# The Study of Critical Heat Flux in Upflow Boiling Vertical Round Tube under High Pressure

Authors: Wei Liu; Jianqiang Shan; Shinian Peng; Guangming Jiang; Yu Liu
Journal: Science and Technology of Nuclear Installations
 (2019)

---

## Abstract

The Critical Heat Flux (CHF) prediction under high pressure condition, even close to the vicinity of the critical pressure of water, is an important issue. Although there are many empirical CHF correlations, most of them have covered the pressure under 15MPa. In this study, based on the CHF experiment database of upflow boiling in vertical round tube from 15MPa to the vicinity of the critical pressure of water, the Katto, Bowring, Hall-Mudawar, Alekseev correlations, and Groeneveld LUT-2006 are comparatively studied. With an error analysis of the predicted CHF to the experiment database, the prediction capability and the applicability of these correlations are evaluated and the parametric trends of CHF varying with pressure from 15MPa to critical pressure are proposed. Simultaneously, according to the characteristics of Departure from Nucleate Boiling (DNB) type CHF under high pressure condition, the constitutive correlations of Weisman & Pei model are proposed. The prediction results of three entrainment and deposition correlations of Kataoka, Celata, and Hewitt corresponding to the Dry-Out (DO) type CHF are analyzed. Based on the two improved models above, a comprehensive CHF mechanistic model under high pressure condition combining the DNB and DO type CHF is established. The verification based on the experiment database of upflow boiling in vertical round tube and the parametric trends analysis of CHF varying with thermal-hydraulic and geometric parameters are carried out. Findings of this study have a positive effect on further development of CHF prediction method for universal CHF mechanism, especially under high pressure region.


---

## Body

## 1. Introduction

The accurate prediction of critical heat flux (CHF) in flow boiling is important in the design and safety analysis of nuclear reactor. The occurrence of CHF results in a sharp degradation of the convective heat transfer between the fuel rod cladding and the reactor coolant which may result in cladding failure.The supercritical water cooled reactor (SCWR) has high operating pressure and temperature, and, during sliding pressure start-up procedure from subcritical pressure to supercritical pressure, the thermophysical properties and transport properties of the coolant in the core would change greatly [1]. Thus, the CHF prediction under high pressure condition, even close to the vicinity of the critical pressure of water, is an important issue for SCWR.Although the CHF phenomenon has been extensively investigated over the last five decades, knowledge of the physical nature of CHF is still incomplete and the mechanisms of boiling crisis are still not well understood.Methods for predicting CHF can be categorized as empirical correlations, look-up tables, and mechanistic models. According to the statistics [2], the number of published CHF correlations for water-cooled round tubes has increased to well over 500 and there are also over 50 CHF models available. Based on the difference of flow regime and heat transfer characteristics on the occurrence of CHF, the boiling crisis can be generally divided into DNB (Departure from Nucleate Boiling) type and DO (Dry-Out) type [3].Although there are many empirical CHF correlations, there is no valid and accurate CHF correlation verified by experiment database in the range from 15 MPa to the vicinity of critical pressure, in which the parametric trend of CHF varying with pressure is also unknown.Furthermore, different CHF mechanistic models usually are proposed for specific flow regime, and one CHF model is only effective in one flow regime resulting in a narrow prediction scope. This cannot satisfy the prediction of CHF when different flow regimes appear successively in the same channel.In this study, based on the CHF experiment database of upflow boiling in vertical round tube from 15MPa to the vicinity of the critical pressure of water, the prediction capability, and the applicability of Katto [4], Bowring [5], Hall-Mudawar [6], Alekseev [7] correlations, and LUT-2006 [8] will be evaluated. The parametric trend of CHF varying with pressure from 15MPa to the vicinity of critical pressure will be discussed. Simultaneously, combining the DNB and DO type CHF mechanism, a comprehensive CHF model under high pressure condition will be established. Finally, the verification of the CHF mechanistic model based on the experiment database and the parametric trends analysis of CHF varying with thermal-hydraulic and geometric parameters will be carried out.


## 2. The Experiment Database

Based on the existing CHF database of upflow boiling vertical round tube, the evaluation and screening of experiment database are carried out. As a result, 18 different sets of CHF experiment database were obtained from 1951 to 1999, in total of 4355 data points are applied in this study. The database sources and distribution are shown in Table1, among which 2735 experiment data points belong to the DNB type CHF of subcooled bubbly flow and 1620 experiment data points belong to the DO type CHF of saturated annular flow. Table 2 shows the experimental ranges of database.Table 1
CHF experiment database under high pressure region.
Reference Time No. of data McGill & Sibbitt [17] 1951 9 Epstein et al. [18] 1956 90 Ornatskii & Kichigin [19] 1962 31 Ornatskii [20] 1963 69 Alekseev et al. [21] 1964 508 Bailey & Lee [22] 1969 41 Peskov et al. [23] 1969 127 Zenkevich et al. [24] 1969 1298 Zenkevich et al. [25] 1971 152 Zenkevich [26] 1974 192 Belyakov et al. [27] 1976 360 Smolin et al. [28] 1979 393 Williams & Beus [29] 1980 25 Kirillov et al. [30] 1984 506 Groeneveld [31] 1985 8 Yin et al. [32] 1988 130 Soderquist [33] 1994 412 Mudawar & Bowers [34] 1999 4 Total 4355Table 2
Experimental ranges of database.
Variable Δ h i n(kJ/kg) G (kg/m2s) P(MPa) D (m) L(m) x e The DNB type CHF  Maximum value 1811 7530 21.20 0.016 7 0.54 Minimum value 36 256 15.17 0.0055 0.0254 -2.41  The DO type CHF  Maximum value 1549 6578 20.10 0.016 20 0.97 Minimum value 36 156 15.17 0.0055 0.79 0.02


## 3. The Comparative Study of CHF Correlations

### 3.1. CHF Correlations

Through the comparison and analysis of dozens of round tube CHF correlations, this study has selected the Katto [4], Bowring [5], Hall-Mudawar [6], Alekseev [7] correlations, and LUT-2006 [8] which all cover the high-pressure region for further analysis. The parameter range of each correlation is shown in Table 3.Table 3
Parameter ranges of CHF correlations.
Variable Katto Bowring Hall-Mudawar Alekseev LUT-2006 P (MPa) 0.5~20 0.2~19.0 1~20 9.81~19.62 0.1~20 G (kg/m 2s) 10.5~8800 136~18600 300~30000 1000~5000 0~8000 x e - - -1.00~-0.05 0~0.4 0.5~1 D (m) 0.001~0.038 0.002~0.045 0.00025~0.015 0.004~0.012 0.002~0.016 L (m) 0.01~8.8 0.15~3.7 0.3~8 ⩾0.2 -


### 3.2. Comparative Analysis

For analysis, the errorE, mean errorME, mean absolute errorMAE, and root mean square errorRMS are defined as follows:ErrorE:(1)E=qCHF,pre-qCHF,expqCHF,expMean errorME:(2)ME=1N∑qCHF,pre-qCHF,expqCHF,exp×100%Mean absolute errorMAE:(3)MAE=1N∑qCHF,pre-qCHF,expqCHF,exp×100%Root mean square errorRMS:(4)RMS=1N∑qCHF,pre-qCHF,expqCHF,exp2×100%where qCHF,pre is the predictive value of CHF, while qCHF,exp is the experimental value of CHF.


#### 3.2.1. Prediction Error Analysis

In evaluation of the correlations and LUT-2006 accuracy, the Heat Balance Method (HBM) has been used. The prediction errors of Katto, Bowring, Hall-Mudawar, Alekseev correlations, and LUT-2006 are shown in Table4. It demonstrates that the prediction errors of LUT-2006 and Alekseev correlation are relatively smaller in the high pressure region above 15MPa, while the prediction errors of Katto and Bowring correlation are larger. Although the Hall-Mudawar correlation is only applicable for subcooled boiling, its prediction error is also not large which belongs to the middle level between the LUT-2006 and Alekseev and Katto and Bowring correlations.Table 4
Prediction error of each CHF correlation.
CHF correlations ME MAE RMS P > 15 MPa P > 19 MPa P > 15 MPa P > 19 MPa P > 15 MPa P > 19 MPa Katto 14.2% 6.7% 15.6% 8.0% 23.4% 14.1% Bowring -14.2% -16.6% 15.8% 18.5% 20.4% 23.2% Hall-Mudawar -8.3% -6.1% 9.4% 7.3% 12.8% 9.8% Alekseev 0.1% 0.7% 6.3% 8.6% 9.6% 9.0% LUT-2006 1.3% 1.1% 4.4% 5.5% 7.9% 8.4%The prediction errors of Katto, Alekseev correlation, and LUT-2006 varying with pressure are shown in Figure1.Figure 1
Prediction error of Katto, Alekseev, and LUT-2006 varying with pressure.It indicates that, in the range of 19MPa to 20.3MPa, the Katto correlation overpredicts the experimental value, but when the pressure is higher than 20.3MPa, the predictive value and the experimental value become closer.Although theRMS errors of the predictive value and the experimental value for Alekseev correlation and LUT-2006 are smaller, in the range of 20.3MPa to 21.3MPa, these two correlations underpredict the experimental values.When the pressure is higher than 21.3MPa to the vicinity of critical pressure, the prediction capabilities of the three correlations could not be evaluated due to the lack of CHF experiment database.


#### 3.2.2. Parametric Trend Analysis

In the vicinity of critical pressure, the physical properties and heat transfer characteristics of water have greatly changed; consequently, the CHF has become more sensitive to pressure. For simplicity, Figure2 shows the predictive value of different CHF correlation varying with pressure when other parametersG, △hin,D, andLare fixed constant.Figure 2
q C H F as a function of pressure (G=2000 kg/m2s,△hin=450 kJ/kg,D=0.013m,L=3m).Combined with the error comparison analysis in Section3.2.1, the parametric trend of CHF in high pressure region can be summarized as follows (identified by the red dotted line on Figure 2): in the range of 19MPa to 20.3MPa, it is close to the prediction trend of LUT-2006; when the pressure is higher than 20.3MPa, it should be closer to the prediction trend of Katto correlation; and when approaching critical pressure, the CHF quickly goes down to zero.


## 3.1. CHF Correlations

Through the comparison and analysis of dozens of round tube CHF correlations, this study has selected the Katto [4], Bowring [5], Hall-Mudawar [6], Alekseev [7] correlations, and LUT-2006 [8] which all cover the high-pressure region for further analysis. The parameter range of each correlation is shown in Table 3.Table 3
Parameter ranges of CHF correlations.
Variable Katto Bowring Hall-Mudawar Alekseev LUT-2006 P (MPa) 0.5~20 0.2~19.0 1~20 9.81~19.62 0.1~20 G (kg/m 2s) 10.5~8800 136~18600 300~30000 1000~5000 0~8000 x e - - -1.00~-0.05 0~0.4 0.5~1 D (m) 0.001~0.038 0.002~0.045 0.00025~0.015 0.004~0.012 0.002~0.016 L (m) 0.01~8.8 0.15~3.7 0.3~8 ⩾0.2 -


## 3.2. Comparative Analysis

For analysis, the errorE, mean errorME, mean absolute errorMAE, and root mean square errorRMS are defined as follows:ErrorE:(1)E=qCHF,pre-qCHF,expqCHF,expMean errorME:(2)ME=1N∑qCHF,pre-qCHF,expqCHF,exp×100%Mean absolute errorMAE:(3)MAE=1N∑qCHF,pre-qCHF,expqCHF,exp×100%Root mean square errorRMS:(4)RMS=1N∑qCHF,pre-qCHF,expqCHF,exp2×100%where qCHF,pre is the predictive value of CHF, while qCHF,exp is the experimental value of CHF.


### 3.2.1. Prediction Error Analysis

In evaluation of the correlations and LUT-2006 accuracy, the Heat Balance Method (HBM) has been used. The prediction errors of Katto, Bowring, Hall-Mudawar, Alekseev correlations, and LUT-2006 are shown in Table4. It demonstrates that the prediction errors of LUT-2006 and Alekseev correlation are relatively smaller in the high pressure region above 15MPa, while the prediction errors of Katto and Bowring correlation are larger. Although the Hall-Mudawar correlation is only applicable for subcooled boiling, its prediction error is also not large which belongs to the middle level between the LUT-2006 and Alekseev and Katto and Bowring correlations.Table 4
Prediction error of each CHF correlation.
CHF correlations ME MAE RMS P > 15 MPa P > 19 MPa P > 15 MPa P > 19 MPa P > 15 MPa P > 19 MPa Katto 14.2% 6.7% 15.6% 8.0% 23.4% 14.1% Bowring -14.2% -16.6% 15.8% 18.5% 20.4% 23.2% Hall-Mudawar -8.3% -6.1% 9.4% 7.3% 12.8% 9.8% Alekseev 0.1% 0.7% 6.3% 8.6% 9.6% 9.0% LUT-2006 1.3% 1.1% 4.4% 5.5% 7.9% 8.4%The prediction errors of Katto, Alekseev correlation, and LUT-2006 varying with pressure are shown in Figure1.Figure 1
Prediction error of Katto, Alekseev, and LUT-2006 varying with pressure.It indicates that, in the range of 19MPa to 20.3MPa, the Katto correlation overpredicts the experimental value, but when the pressure is higher than 20.3MPa, the predictive value and the experimental value become closer.Although theRMS errors of the predictive value and the experimental value for Alekseev correlation and LUT-2006 are smaller, in the range of 20.3MPa to 21.3MPa, these two correlations underpredict the experimental values.When the pressure is higher than 21.3MPa to the vicinity of critical pressure, the prediction capabilities of the three correlations could not be evaluated due to the lack of CHF experiment database.


### 3.2.2. Parametric Trend Analysis

In the vicinity of critical pressure, the physical properties and heat transfer characteristics of water have greatly changed; consequently, the CHF has become more sensitive to pressure. For simplicity, Figure2 shows the predictive value of different CHF correlation varying with pressure when other parametersG, △hin,D, andLare fixed constant.Figure 2
q C H F as a function of pressure (G=2000 kg/m2s,△hin=450 kJ/kg,D=0.013m,L=3m).Combined with the error comparison analysis in Section3.2.1, the parametric trend of CHF in high pressure region can be summarized as follows (identified by the red dotted line on Figure 2): in the range of 19MPa to 20.3MPa, it is close to the prediction trend of LUT-2006; when the pressure is higher than 20.3MPa, it should be closer to the prediction trend of Katto correlation; and when approaching critical pressure, the CHF quickly goes down to zero.


## 3.2.1. Prediction Error Analysis

In evaluation of the correlations and LUT-2006 accuracy, the Heat Balance Method (HBM) has been used. The prediction errors of Katto, Bowring, Hall-Mudawar, Alekseev correlations, and LUT-2006 are shown in Table4. It demonstrates that the prediction errors of LUT-2006 and Alekseev correlation are relatively smaller in the high pressure region above 15MPa, while the prediction errors of Katto and Bowring correlation are larger. Although the Hall-Mudawar correlation is only applicable for subcooled boiling, its prediction error is also not large which belongs to the middle level between the LUT-2006 and Alekseev and Katto and Bowring correlations.Table 4
Prediction error of each CHF correlation.
CHF correlations ME MAE RMS P > 15 MPa P > 19 MPa P > 15 MPa P > 19 MPa P > 15 MPa P > 19 MPa Katto 14.2% 6.7% 15.6% 8.0% 23.4% 14.1% Bowring -14.2% -16.6% 15.8% 18.5% 20.4% 23.2% Hall-Mudawar -8.3% -6.1% 9.4% 7.3% 12.8% 9.8% Alekseev 0.1% 0.7% 6.3% 8.6% 9.6% 9.0% LUT-2006 1.3% 1.1% 4.4% 5.5% 7.9% 8.4%The prediction errors of Katto, Alekseev correlation, and LUT-2006 varying with pressure are shown in Figure1.Figure 1
Prediction error of Katto, Alekseev, and LUT-2006 varying with pressure.It indicates that, in the range of 19MPa to 20.3MPa, the Katto correlation overpredicts the experimental value, but when the pressure is higher than 20.3MPa, the predictive value and the experimental value become closer.Although theRMS errors of the predictive value and the experimental value for Alekseev correlation and LUT-2006 are smaller, in the range of 20.3MPa to 21.3MPa, these two correlations underpredict the experimental values.When the pressure is higher than 21.3MPa to the vicinity of critical pressure, the prediction capabilities of the three correlations could not be evaluated due to the lack of CHF experiment database.


## 3.2.2. Parametric Trend Analysis

In the vicinity of critical pressure, the physical properties and heat transfer characteristics of water have greatly changed; consequently, the CHF has become more sensitive to pressure. For simplicity, Figure2 shows the predictive value of different CHF correlation varying with pressure when other parametersG, △hin,D, andLare fixed constant.Figure 2
q C H F as a function of pressure (G=2000 kg/m2s,△hin=450 kJ/kg,D=0.013m,L=3m).Combined with the error comparison analysis in Section3.2.1, the parametric trend of CHF in high pressure region can be summarized as follows (identified by the red dotted line on Figure 2): in the range of 19MPa to 20.3MPa, it is close to the prediction trend of LUT-2006; when the pressure is higher than 20.3MPa, it should be closer to the prediction trend of Katto correlation; and when approaching critical pressure, the CHF quickly goes down to zero.


## 4. The Development of CHF Mechanistic Model

### 4.1. DNB Type CHF Mechanistic Model

Weisman & Pei [9] developed a phenomenological model for CHF at low void fractions or subcooled conditions. The model is generally applicable in the bubbly flow regime, where it is assumed that a bubbly layer exists adjacent to the heater surface. At high heat fluxes, the local vapour generation rate becomes so high that it prevents the liquid from reaching and cooling the heated surface, leading to CHF, as shown in Figure 3.Figure 3
Schematic diagram of Weisman & Pei model.Weisman & Pei utilized a number of assumptions in the development of the model and obtained(5)q′′DNBhfgG′=x2-x1hf-hldhl-hldThe quantityG’ represents the total mass velocity into the bubbly layer, due to turbulent interchange at the edge of the bubbly layer.(6)G′=ψIbGThe parameterIb represents the turbulent intensity at the bubbly layer/core interface. (7)Ib=0.462k0.6Re-0.1Dbd0.61+aρl-ρvρvThe parametersk (= 2.4) anda (which depends on the mass velocity) were empirically determined by fitting a large number of uniform heat flux experiment data. Therefore, the validity of the Weisman & Pei model is limited by the ranges of the databases, from which these empirical coefficients were obtained.In this study, based on the experiment database of upflow boiling in vertical round tube under high pressure condition, the new parameterk=1.8 is obtained and parametera is associated with the velocity change and pressure effect (through ρv/ρl).The final expression is as follows:(8)a=1.23-1.25u+5.47×ρvρl2.3u<1.2m/s-0.15×3-u+5.47×ρvρl2.31.2m/s≤u<3m/s5.47×ρvρl2.3×u30.3u≥3m/s


### 4.2. DO Type CHF Mechanistic Model

The majority of the available annular flow DO models are based on that of Whalley et al. [10] which is a three-field model representing two-phase interactions between vapour, liquid film, and entrained droplets, as shown in Figure 4. The difference from each model is in the constitutive correlations representing the mechanisms of entrainment and deposition. In fact, it is these correlations that distinguish one phenomenological DO model from another, since the basic conservation equations are the same for any DO model in the annular flow regime.Figure 4
Schematic diagram of DO model.In this study, three different entrainment and deposition correlations of Kataoka, Celata, and Hewitt are compared and analyzed.The droplet deposition rate is calculated from(9)D=kCwhereC is the concentration of droplets in the vapour core andk is the deposition mass transfer coefficient.(  1) Kataoka Correlation. Kataoka et al. [11] developed correlations for entrainment rate covering both entrance region and equilibrium region from a simple model in collaboration with data.The deposition mass transfer coefficient k is determined by Paleev & Filippovich correlation [12].(10)kjg=0.022Reg-0.25Cρf-0.26ρgρf0.26The entrainment fraction is calculated by Ishii & Mishima [13] correlation which was developed based on the mechanistic model of shearing-off of roll wave crest by a streaming gas.(  2) Celata Correlation. In developing of Celata et al.’s [14] DO type CHF model, liquid film flow rate is obtained by a balance of liquid entrainment and droplet deposition.The droplets deposition rate is calculated with the equation given by Kataoka & Ishii [15].(11)dDμl=0.22Rel0.74μvμl0.26E0.74The droplets entrainment rate is calculated considering the contribution of two different mechanisms of droplets formation: breakup of disturbance waves (Ew) and boiling in the liquid film (EB).(12)E=Ew+EB(  3) Hewitt Correlation. Hewitt et al. [16] derived improved models for deposition and entrainment in annular flow. The new models successfully predicted a wide range of equilibrium and non-equilibrium data.The correlation for the deposition rate coefficient is as follows:(13)k=0.18σρvd,Cρv≤0.30.083σρvdCρv-0.65,Cρv>0.3The entrainment correlation is(14)EGv=5.75×10-5GLF-GLFC2dρlσρv20.316,GLF>GLFC0,GLF≤GLFCwhere GLFC is the critical film mass velocity for the onset of entrainment.


### 4.3. The Comprehensive CHF Mechanistic Model

According to the two improved CHF mechanistic model types above, a comprehensive CHF mechanistic model under high pressure condition combining the DNB and DO type CHF is established. The detailed calculation process of the present mechanistic model is shown in Figure5.Figure 5
Flow chart of detailed calculation.The void fractionαann is the transition point of bubble flow and annular flow. When α<αann, the flow regime is bubble flow and the CHF is calculated by the DNB model. When α>αann, the flow regime is annular flow and the CHF calculation is divided into two situations: (1) if α<αDNB, the DNB model is still used; (2) if no DNB occurrence or α>αDNB, then the DO model is used to calculate the flow rate of liquid film.


### 4.4. Results Analysis

#### 4.4.1. Prediction Error Analysis

In evaluation of the present mechanistic model accuracy, the Heat Balance Method (HBM) has been used. Table5 shows the prediction error of the present mechanistic model for the whole experiment database and the DNB type CHF, respectively. Table 6 shows the prediction results of three different entrainment and deposition correlations.Table 5
Prediction error of the present mechanistic model.
Data points Maximum error Minimum error The fraction of error within 10% The fraction of error within 20% ME RMS The whole experiment database  4355 44.2% -38.9% 82.2% 96.0% 0.5% 8.3%  The DNB type CHF  2735 44.5% -46.0% 85.8% 96.4% 0.6% 7.0%Table 6
Prediction results of three different entrainment and deposition correlations.
Data points Different correlation Maximum error Minimum error The fraction of error within 10% The fraction of error within 20% ME RMS 1620 Kataoka 34.7% -38.9% 72.1% 94.5% 0.4% 9.9% Celata 40.3% -61.3% 36.2% 49.0% 27.9% 40.1% Hewitt 14.4% -86.1% 2.9% 15.8% 40.6% 45.0%It demonstrates that the present mechanistic model is applicable for the CHF prediction of upflow boiling in vertical round tube under high pressure conditions, and theRMS of the DNB model is the lowest. For the DO type CHF prediction, the results of the Kataoka correlation are the best.The comparison of predicted CHF and measured CHF is shown in Figure6.Figure 6
Predicted CHF as a function of measured CHF.Figure7 shows the prediction results of the present mechanistic model, Katto correlation, and LUT-2006. It indicates that the present mechanistic model is more accurate than the other two predictions, especially when pressure is higher than 19MPa.Figure 7
RMS histograms of each method.


#### 4.4.2. Continuity of the Developed Model

The significant characteristic of the present mechanistic model is that, for a certain flow regime, it can automatically judge and select the particular model to calculate the CHF value.Figures8 and 9 show the continuous variation of DNB and DO type CHF predicted by the present mechanistic model with mass velocity and inlet subcooled enthalpy, respectively. It indicates that the DO type CHF occurs at low mass velocity and low inlet subcooled enthalpy. With the increase of mass velocity and inlet subcooled enthalpy, the DNB type CHF occurs. In this study, the DNB and DO type CHF can be smoothly joined together with the present mechanistic model.Figure 8
Continuous variation of CHF with mass velocity.Figure 9
Continuous variation of CHF with inlet subcooled enthalpy.


#### 4.4.3. Parametric Trend Analysis

The parametric trends of the CHF vary according to the thermal-hydraulics conditions determined by the combination of the various ranges of pressure, mass velocity, inlet subcooled enthalpy, and geometric parameters.The CHF predicted by the present mechanistic model as a function of independent variables pressure, mass velocity, inlet subcooled enthalpy, tube diameter, and tube length are shown in Figures10–14, respectively. It indicates that the predicted CHF decreases with the increase of pressure. When approaching the critical pressure, CHF rapidly drops to zero. The predicted CHF almost linearly increases with the increase of mass velocity and inlet subcooled enthalpy. As for the geometric parameters, the predicted CHF increases with the increase of tube diameter and decreases with the increase of tube length. While beyond the threshold ofD orL, the influence of theD andL is small.Figure 10
CHF as a function of pressure (G=1000kg/m2s).Figure 11
CHF as a function of mass velocity (Δhin=500kJ/kg).Figure 12
CHF as a function of inlet subcooled enthalpy (P=22MPa).Figure 13
CHF as a function of tube diameter (P=17MPa,G=1000kg/m2s, L=1m).Figure 14
CHF as a function of tube length (P=21MPa, Δhin=100kJ/kg, D=0.008m).


## 4.1. DNB Type CHF Mechanistic Model

Weisman & Pei [9] developed a phenomenological model for CHF at low void fractions or subcooled conditions. The model is generally applicable in the bubbly flow regime, where it is assumed that a bubbly layer exists adjacent to the heater surface. At high heat fluxes, the local vapour generation rate becomes so high that it prevents the liquid from reaching and cooling the heated surface, leading to CHF, as shown in Figure 3.Figure 3
Schematic diagram of Weisman & Pei model.Weisman & Pei utilized a number of assumptions in the development of the model and obtained(5)q′′DNBhfgG′=x2-x1hf-hldhl-hldThe quantityG’ represents the total mass velocity into the bubbly layer, due to turbulent interchange at the edge of the bubbly layer.(6)G′=ψIbGThe parameterIb represents the turbulent intensity at the bubbly layer/core interface. (7)Ib=0.462k0.6Re-0.1Dbd0.61+aρl-ρvρvThe parametersk (= 2.4) anda (which depends on the mass velocity) were empirically determined by fitting a large number of uniform heat flux experiment data. Therefore, the validity of the Weisman & Pei model is limited by the ranges of the databases, from which these empirical coefficients were obtained.In this study, based on the experiment database of upflow boiling in vertical round tube under high pressure condition, the new parameterk=1.8 is obtained and parametera is associated with the velocity change and pressure effect (through ρv/ρl).The final expression is as follows:(8)a=1.23-1.25u+5.47×ρvρl2.3u<1.2m/s-0.15×3-u+5.47×ρvρl2.31.2m/s≤u<3m/s5.47×ρvρl2.3×u30.3u≥3m/s


## 4.2. DO Type CHF Mechanistic Model

The majority of the available annular flow DO models are based on that of Whalley et al. [10] which is a three-field model representing two-phase interactions between vapour, liquid film, and entrained droplets, as shown in Figure 4. The difference from each model is in the constitutive correlations representing the mechanisms of entrainment and deposition. In fact, it is these correlations that distinguish one phenomenological DO model from another, since the basic conservation equations are the same for any DO model in the annular flow regime.Figure 4
Schematic diagram of DO model.In this study, three different entrainment and deposition correlations of Kataoka, Celata, and Hewitt are compared and analyzed.The droplet deposition rate is calculated from(9)D=kCwhereC is the concentration of droplets in the vapour core andk is the deposition mass transfer coefficient.(  1) Kataoka Correlation. Kataoka et al. [11] developed correlations for entrainment rate covering both entrance region and equilibrium region from a simple model in collaboration with data.The deposition mass transfer coefficient k is determined by Paleev & Filippovich correlation [12].(10)kjg=0.022Reg-0.25Cρf-0.26ρgρf0.26The entrainment fraction is calculated by Ishii & Mishima [13] correlation which was developed based on the mechanistic model of shearing-off of roll wave crest by a streaming gas.(  2) Celata Correlation. In developing of Celata et al.’s [14] DO type CHF model, liquid film flow rate is obtained by a balance of liquid entrainment and droplet deposition.The droplets deposition rate is calculated with the equation given by Kataoka & Ishii [15].(11)dDμl=0.22Rel0.74μvμl0.26E0.74The droplets entrainment rate is calculated considering the contribution of two different mechanisms of droplets formation: breakup of disturbance waves (Ew) and boiling in the liquid film (EB).(12)E=Ew+EB(  3) Hewitt Correlation. Hewitt et al. [16] derived improved models for deposition and entrainment in annular flow. The new models successfully predicted a wide range of equilibrium and non-equilibrium data.The correlation for the deposition rate coefficient is as follows:(13)k=0.18σρvd,Cρv≤0.30.083σρvdCρv-0.65,Cρv>0.3The entrainment correlation is(14)EGv=5.75×10-5GLF-GLFC2dρlσρv20.316,GLF>GLFC0,GLF≤GLFCwhere GLFC is the critical film mass velocity for the onset of entrainment.


## 4.3. The Comprehensive CHF Mechanistic Model

According to the two improved CHF mechanistic model types above, a comprehensive CHF mechanistic model under high pressure condition combining the DNB and DO type CHF is established. The detailed calculation process of the present mechanistic model is shown in Figure5.Figure 5
Flow chart of detailed calculation.The void fractionαann is the transition point of bubble flow and annular flow. When α<αann, the flow regime is bubble flow and the CHF is calculated by the DNB model. When α>αann, the flow regime is annular flow and the CHF calculation is divided into two situations: (1) if α<αDNB, the DNB model is still used; (2) if no DNB occurrence or α>αDNB, then the DO model is used to calculate the flow rate of liquid film.


## 4.4. Results Analysis

### 4.4.1. Prediction Error Analysis

In evaluation of the present mechanistic model accuracy, the Heat Balance Method (HBM) has been used. Table5 shows the prediction error of the present mechanistic model for the whole experiment database and the DNB type CHF, respectively. Table 6 shows the prediction results of three different entrainment and deposition correlations.Table 5
Prediction error of the present mechanistic model.
Data points Maximum error Minimum error The fraction of error within 10% The fraction of error within 20% ME RMS The whole experiment database  4355 44.2% -38.9% 82.2% 96.0% 0.5% 8.3%  The DNB type CHF  2735 44.5% -46.0% 85.8% 96.4% 0.6% 7.0%Table 6
Prediction results of three different entrainment and deposition correlations.
Data points Different correlation Maximum error Minimum error The fraction of error within 10% The fraction of error within 20% ME RMS 1620 Kataoka 34.7% -38.9% 72.1% 94.5% 0.4% 9.9% Celata 40.3% -61.3% 36.2% 49.0% 27.9% 40.1% Hewitt 14.4% -86.1% 2.9% 15.8% 40.6% 45.0%It demonstrates that the present mechanistic model is applicable for the CHF prediction of upflow boiling in vertical round tube under high pressure conditions, and theRMS of the DNB model is the lowest. For the DO type CHF prediction, the results of the Kataoka correlation are the best.The comparison of predicted CHF and measured CHF is shown in Figure6.Figure 6
Predicted CHF as a function of measured CHF.Figure7 shows the prediction results of the present mechanistic model, Katto correlation, and LUT-2006. It indicates that the present mechanistic model is more accurate than the other two predictions, especially when pressure is higher than 19MPa.Figure 7
RMS histograms of each method.


### 4.4.2. Continuity of the Developed Model

The significant characteristic of the present mechanistic model is that, for a certain flow regime, it can automatically judge and select the particular model to calculate the CHF value.Figures8 and 9 show the continuous variation of DNB and DO type CHF predicted by the present mechanistic model with mass velocity and inlet subcooled enthalpy, respectively. It indicates that the DO type CHF occurs at low mass velocity and low inlet subcooled enthalpy. With the increase of mass velocity and inlet subcooled enthalpy, the DNB type CHF occurs. In this study, the DNB and DO type CHF can be smoothly joined together with the present mechanistic model.Figure 8
Continuous variation of CHF with mass velocity.Figure 9
Continuous variation of CHF with inlet subcooled enthalpy.


### 4.4.3. Parametric Trend Analysis

The parametric trends of the CHF vary according to the thermal-hydraulics conditions determined by the combination of the various ranges of pressure, mass velocity, inlet subcooled enthalpy, and geometric parameters.The CHF predicted by the present mechanistic model as a function of independent variables pressure, mass velocity, inlet subcooled enthalpy, tube diameter, and tube length are shown in Figures10–14, respectively. It indicates that the predicted CHF decreases with the increase of pressure. When approaching the critical pressure, CHF rapidly drops to zero. The predicted CHF almost linearly increases with the increase of mass velocity and inlet subcooled enthalpy. As for the geometric parameters, the predicted CHF increases with the increase of tube diameter and decreases with the increase of tube length. While beyond the threshold ofD orL, the influence of theD andL is small.Figure 10
CHF as a function of pressure (G=1000kg/m2s).Figure 11
CHF as a function of mass velocity (Δhin=500kJ/kg).Figure 12
CHF as a function of inlet subcooled enthalpy (P=22MPa).Figure 13
CHF as a function of tube diameter (P=17MPa,G=1000kg/m2s, L=1m).Figure 14
CHF as a function of tube length (P=21MPa, Δhin=100kJ/kg, D=0.008m).


## 4.4.1. Prediction Error Analysis

In evaluation of the present mechanistic model accuracy, the Heat Balance Method (HBM) has been used. Table5 shows the prediction error of the present mechanistic model for the whole experiment database and the DNB type CHF, respectively. Table 6 shows the prediction results of three different entrainment and deposition correlations.Table 5
Prediction error of the present mechanistic model.
Data points Maximum error Minimum error The fraction of error within 10% The fraction of error within 20% ME RMS The whole experiment database  4355 44.2% -38.9% 82.2% 96.0% 0.5% 8.3%  The DNB type CHF  2735 44.5% -46.0% 85.8% 96.4% 0.6% 7.0%Table 6
Prediction results of three different entrainment and deposition correlations.
Data points Different correlation Maximum error Minimum error The fraction of error within 10% The fraction of error within 20% ME RMS 1620 Kataoka 34.7% -38.9% 72.1% 94.5% 0.4% 9.9% Celata 40.3% -61.3% 36.2% 49.0% 27.9% 40.1% Hewitt 14.4% -86.1% 2.9% 15.8% 40.6% 45.0%It demonstrates that the present mechanistic model is applicable for the CHF prediction of upflow boiling in vertical round tube under high pressure conditions, and theRMS of the DNB model is the lowest. For the DO type CHF prediction, the results of the Kataoka correlation are the best.The comparison of predicted CHF and measured CHF is shown in Figure6.Figure 6
Predicted CHF as a function of measured CHF.Figure7 shows the prediction results of the present mechanistic model, Katto correlation, and LUT-2006. It indicates that the present mechanistic model is more accurate than the other two predictions, especially when pressure is higher than 19MPa.Figure 7
RMS histograms of each method.


## 4.4.2. Continuity of the Developed Model

The significant characteristic of the present mechanistic model is that, for a certain flow regime, it can automatically judge and select the particular model to calculate the CHF value.Figures8 and 9 show the continuous variation of DNB and DO type CHF predicted by the present mechanistic model with mass velocity and inlet subcooled enthalpy, respectively. It indicates that the DO type CHF occurs at low mass velocity and low inlet subcooled enthalpy. With the increase of mass velocity and inlet subcooled enthalpy, the DNB type CHF occurs. In this study, the DNB and DO type CHF can be smoothly joined together with the present mechanistic model.Figure 8
Continuous variation of CHF with mass velocity.Figure 9
Continuous variation of CHF with inlet subcooled enthalpy.


## 4.4.3. Parametric Trend Analysis

The parametric trends of the CHF vary according to the thermal-hydraulics conditions determined by the combination of the various ranges of pressure, mass velocity, inlet subcooled enthalpy, and geometric parameters.The CHF predicted by the present mechanistic model as a function of independent variables pressure, mass velocity, inlet subcooled enthalpy, tube diameter, and tube length are shown in Figures10–14, respectively. It indicates that the predicted CHF decreases with the increase of pressure. When approaching the critical pressure, CHF rapidly drops to zero. The predicted CHF almost linearly increases with the increase of mass velocity and inlet subcooled enthalpy. As for the geometric parameters, the predicted CHF increases with the increase of tube diameter and decreases with the increase of tube length. While beyond the threshold ofD orL, the influence of theD andL is small.Figure 10
CHF as a function of pressure (G=1000kg/m2s).Figure 11
CHF as a function of mass velocity (Δhin=500kJ/kg).Figure 12
CHF as a function of inlet subcooled enthalpy (P=22MPa).Figure 13
CHF as a function of tube diameter (P=17MPa,G=1000kg/m2s, L=1m).Figure 14
CHF as a function of tube length (P=21MPa, Δhin=100kJ/kg, D=0.008m).


## 5. Conclusion

In this study, based on the CHF experiment database of upflow boiling in vertical round tube from 15MPa to the vicinity of the critical pressure of water, five CHF correlations under high pressure conditions are selected and the prediction results have been comparatively analyzed. Simultaneously, a comprehensive CHF mechanistic model under high pressure condition combined the DNB and DO type CHF has been established. The verification of the present mechanistic model based on the experiment database and the parametric trends analysis of CHF varying with thermal-hydraulic and geometric parameters have been carried out. The conclusions can be briefly summarized as follows:(1)
In the high pressure range of 15MPa to the vicinity of critical pressure, the Alekseev correlation and LUT-2006 are recommended for their smaller prediction error to predict CHF in round tube.(2)
The parametric trend of CHF varying with pressure in the range from 15 MPa to the vicinity of critical pressure is obtained. In the transition point of 20.3MPa, the CHF parametric trend varies from the LUT-2006 curve to the Katto correlation curve, and when approaching critical pressure, the CHF quickly goes down to zero.(3)
The present mechanistic model is applicable for the CHF prediction of upflow boiling in vertical round tube under high pressure conditions, and the fraction of error within ±20% is 96.0% of total data points. For the DO type CHF prediction, the prediction results of Kataoka correlation are more accurate than Celata and Hewitt methods, and the wholeRMS is 9.9%.(4)
For a specific flow regime, the present mechanistic model can automatically judge and select the particular model to calculate the CHF value, which can smoothly join the DNB and DO type CHF.(5)
The parametric trends of predicted CHF varying with thermal-hydraulic and geometric parameters are quite similar to those for the conventional models, which are consistent with the physical mechanism and experimental phenomena.



---

*Source: 3695685-2019-06-04.xml*