**3.2 Vertical annular channel, and three- and seven-rod bundles cooled with SCW**

In future SCWRs the main flow geometry will be bundles of various designs [6, 10]. Therefore, a limited number of experiments have been performed in simplified bundle simulators cooled with SCW and heated with an electrical current [10, 35–44].

### **Figure 14.**

*Profiles of bulk-fluid and inside-wall temperatures, and HTC along heated length of vertical bare tube with upward flow of SCW at various heat fluxes: (a) q = 944 kW/m<sup>2</sup> ; Tb in = 313°C (entrance region can be identified within Lh = 0–150 mm) and (b) q = 2079 kW/m<sup>2</sup> ; Tb in = 308°C (data by Razumovskiy et al.). For both graphs, qdht = 1575 kW/m<sup>2</sup> at G = 2193 kg/m<sup>2</sup> s (based on Eq. (5) [51]: P = 23.5 MPa; G = 2193 kg/m<sup>2</sup> s; and. Points—experimental data; curves—calculated data; curves for HTC and Tw are calculated through Dittus-Boelter correlation (Eq. (1)). Uncertainties of primary parameters are similar to those listed in* **Table 6***.*

**Figure 16.**

*[21])).*

**Figure 17.**

*[21])).*

**17**

*Thermal-conductivity profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and*

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*Dynamic-viscosity profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and*

### **Figure 15.**

*Density profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and [21])).*

### *Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering… DOI: http://dx.doi.org/10.5772/intechopen.91474*

### **Figure 16.**

**3.2 Vertical annular channel, and three- and seven-rod bundles cooled**

Therefore, a limited number of experiments have been performed in simplified bundle simulators cooled with SCW and heated with an electrical current [10, 35–44].

*Profiles of bulk-fluid and inside-wall temperatures, and HTC along heated length of vertical bare tube with*

*and. Points—experimental data; curves—calculated data; curves for HTC and Tw are calculated through Dittus-Boelter correlation (Eq. (1)). Uncertainties of primary parameters are similar to those listed in* **Table 6***.*

*; Tb in = 313°C (entrance region can be*

*s (based on Eq. (5) [51]: P = 23.5 MPa; G = 2193 kg/m<sup>2</sup>*

*; Tb in = 308°C (data by Razumovskiy et al.). For*

*s;*

*upward flow of SCW at various heat fluxes: (a) q = 944 kW/m<sup>2</sup>*

*identified within Lh = 0–150 mm) and (b) q = 2079 kW/m<sup>2</sup>*

*both graphs, qdht = 1575 kW/m<sup>2</sup> at G = 2193 kg/m<sup>2</sup>*

*Density profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and [21])).*

In future SCWRs the main flow geometry will be bundles of various designs [6, 10].

**with SCW**

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**Figure 15.**

**16**

**Figure 14.**

*Thermal-conductivity profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and [21])).*

### **Figure 17.**

*Dynamic-viscosity profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and [21])).*

**Figure 18.**

*Specific-heat profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and [21])).*

**Figure 20.**

*[21])).*

**Figure 21.**

**19**

*fluxes (data from Yamagata et al. [46]).*

*Prandtl-Number profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and*

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*Heat transfer coefficient vs. bulk-fluid enthalpy in vertical tube with upward flow of SCW at various heat*

### **Figure 19.**

*Specific-enthalpy profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and [21])).*

### *Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering… DOI: http://dx.doi.org/10.5772/intechopen.91474*

### **Figure 20.**

**Figure 18.**

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*[21])).*

**Figure 19.**

*[21])).*

**18**

*Specific-heat profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and*

*Specific-enthalpy profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and*

*Prandtl-Number profiles vs. reduced temperature and temperature for water, carbon dioxide, ethanol, and methanol (based on NIST [25]) (prepared by D. Mann): (a) at critical pressures; and (b) at 25 MPa for water and equivalent pressures for other SCFs (based on reduced-pressure scaling (for details, see* **Table 4** *and [21])).*

### **Figure 21.**

*Heat transfer coefficient vs. bulk-fluid enthalpy in vertical tube with upward flow of SCW at various heat fluxes (data from Yamagata et al. [46]).*

An annulus or a one-rod (single-rod) bundle is the simplest bundle geometry (see **Figures 22a** and **23**), and **Figure 24** shows profiles of bulk-fluid and wall temperatures, and HTC along heated length of vertical annular channel (one-rod bundle). **Figures 22b** and **23** show three-rod-bundle flow geometry, and **Figure 25** shows profiles of bulk-fluid and wall temperatures, and HTC along heated length of vertical three-rod bundle. **Figure 26** shows seven-rod-bundle flow geometry, and **Figure 27** shows profiles of bulk-fluid and wall temperatures, and HTC along heated length of the vertical seven-rod bundle.

**3.3 Vertical seven-rod bundle cooled with SC R-12**

*DOI: http://dx.doi.org/10.5772/intechopen.91474*

**Figure 24.**

**Figure 25.**

**Table 6***.*

**21**

*qave = 3.07 MW/m2*

*; G = 1500 kg/m2*

*and (b) Tin = 212°C. Bare tube qdht = 1431 kW/m<sup>2</sup> at G = 2000 kg/m2*

*G = 2000 kg/m2*

**Figures 28** and **29** show a seven-rod bundle test section, which can be considered as a bare bundle, and **Figures 30** and **31** show profiles of bulk-fluid and wall temperatures, and HTC vs. heated length of the central rod at three circumferential locations. Analysis of **Figures 30** and **31** shows that we also have here all three HT

*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering…*

*Profiles of bulk-fluid and wall temperatures, and HTC along heated length of vertical annular channel (one-rod bundle; rod with four helical ribs) cooled with upward flow of SCW ([36])—P = 22.6 MPa and*

*Tin = 210°C; and (b) qave = 2.547 MW/m<sup>2</sup> and Tin = 214°C). For details of test section, see* **Figure 23***. Points are experimental data; curves are calculated data; curves for HTC and* T*<sup>w</sup> are calculated through Dittus-Boelter correlation (Eq. (1)). Uncertainties of primary parameters are listed in* **Table 6***.*

*Profiles of bulk-fluid and wall temperatures, and HTC along heated length of vertical annular channel (three-rod bundle; each rods with 4 helical ribs) cooled with upward flow of SCW ([36])—P = 27.5 MPa;*

*section, see* **Figure 23***). Points are experimental data; curves are calculated data; curves for HTC and Tw are calculated through Dittus-Boelter correlation (Eq. (5)). Uncertainties of primary parameters are listed in*

*s (bare tube qdht = 1059 kW/m<sup>2</sup> (based on Eq. (5)): (a) Tin = 166°C*

*s (based on Eq. (5)); for details of test*

*s (bare tube qdht = 1431 kW/m<sup>2</sup> (based on Eq. (5)): (a) qave = 2.244 MW/m2 and*

Analysis of data in **Figures 25b** and **27b** shows that all three HT regimes, which were noticed in bare circular tubes, are also possible in annuli and bundle flow geometries. **Figures 24** and **25** show a comparison between the HTC experimental data obtained in annulus and three-rod bundle with those calculated through the Dittus-Boelter correlation (Eq. (1)). The comparison showed that, in general, there is no significant difference between calculated HTC values and experimental ones. This finding means that in spite of the presence of rod(s) with four helical ribs in SCW flow, which can be considered as an HT enhancement surface(s), there is no significant increase in HTC. However, when *q*dht values reached in SCW-cooled annulus and 3- and seven-rod bundles were compared to those obtained in bare tubes, it was found that *q*dht in bare tubes were 1.6–1.8 times lower (see **Table 7**).

**Figure 22.**

*3-D image of vertical annular channel (a) and three-rod bundle (b) cooled with upward flow of SCW (for other details, see* **Figure 23***) [35]: heated rods equipped with four helical ribs.*

### **Figure 23.**

*Radial cross-sections of annular channel (single rod) and three-rod bundle (for other details, see* **Figure 22***) [35]: heated rods equipped with four helical ribs; all dimensions in mm; and Ukrainian stainless steel has been used for heated rods, by content and other parameters, this steel is very close to those of SS-304.*

*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering… DOI: http://dx.doi.org/10.5772/intechopen.91474*

### **3.3 Vertical seven-rod bundle cooled with SC R-12**

**Figures 28** and **29** show a seven-rod bundle test section, which can be considered as a bare bundle, and **Figures 30** and **31** show profiles of bulk-fluid and wall temperatures, and HTC vs. heated length of the central rod at three circumferential locations. Analysis of **Figures 30** and **31** shows that we also have here all three HT

### **Figure 24.**

An annulus or a one-rod (single-rod) bundle is the simplest bundle geometry

vertical seven-rod bundle.

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(see **Table 7**).

**Figure 22.**

**Figure 23.**

**20**

(see **Figures 22a** and **23**), and **Figure 24** shows profiles of bulk-fluid and wall temperatures, and HTC along heated length of vertical annular channel (one-rod bundle). **Figures 22b** and **23** show three-rod-bundle flow geometry, and **Figure 25** shows profiles of bulk-fluid and wall temperatures, and HTC along heated length of vertical three-rod bundle. **Figure 26** shows seven-rod-bundle flow geometry, and **Figure 27** shows profiles of bulk-fluid and wall temperatures, and HTC along heated length of the

Analysis of data in **Figures 25b** and **27b** shows that all three HT regimes, which

obtained in bare tubes, it was found that *q*dht in bare tubes were 1.6–1.8 times lower

*3-D image of vertical annular channel (a) and three-rod bundle (b) cooled with upward flow of SCW (for*

*Radial cross-sections of annular channel (single rod) and three-rod bundle (for other details, see* **Figure 22***) [35]: heated rods equipped with four helical ribs; all dimensions in mm; and Ukrainian stainless steel has been*

*used for heated rods, by content and other parameters, this steel is very close to those of SS-304.*

*other details, see* **Figure 23***) [35]: heated rods equipped with four helical ribs.*

were noticed in bare circular tubes, are also possible in annuli and bundle flow geometries. **Figures 24** and **25** show a comparison between the HTC experimental data obtained in annulus and three-rod bundle with those calculated through the Dittus-Boelter correlation (Eq. (1)). The comparison showed that, in general, there is no significant difference between calculated HTC values and experimental ones. This finding means that in spite of the presence of rod(s) with four helical ribs in SCW flow, which can be considered as an HT enhancement surface(s), there is no significant increase in HTC. However, when *q*dht values reached in SCW-cooled annulus and 3- and seven-rod bundles were compared to those

*Profiles of bulk-fluid and wall temperatures, and HTC along heated length of vertical annular channel (one-rod bundle; rod with four helical ribs) cooled with upward flow of SCW ([36])—P = 22.6 MPa and G = 2000 kg/m2 s (bare tube qdht = 1431 kW/m<sup>2</sup> (based on Eq. (5)): (a) qave = 2.244 MW/m2 and Tin = 210°C; and (b) qave = 2.547 MW/m<sup>2</sup> and Tin = 214°C). For details of test section, see* **Figure 23***. Points are experimental data; curves are calculated data; curves for HTC and* T*<sup>w</sup> are calculated through Dittus-Boelter correlation (Eq. (1)). Uncertainties of primary parameters are listed in* **Table 6***.*

### **Figure 25.**

*Profiles of bulk-fluid and wall temperatures, and HTC along heated length of vertical annular channel (three-rod bundle; each rods with 4 helical ribs) cooled with upward flow of SCW ([36])—P = 27.5 MPa; qave = 3.07 MW/m2 ; G = 1500 kg/m2 s (bare tube qdht = 1059 kW/m<sup>2</sup> (based on Eq. (5)): (a) Tin = 166°C and (b) Tin = 212°C. Bare tube qdht = 1431 kW/m<sup>2</sup> at G = 2000 kg/m2 s (based on Eq. (5)); for details of test section, see* **Figure 23***). Points are experimental data; curves are calculated data; curves for HTC and Tw are calculated through Dittus-Boelter correlation (Eq. (5)). Uncertainties of primary parameters are listed in* **Table 6***.*

### **Figure 26.**

*3-D view (a) and cross-sectional view of vertical seven-rod bundle (b) cooled with upward flow of SCW [41, 42]: heated rods equipped with four helical ribs; all dimensions in mm; and Ukrainian stainless steel has been used for heated rods, by content and other parameters this steel is very close to those of SS-304.*

### **Figure 27.**

*Profiles of bulk-fluid and wall temperatures, and HTC vs. heated length; vertical seven-rod bundle (see* **Figure 26***) cooled with upward flow of SCW [42]: P = 22.6 MPa. Uncertainties of primary parameters are listed in* **Table 6***. (a) G = 1000 kg/m2 s; qave = 1.29 MW/m2 (bare tube qdht = 0.69 MW/m2 ); Tin = 178ºC; and central and peripheral rods; (b) G = 1000; qave = 1.29 MW/m2 (bare tube qdht = 0.69 MW/m2 ); Tin = 178ºC; and G = 800 kg/m2 s; qave = 1.18 MW/m2 (bare tube qdht = 0.54 MW/m2 ); Tin = 210ºC; and central rod.*

dataset, which was used to derive the correlation, but show a significant deviation in predicting other experimental data. Therefore, only selected correlations are

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In general, many of these correlations are based on the conventional Dittus-Boelter-type correlation (see Eq. (1)) in which the "regular" specific heat (i.e., based on bulk-fluid temperature) is replaced with the cross-sectional averaged

significant variations in thermophysical properties within a cross-section due to a

It should be noted that usually generalized correlations, which contain fluid properties at a wall temperature, require iterations to be solved, because there are two unknowns: (1) HTC and (2) the corresponding wall temperature. Therefore, the initial wall temperature value at which fluid properties will be estimated should

The most widely used heat transfer correlation at subcritical pressures for forced convection is the Dittus-Boelter [49] correlation. In 1942, McAdams [50] proposed to use the Dittus-Boelter correlation in the following form, for forced-convective

*Tw*�*Tb*

; etc., can be added into correlations to account for

, J/kg K. Also, additional terms,

specific heat within the range of (*Tw* � *Tb*); *Hw*�*Hb*

*Spacer grid locations and dimensions (all dimensions are in mm) [43].*

nonuniform temperature profile, that is, due to heat flux.

; *<sup>ρ</sup><sup>b</sup> ρw <sup>n</sup>*

considered below.

*kw <sup>k</sup>* ; *<sup>μ</sup><sup>b</sup> μw <sup>m</sup>*

be "guessed" to start iterations.

heat transfer in turbulent flows:

such as: *kb*

**23**

**Figure 28.**

regimes plus sometimes quite significant differences in local HTC values and wall temperatures around the central rod circumference.

## **4. Practical prediction methods for forced-convection heat transfer at supercritical pressures**

### **4.1 Supercritical water (SCW)**

Unfortunately, satisfactory analytical methods for practical prediction of forcedconvection heat transfer at SCPs have not yet been developed due to the difficulty in dealing with steep property variations, especially, in turbulent flows and at high heat fluxes [10, 48]. Therefore, generalized correlations based on experimental data are used for HTC calculations at SCPs.

There are numerous correlations for convective heat transfer in circular tubes at SCPs (for details, see in Pioro and Duffey [9]). However, an analysis of these correlations has shown that they are more or less accurate only within the particular *Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering… DOI: http://dx.doi.org/10.5772/intechopen.91474*

dataset, which was used to derive the correlation, but show a significant deviation in predicting other experimental data. Therefore, only selected correlations are considered below.

In general, many of these correlations are based on the conventional Dittus-Boelter-type correlation (see Eq. (1)) in which the "regular" specific heat (i.e., based on bulk-fluid temperature) is replaced with the cross-sectional averaged specific heat within the range of (*Tw* � *Tb*); *Hw*�*Hb Tw*�*Tb* , J/kg K. Also, additional terms, such as: *kb kw <sup>k</sup>* ; *<sup>μ</sup><sup>b</sup> μw <sup>m</sup>* ; *<sup>ρ</sup><sup>b</sup> ρw <sup>n</sup>* ; etc., can be added into correlations to account for significant variations in thermophysical properties within a cross-section due to a nonuniform temperature profile, that is, due to heat flux.

It should be noted that usually generalized correlations, which contain fluid properties at a wall temperature, require iterations to be solved, because there are two unknowns: (1) HTC and (2) the corresponding wall temperature. Therefore, the initial wall temperature value at which fluid properties will be estimated should be "guessed" to start iterations.

The most widely used heat transfer correlation at subcritical pressures for forced convection is the Dittus-Boelter [49] correlation. In 1942, McAdams [50] proposed to use the Dittus-Boelter correlation in the following form, for forced-convective heat transfer in turbulent flows:

regimes plus sometimes quite significant differences in local HTC values and wall

*Profiles of bulk-fluid and wall temperatures, and HTC vs. heated length; vertical seven-rod bundle (see* **Figure 26***) cooled with upward flow of SCW [42]: P = 22.6 MPa. Uncertainties of primary parameters are listed in* **Table 6***.*

*3-D view (a) and cross-sectional view of vertical seven-rod bundle (b) cooled with upward flow of SCW [41, 42]: heated rods equipped with four helical ribs; all dimensions in mm; and Ukrainian stainless steel has been used for heated rods, by content and other parameters this steel is very close to those of SS-304.*

*); Tin = 178ºC; and central and*

*); Tin = 210ºC; and central rod.*

*); Tin = 178ºC; and*

*s; qave = 1.29 MW/m2 (bare tube qdht = 0.69 MW/m2*

*peripheral rods; (b) G = 1000; qave = 1.29 MW/m2 (bare tube qdht = 0.69 MW/m2*

*s; qave = 1.18 MW/m2 (bare tube qdht = 0.54 MW/m2*

**4. Practical prediction methods for forced-convection heat transfer at**

Unfortunately, satisfactory analytical methods for practical prediction of forcedconvection heat transfer at SCPs have not yet been developed due to the difficulty in dealing with steep property variations, especially, in turbulent flows and at high heat fluxes [10, 48]. Therefore, generalized correlations based on experimental data

There are numerous correlations for convective heat transfer in circular tubes at

SCPs (for details, see in Pioro and Duffey [9]). However, an analysis of these correlations has shown that they are more or less accurate only within the particular

temperatures around the central rod circumference.

**supercritical pressures**

**Figure 27.**

**Figure 26.**

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*(a) G = 1000 kg/m2*

*G = 800 kg/m2*

**22**

**4.1 Supercritical water (SCW)**

are used for HTC calculations at SCPs.

**Figure 29.** *Photo of central part of 7-element bundle with spacer grid [43].*

$$\text{Nu}\_{\text{b}} = 0.0243 \cdot \text{Re}\_{\text{b}}^{0.8} \text{Pr}\_{\text{b}}^{0.4} \tag{1}$$

that time (for example, a peak in thermal conductivity in critical and pseudocritical points within a range of pressures from 22.1 to 25 MPa for water was not officially

*s;*

*Bulk-fluid and wall temperatures, and HTC profiles along heated length of vertical bare 7-element bundle (Dhy = 4.7 mm) cooled with upward flow of SC R-12 [43, 44]: Run 3: Pin = 4.65 MPa; G = 508 kg/m2*

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Therefore, new correlations within the SCWRs operating range, were developed and evaluated by I. Pioro with his students (mainly, by S. Mokry et al. (bulk-fluidtemperature approach) and S. Gupta et al. (wall temperature approach)) using the

recognized until the 1990s).

*, and Tin = 74°C.*

**Figure 30.**

**25**

*qave = 19.4 kW/m2*

However, it was noted that Eq. (1) might produce unrealistic results at SCPs within some flow conditions (see **Figure 12**), especially, near the critical and pseudocritical points, because it is very sensitive to properties variations.

In general, experimental HTC values show just a moderate increase within the pseudocritical region. This increase depends on mass flux and heat flux: higher heat flux—less increase. Thus, the bulk-fluid temperature might not be the best characteristic temperature at which all thermophysical properties should be evaluated. Therefore, the cross-sectional averaged Prandtl number, which accounts for thermophysical-properties variations within a cross-section due to heat flux, was proposed to be used in many SC HT correlations instead of the regular Prandtl number. Nevertheless, this classical correlation (Eq. (1)) was used extensively as a basis for various SC HT correlations [9].

The majority of empirical correlations were proposed in the 1960s–1970s [9], when experimental techniques were not at the same level (i.e., advanced level) as they are today. Also, thermophysical properties of SCW have been updated since

*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering… DOI: http://dx.doi.org/10.5772/intechopen.91474*

### **Figure 30.**

Nub <sup>¼</sup> <sup>0</sup>*:*0243 Re <sup>0</sup>*:*<sup>8</sup>

basis for various SC HT correlations [9].

*Photo of central part of 7-element bundle with spacer grid [43].*

*Advanced Supercritical Fluids Technologies*

**Figure 29.**

**24**

<sup>b</sup> . (1)

However, it was noted that Eq. (1) might produce unrealistic results at SCPs within some flow conditions (see **Figure 12**), especially, near the critical and pseudocritical points, because it is very sensitive to properties variations.

In general, experimental HTC values show just a moderate increase within the pseudocritical region. This increase depends on mass flux and heat flux: higher heat flux—less increase. Thus, the bulk-fluid temperature might not be the best characteristic temperature at which all thermophysical properties should be evaluated. Therefore, the cross-sectional averaged Prandtl number, which accounts for thermophysical-properties variations within a cross-section due to heat flux, was proposed to be used in many SC HT correlations instead of the regular Prandtl number. Nevertheless, this classical correlation (Eq. (1)) was used extensively as a

The majority of empirical correlations were proposed in the 1960s–1970s [9], when experimental techniques were not at the same level (i.e., advanced level) as they are today. Also, thermophysical properties of SCW have been updated since

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

*Bulk-fluid and wall temperatures, and HTC profiles along heated length of vertical bare 7-element bundle (Dhy = 4.7 mm) cooled with upward flow of SC R-12 [43, 44]: Run 3: Pin = 4.65 MPa; G = 508 kg/m2 s; qave = 19.4 kW/m2 , and Tin = 74°C.*

that time (for example, a peak in thermal conductivity in critical and pseudocritical points within a range of pressures from 22.1 to 25 MPa for water was not officially recognized until the 1990s).

Therefore, new correlations within the SCWRs operating range, were developed and evaluated by I. Pioro with his students (mainly, by S. Mokry et al. (bulk-fluidtemperature approach) and S. Gupta et al. (wall temperature approach)) using the

Nub <sup>¼</sup> <sup>0</sup>*:*0061 Re <sup>0</sup>*:*<sup>904</sup>

2.Pioro-Gupta correlation (wall temperature approach) [53]:

and heat flux—70–1250 kW/m<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91474*

**Figure 32.**

*(NHT) regime.*

**27**

*Kirillov et al. [26]) [54]: Pin* ≈ *24 MPa, G = 500 kg/m2*

can be less or even significantly less in these cases.

**Nuw** <sup>¼</sup> <sup>0</sup>*:*<sup>0033</sup> **Re**<sup>0</sup>*:*<sup>941</sup>

<sup>b</sup> Prb

*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering…*

The Pioro-Mokry correlation (Eq. (2)) was verified within the following operating conditions (only for NHT and IHT regimes (see **Figures 32** and **33**), but not for the DHT regime): SCW, upward flow, vertical bare circular tubes with inside diameters of 3–38 mm, pressure—22.8–29.4 MPa, mass flux—200–3000 kg/m<sup>2</sup>

lated according to NIST REFPROP software [25]. This correlation has accuracy of �25% for HTCs and �15 for wall temperatures (**Figure 34**). Eventually, this nondimensional correlation can be also used for other SCFs. However, its accuracy

**<sup>w</sup> Prw**

**Nuw** <sup>¼</sup> **Nu**<sup>w</sup> <sup>1</sup> <sup>þ</sup> *exp* � *<sup>x</sup>*

*Temperature and HTC profiles along 4-m circular tube (D = 10 mm) with upward flow of SCW (data by*

*values through the "proposed correlation"—Eq. (2) with experimental data within Normal Heat Transfer*

*s; qave = 287 kW/m<sup>2</sup>*

Eq. (3) has an uncertainty of about �25% for HTC values and about �15% for calculated wall temperatures within the same ranges as those for Eq. (2). Also, it was decided to add an entrance effect to make this correlation even more accurate. This entrance effect was modeled by an exponentially-decreasing term as shown below:

<sup>0</sup>*:*<sup>764</sup> *μ<sup>w</sup> μb* � �<sup>0</sup>*:*<sup>398</sup>

h i � � <sup>0</sup>*:*<sup>3</sup>

24*D*

<sup>0</sup>*:*<sup>684</sup> *ρ<sup>w</sup> ρb* � �0*:*<sup>564</sup>

. All thermophysical properties of SCW were calcu-

*ρw ρb* � �<sup>0</sup>*:*<sup>156</sup>

*:* (2)

s,

(3)

, (4)

*; comparison of calculated HTC*

**Figure 31.**

*Bulk-fluid and wall temperatures, and HTC profiles along heated length of vertical bare 7-element bundle (Dhy = 4.7 mm) cooled with upward glow of SC R-12 [43, 44]: Run 7: Pin = 4.64 MPa; G = 517 kg/m2 s; qave = 33.4 kW/m2 , and Tin = 112°C.*

best SCW dataset by P.L. Kirillov and his co-workers and adding smaller datasets by other researchers:

1.Pioro-Mokry correlation (bulk-fluid-temperature approach) [51, 52]:

*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering… DOI: http://dx.doi.org/10.5772/intechopen.91474*

$$\mathrm{Nu}\_{\mathrm{b}} = 0.0061 \quad \mathrm{Re}\_{\mathrm{b}}^{0.904} \overline{\mathrm{Pr}\_{\mathrm{b}}}^{0.684} \left(\frac{\rho\_w}{\rho\_b}\right)^{0.564}.\tag{2}$$

The Pioro-Mokry correlation (Eq. (2)) was verified within the following operating conditions (only for NHT and IHT regimes (see **Figures 32** and **33**), but not for the DHT regime): SCW, upward flow, vertical bare circular tubes with inside diameters of 3–38 mm, pressure—22.8–29.4 MPa, mass flux—200–3000 kg/m<sup>2</sup> s, and heat flux—70–1250 kW/m<sup>2</sup> . All thermophysical properties of SCW were calculated according to NIST REFPROP software [25]. This correlation has accuracy of �25% for HTCs and �15 for wall temperatures (**Figure 34**). Eventually, this nondimensional correlation can be also used for other SCFs. However, its accuracy can be less or even significantly less in these cases.

2.Pioro-Gupta correlation (wall temperature approach) [53]:

$$\mathbf{Nu\_w} = 0.0033 \ \mathbf{Re\_w^{0.941}} \overline{\mathbf{Pr\_w}}^{0.764} \left(\frac{\mu\_w}{\mu\_b}\right)^{0.398} \left(\frac{\rho\_w}{\rho\_b}\right)^{0.156} \tag{3}$$

Eq. (3) has an uncertainty of about �25% for HTC values and about �15% for calculated wall temperatures within the same ranges as those for Eq. (2). Also, it was decided to add an entrance effect to make this correlation even more accurate. This entrance effect was modeled by an exponentially-decreasing term as shown below:

$$\mathbf{Nu\_w} = \mathbf{Nu\_w} \left[ 1 + \exp\left( -\frac{\varkappa}{24D} \right) \right]^{0.3},\tag{4}$$

### **Figure 32.**

*Temperature and HTC profiles along 4-m circular tube (D = 10 mm) with upward flow of SCW (data by Kirillov et al. [26]) [54]: Pin* ≈ *24 MPa, G = 500 kg/m2 s; qave = 287 kW/m<sup>2</sup> ; comparison of calculated HTC values through the "proposed correlation"—Eq. (2) with experimental data within Normal Heat Transfer (NHT) regime.*

best SCW dataset by P.L. Kirillov and his co-workers and adding smaller datasets by

*s;*

*Bulk-fluid and wall temperatures, and HTC profiles along heated length of vertical bare 7-element bundle (Dhy = 4.7 mm) cooled with upward glow of SC R-12 [43, 44]: Run 7: Pin = 4.64 MPa; G = 517 kg/m2*

1.Pioro-Mokry correlation (bulk-fluid-temperature approach) [51, 52]:

other researchers:

*qave = 33.4 kW/m2*

*, and Tin = 112°C.*

*Advanced Supercritical Fluids Technologies*

**Figure 31.**

**26**

### **Figure 33.**

*Temperature and HTC profiles along circular tube (D = 7.5 mm) with upward flow of SCW (data by Yamagata et al. [46]) [54]: Pin = 24.5 MPa; G = 1260 kg/m<sup>2</sup> s; qave = 233 kW/m2 ; comparison of calculated HTC values through the "proposed correlation"—Eq. (2) with experimental data within normal and improved heat transfer (NHT and IHT) regimes.*

Pioro-Mokry correlation for *q*dht [51]:

*DOI: http://dx.doi.org/10.5772/intechopen.91474*

*flow of SC CO2 (data by I. Pioro): P = 8.8 MPa; G = 2000 kg/m2*

Uncertainty is about �15% for the DHT heat flux.

24 MPa, mass flux 200–1500 kg/m<sup>2</sup>

estimations."

**Table 5.**

**29**

*Uncertainties of primary parameters [51].*

**Figure 35.**

*<sup>q</sup>*dht ¼ �58*:*<sup>97</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>745</sup> � *<sup>G</sup>*, kW*=*m<sup>2</sup>

*Wall temperature and HTC profiles along vertical circular tube (D = 8 mm and L = 2.208 m) with upward*

*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering…*

Wang et al. [33] have evaluated 15 *q*dht correlations for SCW, and they have concluded that Pioro-Mokry correlation (Eq. (5)) "may be used for preliminary

transfer look-up table for the critical/SCPs. An extensive literature review was conducted, which included 28 datasets and 6663 trans-critical heat transfer data (**Figure 35**). **Tables 8** and **9** list results from this study in the form of the overallweighted average and root-mean-square (RMS) errors: (a) within three SC subregions; and (b) for subcritical liquid and superheated steam. Many of the correlations listed in these tables can be found in Zahlan et al. [55, 56] and Pioro and Duffey [9]. In their conclusions, Zahlan et al. [55, 56] determined that within the SC

region, the latest correlation by Pioro-Mokry [51] (Eq. (2)) showed the best

**Parameters Uncertainty** Test-section power �1.0% Inlet pressure �0.25% Wall temperature �3.0% Mass-flow rate �1.5% Heat loss ≤3.0%

A recent study was conducted by Zahlan et al. [55, 56] in order to develop a heat

Correlation (Eq. (5)) is valid within the following range of experimental parameters: SCW, upward flow, vertical bare tube with inside diameter 10 mm, pressure

*:* (5)

*, and* T*in = 29°C.*

s, and bulk-fluid inlet temperature 320–350°C.

*s; q = 428 kW/m<sup>2</sup>*

### **Figure 34.**

*Wall temperature and HTC profiles along vertical circular tube (D = 8 mm and L = 2.208 m) with upward flow of SC CO2 (data by I. Pioro): P = 8.8 MPa; G = 940 kg/m2 s; q = 225 kW/m<sup>2</sup> , and Tin = 30°C.*

where, **Nu**<sup>w</sup> is calculated using Eq. (3). It should be noted that this HT correlation is also intended only for NHT and IHT regimes.

The following empirical correlation was proposed by I. Pioro and S. Mokry for calculating the minimum heat flux at which the DHT regime appears in vertical bare circular tubes:

*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering… DOI: http://dx.doi.org/10.5772/intechopen.91474*

### **Figure 35.**

*Wall temperature and HTC profiles along vertical circular tube (D = 8 mm and L = 2.208 m) with upward flow of SC CO2 (data by I. Pioro): P = 8.8 MPa; G = 2000 kg/m2 s; q = 428 kW/m<sup>2</sup> , and* T*in = 29°C.*

Pioro-Mokry correlation for *q*dht [51]:

$$q\_{\rm dht} = -58.97 + 0.745 \cdot G, \quad \text{kW/m}^2. \tag{5}$$

Correlation (Eq. (5)) is valid within the following range of experimental parameters: SCW, upward flow, vertical bare tube with inside diameter 10 mm, pressure 24 MPa, mass flux 200–1500 kg/m<sup>2</sup> s, and bulk-fluid inlet temperature 320–350°C. Uncertainty is about �15% for the DHT heat flux.

Wang et al. [33] have evaluated 15 *q*dht correlations for SCW, and they have concluded that Pioro-Mokry correlation (Eq. (5)) "may be used for preliminary estimations."

A recent study was conducted by Zahlan et al. [55, 56] in order to develop a heat transfer look-up table for the critical/SCPs. An extensive literature review was conducted, which included 28 datasets and 6663 trans-critical heat transfer data (**Figure 35**). **Tables 8** and **9** list results from this study in the form of the overallweighted average and root-mean-square (RMS) errors: (a) within three SC subregions; and (b) for subcritical liquid and superheated steam. Many of the correlations listed in these tables can be found in Zahlan et al. [55, 56] and Pioro and Duffey [9]. In their conclusions, Zahlan et al. [55, 56] determined that within the SC region, the latest correlation by Pioro-Mokry [51] (Eq. (2)) showed the best


**Table 5.** *Uncertainties of primary parameters [51].*

where, **Nu**<sup>w</sup> is calculated using Eq. (3). It should be noted that this HT correla-

*s; q = 225 kW/m<sup>2</sup>*

*s; qave = 233 kW/m2*

*, and Tin = 30°C.*

*; comparison of calculated*

*Wall temperature and HTC profiles along vertical circular tube (D = 8 mm and L = 2.208 m) with upward*

*Temperature and HTC profiles along circular tube (D = 7.5 mm) with upward flow of SCW (data by*

*HTC values through the "proposed correlation"—Eq. (2) with experimental data within normal and improved*

The following empirical correlation was proposed by I. Pioro and S. Mokry for calculating the minimum heat flux at which the DHT regime appears in vertical

tion is also intended only for NHT and IHT regimes.

*flow of SC CO2 (data by I. Pioro): P = 8.8 MPa; G = 940 kg/m2*

*Yamagata et al. [46]) [54]: Pin = 24.5 MPa; G = 1260 kg/m<sup>2</sup>*

*heat transfer (NHT and IHT) regimes.*

*Advanced Supercritical Fluids Technologies*

bare circular tubes:

**Figure 34.**

**28**

**Figure 33.**


**No. Correlation Regions**

*DOI: http://dx.doi.org/10.5772/intechopen.91474*

*In bold—minimum values.*

*In bold—minimum values.*

*Overall average and RMS error within subcritical region [55, 56].*

*Ranges of parameters of dataset used to develop Eq. (6).*

**Table 9.**

**Table 10.**

**31**

**Table 8.**

1 Dittus-Boelter [49] 24 44 90 127 ‑ ‑ 2 Sieder and Tate [59] 46 65 97 132 ‑ ‑ 3 Bishop et al. [60] 5 28 5 20 23 31 4 Swenson et al. [61] **<sup>1</sup>** <sup>31</sup> ‑16 21 4 <sup>23</sup> 5 Krasnoshchekov et al. [62] 18 40 ‑30 32 24 <sup>65</sup> 6 Hadaller and Banerjee [63] 34 53 14 24 ‑ ‑ <sup>7</sup> Gnielinski [64] 10 36 99 139 ‑ ‑ 8 Watts and Chou [65], NHT 6 30 ‑6 21 11 <sup>28</sup> 9 Watts and Chou [65], DHT 2 **26** 9 24 17 30 10 Griem [66] 2 28 11 28 9 35 11 Koshizuka and Oka [67] 26 47 27 54 39 83 12 Jackson [68] 15 36 15 32 30 49 13 Mokry et al. [51, 52] ‑<sup>5</sup> **<sup>26</sup>** ‑<sup>9</sup> **<sup>18</sup> ‑<sup>1</sup> <sup>17</sup>** 14 Kuang et al. [69] ‑6 27 10 24 ‑<sup>3</sup> <sup>26</sup> 15 Cheng et al. [70] 4 30 **2** 28 21 85 <sup>16</sup> Gupta et al. [53] ‑26 33 ‑12 20 **‑<sup>1</sup>** <sup>18</sup>

*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering…*

*Overall-weighted average and RMS errors within three supercritical sub-regions (correlations are listed*

**No. Correlation Subcritical liquid Superheated steam**

 Dittus and Boelter [49] 10 23 75 127 Sieder and Tate [59] 28 37 84 138 3 Hadaller and Banerjee [63] 27 36 19 34 Gnielinski [64] ‑<sup>4</sup> **<sup>18</sup>** <sup>80</sup> <sup>130</sup> Mokry et al. [51] **‑<sup>1</sup>** <sup>19</sup> **‑<sup>5</sup> <sup>20</sup>**

*P***, MPa** *Tin***, °C** *Tout***, °C** *Tw***, °C** *q***, kW/m<sup>2</sup>** *G***, kg/m2**

7.57‑8.8 20‑<sup>40</sup> <sup>29</sup>‑136 29‑224 9.3‑616.6 <sup>706</sup>‑<sup>3169</sup>

**Error, % Ave. RMS Ave. RMS**

**s**

*according to the year of publication, that is, from early ones to the latest ones) [55, 56].*

**Liquid-like Gas-like Critical or pseudocritical Errors, % Ave. RMS Ave. RMS Ave. RMS**

### **Table 6.**

*Maximum uncertainties of measured and calculated parameters [35–40].*


### **Table 7.**

*Comparison of DHT values in bare-tube, annular channel (one-rod), and three-rod and seven-rod bundles [35, 42].*

prediction for the data within all three sub-regions investigated (based on RMS error) (see **Table 8**). Also, the Pioro-Mokry correlation showed quite good predictions for subcritical-pressure water and superheated steam compared to other several correlations (see **Table 9**). Also, it was concluded that Pioro-Gupta correlation (Eq. (3)) was quite close by RMS errors to the Pioro-Mokry correlation.

Chen et al. [57] has also concluded that the Pioro-Mokry correlation for SCW HT "performs best" compared to other 14 correlations.

### **4.2 Supercritical carbon dioxide**

The following correlation was proposed by S. Gupta (an MASc student of I. Pioro) [21] for SC carbon dioxide flowing inside vertical bare tubes:

$$\mathbf{Nu\_w} = 0.0038 \mathbf{Re\_{w}^{0.957} \mathbf{Pr\_{w}}}^{0.957} \left(\frac{\rho\_w}{\rho\_b}\right)^{0.84} \left(\frac{k\_w}{k\_b}\right)^{-0.75} \left(\frac{\mu\_w}{\mu\_b}\right)^{-0.22} \tag{6}$$

Uncertainties associated with this correlation are �30% for HTC values and � 20% for calculated wall temperatures (see **Figures 36** and **37**). Ranges of parameters for the dataset used to develop Eq. (6) are listed in **Table 10**.

**Table 11** list mean and root-mean square (RMS) errors in HTC and *Tw* for proposed correlations using equations shown below:

It was also decided to develop the *q*dht correlation for SC carbon dioxide based on the dataset obtained by I. Pioro in vertical bare tube with upward flow, which ranges are listed in **Table 10** [58]. Therefore, based on the identified 41 cases of


*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering… DOI: http://dx.doi.org/10.5772/intechopen.91474*

### **Table 8.**

prediction for the data within all three sub-regions investigated (based on RMS error) (see **Table 8**). Also, the Pioro-Mokry correlation showed quite good predictions for subcritical-pressure water and superheated steam compared to other several correlations (see **Table 9**). Also, it was concluded that Pioro-Gupta correlation (Eq. (3)) was quite close by RMS errors to the Pioro-Mokry correlation.

*Comparison of DHT values in bare-tube, annular channel (one-rod), and three-rod and seven-rod bundles*

**Parameters Maximum uncertainty**

**No. Test section Operating conditions** *q***dht, MW/m2 Increase in** *q***dht value**

Bulk-fluid temperature �3.4% Wall temperature �3.2%

> Heat flux �3.5% HTC �12.7% Heat loss ≤3.4%

> > s 2.55

s 3.20

s 0.96

s 1.43 1.8

s 1.95 1.6

s 0.54 1.8

Measured Inlet pressure �0.2%

Calculated Mass-flow rate �2.3%

*Maximum uncertainties of measured and calculated parameters [35–40].*

1 Bare tube *P* = 24.1 MPa and *G* = 2000 kg/m<sup>2</sup>

*Advanced Supercritical Fluids Technologies*

2 Annulus *P* = 22.6 MPa and *G* = 2000 kg/m2

3 Bare tube *P* = 24.1 MPa and *G* = 2700 kg/m<sup>2</sup>

4 Three-rod bundle *P* = 22.6 MPa and *G* = 2700 kg/m<sup>2</sup>

5 Bare tube *P* = 24.5 MPa and *G* = 800 kg/m2

6 Seven-rod bundle *P* = 24.5 MPa and *G* = 800 kg/m2

"performs best" compared to other 14 correlations.

**Nuw** <sup>¼</sup> <sup>0</sup>*:*0038**Re**<sup>0</sup>*:*<sup>957</sup>

proposed correlations using equations shown below:

**4.2 Supercritical carbon dioxide**

**Table 6.**

**Table 7.**

*[35, 42].*

**30**

Chen et al. [57] has also concluded that the Pioro-Mokry correlation for SCW HT

The following correlation was proposed by S. Gupta (an MASc student of I.

�0*:*<sup>14</sup> *ρ<sup>w</sup> ρb*

Uncertainties associated with this correlation are �30% for HTC values and � 20% for calculated wall temperatures (see **Figures 36** and **37**). Ranges of parameters for the dataset used to develop Eq. (6) are listed in **Table 10**.

**Table 11** list mean and root-mean square (RMS) errors in HTC and *Tw* for

the dataset obtained by I. Pioro in vertical bare tube with upward flow, which ranges are listed in **Table 10** [58]. Therefore, based on the identified 41 cases of

It was also decided to develop the *q*dht correlation for SC carbon dioxide based on

<sup>0</sup>*:*<sup>84</sup> *kw*

*kb* �0*:*<sup>75</sup>

*μw μb* �0*:*<sup>22</sup>

**compared to that of bare tube**

(6)

Pioro) [21] for SC carbon dioxide flowing inside vertical bare tubes:

**<sup>w</sup> Prw**

*Overall-weighted average and RMS errors within three supercritical sub-regions (correlations are listed according to the year of publication, that is, from early ones to the latest ones) [55, 56].*


### **Table 9.**

*Overall average and RMS error within subcritical region [55, 56].*


### **Table 10.**

*Ranges of parameters of dataset used to develop Eq. (6).*

DHT within the SC carbon dioxide dataset, the following correlation for the mini-

*Mean and RMS errors for HTC values of proposed correlations (values in bold represent minimum errors) [21].*

Proposed new correlation (*T*<sup>b</sup> approach) 0.9% 22.4% Proposed new correlation (*T*film approach) **0.2%** 21.7% Proposed new correlation (*T*<sup>w</sup> approach—Eq. (6)) 0.8% **20.3%** Swenson et al. [61] correlation 89% 132% Mokry et al. [51] correlation for SCW 68% 123% Gupta et al. [53] correlation for SCW 78% 130%

*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering…*

In general, the total pressure drop for forced convection inside a channel can be calculated according to expressions listed in Pioro and Duffey [9] and Pioro et al. [71].

Supercritical fluids are used quite intensively in various industries. Therefore, understanding specifics of thermophysical properties, heat transfer, and pressure drop in various flow geometries at supercritical pressures is an important task.

In general, three major heat transfer regimes were noticed at critical and supercritical pressures in various flow geometries (vertical bare tubes, annulus, threeand seven-rod bundles) and several SCFs (SCW, SC carbon dioxide, and SC R-12): (1) normal heat transfer; (2) improved heat transfer; and (3) deteriorated heat transfer. Also, two special phenomena may appear along a heated channel: (1) pseudo-boiling; and (2) pseudo-film boiling. These heat transfer regimes and special phenomena appear to be due to significant variations of thermophysical properties near the critical and pseudocritical points and due to operating conditions. Comparison of heat transfer-coefficient values obtained in bare circular tubes

with those obtained in annulus (one-rod bundle)/three-rod bundle (rod(s)

equipped with four helical ribs) shows that there are almost no differences between these values. However, the minimal heat flux at which deterioration occurs (*q*dht) in annulus, and three- and seven-rod bundles are in 1.6–1.8 times higher compared to

The current analysis of a number of well-known heat transfer correlations for supercritical fluids showed that the Dittus-Boelter correlation [49] significantly overestimates experimental HTC values within the pseudocritical range. The Bishop et al. [60] and Jackson [68] correlations tend also to deviate substantially from the experimental data within the pseudocritical range. The Swenson et al. [61] correlation provided a better fit for the experimental data than the previous three correlations within some flow conditions, but does not follow up closely the experimental

Therefore, new correlations were developed by Pioro with his students Mokry et al.

[51] (bulk-fluid-temperature approach) and Gupta et al. [21] (wall temperature approach), which showed the best fit for the experimental data within a wide range of

*q*min ¼ 66*:*81 þ 0*:*18 � *G* (7)

**Mean Error RMS**

mal heat flux at which deterioration occurs was proposed:

**Errors in HTC (for the reference dataset), %**

*DOI: http://dx.doi.org/10.5772/intechopen.91474*

**5. Conclusions**

**Table 11.**

that recorded in bare tubes.

data within others.

**33**

**Figure 36.**

*HTC and* Tw *variations along* L *= 2.208 m circular tube (*D *= 8 mm):* q *= 90.7 kW/m<sup>2</sup>* P *= 8.4 MPa, and* G *= 1608 kg/m<sup>2</sup> s. Wall Approach Corr. is Eq. (6) and Mokry et al. Corr. – Eq. (2).*

### **Figure 37.**

*HTC and* Tw *variations along* L *= 2.208 m circular tube (*D *= 8 mm):* q *= 161.2 kW/m<sup>2</sup>* P *= 8.8 MPa, and* G *= 2000 kg/m<sup>2</sup> s. Wall Approach Corr. is Eq. (6) and Mokry et al. Corr. – Eq. (2).*


**Table 11.**

*Mean and RMS errors for HTC values of proposed correlations (values in bold represent minimum errors) [21].*

DHT within the SC carbon dioxide dataset, the following correlation for the minimal heat flux at which deterioration occurs was proposed:

$$q\_{\rm min} = 66.81 + 0.18 \cdot G \tag{7}$$

In general, the total pressure drop for forced convection inside a channel can be calculated according to expressions listed in Pioro and Duffey [9] and Pioro et al. [71].

### **5. Conclusions**

**Figure 36.**

*Advanced Supercritical Fluids Technologies*

*1608 kg/m<sup>2</sup>*

**Figure 37.**

*2000 kg/m<sup>2</sup>*

**32**

*HTC and* Tw *variations along* L *= 2.208 m circular tube (*D *= 8 mm):* q *= 90.7 kW/m<sup>2</sup>* P *= 8.4 MPa, and* G *=*

*HTC and* Tw *variations along* L *= 2.208 m circular tube (*D *= 8 mm):* q *= 161.2 kW/m<sup>2</sup>* P *= 8.8 MPa, and* G *=*

*s. Wall Approach Corr. is Eq. (6) and Mokry et al. Corr. – Eq. (2).*

*s. Wall Approach Corr. is Eq. (6) and Mokry et al. Corr. – Eq. (2).*

Supercritical fluids are used quite intensively in various industries. Therefore, understanding specifics of thermophysical properties, heat transfer, and pressure drop in various flow geometries at supercritical pressures is an important task.

In general, three major heat transfer regimes were noticed at critical and supercritical pressures in various flow geometries (vertical bare tubes, annulus, threeand seven-rod bundles) and several SCFs (SCW, SC carbon dioxide, and SC R-12): (1) normal heat transfer; (2) improved heat transfer; and (3) deteriorated heat transfer. Also, two special phenomena may appear along a heated channel: (1) pseudo-boiling; and (2) pseudo-film boiling. These heat transfer regimes and special phenomena appear to be due to significant variations of thermophysical properties near the critical and pseudocritical points and due to operating conditions.

Comparison of heat transfer-coefficient values obtained in bare circular tubes with those obtained in annulus (one-rod bundle)/three-rod bundle (rod(s) equipped with four helical ribs) shows that there are almost no differences between these values. However, the minimal heat flux at which deterioration occurs (*q*dht) in annulus, and three- and seven-rod bundles are in 1.6–1.8 times higher compared to that recorded in bare tubes.

The current analysis of a number of well-known heat transfer correlations for supercritical fluids showed that the Dittus-Boelter correlation [49] significantly overestimates experimental HTC values within the pseudocritical range. The Bishop et al. [60] and Jackson [68] correlations tend also to deviate substantially from the experimental data within the pseudocritical range. The Swenson et al. [61] correlation provided a better fit for the experimental data than the previous three correlations within some flow conditions, but does not follow up closely the experimental data within others.

Therefore, new correlations were developed by Pioro with his students Mokry et al. [51] (bulk-fluid-temperature approach) and Gupta et al. [21] (wall temperature approach), which showed the best fit for the experimental data within a wide range of operating conditions. These correlations have uncertainties of about �25% for HTC values and about �15% for calculated wall temperature. Also, based on an independent study performed by Zahlan et al. [55, 56], Pioro-Mokry correlation (given as Eq. (2)) is the best for superheated steam compared to other well-known correlations. Also, this correlation showed quite good predictions for subcritical-pressure fluids.

Pr cross-sectional average Prandtl number within the

*Supercritical-Fluids Thermophysical Properties and Heat Transfer in Power-Engineering…*

**Re** Reynolds number; *<sup>G</sup>* �*<sup>D</sup>*

dht deteriorated heat transfer

hy hydraulic-equivalent

AECL Atomic Energy of Canada Limited AGR advanced gas-cooled reactor

ASME American Society of Mechanical Engineers

BN fast sodium (reactor; in Russian abbreviations)

pc pseudocritical sat saturation th thermal w wall

**Subscripts or superscripts**

ave. average b bulk cal calculated corr. correlation cr critical

*DOI: http://dx.doi.org/10.5772/intechopen.91474*

fl flow h heated

in inlet max maximum min minimum out outlet

**Abbreviations and acronyms**

Ave. average

corr. correlation

HT heat transfer

ID inside diameter

BWR boiling water reactor CHF critical heat flux

CFD computational fluid dynamics

DHT deteriorated heat transfer GFR Gas-cooled fast reactor

HTC heat transfer coefficient HTR high-temperature reactor HPT high-pressure turbine

IHT improved heat transfer IHX intermediate heat exchanger LFR lead-cooled fast reactor

LNG liquified natural gas LPT low-pressure turbine

**35**

CRL Chalk River Laboratotries (AECL)

GIF Generation-IV International Forum

IAEA International Atomic Energy Agency

LGR light-water-cooled graphite-moderated reactor

range of (*Tw* – *Tb*); *<sup>μ</sup>* �*cp*

*k* 

*μ* 
