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An assessment of correlations of forced convection heat transfer to water at supercritical pressure
Annals of Nuclear Energy 76 (2015) 451–460
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
An assessment of correlations of forced convection heat transfer to water at supercritical pressure Weiwei Chen, Xiande Fang ⇑, Yu Xu, Xianghui Su Institute of Air Conditioning and Refrigeration, Nanjing University of Aeronautics and Astronautics, 29 Yudao St, Nanjing 210016, China
a r t i c l e
i n f o
Article history: Received 25 February 2014 Received in revised form 18 June 2014 Accepted 22 October 2014
Keywords: Supercritical water Heat transfer Correlation Vertical tubes Nuclear reactor
a b s t r a c t The heat transfer of supercritical water is essential for supercritical water-cooled nuclear reactors. Many empirical correlations for heat transfer to supercritical water were proposed over the past few decades. Some evaluations of the correlations were conducted, and inconsistent conclusions appeared owing to limited correlations or experimental data. This work presents an extensive survey of the literature of correlations and experiments of forced convection heat transfer to water flowing upward in vertical tubes at supercritical pressure. There are 26 correlations found, and an experimental database containing 3220 data points from vertical tubes are compiled from nine independent laboratories. All available correlations are assessed against the experimental database. The results show that the best correlation has a mean absolute deviation of 12.8%, predicting 82.3% of the database within ±20%. The entire database is divided into three categories, and the correlations which can give the most accurate predictions of the experimental data from different categories are also identified. The results provide a guide to choosing a proper correlation for engineering practice. Some topics worthy of attention for future studies are indicated. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Supercritical water is of great interest for its applications in nuclear reactor cooling since it has unique properties and favorable heat and mass transfer characteristics. A supercritical water-cooled nuclear reactor (SCWR) is a high pressure (about 25 MPa) and high temperature (up to 625 °C) reactor that operates above the critical point of water (22.064 MPa and 373.95 °C). The SCWR offers the potential for high thermal efficiencies, considerable plant simplifications, and better safety and economy (Mokry et al., 2010a). Empirical correlations with good predictions of heat transfer for supercritical water are of considerable significance for developing a SCWR. Due to the strong variation of thermophysical properties in the vicinity of the critical and pseudo-critical point, water at supercritical pressure shows different heat transfer behaviors than at subcritical pressure, and conventional single-phase correlations cannot predict it (Song et al., 2008; Cheng et al., 2009). The investigations of heat transfer of supercritical water have been carried out since the 1930s. Detailed reviews on the existing experimental and theoretical studies were performed by several ⇑ Corresponding author. Tel./fax: +86 25 8489 6381. E-mail address: [email protected] (X. Fang). http://dx.doi.org/10.1016/j.anucene.2014.10.027 0306-4549/Ó 2014 Elsevier Ltd. All rights reserved.
authors (Petukhov, 1970; Jackson and Hall, 1979; Polyakov, 1991; Cheng and Schulenberg, 2001; Pioro et al., 2004; Pioro and Duffey, 2005; Pioro and Duffey, 2007). As the prediction of the heat transfer coefficient for supercritical water is mainly conducted using empirical approaches, a number of empirical correlations exist in the open literature, which were derived based on experimental data with limited parameter ranges (Bishop et al., 1964; Swenson et al., 1965; Krasnoshchekov et al., 1967; Yamagata et al., 1972; Griem, 1996; Mokry et al., 2010a). Subsequently, some evaluations were carried out to find out the best correlations. Cheng and Schulenberg (2001) conducted a thorough review on heat transfer of supercritical water at the HPLWR condition. The HPLWR means the High Performance Light Water Reactor, a joined research project in Europe. Five heat transfer correlations for supercritical water (Bishop et al., 1964; Swenson et al., 1965; Yamagata et al., 1972; Krasnoshchekov et al., 1967; Griem, 1996) were implemented into the sub-channel analysis code to determine their applicability to the HPLWR fuel assembly. The number of the experimental data points used for their analysis was not given. As a result, the Bishop et al. (1964) correlation was recommended for calculating the heat transfer coefficient in an HPLWR fuel assembly, and the Yamagata et al. (1972) correlation was suggested to be used for determining the onset of heat transfer deterioration.
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Nomenclature Bu cp cp D G Gr Gr⁄ Gr g h L Nu Pr Pr p q q+ Re
buoyancy parameter Gr= Re2:7 Pr 0:5 specific heat at constant pressure (J/kg K) average specific heat at constant pressure (J/kg K), (hw hb)/(tw tb) inner tube diameter (m) mass flux (kg/m2 s) Grashof number (gD3(qb qw)/qm2) 2 Grashof number based onheat flux gbD4 q=k m average Grashof number gD3 ðqb qÞ=qm2 acceleration due to gravity (m/s2) specific enthalpy (J/kg) tube length (m) Nusselt number (aD=k) Prandtl number (lcp =k) average Prandtl number (lcp =k) pressure (Pa) heat flux (W/m2) non-dimensional heat flux (qb=ðGcp Þ) Reynolds number (GD/l)
Jackson (2002) evaluated nine heat transfer correlations for water flowing in vertical tubes based on 1500 experimental data points. They modified the Krasnoshchekov and Protopopov (1959) correlation for forced convective heat transfer in water and carbon dioxide at supercritical pressures, capturing 97% of the experimental data within ±25%. The Results showed that the Krasnoshchekov et al. (1967) correlation and the newly modified correlation were the most accurate ones. Pioro et al. (2004) conducted the literature survey of the work in the area of heat transfer at supercritical pressures. Eight correlations were compared based on the Shitsman (1963) experimental data for supercritical heat transfer in tubes and bundles to choose the most reliable ones. The comparisons showed that there was a significant difference in heat transfer coefficient values calculated according to various correlations. Only some correlations showed similar results, which were quite close to the experimental data for normal supercritical heat transfer in water. Also, no one correlation was able to accurately predict deteriorated or improved heat transfer in tubes. Based on the eight chosen correlations, the heat transfer coefficients and temperature profiles in the CANDU-X reactor cooled with supercritical water were calculated. Licht et al. (2008) compared four selected heat transfer correlations with their own experimental results and found that the Jackson (2002) correlation predicted the test data best, capturing 86% of the data within ±25%. The Watts and Chou (1982) correlation showed a similar trend but under-predicted the measurements by 10% relative to the Jackson (2002) correlation. Yu et al. (2009a) verified 14 supercritical heat transfer correlations based on 1142 experimental data points, and Yu et al. (2009b) compared 16 supercritical heat transfer correlations with the Styrikovich et al. (1967) data. The results showed that the Bishop et al. (1964) correlation performed best. Zhu et al. (2009) compared five selected heat transfer correlations based on their own experimental results of the supercritical heat transfer of water and found that their own correlation and the Swenson et al. (1965) correlation were the best. Mokry et al. (2010a) verified five selected heat transfer correlations and found that all of them deviated substantially from the
T t
temperature (K) temperature (°C)
Greek symbols a heat transfer coefficient (W/m2 K) b thermal expansion coefficient (1/K) k thermal conductivity (W/mK) l dynamic viscosity (Pa s) m kinematic viscosity (m2/s) n friction coefficient q density (kg/m3) q average density (kg/m3) Subscripts b at bulk temperature exp experimental in inlet pc at pseudo-critical temperature pred predicted w at wall temperature
experimental data within the pseudo-critical range. Therefore, they proposed their own correlation and recommended it to be used for SCWRs and supercritical water heat exchangers. Jäger et al. (2011) summarized the activities of the TRACE code validation at the Institute for Neutron Physics and Reactor Technology (Germany) related to supercritical water conditions. The 15 existing heat transfer correlations were reviewed and implemented into TRACE, and six selected experimental data sources were used to identify the most suitable heat transfer correlation(s). The number of the experimental data used was not stated, and the overall performance of each correlation for predicting the entire database was not clear. As a result, they recommended the Bishop et al. (1964) model for design and safety evaluation of SCWRs. The above evaluations presented inconsistent results due to limited experimental data or correlations. The most comprehensive reviews might be those by Jäger et al. (2011) and Yu et al. (2009a). The former evaluated 15 existing supercritical heat transfer correlations with the experimental data from six selected sources, and the latter assessed 14 based on 1142 experimental data points. This paper conducts an all-around survey of the correlations and experimental results, and 26 existing correlations for water supercritical heat transfer in vertical tubes are assessed with the supercritical water heat transfer database containing 3220 data points compiled from nine independent laboratories. The number of the correlations evaluated and the data used is far more than previous ones. Furthermore, the available experimental data are partitioned into three different heat transfer regimes, including the normal heat transfer regime, the enhanced heat transfer regime, and the deteriorated heat transfer regime. The evaluation of the surveyed correlations is implemented for each regime. To the best of the authors’ knowledge, it is the first state-of-the-art review using a multiple-source database consisting of more than 1500 data points to evaluate more than 15 correlations for supercritical heat transfer to water, and it is the first practice to evaluate the correlations for each of the three heat transfer regimes. The evaluation results provide a guide to choosing a proper correlation for engineering practice. Some topics worthy of attention for future studies are indicated.
W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460
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(1990), Griem (1996), and Kitoh et al. (1999), as summarized in Table 1. Most of the correlations were proposed by considering the ratio of density, thermal conductivity, specific heat, or viscosity evaluated at the wall temperature to those at the bulk temperature, such as those of Krasnoshchekov and Protopopov (1959), Bishop et al. (1964), Swenson et al. (1965), Krasnoshchekov et al. (1967), Jackson (2002), Xu et al. (2005), and Mokry et al. (2010a), as summarized in Tables 1 and 2. The correlations of Bringer and Smith (1957), Krasnoshchekov and Protopopov (1959), and Krasnoshchekov et al. (1967) can be used
2. Review of existing correlations In the past few decades, a number of heat transfer correlations based on experimental results of supercritical water were proposed, among which most are the modified forms of the Dittus–Boelter (1930) equation or the Gnielinski (1976) equation. Some correlations were developed by combining the Reynolds number (Re) and the Prandtl number (Pr) with the physical properties at the bulk and wall temperatures, such as those of McAdams (1942), Bringer and Smith (1957), Shitsman (1963), Gorban et al.
Table 1 Correlations taking the form of Dittus–Boelter equation. Author
Correlation
Flow and operating parameters
McAdams (1942)
0:4 Nub ¼ 0:0243Re0:8 b Pr b
High pressures and low heat fluxes
Bringer and Smith (1957)
0:55 Nux ¼ 0:0266Rex0:77 Pr w Nux and Rex are evaluated at temperature tx: tx = tb for E < 0, tx = tpc for 0 5 E 5 1 tx = tw for E > 1, and E = (tpc tb)/(tw tb)
Apart from critical and pseudo-critical regions
Shitsman (1963)
0:8 Nub ¼ 0:023Re0:8 b Pr min Prmin is the smaller one of Prw and Prb
Applied to conditions with Pr 1 D = 7.8, 8.2 mm
Bishop et al. (1964)
Nub ¼ 0:0069Reb0:9 Pr b 0:66 ðqw =qb Þ0:43 ð1 þ 2:4D=xÞ x is the axial location along the heated length (m)
p = 22.8–27.6 MPa, tb = 282–527 °C, G = 651–3662 kg/m2 s, q = 310–3460 kW/m2 and x/D = 30–565
Swenson et al. (1965)
Pr w 0:613 ðqw =qb Þ0:231 Nuw ¼ 0:00459Re0:923 w
p = 22.8–41.4 MPa, tb = 75–576 °C, G = 542–2150 kg/m2 s, tw = 93–649 °C, Re = 7.5 104–3.16 106
Ornatsky et al. (1970)
0:3 0:8 Nub ¼ 0:023Re0:8 b Pr min ðqw =qb Þ
Yamagata et al. (1972)
Pr0:8 Nub ¼ 0:0135Re0:85 b b Fc Fc = 1, for E > 1, F c ¼ ðcp =cp;b Þn2 for E < 0,
p = 22.6–29.4 MPa, tb = 230–540 °C, G = 310–1830 kg/m2 s, q = 116–930 kW/m2 D = 7.5, 10 mm
F c ¼ 0:67Pr 0:05 ðcp =cp;b Þn1 for 0 5 E 5 1, and pc n1 = 0.77(1 + 1/Prpc) + 1.49, n2 = 1.44(1 + 1/Prpc) 0.53 Watts and Chou (1982)
P = 25.0 MPa, tb = 150–350 °C,
0:55 ðqw =qb Þ0:35 / Nub ¼ 0:021Re0:8 b Pr b
G = 130–1000 kg/m2 s, q = 170–450 kW/m2, Pr b = 0.85–2.30 Reb = 6.5 103–3 105
/ ¼ 1 for Bu 5105 ; / ¼ ð7000BuÞ0:295 for Bu =104 / ¼ ð1 3000BuÞ
0:295
for 10
5
4
< Bu < 10
Gorban’ et al. (1990)
Nub ¼ 0:0059Reb0:90 Pr0:12 b
Griem (1996)
Pr 0:432 x Num ¼ 0:0169Re0:8356 b sel Pr sel ¼ lb cp;sel =km , and km ¼ 0:5ðkw þ kb Þ x = 0.82 for hb < 1540 kJ/kg, x = 1 for hb > 1740 kJ/kg x = 0.82 + 9 104(hb 1540) for 1540 5 hb 5 1740 kJ/kg
p = 22.0–27.0 MPa, G = 300–2500 kg/m2 s q = 200–700 kW/m2 and D = 10, 14, 20 mm For detail of cp,sel, please see Griem (1996)
Kitoh et al. (1999)
Pr m Nub ¼ 0:015Re0:85 b b m = 0.69–81000/200G1.2 + fcq –8 fc = 2.9 10 + 0.11/200G1.2 for 0 6 hb 6 1500 kJ/kg fc = 8.7 10–8 0.65/200G1.2 for 1500 < hb < 3300 kJ/kg fc = 9.7 10–7 + 1.3/200G1.2 for 3300 5 hb 5 4000 kJ/kg
tb = 20–550 °C, G = 100–1750 kg/m2 s, and q = 0–1800 kW/m2
Jackson (2002)
0:3 Pr0:5 ðcp =cp;b Þn Nub ¼ 0:0183Re0:82 b b ðqw =qb Þ n = 0.4 for Tb < Tw < Tpc or 1.2Tpc < Tb < Tw n = 0.4 + 0.2(Tw/Tpc 1) for Tb < Tpc < Tw n = 0.4 + 0.2(Tw/Tpc 1)[1–5(Tb/Tpc 1)] for Tpc < Tb < 1.2Tpc and Tb < Tw
p = 23.4–29.3 MPa, G = 700–3600 kg/m2s q = 46–2600 kW/m2, Reb = 8 104–5 105 D = 1.6–20 mm
Xu et al. (2005)
Pr b 0:9213 ðqw =qb Þ0:6638 Nub ¼ 0:02269Re0:8079 b
Kuang et al. (2008)
Nub ¼ 0:0239Re0:759 Pr b 0:833 ðqw =qb Þ0:31 ðkw =kb Þ0:0863 b 0:014
ðlw =lb Þ0:832 ðGr b Þ
lw =lb
0:8687
p = 23.0–30.0 MPa, G = 600–1200 kg/m2 s q = 100–600 kW/m2, and D = 12.0 mm p = 22.8–31.0 MPa, G = 380–3600 kg/m2 s q = 233–3474 kW/m2
0:021
ðqþ Þ b
Nub ¼ F ¼ minðF 1 ; F 2 Þ
p = 22.5–25.0 MPa, G = 700–3500 kg/m2 s q = 300–2000 kW/m2, tb = 300–450 °C D = 10, 20 mm
Zhu et al. (2009)
Nub ¼ 0:0068Reb0:9 Pr b 0:63 ðqw =qb Þ0:17 ðkw =kb Þ029
Yu et al. (2009a)
Nub ¼ 0:01378Re0:9078 Pr 0:6171 ðqw =qb Þ0:4356 ðGr b Þ b b
Mokry et al. (2010a)
ðqw =qb Þ0:564 Nub ¼ 0:0061Reb0:904 Pr0:684 b
Gupta et al. (2010)
Nuw ¼ 0:004Re0:923 Pr w 0:773 ðqw =qb Þ0:186 ðlw =lb Þ0:366 w
Liu and Kuang (2012)
Pr b 0:73 ðqw =qb Þ0:401 ðkw =kb Þ0:24 Nub ¼ 0:01Re0:889 b
p = 22.0–30.0 MPa, G = 600–1200 kg/m2 s q = 200–600 kW/m2, and D = 26.0 mm p = 22.6–41.0 MPa, D = 1.5–38.1 mm G = 90–2441 kg/m2 s, q = 90–1800 kW/m2 p = 24.0 MPa, G = 200–1500 kg/m2 s q = 0–1250 kW/m2, and D = 10.0 mm P = 24.0 MPa, G = 200–1500 kg/m2 s q = 0–1250 kW/m2, and D = 10.0 mm P = 22.4–31.0 MPa, D = 6–38 mm G = 200–3500 kg/m2 s, q = 37–2000 kW/m2
Cheng et al. (2009)
1=3 0:023Re0:8 b Pr b F
2:4 F 1 ¼ 0:85 þ 0:776 1000qþ b qþ 0:48 b F2 ¼ 1:55 þ 1:21 1 qþ pc ð1000qþpc Þ
ðlw =lb Þ0:153 ðcp =cp;b Þ0:014 ðGr b Þ
0:007
0:041
ðqþ Þ b
0:012
0:0605
ðqþ Þ b
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Table 2 Correlations taking the form of Gnielinski equation. Author
Correlation
Krasnoshchekov and Protopopov (1959)
Flow and operating parameters 0:11
Nub ¼ Nu0 ðlw =lb Þ Nu0 ¼
0:33
ðkb =kw Þ
ðcp =cp;b Þ
p = 22.3–32 MPa, Reb = 2 104–8.6 105,
0:35
Pr b = 0:85 65; lb =lw = 0.9–3.6, kb =kw = 1–6 cp =cp;b = 0.07–4.5
ðn0 =8ÞReb Prb
pffiffiffiffiffiffiffiffi
1:07þ12:7
n0 =8ðPrb 2=3 1Þ 2
n0 ¼ ½1:82log10 ðReb Þ 1:64 Krasnoshchekov et al. (1967)
Nub ¼ Nu0 ðqw =qb Þ0:3 ðcp =cp;b Þn n = 0.4 for Tb < Tw < Tpc or 1.2Tpc < Tb < Tw n = n1 = 0.22 + 0.18Tw/Tpc for 1 < Tw/Tpc < 2.5n = n1 + (5n1 2)(1 Tb/Tpc) for Tpc < Tb < 1.2Tpc and Tb < Tw Nu0 is evaluated as above
Grass et al. (1971)
Nub ¼ Pr G ¼ Nub ¼
Prb 0 =8ÞRe pðnffiffiffiffiffiffiffi ffi b2=3 n0 =8ðPrG cp;b =cp;w 1Þ Pr b ; Pr b < 0:5Pr w Pr w ; Prb > 0:5Pr w
1:07þ12:7
Petukhov et al. (1983)
ðn=8ÞReb Prb
pffiffiffiffiffiffi
Nub ¼
n=8ðPrb 2=3 1Þ
1þ900=Reb þ12:7
n ¼ n0 ðqw =qb Þ Razumovskiy et al. (1990)
0:4
ðlw =lb Þ0:2
ðnr =8ÞReb Pr b
pffiffiffiffiffiffiffi 2=3 ðcp =cp;b Þ0:65 nr =8ðPrb 1Þ
1:07þ12:7
nr ¼ n0 ðqw =qb Þ0:18 ðlw =lb Þ0:18 Nub ¼ Nu0 ðqw =qb Þ0:4 ðcp =cp;b Þn uðk Þ
Kirillov et al. (1990)
Nu0 ¼
P = 23.4–29.3 MPa, G = 700–3600 kg/m2 s, q = 46–2600 kW/m2, Reb = 8 104–5 105, D = 1.6–20 mm
ðn0 =8ÞReb Prb
pffiffiffiffiffiffiffiffi 2=3 n0 =8ðPrb 1Þ
1þ900=Reb þ12:7
p = 22.3–29.3 MPa, Reb = 2 104–8 105, Pr b = 0.85–65, q = 23–2600 kW/m2, For details of uðk Þ please see Kirillov et al. (1990)
k ¼ ð1 qw =qb ÞGr b =Re2b , n = 0.7 for cp cp;b n is defined as following for cp < cp;b : n = 0.4 for Tb < Tw < Tpc or 1.2Tpc < Tb < Tw n = 0.22 + 0.18Tw/Tpc for Tb < Tpc < Tw n = 0.9 Tb/Tpc(1- Tw/Tpc) + 1.08 Tw/Tpc-0.68 for Tpc < Tb < 1.2Tpc and Tpc < Tw
Table 3 Experimental data sources for vertical upward tubes. Data source
Flow range: t (°C)/pin (MPa)/G (kg/m2 s)/q (kW/m2)
Geometry range: D (mm)/(mm)/material
Data points
Alekseev et al. (1976) Griem (1996) Mokry et al. (2010b) Pan et al. (2011) Shitsman (1968) Swenson et al. (1965) Vikhrev et al. (1967) Yamagata et al. (1972) Zhu et al. (2009)
100–350(tin)/24.5/380–820/100–900 343–421(tb)/22–27/300–2500/200–700 320–350(tin)/24/200–1500/0–884 330–550(tb)/22.5–30/1009–1626/216–822 100–250(tin)/24.5–34.3/350–600/270–700 75–576(tb)/23–41/542–2150/200–1800 50–425(tb)/26.5/500–1900/230–1250 230–540(tb)/22.6–29.4/310–1830/116–930 282–440(tb)/23–30/600–1200/200–600
10.4/750/Kh18N10T steel 14/unmentioned/unmentioned 10/4000/12Cr18Ni10Ti stainless steel 17/2000/1Gr18Ni9Ti stainless steel 8,16/800,1600,3200/1Gr18Ni9Ti steel 9.42/1830/AISI-304 stainless steel 20.4/6000/1Gr18Ni10Ti stainless steel 7.5,10/1500,2000/AISI-316 stainless steel 26/1000/1Gr18Ni9Ti stainless steel
163 259 1323 231 291 159 424 250 120
both for water and CO2, and the Gorban et al. (1990) correlation can be used both for water and Freon. However, some constants in the correlations for water are different from those for CO2 (or Freon). The constants of these four correlations shown in Tables 1 and 2 are for water. In particular, large differences in density between bulk and wall positions and along the flow direction can change the flow structure. The effects of the density variation were represented as the buoyancy and flow acceleration, and some authors presented correlations by adding non-dimensional parameter concerning buoyancy and flow acceleration to account for the effects, such as Watts and Chou (1982), Kuang et al. (2008), Cheng et al. (2009), Yu et al. (2009a), and Liu and Kuang (2012), as summarized in Table 1.
3. Experimental data Through the comprehensive survey of the experimental studies of the forced convective heat transfer to supercritical water flowing in vertical tubes, nine available experimental data sources are identified as listed in Table 3, and a database containing 3220
Fig. 1. Partition of total experimental data.
W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460 Table 4 Statistics of the top eight correlations against the entire database (3220 data points).
a b
Correlations
MAD
MRD
RMSD
SD
R20a
R30b
Mokry et al. (2010a) Petukhov et al. (1983) Swenson et al. (1965) Liu and Kuang (2012) Gupta et al. (2010) Watts and Chou (1982) Kuang et al. (2008) Zhu et al. (2009)
12.8 15.1 15.8 16.8 17.0 17.0 17.4 19.2
0.9 6.7 3.5 12.2 7.0 6.4 2.9 15.0
19.0 22.6 23.2 24.8 22.4 25.7 25.0 28.8
19.0 21.6 23.0 21.6 21.3 24.9 24.8 24.6
82.3 75.2 73.5 73.0 66.4 70.8 66.3 67.1
92.7 89.2 87.4 85.5 87.7 86.8 84.4 81.1
Percentage of the data points within ±20% error band. Percentage of the data points within ±30% error band.
experimental data points are established. The parameter ranges cover bulk enthalpy from 278.1 to 3169 kJ/kg, mass fluxes from 201 to 2500 kg/m2 s, heat fluxes from 129 to 1735 kW/m2, pressures from 22 to 34.3 MPa, and tube hydraulic diameters from 7.5 to 26 mm.
455
For forced convective heat transfer to water at supercritical pressures, there are three heat transfer regimes characterized as the normal heat transfer regime, the enhanced heat transfer regime and the deteriorated heat transfer regime (Mokry et al., 2010b). Many different criteria defining the three heat transfer regimes were presented in the literature (Cheng and Schulenberg, 2001), among which the one proposed by Koshizuka et al. (1995) as the following has been frequently mentioned:
R ¼ aexp =a0
ð1Þ
where a0 is the calculated heat transfer coefficient using the Dittus and Boelter (1930) correlation, and aexp is the experimental heat transfer coefficient. This criterion classifies the heat transfer of supercritical water flow as normal when 0.3 6 R 6 1, enhanced when R > 1, and deteriorated when R < 0.3. According to the criterion, all the 3220 experimental data points are divided into three regimes as showed in Fig. 1, with the normal heat transfer regime having 2145 points (66.6%), the enhanced heat transfer
Fig. 2. Comparison of the top three correlations with the entire experimental data.
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Table 5 Comparison of the top three correlations in different heat transfer regimes. Correlations
Normal data
Mokry et al. (2010a) Petukhov et al. (1983) Swenson et al. (1965)
Enhanced data
Deteriorated data
MAD
RMSD
MAD
RMSD
MAD
RMSD
10.6 12.0 14.0
14.8 16.4 20.4
14.7 13.8 19.6
18.4 16.4 25.2
24.0 43.8 18.4
39.2 55.7 35.2
Table 6 Statistics of the top twelve correlations against the normal database (2145 data points). Correlations
MAD
MRD
RMSD
SD
R20
R30
Mokry et al. (2010a) Petukhov et al. (1983) Watts and Chou (1982) Swenson et al. (1965) Liu and Kuang (2012) Griem (1996) Kuang et al. (2008) Gupta et al. (2010) Zhu et al. (2009) Yu et al. (2009a) Ornatsky et al. (1970) Bishop et al. (1964)
10.6 12.0 13.6 14.0 14.1 14.8 15.2 15.7 16.2 17.4 17.6 19.2
0.3 6.9 6.4 2.3 10.2 7.0 0.9 9.5 13.5 16.1 8.7 17.7
14.8 16.4 18.3 20.4 20.3 22.1 19.3 19.3 23.0 22.7 23.6 25.0
14.8 14.9 17.1 20.3 17.6 20.9 19.3 16.8 18.6 16.0 21.9 17.6
88.2 82.1 78.3 78.4 79.7 76.0 70.4 68.5 74.6 67.6 68.0 62.8
95.4 93.1 91.8 90.7 89.4 88.0 87.8 91.6 85.7 85.3 83.8 82.9
regime having 803 points (24.9%), and the deteriorated heat transfer regime having 272 points (8.5%). From Fig. 1 it can be seen that most of the enhanced and deteriorated heat transfer phenomena appeared near the pseudo-critical points. 4. Assessment of the correlations The 26 reviewed heat transfer correlations are assessed with the database of the entire, normal, enhanced and deteriorated heat transfer data points, respectively. For the standard statistical procedure, criterion MAD, the root mean square deviation (RMSD), the standard deviation (SD), and the mean relative deviation (MRD) are often used.
MAD ¼
N 1X jRDi j N i¼1
Fig. 3. Deviations of the predictions of the top three correlations from the normal heat transfer data.
ð2Þ
457
W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460 Table 7 Statistics of the top fourteen correlations against the enhanced database (803 data points).
Table 8 Statistics of the top four correlations against the deteriorated database (272 data points).
Correlations
MAD
MRD
RMSD
SD
R20
R30
Correlations
MAD
MRD
RMSD
SD
R20
R30
McAdams (1942) Jackson (2002) Shitsman (1963) Kitoh et al. (1999) Petukhov et al. (1983) Griem (1996) Mokry et al. (2010a) Watts and Chou (1982) Yu et al. (2009a) Kuang et al. (2008) Xu et al. (2005) Bishop et al. (1964) Ornatsky et al. (1970) Gupta et al. (2010)
10.3 11.5 11.5 12.3 13.8 14.5 14.7 15.3 15.5 16.7 19.0 19.0 19.8 20.0
8.9 5.5 0.5 1.8 6.6 12.2 3.5 8.1 5.2 3.2 11.3 11.2 18.9 6.7
14.2 14.2 14.9 15.5 16.4 17.5 18.4 18.4 19.8 20.5 24.2 25.0 22.3 24.5
11.1 13.1 14.9 15.4 15.0 12.6 18.0 16.5 19.1 20.2 21.4 22.4 11.9 23.6
82.8 83.8 82.8 80.9 75.8 73.3 72.4 69.5 69.6 64.1 61.5 62.9 52.6 56.8
94.4 96.9 95.4 95.3 94.3 91.7 89.9 89.9 88.2 84.8 74.3 79.8 81.6 77.7
Gupta et al. (2010) Swenson et al. (1965) Xu et al. (2005) Mokry et al. (2010a)
18.1 18.4 20.5 24.0
11.6 12.0 4.1 23.2
35.2 35.2 31.3 39.2
33.3 33.1 31.1 31.7
77.6 76.8 66.2 65.4
86.8 87.9 84.6 79.8
RMSD ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN RD2i i¼1 N
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN SD ¼ ðRDi MRDÞ2 i¼1 N1 MRD ¼
N 1X RDi N i¼1
ð3Þ
where
RDi ¼
apred ðiÞ aexp ðiÞ 100 aexp ðiÞ
ð6Þ
The MAD and the RMSD are used to gauge prediction accuracy, the SD is used to measure the dispersion degree of the prediction, and the MRD is used to check if a correlation has an over-prediction or under-prediction on average. 4.1. Evaluation of correlations with the entire data
ð4Þ
ð5Þ
For the entire data without partition, all the reviewed correlations are assessed and eight correlations have the MAD less than 20% and RMSD less than 30%, as shown in Table 4. The top three correlations are those of Mokry et al. (2010a), Petukhov et al. (1983) and Swenson et al. (1965). The Mokry et al. (2010a)
Fig. 4. Comparison of the top three correlations with the enhanced experimental data.
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Fig. 5. Comparison of the top three correlations with the deteriorated experimental data.
correlation performs best with an MAD of 12.8%, a RMSD of 19.0%, and an SD of 19.0%, respectively, predicting 82.3% of the entire data within ±20% and 92.7% of those within ±30%. The MRD values indicate that all the top three correlations give an over-prediction on average, with the Petukhov et al. (1983) correlation most noticeable. Fig. 2 shows the deviations of the predictions of the top three correlations from the entire data. Table 5 compared the top three correlations in different heat transfer regimes, from which it can be seen that the prediction performances of the top three correlations depend on the heat transfer regimes. They all perform best in the normal heat transfer regime and worst in the deteriorated heat transfer regime. Moreover, the Mokry et al. (2010a) correlation has the highest accuracy in the normal heat transfer regime, the Petukhov et al. (1983) correlation has the highest accuracy in the enhanced heat transfer regime, and the Swenson et al. (1965) correlation performs best in the deteriorated heat transfer regime. Therefore, it is necessary to ascertain top correlations in different heat transfer regimes.
and Chou (1982) are the most agreeable ones. The Mokry et al. (2010a) correlation performs best with the MAD, RMSD, and SD of 10.6%, 14.8%, and 14.8%, respectively, predicting 88.2% of the normal heat transfer data within ±20% and 95.4% of those within ±30%. The Petukhov et al. (1983) and Watts and Chou (1982) correlations give noticeable over-predictions on average. Fig. 3 shows the deviations of the predictions of the top three correlations from the experimental data in the normal heat transfer regime, from which it can be seen that the top three correlations predict well the experimental data in the normal heat transfer regime only if the data are away from the pseudo-critical points. Unacceptable deviations emerge in the vicinity of the pseudo-critical points due to the strong variation of thermophysical properties. Thus, the predictions of normal heat transfer near the pseudo-critical points need to be improved.
4.2. Evaluation of correlations with data from the normal heat transfer regime
Table 7 shows the evaluation results with the experimental data in the enhanced heat transfer regime, where fourteen correlations with the MAD less than 20% and RMSD less than 25% are displayed. The McAdams (1942) correlation is the best with the smallest MAD (10.3%), RMSD (14.2%), and SD (11.1%), predicting 82.8% of the experimental data in the enhanced heat transfer regime within ±20% and 94.4% of those within ±30%. The Jackson (2002) and
For the data in the normal heat transfer regime, there are twelve correlations having the MAD less than 20% and RMSD less than 25%, as presented in Table 6. The statistics indicate that the correlations of Mokry et al. (2010a), Petukhov et al. (1983) and Watts
4.3. Evaluation of correlations with data from the enhanced heat transfer regime
W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460
Shitsman (1963) are the next two best ones. Among the top three correlations, those of McAdams (1942) and Jackson (2002) give noticeable under-predictions on average. Fig. 4 illustrates the deviations of the predictions of the top three correlations from the experimental data in the enhanced heat transfer regime. As can be seen in Fig. 4, away from the pseudo-critical points, the top three correlations perform well. Similar to the situation in the normal heat transfer regime, large deviations appear near the pseudo-critical points owing to the intense variation of the thermophysical properties. Although the vast majority of the experimental data in the enhanced heat transfer regime congregate near the pseudo-critical points, fourteen correlations can provide the MADs less than 20%, and the performances of the top correlations are a little better than those in the normal heat transfer regime. 4.4. Evaluation of correlations with data from the deteriorated heat transfer regime For the deteriorated heat transfer regime, there are four correlations with the MAD less than 30% and RMSD less than 40%, as shown in Table 8. The accuracies of the Gupta et al. (2010) and Swenson et al. (1965) correlations are very close, having the MAD less than 18.5% and predicting about 80% of the experimental data in the deteriorated heat transfer regime within ±20%. The Xu et al. (2005) correlation has the smallest RMSD (31.3%) and SD (31.1%) and predicts 66.2% of the deteriorated data within ±20%. All of the four correlations over-predict the data on average. Fig. 5 shows the comparisons of the predictions of the top three correlations with the experimental data in the deteriorated heat transfer regime. It can be seen that almost all the deteriorated data points are near the pseudo-critical points, and that even the top correlations are not satisfactory. Tables 6–8 indicate that the prediction of the deteriorated heat transfer is the most challenging and that of the enhanced heat transfer is the easiest. Figs. 3–5 illustrate that the deviations in the vicinity of the pseudo-critical points are remarkably larger for all of the three heat transfer regimes, and thus it is essential to improve the prediction accuracy near the pseudo-critical points. 5. Conclusions An extensive survey of the investigations of the correlations and experiments of the forced convective heat transfer to water at supercritical pressure has been carried out. Twenty six correlations are evaluated with 3220 experimental data points from vertical tubes compiled from nine published articles. The experimental ranges cover bulk enthalpies from 278.1 to 3169 kJ/kg, mass fluxes from 201 to 2500 kg/m2 s, heat fluxes from 129 to 1735 kW/m2, pressures from 22 to 34.3 MPa, and tube hydraulic diameters from 7.5 to 26 mm. The entire database is partitioned into three heat transfer regimes as per the Koshizuka et al. (1995) method, with 66.6% data points (2145) in the normal heat transfer regime, 24.9% (803) in the enhanced heat transfer regime, and 8.5% (272) in the deteriorated heat transfer regime. The following conclusions can be drawn from this study: (1) For the entire database, the top three correlations are those of Mokry et al. (2010a), Petukhov et al. (1983), and Swenson et al. (1965). The Mokry et al. (2010a) correlation performs best, which has the smallest MAD (12.8%), RMSD (19.0%), and SD (19.0%) and predicts 82.3% of the entire data within ±20% and 92.7% of those within ±30%. The prediction performances of the correlations depend on the heat transfer regimes, and none of the reviewed correlations can performs
(2)
(3)
(4)
(5)
(6)
459
best in all the three heat transfer regimes. Therefore, it is necessary to ascertain top correlations in different heat transfer regimes. For the normal heat transfer regime, the correlations of Mokry et al. (2010a), Petukhov et al. (1983), and Watts and Chou (1982) have the highest accuracy. The Mokry et al. (2010a) correlation has the smallest MAD (10.6%), RMSD (14.8%), and SD (14.8%), and predicts 88.2% of the experimental data in the normal heat transfer regime within ±20% and 95.4% within ±30%. For the enhanced heat transfer regime, the correlations of McAdams (1942), Jackson (2002), and Shitsman (1963) have the highest accuracy. The McAdams (1942) correlation has the smallest MAD (10.3%), RMSD (14.2%), and SD (11.1%), predicting 82.8% of the experimental data in the enhanced heat transfer regime within ±20% and 94.4% within ±30%. For the deteriorated heat transfer regime, the correlations of Gupta et al. (2010), Swenson et al. (1965), and Xu et al. (2005) are the most promising ones. The Gupta et al. (2010) correlation has the smallest MAD of 18.1%, predicting 77.6% of the experimental data in the deteriorated heat transfer regime within ±20%, and 86.8% within ±30%. The Xu et al. (2005) correlation has the smallest RMSD (31.3%) and SD (31.1%), predicting 66.2% of the deteriorated heat transfer data within ±20% and 84.6% within ±30%. The prediction of the deteriorated heat transfer is the most challenging and that of the enhanced heat transfer is the easiest. All of the mentioned top correlations have remarkably larger deviations in the vicinity of the pseudo-critical points than where away from the pseudo-critical points because the thermophysical properties of water vary drastically near the pseudo-critical point. A more accurate correlation needs to be developed. Efforts should be made to understand the mechanisms of the deteriorated heat transfer and the heat transfer near the pseudocritical point so that the prediction performances in these two aspects can be improved.
References Alekseev, G.V., Silin, V.A., Smirnov, A.M., Subbotin, V.I., 1976. Study of the thermal conditions on the wall of a pipe during the removal of heat by water at a supercritical pressure. High Temp. 14, 683–687. Bringer, R.P., Smith, J.M., 1957. Heat transfer in the critical region. AIChE J. 3, 49–55. Bishop, A.A., Sandberg, R.O. Tong, L.S., 1964. Forced convection heat transfer to water at near-critical temperatures and super-critical pressures. Report WCAP– 2056, Westinghouse Electric Corporation, Atomic Power Division, Pittsburgh, PA, USA, December, 85 pages. Cheng, X., Schulenberg, T., 2001. Heat transfer at supercritical pressures-literature review and application to a HPLWR. Forschungszentrum Karlsruhe, Technik und Umwelt, Wissenschaftliche Berichte, FZKA 6609, Institute für Kernund Energietechnik, Mai 2001. Cheng, X., Yang, Y.H., Huang, S.F., 2009. A simplified method for heat transfer prediction of supercritical fluids in circular tubes. Ann. Nucl. Energy 36, 1120– 1128. Dittus, FW., Boelter, L.M.K., 1930. Heat transfer in automobile radiators of the tubular type. University of California Publications in English, Berkeley, 2, 443– 461. Grass, G., Herkenrath, H., Hufschmidt, W., 1971. Anwendung des Prandtlschen Grenzschichtmodells auf den Waermeuebergang an Fluessigkeiten mitstark temperaturabhaengigen Stoffeigenschaften bei erzwungener Stroemung. Waerme-und Stoffuebertragung 4, 113–119 (In German). Gnielinski, V., 1976. New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chem. Eng. 16, 359–368. Gorban, L.M., Pomet’ko, R.S., Khryaschev, O.A., 1990. Modeling of water heat transfer with Freon of supercritical pressure (in Russian). Institute of Physics and Power Engineering, Obninsk, Russia. Griem, H., 1996. A new procedure for the prediction of forced convection heat transfer at near-and supercritical pressure. Heat Mass Transf. 31, 301–305. Gupta, S., Farah, A., King, K., Mokry, S., Pioro, I., 2010. Developing new heat-transfer correlation for supercritical-water flow in vertical bare tubes. In: Proceedings of the 18th International Conference on Nuclear Engineering (ICONE–18), Xi’an, China, May 17–21, Paper No. 30024.
460
W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460
Jackson, J.D., Hall, W.B., 1979. Forced convection heat transfer to fluids at supercritical pressure. Turbulent Forced Convection in Channels and Bundles 2, 563–611. Jackson, J.D., 2002. Consideration of the heat transfer properties of supercritical pressure water in connection with the cooling of advanced nuclear reactors. In: Proceedings of the 13th Pacific Basin Nuclear Conference, Shenzhen City, China, October 21–25. Jäger, W., Sánchez Espinoza, V.H., Hurtado, A., 2011. Review and proposal for heat transfer predictions at supercritical water conditions using existing correlations and experiments. Nucl. Eng. Des. 241, 2184–2203. Krasnoshchekov, E.A., Protopopov, V.S., 1959. Heat transfer at supercritical region in flow of carbon dioxide and water in tubes (in Russian). Therm. Eng. 12, 26–30. Krasnoshchekov, E.A., Protopopov, V.S., Van Fen, Kuraeva, I.V., 1967. Experimental investigation of heat transfer for carbon dioxide in the supercritical region. In: Gazley, Jr., C., Hartnett, J.P., Ecker, E.R.C. (Eds.), Proceedings of the Second All-Soviet Union Conference on Heat and Mass Transfer, Minsk, Belarus, May, Published as Rand Report R–451–PR,1, 26–35. Kirillov, P.L., Yur’ev, Yu, S., Bobkov, V.P., 1990. Handbook of Thermal-Hydraulics Calculations (in Russian). Energoatomizdat Publishing House, Moscow, Russia, 66-67,130–132. Kitoh, K., Koshizuka, S., Oka, Yo., 1999. Refinement of transient criteria and safety analysis for a high temperature reactor cooled by supercritical water. In: Proceedings of the Seventh International Conference on Nuclear Engineering (ICONE–7), Tokyo, Japan, April 19–23, Paper–7234. Koshizuka, S., Takano, N., Oka, Y., 1995. Numerical analysis of deterioration phenomena in heat transfer to supercritical water. Int. J. Heat Mass Transf. 38, 3077–3084. Kuang, B., Zhang, Y., Cheng, X., 2008. A new, wide-ranged heat transfer correlation of water at supercritical pressures in vertical upward ducts, NUTHOS–7, Seoul, Korea, October 5–9. Licht, J., Anderson, M., Corradini, M., 2008. Heat transfer to water at supercritical pressures in a circular and square annular flow geometry. Int. J. Heat Fluid Flow 29, 156–166. Liu, X., Kuang, B., 2012. Wide-ranged heat transfer correlations of supercritical water in vertical upward channels. Nucl. Sci. Eng. 32, 344–354 (in Chinese). McAdams, W.H., 1942. Heat transmission, 2nd ed. McGraw-Hill, New York, NY, USA, 459. Mokry, S., Farah, A., King, K., Gupta, S., Pioro, I., 2010a. Development of a heattransfer correlation for supercritical water flowing in a vertical bare tube. In: Proceedings of the 14th International Heat Transfer Conference (IHTC–14), Washington, DC, USA, August 8–13, Paper No. 22908. Mokry, S., Pioro, I., Kirillov, P., Gospodinov, Y., 2010. Supercritical-water heat transfer in a vertical bare tube. Nucl. Eng. Des. 240, 568–576. Ornatsky, A.P., Glushchenko, L.P., Siomin, E.T., 1970. The research of temperature conditions of small diameter parallel tubes cooled by water under supercritical pressures. In: Proceedings of the Fourth International Heat Transfer Conference, Paris-Versailles, France, vol. VI. Elsevier, Amsterdam, Paper B 8.11. Pan, J., Yang, D., Dong, Z.C., Zhu, T., Bi, Q.C., 2011. Experimental investigation on heat transfer characteristics of water in vertical upward tube under supercritical pressure. Nucl. Power Eng. 32, 75–79 (in Chinese).
Petukhov, B.S., 1970. Heat transfer and friction in turbulent pipe flow with variable physical properties. Adv. Heat Transfer 6, 503–564. Petukhov, B.S., Kurganov, V.A., Ankudinov, V.B., 1983. Heat transfer and flow resistance in the turbulent pipe flow of a fluid with near-critical state parameters. High Temp. Sci. 21, 81–89. Polyakov, A.F., 1991. Heat transfer under supercritical pressures. Adv. Heat Transfer 21, 1–53. Pioro, I.L., Khartabil, H.F., Duffey, R.B., 2004. Heat transfer to supercritical fluids flowing in channels–empirical correlations (survey). Nucl. Eng. Des. 230, 69–91. Pioro, I.L., Duffey, R.B., 2005. Experimental heat transfer in supercritical water flowing inside channels (survey). Nucl. Eng. Des. 235, 2407–2430. Pioro, I.L., Duffey, R.B., 2007. Heat transfer and hydraulic resistance at supercritical pressures in power engineering applications. ASME Press, New York, NY, USA, 334. Razumovskiy, V.G., Ornatskiy, A.P., Mayevskiy, Y.M., 1990. Local heat transfer and hydraulic behavior in turbulent channel flow of water at supercritical pressure. Heat Transfer-Soviet Res. 22 (1), 91–102. Shitsman, M.E., 1963. Impairment of the heat transmission at supercritical pressures. High Temp. 1, 237–244. Shitsman, M.E., 1968. Temperature conditions in tubes at supercritical pressures. Therm. Eng. 15 (5), 72–77. Swenson, H.S., Carver, J.R., Kakarala, C.R., 1965. Heat transfer to supercritical water in smooth-bore tubes. J. Heat Transfer, Trans. ASME, Ser. C 87, 477–484. Song, J.H., Kim, H.Y., Kim, H., Bae, Y.Y., 2008. Heat transfer characteristics of a supercritical fluid flow in a vertical pipe. J. Supercrit. Fluids 44, 164–171. Styrikovich, M.A., Margulova, Kh.T., Miropol’skii, Z.L., 1967. Problems in the development of designs of supercritical boilers (in Russian). Therm. Eng. 14 (6), 5–9. Vikhrev, Yu.V., Barulin, Yu.D., Kon’kov, A.S., 1967. A study of heat transfer in vertical tubes at supercritical pressures. Therm. Eng. 14, 116–119. Watts, M.J., Chou, C.T., 1982. Mixed convection heat transfer to supercritical pressure water. In: Proceedings of the 7th IHTC. Munchen, Germany, pp. 495– 500. Xu, F., Guo, L.J., Mao, Y.F., Jiang, X.E., 2005. Experimental investigation to the heat transfer characteristics of water in vertical pipes under supercritical pressure. J. Xi’an Jiaotong Univ. 39, 468–471 (In Chinese). Yamagata, K., Nishikawa, K., Hasegawa, S., et al., 1972. Forced convective heat transfer to supercritical water flowing in tubes. Int. J. Heat Mass Transf. 15, 2575–2593. Yu, J., Jia, B., Wu, D., Wang, D., 2009a. Optimization of heat transfer coefficient correlation at supercritical pressure using genetic algorithms. Heat Mass Transf. 45, 757–766. Yu, J., Liu, H., Jia, B., 2009b. Sub-channel analysis of CANDU–SCWR and review of heat-transfer correlations. Prog. Nucl. Energy 51, 246–252. Zhu, X., Bi, Q., Yang, D., Chen, T., 2009. An investigation on heat transfer characteristics of different pressure steam-water in vertical upward tube. Nucl. Eng. Des. 239, 381–388.
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
An assessment of correlations of forced convection heat transfer to water at supercritical pressure Weiwei Chen, Xiande Fang ⇑, Yu Xu, Xianghui Su Institute of Air Conditioning and Refrigeration, Nanjing University of Aeronautics and Astronautics, 29 Yudao St, Nanjing 210016, China
a r t i c l e
i n f o
Article history: Received 25 February 2014 Received in revised form 18 June 2014 Accepted 22 October 2014
Keywords: Supercritical water Heat transfer Correlation Vertical tubes Nuclear reactor
a b s t r a c t The heat transfer of supercritical water is essential for supercritical water-cooled nuclear reactors. Many empirical correlations for heat transfer to supercritical water were proposed over the past few decades. Some evaluations of the correlations were conducted, and inconsistent conclusions appeared owing to limited correlations or experimental data. This work presents an extensive survey of the literature of correlations and experiments of forced convection heat transfer to water flowing upward in vertical tubes at supercritical pressure. There are 26 correlations found, and an experimental database containing 3220 data points from vertical tubes are compiled from nine independent laboratories. All available correlations are assessed against the experimental database. The results show that the best correlation has a mean absolute deviation of 12.8%, predicting 82.3% of the database within ±20%. The entire database is divided into three categories, and the correlations which can give the most accurate predictions of the experimental data from different categories are also identified. The results provide a guide to choosing a proper correlation for engineering practice. Some topics worthy of attention for future studies are indicated. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Supercritical water is of great interest for its applications in nuclear reactor cooling since it has unique properties and favorable heat and mass transfer characteristics. A supercritical water-cooled nuclear reactor (SCWR) is a high pressure (about 25 MPa) and high temperature (up to 625 °C) reactor that operates above the critical point of water (22.064 MPa and 373.95 °C). The SCWR offers the potential for high thermal efficiencies, considerable plant simplifications, and better safety and economy (Mokry et al., 2010a). Empirical correlations with good predictions of heat transfer for supercritical water are of considerable significance for developing a SCWR. Due to the strong variation of thermophysical properties in the vicinity of the critical and pseudo-critical point, water at supercritical pressure shows different heat transfer behaviors than at subcritical pressure, and conventional single-phase correlations cannot predict it (Song et al., 2008; Cheng et al., 2009). The investigations of heat transfer of supercritical water have been carried out since the 1930s. Detailed reviews on the existing experimental and theoretical studies were performed by several ⇑ Corresponding author. Tel./fax: +86 25 8489 6381. E-mail address: [email protected] (X. Fang). http://dx.doi.org/10.1016/j.anucene.2014.10.027 0306-4549/Ó 2014 Elsevier Ltd. All rights reserved.
authors (Petukhov, 1970; Jackson and Hall, 1979; Polyakov, 1991; Cheng and Schulenberg, 2001; Pioro et al., 2004; Pioro and Duffey, 2005; Pioro and Duffey, 2007). As the prediction of the heat transfer coefficient for supercritical water is mainly conducted using empirical approaches, a number of empirical correlations exist in the open literature, which were derived based on experimental data with limited parameter ranges (Bishop et al., 1964; Swenson et al., 1965; Krasnoshchekov et al., 1967; Yamagata et al., 1972; Griem, 1996; Mokry et al., 2010a). Subsequently, some evaluations were carried out to find out the best correlations. Cheng and Schulenberg (2001) conducted a thorough review on heat transfer of supercritical water at the HPLWR condition. The HPLWR means the High Performance Light Water Reactor, a joined research project in Europe. Five heat transfer correlations for supercritical water (Bishop et al., 1964; Swenson et al., 1965; Yamagata et al., 1972; Krasnoshchekov et al., 1967; Griem, 1996) were implemented into the sub-channel analysis code to determine their applicability to the HPLWR fuel assembly. The number of the experimental data points used for their analysis was not given. As a result, the Bishop et al. (1964) correlation was recommended for calculating the heat transfer coefficient in an HPLWR fuel assembly, and the Yamagata et al. (1972) correlation was suggested to be used for determining the onset of heat transfer deterioration.
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Nomenclature Bu cp cp D G Gr Gr⁄ Gr g h L Nu Pr Pr p q q+ Re
buoyancy parameter Gr= Re2:7 Pr 0:5 specific heat at constant pressure (J/kg K) average specific heat at constant pressure (J/kg K), (hw hb)/(tw tb) inner tube diameter (m) mass flux (kg/m2 s) Grashof number (gD3(qb qw)/qm2) 2 Grashof number based onheat flux gbD4 q=k m average Grashof number gD3 ðqb qÞ=qm2 acceleration due to gravity (m/s2) specific enthalpy (J/kg) tube length (m) Nusselt number (aD=k) Prandtl number (lcp =k) average Prandtl number (lcp =k) pressure (Pa) heat flux (W/m2) non-dimensional heat flux (qb=ðGcp Þ) Reynolds number (GD/l)
Jackson (2002) evaluated nine heat transfer correlations for water flowing in vertical tubes based on 1500 experimental data points. They modified the Krasnoshchekov and Protopopov (1959) correlation for forced convective heat transfer in water and carbon dioxide at supercritical pressures, capturing 97% of the experimental data within ±25%. The Results showed that the Krasnoshchekov et al. (1967) correlation and the newly modified correlation were the most accurate ones. Pioro et al. (2004) conducted the literature survey of the work in the area of heat transfer at supercritical pressures. Eight correlations were compared based on the Shitsman (1963) experimental data for supercritical heat transfer in tubes and bundles to choose the most reliable ones. The comparisons showed that there was a significant difference in heat transfer coefficient values calculated according to various correlations. Only some correlations showed similar results, which were quite close to the experimental data for normal supercritical heat transfer in water. Also, no one correlation was able to accurately predict deteriorated or improved heat transfer in tubes. Based on the eight chosen correlations, the heat transfer coefficients and temperature profiles in the CANDU-X reactor cooled with supercritical water were calculated. Licht et al. (2008) compared four selected heat transfer correlations with their own experimental results and found that the Jackson (2002) correlation predicted the test data best, capturing 86% of the data within ±25%. The Watts and Chou (1982) correlation showed a similar trend but under-predicted the measurements by 10% relative to the Jackson (2002) correlation. Yu et al. (2009a) verified 14 supercritical heat transfer correlations based on 1142 experimental data points, and Yu et al. (2009b) compared 16 supercritical heat transfer correlations with the Styrikovich et al. (1967) data. The results showed that the Bishop et al. (1964) correlation performed best. Zhu et al. (2009) compared five selected heat transfer correlations based on their own experimental results of the supercritical heat transfer of water and found that their own correlation and the Swenson et al. (1965) correlation were the best. Mokry et al. (2010a) verified five selected heat transfer correlations and found that all of them deviated substantially from the
T t
temperature (K) temperature (°C)
Greek symbols a heat transfer coefficient (W/m2 K) b thermal expansion coefficient (1/K) k thermal conductivity (W/mK) l dynamic viscosity (Pa s) m kinematic viscosity (m2/s) n friction coefficient q density (kg/m3) q average density (kg/m3) Subscripts b at bulk temperature exp experimental in inlet pc at pseudo-critical temperature pred predicted w at wall temperature
experimental data within the pseudo-critical range. Therefore, they proposed their own correlation and recommended it to be used for SCWRs and supercritical water heat exchangers. Jäger et al. (2011) summarized the activities of the TRACE code validation at the Institute for Neutron Physics and Reactor Technology (Germany) related to supercritical water conditions. The 15 existing heat transfer correlations were reviewed and implemented into TRACE, and six selected experimental data sources were used to identify the most suitable heat transfer correlation(s). The number of the experimental data used was not stated, and the overall performance of each correlation for predicting the entire database was not clear. As a result, they recommended the Bishop et al. (1964) model for design and safety evaluation of SCWRs. The above evaluations presented inconsistent results due to limited experimental data or correlations. The most comprehensive reviews might be those by Jäger et al. (2011) and Yu et al. (2009a). The former evaluated 15 existing supercritical heat transfer correlations with the experimental data from six selected sources, and the latter assessed 14 based on 1142 experimental data points. This paper conducts an all-around survey of the correlations and experimental results, and 26 existing correlations for water supercritical heat transfer in vertical tubes are assessed with the supercritical water heat transfer database containing 3220 data points compiled from nine independent laboratories. The number of the correlations evaluated and the data used is far more than previous ones. Furthermore, the available experimental data are partitioned into three different heat transfer regimes, including the normal heat transfer regime, the enhanced heat transfer regime, and the deteriorated heat transfer regime. The evaluation of the surveyed correlations is implemented for each regime. To the best of the authors’ knowledge, it is the first state-of-the-art review using a multiple-source database consisting of more than 1500 data points to evaluate more than 15 correlations for supercritical heat transfer to water, and it is the first practice to evaluate the correlations for each of the three heat transfer regimes. The evaluation results provide a guide to choosing a proper correlation for engineering practice. Some topics worthy of attention for future studies are indicated.
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(1990), Griem (1996), and Kitoh et al. (1999), as summarized in Table 1. Most of the correlations were proposed by considering the ratio of density, thermal conductivity, specific heat, or viscosity evaluated at the wall temperature to those at the bulk temperature, such as those of Krasnoshchekov and Protopopov (1959), Bishop et al. (1964), Swenson et al. (1965), Krasnoshchekov et al. (1967), Jackson (2002), Xu et al. (2005), and Mokry et al. (2010a), as summarized in Tables 1 and 2. The correlations of Bringer and Smith (1957), Krasnoshchekov and Protopopov (1959), and Krasnoshchekov et al. (1967) can be used
2. Review of existing correlations In the past few decades, a number of heat transfer correlations based on experimental results of supercritical water were proposed, among which most are the modified forms of the Dittus–Boelter (1930) equation or the Gnielinski (1976) equation. Some correlations were developed by combining the Reynolds number (Re) and the Prandtl number (Pr) with the physical properties at the bulk and wall temperatures, such as those of McAdams (1942), Bringer and Smith (1957), Shitsman (1963), Gorban et al.
Table 1 Correlations taking the form of Dittus–Boelter equation. Author
Correlation
Flow and operating parameters
McAdams (1942)
0:4 Nub ¼ 0:0243Re0:8 b Pr b
High pressures and low heat fluxes
Bringer and Smith (1957)
0:55 Nux ¼ 0:0266Rex0:77 Pr w Nux and Rex are evaluated at temperature tx: tx = tb for E < 0, tx = tpc for 0 5 E 5 1 tx = tw for E > 1, and E = (tpc tb)/(tw tb)
Apart from critical and pseudo-critical regions
Shitsman (1963)
0:8 Nub ¼ 0:023Re0:8 b Pr min Prmin is the smaller one of Prw and Prb
Applied to conditions with Pr 1 D = 7.8, 8.2 mm
Bishop et al. (1964)
Nub ¼ 0:0069Reb0:9 Pr b 0:66 ðqw =qb Þ0:43 ð1 þ 2:4D=xÞ x is the axial location along the heated length (m)
p = 22.8–27.6 MPa, tb = 282–527 °C, G = 651–3662 kg/m2 s, q = 310–3460 kW/m2 and x/D = 30–565
Swenson et al. (1965)
Pr w 0:613 ðqw =qb Þ0:231 Nuw ¼ 0:00459Re0:923 w
p = 22.8–41.4 MPa, tb = 75–576 °C, G = 542–2150 kg/m2 s, tw = 93–649 °C, Re = 7.5 104–3.16 106
Ornatsky et al. (1970)
0:3 0:8 Nub ¼ 0:023Re0:8 b Pr min ðqw =qb Þ
Yamagata et al. (1972)
Pr0:8 Nub ¼ 0:0135Re0:85 b b Fc Fc = 1, for E > 1, F c ¼ ðcp =cp;b Þn2 for E < 0,
p = 22.6–29.4 MPa, tb = 230–540 °C, G = 310–1830 kg/m2 s, q = 116–930 kW/m2 D = 7.5, 10 mm
F c ¼ 0:67Pr 0:05 ðcp =cp;b Þn1 for 0 5 E 5 1, and pc n1 = 0.77(1 + 1/Prpc) + 1.49, n2 = 1.44(1 + 1/Prpc) 0.53 Watts and Chou (1982)
P = 25.0 MPa, tb = 150–350 °C,
0:55 ðqw =qb Þ0:35 / Nub ¼ 0:021Re0:8 b Pr b
G = 130–1000 kg/m2 s, q = 170–450 kW/m2, Pr b = 0.85–2.30 Reb = 6.5 103–3 105
/ ¼ 1 for Bu 5105 ; / ¼ ð7000BuÞ0:295 for Bu =104 / ¼ ð1 3000BuÞ
0:295
for 10
5
4
< Bu < 10
Gorban’ et al. (1990)
Nub ¼ 0:0059Reb0:90 Pr0:12 b
Griem (1996)
Pr 0:432 x Num ¼ 0:0169Re0:8356 b sel Pr sel ¼ lb cp;sel =km , and km ¼ 0:5ðkw þ kb Þ x = 0.82 for hb < 1540 kJ/kg, x = 1 for hb > 1740 kJ/kg x = 0.82 + 9 104(hb 1540) for 1540 5 hb 5 1740 kJ/kg
p = 22.0–27.0 MPa, G = 300–2500 kg/m2 s q = 200–700 kW/m2 and D = 10, 14, 20 mm For detail of cp,sel, please see Griem (1996)
Kitoh et al. (1999)
Pr m Nub ¼ 0:015Re0:85 b b m = 0.69–81000/200G1.2 + fcq –8 fc = 2.9 10 + 0.11/200G1.2 for 0 6 hb 6 1500 kJ/kg fc = 8.7 10–8 0.65/200G1.2 for 1500 < hb < 3300 kJ/kg fc = 9.7 10–7 + 1.3/200G1.2 for 3300 5 hb 5 4000 kJ/kg
tb = 20–550 °C, G = 100–1750 kg/m2 s, and q = 0–1800 kW/m2
Jackson (2002)
0:3 Pr0:5 ðcp =cp;b Þn Nub ¼ 0:0183Re0:82 b b ðqw =qb Þ n = 0.4 for Tb < Tw < Tpc or 1.2Tpc < Tb < Tw n = 0.4 + 0.2(Tw/Tpc 1) for Tb < Tpc < Tw n = 0.4 + 0.2(Tw/Tpc 1)[1–5(Tb/Tpc 1)] for Tpc < Tb < 1.2Tpc and Tb < Tw
p = 23.4–29.3 MPa, G = 700–3600 kg/m2s q = 46–2600 kW/m2, Reb = 8 104–5 105 D = 1.6–20 mm
Xu et al. (2005)
Pr b 0:9213 ðqw =qb Þ0:6638 Nub ¼ 0:02269Re0:8079 b
Kuang et al. (2008)
Nub ¼ 0:0239Re0:759 Pr b 0:833 ðqw =qb Þ0:31 ðkw =kb Þ0:0863 b 0:014
ðlw =lb Þ0:832 ðGr b Þ
lw =lb
0:8687
p = 23.0–30.0 MPa, G = 600–1200 kg/m2 s q = 100–600 kW/m2, and D = 12.0 mm p = 22.8–31.0 MPa, G = 380–3600 kg/m2 s q = 233–3474 kW/m2
0:021
ðqþ Þ b
Nub ¼ F ¼ minðF 1 ; F 2 Þ
p = 22.5–25.0 MPa, G = 700–3500 kg/m2 s q = 300–2000 kW/m2, tb = 300–450 °C D = 10, 20 mm
Zhu et al. (2009)
Nub ¼ 0:0068Reb0:9 Pr b 0:63 ðqw =qb Þ0:17 ðkw =kb Þ029
Yu et al. (2009a)
Nub ¼ 0:01378Re0:9078 Pr 0:6171 ðqw =qb Þ0:4356 ðGr b Þ b b
Mokry et al. (2010a)
ðqw =qb Þ0:564 Nub ¼ 0:0061Reb0:904 Pr0:684 b
Gupta et al. (2010)
Nuw ¼ 0:004Re0:923 Pr w 0:773 ðqw =qb Þ0:186 ðlw =lb Þ0:366 w
Liu and Kuang (2012)
Pr b 0:73 ðqw =qb Þ0:401 ðkw =kb Þ0:24 Nub ¼ 0:01Re0:889 b
p = 22.0–30.0 MPa, G = 600–1200 kg/m2 s q = 200–600 kW/m2, and D = 26.0 mm p = 22.6–41.0 MPa, D = 1.5–38.1 mm G = 90–2441 kg/m2 s, q = 90–1800 kW/m2 p = 24.0 MPa, G = 200–1500 kg/m2 s q = 0–1250 kW/m2, and D = 10.0 mm P = 24.0 MPa, G = 200–1500 kg/m2 s q = 0–1250 kW/m2, and D = 10.0 mm P = 22.4–31.0 MPa, D = 6–38 mm G = 200–3500 kg/m2 s, q = 37–2000 kW/m2
Cheng et al. (2009)
1=3 0:023Re0:8 b Pr b F
2:4 F 1 ¼ 0:85 þ 0:776 1000qþ b qþ 0:48 b F2 ¼ 1:55 þ 1:21 1 qþ pc ð1000qþpc Þ
ðlw =lb Þ0:153 ðcp =cp;b Þ0:014 ðGr b Þ
0:007
0:041
ðqþ Þ b
0:012
0:0605
ðqþ Þ b
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W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460
Table 2 Correlations taking the form of Gnielinski equation. Author
Correlation
Krasnoshchekov and Protopopov (1959)
Flow and operating parameters 0:11
Nub ¼ Nu0 ðlw =lb Þ Nu0 ¼
0:33
ðkb =kw Þ
ðcp =cp;b Þ
p = 22.3–32 MPa, Reb = 2 104–8.6 105,
0:35
Pr b = 0:85 65; lb =lw = 0.9–3.6, kb =kw = 1–6 cp =cp;b = 0.07–4.5
ðn0 =8ÞReb Prb
pffiffiffiffiffiffiffiffi
1:07þ12:7
n0 =8ðPrb 2=3 1Þ 2
n0 ¼ ½1:82log10 ðReb Þ 1:64 Krasnoshchekov et al. (1967)
Nub ¼ Nu0 ðqw =qb Þ0:3 ðcp =cp;b Þn n = 0.4 for Tb < Tw < Tpc or 1.2Tpc < Tb < Tw n = n1 = 0.22 + 0.18Tw/Tpc for 1 < Tw/Tpc < 2.5n = n1 + (5n1 2)(1 Tb/Tpc) for Tpc < Tb < 1.2Tpc and Tb < Tw Nu0 is evaluated as above
Grass et al. (1971)
Nub ¼ Pr G ¼ Nub ¼
Prb 0 =8ÞRe pðnffiffiffiffiffiffiffi ffi b2=3 n0 =8ðPrG cp;b =cp;w 1Þ Pr b ; Pr b < 0:5Pr w Pr w ; Prb > 0:5Pr w
1:07þ12:7
Petukhov et al. (1983)
ðn=8ÞReb Prb
pffiffiffiffiffiffi
Nub ¼
n=8ðPrb 2=3 1Þ
1þ900=Reb þ12:7
n ¼ n0 ðqw =qb Þ Razumovskiy et al. (1990)
0:4
ðlw =lb Þ0:2
ðnr =8ÞReb Pr b
pffiffiffiffiffiffiffi 2=3 ðcp =cp;b Þ0:65 nr =8ðPrb 1Þ
1:07þ12:7
nr ¼ n0 ðqw =qb Þ0:18 ðlw =lb Þ0:18 Nub ¼ Nu0 ðqw =qb Þ0:4 ðcp =cp;b Þn uðk Þ
Kirillov et al. (1990)
Nu0 ¼
P = 23.4–29.3 MPa, G = 700–3600 kg/m2 s, q = 46–2600 kW/m2, Reb = 8 104–5 105, D = 1.6–20 mm
ðn0 =8ÞReb Prb
pffiffiffiffiffiffiffiffi 2=3 n0 =8ðPrb 1Þ
1þ900=Reb þ12:7
p = 22.3–29.3 MPa, Reb = 2 104–8 105, Pr b = 0.85–65, q = 23–2600 kW/m2, For details of uðk Þ please see Kirillov et al. (1990)
k ¼ ð1 qw =qb ÞGr b =Re2b , n = 0.7 for cp cp;b n is defined as following for cp < cp;b : n = 0.4 for Tb < Tw < Tpc or 1.2Tpc < Tb < Tw n = 0.22 + 0.18Tw/Tpc for Tb < Tpc < Tw n = 0.9 Tb/Tpc(1- Tw/Tpc) + 1.08 Tw/Tpc-0.68 for Tpc < Tb < 1.2Tpc and Tpc < Tw
Table 3 Experimental data sources for vertical upward tubes. Data source
Flow range: t (°C)/pin (MPa)/G (kg/m2 s)/q (kW/m2)
Geometry range: D (mm)/(mm)/material
Data points
Alekseev et al. (1976) Griem (1996) Mokry et al. (2010b) Pan et al. (2011) Shitsman (1968) Swenson et al. (1965) Vikhrev et al. (1967) Yamagata et al. (1972) Zhu et al. (2009)
100–350(tin)/24.5/380–820/100–900 343–421(tb)/22–27/300–2500/200–700 320–350(tin)/24/200–1500/0–884 330–550(tb)/22.5–30/1009–1626/216–822 100–250(tin)/24.5–34.3/350–600/270–700 75–576(tb)/23–41/542–2150/200–1800 50–425(tb)/26.5/500–1900/230–1250 230–540(tb)/22.6–29.4/310–1830/116–930 282–440(tb)/23–30/600–1200/200–600
10.4/750/Kh18N10T steel 14/unmentioned/unmentioned 10/4000/12Cr18Ni10Ti stainless steel 17/2000/1Gr18Ni9Ti stainless steel 8,16/800,1600,3200/1Gr18Ni9Ti steel 9.42/1830/AISI-304 stainless steel 20.4/6000/1Gr18Ni10Ti stainless steel 7.5,10/1500,2000/AISI-316 stainless steel 26/1000/1Gr18Ni9Ti stainless steel
163 259 1323 231 291 159 424 250 120
both for water and CO2, and the Gorban et al. (1990) correlation can be used both for water and Freon. However, some constants in the correlations for water are different from those for CO2 (or Freon). The constants of these four correlations shown in Tables 1 and 2 are for water. In particular, large differences in density between bulk and wall positions and along the flow direction can change the flow structure. The effects of the density variation were represented as the buoyancy and flow acceleration, and some authors presented correlations by adding non-dimensional parameter concerning buoyancy and flow acceleration to account for the effects, such as Watts and Chou (1982), Kuang et al. (2008), Cheng et al. (2009), Yu et al. (2009a), and Liu and Kuang (2012), as summarized in Table 1.
3. Experimental data Through the comprehensive survey of the experimental studies of the forced convective heat transfer to supercritical water flowing in vertical tubes, nine available experimental data sources are identified as listed in Table 3, and a database containing 3220
Fig. 1. Partition of total experimental data.
W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460 Table 4 Statistics of the top eight correlations against the entire database (3220 data points).
a b
Correlations
MAD
MRD
RMSD
SD
R20a
R30b
Mokry et al. (2010a) Petukhov et al. (1983) Swenson et al. (1965) Liu and Kuang (2012) Gupta et al. (2010) Watts and Chou (1982) Kuang et al. (2008) Zhu et al. (2009)
12.8 15.1 15.8 16.8 17.0 17.0 17.4 19.2
0.9 6.7 3.5 12.2 7.0 6.4 2.9 15.0
19.0 22.6 23.2 24.8 22.4 25.7 25.0 28.8
19.0 21.6 23.0 21.6 21.3 24.9 24.8 24.6
82.3 75.2 73.5 73.0 66.4 70.8 66.3 67.1
92.7 89.2 87.4 85.5 87.7 86.8 84.4 81.1
Percentage of the data points within ±20% error band. Percentage of the data points within ±30% error band.
experimental data points are established. The parameter ranges cover bulk enthalpy from 278.1 to 3169 kJ/kg, mass fluxes from 201 to 2500 kg/m2 s, heat fluxes from 129 to 1735 kW/m2, pressures from 22 to 34.3 MPa, and tube hydraulic diameters from 7.5 to 26 mm.
455
For forced convective heat transfer to water at supercritical pressures, there are three heat transfer regimes characterized as the normal heat transfer regime, the enhanced heat transfer regime and the deteriorated heat transfer regime (Mokry et al., 2010b). Many different criteria defining the three heat transfer regimes were presented in the literature (Cheng and Schulenberg, 2001), among which the one proposed by Koshizuka et al. (1995) as the following has been frequently mentioned:
R ¼ aexp =a0
ð1Þ
where a0 is the calculated heat transfer coefficient using the Dittus and Boelter (1930) correlation, and aexp is the experimental heat transfer coefficient. This criterion classifies the heat transfer of supercritical water flow as normal when 0.3 6 R 6 1, enhanced when R > 1, and deteriorated when R < 0.3. According to the criterion, all the 3220 experimental data points are divided into three regimes as showed in Fig. 1, with the normal heat transfer regime having 2145 points (66.6%), the enhanced heat transfer
Fig. 2. Comparison of the top three correlations with the entire experimental data.
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Table 5 Comparison of the top three correlations in different heat transfer regimes. Correlations
Normal data
Mokry et al. (2010a) Petukhov et al. (1983) Swenson et al. (1965)
Enhanced data
Deteriorated data
MAD
RMSD
MAD
RMSD
MAD
RMSD
10.6 12.0 14.0
14.8 16.4 20.4
14.7 13.8 19.6
18.4 16.4 25.2
24.0 43.8 18.4
39.2 55.7 35.2
Table 6 Statistics of the top twelve correlations against the normal database (2145 data points). Correlations
MAD
MRD
RMSD
SD
R20
R30
Mokry et al. (2010a) Petukhov et al. (1983) Watts and Chou (1982) Swenson et al. (1965) Liu and Kuang (2012) Griem (1996) Kuang et al. (2008) Gupta et al. (2010) Zhu et al. (2009) Yu et al. (2009a) Ornatsky et al. (1970) Bishop et al. (1964)
10.6 12.0 13.6 14.0 14.1 14.8 15.2 15.7 16.2 17.4 17.6 19.2
0.3 6.9 6.4 2.3 10.2 7.0 0.9 9.5 13.5 16.1 8.7 17.7
14.8 16.4 18.3 20.4 20.3 22.1 19.3 19.3 23.0 22.7 23.6 25.0
14.8 14.9 17.1 20.3 17.6 20.9 19.3 16.8 18.6 16.0 21.9 17.6
88.2 82.1 78.3 78.4 79.7 76.0 70.4 68.5 74.6 67.6 68.0 62.8
95.4 93.1 91.8 90.7 89.4 88.0 87.8 91.6 85.7 85.3 83.8 82.9
regime having 803 points (24.9%), and the deteriorated heat transfer regime having 272 points (8.5%). From Fig. 1 it can be seen that most of the enhanced and deteriorated heat transfer phenomena appeared near the pseudo-critical points. 4. Assessment of the correlations The 26 reviewed heat transfer correlations are assessed with the database of the entire, normal, enhanced and deteriorated heat transfer data points, respectively. For the standard statistical procedure, criterion MAD, the root mean square deviation (RMSD), the standard deviation (SD), and the mean relative deviation (MRD) are often used.
MAD ¼
N 1X jRDi j N i¼1
Fig. 3. Deviations of the predictions of the top three correlations from the normal heat transfer data.
ð2Þ
457
W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460 Table 7 Statistics of the top fourteen correlations against the enhanced database (803 data points).
Table 8 Statistics of the top four correlations against the deteriorated database (272 data points).
Correlations
MAD
MRD
RMSD
SD
R20
R30
Correlations
MAD
MRD
RMSD
SD
R20
R30
McAdams (1942) Jackson (2002) Shitsman (1963) Kitoh et al. (1999) Petukhov et al. (1983) Griem (1996) Mokry et al. (2010a) Watts and Chou (1982) Yu et al. (2009a) Kuang et al. (2008) Xu et al. (2005) Bishop et al. (1964) Ornatsky et al. (1970) Gupta et al. (2010)
10.3 11.5 11.5 12.3 13.8 14.5 14.7 15.3 15.5 16.7 19.0 19.0 19.8 20.0
8.9 5.5 0.5 1.8 6.6 12.2 3.5 8.1 5.2 3.2 11.3 11.2 18.9 6.7
14.2 14.2 14.9 15.5 16.4 17.5 18.4 18.4 19.8 20.5 24.2 25.0 22.3 24.5
11.1 13.1 14.9 15.4 15.0 12.6 18.0 16.5 19.1 20.2 21.4 22.4 11.9 23.6
82.8 83.8 82.8 80.9 75.8 73.3 72.4 69.5 69.6 64.1 61.5 62.9 52.6 56.8
94.4 96.9 95.4 95.3 94.3 91.7 89.9 89.9 88.2 84.8 74.3 79.8 81.6 77.7
Gupta et al. (2010) Swenson et al. (1965) Xu et al. (2005) Mokry et al. (2010a)
18.1 18.4 20.5 24.0
11.6 12.0 4.1 23.2
35.2 35.2 31.3 39.2
33.3 33.1 31.1 31.7
77.6 76.8 66.2 65.4
86.8 87.9 84.6 79.8
RMSD ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN RD2i i¼1 N
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN SD ¼ ðRDi MRDÞ2 i¼1 N1 MRD ¼
N 1X RDi N i¼1
ð3Þ
where
RDi ¼
apred ðiÞ aexp ðiÞ 100 aexp ðiÞ
ð6Þ
The MAD and the RMSD are used to gauge prediction accuracy, the SD is used to measure the dispersion degree of the prediction, and the MRD is used to check if a correlation has an over-prediction or under-prediction on average. 4.1. Evaluation of correlations with the entire data
ð4Þ
ð5Þ
For the entire data without partition, all the reviewed correlations are assessed and eight correlations have the MAD less than 20% and RMSD less than 30%, as shown in Table 4. The top three correlations are those of Mokry et al. (2010a), Petukhov et al. (1983) and Swenson et al. (1965). The Mokry et al. (2010a)
Fig. 4. Comparison of the top three correlations with the enhanced experimental data.
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W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460
Fig. 5. Comparison of the top three correlations with the deteriorated experimental data.
correlation performs best with an MAD of 12.8%, a RMSD of 19.0%, and an SD of 19.0%, respectively, predicting 82.3% of the entire data within ±20% and 92.7% of those within ±30%. The MRD values indicate that all the top three correlations give an over-prediction on average, with the Petukhov et al. (1983) correlation most noticeable. Fig. 2 shows the deviations of the predictions of the top three correlations from the entire data. Table 5 compared the top three correlations in different heat transfer regimes, from which it can be seen that the prediction performances of the top three correlations depend on the heat transfer regimes. They all perform best in the normal heat transfer regime and worst in the deteriorated heat transfer regime. Moreover, the Mokry et al. (2010a) correlation has the highest accuracy in the normal heat transfer regime, the Petukhov et al. (1983) correlation has the highest accuracy in the enhanced heat transfer regime, and the Swenson et al. (1965) correlation performs best in the deteriorated heat transfer regime. Therefore, it is necessary to ascertain top correlations in different heat transfer regimes.
and Chou (1982) are the most agreeable ones. The Mokry et al. (2010a) correlation performs best with the MAD, RMSD, and SD of 10.6%, 14.8%, and 14.8%, respectively, predicting 88.2% of the normal heat transfer data within ±20% and 95.4% of those within ±30%. The Petukhov et al. (1983) and Watts and Chou (1982) correlations give noticeable over-predictions on average. Fig. 3 shows the deviations of the predictions of the top three correlations from the experimental data in the normal heat transfer regime, from which it can be seen that the top three correlations predict well the experimental data in the normal heat transfer regime only if the data are away from the pseudo-critical points. Unacceptable deviations emerge in the vicinity of the pseudo-critical points due to the strong variation of thermophysical properties. Thus, the predictions of normal heat transfer near the pseudo-critical points need to be improved.
4.2. Evaluation of correlations with data from the normal heat transfer regime
Table 7 shows the evaluation results with the experimental data in the enhanced heat transfer regime, where fourteen correlations with the MAD less than 20% and RMSD less than 25% are displayed. The McAdams (1942) correlation is the best with the smallest MAD (10.3%), RMSD (14.2%), and SD (11.1%), predicting 82.8% of the experimental data in the enhanced heat transfer regime within ±20% and 94.4% of those within ±30%. The Jackson (2002) and
For the data in the normal heat transfer regime, there are twelve correlations having the MAD less than 20% and RMSD less than 25%, as presented in Table 6. The statistics indicate that the correlations of Mokry et al. (2010a), Petukhov et al. (1983) and Watts
4.3. Evaluation of correlations with data from the enhanced heat transfer regime
W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460
Shitsman (1963) are the next two best ones. Among the top three correlations, those of McAdams (1942) and Jackson (2002) give noticeable under-predictions on average. Fig. 4 illustrates the deviations of the predictions of the top three correlations from the experimental data in the enhanced heat transfer regime. As can be seen in Fig. 4, away from the pseudo-critical points, the top three correlations perform well. Similar to the situation in the normal heat transfer regime, large deviations appear near the pseudo-critical points owing to the intense variation of the thermophysical properties. Although the vast majority of the experimental data in the enhanced heat transfer regime congregate near the pseudo-critical points, fourteen correlations can provide the MADs less than 20%, and the performances of the top correlations are a little better than those in the normal heat transfer regime. 4.4. Evaluation of correlations with data from the deteriorated heat transfer regime For the deteriorated heat transfer regime, there are four correlations with the MAD less than 30% and RMSD less than 40%, as shown in Table 8. The accuracies of the Gupta et al. (2010) and Swenson et al. (1965) correlations are very close, having the MAD less than 18.5% and predicting about 80% of the experimental data in the deteriorated heat transfer regime within ±20%. The Xu et al. (2005) correlation has the smallest RMSD (31.3%) and SD (31.1%) and predicts 66.2% of the deteriorated data within ±20%. All of the four correlations over-predict the data on average. Fig. 5 shows the comparisons of the predictions of the top three correlations with the experimental data in the deteriorated heat transfer regime. It can be seen that almost all the deteriorated data points are near the pseudo-critical points, and that even the top correlations are not satisfactory. Tables 6–8 indicate that the prediction of the deteriorated heat transfer is the most challenging and that of the enhanced heat transfer is the easiest. Figs. 3–5 illustrate that the deviations in the vicinity of the pseudo-critical points are remarkably larger for all of the three heat transfer regimes, and thus it is essential to improve the prediction accuracy near the pseudo-critical points. 5. Conclusions An extensive survey of the investigations of the correlations and experiments of the forced convective heat transfer to water at supercritical pressure has been carried out. Twenty six correlations are evaluated with 3220 experimental data points from vertical tubes compiled from nine published articles. The experimental ranges cover bulk enthalpies from 278.1 to 3169 kJ/kg, mass fluxes from 201 to 2500 kg/m2 s, heat fluxes from 129 to 1735 kW/m2, pressures from 22 to 34.3 MPa, and tube hydraulic diameters from 7.5 to 26 mm. The entire database is partitioned into three heat transfer regimes as per the Koshizuka et al. (1995) method, with 66.6% data points (2145) in the normal heat transfer regime, 24.9% (803) in the enhanced heat transfer regime, and 8.5% (272) in the deteriorated heat transfer regime. The following conclusions can be drawn from this study: (1) For the entire database, the top three correlations are those of Mokry et al. (2010a), Petukhov et al. (1983), and Swenson et al. (1965). The Mokry et al. (2010a) correlation performs best, which has the smallest MAD (12.8%), RMSD (19.0%), and SD (19.0%) and predicts 82.3% of the entire data within ±20% and 92.7% of those within ±30%. The prediction performances of the correlations depend on the heat transfer regimes, and none of the reviewed correlations can performs
(2)
(3)
(4)
(5)
(6)
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best in all the three heat transfer regimes. Therefore, it is necessary to ascertain top correlations in different heat transfer regimes. For the normal heat transfer regime, the correlations of Mokry et al. (2010a), Petukhov et al. (1983), and Watts and Chou (1982) have the highest accuracy. The Mokry et al. (2010a) correlation has the smallest MAD (10.6%), RMSD (14.8%), and SD (14.8%), and predicts 88.2% of the experimental data in the normal heat transfer regime within ±20% and 95.4% within ±30%. For the enhanced heat transfer regime, the correlations of McAdams (1942), Jackson (2002), and Shitsman (1963) have the highest accuracy. The McAdams (1942) correlation has the smallest MAD (10.3%), RMSD (14.2%), and SD (11.1%), predicting 82.8% of the experimental data in the enhanced heat transfer regime within ±20% and 94.4% within ±30%. For the deteriorated heat transfer regime, the correlations of Gupta et al. (2010), Swenson et al. (1965), and Xu et al. (2005) are the most promising ones. The Gupta et al. (2010) correlation has the smallest MAD of 18.1%, predicting 77.6% of the experimental data in the deteriorated heat transfer regime within ±20%, and 86.8% within ±30%. The Xu et al. (2005) correlation has the smallest RMSD (31.3%) and SD (31.1%), predicting 66.2% of the deteriorated heat transfer data within ±20% and 84.6% within ±30%. The prediction of the deteriorated heat transfer is the most challenging and that of the enhanced heat transfer is the easiest. All of the mentioned top correlations have remarkably larger deviations in the vicinity of the pseudo-critical points than where away from the pseudo-critical points because the thermophysical properties of water vary drastically near the pseudo-critical point. A more accurate correlation needs to be developed. Efforts should be made to understand the mechanisms of the deteriorated heat transfer and the heat transfer near the pseudocritical point so that the prediction performances in these two aspects can be improved.
References Alekseev, G.V., Silin, V.A., Smirnov, A.M., Subbotin, V.I., 1976. Study of the thermal conditions on the wall of a pipe during the removal of heat by water at a supercritical pressure. High Temp. 14, 683–687. Bringer, R.P., Smith, J.M., 1957. Heat transfer in the critical region. AIChE J. 3, 49–55. Bishop, A.A., Sandberg, R.O. Tong, L.S., 1964. Forced convection heat transfer to water at near-critical temperatures and super-critical pressures. Report WCAP– 2056, Westinghouse Electric Corporation, Atomic Power Division, Pittsburgh, PA, USA, December, 85 pages. Cheng, X., Schulenberg, T., 2001. Heat transfer at supercritical pressures-literature review and application to a HPLWR. Forschungszentrum Karlsruhe, Technik und Umwelt, Wissenschaftliche Berichte, FZKA 6609, Institute für Kernund Energietechnik, Mai 2001. Cheng, X., Yang, Y.H., Huang, S.F., 2009. A simplified method for heat transfer prediction of supercritical fluids in circular tubes. Ann. Nucl. Energy 36, 1120– 1128. Dittus, FW., Boelter, L.M.K., 1930. Heat transfer in automobile radiators of the tubular type. University of California Publications in English, Berkeley, 2, 443– 461. Grass, G., Herkenrath, H., Hufschmidt, W., 1971. Anwendung des Prandtlschen Grenzschichtmodells auf den Waermeuebergang an Fluessigkeiten mitstark temperaturabhaengigen Stoffeigenschaften bei erzwungener Stroemung. Waerme-und Stoffuebertragung 4, 113–119 (In German). Gnielinski, V., 1976. New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chem. Eng. 16, 359–368. Gorban, L.M., Pomet’ko, R.S., Khryaschev, O.A., 1990. Modeling of water heat transfer with Freon of supercritical pressure (in Russian). Institute of Physics and Power Engineering, Obninsk, Russia. Griem, H., 1996. A new procedure for the prediction of forced convection heat transfer at near-and supercritical pressure. Heat Mass Transf. 31, 301–305. Gupta, S., Farah, A., King, K., Mokry, S., Pioro, I., 2010. Developing new heat-transfer correlation for supercritical-water flow in vertical bare tubes. In: Proceedings of the 18th International Conference on Nuclear Engineering (ICONE–18), Xi’an, China, May 17–21, Paper No. 30024.
460
W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460
Jackson, J.D., Hall, W.B., 1979. Forced convection heat transfer to fluids at supercritical pressure. Turbulent Forced Convection in Channels and Bundles 2, 563–611. Jackson, J.D., 2002. Consideration of the heat transfer properties of supercritical pressure water in connection with the cooling of advanced nuclear reactors. In: Proceedings of the 13th Pacific Basin Nuclear Conference, Shenzhen City, China, October 21–25. Jäger, W., Sánchez Espinoza, V.H., Hurtado, A., 2011. Review and proposal for heat transfer predictions at supercritical water conditions using existing correlations and experiments. Nucl. Eng. Des. 241, 2184–2203. Krasnoshchekov, E.A., Protopopov, V.S., 1959. Heat transfer at supercritical region in flow of carbon dioxide and water in tubes (in Russian). Therm. Eng. 12, 26–30. Krasnoshchekov, E.A., Protopopov, V.S., Van Fen, Kuraeva, I.V., 1967. Experimental investigation of heat transfer for carbon dioxide in the supercritical region. In: Gazley, Jr., C., Hartnett, J.P., Ecker, E.R.C. (Eds.), Proceedings of the Second All-Soviet Union Conference on Heat and Mass Transfer, Minsk, Belarus, May, Published as Rand Report R–451–PR,1, 26–35. Kirillov, P.L., Yur’ev, Yu, S., Bobkov, V.P., 1990. Handbook of Thermal-Hydraulics Calculations (in Russian). Energoatomizdat Publishing House, Moscow, Russia, 66-67,130–132. Kitoh, K., Koshizuka, S., Oka, Yo., 1999. Refinement of transient criteria and safety analysis for a high temperature reactor cooled by supercritical water. In: Proceedings of the Seventh International Conference on Nuclear Engineering (ICONE–7), Tokyo, Japan, April 19–23, Paper–7234. Koshizuka, S., Takano, N., Oka, Y., 1995. Numerical analysis of deterioration phenomena in heat transfer to supercritical water. Int. J. Heat Mass Transf. 38, 3077–3084. Kuang, B., Zhang, Y., Cheng, X., 2008. A new, wide-ranged heat transfer correlation of water at supercritical pressures in vertical upward ducts, NUTHOS–7, Seoul, Korea, October 5–9. Licht, J., Anderson, M., Corradini, M., 2008. Heat transfer to water at supercritical pressures in a circular and square annular flow geometry. Int. J. Heat Fluid Flow 29, 156–166. Liu, X., Kuang, B., 2012. Wide-ranged heat transfer correlations of supercritical water in vertical upward channels. Nucl. Sci. Eng. 32, 344–354 (in Chinese). McAdams, W.H., 1942. Heat transmission, 2nd ed. McGraw-Hill, New York, NY, USA, 459. Mokry, S., Farah, A., King, K., Gupta, S., Pioro, I., 2010a. Development of a heattransfer correlation for supercritical water flowing in a vertical bare tube. In: Proceedings of the 14th International Heat Transfer Conference (IHTC–14), Washington, DC, USA, August 8–13, Paper No. 22908. Mokry, S., Pioro, I., Kirillov, P., Gospodinov, Y., 2010. Supercritical-water heat transfer in a vertical bare tube. Nucl. Eng. Des. 240, 568–576. Ornatsky, A.P., Glushchenko, L.P., Siomin, E.T., 1970. The research of temperature conditions of small diameter parallel tubes cooled by water under supercritical pressures. In: Proceedings of the Fourth International Heat Transfer Conference, Paris-Versailles, France, vol. VI. Elsevier, Amsterdam, Paper B 8.11. Pan, J., Yang, D., Dong, Z.C., Zhu, T., Bi, Q.C., 2011. Experimental investigation on heat transfer characteristics of water in vertical upward tube under supercritical pressure. Nucl. Power Eng. 32, 75–79 (in Chinese).
Petukhov, B.S., 1970. Heat transfer and friction in turbulent pipe flow with variable physical properties. Adv. Heat Transfer 6, 503–564. Petukhov, B.S., Kurganov, V.A., Ankudinov, V.B., 1983. Heat transfer and flow resistance in the turbulent pipe flow of a fluid with near-critical state parameters. High Temp. Sci. 21, 81–89. Polyakov, A.F., 1991. Heat transfer under supercritical pressures. Adv. Heat Transfer 21, 1–53. Pioro, I.L., Khartabil, H.F., Duffey, R.B., 2004. Heat transfer to supercritical fluids flowing in channels–empirical correlations (survey). Nucl. Eng. Des. 230, 69–91. Pioro, I.L., Duffey, R.B., 2005. Experimental heat transfer in supercritical water flowing inside channels (survey). Nucl. Eng. Des. 235, 2407–2430. Pioro, I.L., Duffey, R.B., 2007. Heat transfer and hydraulic resistance at supercritical pressures in power engineering applications. ASME Press, New York, NY, USA, 334. Razumovskiy, V.G., Ornatskiy, A.P., Mayevskiy, Y.M., 1990. Local heat transfer and hydraulic behavior in turbulent channel flow of water at supercritical pressure. Heat Transfer-Soviet Res. 22 (1), 91–102. Shitsman, M.E., 1963. Impairment of the heat transmission at supercritical pressures. High Temp. 1, 237–244. Shitsman, M.E., 1968. Temperature conditions in tubes at supercritical pressures. Therm. Eng. 15 (5), 72–77. Swenson, H.S., Carver, J.R., Kakarala, C.R., 1965. Heat transfer to supercritical water in smooth-bore tubes. J. Heat Transfer, Trans. ASME, Ser. C 87, 477–484. Song, J.H., Kim, H.Y., Kim, H., Bae, Y.Y., 2008. Heat transfer characteristics of a supercritical fluid flow in a vertical pipe. J. Supercrit. Fluids 44, 164–171. Styrikovich, M.A., Margulova, Kh.T., Miropol’skii, Z.L., 1967. Problems in the development of designs of supercritical boilers (in Russian). Therm. Eng. 14 (6), 5–9. Vikhrev, Yu.V., Barulin, Yu.D., Kon’kov, A.S., 1967. A study of heat transfer in vertical tubes at supercritical pressures. Therm. Eng. 14, 116–119. Watts, M.J., Chou, C.T., 1982. Mixed convection heat transfer to supercritical pressure water. In: Proceedings of the 7th IHTC. Munchen, Germany, pp. 495– 500. Xu, F., Guo, L.J., Mao, Y.F., Jiang, X.E., 2005. Experimental investigation to the heat transfer characteristics of water in vertical pipes under supercritical pressure. J. Xi’an Jiaotong Univ. 39, 468–471 (In Chinese). Yamagata, K., Nishikawa, K., Hasegawa, S., et al., 1972. Forced convective heat transfer to supercritical water flowing in tubes. Int. J. Heat Mass Transf. 15, 2575–2593. Yu, J., Jia, B., Wu, D., Wang, D., 2009a. Optimization of heat transfer coefficient correlation at supercritical pressure using genetic algorithms. Heat Mass Transf. 45, 757–766. Yu, J., Liu, H., Jia, B., 2009b. Sub-channel analysis of CANDU–SCWR and review of heat-transfer correlations. Prog. Nucl. Energy 51, 246–252. Zhu, X., Bi, Q., Yang, D., Chen, T., 2009. An investigation on heat transfer characteristics of different pressure steam-water in vertical upward tube. Nucl. Eng. Des. 239, 381–388.