Annals of Nuclear Energy 76 (2015) 451–460

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An assessment of correlations of forced convection heat transfer to waterat supercritical pressure

http://dx.doi.org/10.1016/j.anucene.2014.10.0270306-4549/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel./fax: +86 25 8489 6381.E-mail address: xd_fang@yahoo.com (X. Fang).

Weiwei Chen, Xiande Fang ⇑, Yu Xu, Xianghui SuInstitute 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 2014Received in revised form 18 June 2014Accepted 22 October 2014

Keywords:Supercritical waterHeat transferCorrelationVertical tubesNuclear reactor

a b s t r a c t

The heat transfer of supercritical water is essential for supercritical water-cooled nuclear reactors. Manyempirical 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 tolimited correlations or experimental data. This work presents an extensive survey of the literature of cor-relations and experiments of forced convection heat transfer to water flowing upward in vertical tubes atsupercritical pressure. There are 26 correlations found, and an experimental database containing 3220data points from vertical tubes are compiled from nine independent laboratories. All available correla-tions are assessed against the experimental database. The results show that the best correlation has amean absolute deviation of 12.8%, predicting 82.3% of the database within ±20%. The entire database isdivided into three categories, and the correlations which can give the most accurate predictions of theexperimental data from different categories are also identified. The results provide a guide to choosinga proper correlation for engineering practice. Some topics worthy of attention for future studies areindicated.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Supercritical water is of great interest for its applications innuclear reactor cooling since it has unique properties and favorableheat and mass transfer characteristics. A supercritical water-coolednuclear reactor (SCWR) is a high pressure (about 25 MPa) and hightemperature (up to 625 �C) reactor that operates above the criticalpoint of water (22.064 MPa and 373.95 �C). The SCWR offers thepotential for high thermal efficiencies, considerable plant simplifi-cations, and better safety and economy (Mokry et al., 2010a).Empirical correlations with good predictions of heat transfer forsupercritical water are of considerable significance for developinga SCWR.

Due to the strong variation of thermophysical properties in thevicinity of the critical and pseudo-critical point, water at supercrit-ical pressure shows different heat transfer behaviors than atsubcritical pressure, and conventional single-phase correlationscannot predict it (Song et al., 2008; Cheng et al., 2009).

The investigations of heat transfer of supercritical water havebeen carried out since the 1930s. Detailed reviews on the existingexperimental and theoretical studies were performed by several

authors (Petukhov, 1970; Jackson and Hall, 1979; Polyakov,1991; Cheng and Schulenberg, 2001; Pioro et al., 2004; Pioro andDuffey, 2005; Pioro and Duffey, 2007). As the prediction of the heattransfer coefficient for supercritical water is mainly conductedusing empirical approaches, a number of empirical correlationsexist in the open literature, which were derived based on experi-mental data with limited parameter ranges (Bishop et al., 1964;Swenson et al., 1965; Krasnoshchekov et al., 1967; Yamagataet al., 1972; Griem, 1996; Mokry et al., 2010a). Subsequently, someevaluations were carried out to find out the best correlations.

Cheng and Schulenberg (2001) conducted a thorough review onheat transfer of supercritical water at the HPLWR condition. TheHPLWR means the High Performance Light Water Reactor, a joinedresearch project in Europe. Five heat transfer correlations forsupercritical 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 deter-mine their applicability to the HPLWR fuel assembly. The numberof the experimental data points used for their analysis was notgiven. As a result, the Bishop et al. (1964) correlation was recom-mended for calculating the heat transfer coefficient in an HPLWRfuel assembly, and the Yamagata et al. (1972) correlation wassuggested to be used for determining the onset of heat transferdeterioration.

Nomenclature

Bu buoyancy parameter Gr= Re2:7Pr0:5� �� �

cp specific heat at constant pressure (J/kg K)cp average specific heat at constant pressure (J/kg K),

(hw � hb)/(tw � tb)D inner tube diameter (m)G mass flux (kg/m2 s)Gr Grashof number (gD3(qb � qw)/qm2)

Gr⁄ Grashof number based on heat flux gbD4q=km2� �

Gr average Grashof number gD3 qb � qð Þ=qm2� �

g acceleration due to gravity (m/s2)h specific enthalpy (J/kg)L tube length (m)Nu Nusselt number (aD=k)Pr Prandtl number (lcp=k)Pr average Prandtl number (lcp=k)p pressure (Pa)q heat flux (W/m2)q+ non-dimensional heat flux (qb=ðGcpÞ)Re Reynolds number (GD/l)

T temperature (K)t temperature (�C)

Greek symbolsa 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 coefficientq density (kg/m3)q average density (kg/m3)

Subscriptsb at bulk temperatureexp experimentalin inletpc at pseudo-critical temperaturepred predictedw at wall temperature

452 W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460

Jackson (2002) evaluated nine heat transfer correlations forwater flowing in vertical tubes based on 1500 experimental datapoints. They modified the Krasnoshchekov and Protopopov(1959) correlation for forced convective heat transfer in waterand carbon dioxide at supercritical pressures, capturing 97% ofthe experimental data within ±25%. The Results showed that theKrasnoshchekov et al. (1967) correlation and the newly modifiedcorrelation were the most accurate ones.

Pioro et al. (2004) conducted the literature survey of the workin the area of heat transfer at supercritical pressures. Eight corre-lations were compared based on the Shitsman (1963) experimen-tal data for supercritical heat transfer in tubes and bundles tochoose the most reliable ones. The comparisons showed thatthere was a significant difference in heat transfer coefficientvalues calculated according to various correlations. Only somecorrelations showed similar results, which were quite close tothe experimental data for normal supercritical heat transfer inwater. Also, no one correlation was able to accurately predictdeteriorated or improved heat transfer in tubes. Based on theeight chosen correlations, the heat transfer coefficients andtemperature profiles in the CANDU-X reactor cooled with super-critical water were calculated.

Licht et al. (2008) compared four selected heat transfer correla-tions with their own experimental results and found that theJackson (2002) correlation predicted the test data best, capturing86% of the data within ±25%. The Watts and Chou (1982)correlation showed a similar trend but under-predicted themeasurements by 10% relative to the Jackson (2002) correlation.

Yu et al. (2009a) verified 14 supercritical heat transfer correla-tions based on 1142 experimental data points, and Yu et al.(2009b) compared 16 supercritical heat transfer correlations withthe Styrikovich et al. (1967) data. The results showed that theBishop et al. (1964) correlation performed best.

Zhu et al. (2009) compared five selected heat transfer correla-tions based on their own experimental results of the supercriticalheat transfer of water and found that their own correlation andthe Swenson et al. (1965) correlation were the best.

Mokry et al. (2010a) verified five selected heat transfer correla-tions and found that all of them deviated substantially from the

experimental data within the pseudo-critical range. Therefore,they proposed their own correlation and recommended it to beused for SCWRs and supercritical water heat exchangers.

Jäger et al. (2011) summarized the activities of the TRACE codevalidation at the Institute for Neutron Physics and ReactorTechnology (Germany) related to supercritical water conditions.The 15 existing heat transfer correlations were reviewed andimplemented into TRACE, and six selected experimental datasources were used to identify the most suitable heat transfercorrelation(s). The number of the experimental data used wasnot stated, and the overall performance of each correlation forpredicting the entire database was not clear. As a result, theyrecommended the Bishop et al. (1964) model for design and safetyevaluation of SCWRs.

The above evaluations presented inconsistent results due tolimited experimental data or correlations. The most comprehen-sive reviews might be those by Jäger et al. (2011) and Yu et al.(2009a). The former evaluated 15 existing supercritical heat trans-fer correlations with the experimental data from six selectedsources, and the latter assessed 14 based on 1142 experimentaldata points. This paper conducts an all-around survey of the corre-lations and experimental results, and 26 existing correlations forwater supercritical heat transfer in vertical tubes are assessed withthe supercritical water heat transfer database containing 3220 datapoints compiled from nine independent laboratories. The numberof the correlations evaluated and the data used is far more thanprevious ones. Furthermore, the available experimental data arepartitioned into three different heat transfer regimes, includingthe normal heat transfer regime, the enhanced heat transferregime, and the deteriorated heat transfer regime. The evaluationof the surveyed correlations is implemented for each regime. Tothe best of the authors’ knowledge, it is the first state-of-the-artreview using a multiple-source database consisting of more than1500 data points to evaluate more than 15 correlations for super-critical heat transfer to water, and it is the first practice to evaluatethe correlations for each of the three heat transfer regimes. Theevaluation results provide a guide to choosing a proper correlationfor engineering practice. Some topics worthy of attention for futurestudies are indicated.

W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460 453

2. Review of existing correlations

In the past few decades, a number of heat transfer correlationsbased on experimental results of supercritical water wereproposed, among which most are the modified forms of theDittus–Boelter (1930) equation or the Gnielinski (1976) equation.

Some correlations were developed by combining the Reynoldsnumber (Re) and the Prandtl number (Pr) with the physical proper-ties at the bulk and wall temperatures, such as those of McAdams(1942), Bringer and Smith (1957), Shitsman (1963), Gorban et al.

Table 1Correlations taking the form of Dittus–Boelter equation.

Author Correlation

McAdams (1942) Nub ¼ 0:0243Re0:8b Pr0:4

b

Bringer and Smith (1957) Nux ¼ 0:0266Re0:77x Pr0:55

w

Nux and Rex are evaluated at temperature tx:tx = tb for E < 0, tx = tpc for 0 5 E 5 1tx = tw for E > 1, and E = (tpc � tb)/(tw � tb)

Shitsman (1963) Nub ¼ 0:023Re0:8b Pr0:8

min

Prmin is the smaller one of Prw and Prb

Bishop et al. (1964) Nub ¼ 0:0069Re0:9b Prb

0:66 qw=qbð Þ0:43ð1þ 2:4D=xÞx is the axial location along the heated length (m)

Swenson et al. (1965) Nuw ¼ 0:00459Re0:923w Prw

0:613ðqw=qbÞ0:231

Ornatsky et al. (1970) Nub ¼ 0:023Re0:8b Pr0:8

minðqw=qbÞ0:3

Yamagata et al. (1972) Nub ¼ 0:0135Re0:85b Pr0:8

b Fc

Fc = 1, for E > 1, Fc ¼ ðcp=cp;bÞn2 for E < 0,

Fc ¼ 0:67Pr�0:05pc ðcp=cp;bÞn1 for 0 5 E 5 1, and

n1 = �0.77(1 + 1/Prpc) + 1.49, n2 = 1.44(1 + 1/Prpc) � 0.5Watts and Chou (1982) Nub ¼ 0:021Re0:8

b Prb0:55ðqw=qbÞ

0:35/

/ ¼ 1 for Bu 510�5;/ ¼ ð7000BuÞ0:295 for Bu =10�4

/ ¼ ð1� 3000BuÞ0:295 for 10�5 < Bu < 10�4

Gorban’ et al. (1990) Nub ¼ 0:0059Re0:90b Pr�0:12

b

Griem (1996) Num ¼ 0:0169Re0:8356b Pr0:432

sel xPrsel ¼ lbcp;sel=km , and km ¼ 0:5ðkw þ kbÞx = 0.82 for hb < 1540 kJ/kg,x = 1 for hb > 1740 kJ/kgx = 0.82 + 9 � 10�4(hb � 1540) for 1540 5 hb 5 1740 k

Kitoh et al. (1999) Nub ¼ 0:015Re0:85b Prm

b

m = 0.69–81000/200G1.2 + fcqfc = 2.9 � 10–8 + 0.11/200G1.2 for 0 6 hb 6 1500 kJ/kgfc = �8.7 � 10–8 � 0.65/200G1.2 for 1500 < hb < 3300 kJ/fc = �9.7 � 10–7 + 1.3/200G1.2 for 3300 5 hb 5 4000 kJ/k

Jackson (2002) Nub ¼ 0:0183Re0:82b Pr0:5

b ðqw=qbÞ0:3ðcp=cp;bÞn

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

Xu et al. (2005) Nub ¼ 0:02269Re0:8079b Prb

0:9213 qw=qbð Þ0:6638 lw=lb

� �0:86

Kuang et al. (2008) Nub ¼ 0:0239Re0:759b Prb

0:833 qw=qbð Þ0:31 kw=kbð Þ0:0863

�ðlw=lbÞ0:832ðGr�bÞ

0:014ðqþb Þ�0:021

Cheng et al. (2009) Nub ¼ 0:023Re0:8b Pr1=3

b FF ¼ minðF1; F2ÞF1 ¼ 0:85þ 0:776 1000qþb

� �2:4

F2 ¼ 0:481000qþpcð Þ1:55 þ 1:21 1� qþ

bqþpc

� �

Zhu et al. (2009) Nub ¼ 0:0068Re0:9b Prb

0:63 qw=qbð Þ0:17 kw=kbð Þ029

Yu et al. (2009a) Nub ¼ 0:01378Re0:9078b Pr0:6171

b qw=qbð Þ0:4356 � ðGr�bÞ�0:012

Mokry et al. (2010a) Nub ¼ 0:0061Re0:904b Pr0:684

b qw=qbð Þ0:564

Gupta et al. (2010) Nuw ¼ 0:004Re0:923w Prw

0:773ðqw=qbÞ0:186ðlw=lbÞ

0:366

Liu and Kuang (2012) Nub ¼ 0:01Re0:889b Prb

0:73ðqw=qbÞ0:401ðkw=kbÞ0:24

�ðlw=lbÞ0:153ðcp=cp;bÞ0:014ðGr�bÞ

0:007ðqþb Þ0:041

(1990), Griem (1996), and Kitoh et al. (1999), as summarized inTable 1. Most of the correlations were proposed by consideringthe ratio of density, thermal conductivity, specific heat, orviscosity evaluated at the wall temperature to those at the bulktemperature, 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 cor-relations of Bringer and Smith (1957), Krasnoshchekov andProtopopov (1959), and Krasnoshchekov et al. (1967) can be used

Flow and operating parameters

High pressures and low heat fluxes

Apart from critical and pseudo-critical regions

Applied to conditions with Pr � 1D = 7.8, 8.2 mm

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

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

3

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

P = 25.0 MPa, tb = 150–350 �C,G = 130–1000 kg/m2 s, q = 170–450 kW/m2, Prb = 0.85–2.30Reb = 6.5 � 103–3 � 105

J/kg

p = 22.0–27.0 MPa, G = 300–2500 kg/m2 sq = 200–700 kW/m2 and D = 10, 14, 20 mmFor detail of cp,sel, please see Griem (1996)

kgg

tb = 20–550 �C, G = 100–1750 kg/m2 s, andq = 0–1800 kW/m2

p = 23.4–29.3 MPa, G = 700–3600 kg/m2sq = 46–2600 kW/m2, Reb = 8 � 104–5 � 105

D = 1.6–20 mm

87 p = 23.0–30.0 MPa, G = 600–1200 kg/m2 sq = 100–600 kW/m2, andD = 12.0 mmp = 22.8–31.0 MPa, G = 380–3600 kg/m2 sq = 233–3474 kW/m2

p = 22.5–25.0 MPa, G = 700–3500 kg/m2 sq = 300–2000 kW/m2, tb = 300–450 �CD = 10, 20 mm

p = 22.0–30.0 MPa, G = 600–1200 kg/m2 sq = 200–600 kW/m2, and D = 26.0 mm

ðqþb Þ�0:0605 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 sq = 0–1250 kW/m2, andD = 10.0 mmP = 24.0 MPa, G = 200–1500 kg/m2 sq = 0–1250 kW/m2, and D = 10.0 mmP = 22.4–31.0 MPa, D = 6–38 mmG = 200–3500 kg/m2 s, q = 37–2000 kW/m2

Table 2Correlations taking the form of Gnielinski equation.

Author Correlation Flow and operating parameters

Krasnoshchekov and Protopopov (1959) Nub ¼ Nu0ðlw=lbÞ0:11ðkb=kwÞ�0:33ðcp=cp;bÞ0:35

Nu0 ¼ ðn0=8ÞReb Prb

1:07þ12:7ffiffiffiffiffiffiffiffin0=8p

ðPrb2=3�1Þ

n0 ¼ 1:82log10ðRebÞ � 1:64½ ��2

p = 22.3–32 MPa, Reb = 2 � 104–8.6 � 105,Prb = 0:85� 65;lb=lw = 0.9–3.6, kb=kw = 1–6cp=cp;b = 0.07–4.5

Krasnoshchekov et al. (1967) Nub ¼ Nu0ðqw=qbÞ0:3ðcp=cp;bÞn

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) forTpc < Tb < 1.2Tpc and Tb < Tw

Nu0 is evaluated as above

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

Grass et al. (1971) Nub ¼ ðn0=8ÞReb Prb

1:07þ12:7ffiffiffiffiffiffiffiffin0=8p

Pr2=3G cp;b=cp;w�1ð Þ

PrG ¼Prb; Prb < 0:5Prw

Prw; Prb > 0:5Prw

�

Petukhov et al. (1983) Nub ¼ ðn=8ÞReb Prb

1þ900=Rebþ12:7ffiffiffiffiffiffin=8p

ðPrb2=3�1Þ

n ¼ n0ðqw=qbÞ0:4ðlw=lbÞ

0:2

Razumovskiy et al. (1990) Nub ¼ ðnr=8ÞReb Prb

1:07þ12:7ffiffiffiffiffiffiffinr=8p

Prb2=3�1ð Þ ðcp=cp;bÞ0:65

nr ¼ n0ðqw=qbÞ0:18ðlw=lbÞ

0:18

Kirillov et al. (1990) Nub ¼ Nu0ðqw=qbÞ0:4ðcp=cp;bÞnuðk�Þ

Nu0 ¼ n0=8ð ÞReb Prb

1þ900=Rebþ12:7ffiffiffiffiffiffiffiffin0=8p

Prb2=3�1ð Þ

k� ¼ ð1� qw=qbÞGrb=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-0.68for Tpc < Tb < 1.2Tpc and Tpc < Tw

p = 22.3–29.3 MPa, Reb = 2 � 104–8 � 105,Prb = 0.85–65, q = 23–2600 kW/m2,For details of uðk�Þ please see Kirillov et al. (1990)

Table 3Experimental 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) 100–350(tin)/24.5/380–820/100–900 10.4/750/Kh18N10T steel 163Griem (1996) 343–421(tb)/22–27/300–2500/200–700 14/unmentioned/unmentioned 259Mokry et al. (2010b) 320–350(tin)/24/200–1500/0–884 10/4000/12Cr18Ni10Ti stainless steel 1323Pan et al. (2011) 330–550(tb)/22.5–30/1009–1626/216–822 17/2000/1Gr18Ni9Ti stainless steel 231Shitsman (1968) 100–250(tin)/24.5–34.3/350–600/270–700 8,16/800,1600,3200/1Gr18Ni9Ti steel 291Swenson et al. (1965) 75–576(tb)/23–41/542–2150/200–1800 9.42/1830/AISI-304 stainless steel 159Vikhrev et al. (1967) 50–425(tb)/26.5/500–1900/230–1250 20.4/6000/1Gr18Ni10Ti stainless steel 424Yamagata et al. (1972) 230–540(tb)/22.6–29.4/310–1830/116–930 7.5,10/1500,2000/AISI-316 stainless steel 250Zhu et al. (2009) 282–440(tb)/23–30/600–1200/200–600 26/1000/1Gr18Ni9Ti stainless steel 120

454 W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460

both for water and CO2, and the Gorban et al. (1990) correlationcan be used both for water and Freon. However, some constantsin the correlations for water are different from those for CO2

(or Freon). The constants of these four correlations shown inTables 1 and 2 are for water.

In particular, large differences in density between bulk and wallpositions and along the flow direction can change the flow struc-ture. The effects of the density variation were represented as thebuoyancy and flow acceleration, and some authors presented cor-relations by adding non-dimensional parameter concerning buoy-ancy and flow acceleration to account for the effects, such as Wattsand Chou (1982), Kuang et al. (2008), Cheng et al. (2009), Yu et al.(2009a), and Liu and Kuang (2012), as summarized in Table 1.

Fig. 1. Partition of total experimental data.

3. Experimental data

Through the comprehensive survey of the experimental studiesof the forced convective heat transfer to supercritical water flowingin vertical tubes, nine available experimental data sources areidentified as listed in Table 3, and a database containing 3220

Table 4Statistics of the top eight correlations against the entire database (3220 data points).

Correlations MAD MRD RMSD SD R20a R30

b

Mokry et al. (2010a) 12.8 0.9 19.0 19.0 82.3 92.7Petukhov et al. (1983) 15.1 6.7 22.6 21.6 75.2 89.2Swenson et al. (1965) 15.8 3.5 23.2 23.0 73.5 87.4Liu and Kuang (2012) 16.8 12.2 24.8 21.6 73.0 85.5Gupta et al. (2010) 17.0 �7.0 22.4 21.3 66.4 87.7Watts and Chou (1982) 17.0 6.4 25.7 24.9 70.8 86.8Kuang et al. (2008) 17.4 2.9 25.0 24.8 66.3 84.4Zhu et al. (2009) 19.2 15.0 28.8 24.6 67.1 81.1

a Percentage of the data points within ±20% error band.b Percentage of the data points within ±30% error band.

W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460 455

experimental data points are established. The parameter rangescover bulk enthalpy from 278.1 to 3169 kJ/kg, mass fluxes from201 to 2500 kg/m2 s, heat fluxes from 129 to 1735 kW/m2,pressures from 22 to 34.3 MPa, and tube hydraulic diameters from7.5 to 26 mm.

Fig. 2. Comparison of the top three correlat

For forced convective heat transfer to water at supercriticalpressures, there are three heat transfer regimes characterized asthe normal heat transfer regime, the enhanced heat transfer regimeand the deteriorated heat transfer regime (Mokry et al., 2010b).Many different criteria defining the three heat transfer regimeswere presented in the literature (Cheng and Schulenberg, 2001),among which the one proposed by Koshizuka et al. (1995) as thefollowing has been frequently mentioned:

R ¼ aexp=a0 ð1Þ

where a0 is the calculated heat transfer coefficient using the Dittusand Boelter (1930) correlation, and aexp is the experimental heattransfer coefficient. This criterion classifies the heat transfer ofsupercritical water flow as normal when 0.3 6 R 6 1, enhancedwhen R > 1, and deteriorated when R < 0.3. According to thecriterion, all the 3220 experimental data points are divided intothree regimes as showed in Fig. 1, with the normal heat transferregime having 2145 points (66.6%), the enhanced heat transfer

ions with the entire experimental data.

Table 5Comparison of the top three correlations in different heat transfer regimes.

Correlations Normal data Enhanced data Deteriorated data

MAD RMSD MAD RMSD MAD RMSD

Mokry et al. (2010a) 10.6 14.8 14.7 18.4 24.0 39.2Petukhov et al. (1983) 12.0 16.4 13.8 16.4 43.8 55.7Swenson et al. (1965) 14.0 20.4 19.6 25.2 18.4 35.2

Table 6Statistics of the top twelve correlations against the normal database (2145 datapoints).

Correlations MAD MRD RMSD SD R20 R30

Mokry et al. (2010a) 10.6 �0.3 14.8 14.8 88.2 95.4Petukhov et al. (1983) 12.0 6.9 16.4 14.9 82.1 93.1Watts and Chou (1982) 13.6 6.4 18.3 17.1 78.3 91.8Swenson et al. (1965) 14.0 2.3 20.4 20.3 78.4 90.7Liu and Kuang (2012) 14.1 10.2 20.3 17.6 79.7 89.4Griem (1996) 14.8 7.0 22.1 20.9 76.0 88.0Kuang et al. (2008) 15.2 0.9 19.3 19.3 70.4 87.8Gupta et al. (2010) 15.7 �9.5 19.3 16.8 68.5 91.6Zhu et al. (2009) 16.2 13.5 23.0 18.6 74.6 85.7Yu et al. (2009a) 17.4 16.1 22.7 16.0 67.6 85.3Ornatsky et al. (1970) 17.6 8.7 23.6 21.9 68.0 83.8Bishop et al. (1964) 19.2 17.7 25.0 17.6 62.8 82.9

Fig. 3. Deviations of the predictions of the top three

456 W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460

regime having 803 points (24.9%), and the deteriorated heat trans-fer regime having 272 points (8.5%). From Fig. 1 it can be seen thatmost of the enhanced and deteriorated heat transfer phenomenaappeared near the pseudo-critical points.

4. Assessment of the correlations

The 26 reviewed heat transfer correlations are assessed withthe database of the entire, normal, enhanced and deteriorated heattransfer data points, respectively. For the standard statistical pro-cedure, criterion MAD, the root mean square deviation (RMSD),the standard deviation (SD), and the mean relative deviation(MRD) are often used.

MAD ¼ 1N

XN

i¼1

jRDij ð2Þ

correlations from the normal heat transfer data.

Table 7Statistics of the top fourteen correlations against the enhanced database (803 datapoints).

Correlations MAD MRD RMSD SD R20 R30

McAdams (1942) 10.3 �8.9 14.2 11.1 82.8 94.4Jackson (2002) 11.5 �5.5 14.2 13.1 83.8 96.9Shitsman (1963) 11.5 �0.5 14.9 14.9 82.8 95.4Kitoh et al. (1999) 12.3 �1.8 15.5 15.4 80.9 95.3Petukhov et al. (1983) 13.8 �6.6 16.4 15.0 75.8 94.3Griem (1996) 14.5 �12.2 17.5 12.6 73.3 91.7Mokry et al. (2010a) 14.7 �3.5 18.4 18.0 72.4 89.9Watts and Chou (1982) 15.3 �8.1 18.4 16.5 69.5 89.9Yu et al. (2009a) 15.5 5.2 19.8 19.1 69.6 88.2Kuang et al. (2008) 16.7 �3.2 20.5 20.2 64.1 84.8Xu et al. (2005) 19.0 �11.3 24.2 21.4 61.5 74.3Bishop et al. (1964) 19.0 11.2 25.0 22.4 62.9 79.8Ornatsky et al. (1970) 19.8 �18.9 22.3 11.9 52.6 81.6Gupta et al. (2010) 20.0 �6.7 24.5 23.6 56.8 77.7

Table 8Statistics of the top four correlations against the deteriorated database (272 datapoints).

Correlations MAD MRD RMSD SD R20 R30

Gupta et al. (2010) 18.1 11.6 35.2 33.3 77.6 86.8Swenson et al. (1965) 18.4 12.0 35.2 33.1 76.8 87.9Xu et al. (2005) 20.5 4.1 31.3 31.1 66.2 84.6Mokry et al. (2010a) 24.0 23.2 39.2 31.7 65.4 79.8

W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460 457

RMSD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

XN

i¼1RD2

i

rð3Þ

SD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1N � 1

XN

i¼1RDi �MRDð Þ2

rð4Þ

MRD ¼ 1N

XN

i¼1

RDi ð5Þ

Fig. 4. Comparison of the top three correlatio

where

RDi ¼apredðiÞ � aexpðiÞ

aexpðiÞ� 100 ð6Þ

The MAD and the RMSD are used to gauge prediction accuracy, theSD is used to measure the dispersion degree of the prediction, andthe MRD is used to check if a correlation has an over-predictionor under-prediction on average.

4.1. Evaluation of correlations with the entire data

For the entire data without partition, all the reviewed correla-tions are assessed and eight correlations have the MAD less than20% and RMSD less than 30%, as shown in Table 4. The top threecorrelations are those of Mokry et al. (2010a), Petukhov et al.(1983) and Swenson et al. (1965). The Mokry et al. (2010a)

ns with the enhanced experimental data.

Fig. 5. Comparison of the top three correlations with the deteriorated experimental data.

458 W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460

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 datawithin ±20% and 92.7% of those within ±30%. The MRD valuesindicate that all the top three correlations give an over-predictionon average, with the Petukhov et al. (1983) correlation mostnoticeable. Fig. 2 shows the deviations of the predictions of thetop three correlations from the entire data. Table 5 compared thetop three correlations in different heat transfer regimes, fromwhich it can be seen that the prediction performances of the topthree correlations depend on the heat transfer regimes. They allperform best in the normal heat transfer regime and worst in thedeteriorated heat transfer regime. Moreover, the Mokry et al.(2010a) correlation has the highest accuracy in the normal heattransfer regime, the Petukhov et al. (1983) correlation has thehighest accuracy in the enhanced heat transfer regime, and theSwenson et al. (1965) correlation performs best in the deterioratedheat transfer regime. Therefore, it is necessary to ascertain topcorrelations in different heat transfer regimes.

4.2. Evaluation of correlations with data from the normal heat transferregime

For the data in the normal heat transfer regime, there are twelvecorrelations having the MAD less than 20% and RMSD less than25%, as presented in Table 6. The statistics indicate that the corre-lations of Mokry et al. (2010a), Petukhov et al. (1983) and Watts

and Chou (1982) are the most agreeable ones. The Mokry et al.(2010a) correlation performs best with the MAD, RMSD, and SDof 10.6%, 14.8%, and 14.8%, respectively, predicting 88.2% of thenormal heat transfer data within ±20% and 95.4% of those within±30%. The Petukhov et al. (1983) and Watts and Chou (1982) cor-relations give noticeable over-predictions on average. Fig. 3 showsthe deviations of the predictions of the top three correlations fromthe experimental data in the normal heat transfer regime, fromwhich it can be seen that the top three correlations predict wellthe experimental data in the normal heat transfer regime only ifthe data are away from the pseudo-critical points. Unacceptabledeviations emerge in the vicinity of the pseudo-critical pointsdue to the strong variation of thermophysical properties. Thus,the predictions of normal heat transfer near the pseudo-criticalpoints need to be improved.

4.3. Evaluation of correlations with data from the enhanced heattransfer regime

Table 7 shows the evaluation results with the experimental datain the enhanced heat transfer regime, where fourteen correlationswith 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 theexperimental data in the enhanced heat transfer regime within±20% and 94.4% of those within ±30%. The Jackson (2002) and

W. Chen et al. / Annals of Nuclear Energy 76 (2015) 451–460 459

Shitsman (1963) are the next two best ones. Among the top threecorrelations, those of McAdams (1942) and Jackson (2002) givenoticeable under-predictions on average. Fig. 4 illustrates thedeviations of the predictions of the top three correlations fromthe experimental data in the enhanced heat transfer regime. Ascan be seen in Fig. 4, away from the pseudo-critical points, thetop three correlations perform well. Similar to the situation inthe normal heat transfer regime, large deviations appear nearthe pseudo-critical points owing to the intense variation of thethermophysical properties. Although the vast majority of theexperimental data in the enhanced heat transfer regime congre-gate near the pseudo-critical points, fourteen correlations canprovide the MADs less than 20%, and the performances of thetop correlations are a little better than those in the normal heattransfer regime.

4.4. Evaluation of correlations with data from the deteriorated heattransfer regime

For the deteriorated heat transfer regime, there are four corre-lations with the MAD less than 30% and RMSD less than 40%, asshown in Table 8. The accuracies of the Gupta et al. (2010) andSwenson et al. (1965) correlations are very close, having theMAD less than 18.5% and predicting about 80% of the experimentaldata in the deteriorated heat transfer regime within ±20%. The Xuet 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 threecorrelations with the experimental data in the deteriorated heattransfer regime. It can be seen that almost all the deteriorated datapoints are near the pseudo-critical points, and that even the topcorrelations are not satisfactory.

Tables 6–8 indicate that the prediction of the deteriorated heattransfer is the most challenging and that of the enhanced heattransfer is the easiest.

Figs. 3–5 illustrate that the deviations in the vicinity of thepseudo-critical points are remarkably larger for all of the threeheat transfer regimes, and thus it is essential to improve theprediction accuracy near the pseudo-critical points.

5. Conclusions

An extensive survey of the investigations of the correlations andexperiments of the forced convective heat transfer to water atsupercritical pressure has been carried out. Twenty six correlationsare evaluated with 3220 experimental data points from verticaltubes compiled from nine published articles. The experimentalranges cover bulk enthalpies from 278.1 to 3169 kJ/kg, mass fluxesfrom 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 from7.5 to 26 mm. The entire database is partitioned into three heattransfer regimes as per the Koshizuka et al. (1995) method, with66.6% 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 thoseof Mokry et al. (2010a), Petukhov et al. (1983), and Swensonet al. (1965). The Mokry et al. (2010a) correlation performsbest, 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 perfor-mances of the correlations depend on the heat transferregimes, and none of the reviewed correlations can performs

best in all the three heat transfer regimes. Therefore, it isnecessary to ascertain top correlations in different heattransfer regimes.

(2) For the normal heat transfer regime, the correlations ofMokry et al. (2010a), Petukhov et al. (1983), and Watts andChou (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 experi-mental data in the normal heat transfer regime within±20% and 95.4% within ±30%.

(3) For the enhanced heat transfer regime, the correlations ofMcAdams (1942), Jackson (2002), and Shitsman (1963) havethe highest accuracy. The McAdams (1942) correlation hasthe smallest MAD (10.3%), RMSD (14.2%), and SD (11.1%),predicting 82.8% of the experimental data in the enhancedheat transfer regime within ±20% and 94.4% within ±30%.

(4) For the deteriorated heat transfer regime, the correlations ofGupta 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%, predicting77.6% of the experimental data in the deteriorated heattransfer regime within ±20%, and 86.8% within ±30%. TheXu et al. (2005) correlation has the smallest RMSD (31.3%)and SD (31.1%), predicting 66.2% of the deteriorated heattransfer data within ±20% and 84.6% within ±30%.

(5) The prediction of the deteriorated heat transfer is the mostchallenging and that of the enhanced heat transfer is theeasiest. All of the mentioned top correlations have remark-ably larger deviations in the vicinity of the pseudo-criticalpoints than where away from the pseudo-critical pointsbecause the thermophysical properties of water vary drasti-cally near the pseudo-critical point.

(6) A more accurate correlation needs to be developed. Effortsshould be made to understand the mechanisms of the dete-riorated heat transfer and the heat transfer near the pseudo-critical point so that the prediction performances in thesetwo aspects can be improved.

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