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

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 correlations.

**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

### **Figure 1.**

*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.*

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

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

*F*<sup>1</sup> ¼ 0*:*62 þ 0*:*06 ln ð Þ *π<sup>B</sup> <sup>F</sup>*<sup>2</sup> <sup>¼</sup> <sup>11</sup>*:*46 ln ð Þ *<sup>π</sup><sup>B</sup>* �1*:*<sup>04</sup> *<sup>π</sup><sup>B</sup>* <sup>¼</sup> *<sup>β</sup>*b*qwD λb*

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

**Author Year Correlation**

Zhao et al. [45] 2014 Nu <sup>¼</sup> <sup>0</sup>*:*023 Re <sup>0</sup>*:*<sup>8</sup>

*Advanced Supercritical Fluids Technologies*

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

**3. Assessment of the prediction performance of the existing heat**

As seen in **Table 3**, the heat transfer correlations for SCW have been proposed in different years and might be developed on different basis of experimental data. As a result, the applicability of each correlation might be different. As reported by Pioro et al. [8] and Lei et al. [2], there exist distinct discrepancies between the results predicted by different correlations. It is necessary to quantitatively evaluate the

In this part, the prediction performance of the existing heat transfer correlations are quantitatively estimated by introducing four parameters, i.e., *σ*<sup>1</sup> (mean relative deviation, MRD), *σ*<sup>2</sup> (mean absolute deviation, MAD), *σ*<sup>3</sup> (standard deviation, SD), and *ρ*xy (correlation coefficient between the predicted values and experimental

*ei=N* (1)

∣*ei*∣*=N* (2)

� �*=*Nuexp (4)

*D X*ð Þ p ffiffiffiffiffiffiffiffiffiffiffi *D Y*ð Þ <sup>p</sup> (5)

(3)

*<sup>σ</sup>*<sup>1</sup> <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼1

*<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼1

X*<sup>n</sup> i*¼1

*ei* <sup>¼</sup> Nucal‐Nuexp

*<sup>ρ</sup>xy* <sup>¼</sup> *Cov X*ð Þ , *<sup>Y</sup>* ffiffiffiffiffiffiffiffiffiffiffi

where *D*(*X*) refers to the variance of the experimental data *X*, *D*(*Y*) refers to the variance of the calculated results Y, and *Cov*(*X*, *Y*) is the covariation of *X* and *Y* [9].

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ *ei* � *σ*<sup>1</sup> *=*ð Þ *N* � 1

updated properties database.

**Table 3.**

**transfer correlations**

prediction performance of the existing correlations.

values), as defined by Eq. (1) through Eq. (5).

*σ*<sup>3</sup> ¼

The closer the *ρ*xy is to 1.0, the better the correlation is [9].

**3.1 Assessment method**

where *e*<sup>i</sup> is

**50**

et al.'s correlation [8] here. It is seen from **Figure 1(a)** that most of the **Nu** values predicted by the correlations are out of 20% error band, indicating a generally low prediction accuracy of this type of correlations. As explained in the preceding sections, no consideration in these correlations of the dramatic change of thermophysical properties of SCW in the vicinity of its pseudocritical point might be responsible for the low prediction accuracy of the correlations.

**Figure 1(b)** gives the representative comparison of the results calculated by the second type of correlations for SCW to the experimental data. This type of correlations is represented by the Mokry et al. [3] and Xu [32] correlations here. It can be seen from **Figure 1(b)** that most of predicted results by the correlations concentrate around the line of **Nucal** = **Nuexp**, suggesting a remarkable improvement in the prediction accuracy of the correlations in comparison to that of the first type of the correlations. This improvement in the prediction accuracy might due to introducing thermophysical properties correction terms into the correlations.

Comparison of the results calculated by the third type correlations to the experimental data are illustrated in **Figure 1(c)**. Here, the third type correlations are represented by the Kuang et al.'s correlation [60] and Liu's correlation [66].

It is seen in **Figure 1(b)**, (c) that the prediction accuracy of the Mokry et al.'s correlation [3], Xu's correlation [32], and Liu's correlation [66] are in roughly the same level. It is surprising that although other two more correction factors (i.e., **Gr\*** and *q***<sup>+</sup>** ) are introduced into the correlations, the prediction accuracy of Kuang et al.'s correlation [60] is unexpectedly worse than that of the Mokry et al.'s correlation [3] and Xu's correlation [32] in the present study. Another two correlations, i.e., Yu et al.'s correlation [62] and Liao's correlation [9], which are of forms similar to that of Kuang et al.'s correlation [60], provide similar prediction performance to that of Kuang et al.'s correlation [60] in this study. This result indicates that adding more correction factors in the correlations does not always produce better accuracy, and the correction terms added to the correlations should be selected carefully.

**Figures 2**, **3** depict the prediction performance of the 34 existing correlations listed in **Table 3**, under the EHT condition (experimental data from Yamagata et al. [25] is used) and the DHT condition (experimental data from Herkenrath et al. [21] is used), respectively.

It is seen from **Figure 2** that under the EHT conditions, most of existing correlations could provide relatively good prediction accuracy in the enthalpy region lower than 1600 kJ/kg and the region higher than 2800 kJ/kg of SCW (i.e., in the regions far away from the pseudocritical point). However, in the enthalpy region of 1600–2300 kJ/kg (a region around the pseudocritical point, named in lots of papers as the large specific heat region, and is hereafter abbreviated as LSHR), the predicated values of many heat transfer correlations, such as Yamagata et al.'s correlation [25], Domin et al. correlation [51] and Swenson et al. correlation [17], are much higher than the corresponding experimental values, implying low prediction capability of these correlations in the LSHR. Careful analysis of the forms of Yamagata et al. correlation [25], Domin's correlation [51], and Swenson et al.'s correlation [17] shows that only one thermophysical correction property factor is employed in the abovementioned three correlations. None of the 34 correlations could give good prediction accuracy in the whole enthalpy region of SCW. It is well known that the thermophysical properties of SCW experience dramatic change in the vicinity of pseudocritical point (i.e., in the LSHR), and with this view in mind, it is suggested that one thermophysical property correction factor might not be sufficient, and, however, proper and enough correction factors should be used in the corrections in order to capture the effect of the dramatic variation in thermophysical properties on the heat transfer of SCW in LSHR.

**Figure 3** shows that most of the 34 heat transfer correlations for SCW could not provide high prediction accuracy under DHT conditions, except for Grass et al.'s correlation [53] and Petukhov et al.'s correlation [56]. It is seen from **Figure 3** that the predicted results of Grass et al.'s correlation [53] and Petukhov et al.'s correlation [56] agree pretty well with the corresponding experimental data under DHT conditions through the whole enthalpy region of SCW studied here. Unfortunately, as seen in **Figure 2**, the prediction results of the above two correlations do not agree satisfactorily with the corresponding experimental data under the EHT conditions, especially in the vicinity of pseudocritical point (in the LSHR). The reason for this result might be attributed to that most of the experimental data adopted in the development Petukhov et al.'s correlation [56] were under DHT regimes. Little

*Comparison of the calculated values to the corresponding experimental data (Yamagata et al. [25], EHT case). (a) Prediction results of the first 14 correlations in Table 3 (b)Prediction results of the last 20 correlations in*

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

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

**Figure 2.**

*Table 3.*

**53**

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

### **Figure 2.**

et al.'s correlation [8] here. It is seen from **Figure 1(a)** that most of the **Nu** values predicted by the correlations are out of 20% error band, indicating a generally low prediction accuracy of this type of correlations. As explained in the preceding sections, no consideration in these correlations of the dramatic change of

thermophysical properties of SCW in the vicinity of its pseudocritical point might

**Figure 1(b)** gives the representative comparison of the results calculated by the second type of correlations for SCW to the experimental data. This type of correlations is represented by the Mokry et al. [3] and Xu [32] correlations here. It can be seen from **Figure 1(b)** that most of predicted results by the correlations concentrate around the line of **Nucal** = **Nuexp**, suggesting a remarkable improvement in the prediction accuracy of the correlations in comparison to that of the first type of the correlations. This improvement in the prediction accuracy might due to introducing thermophysical properties correction terms into the

Comparison of the results calculated by the third type correlations to the experimental data are illustrated in **Figure 1(c)**. Here, the third type correlations are represented by the Kuang et al.'s correlation [60] and Liu's correlation [66].

It is seen in **Figure 1(b)**, (c) that the prediction accuracy of the Mokry et al.'s correlation [3], Xu's correlation [32], and Liu's correlation [66] are in roughly the same level. It is surprising that although other two more correction factors (i.e., **Gr\***

) are introduced into the correlations, the prediction accuracy of Kuang et al.'s correlation [60] is unexpectedly worse than that of the Mokry et al.'s correlation [3] and Xu's correlation [32] in the present study. Another two correlations, i.e., Yu et al.'s correlation [62] and Liao's correlation [9], which are of forms similar to that of Kuang et al.'s correlation [60], provide similar prediction performance to that of Kuang et al.'s correlation [60] in this study. This result indicates that adding more correction factors in the correlations does not always produce better accuracy, and the correction terms added to the correlations should be selected carefully. **Figures 2**, **3** depict the prediction performance of the 34 existing correlations listed in **Table 3**, under the EHT condition (experimental data from Yamagata et al. [25] is used) and the DHT condition (experimental data from Herkenrath et al. [21]

It is seen from **Figure 2** that under the EHT conditions, most of existing correlations could provide relatively good prediction accuracy in the enthalpy region lower than 1600 kJ/kg and the region higher than 2800 kJ/kg of SCW (i.e., in the regions far away from the pseudocritical point). However, in the enthalpy region of 1600–2300 kJ/kg (a region around the pseudocritical point, named in lots of papers as the large specific heat region, and is hereafter abbreviated as LSHR), the predicated values of many heat transfer correlations, such as Yamagata et al.'s correlation [25], Domin et al. correlation [51] and Swenson et al. correlation [17], are much higher than the corresponding experimental values, implying low prediction capability of these correlations in the LSHR. Careful analysis of the forms of Yamagata et al. correlation [25], Domin's correlation [51], and Swenson et al.'s correlation [17] shows that only one thermophysical correction property factor is employed in the abovementioned three correlations. None of the 34 correlations could give good prediction accuracy in the whole enthalpy region of SCW. It is well known that the thermophysical properties of SCW experience dramatic change in the vicinity of pseudocritical point (i.e., in the LSHR), and with this view in mind, it is suggested that one thermophysical property correction factor might not be sufficient, and, however, proper and enough correction factors should be used in the corrections in order to capture the effect of the dramatic variation in thermophysical properties on

be responsible for the low prediction accuracy of the correlations.

*Advanced Supercritical Fluids Technologies*

correlations.

and *q***<sup>+</sup>**

is used), respectively.

the heat transfer of SCW in LSHR.

**52**

*Comparison of the calculated values to the corresponding experimental data (Yamagata et al. [25], EHT case). (a) Prediction results of the first 14 correlations in Table 3 (b)Prediction results of the last 20 correlations in Table 3.*

**Figure 3** shows that most of the 34 heat transfer correlations for SCW could not provide high prediction accuracy under DHT conditions, except for Grass et al.'s correlation [53] and Petukhov et al.'s correlation [56]. It is seen from **Figure 3** that the predicted results of Grass et al.'s correlation [53] and Petukhov et al.'s correlation [56] agree pretty well with the corresponding experimental data under DHT conditions through the whole enthalpy region of SCW studied here. Unfortunately, as seen in **Figure 2**, the prediction results of the above two correlations do not agree satisfactorily with the corresponding experimental data under the EHT conditions, especially in the vicinity of pseudocritical point (in the LSHR). The reason for this result might be attributed to that most of the experimental data adopted in the development Petukhov et al.'s correlation [56] were under DHT regimes. Little

### **Figure 3.**

*Comparison of the calculated values to the corresponding experimental data (Herkenrath et al. [21], DHT case). (a) Prediction results of the first 14 correlations in Table 3 (b) Prediction results of the last 20 correlations in Table 3*

information can be found about the experimental data used for developing Grass et al.'s correlation [53].

It is confusing that under EHT regimes, the prediction results of Miropolskii and Shitsman correlation [50] and Yu et al.'s correlation [62] agree relatively well with the corresponding experimental (see **Figure 2**) data, however, under DHT regimes, the prediction results of these two correlations are in quite different trend from that of the experimental data, indicating that these two correlations could not capture the variation characteristics of the experimental data under the DHT regimes (see **Figure 3**). As reported early by Pioro et al. [8] and Lei et al. [2], experiment data under DHT regime were excluded in the development of many of the heat transfer correlations, and this exclusion of the data under DHT regimes results in the low prediction accuracy under DHT regimes. It is seen again from

the above results that the applicable scope of each heat transfer correlation is limited by the scope of the experimental database employed. Thus, it is of great importance to develop a new heat transfer correlation with high prediction accuracy

*Quantitative analysis of the existing heat transfer correlations.*

**Table 4.**

**55**

**Author** *σ***<sup>1</sup> (%)** *σ***<sup>2</sup> (%)** *σ***<sup>3</sup> (%)** *ρ***xy % of data within the error bands**

Dittus-Boelter et al. [10] 63.65 73.22 130.58 0.63 24.09 44.18 56.42 Shitsman et al. [50] 41.26 46.01 62.73 0.87 27.86 47.54 58.34 Petukhov et al. [56] 21.87 30.45 43.89 0.88 33.00 56.49 69.16 Domin et al. [51] 104.07 108.80 103.59 0.66 10.47 20.58 27.29 Swenson et al. [17] 124.64 127.80 122.01 0.67 14.40 24.35 30.17 Krasnoschekov et al. [52] 8.38 38.51 48.32 0.73 15.51 31.10 47.13 Kondratev et al. [22] 3.05 36.60 55.67 0.71 22.39 41.17 57.53 Ornatsky et al. [24] 8.51 25.29 38.31 0.90 35.26 59.41 73.23 Grass et al. [53] 12.90 25.84 35.85 0.87 33.96 58.78 71.51 Yamagata et al. [25] 224.43 226.89 461.56 0.42 15.67 28.81 37.73 Yeroshenko et al. [54] 9.98 21.85 30.41 0.90 34.45 59.81 76.28 Watts-Chou et al. [55] 101.98 106.21 113.46 0.69 14.95 26.94 33.73 Petukhov et al. [56] 2.66 15.03 20.42 0.92 46.26 74.20 87.30 Bringer et al. [63] �8.17 33.42 45.33 0.70 20.84 39.92 56.12 Gorban et al. [8] 13.86 23.62 29.23 0.89 32.34 55.74 71.14 Razumovskiy et al. [41] 9.37 39.78 51.68 0.70 14.83 29.87 47.67 Kirillov et al. [41] 14.23 37.20 58.65 0.77 26.07 47.14 63.61 Griem et al. [28] 9.44 17.16 22.19 0.91 42.71 70.55 82.98 Hu et al. [31] 29.98 40.05 56.33 0.83 21.21 40.03 57.30 Kitoh et al. [57] �14.18 20.45 22.89 0.93 39.15 58.84 72.20 Xu et al. [32] 52.14 69.88 133.56 0.65 22.48 41.18 56.38 Jackson [58] 16.39 24.13 29.53 0.90 31.54 55.12 73.00 Fewster et al. [59] �0.42 16.41 22.62 0.93 45.37 68.69 83.67 Kuang et al. [60] 78.42 85.38 153.65 0.61 21.99 39.65 53.28 Cheng et al. [61] �11.99 17.94 18.53 0.90 30.75 61.90 84.39 Yu et al. [62] 78.11 87.44 103.32 0.59 12.91 25.36 35.84 Gupta et al. [64] 104.79 111.99 111.23 0.70 13.59 25.09 33.42 Mokry et al. [3] �5.03 13.26 16.58 0.93 46.04 78.81 93.30 Wang et al. [65] 22.92 39.26 44.58 0.80 14.26 30.07 49.74 Liu et al. [66] 6.10 14.81 19.29 0.93 45.46 75.86 88.79 Liao et al. [9] �54.71 60.61 37.7 0.69 3.41 7.38 14.33 Wang et al. [42] 6.19 16.14 20.47 0.94 40.22 71.62 86.90 Wang and Li [67] 0.39 13.33 18.45 0.92 50.56 79.01 90.48 Zhao et al. [45] 60.03 76.49 146.38 0.59 23.65 44.20 57.80

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

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

�**10%** �**20%** �**30%**


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

### **Table 4.**

information can be found about the experimental data used for developing Grass

*Comparison of the calculated values to the corresponding experimental data (Herkenrath et al. [21], DHT case). (a) Prediction results of the first 14 correlations in Table 3 (b) Prediction results of the last 20*

It is confusing that under EHT regimes, the prediction results of Miropolskii and Shitsman correlation [50] and Yu et al.'s correlation [62] agree relatively well with the corresponding experimental (see **Figure 2**) data, however, under DHT regimes, the prediction results of these two correlations are in quite different trend from that of the experimental data, indicating that these two correlations could not capture the variation characteristics of the experimental data under the DHT regimes (see **Figure 3**). As reported early by Pioro et al. [8] and Lei et al. [2], experiment data under DHT regime were excluded in the development of many of the heat transfer correlations, and this exclusion of the data under DHT regimes results in the low prediction accuracy under DHT regimes. It is seen again from

et al.'s correlation [53].

*Advanced Supercritical Fluids Technologies*

*correlations in Table 3*

**Figure 3.**

**54**

*Quantitative analysis of the existing heat transfer correlations.*

the above results that the applicable scope of each heat transfer correlation is limited by the scope of the experimental database employed. Thus, it is of great importance to develop a new heat transfer correlation with high prediction accuracy over a wide range of experimental parameters covering the NHT regime, the EHT regime, and the DHT regime [4].

In order to gain comprehensive understanding of prediction performance of the existing correlations, quantitative analyses are conducted by employing four parameters, i.e., *σ*1, *σ*2, *σ*3, and *ρ*xy as defined in the previous section of this chapter, and the results of *σ*1, *σ*2, *σ*3, and *ρ*xy for each correlation are listed in **Table 4**.

It is seen from **Table 4** that the predicted values of most of the existing correlations falling into the �10% error band are lower than 50%, and that falling into the �30% error band are lower than 90%, indicating that no of these heat transfer correlations could give satisfactory predict accuracy under both DHT and EHT regimes. Generally, the prediction performance of Mokry et al.'s correlation [3]and Wang and Li's correlation [67] are compareatively the best among the 34 correlations. More than 90% of the calculated values of these two correlations fall into the �30% error band. However, it should be noted that only about 50% of the calculated values of these two correlations fall into the �10% error band, and the prediction accuracy the correlations needs to be improved further.

The Domin's correlation [51] and the Swenson et al.'s correlation [17] exhibit special prediction features, as seen in **Figure 2**. It is seen from **Table 4** that the predicted values of Domin's correlation [51] and Swenson et al.'s correlation [17] falling into �30% error band are lower than 50%. This result is in accordance with that in **Figure 2**, that is, under EHT conditions, the prediction accuracy of these correlations are quite low in the LSHR of SCW although relatively good prediction performance is observed in the enthalpy region lower than 1600 kJ/kg and higher than 2800 kJ/kg of SCW. As seen in **Figure 3**, the Domin's correlation [51] and Swenson et al.'s correlation [17] could not provide prediction accuracy either under DHT conditions.

### **4. Development of the new heat transfer correlation**

As we discussed earlier, insufficient description of the severe variation of thermophysical properties of SCW in the LSHR is one of the main reasons for the low prediction accuracy of the existing heat transfer correlations. Proper correction terms should be selected carefully to reflect the impact of thermophysical properties of SCW in LSHR on the heat transfer.

Based on the analysis of the existing heat transfer correlations, a general form of heat transfer correlations for SCW is suggested as follows:

$$\mathbf{Nu} = \mathbf{C}\_0 \operatorname{Re}\_b^{C\_1} \mathbf{Pr}\_b^{C\_2} F\_1^{k\_1} \cdots F\_n^{k\_n} \tag{6}$$

[68], four correction terms are finally adopted in the new heat transfer

*Comparison between experimental and calculated Nu using new heat transfer correlation.*

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

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

*ρw ρb* � �

1

Experimental data including both the DHT regimes and the EHT regimes (as listed in **Table 2**) are adopted simultaneously in this study to develop the new heat transfer correlation, in hope to get high prediction accuracy under both the DHT and the EHT regimes. Based on the experimental database (**Table 2**), the constant indices in Eq. (7) are determined via the Levenberg– Marquardt method [69]. The final new heat transfer correlation for SCW is

*<sup>k</sup>*<sup>1</sup> *ν<sup>w</sup> νb*

**Correlation** *σ***1(%)** *σ***2(%)** *σ***3(%)** *ρ***xy % of data within the error bands**

New Correlation 0.77 6.71 9.23 0.991 78.57 95.18 99.39

*νw νb*

**Figure 4** displays the comparison of the calculated **Nu** by using the new heat transfer correlation to the corresponding experimental results. It is seen from **Figure 4** that the relative errors of most of the calculated values are much lower than �20%. **Table 5** gives the quantitative evaluation of the prediction performance

� ��0*:*<sup>97</sup> *Cp*

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

!<sup>0</sup>*:*<sup>86</sup>

� �*<sup>k</sup>*<sup>2</sup> *Cp*

*Cp*,*<sup>b</sup>* !*<sup>k</sup>*<sup>3</sup>

Gr <sup>∗</sup> Grb � �*<sup>k</sup>*<sup>4</sup>

�**10%** �**20%** �**30%**

Gr <sup>∗</sup> Grb � �<sup>0</sup>*:*<sup>92</sup> (7)

(8)

<sup>b</sup> Pr*<sup>C</sup>*<sup>2</sup> b

Nu <sup>¼</sup> *<sup>C</sup>*<sup>0</sup> Re *<sup>C</sup>*<sup>1</sup>

*Quantitative analysis of the new heat transfer correlation.*

<sup>b</sup> Pr�0*:*<sup>2</sup> b

*ρw ρb* � ��0*:*<sup>26</sup>

correlation, i.e.,

**Figure 4.**

**Table 5.**

obtained as Eq. (8):

**57**

Nu <sup>¼</sup> <sup>0</sup>*:*83 Re <sup>0</sup>*:*<sup>062</sup>

where *F*<sup>1</sup> ��� *F*<sup>n</sup> are correction terms, defined as one of the parameters such as *ρ*w/*ρ*b, *μ*w/*μ*b, *λ*w/*λ*b, *C*p, and **Gr\***, and so on, and *C*0, *C*1, *C*2, *k*1, ��� *k*<sup>2</sup> are constant indices for the corresponding correction terms. As mentioned in the previous section, one thermophysical correction factor might not be sufficient to capture the effect of the severe variation of thermophysical properties of SCW on the heat transfer in LSHR. On the other hand, introducing too many correction terms into the correlation does not always mean a high prediction accuracy. It was shown that strong linear correlations existed among the correction terms, and such linear correlation might limit, even reduce, the prediction performance of the correlations [68]. Based on the multicollinearity analysis as conducted in

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

### **Figure 4.**

over a wide range of experimental parameters covering the NHT regime, the EHT

In order to gain comprehensive understanding of prediction performance of the existing correlations, quantitative analyses are conducted by employing four parameters, i.e., *σ*1, *σ*2, *σ*3, and *ρ*xy as defined in the previous section of this chapter, and the results of *σ*1, *σ*2, *σ*3, and *ρ*xy for each correlation are listed in

It is seen from **Table 4** that the predicted values of most of the existing correlations falling into the �10% error band are lower than 50%, and that falling into the �30% error band are lower than 90%, indicating that no of these heat transfer correlations could give satisfactory predict accuracy under both DHT and EHT regimes. Generally, the prediction performance of Mokry et al.'s correlation [3]and Wang and Li's correlation [67] are compareatively the best among the 34 correlations. More than 90% of the calculated values of these two correlations fall into the �30% error band. However, it should be noted that only about 50% of the calculated values of these two correlations fall into the �10% error band, and the prediction accuracy the correlations needs to be

The Domin's correlation [51] and the Swenson et al.'s correlation [17] exhibit special prediction features, as seen in **Figure 2**. It is seen from **Table 4** that the predicted values of Domin's correlation [51] and Swenson et al.'s correlation [17] falling into �30% error band are lower than 50%. This result is in accordance with that in **Figure 2**, that is, under EHT conditions, the prediction accuracy of these correlations are quite low in the LSHR of SCW although relatively good prediction performance is observed in the enthalpy region lower than 1600 kJ/kg and higher than 2800 kJ/kg of SCW. As seen in **Figure 3**, the Domin's correlation [51] and Swenson et al.'s correlation [17] could not provide prediction accuracy either under

As we discussed earlier, insufficient description of the severe variation of thermophysical properties of SCW in the LSHR is one of the main reasons for the low prediction accuracy of the existing heat transfer correlations. Proper correction terms should be selected carefully to reflect the impact of thermophysical properties

Based on the analysis of the existing heat transfer correlations, a general form of

*<sup>b</sup>* Pr*<sup>C</sup>*<sup>2</sup> *<sup>b</sup> F*<sup>1</sup>

where *F*<sup>1</sup> ��� *F*<sup>n</sup> are correction terms, defined as one of the parameters such as *ρ*w/*ρ*b, *μ*w/*μ*b, *λ*w/*λ*b, *C*p, and **Gr\***, and so on, and *C*0, *C*1, *C*2, *k*1, ��� *k*<sup>2</sup> are constant indices for the corresponding correction terms. As mentioned in the previous section, one thermophysical correction factor might not be sufficient to capture the effect of the severe variation of thermophysical properties of SCW on the heat transfer in LSHR. On the other hand, introducing too many correction terms into the correlation does not always mean a high prediction accuracy. It was shown that strong linear correlations existed among the correction terms, and such linear correlation might limit, even reduce, the prediction performance of the correlations [68]. Based on the multicollinearity analysis as conducted in

*<sup>k</sup>*1⋯*Fkn*

*<sup>n</sup>* (6)

**4. Development of the new heat transfer correlation**

heat transfer correlations for SCW is suggested as follows:

Nu <sup>¼</sup> *<sup>C</sup>*<sup>0</sup> Re *<sup>C</sup>*<sup>1</sup>

of SCW in LSHR on the heat transfer.

regime, and the DHT regime [4].

*Advanced Supercritical Fluids Technologies*

**Table 4**.

improved further.

DHT conditions.

**56**

*Comparison between experimental and calculated Nu using new heat transfer correlation.*


**Table 5.**

*Quantitative analysis of the new heat transfer correlation.*

[68], four correction terms are finally adopted in the new heat transfer correlation, i.e.,

$$\mathbf{Nu} = \mathbf{C\_0} \, \text{Re}\_\mathbf{b}^{C\_1} \text{Pr}\_\mathbf{b}^{C\_2} \left(\frac{\rho\_w}{\rho\_b}\right)\_1^{k\_1} \left(\frac{\nu\_w}{\nu\_b}\right)^{k\_2} \left(\frac{\overline{\mathbf{C\_p}}}{\mathbf{C\_{p,b}}}\right)^{k\_3} \left(\frac{\mathbf{Gr}^\*}{\mathbf{Gr\_b}}\right)^{k\_4} \tag{7}$$

Experimental data including both the DHT regimes and the EHT regimes (as listed in **Table 2**) are adopted simultaneously in this study to develop the new heat transfer correlation, in hope to get high prediction accuracy under both the DHT and the EHT regimes. Based on the experimental database (**Table 2**), the constant indices in Eq. (7) are determined via the Levenberg– Marquardt method [69]. The final new heat transfer correlation for SCW is obtained as Eq. (8):

$$\mathbf{Nu} = \mathbf{0.83} \,\mathrm{Re}\_{\mathrm{b}}^{0.062} \mathrm{Pr}\_{\mathrm{b}}^{-0.2} \left(\frac{\rho\_w}{\rho\_b}\right)^{-0.26} \left(\frac{\nu\_w}{\nu\_b}\right)^{-0.97} \left(\frac{\overline{\mathbf{C}\_p}}{\mathbf{C}\_{p,b}}\right)^{0.86} \left(\frac{\mathbf{Gr}^\*}{\mathbf{Gr}\_b}\right)^{0.92} \tag{8}$$

**Figure 4** displays the comparison of the calculated **Nu** by using the new heat transfer correlation to the corresponding experimental results. It is seen from **Figure 4** that the relative errors of most of the calculated values are much lower than �20%. **Table 5** gives the quantitative evaluation of the prediction performance of the new heat transfer correlation. It is clearly seen from **Table 5** that the new correlation has a mean absolute deviation of 9.23%, and 78.57% of the predicted results fall into the 10% error band, and 95.18% of the predicted results fall into the 20% error band, and 99.39% of the predicted results fall into the 30% error band, indicating that the prediction accuracy of the new correlation is better than that of the existing heat transfer correlations.

Nu Nusselt number *P* Pressure, Pa Pr Prandtl number

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

Re Reynolds number

*t* Celsius temperature, °C *σ*<sup>1</sup> mean relative deviation, MRD *σ*<sup>2</sup> mean absolute deviation, MAD

*σ*<sup>3</sup> standard deviation, SD *F* correction terms

*μ* dynamic viscosity, Pas *ν* kinematic viscosity, m<sup>2</sup>

*ρ*xy correlation coefficient

DHT deteriorated heat transfer EHT enhanced heat transfer LSHR the large specific heat region MAD mean absolute deviation MRD mean relative deviation NHT normal heat transfer SD standard deviation

SCFs supercritical pressure fluids

SCWRs supercritical water-cooled reactors

SCW supercritical water

*ρ* density, kg/m<sup>3</sup>

*cal* calculated value *exp* experimental value *pc* pseudo-critical point *w* at inner wall temperature

**Greek symbols**

**Subscripts**

**Abbreviations**

**59**

Pr average Prandtl number, *Cp*,*bμb=λ<sup>b</sup> q\** buoyance correction factor, *qwβ=GCp*,*<sup>b</sup> <sup>q</sup><sup>+</sup>* thermal acceleration parameter, *qw <sup>β</sup>=Cp*

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

*T* thermodynamic temperature, K

*α* heat transfer coefficient, W/(m2

*β* thermal expansion coefficient, K<sup>1</sup> *λ* thermal conductivity W/(mK)

*<sup>ρ</sup>* average density, , kg/m<sup>3</sup>

*b* at bulk temperature experimental

*=ρu*

K)

/s
