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Flow and heat transfer of supercritical water in the vertical helically-coiled tube under half-side heating condition
Accepted Manuscript Flow and heat transfer of supercritical water in the vertical helically- coiled tube under half-side heating condition Fangbo Li, Bofeng Bai PII: DOI: Reference:
S1359-4311(17)36597-3 https://doi.org/10.1016/j.applthermaleng.2018.01.047 ATE 11701
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
14 October 2017 4 January 2018 13 January 2018
Please cite this article as: F. Li, B. Bai, Flow and heat transfer of supercritical water in the vertical helically- coiled tube under half-side heating condition, Applied Thermal Engineering (2018), doi: https://doi.org/10.1016/ j.applthermaleng.2018.01.047
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Flow and heat transfer of supercritical water in the vertical helicallycoiled tube under half-side heating condition Fangbo Li, Bofeng Bai* State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China *Corresponding author E-mail: [email protected]; Tel: +86-29-8266-8260; Fax: +86-29-8266-9033
Abstract Flow and heat transfer of supercritical water in the vertical helically-coiled tube were investigated numerically using RNG k-ε model. The inner side of the coil was heated by constant heat flux and the outer side was adiabatic. Heat transfer performance under half-side heating is obtained and compared with that under uniform heating at the same heat transfer rate. Effect of specific heat, secondary flow velocity and buoyancy are discussed. The decrease of specific heat in the near-wall region, and the attenuated turbulence caused by the strong buoyancy force lead to a decrease in heat transfer for half-side heating. The deviations of heat transfer correlations on predicting half-side heating condition are evaluated, and a new heat transfer correlation for supercritical water in the vertical helically-coiled tube under inner half-side heating is proposed. The investigation into different statistical parameters shows that it is superior to the existing correlations.
Key Words: Supercritical water; Helically-coiled tube; Half-side heating; Heat transfer correlation.
1. Introduction A supercritical fluid is any substance at a temperature and pressure above its critical point, where distinct liquid and gas phases do not exist. Thermo-physical properties of supercritical fluids are quite special, a transition from a liquid-like to a vapor-like substance occurs continuously with the increasing temperature, and during this transition, all the properties of the fluid vary significantly. A peak value for specific heat capacity and thermal conductivity occurs at a point, which is called “pseudo-critical point”, but the density and viscosity drastically decrease near that point [1]. Heat transfer at supercritical pressure is strongly affected by the fluid’s thermophysical properties
or their gradient. For normal heat transfer, a maximum value of heat transfer coefficient would happen in the pseudo-critical region because of the large specific heat. However, heat transfer gradually deteriorates at high heat load. So far many studies on heat transfer of supercritical fluids have been carried out, some typical literatures are listed in Table 1. These studies cover both heat transfer enhancement and deterioration, and the tube type is mainly straight, curved, or helically-coiled tube. Licht et al. [5] numerically investigated supercritical water heat transfer and reported that the integrated effects of specific heat can be used to explain the special heat transfer characteristics near the pseudo-critical region. Xu et al. [9] researched heat transfer of supercritical CO2 in the helical tube, and found that the combined effect of centrifugal force and buoyancy force would cause the peculiar distribution of the flow and heat transfer parameters. Li et al. [10] and Hu [11] reported relevant results of supercritical water heat transfer under half-side heating in straight tubes. They found that the heat transfer in the half-side heated tube is quite different from that under uniform heating. The experiment conducted by Kurganov et al. [2] offered a good insight into the mechanism of heat transfer deterioration, the M-shaped velocity profile was observed and it was considered to be closely associated with heat transfer deterioration. Bae et al. [3, 4] obtained the budget of the turbulent kinetic energy in deteriorated heat transfer by using direct numerical simulation, and the mechanism of buoyancy and flow acceleration were also analyzed. Reference
Approach
Tube Type
Heating Method
Fluid
Kurganov and Kaptil'ny [2]
Experiment
Straight tube
Uniform heating
CO2
Bae et al. [3,4]
Direct numerical simulation
Straight tube
Uniform heating
CO2
Licht et al. [5]
Experiment &CFD
Straight tube
Uniform heating
H2O
Mao et al. [6]
Experiment
Straight tube & Helically-coil ed tube
Uniform heating
H2O
Parameters P=9MPa d=22.7mm G=800~2100kgm-2s-1 qw/G=0.05~0.22kJkg-1 P=8MPa, d=1~3mm Rein=5400,8900 G=111.08~333.24kgm-2s-1 qw=20.58~72.63kWm-2 P=25MPa Tb=175~400℃ G=300~1000kgm-2s-1 qw=220,440kWm-2 Straight tube: P=23~30MPa d=10mm G=600~1200kgm-2s-1 qw=200~600kWm-2.
Zhang et al. [7]
CFD
Straight tube
Uniform heating
H2O
Zhang et al. [8], Xu et al. [9]
Experiment & CFD
Helically-coil ed Tube
Uniform heating
CO2
Li et al. [10]
CFD
Straight tube
Half-side heating
H2O
Hu [11]
Experiment
Straight tube
Half-side heating
H2O
Helical tube: P=23.5~26.5MPa d=10mm G=800~1600kgm-2s-1 qw=100~400kWm-2 P=23, 25MPa Tin=301, 320℃ d=10mm G=596, 2021kgm-2s-1 qw=772, 1385kWm-2 P=8.02~10.05MPa d=9mm G=0~650kgm-2s-1 qw=0.4~50kWm-2 P=24.82MPa d=18.5mm G=404kgm-2s-1 qave=83~203kWm-2 P=23, 26, 30MPa d=26mm G=600, 900kgm-2s-1 qave=150, 200, 250, 300kWm-2
Table 1 Typical literatures on heat transfer of supercritical fluids
There have been very abundant correlations on heat transfer of supercritical fluids, but most correlations are developed for uniform heating condition. Most empirical correlations are formed by adding correction terms to the Dittus-Boelter correlation [12], and the correction terms consist of the ratio of the fluid’s physical properties under wall and bulk temperature. Some typical existing correlations are listed in Ref. [13]. The most frequently-used form of correlations is written as below [14]: Nu b c Rebm Prbn w / b
A
c
p
/ cpb
B
, c p H w H b / Tw Tb
(1)
or:
Nu b c Rebm Pr b w / b , Pr b b c p / b n
A
(2)
Half-side heating condition is very common in boilers. However, the mechanism behind the peculiar heat transfer under half-side heating is still unknown. Moreover, the heat transfer correlation for supercritical water under half-side heating is lacking. In this study, flow and heat transfer of supercritical water in the vertical helically-coiled tube under half-side heating is investigated
numerically, only the normal heat transfer regime is considered here. The mechanism causing the differences between the half-side and uniform heating methods is discussed. Meanwhile a new heat transfer correlation for supercritical water in the helically-coiled tube under half-side heating is proposed and compared with the existing correlations.
2. Physical and mathematical model 2.1 Physical model The vertical helically-coiled tube in this study is the same as the test section in Ref. [8]. The inside diameter is 9.0mm and the outside diameter is 12.0mm, the wall thickness is considered here. The tube is made of 316L stainless steel and the physical properties are: ρ=7980kgm-3, λ=16.3Wm-1 K-1, cp=502Jkg-1 K-1. Fig. 1 shows the structure and parameters of one turn of the tube. The diameter of the coil is 283.0mm and the pitch of the tube is 32.0mm. The height of the helically-coiled tube is 192.0mm, which means the tube has 6 turns. The total length of the tube is 5.34m. The inner side of the coil is heated and the outer side is adiabatic.
Fig. 1. Physical model of the helically-coiled tube
2.2 Turbulence model and boundary conditions The steady RNG k-ε model [15] is chosen as the turbulence model in this paper due to its higher precision in calculating swirl flow and secondary flow. This model has been used by Xu et al. [9] to simulate the supercritical fluids heat transfer in helical tubes. The parameters of boundary conditions are listed in Table 2. The mass flow rate boundary is applied at the inlet. The k and ε at the inlet are calculated according to the following correlations [9]: kin 1.5 uin Iin , Iin 0.16 Rein1/8
(3)
in C0.75kin1.5 / l , l 0.07d , C 0.085
(4)
2
Parameter
P (MPa)
Tin (K)
G (kgm-2s-1)
qave (kWm-2)
Value
23~27
610~630
97.5~1100
0~300
Table 2 The parameters of boundary conditions
The range of Iin is 0.037~0.05. The nonslip and coupled heat transfer boundary are applied on the inside wall of the tube. The inner side of the coil is kept as uniform heat flux and the outer side is set as adiabatic boundary. The outlet uses outflow boundary.
3. Numerical details 3.1 Numerical approaches The governing equations are solved by finite volume method in FLUENT 14.0, the SIMPLEC algorithm is adopted to solve the coupling between pressure and momentum equation. The second-order upwind scheme is used for the discretization of the convective terms in the governing equations. The absolute values of the residuals of all equations are kept lower than 10-5 during the calculation. The enhanced wall function is applied on the fluid-solid interface to improve the numerical precision near the wall. The relationship between thermo-physical properties and temperature of supercritical water is obtained by piecewise liner fitting, and all the properties come from NIST Standard Reference Database. 3.2 Mesh generation The mesh generation is achieved in ICEM 14.0. The structured grid is used and the mesh near the inside wall is refined. The non-dimensional height y+ of the first layer of the mesh near the inside wall is set lower than 1.0. The total number of the grid is 7,776,000 (section×axial=1,440×5,400). The grid independence is checked, and further refinement of the mesh has no improvement for the numerical results, as is shown in Appendix.
4 Results and discussions 4.1 Experimental validation We calculate the fluid bulk temperature and inner wall temperature of supercritical CO 2 under uniform heat load with RNG k-ε model, and the results are compared with the experimental results of Zhang et.al [8], as is shown in Fig. 2. Both numerical Tw and Tb results show good agreement with the experimental data, and the maximum relative deviation is less than ±5%. The calculation results of SST k-ω model are also given in Fig. 2. The results of SST k-ω model have no
improvement compared with those of RNG k-ε model, proving the accuracy and superiority of our numerical approach.
Fig. 2. Comparison of numerical results and experimental data
4.2 Heat transfer performances and mechanism under half-side heating condition in the helically-coiled tube 4.2.1 Heat transfer under half-side heating and the comparison with uniform heating The variations of local heat transfer coefficients under half-side heating with the pressure and heat flux are shown in Fig. 3(a) ~ (b) respectively, in which the comparisons with other heating methods with the same total heat transfer rate are also obtained. The local heat transfer coefficients are defined in Eq. (5).
h
q Tw Tb
(5)
As can be seen, the h of supercritical water reaches the maximum before the pseudo-critical point. Under uniform heating, h significantly decreases with the increasing pressure. However, it can be seen from Fig. 3(a) that quite different from uniform heating, the influence of pressure on h under half-side heating is not obvious. Besides, as is shown in Fig. 3(b), h under half-side heating increases with the decreasing wall heat flux. Moreover, uniform heating has the best heat transfer performance while half-side heating has the poorest, indicating that the heat transfer is gradually weakened as the degree of non-uniform heating increases. The discrepancy of h under different heating conditions reaches the peak at the pseudo-critical enthalpy, and as the pressure increases, the gap between h under the two heating methods gradually narrows.
(a)
(b)
Fig. 3. Heat transfer coefficients under half-side heating and the comparison with other heating methods: (a) at different pressures (b) at different heat fluxes
4.2.2 Effect of specific heat distribution on heat transfer under half-side heating Fig. 4 shows the distributions of the cp from the inner side to the outer side of the coil on different cross-sections under half-side and uniform heating (The left and the right side of x-axis represent the inner and outer side respectively, the same below). Both heating methods have the same average heat flux of 31.25kWm-2 and mass flux of 97.5kgm-2s-1. The inlet pressure and temperature are 25MPa and 610K, respectively. As can be seen, at the entrance section of the tube (Hb=1700kJkg-1), the cp near the inner wall under half-side heating is higher than that under uniform heating. This is because the temperature of the fluid will reach the pseudo-critical value earlier under half-side heating due to the higher heat flux at the heating side of the coil. When Hb=2000kJkg-1 (Hb
Fig. 4. Specific heat distributions under half-side heating and uniform heating (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
4.2.3 Effect of secondary motion on heat transfer under half-side heating The secondary flow velocity distributions from the inner to the outer side of the coil on different cross-sections under half-side heating and uniform heating are given in Fig. 5. The calculation parameters are the same as those discussed in section 4.2.2. The secondary flow velocity near the wall is higher than that in the core region, the peak near the inner-wall results from the buoyancy force while that near the outer wall is caused by the centrifugal force. Compared with uniform heating, the secondary flow velocity under half-side heating is strengthened at the inner side but lower at the outer side due to the non-uniform buoyancy effect. Fig. 6 shows the stramwise distribution of the bulk secondary flow velocity under uniform heating and half-side heating. The bulk secondary flow velocity on the cross-section is defined as Eq. (6) [16].
us
0.5 1 ur2,i ut2,i dAi A A
(6)
As is shown, the us is obviously strengthened with the rising enthalpy due to the expansion and acceleration of the fluid along the streamwise direction. When Hb
Fig. 5. Secondary flow velocity distribution under half-side heating and uniform heating (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
Fig. 6. Average secondary flow velocity versus the fluid’s enthalpy for uniform heating and half-side heating (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
4.2.4 Effect of buoyancy on heat transfer under half-side heating Heat transfer coefficients and buoyancy effects against the enthalpy of the fluid under half-side heating and uniform heating are depicted in Fig. 7. The calculation parameters are the same as those discussed in section 4.2.2. The non-dimensional buoyancy parameter, Bo*, which is defined as Eq. (7), is selected to evaluate the buoyancy effect [17]:
Bo*
Grb Reb2.7
(7)
It can be found that the Bo* reaches the peak near the pseudo-critical region because of the drastic
variation in the physical properties. The Bo * under half-side heating is higher than that under uniform heating, indicating that the buoyancy effect under half-side heating is stronger.
Fig. 7. The variation of heat transfer coefficients and Bo* with the fluid’s enthalpy (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
In Fig. 8, the axial velocity distributions from the inner side to the outer side of the coil under the two heating conditions are compared. Under uniform heating, the velocity at the outer side is obviously higher than that at the inner side due to the centrifugal force. Therefore, the velocity gradient across the tube diameter is clear. Nevertheless, under half-side heating, the buoyancy effect near the inner wall caused by the density gradient is much stronger than that under uniform heating. Therefore, the fluid near the inner side is obviously accelerated and the velocity gradient of the fluid is lower than that under uniform heating. Fig. 9 shows the turbulent kinetic energy distribution from the inner side to the outer side of the coil under the two heating conditions. The turbulent kinetic energy under half-side heating is much lower than that under uniform heating. The formation of the flat velocity profile in half-side heating mentioned above indicates the modification and reduction of the shear stress in the flow field. As a result, the production of turbulent kinetic energy, and the turbulent kinetic energy itself, are diminished, leading to reduced mixing and thus to lower heat transfer coefficients.
Fig. 8. Streamwise velocity distribution under half-side heating and uniform heating (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
Fig. 9. Turbulent kinetic energy distribution under half-side heating and uniform heating (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
4.3 Heat transfer correlation for supercritical water under half-side heating in the helically-coiled tube Fig. 10 gives the relative deviations of five heat transfer correlations of supercritical water (Yamagata correlation [18], Hu correlation [11], Mao correlation [6], Jackson correlation [14] and Bishop correlation [19]) when they are used to predict Nub of supercritical water in the helically-coiled tube under inner half-side heating. Note that these correlations were obtained under
uniform heating. The relative deviations (RD) of Nub given by different correlations are calculated according to Eq. (8):
The
range
of
the
RD
parameters
is:
PRE CAL (8) CAL
d=9mm,
P=23~27MPa,
G=97.5~1100kg·m-2·s-1,
qwi=0~600kW·m-2 (qave=0~300kWm-2), Hb=1500~2900kJ·kg-1, Bo*<1.5×10-5. The RDs of Yamagata correlation, Jackson correlation are very large at the pseudo-critical region (around 2100kJkg-1), reaching about +90%. At low enthalpy region (lower than 1800kJkg-1), predictions of all correlations except Mao correlation are obviously lower than the actual values (RDs<-30%). The deviations of all correlations stay at a relatively low level when Hb>Hpc (within ±30%). The performance of Mao correlation is the best among all correlations, the relative deviations are less than ±20% at the confidence level of 92%.
(a)
(b)
Fig. 10. Relative deviations of the existing correlations versus the fluid’s enthalpy when predicting half-side heating in the helical tube
A new heat transfer correlation of supecritical water in the helical tube under half-side heating is obtained: 0.469
Nu b 0.0289Re0.805 Pr b b
0.545 0.218 w b w / b
(9)
Fig. 11 shows the comparison between the predicted Nub values given by Eq. (9) and the actual Nub values. The relative deviations of Eq. (9) are less than ±20% and 10% at the confidence level of 97.9% and 82.4%, respectively.
Here we adopt 4 statistical parameters: RD, MRD, MARD and RSMRD to evaluate the performance of different correlations on predicting half-side heating. Definitions of the 4 parameters are listed as Eq.(8), (10), (11) and (12). Table 3 compares the statistics of the present and existing correlations. As can be seen, the present correlation (Eq. (9)) gives the smallest value for each statistical parameter, proving its accuracy and superiority. MRD
1 n RDi n i 1
MARD
RMSRD
(10)
1 n RDi n i 1
(11)
1 n RDi2 n i 1
(12)
Fig. 11. Comparison between the Nu values predicted by Eq. (9) and numerical simulation
Correlations
Eq. (9)
Yamagata
Hu
Mao
Jackson
Bishop
+(RD)max (%)
38.4
91.73
56.54
40.87
96.78
65.83
-(RD)max (%)
-27.3
-36.86
-48.38
-30.35
-38.47
-47.51
MRD(%)
0.42
19.07
-1.79
2.07
3.78
3.98
MARD(%)
6.34
27.45
16.25
8.53
17.23
18.15
RSMRD(%)
8.33
28.38
20.07
10.75
23.41
21.72
Table 3 Comparison between present and existing heat transfer correlations
5. Conclusions In this paper, numerical investigation on flow and heat transfer of supercritical water in the vertical helically-coiled tube under inner half-side heating condition is conducted. Results show that the heat transfer coefficient under half-side heating is lower than that under uniform heating at the same heat transfer rate. The specific heat of supercritical water at the near-wall region under half-side heating is obvious lower than that under uniform heating. Additionally, the strong buoyancy effect under half-side heating near the inner wall reduces the axial velocity gradient and the turbulent kinetic energy, causing the weaker heat transfer compared with uniform heating. The majority of the existing correlations of supercritical water cannot obtain satisfying results when predicting half-side heating. A new correlation aiming at heat transfer of supercritical water in the helically-coiled tube under inner half-side heating is proposed.
Nomenclature a
inside radius of the helical tube (mm)
A
cross-section of the tube (mm2)
Bo*
Jackson buoyancy parameter
cp
specific heat (kJkg-1K-1)
cp
average specific heat on the cross-section (=(Hw-Hb)/(Tw-Tb)) (kJkg-1K-1)
D
outside diameter (mm)
d
inside diameter (mm)
G
mass flux (kgm-2s-1)
Gr
Grashof number
h
local heat transfer coefficient (kWm-2 K-1 )
H
enthalpy (kJkg-1)
I
turbulent intensity
k
turbulent kinetic energy (m2 s-2)
l
turbulent length (m)
Nu
Nusselt number
P
pressure (MPa)
Pr
Prantl number
qave
average heat flux of the tube coil (kWm-2)
qwi
heat flux of the inner side of the coil (kWm-2)
qwo
heat flux of the outer side of the coil (kWm-2)
Re
Reynolds number
T
temperature (K)
Tw
inside wall temperature (K)
R
distance from the tube’s central axis (mm)
rin
distance of the inner side of the coil from the tube’s central axis (mm)
ur
radial velocity (ms-1)
ut
tangential velocity (ms-1)
us
secondary flow velocity on the cross-section (ms-1)
Greek symbols λ
thermal conductivity (Wm-1K-1)
ρ
density (kgm-3)
μ
dynamic viscosity (Pas)
θ
circumferential angle of the cross section (deg.)
ε
turbulent dissipation rate (m2s-3)
Subscripts b
bulk
i
mesh unit
in
inlet
pc
pseudo-critical state
wi
inner half-side of the coil
wo
outer half-side of the coil
Abbreviations PRE
results given by correlations
CAL
numerical results
RD
relative deviation
+(RD)max
maximum positive relative deviation
-(RD)max
maximum negative relative deviation
MRD
mean absolute relative deviation
MARD
mean absolute relative deviation
RSMRD
root mean square deviation
Acknowledgement This study is supported by the National Key Research and Development Program of China (No. 2016YFB0600100).
Appendix Different structured meshes with grid numbers ranging from 3,726,000 to 13,149,000 are evaluated to analyze the mesh independence. The average inside wall temperatures and heat transfer coefficients under different grid numbers are given in Fig. 12. When the gird number increases from 7,776,000 to 13,149,000, the relative deviations of inside wall temperature and heat transfer coefficients are less than 2%, indicating that the solutions are mesh-independent. Thus, the grid number of 7,776,000 is selected in this paper.
Fig. 12. Grid independence analysis
References [1] J. Y. Yoo, The Turbulent Flows of Supercritical Fluids with Heat Transfer, Annu. Rev. Fluid. Mech. 45 (2013) 495-525.
[2] V. A. Kurganov, A. G. Kaptil'ny, Velocity and enthalpy fields and eddy diffusivities in a heated supercritical fluid flow, Exp. Therm. Fluid Sci. 5 (1992) 465-478. [3] J. H. Bae, J. Y. Yoo, H. Choi, Direct numerical simulation of turbulent supercritical flows with heat transfer, Phys. Fluids. 17 (2005) (105104). [4] J. H. Bae, J. Y. Yoo, D. M. McEligot, Direct numerical simulation of heated CO2 flows at supercritical pressure in a vertical annulus at Re = 8900, Phys. Fluids. 20 (2008) (055108). [5] J. Licht, M. Anderson, M. Corradini, Heat transfer and fluid flow characteristics in supercritical Pressure water, Trans. ASME J. Heat Transf. 131 (2009) 301-314. [6] Y. F. Mao, L. J. Guo, B. F. Bai, X. M. Zhang, Convective heat transfer in helical coils for constant-property and variable-property flows with high Reynolds numbers, Front. Energy. 4 (2010) 546-552. [7] G. Zhang, H. Zhang, H. Y. Gu, Y. H. Yang, X. Cheng, Experimental and numerical investigation of turbulent convective heat transfer deterioration of supercritical water in vertical tube, Nucl. Eng. Des. 248 (2012) 226-237. [8] W. Zhang, S. X. Wang, C. D. Li, J. L. Xu, Mixed convective heat transfer of CO 2 at supercritical pressures flowing upward through a vertical helically coiled tube, Appl. Therm. Eng. 88 (2015) 61-70. [9] J. L. Xu, C. Yang, W. Zhang, D. L. Sun, Turbulent convective heat transfer of CO2 in a helical tube at near-critical pressure, Int. J. Heat Mass Transf. 80 (2015) 748–758. [10] Z. H. Li, Y. X. Wu, J. F. Lu, D. L. Zhang, H. Zhang, Heat transfer to supercritical water in circular tubes with circumferentially non-uniform heating, Appl. Therm. Eng 70 (2014) 190-200. [11] Z. H. Hu, Heat transfer characteristics of vertical upflow and inclined tubes in the supercritical pressure and near-critical pressure region, Xi’an Jiaotong University, 2001. [12] F. W. Dittus, L. M. K. Boelter, University of California, Publications of Engineering, vol. 2, 1930, pp. 443. [13] I. L. Pioro, H. F. Khartabil, R. B. Duffey, Heat transfer to supercritical fluids flowing in channels - empirical correlations (survey), Nucl. Eng. Des. 230 (2004) 69-91. [14] J. D. Jackson, Fluid flow and convective heat transfer to fluids at supercritical pressure, Nucl. Eng. Des. 264 (2013) 24-40. [15] V. Yakhot, S. A. Orszag, Renormalization group analysis of turbulence: I. Basic theory, J. Sci. Comput. 1 (1986) 1–51. [16] X. L. Lei, H. X. Li, S. Q. Yu, D. L. Ren, Numerical simulation on heterogeneous heat transfer in water at supercritical pressures in inclined upward tubes, Chin. J. Comput. Phys. 27 (2010) 217–228.
[17] J. D. Jackson, Consideration of the heat transfer properties of supercritical pressure water in connection with the cooling of advanced nuclear reactors, in: Proc. 13th Pacific Basin Nuclear Conference, Shenzhen City, China, October 21-25, 2002. [18] K. Yamagata, K. Nishikawa, S. Hasegawa, T. Fujii, S. Yoshida, Forced convective heat transfer to supercritical water flowing in tubes, Int. J. Heat Mass Transf. 15 (1972) 2575–2593. [19] A. A Bishop, R. O. Sandberg, L.S. Tong, High temperature supercritical pressure water loop: Part IV, Forced convection heat transfer to water at near-critical temperatures and super-critical pressures, Westinghouse Electric Corporation, Pittsburgh, PA, 1964.
Highlights:
1. Investigate supercritical water heat transfer in a half-side heated helical tube. 2. Analyze the mechanism of the difference between half-side and uniform heating. 3. Evaluate deviations of heat transfer correlations on predicting half-side heating. 4. Propose a heat transfer correlation in the half-side heated helical tube.
S1359-4311(17)36597-3 https://doi.org/10.1016/j.applthermaleng.2018.01.047 ATE 11701
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
14 October 2017 4 January 2018 13 January 2018
Please cite this article as: F. Li, B. Bai, Flow and heat transfer of supercritical water in the vertical helically- coiled tube under half-side heating condition, Applied Thermal Engineering (2018), doi: https://doi.org/10.1016/ j.applthermaleng.2018.01.047
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Flow and heat transfer of supercritical water in the vertical helicallycoiled tube under half-side heating condition Fangbo Li, Bofeng Bai* State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China *Corresponding author E-mail: [email protected]; Tel: +86-29-8266-8260; Fax: +86-29-8266-9033
Abstract Flow and heat transfer of supercritical water in the vertical helically-coiled tube were investigated numerically using RNG k-ε model. The inner side of the coil was heated by constant heat flux and the outer side was adiabatic. Heat transfer performance under half-side heating is obtained and compared with that under uniform heating at the same heat transfer rate. Effect of specific heat, secondary flow velocity and buoyancy are discussed. The decrease of specific heat in the near-wall region, and the attenuated turbulence caused by the strong buoyancy force lead to a decrease in heat transfer for half-side heating. The deviations of heat transfer correlations on predicting half-side heating condition are evaluated, and a new heat transfer correlation for supercritical water in the vertical helically-coiled tube under inner half-side heating is proposed. The investigation into different statistical parameters shows that it is superior to the existing correlations.
Key Words: Supercritical water; Helically-coiled tube; Half-side heating; Heat transfer correlation.
1. Introduction A supercritical fluid is any substance at a temperature and pressure above its critical point, where distinct liquid and gas phases do not exist. Thermo-physical properties of supercritical fluids are quite special, a transition from a liquid-like to a vapor-like substance occurs continuously with the increasing temperature, and during this transition, all the properties of the fluid vary significantly. A peak value for specific heat capacity and thermal conductivity occurs at a point, which is called “pseudo-critical point”, but the density and viscosity drastically decrease near that point [1]. Heat transfer at supercritical pressure is strongly affected by the fluid’s thermophysical properties
or their gradient. For normal heat transfer, a maximum value of heat transfer coefficient would happen in the pseudo-critical region because of the large specific heat. However, heat transfer gradually deteriorates at high heat load. So far many studies on heat transfer of supercritical fluids have been carried out, some typical literatures are listed in Table 1. These studies cover both heat transfer enhancement and deterioration, and the tube type is mainly straight, curved, or helically-coiled tube. Licht et al. [5] numerically investigated supercritical water heat transfer and reported that the integrated effects of specific heat can be used to explain the special heat transfer characteristics near the pseudo-critical region. Xu et al. [9] researched heat transfer of supercritical CO2 in the helical tube, and found that the combined effect of centrifugal force and buoyancy force would cause the peculiar distribution of the flow and heat transfer parameters. Li et al. [10] and Hu [11] reported relevant results of supercritical water heat transfer under half-side heating in straight tubes. They found that the heat transfer in the half-side heated tube is quite different from that under uniform heating. The experiment conducted by Kurganov et al. [2] offered a good insight into the mechanism of heat transfer deterioration, the M-shaped velocity profile was observed and it was considered to be closely associated with heat transfer deterioration. Bae et al. [3, 4] obtained the budget of the turbulent kinetic energy in deteriorated heat transfer by using direct numerical simulation, and the mechanism of buoyancy and flow acceleration were also analyzed. Reference
Approach
Tube Type
Heating Method
Fluid
Kurganov and Kaptil'ny [2]
Experiment
Straight tube
Uniform heating
CO2
Bae et al. [3,4]
Direct numerical simulation
Straight tube
Uniform heating
CO2
Licht et al. [5]
Experiment &CFD
Straight tube
Uniform heating
H2O
Mao et al. [6]
Experiment
Straight tube & Helically-coil ed tube
Uniform heating
H2O
Parameters P=9MPa d=22.7mm G=800~2100kgm-2s-1 qw/G=0.05~0.22kJkg-1 P=8MPa, d=1~3mm Rein=5400,8900 G=111.08~333.24kgm-2s-1 qw=20.58~72.63kWm-2 P=25MPa Tb=175~400℃ G=300~1000kgm-2s-1 qw=220,440kWm-2 Straight tube: P=23~30MPa d=10mm G=600~1200kgm-2s-1 qw=200~600kWm-2.
Zhang et al. [7]
CFD
Straight tube
Uniform heating
H2O
Zhang et al. [8], Xu et al. [9]
Experiment & CFD
Helically-coil ed Tube
Uniform heating
CO2
Li et al. [10]
CFD
Straight tube
Half-side heating
H2O
Hu [11]
Experiment
Straight tube
Half-side heating
H2O
Helical tube: P=23.5~26.5MPa d=10mm G=800~1600kgm-2s-1 qw=100~400kWm-2 P=23, 25MPa Tin=301, 320℃ d=10mm G=596, 2021kgm-2s-1 qw=772, 1385kWm-2 P=8.02~10.05MPa d=9mm G=0~650kgm-2s-1 qw=0.4~50kWm-2 P=24.82MPa d=18.5mm G=404kgm-2s-1 qave=83~203kWm-2 P=23, 26, 30MPa d=26mm G=600, 900kgm-2s-1 qave=150, 200, 250, 300kWm-2
Table 1 Typical literatures on heat transfer of supercritical fluids
There have been very abundant correlations on heat transfer of supercritical fluids, but most correlations are developed for uniform heating condition. Most empirical correlations are formed by adding correction terms to the Dittus-Boelter correlation [12], and the correction terms consist of the ratio of the fluid’s physical properties under wall and bulk temperature. Some typical existing correlations are listed in Ref. [13]. The most frequently-used form of correlations is written as below [14]: Nu b c Rebm Prbn w / b
A
c
p
/ cpb
B
, c p H w H b / Tw Tb
(1)
or:
Nu b c Rebm Pr b w / b , Pr b b c p / b n
A
(2)
Half-side heating condition is very common in boilers. However, the mechanism behind the peculiar heat transfer under half-side heating is still unknown. Moreover, the heat transfer correlation for supercritical water under half-side heating is lacking. In this study, flow and heat transfer of supercritical water in the vertical helically-coiled tube under half-side heating is investigated
numerically, only the normal heat transfer regime is considered here. The mechanism causing the differences between the half-side and uniform heating methods is discussed. Meanwhile a new heat transfer correlation for supercritical water in the helically-coiled tube under half-side heating is proposed and compared with the existing correlations.
2. Physical and mathematical model 2.1 Physical model The vertical helically-coiled tube in this study is the same as the test section in Ref. [8]. The inside diameter is 9.0mm and the outside diameter is 12.0mm, the wall thickness is considered here. The tube is made of 316L stainless steel and the physical properties are: ρ=7980kgm-3, λ=16.3Wm-1 K-1, cp=502Jkg-1 K-1. Fig. 1 shows the structure and parameters of one turn of the tube. The diameter of the coil is 283.0mm and the pitch of the tube is 32.0mm. The height of the helically-coiled tube is 192.0mm, which means the tube has 6 turns. The total length of the tube is 5.34m. The inner side of the coil is heated and the outer side is adiabatic.
Fig. 1. Physical model of the helically-coiled tube
2.2 Turbulence model and boundary conditions The steady RNG k-ε model [15] is chosen as the turbulence model in this paper due to its higher precision in calculating swirl flow and secondary flow. This model has been used by Xu et al. [9] to simulate the supercritical fluids heat transfer in helical tubes. The parameters of boundary conditions are listed in Table 2. The mass flow rate boundary is applied at the inlet. The k and ε at the inlet are calculated according to the following correlations [9]: kin 1.5 uin Iin , Iin 0.16 Rein1/8
(3)
in C0.75kin1.5 / l , l 0.07d , C 0.085
(4)
2
Parameter
P (MPa)
Tin (K)
G (kgm-2s-1)
qave (kWm-2)
Value
23~27
610~630
97.5~1100
0~300
Table 2 The parameters of boundary conditions
The range of Iin is 0.037~0.05. The nonslip and coupled heat transfer boundary are applied on the inside wall of the tube. The inner side of the coil is kept as uniform heat flux and the outer side is set as adiabatic boundary. The outlet uses outflow boundary.
3. Numerical details 3.1 Numerical approaches The governing equations are solved by finite volume method in FLUENT 14.0, the SIMPLEC algorithm is adopted to solve the coupling between pressure and momentum equation. The second-order upwind scheme is used for the discretization of the convective terms in the governing equations. The absolute values of the residuals of all equations are kept lower than 10-5 during the calculation. The enhanced wall function is applied on the fluid-solid interface to improve the numerical precision near the wall. The relationship between thermo-physical properties and temperature of supercritical water is obtained by piecewise liner fitting, and all the properties come from NIST Standard Reference Database. 3.2 Mesh generation The mesh generation is achieved in ICEM 14.0. The structured grid is used and the mesh near the inside wall is refined. The non-dimensional height y+ of the first layer of the mesh near the inside wall is set lower than 1.0. The total number of the grid is 7,776,000 (section×axial=1,440×5,400). The grid independence is checked, and further refinement of the mesh has no improvement for the numerical results, as is shown in Appendix.
4 Results and discussions 4.1 Experimental validation We calculate the fluid bulk temperature and inner wall temperature of supercritical CO 2 under uniform heat load with RNG k-ε model, and the results are compared with the experimental results of Zhang et.al [8], as is shown in Fig. 2. Both numerical Tw and Tb results show good agreement with the experimental data, and the maximum relative deviation is less than ±5%. The calculation results of SST k-ω model are also given in Fig. 2. The results of SST k-ω model have no
improvement compared with those of RNG k-ε model, proving the accuracy and superiority of our numerical approach.
Fig. 2. Comparison of numerical results and experimental data
4.2 Heat transfer performances and mechanism under half-side heating condition in the helically-coiled tube 4.2.1 Heat transfer under half-side heating and the comparison with uniform heating The variations of local heat transfer coefficients under half-side heating with the pressure and heat flux are shown in Fig. 3(a) ~ (b) respectively, in which the comparisons with other heating methods with the same total heat transfer rate are also obtained. The local heat transfer coefficients are defined in Eq. (5).
h
q Tw Tb
(5)
As can be seen, the h of supercritical water reaches the maximum before the pseudo-critical point. Under uniform heating, h significantly decreases with the increasing pressure. However, it can be seen from Fig. 3(a) that quite different from uniform heating, the influence of pressure on h under half-side heating is not obvious. Besides, as is shown in Fig. 3(b), h under half-side heating increases with the decreasing wall heat flux. Moreover, uniform heating has the best heat transfer performance while half-side heating has the poorest, indicating that the heat transfer is gradually weakened as the degree of non-uniform heating increases. The discrepancy of h under different heating conditions reaches the peak at the pseudo-critical enthalpy, and as the pressure increases, the gap between h under the two heating methods gradually narrows.
(a)
(b)
Fig. 3. Heat transfer coefficients under half-side heating and the comparison with other heating methods: (a) at different pressures (b) at different heat fluxes
4.2.2 Effect of specific heat distribution on heat transfer under half-side heating Fig. 4 shows the distributions of the cp from the inner side to the outer side of the coil on different cross-sections under half-side and uniform heating (The left and the right side of x-axis represent the inner and outer side respectively, the same below). Both heating methods have the same average heat flux of 31.25kWm-2 and mass flux of 97.5kgm-2s-1. The inlet pressure and temperature are 25MPa and 610K, respectively. As can be seen, at the entrance section of the tube (Hb=1700kJkg-1), the cp near the inner wall under half-side heating is higher than that under uniform heating. This is because the temperature of the fluid will reach the pseudo-critical value earlier under half-side heating due to the higher heat flux at the heating side of the coil. When Hb=2000kJkg-1 (Hb
Fig. 4. Specific heat distributions under half-side heating and uniform heating (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
4.2.3 Effect of secondary motion on heat transfer under half-side heating The secondary flow velocity distributions from the inner to the outer side of the coil on different cross-sections under half-side heating and uniform heating are given in Fig. 5. The calculation parameters are the same as those discussed in section 4.2.2. The secondary flow velocity near the wall is higher than that in the core region, the peak near the inner-wall results from the buoyancy force while that near the outer wall is caused by the centrifugal force. Compared with uniform heating, the secondary flow velocity under half-side heating is strengthened at the inner side but lower at the outer side due to the non-uniform buoyancy effect. Fig. 6 shows the stramwise distribution of the bulk secondary flow velocity under uniform heating and half-side heating. The bulk secondary flow velocity on the cross-section is defined as Eq. (6) [16].
us
0.5 1 ur2,i ut2,i dAi A A
(6)
As is shown, the us is obviously strengthened with the rising enthalpy due to the expansion and acceleration of the fluid along the streamwise direction. When Hb
Fig. 5. Secondary flow velocity distribution under half-side heating and uniform heating (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
Fig. 6. Average secondary flow velocity versus the fluid’s enthalpy for uniform heating and half-side heating (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
4.2.4 Effect of buoyancy on heat transfer under half-side heating Heat transfer coefficients and buoyancy effects against the enthalpy of the fluid under half-side heating and uniform heating are depicted in Fig. 7. The calculation parameters are the same as those discussed in section 4.2.2. The non-dimensional buoyancy parameter, Bo*, which is defined as Eq. (7), is selected to evaluate the buoyancy effect [17]:
Bo*
Grb Reb2.7
(7)
It can be found that the Bo* reaches the peak near the pseudo-critical region because of the drastic
variation in the physical properties. The Bo * under half-side heating is higher than that under uniform heating, indicating that the buoyancy effect under half-side heating is stronger.
Fig. 7. The variation of heat transfer coefficients and Bo* with the fluid’s enthalpy (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
In Fig. 8, the axial velocity distributions from the inner side to the outer side of the coil under the two heating conditions are compared. Under uniform heating, the velocity at the outer side is obviously higher than that at the inner side due to the centrifugal force. Therefore, the velocity gradient across the tube diameter is clear. Nevertheless, under half-side heating, the buoyancy effect near the inner wall caused by the density gradient is much stronger than that under uniform heating. Therefore, the fluid near the inner side is obviously accelerated and the velocity gradient of the fluid is lower than that under uniform heating. Fig. 9 shows the turbulent kinetic energy distribution from the inner side to the outer side of the coil under the two heating conditions. The turbulent kinetic energy under half-side heating is much lower than that under uniform heating. The formation of the flat velocity profile in half-side heating mentioned above indicates the modification and reduction of the shear stress in the flow field. As a result, the production of turbulent kinetic energy, and the turbulent kinetic energy itself, are diminished, leading to reduced mixing and thus to lower heat transfer coefficients.
Fig. 8. Streamwise velocity distribution under half-side heating and uniform heating (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
Fig. 9. Turbulent kinetic energy distribution under half-side heating and uniform heating (P=25MPa, G=97.5kgm-2s-1, qave=31.25kWm-2)
4.3 Heat transfer correlation for supercritical water under half-side heating in the helically-coiled tube Fig. 10 gives the relative deviations of five heat transfer correlations of supercritical water (Yamagata correlation [18], Hu correlation [11], Mao correlation [6], Jackson correlation [14] and Bishop correlation [19]) when they are used to predict Nub of supercritical water in the helically-coiled tube under inner half-side heating. Note that these correlations were obtained under
uniform heating. The relative deviations (RD) of Nub given by different correlations are calculated according to Eq. (8):
The
range
of
the
RD
parameters
is:
PRE CAL (8) CAL
d=9mm,
P=23~27MPa,
G=97.5~1100kg·m-2·s-1,
qwi=0~600kW·m-2 (qave=0~300kWm-2), Hb=1500~2900kJ·kg-1, Bo*<1.5×10-5. The RDs of Yamagata correlation, Jackson correlation are very large at the pseudo-critical region (around 2100kJkg-1), reaching about +90%. At low enthalpy region (lower than 1800kJkg-1), predictions of all correlations except Mao correlation are obviously lower than the actual values (RDs<-30%). The deviations of all correlations stay at a relatively low level when Hb>Hpc (within ±30%). The performance of Mao correlation is the best among all correlations, the relative deviations are less than ±20% at the confidence level of 92%.
(a)
(b)
Fig. 10. Relative deviations of the existing correlations versus the fluid’s enthalpy when predicting half-side heating in the helical tube
A new heat transfer correlation of supecritical water in the helical tube under half-side heating is obtained: 0.469
Nu b 0.0289Re0.805 Pr b b
0.545 0.218 w b w / b
(9)
Fig. 11 shows the comparison between the predicted Nub values given by Eq. (9) and the actual Nub values. The relative deviations of Eq. (9) are less than ±20% and 10% at the confidence level of 97.9% and 82.4%, respectively.
Here we adopt 4 statistical parameters: RD, MRD, MARD and RSMRD to evaluate the performance of different correlations on predicting half-side heating. Definitions of the 4 parameters are listed as Eq.(8), (10), (11) and (12). Table 3 compares the statistics of the present and existing correlations. As can be seen, the present correlation (Eq. (9)) gives the smallest value for each statistical parameter, proving its accuracy and superiority. MRD
1 n RDi n i 1
MARD
RMSRD
(10)
1 n RDi n i 1
(11)
1 n RDi2 n i 1
(12)
Fig. 11. Comparison between the Nu values predicted by Eq. (9) and numerical simulation
Correlations
Eq. (9)
Yamagata
Hu
Mao
Jackson
Bishop
+(RD)max (%)
38.4
91.73
56.54
40.87
96.78
65.83
-(RD)max (%)
-27.3
-36.86
-48.38
-30.35
-38.47
-47.51
MRD(%)
0.42
19.07
-1.79
2.07
3.78
3.98
MARD(%)
6.34
27.45
16.25
8.53
17.23
18.15
RSMRD(%)
8.33
28.38
20.07
10.75
23.41
21.72
Table 3 Comparison between present and existing heat transfer correlations
5. Conclusions In this paper, numerical investigation on flow and heat transfer of supercritical water in the vertical helically-coiled tube under inner half-side heating condition is conducted. Results show that the heat transfer coefficient under half-side heating is lower than that under uniform heating at the same heat transfer rate. The specific heat of supercritical water at the near-wall region under half-side heating is obvious lower than that under uniform heating. Additionally, the strong buoyancy effect under half-side heating near the inner wall reduces the axial velocity gradient and the turbulent kinetic energy, causing the weaker heat transfer compared with uniform heating. The majority of the existing correlations of supercritical water cannot obtain satisfying results when predicting half-side heating. A new correlation aiming at heat transfer of supercritical water in the helically-coiled tube under inner half-side heating is proposed.
Nomenclature a
inside radius of the helical tube (mm)
A
cross-section of the tube (mm2)
Bo*
Jackson buoyancy parameter
cp
specific heat (kJkg-1K-1)
cp
average specific heat on the cross-section (=(Hw-Hb)/(Tw-Tb)) (kJkg-1K-1)
D
outside diameter (mm)
d
inside diameter (mm)
G
mass flux (kgm-2s-1)
Gr
Grashof number
h
local heat transfer coefficient (kWm-2 K-1 )
H
enthalpy (kJkg-1)
I
turbulent intensity
k
turbulent kinetic energy (m2 s-2)
l
turbulent length (m)
Nu
Nusselt number
P
pressure (MPa)
Pr
Prantl number
qave
average heat flux of the tube coil (kWm-2)
qwi
heat flux of the inner side of the coil (kWm-2)
qwo
heat flux of the outer side of the coil (kWm-2)
Re
Reynolds number
T
temperature (K)
Tw
inside wall temperature (K)
R
distance from the tube’s central axis (mm)
rin
distance of the inner side of the coil from the tube’s central axis (mm)
ur
radial velocity (ms-1)
ut
tangential velocity (ms-1)
us
secondary flow velocity on the cross-section (ms-1)
Greek symbols λ
thermal conductivity (Wm-1K-1)
ρ
density (kgm-3)
μ
dynamic viscosity (Pas)
θ
circumferential angle of the cross section (deg.)
ε
turbulent dissipation rate (m2s-3)
Subscripts b
bulk
i
mesh unit
in
inlet
pc
pseudo-critical state
wi
inner half-side of the coil
wo
outer half-side of the coil
Abbreviations PRE
results given by correlations
CAL
numerical results
RD
relative deviation
+(RD)max
maximum positive relative deviation
-(RD)max
maximum negative relative deviation
MRD
mean absolute relative deviation
MARD
mean absolute relative deviation
RSMRD
root mean square deviation
Acknowledgement This study is supported by the National Key Research and Development Program of China (No. 2016YFB0600100).
Appendix Different structured meshes with grid numbers ranging from 3,726,000 to 13,149,000 are evaluated to analyze the mesh independence. The average inside wall temperatures and heat transfer coefficients under different grid numbers are given in Fig. 12. When the gird number increases from 7,776,000 to 13,149,000, the relative deviations of inside wall temperature and heat transfer coefficients are less than 2%, indicating that the solutions are mesh-independent. Thus, the grid number of 7,776,000 is selected in this paper.
Fig. 12. Grid independence analysis
References [1] J. Y. Yoo, The Turbulent Flows of Supercritical Fluids with Heat Transfer, Annu. Rev. Fluid. Mech. 45 (2013) 495-525.
[2] V. A. Kurganov, A. G. Kaptil'ny, Velocity and enthalpy fields and eddy diffusivities in a heated supercritical fluid flow, Exp. Therm. Fluid Sci. 5 (1992) 465-478. [3] J. H. Bae, J. Y. Yoo, H. Choi, Direct numerical simulation of turbulent supercritical flows with heat transfer, Phys. Fluids. 17 (2005) (105104). [4] J. H. Bae, J. Y. Yoo, D. M. McEligot, Direct numerical simulation of heated CO2 flows at supercritical pressure in a vertical annulus at Re = 8900, Phys. Fluids. 20 (2008) (055108). [5] J. Licht, M. Anderson, M. Corradini, Heat transfer and fluid flow characteristics in supercritical Pressure water, Trans. ASME J. Heat Transf. 131 (2009) 301-314. [6] Y. F. Mao, L. J. Guo, B. F. Bai, X. M. Zhang, Convective heat transfer in helical coils for constant-property and variable-property flows with high Reynolds numbers, Front. Energy. 4 (2010) 546-552. [7] G. Zhang, H. Zhang, H. Y. Gu, Y. H. Yang, X. Cheng, Experimental and numerical investigation of turbulent convective heat transfer deterioration of supercritical water in vertical tube, Nucl. Eng. Des. 248 (2012) 226-237. [8] W. Zhang, S. X. Wang, C. D. Li, J. L. Xu, Mixed convective heat transfer of CO 2 at supercritical pressures flowing upward through a vertical helically coiled tube, Appl. Therm. Eng. 88 (2015) 61-70. [9] J. L. Xu, C. Yang, W. Zhang, D. L. Sun, Turbulent convective heat transfer of CO2 in a helical tube at near-critical pressure, Int. J. Heat Mass Transf. 80 (2015) 748–758. [10] Z. H. Li, Y. X. Wu, J. F. Lu, D. L. Zhang, H. Zhang, Heat transfer to supercritical water in circular tubes with circumferentially non-uniform heating, Appl. Therm. Eng 70 (2014) 190-200. [11] Z. H. Hu, Heat transfer characteristics of vertical upflow and inclined tubes in the supercritical pressure and near-critical pressure region, Xi’an Jiaotong University, 2001. [12] F. W. Dittus, L. M. K. Boelter, University of California, Publications of Engineering, vol. 2, 1930, pp. 443. [13] I. L. Pioro, H. F. Khartabil, R. B. Duffey, Heat transfer to supercritical fluids flowing in channels - empirical correlations (survey), Nucl. Eng. Des. 230 (2004) 69-91. [14] J. D. Jackson, Fluid flow and convective heat transfer to fluids at supercritical pressure, Nucl. Eng. Des. 264 (2013) 24-40. [15] V. Yakhot, S. A. Orszag, Renormalization group analysis of turbulence: I. Basic theory, J. Sci. Comput. 1 (1986) 1–51. [16] X. L. Lei, H. X. Li, S. Q. Yu, D. L. Ren, Numerical simulation on heterogeneous heat transfer in water at supercritical pressures in inclined upward tubes, Chin. J. Comput. Phys. 27 (2010) 217–228.
[17] J. D. Jackson, Consideration of the heat transfer properties of supercritical pressure water in connection with the cooling of advanced nuclear reactors, in: Proc. 13th Pacific Basin Nuclear Conference, Shenzhen City, China, October 21-25, 2002. [18] K. Yamagata, K. Nishikawa, S. Hasegawa, T. Fujii, S. Yoshida, Forced convective heat transfer to supercritical water flowing in tubes, Int. J. Heat Mass Transf. 15 (1972) 2575–2593. [19] A. A Bishop, R. O. Sandberg, L.S. Tong, High temperature supercritical pressure water loop: Part IV, Forced convection heat transfer to water at near-critical temperatures and super-critical pressures, Westinghouse Electric Corporation, Pittsburgh, PA, 1964.
Highlights:
1. Investigate supercritical water heat transfer in a half-side heated helical tube. 2. Analyze the mechanism of the difference between half-side and uniform heating. 3. Evaluate deviations of heat transfer correlations on predicting half-side heating. 4. Propose a heat transfer correlation in the half-side heated helical tube.