
0.25
, y*  u , u   
 , Ret  , Re y 2  x j xi 

2

Table 2 The check of mesh independency
Cases

Number of cells

The average relative error

Case-1

1969418

0.216 %

Case-22402356

0.0769%Case-335969700%Table 3 Selected numerical investigations on heat transfer of supercritical fluids
ReferenceSupercritical fluid

Dang, et al. [18]CO2Turbulence models

The best model

JL model, LS model,
LS modelMK modelLS model, CH model,
LB model, AKN
He, et al. [40]CO2model, WI model, MK

V2F model

model, YS model,
V2F model
RNG k   model, YS
Jiang, et al. [12]CO2model, AKN model,

YS modelLB modelLB model,YS model,Yang, et al. [41]CO2

LB modelAKN model,
RNG k   modelHe, et al. [42]CO2He, et al. [21]CO2LS modelLS modelAKN model, V2F
AKN model
model

Jaromin, et al. [43]H2O

SST model

SST model

Note: the best model is defined that the model gives the best prediction to theexperimental data