**2. Experimental data and existing heat transfer correlations**

Since 1960s, a series of experimental investigations on heat transfer of SCW have been conducted, and a large amount of experimental data have been accumulated. A comprehensive survey of experimental investigations on heat transfer correlations of SCW has been provided by Pioro et al. [4, 8, 14] and showed that most of previous experimental data for heat transfer of SCW focused primarily on the flow of SCW in vertical circular tubes [2]. In the present study, approximately 12,704 data points and 250 experimental cases about the flow and heat transfer of SCW in vertical circular tubes are retrieved and collected from the published literatures.

pollution products emission. For example, the U.S. Department of Energy has ever proposed a research program (AD 760), in which the operating parameters of the future fossil-fired power plants are suggested to raise to 37.9 MPa/732°C/760°C with an efficiency of up to 47%. A program similar to AD 760, i.e., the AD 700 thermal program, has also been proposed in Europe, and planned to build the ultrasupercritical steam condition at 720°C and 35 MPa in the future thermal power plants so as to obtain a cycle efficiency up to 50% [2]. There is no doubt that the heat transfer characteristics of SCW is an important issue for the thermal power plants with high level parameters. The higher the operating parameters, the higher the thermal efficiency, and as well, accordingly, the more important is the precise prediction of heat transfer characteristics of SCW. It should be noted that deteriorated heat transfer (DHT) is one kind of special heat transfer phenomena of SCW that might happen in the heat transfer tube and even lead to the burst of heat transfer tube or other failure accidents in the plants. On the other hand, the supercritical water-cooled reactors (SCWRs) has been selected as one of the six most promising reactors concepts presented at the Generation IV International Forum due to its high thermal efficiency and other features [3]. Similar to the situation in supercritical pressure thermal power plants, the heat transfer of SCW is an important issue for SCWRs as well. Furthermore, due to the possible risks in release of radioactive medium under conditions with tube or vessel failures, the heat transfer of SCW is of much more concern for the safe operation of SCWRs than that of thermal power stations, and as a result, precise knowledge about the heat transfer characteristics of SCW is one of the major tasks for the design and operation of SCWRs [3, 4]. Besides, heat transfer of SCW is of significance for other applications

*Advanced Supercritical Fluids Technologies*

with SCW as the working fluid, such as SCW gasification system [5].

and engineering.

**Table 1.**

**44**

Two methods have been proposed by different scholars to predict the heat transfer characteristics of SCW: the look-up Table [6, 7] and the heat transfer correlations [8]. The look-up table for heat transfer of SCW is based on real experimental data. The prediction accuracy of such look-up table is pretty high; however, the applicable ranges of the operation parameters (e.g., pressures, mass fluxes, and heat fluxes) of the look-up table are limited by the experimental conditions. In the development of ultra-supercritical pressure boilers or other heat exchangers with high operation parameters, the applicability of the look-up table seems to be limited. Unlike the look-up table, heat transfer correlations have been developed on the basis on experimental data, dimensionless analysis, and theoretical analysis of the heat transfer phenomena. The applicability of heat transfer correlations are much flexible than that of look-up tables, and as a result, heat transfer correlations for SCW have found wide applications in related industries

In the last few decades, many heat transfer correlations for SCW were proposed

represents thermophysical properties correction terms; **Gr\*** represents the buoyancy correction terms; *q*<sup>+</sup> represents the thermal acceleration correlation terms

(see **Table 3** in the later section of this chapter). Generally, the existing heat transfer correlations could be divided into three categories [9], as listed in **Table 1**. The first type heat transfer correlation for SCW was built on the basis of the

<sup>I</sup> Nu <sup>¼</sup> *<sup>C</sup>*<sup>0</sup> � Re *<sup>C</sup>*<sup>1</sup> � Pr*<sup>C</sup>*<sup>2</sup> *<sup>C</sup>*0, *<sup>C</sup>*1, *<sup>C</sup>*2, *<sup>C</sup>*3, *<sup>C</sup>*<sup>4</sup> are constants; *<sup>F</sup>*

**Types Forms Remarks**

III Nu <sup>¼</sup> *<sup>C</sup>*<sup>0</sup> � Re *<sup>C</sup>*<sup>1</sup> � Pr*<sup>C</sup>*<sup>2</sup> � *<sup>F</sup>* � Gr <sup>∗</sup> ð Þ*<sup>C</sup>*<sup>3</sup> � *<sup>q</sup>*<sup>þ</sup> ð Þ*<sup>C</sup>*<sup>4</sup>

II Nu <sup>¼</sup> *<sup>C</sup>*<sup>0</sup> � Re *<sup>C</sup>*<sup>1</sup> � Pr*<sup>C</sup>*<sup>2</sup> � *<sup>F</sup>*

*Three types of heat transfer correlations for SCW.*

**Table 2** outlines the main information about the experimental data collected in this study on the heat transfer of SCW flowing in vertically upward smooth tubes. It is seen that the experimental data used in the present study covers the parameters of pressures from 2.25 <sup>10</sup><sup>7</sup> to 3.103 <sup>10</sup><sup>7</sup> Pa, mass fluxes from 200 kg/(m<sup>2</sup> s) to 2500 kg/(m<sup>2</sup> s), heat fluxes from 1.48 <sup>10</sup><sup>5</sup> to 2 <sup>10</sup><sup>6</sup> W/m2 , and the tube

diameters from 3 to 38 mm. Detailed analysis of the experimental data shows that among the 250 experimental cases collected, including both in this study, 134 cases

are in the EHT regimes, while 116 cases of them are in the DHT regimes. As well known, many heat transfer correlations have been proposed for predicting the heat transfer characteristics of SCW, and **Table 3** listed 34 of them as examples, for SCW in vertically upward tubes. It is seen from **Table 3** that many of the correlations were proposed 40 years ago and old thermophysical property data

*Heat Transfer Correlations for Supercritical Water in Vertically Upward Tubes*

<sup>1985</sup> Nu <sup>¼</sup> <sup>0</sup>*:*023 Re <sup>0</sup>*:*<sup>8</sup>

Nu ¼

8 >>< >>:

0*:*1 Re <sup>0</sup>*:*<sup>66</sup> <sup>b</sup> Pr1*:*<sup>2</sup>

0*:*036 Re <sup>0</sup>*:*<sup>8</sup>

Nu ¼

*ρw ρb* � �<sup>0</sup>*:*<sup>3</sup> *Cp Cp*,*<sup>b</sup>* !<sup>n</sup>

Swenson et al. [17] 1965 Nu <sup>¼</sup> <sup>0</sup>*:*00459 Re <sup>0</sup>*:*<sup>923</sup>

*n* ¼

Grass et al. [53] 1971 Nu <sup>¼</sup> ð Þ *<sup>f</sup> <sup>=</sup>*<sup>8</sup> Re bPrb

Yamagata et al. [25] 1972 Nu <sup>¼</sup> <sup>0</sup>*:*0135 Re <sup>0</sup>*:*<sup>85</sup>

8 >>>< >>>: *n*<sup>1</sup> ¼ �0*:*77 1 þ

*FC* ¼

1981

Ornatsky et al. [24] 1970 Nu <sup>¼</sup> <sup>0</sup>*:*023 Re <sup>0</sup>*:*<sup>8</sup>

Nu ¼ Nu0

8 >>>< >>>:

Kondratev et al. [22] 1969 Nu <sup>¼</sup> <sup>0</sup>*:*02 Re <sup>0</sup>*:*<sup>8</sup>

1967

<sup>b</sup> Pr1*<sup>=</sup>*<sup>3</sup> b

� �

<sup>b</sup> , *Tw* ≥623*:*15K

<sup>w</sup> Pr0*:*<sup>613</sup>

*<sup>n</sup>*<sup>1</sup> <sup>þ</sup> ð Þ <sup>5</sup>*n*<sup>1</sup> � <sup>2</sup> <sup>1</sup> � *Tb=Tpc* � � � � *Tb* <sup>&</sup>lt;*Tw* and 1<sup>≤</sup> *Tb=Tpc* � � <sup>≤</sup>1*:*<sup>2</sup>

Prmin¼ Prb for Prb <Prw, Prmin¼ Prw for Prw <Prb

PrG ¼ Prb for Prb <0*:*5Prw, PrG¼ Prw for Prb >0*:*5Prw

1*:*0 *Tpc* � *Tb*

<sup>b</sup> Pr0*:*<sup>4</sup> b

1 for *Cp* ≤*Cp*,*<sup>b</sup>*

Prpc � � <sup>þ</sup> <sup>1</sup>*:*49, *<sup>n</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*:*44 1 <sup>þ</sup>

b

<sup>G</sup> *Cp*,*<sup>b</sup>=Cp*,*<sup>w</sup>* � 1 � �

> <sup>b</sup> Pr<sup>0</sup>*:*<sup>8</sup> <sup>b</sup> *FC*

0≤ *Tpc* � *Tb*

for *Cp* >*Cp*,*<sup>b</sup>*

� �*=*ð Þ *Tw* � *Tb* <sup>≥</sup><sup>1</sup>

� �*=*ð Þ *Tw* � *Tb* <sup>≤</sup><sup>0</sup>

*F*

� �*=*ð Þ *Tw* � *Tb* <sup>≤</sup><sup>1</sup>

*Tpc* � *Tb*

1 Prpc � � � <sup>0</sup>*:*<sup>53</sup>

2 ð Þ <sup>0</sup>*:*8*<sup>ψ</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>2</sup> <sup>0</sup>*:*<sup>5</sup> <sup>þ</sup> <sup>1</sup> " #<sup>2</sup>

Nu0 <sup>¼</sup> ð Þ *<sup>ξ</sup>=*<sup>8</sup> Re bPr 12*:*7 ffiffiffiffiffiffiffi

� �, 523*:*15K<*Tw* <sup>&</sup>lt;623*:*15K

<sup>w</sup> *<sup>ρ</sup>w=ρ<sup>b</sup>* ð Þ<sup>0</sup>*:*<sup>231</sup>

*<sup>ξ</sup>=*<sup>8</sup> <sup>p</sup> Pr2*<sup>=</sup>*<sup>3</sup> � <sup>1</sup>

� � <sup>þ</sup> <sup>1</sup>*:*<sup>07</sup>

<sup>b</sup> Pr<sup>0</sup>*:*<sup>8</sup> min Prmin¼ Prb for Prb <Prw, Prmin¼ Prw for Prw <Prb

> *f* Re bPrb *<sup>μ</sup> μw* � �<sup>0</sup>*:*<sup>11</sup> *<sup>λ</sup><sup>w</sup> λb* � �<sup>0</sup>*:*<sup>33</sup> *Cp Cp*,*<sup>b</sup>* � �<sup>0</sup>*:*<sup>35</sup> *=*8

12*:*7 ffiffiffi *f* 8 r Pr2*<sup>=</sup>*<sup>3</sup> <sup>b</sup> � 1 � � <sup>þ</sup> <sup>1</sup>*:*<sup>07</sup> " #

*<sup>f</sup>* <sup>¼</sup> <sup>1</sup>*:*82 log <sup>10</sup>ð Þ� Re *<sup>b</sup>* <sup>1</sup>*:*<sup>64</sup> � ��<sup>2</sup>

<sup>b</sup> Pr<sup>0</sup>*:*<sup>4</sup> b *μw μb*

0*:*4 *Tb* <*Tw* <*Tpc* or 1*:*2*Tpc* <*Tb* <*Tw <sup>n</sup>*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>22</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>18</sup> *Tw=Tpc* � � <sup>1</sup><sup>≤</sup> *Tw=Tpc* � � <sup>≤</sup>2*:*<sup>5</sup>

> <sup>b</sup> Pr0*:*<sup>8</sup> min *ρw ρb* � �<sup>0</sup>*:*<sup>3</sup>

*f =*8 p Pr2*<sup>=</sup>*<sup>3</sup>

<sup>1</sup>*:*<sup>07</sup> <sup>þ</sup> <sup>12</sup>*:*<sup>7</sup> ffiffiffiffiffiffiffiffi

*<sup>f</sup>* <sup>¼</sup> <sup>1</sup>*:*82 log <sup>10</sup>ð Þ� Re <sup>b</sup> <sup>1</sup>*:*<sup>64</sup> � �‐<sup>2</sup>

0*:*67Pr‐0*:*<sup>05</sup> pc *Cp=Cp*,*<sup>b</sup>* � �*<sup>n</sup>*<sup>1</sup>

1

Nu <sup>¼</sup> <sup>0</sup>*:*023 Re <sup>0</sup>*:*<sup>8</sup>

*ψ* ¼ 1 þ *β*ð Þ *Tw* � *Tb*

*Cp Cp*,*<sup>b</sup>* � �<sup>0</sup>*:*<sup>28</sup>

8 >< >:

*Cp=Cp*,*<sup>b</sup>* � �*<sup>n</sup>*<sup>1</sup>

*F* ¼

**Author Year Correlation**

Shitsman et al. [50] 1957 Nu <sup>¼</sup> <sup>0</sup>*:*023 Re <sup>0</sup>*:*<sup>8</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.89580*

Dittus-Boelter et al.

Petukhov et al. [8] 1963

Domin et al. [51] 1963

Krasnoschekov et al.

Yeroshenko et al.

[54]

**47**

[52]

[10]


### **Table 2.**

*Experimental data of heat transfer of SCW collected from literatures.*

### *Heat Transfer Correlations for Supercritical Water in Vertically Upward Tubes DOI: http://dx.doi.org/10.5772/intechopen.89580*

diameters from 3 to 38 mm. Detailed analysis of the experimental data shows that among the 250 experimental cases collected, including both in this study, 134 cases are in the EHT regimes, while 116 cases of them are in the DHT regimes.

As well known, many heat transfer correlations have been proposed for predicting the heat transfer characteristics of SCW, and **Table 3** listed 34 of them as examples, for SCW in vertically upward tubes. It is seen from **Table 3** that many of the correlations were proposed 40 years ago and old thermophysical property data


**Table 2** outlines the main information about the experimental data collected in this study on the heat transfer of SCW flowing in vertically upward smooth tubes. It is seen that the experimental data used in the present study covers the parameters of

s) to

**]** *<sup>d</sup>* **<sup>10</sup><sup>3</sup> [m]**

, and the tube

**s)]** *<sup>q</sup>* **<sup>10</sup><sup>3</sup> [W/m<sup>2</sup>**

26.5 495–1400 507–1160 20.4

25–29 600/1200 350/660 17/19.8/26

24.0/24.1/24.2 201–500 148–335 10/38

23/25/26 459.8–1497.5 192–1326.5 7.6

pressures from 2.25 <sup>10</sup><sup>7</sup> to 3.103 <sup>10</sup><sup>7</sup> Pa, mass fluxes from 200 kg/(m<sup>2</sup>

s), heat fluxes from 1.48 <sup>10</sup><sup>5</sup> to 2 <sup>10</sup><sup>6</sup> W/m2

Shitsman et al. [15] 1963 23.3/25.3 430/449 210–386 8 Bishop et al. [16] 1964 24.1/24.2 543/678 252/606 5/38 Swenson et al. [17] 1965 23/31 2150 789 9.42

Styrikovich et al. [20] 1967 24 700 348–872 22 Herkenrath et al. [21] 1967 22.5/24/25 700/1000/1500 300–1410 10/20 Kondratev et al. [22] 1969 25.3 700 581 18 Ackerman et al. [23] 1970 22.75/31.03 542.52–1220 472.9–1261 9.42 Ornatsky et al. [24] 1970 25.5 1500 1810 3 Yamagata et al. [25] 1972 22.6/24.5/29.4 1120–1260 233–930 7.5/10 Lee et al. [26] 1974 24.1/24.5 376/543 252–379 16/38 Polyakov et al. [27] 1991 24.5/29.4 595/675 500/570 8 Griem et al. [28] 1996 25 500/1000 300 14 Koshizuka et al. [29] 2000 31 540/680 473 9.4 Yoshida et al. [30] 2000 24.5 376/410/1180 329/350/698 10/16 Hu et al. [31] 2001 23/26/30 600/900/1200 200–500 26 Xu et al. [32] 2004 23/25/27/30 800/1000/1200 200–600 12 Kirillov et al. [33] 2005 24–24.9 200/1500 227–884 10 Zhu et al. [34] 2009 30 600 250 26

Pan et al. [38] 2011 22.5/27/28/30 1009–1626 216–649 17

Wang et al. [41] 2012 23/25/26 449–1520.6 192–1154.3 7.6/10 Wang et al. [42] 2013 23/24/25/26 450–1500 450–1250 10 Huang et al. [43] 2013 23–25.01 631–1263 420–1102 6 Li et al. [44] 2013 23/24/25 607.5–1263 466–1102 6 Zhao et al. [45] 2014 25 1000 570/760 7.6 Gu et al. [46] 2015 23/25/26 780/1000 700/900 10 Shen et al. [47, 48] 2016 28.5 1536 468 19 Qu et al. [49] 2018 24/24.8/30 404/407/420/690 205/250/284/315 19

**Author Year** *<sup>P</sup>* **<sup>10</sup><sup>6</sup> [Pa]** *<sup>G</sup>* **[kg/(m<sup>2</sup>**

1967

2500 kg/(m<sup>2</sup>

Vikrev et al. [18, 19] 1964

*Advanced Supercritical Fluids Technologies*

Wang et al. [35, 36] 2010

Mokry et al. [3, 37] 2010

Li et al. [39, 40] 2011

**Table 2.**

**46**

2011

2011

2018

*Experimental data of heat transfer of SCW collected from literatures.*


**Author Year Correlation**

*Heat Transfer Correlations for Supercritical Water in Vertically Upward Tubes*

Nu <sup>¼</sup> <sup>0</sup>*:*0183 Re <sup>0</sup>*:*<sup>82</sup>

Gr <sup>∗</sup> <sup>¼</sup> *<sup>g</sup>βd*<sup>4</sup>*q=λν*<sup>2</sup>

*<sup>F</sup>*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>48</sup>

*<sup>π</sup><sup>A</sup>* <sup>¼</sup> *<sup>β</sup>bqw GCp*

Gr <sup>∗</sup> <sup>¼</sup> *<sup>g</sup>βd*<sup>4</sup>*q=λν*2, *<sup>q</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>q</sup><sup>β</sup>*

Gr <sup>∗</sup> <sup>¼</sup> *<sup>g</sup>βd*<sup>4</sup>*q=λν*<sup>2</sup>

Gr <sup>∗</sup> <sup>¼</sup> *<sup>g</sup>βd*<sup>4</sup>*q=λν*2, *<sup>q</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>q</sup><sup>β</sup>*

Nu <sup>¼</sup> <sup>0</sup>*:*032089 Re <sup>0</sup>*:*<sup>88832</sup>

Gr <sup>∗</sup> <sup>¼</sup> *<sup>g</sup>βd*<sup>4</sup>*q=λν*<sup>2</sup>

<sup>2014</sup> Nu <sup>¼</sup> <sup>0</sup>*:*00684 Re <sup>0</sup>*:*<sup>9</sup>

<sup>b</sup> Pr0*:*<sup>73</sup> b *λ*w *λ*b � �<sup>0</sup>*:*<sup>24</sup> *μ*<sup>w</sup> *μ*b � �<sup>0</sup>*:*<sup>153</sup> *ρ*<sup>w</sup>

� Gr <sup>∗</sup> ð Þ<sup>0</sup>*:*<sup>007</sup> *<sup>q</sup>* <sup>∗</sup> ð Þ<sup>0</sup>*:*<sup>041</sup>

Nu <sup>¼</sup> <sup>0</sup>*:*01 Re <sup>0</sup>*:*<sup>889</sup>

Nu <sup>¼</sup> <sup>0</sup>*:*01378 Re <sup>0</sup>*:*<sup>9078</sup>

*<sup>b</sup> Pr*<sup>0</sup>*:*<sup>9213</sup> *b*

*n* ¼ 0*:*4 for *Tb* <*Tw* <*Tpc*, 1*:*2*Tpc* <*Tb* <*Tw <sup>n</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>4</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>2</sup> *Tw=Tpc* � <sup>1</sup> � � for *Tb* <sup>&</sup>lt;*Tpc* <sup>&</sup>lt;*Tw <sup>n</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>4</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>2</sup> *Tw=Tpc* � <sup>1</sup> � � <sup>1</sup> � <sup>5</sup> *Tb=Tpc* � <sup>1</sup> � � � �

<sup>b</sup> Pr0*:*<sup>5</sup> b *ρw ρb* � �<sup>0</sup>*:*<sup>3</sup> *C*<sup>p</sup> *C*p,*<sup>b</sup>* !<sup>n</sup>

<sup>b</sup> Pr<sup>0</sup>*:*<sup>833</sup> b

� Gr <sup>∗</sup> ð Þ<sup>0</sup>*:*<sup>014</sup> *<sup>q</sup>* <sup>∗</sup> ð Þ‐0*:*<sup>021</sup>

*<sup>F</sup>*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>85</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>776</sup> *<sup>π</sup><sup>A</sup>* � <sup>103</sup> � �<sup>2</sup>*:*<sup>4</sup>

<sup>b</sup> Pr0*:*<sup>6171</sup> b

*TX* ¼ *T*<sup>b</sup> *for Tpc* � *Tb*

*TX* ¼ *Tw for Tpc* � *Tb*

*TX* ¼ *Tpc for* 0 ≤ *Tpc* � *Tb*

*GCp*

*<sup>X</sup>* Pr<sup>0</sup>*:*<sup>55</sup> *W*

*<sup>w</sup>* Pr0*:*<sup>773</sup> *w ρw ρb* � �<sup>0</sup>*:*<sup>186</sup> *<sup>μ</sup><sup>w</sup> μb* � �<sup>0</sup>*:*<sup>366</sup>

<sup>b</sup> Pr0*:*<sup>8775</sup> b

� Gr <sup>∗</sup> ð Þ<sup>0</sup>*:*<sup>0157</sup>*KV*‐0*:*<sup>031</sup>

*GCp*

<sup>b</sup> Pr0*:*<sup>38</sup> b *λw λb* � �<sup>0</sup>*:*<sup>39</sup> *<sup>μ</sup><sup>w</sup> μb* � �<sup>1</sup>*:*<sup>47</sup> *<sup>ρ</sup><sup>w</sup> ρb* � �<sup>0</sup>*:*<sup>54</sup>

<sup>b</sup> Pr0*:*<sup>71399</sup> b

> <sup>b</sup> Prb 0*:*69 *λ<sup>w</sup> λb* � �<sup>0</sup>*:*<sup>26</sup> *<sup>ρ</sup><sup>w</sup> ρb* � �<sup>0</sup>*:*<sup>31</sup>

� Gr <sup>∗</sup> ð Þ<sup>0</sup>*:*<sup>00417</sup> *<sup>q</sup>* <sup>∗</sup> ð Þ<sup>0</sup>*:*<sup>18212</sup>

, *<sup>q</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>q</sup><sup>β</sup> GCp*

*λ*w *λ*b � �<sup>0</sup>*:*<sup>03339</sup> *ρ*<sup>w</sup>

� �*=*ð Þ *Tw* � *Tb* <sup>&</sup>lt;<sup>0</sup>

� �*=*ð Þ *Tw* � *Tb* <sup>&</sup>gt;<sup>1</sup>

<sup>b</sup> Pr0*:*<sup>684</sup> b *ρw ρb* � �<sup>0</sup>*:*<sup>564</sup>

, *KV* <sup>¼</sup> <sup>4</sup>*qwDβ<sup>=</sup>* Re <sup>2</sup>

*λw λb* � �<sup>0</sup>*:*<sup>0746</sup> *ρ<sup>w</sup>*

*ρb* � �<sup>0</sup>*:*<sup>0402</sup>

*μCp* � �

> *ρ*b � �<sup>0</sup>*:*<sup>68707</sup>

*<sup>C</sup>*p,b !<sup>0</sup>*:*<sup>014</sup>

*ρ*b � �<sup>0</sup>*:*<sup>401</sup> *C*<sup>p</sup>

� �*=*ð Þ *Tw* � *Tb* <sup>≤</sup><sup>1</sup>

, *<sup>q</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>q</sup><sup>β</sup> GCp*

<sup>b</sup> Pr<sup>1</sup>*=*<sup>3</sup>

*<sup>π</sup><sup>A</sup>*,*pc* � <sup>103</sup> � �<sup>1</sup>*:*<sup>55</sup> <sup>þ</sup> <sup>1</sup>*:*21 1 � *<sup>π</sup>A=π<sup>A</sup>*,*pc* � �

*ρw ρ*b � �<sup>0</sup>*:*<sup>4356</sup>

*μw μb* � �<sup>0</sup>*:*<sup>8687</sup> *<sup>ρ</sup><sup>w</sup> ρb* � �<sup>0</sup>*:*<sup>6638</sup>

for *Tpc* <*Tb* <1*:*2*Tpc* and *Tb* <*Tw*

*ρb* � �<sup>0</sup>*:*<sup>31</sup> *<sup>μ</sup><sup>w</sup> μb* � �<sup>0</sup>*:*<sup>1</sup>

<sup>b</sup> *F*, *F* ¼ min ð Þ *F*1, *F*<sup>2</sup>

Gr <sup>∗</sup> ð Þ‐0*:*<sup>012</sup> *<sup>q</sup>* <sup>∗</sup> ð Þ<sup>0</sup>*:*<sup>0605</sup>

<sup>b</sup> Prb 0*:*5 *ρ*<sup>w</sup> *ρ*b � �<sup>0</sup>*:*<sup>3</sup>

*λw λb* � �<sup>0</sup>*:*<sup>0863</sup> *<sup>ρ</sup><sup>w</sup>*

Xu et al. [32] 2001 *Nu* <sup>¼</sup> <sup>0</sup>*:*02269 Re <sup>0</sup>*:*<sup>8079</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.89580*

Fewster et al. [59] 2004 Nu <sup>¼</sup> <sup>0</sup>*:*0183 Re <sup>0</sup>*:*<sup>82</sup>

Kuang et al. [60] 2008 Nu <sup>¼</sup> <sup>0</sup>*:*0239 Re <sup>0</sup>*:*<sup>759</sup>

Cheng et al. [61] 2009 Nu <sup>¼</sup> <sup>0</sup>*:*023 Re <sup>0</sup>*:*<sup>8</sup>

Bringer et al. [63] 2010 Nu <sup>¼</sup> <sup>0</sup>*:*0266 Re <sup>0</sup>*:*<sup>77</sup>

Gupta et al. [64] 2010 Nu <sup>¼</sup> <sup>0</sup>*:*004 Re <sup>0</sup>*:*<sup>923</sup>

Wang et al. [65] 2012 Nu <sup>¼</sup> <sup>0</sup>*:*0122 Re <sup>0</sup>*:*<sup>752</sup>

Wang et al. [42] 2013 Nu <sup>¼</sup> <sup>0</sup>*:*0182 Re <sup>0</sup>*:*<sup>8</sup>

Mokry et al. [3] 2010 Nu <sup>¼</sup> <sup>0</sup>*:*0061 Re <sup>0</sup>*:*<sup>904</sup>

Jackson [58] 2002

Yu et al. [62] 2009

Liu et al. [66] 2012

Liao et al. [9] 2014

Wang and Li et al.

[67]

**49**


*Heat Transfer Correlations for Supercritical Water in Vertically Upward Tubes DOI: http://dx.doi.org/10.5772/intechopen.89580*

**Author Year Correlation**

Nu ¼ Nu0

8

>>>>>>>>>>>>>><

>>>>>>>>>>>>>>:

*<sup>φ</sup> <sup>k</sup>* <sup>∗</sup> ð Þ¼

*n* ¼

*ϕ* ¼

*ρw ρb* � �<sup>0</sup>*:*<sup>4</sup> *Cp Cp*,*<sup>b</sup>* !*<sup>n</sup>*

*f* ¼ 1*:*82 log <sup>10</sup> Re <sup>b</sup> � ��<sup>2</sup>

> <sup>0</sup>*:*4 for *Tw Tpc*

0*:*22 þ 0*:*18

<sup>0</sup>*:*<sup>9</sup> *Tb Tpc*

> 8 >>>>>>>><

> >>>>>>>>:

Nu ¼ Re bPrbð Þ *f =*8

8 >>< >>:

*ω* ¼

Gorban et al. [8] 1990 Nu <sup>¼</sup> <sup>0</sup>*:*0059 Re <sup>0</sup>*:*<sup>9</sup>

1990

Griem et al. [28] 1996 Nu <sup>¼</sup> <sup>0</sup>*:*0169 Re <sup>0</sup>*:*<sup>84</sup>

Hu et al. [31] 2001 Nu <sup>¼</sup> <sup>0</sup>*:*<sup>0068</sup> � Re <sup>0</sup>*:*<sup>9</sup>

Kitoh et al. [57] 2001 Nu <sup>¼</sup> <sup>0</sup>*:*015 Re <sup>0</sup>*:*<sup>85</sup>

Nu <sup>¼</sup> <sup>0</sup>*:*021 Re <sup>0</sup>*:*<sup>8</sup>

Petukhov et al. [56] 1983 Nu <sup>¼</sup> Re bPrb ð Þ *<sup>f</sup> <sup>=</sup>*<sup>8</sup> *<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>900</sup>*<sup>=</sup>* Re *<sup>b</sup>* <sup>þ</sup> <sup>12</sup>*:*7ð Þ *<sup>f</sup> <sup>=</sup>*<sup>8</sup> <sup>0</sup>*:*<sup>5</sup> Pr<sup>2</sup>*=*<sup>3</sup> � <sup>1</sup>

*<sup>f</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>0</sup> *<sup>ρ</sup>w=ρ<sup>b</sup>* ð Þ<sup>0</sup>*:*<sup>4</sup> *<sup>μ</sup>w=μ<sup>b</sup>* ð Þ<sup>0</sup>*:*<sup>2</sup>

*<sup>f</sup>* <sup>0</sup> <sup>¼</sup> <sup>1</sup>*:*82 log <sup>10</sup>ð Þ Re *<sup>b</sup>=*<sup>8</sup> � ��<sup>2</sup> *<sup>ρ</sup>w=ρ<sup>b</sup>* ð Þ<sup>0</sup>*:*<sup>4</sup>

*<sup>φ</sup> <sup>k</sup>* <sup>∗</sup> ð Þ

Nu0 <sup>¼</sup> Re bPrbð Þ *<sup>f</sup> <sup>=</sup>*<sup>8</sup> *<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>900</sup>*<sup>=</sup>* Re <sup>b</sup> <sup>þ</sup> <sup>12</sup>*:*7ð Þ *<sup>f</sup> <sup>=</sup>*<sup>8</sup> <sup>0</sup>*:*<sup>5</sup> Pr2*<sup>=</sup>*<sup>3</sup>

<sup>&</sup>lt;1*:*0 and *Tb Tpc* >1*:*2

> for *Tw Tpc*

0*:*7 for *Cp=Cp*,*<sup>b</sup>* ≥1*:*0 other conditions

<sup>1</sup>*:*<sup>4</sup> *<sup>k</sup>* <sup>∗</sup> ð Þ<sup>0</sup>*:*<sup>37</sup> for *<sup>k</sup>* <sup>∗</sup> <sup>&</sup>gt;0*:*<sup>4</sup>

*C*<sup>p</sup> *Cp*,*<sup>b</sup>* !<sup>0</sup>*:*<sup>65</sup>

*<sup>f</sup>* <sup>¼</sup> <sup>1</sup>*:*<sup>82</sup> � log <sup>10</sup>ð Þ Re *<sup>b</sup>=*<sup>8</sup> � �‐<sup>2</sup> *<sup>ρ</sup>*w*=ρ*<sup>b</sup> ð Þ� *<sup>μ</sup>*w*=μ*<sup>b</sup> ð Þ ð Þ <sup>0</sup>*:*<sup>18</sup>

<sup>b</sup> Pr<sup>0</sup>*:*<sup>432</sup> *sel ω*

0*:*82 for *hb* <1540kJ*=*kg 1 for *hb* >1740kJ*=*kg <sup>0</sup>*:*<sup>82</sup> <sup>þ</sup> <sup>9</sup> � <sup>10</sup>‐<sup>4</sup> for 1540kJ*=*kg≤*h*<sup>b</sup> <sup>≤</sup>1740kJ*=*kg

<sup>b</sup> � Pr<sup>0</sup>*:*<sup>63</sup>

Pr*sel* ¼ *μbC*p,sel*=λm*, *λ<sup>m</sup>* ¼ 0*:*5ð Þ *λ<sup>w</sup>* þ *λ<sup>b</sup>*

For detailed of *C*p,sel, please see 64 ½ �

<sup>b</sup> Pr*<sup>m</sup>*

*<sup>f</sup> <sup>c</sup>* <sup>¼</sup> <sup>2</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>8</sup> <sup>þ</sup> <sup>0</sup>*:*11*=*200*G*<sup>1</sup>*:*<sup>2</sup>

*<sup>f</sup> <sup>c</sup>* ¼ �8*:*<sup>7</sup> � <sup>10</sup>�<sup>8</sup> � <sup>0</sup>*:*65*=*200*G*<sup>1</sup>*:*<sup>2</sup>

*<sup>f</sup> <sup>c</sup>* ¼ �9*:*<sup>7</sup> � <sup>10</sup>�<sup>7</sup> � <sup>1</sup>*:*3*=*200*G*<sup>1</sup>*:*<sup>2</sup>

*Tw Tpc*

<sup>1</sup> � *Tw Tpc* � �

, *<sup>k</sup>* <sup>∗</sup> <sup>¼</sup> <sup>1</sup> � *<sup>ρ</sup><sup>w</sup> ρb* � � Grb Re <sup>2</sup> b

> <sup>þ</sup> <sup>1</sup>*:*<sup>08</sup> *Tw Tpc*

<sup>0</sup>*:*<sup>8</sup> � <sup>1</sup>*:*65 ln *<sup>k</sup>* <sup>∗</sup> � <sup>2</sup>*:*75 ln *<sup>k</sup>* <sup>∗</sup> ð Þ<sup>2</sup> � <sup>1</sup>*:*74 ln *<sup>k</sup>* <sup>∗</sup> ð Þ<sup>3</sup>

�0*:*55 ln *<sup>k</sup>* <sup>∗</sup> ð Þ<sup>4</sup> � <sup>0</sup>*:*09 ln *<sup>k</sup>* <sup>∗</sup> ð Þ<sup>5</sup> � <sup>0</sup>*:*0056 ln *<sup>k</sup>* <sup>∗</sup> ð Þ<sup>6</sup>

8 >>< >>: Bu <sup>¼</sup> Gr Re <sup>2</sup>*:*<sup>7</sup> Pr<sup>0</sup>*:*<sup>5</sup>

<sup>b</sup> Prb

<sup>0</sup>*:*<sup>55</sup> *ρ<sup>w</sup> ρb* � �<sup>0</sup>*:*<sup>35</sup> *ϕ*

1 for Bu≤10�<sup>5</sup> ð Þ <sup>1</sup> � 3000Bu <sup>0</sup>*:*<sup>295</sup> for 10�<sup>5</sup> <sup>&</sup>lt;Bu <sup>≤</sup>10�<sup>4</sup> ð Þ 7000Bu <sup>0</sup>*:*<sup>295</sup> for Bu≥10�<sup>4</sup>

� � � �

<sup>&</sup>lt;1*:*0 and *Tb Tpc* <1*:*0

> � <sup>0</sup>*:*68 for *Tw Tpc*

> > <sup>b</sup> Pr�0*:*<sup>12</sup> <sup>b</sup>

8 � �<sup>0</sup>*:*<sup>5</sup>

<sup>b</sup> � *<sup>ρ</sup>w=ρ<sup>b</sup>* ð Þ<sup>0</sup>*:*<sup>17</sup> � ð Þ *<sup>λ</sup>w=λ<sup>b</sup>* <sup>0</sup>*:*<sup>29</sup>

for 1*:*<sup>5</sup> � 106J*=*kg≤*hb* <sup>≤</sup>3*:*<sup>3</sup> � 106J*=*kg

for 3*:*<sup>3</sup> � <sup>10</sup>6J*=*kg≤*hb* <sup>≤</sup><sup>4</sup> � 106J*=*kg

<sup>b</sup> , *<sup>m</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>69</sup> � <sup>81000</sup>*=*200*G*<sup>1</sup>*:*<sup>2</sup> <sup>þ</sup> *<sup>f</sup> <sup>c</sup><sup>q</sup>*

for 0 <sup>≤</sup>*hb* <sup>≤</sup>1*:*<sup>5</sup> � 106J*=*kg

� � " #

� *Pr*<sup>2</sup>*=*<sup>3</sup> *<sup>b</sup>* � 1

*<sup>=</sup>* <sup>1</sup>*:*<sup>07</sup> <sup>þ</sup> <sup>12</sup>*:*<sup>7</sup> � *<sup>f</sup>*

h i � �

<sup>b</sup> � 1

<sup>&</sup>lt;1*:*0 and *Tb Tpc* <1*:*2

for *k* <sup>∗</sup> ≤0*:*4

, Grb <sup>¼</sup> *<sup>g</sup>* <sup>1</sup> � *<sup>ρ</sup>w=ρ<sup>b</sup>* ð Þ*D*<sup>3</sup> *ν*2 *b*

1982

*Advanced Supercritical Fluids Technologies*

Watts-Chou et al.

Kirillov et al. [41] 1990

Razumovskiy et al.

[41]

**48**

[55]


**3.2 Comparison of the predicted values by using existing correlations to the**

*Heat Transfer Correlations for Supercritical Water in Vertically Upward Tubes*

As described in the preceding section, the existing heat transfer correlations listed in **Table 3** could generally be divided into three categories (see **Table 1**). For simplicity, two representative correlations are selected from each category of the

**Figure 1(a)** shows the comparison of the results calculated by the first type of correlations for heat transfer of SCW to the experimental data. This type of correlations is represented by the Dittus-Boelter et al.'s correlation [10] and Gorban

*Comparison of predicted values of correlations with all experimental data. (a) First type of correlations,*

*(b) second type of correlations, (c) third type of correlations.*

**experimental data**

*DOI: http://dx.doi.org/10.5772/intechopen.89580*

correlations.

**Figure 1.**

**51**

**Table 3.**

*Existing heat transfer correlations for SCW in vertical upward tubes.*

might have been adopted in the development of those correlations. Kurganov et al. [13] have pointed out that some correlations developed on the basis of the old thermophysical property standard for SCW have become quite impractical when the thermophysical property shifts to IAPWS-97 Standard. From this point of view, accurate correlations for heat transfer of SCW must be developed by using the updated properties database.
