COMPARISON OF CFD AND EMPIRICAL MODELS FOR PREDICTING WALL TEMPERATURE AT SUPERCRITICAL CONDITIONS OF WATER

The present work investigates the wall temperature prediction at supercritical pressure of water by CFD and compares the prediction of CFD and that of 11 empirical correlations available in literature. Supercritical-water heat transfer experimental data, covering a mass flux range of 400-1500 kg/m2s, heat flux range of 150-1000 kW/m2, at pressure 241 bar and diameter 10 mm tube, were obtained from literature. CFD simulations have been carried out for those operating conditions and compared with experimental data. Around 362 experimental wall temperature data of both heat transfer enhancement and heat transfer deterioration region have been taken for comparison. A visual Basic program has also been developed to predict wall temperature for the selected empirical correlations and compared with those of 362 experimental data. Ranking based on the deviation with experimental data is also listed.


INTRODUCTION
The Advanced Ultra Supercritical Boilers will be the next generation of boilers because it possesses various advantages like relatively low energy cost, low pollutant emission and high efficiency. The Advanced Ultra Supercritical Boilers are aimed to operate at a pressure and temperature of 300 bar and 700 o C respectively with a higher thermal efficiency up to 46%. As the water walls operate at conditions above the critical point of water, which is 221 bar and 374 o C, the fluid will remain as a single phase. Thus no departure from nucleate boiling or dry-out can occur. However in the vicinity of this critical point, strong variations of water properties combined with a high heat flux or low mass flux can lead to heat transfer deterioration (HTD), which consequently causes a severe increase of wall temperature (Cheng et.al., 2003). Thus, the study of heat transfer behavior at pseudocritical region is extremely important. It has been found that the fluid temperature and pressure in the evaporator region varies between 342°C & 440°C and 270 & 290 bar respectively at Boiler Maximum Continuous Rating condition. The pseudocritical temperature at 270 and 290 bar is found to be 392 o C (2177.39 kJ/kg) and 399 o C (2204.21 kJ/kg). This indicates that somewhere in the evaporator region the fluid crosses the pseudocritical temperature. Therefore, it is indeed necessary to study the heat transfer behavior at pseudocritical temperature at the desired pressure range and determine the inner wall temperature.
Various experimental investigations have been performed at supercritical conditions with water flowing upward in circular pipes under intense electrical heating (Bishop et al., 1964;Yamagata et al., 1972;Jackson and Hall, 1979;Zhu et al., 2009) and published empirical correlations applicable for predicting the heat transfer at supercritical conditions of water. Each correlation is applicable for the specified range of operating parameters. These correlations are used for determining the inner wall temperature. Several established dimensionless correlations for heat transfer of supercritical water namely, Dittus-Boelter, 1930;McAdams, 1942, Bishop et al., 1964Shitsman, 1968;Ornatsky et al., 1971;Yamagata et al., 1972;Jackson * and Hall, 1979;Watts and Chou, 1982;Griem, 1996;Zhu et al., 2009;Mokry et al., 2011 etc., are available in the literature to predict the wall temperature. Nevertheless, the comparisons of experimental data and correlations still show considerable disagreement at the regimes of the heat transfer deterioration. This is because the derivation of all the existing correlations was based on their own experimental dataset. This is one of the main drawbacks of empirical correlations. Numerical studies indeed become necessary in order to obtain more insight into the heat transfer behavior in supercritical fluids.
The accuracy of numerical simulations of heat transfer in supercritical may depend on the choice of turbulence models. A number of investigations about the application of turbulence models in the numerical simulation of flow and heat transfer for supercritical water have been carried out. But, standard wall functions of the turbulence models are not capable to predict heat transfer deterioration. Jaromin and Anglart (2013) found that SST k-ω model is capable to predict heat transfer deterioration close to the experimental results. Wen and Gu (2011) also validated few turbulent models and found that SST k-ω is more accurate than other models. Zhi et al. (2016) used SST k-ω model for predicting convective heat transfer to hydro carbon fuel at supercritical pressure and assured that it performs well compared to all other turbulence models under supercritical pressure. He also employed the SST k-ω model in his analysis and validated with experimental results and showed that SST k-ω model is capable for predicting the heat transfer enhancement and heat transfer deterioration. In the present work, simulations were carried out using various turbulence models available in Ansys-Fluent to gain confidence.
The purpose of the present work is to numerically simulate the experimental conditions consisting of 362 data points available in literature for a range of parameters such as heat flux, mass flux, bulk fluid enthalpy, pressure, and tube diameter and compare the wall temperature predicted by CFD and 11 empirical correlations with the experimental data.

Definitions of terms and properties of Supercritical water
Critical point is the point where the distinction between the liquid and gas (or vapor) phases disappears, i.e., both phases have the same temperature, pressure and volume (Fig. 1, Pioro and Duffey, 2005). The critical point is characterized by the phase state parameters Tcr, Pcr and Vcr, which have unique values for each pure substance. Pseudocritical point (characterized with Ppc and Tpc) is a point at a pressure above the critical pressure and at a temperature (Tpc > Tcr) corresponding to the maximum value of the specific heat for this particular pressure.
Supercritical fluid is a fluid at pressures and temperatures that are higher than the critical pressure and critical temperature. Supercritical steam ("steam") is actually supercritical water because at supercritical pressures there is no difference between phases. However, this term is widely used in literature in relation to supercritical steam generators and turbines.
Normal heat transfer (NHT) is characterized by the Heat Transfer Coefficient (HTC) is similar to those of convective heat transfer at subcritical condition that occurs far away from pseudocritical regime and are closely matches with the HTC calculated using Dittus-Boelter equation "Eq. (1)" = 0.0243 0.8 0.4 (1) Heat transfer enhancement (HTE) is characterized by higher values of the wall heat transfer coefficient compared to those at the normal heat transfer regime and hence lower values of wall temperature within some part of a test section or within the entire test section.
Heat transfer deterioration (HTD) is characterized by lower values of the wall heat transfer coefficient compared to those at the normal heat transfer regime and hence has higher values of wall temperature within some part of a test section or within the entire test section. Fig. 2 (Pioro and Duffey, 2005) shows the variation of thermophysical properties for water at 250 bar. The large variation in properties like density, specific heat, viscosity and thermal conductivity occur within +/-25 o C from the pseudo-critical temperature (384.9 o C). This large variation influence heat transfer leading to initial heat transfer enhancement or deterioration depending on local conditions.

Geometry
In the present work, vertical smooth tube of ID 10 mm and length 4 m has been chosen for validation for which experimental results are available in the literature Mokry et al., 2010 andMokry et al., 2011. Therefore, the computational test parameters considered in the present work are same as experiments conducted in Mokry et al., 2010 andMokry et al., 2011. All the simulations in the present work are carried out using ANSYS Fluent 17.2. A 2D axis symmetry geometry has been modeled and shown in Fig. 3.

Fig. 3 Computational geometry
Since the wall temperature is uniform around the circumference of the vertical tube, a 2D model with axis-symmetry has been chosen for simulation in order to reduce the computational time. To take care of entrance effects, a 0.5m of additional length is also provided without heat flux to make the flow fully developed. The physical boundary conditions of the geometry are as follows: a uniform mass flux with inlet fluid temperature is specified at the inlet and a uniform heat flux is applied around the wall boundary for the heated length and zero heat flux is applied on the unheated length of wall boundary. The pressure outlet setting in the Fluent is used as outlet boundary condition and the symmetry condition is used for the axis.

Governing Equations
The basic governing equations, including the conservations of mass (continuity equation), momentum and energy, together with SST k-ω method is used to simulate the unique and complicated turbulent heat transfer characteristics at supercritical pressure (Marcin et al., 2017 andChen, 2011). where, the Reynolds stress term , �����ℎ , � can be presented by turbulence models. By using Boussinesq approximation, the turbulent shear stress can be found from the following equation in which Reynolds stresses are related to the average velocity gradient where, is turbulent viscosity which is flow property; not a fluid property. In the present work, SST k-ω model is used, here, = k-equations are derived from transport equations empirically for turbulent kinetic energy(k) and specific turbulent dissipation rate (ω).
-generation of turbulence kinetic energy due to mean velocity gradients, -generation of turbulence kinetic energy at ω, Y M and Y ω -dissipation of k and ω, Г k and Г ω − effective diffusivity of k and ω, , -user defined source terms. The governing differential equations are solved using the finite volume method. The QUICK scheme is used for approximating the convection terms in momentum and energy equations. The SIMPLE procedure is chosen to couple pressure and velocity. The algebraic equations are solved with ADI methodology. As already mentioned, fluid properties also abruptly change with pressure and temperature, therefore NIST Refprop which is an inbuilt program in Fluent has been used to compute fluid properties. The simulations are stopped when the convergence criteria become less than 10 -6 so as to assure the enough accuracy level. In the present work, a number of turbulence models like SST k-ω, Low re -kε, RNG k-ε, Standard k-ε and Realizable models have been examined. Two numerical case studies were conducted for choosing the best turbulent models, Case I low q/G=0.27 & Case II high q/G=0.67. In case I, heat flux 141 kW/m², mass flux 504 kg/m²s and pressure 241 bar and Case II heat flux 334 kW/m², mass flux 499 kg/m²s and pressure 241 bar were used (Mokry et al., 2011). In both studies, Mokry's et al., 2011 experimental wall temperatures was compared with wall temperature predicted by various turbulence models. Fig. 4 In case I, all the models were closely matches with experiment data. Fig. 5 In case II, where q/G is high, causes heat transfer deterioration, only SST k-ω model follows the wall temperature trend with experiment data. All other models not able to predict sharp rise in wall temperature. Therefore, in the present work SST k-ω model has been used for all the computations.

Grid independence study and validation
As the accuracy of results depends upon the fineness of the grid, great care is required for selecting the grid size. More fineness of the grid increases the computational time. Therefore, grid independence study has been carried out to select the appropriate size of the grid. Any further refinement of the mesh doesn't change the solution. The test has been conducted for the geometry shown in Fig. 3 with various grid size of 60×1200, 80×1200, 100×1200, 120×1200, 140×1200 (radial nodes × axial nodes). Since the change in the parameters in radial direction is larger than the axial direction, non-uniform nodes with a successive ratio of 1.02 in the radial direction to have dense mesh near the wall and uniform nodes in the axial direction were used. Fig. 6 shows the zoomed view of computational mesh to represent fine mesh near the wall and coarse mesh near the axis. The additional 0.5 m length (shown in Fig. 3) is separately divided in to 120×300 grid nodes. In order to choose the appropriate mesh, simulation has been carried out for the experimental operating condition of with pressure 241 bar, heat flux 141 kW/m 2 , mass flux of 504 kg/m 2 with various mesh sizes (Mokry et al., 2011). The obtained wall temperature for various meshes are plotted and compared with experimental data as shown in Fig. 7. It is found that the temperature for meshes 120×1200 and 140×1200 closely matches with experimental data. Also, any further refinement of mesh does not alter the solution. Therefore 120×1200 mesh has been chosen for all the computations. In order to gain confidence, two validations have also been carried out for the pressure

Experiment based empirical correlations in literature
For predicting the heat transfer in turbulent convective heat transfer and for heat transfer at supercritical conditions, following experiment based empirical correlations are available in literature. Dittus -Boelter (1930) introduced a heat-transfer correlation at subcritical pressure for forced convection which is still universally used and is given by Shitsman, (1968) analyzed the heat transfer experimental data of supercritical water flowing inside tubes and then generalized these data with the Dittus-Boelter type correlation: The subscript "min" means minimum Pr value, i.e., either the Pr value evaluated at the bulk fluid temperature or the Pr value evaluated at the wall temperature.

Comparison of metal temperature predictions by CFD and empirical correlations
In an effort to make the evaluation of CFD and correlations, an experimental data from Mokry et al., 2010& Mokry et al., 2011 having 362 data points were selected. The selected data points cover a mass flux range of 300 -1600 kg/m 2 s, a heat flux range of 150-1000 kW/m 2 , a pressure 240 bar and a diameter 10 mm. Out of 362 data points, 141 data points belongs to heat transfer enhancement and 221 data points belong to heat transfer deterioration. First, the experimental conditions are numerically simulated using CFD and 11 correlations identified from the literature are evaluated in the interest of determining the best correlation for the upward vertical flow at the supercritical pressure.

Comparison of CFD and correlations predictions with experimental data for heat transfer enhancement zone.
The selected 141 experimental data obtained from the literature for the heat transfer enhancement zone is shown in Table 1. It covers the range of the heat flux and mass flux ratio from 0.27 to 0.48. The wall temperature predictions obtained by CFD and correlations were compared with the 141 experimental data points belonging to heat transfer enhancement zone as summarized in Table 2 (shown in Appendix). In the present work, 11 established correlations such as Dittus-Boelter, 1930;McAdams, 1942;Griem, 1996;Jackson and Hall, 1979;Shitsman 1968;Bishop et al., 1964;Yamagata et al., 1972;Mokry et al., 2011;Zhu et al., 2009;Watts &Chou, 1982 andOrnatsky et al., 1971 were chosen for comparison. A visual basic code had been developed for predicting the wall temperature using correlations and for ranking the correlations and CFD predictions based on their prediction accuracies. Table 2 provides the wall temperature predicted by CFD and six correlations during heat transfer enhancement conditions. Table 3 provides the ranking information of CFD and empirical correlations on wall temperature prediction accuracies based on number of wall temperature points deviated with 141 experimental data points covering pseudocritical region where there is heat transfer enhancement (no heat transfer deterioration). Among the 11 correlations, Zhu et al. (2009), Jackson andHall, (1974) and Mokry et al. (2011) shows the best agreement with 141 nos of data points, followed by Watts andChou, (1982), Shitsman, (1968) and CFD correlations with 139, 138 and 137 number of data points at less than +/-3°C error level respectively. It is found that the CFD predictions have better agreement with 141 number of experimental data points with 100% of the predictions at less than +/-5°C error level. Fig. 10 provides the histogram of CFD and empirical correlations on wall temperature prediction accuracy based on percentage of error deviation in comparison with 141 experimental data points covering pseudo critical region where there is heat transfer enhancement (no heat transfer deterioration). Among the 11 correlations, CFD shows the best agreement with 141 number of experimental data points at less than +/5 °C error level, along with Yamagata et al. (1972); Bishop et al. (1964); Watts and Chou (1982); Shitsman (1968); McAdams (1942) and Griem (1996) correlations with 100 % of predictions. Fig. 11 depicts the comparison of wall temperature predicted by CFD and Mokry et al., (2011) correlation with experimental data for a typical heat transfer enhancement case. It is found that, wall temperature predicted by both correlation and CFD are closely matches with experimental data. This is due to absence of non-linearity behavior of wall temperature.

Comparison of CFD and correlations predictions with experimental data for heat transfer deterioration zone
The selected 221 experimental data points obtained from the literature for the heat transfer deteriorated zone is shown in Table 4. It covers the range of the heat flux and mass flux ratio from 0.58 to 0.83. The wall temperature predictions obtained by CFD and 11 correlations were compared with the 221 experimental data points belonging to heat transfer deterioration zone as summarized in Table  5 (shown in Appendix). Table 6 provides the ranking information of CFD and empirical correlations on wall temperature prediction accuracies based on number of wall temperature points deviated with 221 experimental data points covering pseudocritical region where there is heat transfer deterioration. Table 6 provides the histogram ranking information of CFD and empirical correlations on wall temperature prediction accuracy based on percentage of error deviation in comparison with 221 experimental data points covering pseudocritical region where there is heat transfer deterioration. Among the 11 correlations, CFD shows the best agreement at less than +/-1°C error level followed by Zhu et al. (2009);Ornatsky et al. (1971); Jackson and Hall (1979) and Mokry et al. (2011) correlations. At less than +/-3°C level, Zhu et al. (2009) predicts 147 data points. At less than +/-5°C level, Zhu et al. (2009) predicts 192 number of data points accurately and followed by CFD and Watts and Chou (1982) with 171 & 165 data points respectively. At less than +/-7°C level, Zhu et al. (2009) predicts 205 number of data points accurately and followed by CFD, Watts and Chou (1982) with 190 & 188 data points respectively. At less than +/-10°C level, Zhu et al. (2009) predicts 211 number of data points accurately and followed by CFD, Watts and Chou (1982) with 208 data points. Among the 11 correlations, Zhu et al. (2009) shows the best agreement at less than +/-10°C, +/-7°C, +/-5°C and +/-3°C error level followed by CFD, Watts and Chou (1982) and Mokry et al. (2011) correlations. At less than +/-3°C level, Zhu et al. (2009) predicts 147 of data points accurately and followed by CFD and Watts & Chou with 124 & 117 data points respectively. It is also found that the CFD predictions have best agreement with experimental data at less than +/-1°C error level with 54 data points of the predictions and followed by Zhu et al. (2009) andOrnatsky et al. (1971) with 46 and 45 data points respectively. Fig. 12 provides the histogram of CFD and empirical correlations on wall temperature prediction accuracy based on percentage of error deviation in comparison with 221 experimental data points covering pseudocritical region where there is heat transfer deterioration. CFD shows best agreement with experimental data at less than +/-1 % and better agreement with experiment data at +/-10°C, +/-7 °C, +/-5 °C and +/-3°C . Fig. 13 depicts the comparison of wall temperature predicted by CFD and Zhu et al. (2009)'s correlation with experimental data (Mokry et al., 2011) for a typical heat transfer deterioration case. It is found that, unlike correlations prediction, the sudden rise in temperature is clearly predicted by CFD.

CONCLUSIONS
The present work investigates the heat transfer in supercritical fluids by CFD and compares its prediction with various correlations available in literature. A two-dimensional axis-symmetric model has been considered. In order to evaluate the accuracy of the present model, the experimental data available in literature has been selected for validation. The computational domain is discretized with a nonuniform mesh of 120 nodes along the radial direction and 1200 uniform nodes along the axial direction after performing grid independency test.
and Jackson and Hall (1974). At heat transfer deterioration zone, the prediction accuracy by CFD based on wall temperature