review article review of the wall temperature prediction...

Hindawi Publishing CorporationJournal of EnergyVolume 2013, Article ID 159098, 13 pageshttp://dx.doi.org/10.1155/2013/159098

Review ArticleReview of the Wall Temperature PredictionCapability of Available Correlations for HeatTransfer at Supercritical Conditions of Water

Dhanuskodi Ramasamy, Arunagiri Appusamy, and Anantharaman Narayanan

Department of Chemical Engineering, National Institute of Technology, Tiruchirappalli, Tamil Nadu 620015, India

Correspondence should be addressed to Dhanuskodi Ramasamy; [email protected]

Received 17 January 2013; Revised 4 August 2013; Accepted 1 September 2013

Academic Editor: S. A. Kalogirou

Copyright © 2013 Dhanuskodi Ramasamy et al.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The validity of the wall temperature predictions by 18 correlations available in the literature for supercritical heat-transfer regimesof water was verified for 12 experimental datasets consisting of 355 data points available in the literature. The correlations wereranked based on criteria like % data with

2 Journal of Energy

the respective theoretical curves which correspond with theusual formula given as follows for convective heat transfer:

Nu = 0.023Re0.8Pr0.4. (1)

Many researchers have conducted experiments and publishedempirical correlations applicable for predicting the heattransfer at supercritical conditions of water. Each correlationis applicable for the specified range of operating parameters.Theobjective of the present study is to compare the predictioncapability of various correlations for the experimental dataavailable in the literature.

2. Thermophysical Properties of Fluids NearPseudocritical Conditions

Heat transfer in convective heat transfer of fluids is influencedby their thermophysical properties for the given conditions.The drastic variations of thermophysical properties nearpseudocritical temperatures make the prediction of heattransfer a difficult task. The thermophysical properties at anypressure and temperature or enthalpy condition for manyfluids are well established and are provided by NIST/ASMEsteam property tables based on IAPWS 1997.This is especiallyimportant for the creation of generalized correlations innondimensional form, which allows the experimental datafor several working fluids to be combined into one set. Themost significant thermophysical property variations occurnear the critical and pseudocritical points [2, 7].

The thermophysical properties of water at different pres-sure and temperature, including the supercritical region, canbe calculated using the NIST software (1996, 1997). Also,the latest NIST software (2002) calculates the thermophysicalproperties of different gases and refrigerants for wide rangesof pressure and temperature [7].

Figure 1 displays the trend of thermal conductivity, spe-cific heat, density, and dynamic viscosity with increase intemperature for water at 230 bar taking the peak property inthe temperature range as unity and the same at other temper-atures as a ratio of the property at the given temperature tothe peak value. It is seen that each property has a peak valueat one particular temperature. This particular temperatureabove critical temperature at which the specific heat peaksis termed as pseudocritical temperature. For the selected230 bar, the pseudocritical temperature is 377.63∘C. Thepseudocritical temperature keeps increasing with increase inpressure. With increase in pressure, the peaks in propertyvariation come down. Near critical and pseudocritical tem-peratures, density and viscosity drop significantly, enthalpyand kinematic viscosity increases sharply, and specific heat,thermal conductivity, and Prandtl’s number sharply peak andfall. Similar trends are observed in all other fluids.

3. Predictive Methods for Heat Transfer atSupercritical Pressures

Development of analytical models for predicting heat trans-fers in turbulent flow and at supercritical conditions wasnot successful due to the complex nature of flow and

the abrupt changes in fluid properties. Cheng et al. [8]have plotted the ratio of actual heat-transfer coefficient toheat-transfer coefficient calculated as per (1) with respectto fluid temperature. It is observed that the ratio peaksnear pseudocritical temperature at low-heat fluxes, and thesame is dipping near pseudocritical temperature at high-heatfluxes. If the Dittus and Boelter correlation is valid, the ratiowould be constant at 1 throughout the temperature range.The peaks and dips indicate heat-transfer enhancement anddeterioration and the heat-transfer behavior at supercriticalcondition is different. The wall temperature increase dueto the deterioration in heat-transfer coefficient is smoothcompared to abrupt increase that is taking place due toboiling crisis in subcritical pressures, and, hence, there is nounique definition for the onset of heat-transfer deteriorationso far. For the prediction of heat transfer under such complexconditions, conducting experiments and developing empiri-cal correlations of nondimensional numbers are followed.

Most of the empirical correlations have the general formof a modified Dittus and Boelter equation [8] as given in thefollowing:

Nu𝑥= 𝐶Re𝑚

𝑥

Pr𝑛𝑥

𝐹. (2)

The correction factor 𝐹 takes into account the effect ofproperty variation and the entrance effect; that is, 𝐹 is afunction of {(𝜌

𝑤/𝜌𝑏), ((Cp)av/Cp), (𝐿/𝐷)}.

A more detailed general form of correlations for calculat-ing heat transfer at super-critical pressures in water and otherfluids is given as follows [7]:

Nu𝑡,𝑥= 𝐶1Re𝑚1𝑡,𝑥

Pr𝑚2𝑡,𝑥

(

𝜌𝑡

𝜌𝑡

)

𝑚3

𝑥

(

𝜇𝑡

𝜇𝑡

)

𝑚4

𝑥

(

𝑘𝑡

𝑘𝑡

)

𝑚5

𝑥

× (

(Cp)av,𝑡Cp𝑡

)

𝑚6

𝑥

(1 + 𝐶2(

𝐷hy

𝐿ℎ

))

𝑚7

𝑥

.

(3)

The subscript “𝑡” in (3) refers to either wall temperature orbulk fluid temperatures or average of wall and bulk fluidtemperature or their combinations. Around 18 such equationsproposed by various authors as given in [1, 3–5, 7, 8, 11, 12, 15,19, 20] are listed as (B.1) to (B.18) in Table 1.

4. Comparison of the Wall TemperaturePredictions of Different Correlations

In order to evaluate the prediction capability of the variouscorrelations listed in Table 1, the wall temperature dataalong with the operating conditions given in the literature[3, 7, 10, 12, 14, 15, 18] were used. Twelve experimentalwall temperature datasets available in these literatures wereconsidered. The test parameters and tube inner diameterapplicable for each graph were collected. An excel macropro-gram was developed for computing wall temperature as perdifferent correlations. X steam excel version of IAPWS 1997was used for steam properties. Wall temperatures for theseexperimental conditions containing 355 data points werecalculated using 18 correlations. Absolute and percentagetemperature deviations of prediction by each correlation for

Journal of Energy 3

Table 1: List of correlations used for comparing supercritical water heat transfer in vertical pipe.

Equationno.

Name ofcorrelation Correlation Conditions

Equation(B.1)

Dittus andBoelter (1930)

Nu𝑏

= 0.023Re0.8𝑏

Pr0.4𝑏

[8, 13, 19].

Valid for fully developed turbulent flow in smoothtubes for fluid with Prandtl numbers ranging fromabout 0.6 to 100 and with moderate temperaturedifference between wall and fluid conditions.

Equation(B.2)

Mc Adams(1942)

Nu𝑏

= 0.0243Re0.8𝑏

Pr0.4𝑏

[5, 7, 13, 15].Modified version of (B.1) used for supercriticalcondition.

Equation(B.3)

Bringer andSmith (1957)

Nu𝑥

= 0.0266Re0.77𝑥

Pr0.55𝑤

[5, 7].

Nu𝑥

and Re𝑥

are evaluated at 𝑡𝑥

. Temperature 𝑡𝑥

isdefined as 𝑡

𝑏

if (𝑡pc − 𝑡𝑏)/(𝑡𝑤 − 𝑡𝑏) < 0, as 𝑡pc if 0 ≤(𝑡pc − 𝑡𝑏)/(𝑡𝑤 − 𝑡𝑏) ≤ 1, and as 𝑡𝑤 if (𝑡pc − 𝑡𝑏)/(𝑡𝑤 − 𝑡𝑏) > 1for supercritical water up to 𝑝 = 34.5MPa.

Equation(B.4)

Shitsman (1959,1974)

Nu𝑏

= 0.023Re0.8𝑏

Pr0.8min[5, 7, 19].

“min” means minimum Pr value, that is, either the Prvalue is evaluated at the bulk fluid temperature or thePr value is evaluated at the wall temperature, whicheveris less. Assumption: thermal conductivity is a smoothlydecreasing function of temperature near the critical andpseudo-critical points.

Equation(B.5)

Bishop et al.(1964)

Nu𝑏

= 0.0069Re0.9𝑏

(Pr)0.66av(𝜌𝑤

/𝜌𝑏

)0.43 (1 + 2.4 (𝐷/𝑥))[5, 7, 8, 11, 13, 15].

𝑥 is the axial location along the heated length.Pressure = 22.8–27.6MPa, and bulk fluid temperature =282–527∘C.Mass flux = 651–3662 kg/m2s, and heat flux =0.31–3.46MW/m2.

Equation(B.6)

Swenson et al.[1]

Nu𝑤

= 0.00459Re0.923𝑤

(Pr𝑤

)0.613

av(𝜌𝑤

/𝜌𝑏

)0.231hD/𝑘𝑤

= 0.00459 (GD/𝜇𝑤

)0.923[((𝐻𝑤

− 𝐻𝑏

) (𝜇𝑤

))/((𝑇𝑤

− 𝑇𝑏

) (𝑘𝑤

))]0.613(𝜌𝑤

/𝜌𝑏

)0.231 [1, 5–8, 10, 15, 21].

Pressure = 22.8–41.4MPa, and bulk fluid temperature =75–576∘C.Mass flux = 542–2150 kg/m2s.Assumption: thermal conductivity is a smoothlydecreasing function of temperature near the critical andpseudo-critical points.

Equation(B.7)

Krasnoshchekovet al. (1967)

Nu = Nu0 (𝜌𝑤/𝜌𝑏)0.3 [(Cp)av/Cp𝑏]

𝑛,where, according to Petukhov andKirillov (1958), Nu0 = [(𝜉/8) Re𝑏

(Pr)av]/[12.7 Sqrt (𝜉/8)((Pr)(2/3)av − 1) + 1.07] and 𝜉 = 1/(1.82

log10Re𝑏 − 1.64)2.

Later, Krasnoshchekov et al. (1971) addeda correction factor to the above equationfor the tube entrance region in the form

of 𝑓(𝑥/𝐷) = 0.95 + 0.95(𝑥/𝐷)0.8.Also, this correction factor can be usedfor a heated tube abrupt inlet within

2 ≤ 𝑥/𝐷 ≤ 15 [10].

Exponent 𝑛 = 0.4 at 𝑇𝑤

/𝑇pc ≤ 1 or 𝑇𝑏/𝑇pc > 1.2 ≤ 1,𝑛 = 𝑛

1

= 0.22 + 0.18 𝑇𝑤

/𝑇pc at 1 ≤ 𝑇𝑤/𝑇pc ≤ 2.5, and𝑛 = 𝑛

1

+ (5 ⋅ 𝑛1

− 2) × (1 − 𝑇𝑏

/𝑇pc) at 1 ≤ 𝑇𝑏/𝑇pc ≤ 1.2.Valid within the following range:8 × 10

4

< Re𝑏

< 5 × 105, 0.85 < (Pr

𝑏

)av < 65,0.90 < (𝜌

𝑤

/𝜌𝑏

) < 1.0, 0.02 < (Cp)av/Cp𝑏 < 4.0,0.9 < 𝑇

𝑤

/𝑇pc < 2.5, 4.6 × 104

< 𝑞 < 2.6 × 106, where 𝑞 is

in W/m2 and 𝑥/𝐷 ≥ 15 [5–8].

Equation(B.8) Kondrat’ev [20]

Nu𝑏

= 0.020Re0.8𝑏

[5, 7, 20].

Valid within the range of 104 < Re < 4 × 105 and𝑡𝑏

= 130–600∘C.This equation is not valid within the pseudo-criticalregion.

Equation(B.9)

Ornatsky et al.(1970) Nu𝑏 = 0.023Re

0.8

𝑏

Pr0.8min (𝜌𝑤/𝜌𝑏)0.3[5, 7]. Prmin is in the minimum value of Pr𝑤 or Pr𝑏.

Equation(B.10)

Yamagata et al.[3]

Nu𝑏

= 0.0135Re0.85𝑏

Pr0.8𝑏

𝐹𝑐

[3, 5, 7, 8].

𝐹𝑐

= 1 for 𝐸 > 1, 𝐹𝑐

= 0.67Pr−0.05pc ((Cp)av/Cp𝑏)𝑛1 for

0 ≤ 𝐸 ≤ 1,𝐹𝑐

= ((Cp)av/Cp𝑏)𝑛2 for 𝐸 < 0,

𝐸 = (𝑇pc-𝑇𝑏)/(𝑇𝑤-𝑇𝑏),𝑛1 = −0.77 ( 1 + (1/Prpc))+ 1.49,𝑛2 = −1.44 ( 1 + (1/Prpc)) − 0.53.Pressure = 226–294 bar, and bulk fluid temperature =230–540∘C.Mass flux = 310–1830 kg/m2s, heat flux =116–930 kW/m2.

4 Journal of Energy

Table 1: Continued.

Equationno.

Name ofcorrelation Correlation Conditions

Equation(B.11)

Watts and Chouet al. (1982)

For (Gr𝑏

)av/(Re2.7

𝑏

(Pr𝑏

)0.5

av ) ≤ 10−4,

Nu/Nuvarp =[1–3000(Gr

𝑏

)av/(Re2.7

𝑏

(Pr𝑏

)0.5

av )]0.295.

For (Gr𝑏

)av/(Re2.7

𝑏

(Pr𝑏

)0.5

av ) ≥ 10−4,

Nu/Nuvarp =[7000(Gr

𝑏

)av/(Re2.7

𝑏

(Pr𝑏

)0.5

av )]0.295

[4, 11].

Nu = 𝛼D/𝜆𝑏

,Nuvarp = 0.021 Re

0.8

𝑏

(Pr𝑏

)0.55

av (𝜌𝑤/𝜌𝑏)0.35,

(Gr𝑏

)av =[𝜌𝑏(𝜌𝑏 − 𝜌av)gD3]/𝜇2𝑏

,Re𝑏

= GD/𝜇,(Pr𝑏

)av = Cpav 𝜇𝑏/𝜆𝑏,Cpav = (𝐻𝑤 − 𝐻𝑏)/(𝑇𝑤 − 𝑇𝑏).𝜌av = [Integral (𝜌dT)] with limits 𝑇𝑤 and 𝑇𝑏/[𝑇𝑤 − 𝑇𝑏]

Equation(B.12)

Gorban et al.(1990)

Nu𝑏

= 0.0059 Re0.90𝑏

Pr−0.12𝑏

[5, 7].

Equation(B.13) Griem (1996)

Nu𝑏

= 0.0169Re0.8356𝑏

Pr0.432𝑏

[5, 7, 8].

It covers the entire enthalpy range due to a new methodfor determining a representative specific heat capacity.Heat capacities were computed with semiempiricalequations at five reference temperatures.

Equation(B.14)

Kitoh et al.(1999)

Nu𝑏

= 0.015Re0.85𝑏

Pr𝑚𝑏

[5, 7, 10].

𝑚 = 0.69–81000/𝑞dht + 𝑓𝑐𝑞.The heat flux (𝑞dht) is that at which deterioration-ratedheat transfer occurs (W/m2).The heat flux is calculated according to 𝑞dth = 200𝐺

1.2.The coefficient 𝑓

𝑐

is calculated according to𝑓𝑐

= 29 × 10−8 + 0.11/𝑞dht for 0 ≤ 𝐻𝑏 ≤ 1500 kJ/kg,𝑓𝑐

= −8.7 × 10−8 − 0.65/𝑞dht for 1500 ≤ 𝐻𝑏 ≤ 3300 kJ/kg,𝑓𝑐

= −9.7 × 10−7 − 1.30/𝑞dht for 3300 ≤ 𝐻𝑏 ≤ 4000 kJ/kg.Valid for 𝑇

𝑏

from 20∘C to 550∘C (bulk fluid enthalpyfrom 100 to 3300 kJ/kg), 𝐺 from 100 to 1750 kg/m2s, and𝑞 from 0 to 1.8MW/m2.

Equation(B.15) Jackson (2002)

Nu𝑏

= 0.0183Re0.82𝑏

(Pr𝑏

)0.5

av (𝜌𝑤/𝜌𝑏)0. 3

[(Cp)av/Cp𝑏)]𝑛

[5–7, 11, 15].

Exponent𝑛 = 0.4 for 𝑇

𝑏

< 𝑇𝑤

< 𝑇pc and for 1.2 𝑇pc < 𝑇𝑏 < 𝑇𝑤,𝑛 = 0.4 + 0.2 ((𝑇

𝑤

/𝑇pc)−1) for 𝑇𝑏 < 𝑇pc < 𝑇𝑤,𝑛 = 0.4 + 0.2 ((𝑇

𝑤

/𝑇pc)−1) [1–5((𝑇𝑏/𝑇pc)−1) for𝑇pc < 𝑇𝑏 < 1.2 𝑇pc, and 𝑇𝑏 < 𝑇𝑤.𝑇𝑏

, 𝑇pc, and 𝑇𝑤 are in 𝐾.Valid for forced convection heat transfer in water andcarbon dioxide at supercritical pressures.

Equation(B.16)

Kang and Changet al. [11]

Nu𝑏

= 0.0244Re0.762𝑏

Pr0.552av (𝜌𝑤/𝜌𝑏)0.0293

[11].

Fluidfreon, HFC134a.Pressure: 4.1 to 4.5MPa, mass flux: 600 to 2000 kg/m2s,and heat flux: up to 160 kW/m2.

Equation(B.17) Zhu et al. [12]

Nu𝑏

= 0.0068Re0.90𝑏

(Pr𝑏

)0.63

av(𝜌𝑤

/𝜌𝑏

)0.17(𝑘𝑤

/𝑘𝑏

)0.29 [12].Pressure: 90–300 bar, mass flux: 600–1200 kg/m2s, andheat flux: 200–600 kW/m2.

Equation(B.18) Mokry et al. [15]

Nu𝑏

= 0.0061Re0.904𝑏

(Pr𝑏

)0.684

av(𝜌𝑤

/𝜌𝑏

)0.564[15].

Pressure: 24MPa, inlet fluid temperature: 320–350∘C,mass flux: 200–1500 kg/m2s, and heat flux ≤1250 kW/m2.

each point in the graph were calculated with experimentalwall temperature as base. The correlations were rankedbased on criteria like % predictions with

Journal of Energy 5

Table2:Con

solid

ated

details

ofallexp

erim

entald

ataa

ndtheb

estfi

tting

correlations

forthe

casesw

ithdeterio

ratedheattransfe

r.

Sl.

no.Parameter

Vikh

revetal.[18]

Dataset1

(Figure2

)

Vikh

revetal.[18]

Dataset2

(Figure3

)

Zhuetal.[12]

Dataset1

(Figure8

)

Zhuetal.[12]

Dataset2

(Figure9

)

Mok

ryetal.[15]

Dataset1

(Figure10)

Wangetal.[14]

Dataset1

(Figure12)

Wangetal.[14]

Dataset2

(Figure13)

01Tu

beID

(mm)

20.4

20.4

2626

1019.8

2602

Pressure

(bar

(a))

265

265

260

300

241

250

260

03Heatfl

ux(kW/m

2 )570

1160

300

300

148

660

350

04Massfl

ux(kg/m

2 s)

495

1400

600

600

201

1200

600

05Heatfl

uxto

massfl

uxratio

(kJ/k

g)[5]

1.152

0.829

0.5

0.5

0.736

0.55

0.583

06Limiting

heatflu

xto

avoid

deterio

ratio

n(kW/m

2 )[3,15]

342.4,309.8

1192,98

4431.3

,388

431.3

,388

116,90.7

990.9,835

431.3

,388

07Fluidtemperature

range(∘

C)60

to340

147to

403.8

336.8to

440.6

336.8to

408.1

321.9

4to

397.9

3244.21

to385.29

335to

437

08Pseudo

criticaltem

perature

atthe

givenpressure

(∘ C)

390.53

390.53

388.8

402.3

381.8

3385.17

388.77

09Fluidtemperature

atwhich

peak

change

inmetaltemperature

isob

served

(∘ C)

91.7and329.9

356.9,387.6

,392.3,

and403.8

408.4

408.1

330.7and381.5

385.29

387.53

10(𝑇𝑤

−𝑇𝑏

)atn

ormalwalltem

perature

zone-Exp

erim

ental

116.3to

152

98.8to

111.6

30.7to

50.9

31.3to

41.8

39.5to

68.7

53.7to

64.7

37.6to

116.5

11(𝑇𝑤

−𝑇𝑏

)atp

eakwalltem

perature

zone,exp

erim

ental

282.2and278.4

179.9

191.8

192.2

60.5and70.3

119.1

264.8

12Nam

eofcorrelatio

nthem

etal

temperature

predictio

nof

which

has

thec

losestfit

with

experim

entald

ata

Xiaojin

gZh

uWattsandCh

ouXiaojin

gZh

uWattsandCh

ouGorban

Gorban

Mokry

13(𝑇𝑤

−𝑇𝑏

)atn

ormalwalltem

perature

zone,bestcorrelation

92.2to

143.1

86.9to

111.6

25.41to59.85

32.93to

45.53

52.35

to79.6

50.1to

55.7

42.9to

99.7

14(𝑇𝑤

−𝑇𝑏

)atp

eakwalltem

perature


149and123.8

97.1

36.94

127.6

252.68and48.86

50.1

42.8

15To

taln

umbero

fdatap

oints

2920

168

8026

15

16%datawith<5%

errorinmetal

temperature

predictio

n48.28

7068.75

7598.75

88.46

40

17%datawith<10∘

Cerrorinmetal

temperature

predictio

n37.93

2056.25

5048.75

42.31

20

18Minim

umandmaxim

umdeviation

from

experim

entalm

etaltemperature

(∘ C)

−154.6to

+9.9

−82.78to

+23.33

−155to

+10

−64

.56to

+10.72

−21.39

to+2

4.41

−68.98to

+0.42

−222to

+10

6 Journal of Energy

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500 600 700 800

DensitySpecific heatThermal conductivity

Viscosity

Before pseudocritical region After pseudocritical

region

Pseudocritical region

Temperature (∘C)

Ratio

of p

rope

rty

at th

e ins

tant

to th

e pea

k va

lue

Figure 1: Trends of the thermophysical properties of water at230 bar.

0

100

200

300

400

500

600

700

0 50 100 150 200 250 300 350 400

Experimental Tiw-Vikhrev

Predicted Tiw-Xiaojing Zhu

Bulk fluid temperature (∘C)

Bulk fluid temperature

tem

pera

ture

(∘C)

Bulk

flui

d an

d in

ner w

all ID = 20.4mm

P = 265bar (a)q = 570kW/m2G = 495kg/m2s

Figure 2: Trends of experimental (Vikhrev et al. [18]) wall tem-perature and that calculated using the best fitting Xiaojing Zhucorrelation for dataset 1 in Table 2.

temperature seen in Figure 3 are not predicted even by thebest fitting correlation.

4.2. Experiments of Yamagata et al. (1972) [3]. Figures 4 and5 display the trends of inner wall temperatures calculated bythe best fitting correlations alongwith reference experimentalwall temperatures for two different operating conditionsgiven in [3]. Table 3 details the parameter conditions and thecomparative information on the predicted wall temperatureof the best fitting correlationwith reference to the experimen-tal wall temperature.

Observations of Figure 4 and Table 3 for dataset 1 indicatevery good agreement of the best fitting correlation even atpseudocritical temperature. Also, many correlations predict100% agreement for the first two considerations of % datawith

Journal of Energy 7

Table3:Con

solid

ated

details

ofallexp

erim

entald

ataa

ndtheb

estfi

tting

correlations

forc

ases

with

norm

alheattransfe

r.

Sl.

no.

Parameter

Yamagatae

tal.[3]

Dataset1

(Figure4

)

Yamagatae

tal.[3]

Dataset2

(Figure5

)

Loew

enberg

etal.[10]

Dataset1

(Figure6

)

Loew

enberg

etal.[10]

Dataset2

(Figure7

)

Mok

ryetal.[15]

Dataset2

(Figure11)

01Tu

beID

(mm)

7.57.5

2020

3802

Pressure

(bar

(a))

245

245

250

235

241

03Heatfl

ux(kW/m

2 )233

930

300

1200

252

04Massfl

ux(kg/m

2 s)

1260

1260

1000

2250

543

05Heatfl

uxto

massfl

uxratio

(kJ/k

g)[5]

0.185

0.738

0.300

0.533

0.46

4

06Limiting

heatflu

xto

avoiddeterio

ratio

n(kW/m

2 )[3,15]

1050.7,

879.7

1050.7,

879.7

796.2,686

2106.9,

1617.2

382.6,345.5

07Fluidtemperature

range(∘

C)334.01

to391.8

6341to381.9

273.78

to40

6.41

273.78

to40

6.41

266.33

to380.1

08Pseudo

criticaltem

perature

attheg

iven

pressure

(∘ C)

383.33

383.33

385.17

379.5

6381.8

3

09Fluidtemperature

atwhich

peak

change

inmetaltemperature

isob

served

(∘ C)

381.6

1381.9

376.4

379.5

9373.1

10(𝑇𝑤

−𝑇𝑏

)atn

ormalwalltem

perature

zone,exp

erim

ental

8.6to

14.6

37.1to

51.5

18.2to

36.6

47.95to

71.59

38.7to

57.6

11(𝑇𝑤

−𝑇𝑏

)atp

eakwalltem

perature

zone,

experim

ental

2.9

39.4

15.5

27.41

18.8

12Nam

eofcorrelatio

nthem

etal

temperature

predictio

nof

which

hasthe

closestfit

with

experim

entald

ata

Ornatsky

Shitsman

WattsandCh

ouJackson

Mok

ry

13(𝑇𝑤

−𝑇𝑏

)atn

ormalwalltem

perature


6.2to

13.7

42.2to

51.7

16.5to

3946

.00to

88.41

40.47to

48.81

14(𝑇𝑤

−𝑇𝑏

)atp

eakwalltem

perature

zone,

bestcorrelation

2.8

32.4

1525.74

30

15To

taln

umbero

fdatap

oints

3517

1515

79

16%datawith<5%

errorinmetal

temperature

predictio

n100

100

100

100

100

17%datawith<10∘

Cerrorinmetal

temperature

predictio

n100

100

100

8096.2

18Minim

umandmaxim

umdeviationfro

mexperim

entalm

etaltemperature

(∘ C)

−2.4to

+0.73

−7to

+8.1

−7.59to

+2.39

−3.98

to+16.82

−10.2to

+11.2

8 Journal of Energy

330340350360370380390400410420430

335 340 345 350 355 360 365 370 375 380 385

Predicted Tiw-Shitsman


tem

pera

ture

(∘C)

Experimental Tiw-YamagataBulk fluid temperature

Bulk

flui

d an

d in

ner w

all

ID = 7.5mmP = 245bar (a)q = 930kW/m2

G = 1260kg/m2s

Figure 5: Trends of experimental (Yamagata et al. [3]) wall temper-ature and that calculated using the best fitting Shitsman correlationfor dataset 2 in Table 3.


tem

pera

ture

(∘C)

Fluid temperature (∘C)

250

300

350

400

450

260 280 300 320 340 360 380 400 420

Reference Tiw-LoewenbergPredicted Tiw-Watts and Chou

ID = 20mmP = 250bar (a)q = 300kW/m2G = 1000 kg/m2s

Bulk

flui

d an

d in

ner w

all

Figure 6: Trends of the lookup table (Loewenberg et al. [10]) walltemperature and that calculated using the best fitting Watts andChou correlation for dataset 1 in Table 3.

comparatively higher deviation inwall temperature and lower% agreement for the selected criteria even for the best fittingcorrelation.

4.4. Experiments of Zhu et al. (2009) [12]. Figures 8 and 9display the trends of inner wall temperatures calculated bythe best fitting correlations alongwith experimental wall tem-peratures for two different operating conditions given in [12].Table 2 details the parameter conditions and the comparativeinformation on the predicted wall temperature of the bestfitting correlation with reference to the experimental walltemperature.

As per Figure 8 and dataset 1 in Table 2, the experimentalwall temperature indicates a sharp peak of 600.2∘C at afluid temperature of 408.4∘C with (𝑇

𝑤− 𝑇𝑏) of 191.8∘C.

None of the correlations indicate this peak. Even the bestfitting correlation has only 68.5% of data with

Journal of Energy 9

330

380

430

480

530

580

330 350 370 390 410 430 450

Predicted Tiw-Xiaojing Zhu

tem

pera

ture

(∘C)



P= 260bar (a)ID = 26mm

Bulk

flui

d an

d in

ner w

all

q = 300kW/m2

G = 600kg/m2s

Experimental Tiw-Xiaojing Zhu

Figure 8: Trends of experimental (Zhu et al. [12]) wall temperatureand that calculated using the best fitting Xiaojing Zhu correlationfor dataset 1 in Table 2.

330

380

430

480

530

580

Bulk

flui

d an

d in

ner w

all

tem

pera

ture

(∘C)


Bulk fluid temperature (∘C)330 340 350 360 370 380 390 400 410

Experimental Tiw-Xiaojing Zhu

Predicted Tiw-Watt and Chou

P= 300bar (a)q = 300kW/m2

ID = 26mm

G = 600kg/m2s

Figure 9: Trends of experimental (Zhu et al. [12]) wall temperatureand that calculated using the best fittingWatts and Chou correlationfor dataset 2 in Table 2.

300320340360380400420440460480500

320 330 340 350 360 370 380 390 400

Experimental Tiw-MokryBulk fluid temperaturePredicted Tiw-Gorban

tem

pera

ture

(∘C)

Bulk

flui

d an

d in

ner w

all


ID = 10mmP = 241bar (a)q = 480kW/m2G = 201 kg/m2s

Figure 10: Trends of experimental (Mokry et al. [15]) wall tempera-ture and that calculated using the best fitting Gorban correlation fordataset 1 in Table 2.

250270290

310330350370390410

250 280 310 340 370 400

Bulk

flui

d an

d in

ner w

all

tem

pera

ture

(∘C)


Experimental Tiw-MokryBulk fluid temperaturePredicted Tiw-Mokry

ID = 38mmP = 241bar (a)q = 252kW/m2G = 543kg/m2s

Figure 11: Trends of experimental (Mokry et al. [15]) wall tem-perature and that calculated using the best fitting Sarah Mokrycorrelation for dataset 2 in Table 3.

ID = 19.8mmP = 250bar (a)q = 660kW/m2

G = 1200 kg/m2s

220

270

320

370

420

470

520

240 260 280 300 320 340 360 380

Experimental Tiw-Jianguo Wang

Predicted Tiw-GorbanBulk fluid temperature

tem

pera

ture

(∘C)

Bulk

flui

d an

d in

ner w

all


Figure 12: Trends of experimental (Wang et al. [14]) wall tempera-ture and that calculated using the best fitting Gorban correlation fordataset 1 in Table 2.

the best fitting correlation with reference to the experimentalwall temperature.

Dataset 1 in Table 2 and Figure 12 indicate sharp increasein wall temperature near the exit where the fluid temperatureis just above pseudocritical temperature. None of the corre-lations, including the best one, indicate this peak, and all ofthem show a dip in temperature near the exit contradictingthe experimental observation. The prediction by Gorbancorrelation has 88.46% of predictions with

10 Journal of Energy

320

370

420

470

520

570

620

670

320 340 360 380 400 420 440

Bulk fluid temperatureExperimental Tiw-Jianguo Wang

Predicted Tiw-Mokry

tem

pera

ture

(∘C)

Bulk

flui

d an

d in

ner w

all


ID = 26mmP = 260bar (a)q = 350kW/m2G = 600kg/m2s

Figure 13: Trends of experimental (Wang et al. [14]) wall tem-perature and that calculated using the best fitting Sarah Mokrycorrelation for dataset 2 in Table 2.

The analysis of the above wall temperature figures andthe values in the tables indicate that even the best fittingcorrelations agree well only in the normal heat-transfer zonesand not in the deteriorated heat-transfer zones.

5. Results and Discussion

5.1. Wall Temperature Prediction Capability of Heat-TransferCorrelations. Analyses of the 12 experimental datasets indi-cate that there are 7 deteriorated and 5 normal heat-transferconditions in the group. The consolidated information of theselected experimental works and the data on the predictionlevel of the best fitting correlation for each dataset are listedin Table 2 for the deteriorated and in Table 3 for the normalheat-transfer conditions. As seen in the tables, the Watts andChou correlation is the best for 2 deteriorated and 1 normalheat-transfer cases, theGorban andXiaojingZhu correlationsare the best for each 2 deteriorated heat-transfer cases, theMokry correlation is the best for 1 deteriorated and 1 normalheat-transfer cases, and the Ornatsky, Shitsman and Jacksoncorrelations are the best for each 1 normal heat-transfer case.

It is observed that the agreement of the predictions ofthe best fitting correlations for 5 normal heat transfer casesindicates almost 100% of the prediction satisfying the

Journal of Energy 11

Table 4: Average and RMS errors of wall temperature prediction by various correlations for the selected data points.

Equation no. Correlation Average error (%) RMS error (%) No. of applicable data pointsEquation (B.1) Dittus and Boelter (1930) −5.2500 9.4837 All 355Equation (B.2) Mc Adams (1942) −5.7839 9.8609 All 355Equation (B.3) Bringer and Smith (1957) −5.1090 9.6819 All 355Equation (B.4) Shitsman (1959, 1974) −3.7453 8.5276 All 355Equation (B.5) Bishop et al. (1964) −3.8194 6.5453 66Equation (B.6) Swenson et al. [1] 6.5411 14.4906 90Equation (B.7) Krasnoshchekov et al. (1967) 0.0237 0.1560 9Equation (B.8) Kondrat’ev [20] −1.2354 6.4086 282Equation (B.9) Ornatsky et al. (1970) −1.2940 8.9133 All 355Equation (B.10) Yamagata et al. [3] −7.0392 11.0684 All 355Equation (B.11) Watts and Chou et al. (1982) −3.6744 8.3295 All 355Equation (B.12) Gorban et al. (1990) −1.0198 7.9306 169Equation (B.13) Griem (1996) −6.4152 10.2791 All 355Equation (B.14) Kitoh et al. (1999) −6.5683 10.5762 340Equation (B.15) Jackson (2002) −4.4797 8.6144 All 355Equation (B.16) Kang and Chang et al. [11] −1.6150 7.8109 All 355Equation (B.17) Zhu et al. [12] −3.7481 7.7072∗ All 355Equation (B.18) Mokry et al. [15] −0.0489∗ 8.8816 All 355∗The lowest of the correlations applied to all 355 data points.

Article [15] reports (6) given below for predicting thestarting heat flux for deteriorated heat transfer as a functionof mass flux:

𝑞dht in kW/m2

= −58.97 + 0.745𝐺 (6)(see [15]).

For the same mass flux conditions, the starting heat fluxfor deteriorated heat transfer predicted by (6) is lower thanthat calculated by (4) and (5). Analysis of the 12 experimentsand the details in Tables 2 and 3 indicate that there arefour experiments in which the heat flux is higher than thatcalculated by (6). The wall temperatures for three of thesefour cases indicate deterioration, whereas no deteriorationis observed for one case. Though the experimental heatflux is less than that calculated by (6), the experimentalwall temperatures indicate deteriorated condition for fourmore cases. The overall agreement of the prediction ofthe deteriorated heat transfer by (6) is 58.33% for the 12experiments selected in this paper.

6. Conclusion

The validity of the wall temperature predictions by 18 corre-lations for 12 supercritical experimental datasets consistingof 355 data points available in the literature was verified. Thecorrelations were ranked based on criteria like % data with

12 Journal of Energy

𝐺: Mass flux, kg/m2s𝑔: Acceleration due to gravity, m/s2𝐻: Enthalpy, J/kgℎ: Convection heat-transfer coefficient,

W/m2 K𝑘: Thermal conductivity, W/mK𝐿: Length, m𝑝: Pressure, MPa, bar𝑞: Heat flux, kW/m2𝑇, 𝑡: Temperature, ∘CNu: Nusselt numberRe: Reynolds numberPr: Prandtl numberPrav: Average Prandtl number = Cpav𝜇𝑏/𝑘𝑏.

Greek Letters

𝜇: Dynamic viscosity, Pa s𝜌: Density, kg/m3𝜉: Friction coefficient𝛼: Heat-transfer coefficient𝜆: Thermal conductivity, W/mK.

Subscript

av: Average𝑏: Bulkdht: Deteriorated heat transfer𝑤: Walliw: Inner wallpc: Pseudocritical𝑡: Temperature𝑥: Axial locationhy: Hydraulicmin: Minimum.

Superscript

𝑚: Constant𝑛: Constant.

References

[1] H. S. Swenson, J. R. Carver, and C. R. Kakarala, “Heat transferto supercritical water in smooth bore tubes,” Journal of HeatTransfer, vol. 87, no. 4, pp. 477–484, 1965.

[2] A. W. Ackerman, “Pseudo boiling heat transfer to supercriticalpressure water in smooth and ribbed tubes,” Transactions of theASME, vol. 92, pp. 490–497, 1970.

[3] K. Yamagata, K. Nishikawa, S. Hasegawa, T. Fujii, and S.Yoshida, “Forced convective heat transfer to supercritical waterflowing in tubes,” International Journal of Heat and MassTransfer, vol. 15, no. 12, pp. 2575–2593, 1972.

[4] T. Yamashita, S. Yoshida, H. Mori, S. Morooka, H. Komita, andK. Nishida, “Heat transfer study under supercritical pressureconditions,” GENE4/ANP2003, Paper 1119, Kyoto, Japan, 2003.

[5] I. L. Pioro, H. F. Khartabil, and R. B. Duffey, “Heat transfer tosupercritical fluids flowing in channels—empirical correlations

(survey),” Nuclear Engineering and Design, vol. 230, no. 1–3, pp.69–91, 2004.

[6] F. Yin, T.-K. Chen, and H.-X. Li, “An investigation on heattransfer to supercritical water in inclined upward smoothtubes,”Heat Transfer Engineering, vol. 27, no. 9, pp. 44–52, 2006.

[7] I. L. Pioro and R. B. Duffey, Heat Transfer and Hydraulic Resis-tance at Supercritical Pressures in Power-Engineering Applica-tions, ASME Press, 2006-2007.

[8] X. Cheng, B. Kuang, and Y.H. Yang, “Numerical analysis of heattransfer in supercritical water cooled flow channels,” NuclearEngineering and Design, vol. 237, no. 3, pp. 240–252, 2007.

[9] P.-X. Jiang, Y. Zhang, C.-R. Zhao, and R.-F. Shi, “Convectionheat transfer of CO

2

at supercritical pressures in a vertical minitube at relatively low reynolds numbers,” ExperimentalThermaland Fluid Science, vol. 32, no. 8, pp. 1628–1637, 2008.

[10] M. F. Loewenberg, E. Laurien, A. Class, and T. Schulenberg,“Supercritical water heat transfer in vertical tubes: a look-uptable,” Progress in Nuclear Energy, vol. 50, no. 2–6, pp. 532–538,2008.

[11] K.-H. Kang and S.-H. Chang, “Experimental study on the heattransfer characteristics during the pressure transients undersupercritical pressures,” International Journal of Heat and MassTransfer, vol. 52, no. 21-22, pp. 4946–4955, 2009.

[12] X. Zhu, Q. Bi, D. Yang, and T. Chen, “An investigation on heattransfer characteristics of different pressure steam-water invertical upward tube,”Nuclear Engineering and Design, vol. 239,no. 2, pp. 381–388, 2009.

[13] S.Mokry, I. Pioro, P. Kirillov, andY.Gospodinov, “Supercritical-water heat transfer in a vertical bare tube,” Nuclear Engineeringand Design, vol. 240, no. 3, pp. 568–576, 2010.

[14] J. Wang, H. Li, S. Yu, and T. Chen, “Comparison of the heattransfer characteristics of supercritical pressure water to thatof subcritical pressure water in vertically-upward tubes,” Inter-national Journal of Multiphase Flow, vol. 37, no. 7, pp. 769–776,2011.

[15] S. Mokry, I. Pioro, A. Farah et al., “Development of supercriticalwater heat-transfer correlation for vertical bare tubes,” NuclearEngineering and Design, vol. 241, no. 4, pp. 1126–1136, 2011.

[16] S. B. Shiralkar and G. P. Griffith, “The effect of swirl, inletconditions, flow direction, and tube diameter on the heattransfer to fluids at supercritical pressure,” Journal of HeatTransfer, vol. 92, pp. 465–471, 1970.

[17] R. Dhanuskodi, A. Arunagiri, and N. Anantharaman, “Analysisof variation in properties and its impact on heat transfer insub and supercritical conditions of water/steam,” InternationalJournal of Chemical Engineering and Applications, vol. 2, no. 5,pp. 320–325, 2011.

[18] Y. V. Vikhrev, Y. D. Barulin, and A. S. Kon’Kov, “A studyof heat transfer in vertical tubes at supercritical pressures,”Teploenergetika, vol. 14, no. 9, pp. 80–82, 1967.

[19] M. E. Shitsman, “Temperature conditions in tubes at supercrit-ical pressures,” Teploenergetika, vol. 15, no. 5, pp. 57–61, 1968.

[20] N. S. Kondrat’ev, “Heat transfer and hydraulic resistance withsupercritical water flowing in tubes,” Teploeregetia, vol. 16, no.8, pp. 49–51, 1969.

[21] M. Sharabi and W. Ambrosini, “Discussion of heat transferphenomena in fluids at supercritical pressure with the aid ofCFDmodels,”Annals of Nuclear Energy, vol. 36, no. 1, pp. 60–71,2009.

Journal of Energy 13

[22] H. Zahalan, D. C. Groneveld, and S. Tavoularis, “Look-uptable for trans-critical heat transfer,” in Proceedings of the 2ndCanada-China JointWorkshop on on SupercriticalWater-CooledReactors (CCSC ’10), p. 18, Toronto, Canada, April 2010.

[23] H. Zahalan, D. C. Groneveld, S. Tavoularis, S. Mokry, andI. Pioro, “Assessment of supercritical heat transfer predictionmethods,” in Proceedings of the 5th International Symposiumon on Supercritical Water-Cooled Reactors (ISSWR ’11), p. 20,British Columbia, Canada, March 2011, Paper P008.

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