Evaluation of the Heat Transfer Correlations for Supercritical Pressure Water in Vertical Tubes

A large number of studies have been carried out on the flow and heat transfer of supercritical pressure fluids in the past decades. However, there are still some uncertainties and deficiencies in the accurate prediction for supercritical fluid heat transfer coefficient due to the large and fast variations of fluids properties in the so-called pseudo-critical region. In this paper, 15 correlations were selected from the literature and were compared with each other to verify their capability in predicting heat transfer coefficient of supercritical pressure water in vertical tubes. Based on the comparison between the calculation results of the existing heat transfer correlations and the experimental data obtained from the open literature, it was found that the Swenson et al. correlation and the Hu correlation can reasonably predict the heat transfer coefficient of supercritical water in the pseudo-critical region. After evaluating these correlations, the authors conducted polynomial fitting for the collected experimental data and got a new correlation for heat transfer coefficient of water at supercritical pressures. The new correlation can fit well with the experimental data even in the neighborhood of pseudo-critical temperature.     

A new heat transfer correlation for supercritical fluids

A new method of heat transfer prediction in supercritical fluids is presented. Emphasis is put on the simplicity of the correlation structure and its explicit coupling with physical phenomena. Assessment of qualitative behaviour of heat transfer is conducted based on existing test data and experience gathered from open literature. Based on phenomenological analysis and test data evaluation, a single dimensionless number, the acceleration number, is introduced to correct the deviation of heat transfer from its conventional behaviour, which is predicted by the Dittus-Boelter equation. The new correlation structure excludes direct dependence of heat transfer coefficient on wall surface temperature and eliminates possible numerical convergence. The uncertainty analysis of test data provides information about the sources and the levels of uncertainties of various parameters and is highly required for the selection of both the dimensionless parameters implemented into the heat transfer correlation and the test data for the development and validation of new correlations. Comparison of various heat transfer correlations with the selected test data shows that the new correlation agrees better with the test data than other correlations selected from the open literature.     

Development of Empirical Correlation for Heat Transfer to Nanofluids

Nanofluids are emerging as alternative fluids for heat transfer applications due to enhanced thermal properties. Several correlations are available in open literature for heat transfer coefficient (HTC) and thermophysical properties of nanofluids. Reliability of correlations that use effective properties for estimation of HTC needs to be checked. Comparison of experimental HTC and that estimated from existing correlations is the main objective of the present study. An empirical correlation is developed with experimental data of the HTC for zinc–water and zinc oxide–water nanofluids. Experimental HTC is compared with that estimated from developed correlation and existing correlations. The range of Re considered for the study is 4000 to 18,000. Comparison indicated large deviation in experimental values and the values estimated from existing correlations. Based on comparison results, it can be concluded that the single‐phase models of forced convective heat transfer cannot be extended to nanofluids.     

Experimental study on heat transfer characteristics of supercritical carbon dioxide in horizontal tube

The heat transfer characteristics of supercritical carbon dioxide in a horizontal tube with water in the vertical cross flow form were experimentally investigated. The results indicate that the changes of inlet pressure, mass flow rate, and cooling water flow rate have major effects on heat transfer performance. The variations of Reynolds number and Prandtl number were obtained in counter flow and vertical cross flow. The four conventional correlations for convection heat transfer of supercritical carbon dioxide were verified by the experimental data in this study and the correlation agree with this experimental condition was determined.     

Investigation on the characteristics and mechanisms of unusual heat transfer of supercritical pressure water in vertically-upward tubes

The heat transfer characteristics of supercritical pressure water in a vertically-upward optimized internally-ribbed tube was investigated experimentally to study the mechanisms of unusual heat transfer of supercritical pressure water in the so-called large specific heat region. The experimental parameters were as follows. The pressure at the inlet of the test section ranged from 22.5 to 29.0 MPa, and the mass flux of the fluid was from 650 to 1200 kg/m² s, and the heat flux on the inside wall of the tube varied from 200 to 660 kW/m². According to experimental data, the characteristics of heat transfer enhancement and also the heat transfer deterioration of supercritical pressure water in the large specific heat region was analyzed and based on the comparison and analysis of the current major theories that were used to explain the reasons for unusual heat transfer to occur, the mechanisms of heat transfer enhancement and deterioration were discussed, respectively. The enhanced heat transfer was characterized by the gently changing wall temperature, the small temperature difference between the inside-tube-wall and the bulk fluid and the high heat transfer coefficient in comparison to the normal heat transfer. The deteriorated heat transfer could be characterized by the sharply increasing wall temperature, the large temperature difference and a sudden decrease in heat transfer coefficient in comparison to the normal heat transfer. The heat transfer enhancement of the supercritical pressure water in the large specific heat region was suggested to be a result of combined effect caused by the rapid variations of thermophysical properties of the supercritical pressure water in the large specific heat region, and the same was true of the heat transfer deterioration. The drastic changes in thermophysical properties near the pseudocritical points, especially the sudden rise in the specific heat of water at supercritical pressures, might result in the occurrence of the heat transfer enhancement, while the covering of the heat transfer surface by fluids lighter and hotter than the bulk fluid made the heat transfer deteriorated eventually and explained how this lighter fluid layer formed.     

Forced convective heat transfer to supercritical water flowing in tubes

Experimental investigations were made of heat transfer to supercritical water flowing in a horizontal tube and vertical tubes. A comprehensive set of data was obtained for pressures from 226 to 294 bar, bulk temperatures from 230 to 540°C, heat fluxes from 116 to 930 kW/m² and mass velocities from 310 to 1830 kg/m²s. Because the physical properties of supercritical fluids change rapidly with temperature in the pseudocritical region, the heat transfer coefficients show unusual behavior depending upon the heat flux. At low or modetate heat fluxes relatively to the flow rate, a satisfactory correlation was obtained, which predicts reasonably well the enhanced heat transfer coefficients near the pseudocritical point. The several characteristics of the deterioration in heat transfer which occurs at high heat fluxes were clarified, and the limit heat flux for the occurrence of the deterioration was determined in connection with the flow rate.     

Experimental study on heat transfer characteristics of supercritical carbon dioxide in horizontal tube

The heat transfer characteristics of supercritical carbon dioxide in a horizontal tube with water in the vertical cross flow form were experimentally investigated. The results indicate that the changes of inlet pressure, mass flow rate, and cooling water flow rate have major effects on heat transfer performance. The variations of Reynolds number and Prandtl number were obtained in counter flow and vertical cross flow. The four conventional correlations for convection heat transfer of supercritical carbon dioxide were verified by the experimental data in this study and the correlation agree with this experimental condition was determined. __________ Translated from Journal of Refrigeration, 2007, 28(1): 8–11 [译自:制 冷学报]     

Heat transfer with water flowing upward in a tube for pressures up to supercritical region

A heat transfer experiment was conducted in a tube of 6.07 mm in diameter with water flowing upward, covering the ranges of pressure of 10–23 MPa, mass flux of 288–1298 kg/(m²·s), local water temperature of 78°C–270°C, heat flux of 0.23–1.18 MW/m² and Reynolds number of 5.5 × 10³−3.9 × 10⁴. The experimental results were compared with the predictions of the Dittus-Boelter correlation, Jackson correlation, Bishop correlation, Swenson correlation and Yamagata correlation. Significant deterioration in heat transfer was observed in both subcritical and supercritical region due to the effect of buoyancy force, but it was not predicted reasonably by the existing correlations.     

Comparison of Six Typical Correlations for Upward Flow Boiling Heat Transfer with Kerosene in a Vertical Smooth Tube

The present article is aimed at evaluating six typical flow boiling heat transfer correlations selected from the open literature with experimental results. The selected correlations are correlations of Chen, Shah, Gungor and Winterton, Liu and Winterton, Klimenko, and Kandlikar. Experiments of upward flow boiling heat transfer with kerosene in a vertical smooth tube were conducted. The test tube has a length of 2.5 m and its outer and inner diameters are 19 mm and 15 mm, respectively. The experiments were performed at an absolute atmospheric pressure of 3. The input heat flux ranged from 28.5 to 93.75 kW/m² and the mass fluxes were selected at 410, 610, and 810 kg/m² s, respectively. The experimental flow boiling heat transfer coefficients were compared with flow boiling heat transfer coefficients calculated with the six typical correlations. By comparison, the most suitable correlations are recommended for the calculation of flow boiling heat transfer coefficients with kerosene in a smooth tube.     

Steady-State Behavior of Natural Circulation Loops Operating With Supercritical Fluids for Open and Closed Loop Boundary Conditions

Supercritical water (SCW) exhibits excellent heat transfer characteristics and a high volumetric expansion coefficient (hence high mass flow rates in natural circulation systems) near the critical temperature. SCW is being considered as a coolant in some advanced nuclear reactor designs on account of its potential to offer high thermal efficiency, compact size, and elimination of steam generator, separator, and dryer, making it economically competitive. The elimination of phase change results in elimination of the critical heat flux phenomenon. Cooling a reactor at full power with natural instead of forced circulation is generally considered an enhancement of passive safety. In view of this, it is essential to study natural circulation behavior at supercritical conditions. Carbon dioxide can be considered to be a good simulant of water for natural circulation at supercritical conditions, since the density and viscosity variation of carbon dioxide follows a curve parallel to that of water at supercritical conditions. Hence, experiments were conducted in a closed supercritical pressure natural circulation loop (SPNCL) with supercritical carbon dioxide as working fluid. A nonlinear stability analysis code (NOLSTA) has been developed to carry out steady-state and stability analysis of open and closed loop natural circulation at supercritical conditions. The code has been validated for steady-state predictions with experimental data available in open literature and experiments conducted in SPNCL.     

Heat transfer with water flowing upward in a tube for pressures up to supercritical region

A heat transfer experiment was conducted in a tube of 6.07mm in diameter with water flowing upward, covering the ranges of pressure of 10―23MPa, mass flux of 288―1298kg/(m²·s), local water temperature of 78°C―270°C, heat flux of 0.23―1.18MW/m² and Reynolds number of 5.5×10³―3.9×10⁴. The experimental results were compared with the predictions of the Dittus-Boelter correlation, Jackson correlation, Bishop correlation, Swenson correlation and Yamagata correlation. Significant deterioration in heat transfer was observed in both subcritical and supercritical region due to the effect of buoyancy force, but it was not predicted reasonably by the existing correlations.     

Study of Variable Turbulent Prandtl Number Model for Heat Transfer to Supercritical Fluids in Vertical Tubes

In order to improve the prediction performance of the numerical simulations for heat transfer of supercritical pressure fluids, a variable turbulent Prandtl number (Pr_t) model for vertical upward flow at supercritical pressures was developed in this study. The effects of Pr_t on the numerical simulation were analyzed, especially for the heat transfer deterioration conditions. Based on the analyses, the turbulent Prandtl number was modeled as a function of the turbulent viscosity ratio and molecular Prandtl number. The model was evaluated using experimental heat transfer data of CO₂, water and Freon. The wall temperatures, including the heat transfer deterioration cases, were more accurately predicted by this model than by traditional numerical calculations with a constant Pr_t. By analyzing the predicted results with and without the variable Pr_t model, it was found that the predicted velocity distribution and turbulent mixing characteristics with the variable Pr_t model are quite different from that predicted by a constant Pr_t. When heat transfer deterioration occurs, the radial velocity profile deviates from the log-law profile and the restrained turbulent mixing then leads to the deteriorated heat transfer.     

An Investigation on Heat Transfer to Supercritical Water in Inclined Upward Smooth Tubes

Within the range of pressures from 23 to 30 MPa, mass velocities from 600 to 1200 kg/(m²s), and heat fluxes from 200 to 600 kW/m², experiments have been performed for an investigation on heat transfer to supercritical water in inclined upward smooth tubes with an inner diameter of 26 mm and an inclined angle of 20° from the horizon. The results indicated that heat transfer characteristics of supercritical water are not uniform along the circumference of the inclined tube. An increase in the mass velocity of the working fluid can decrease and even eliminate the non-uniformity. Properties of supercritical fluid acutely vary with the temperature near the pseudocritical point. While the ratio of the mass velocity to the heat flux exceeded 2.16 kg/(kWs), heat transfer enhancement occurred near the pseudocritical point; conversely, heat transfer deterioration occurred while the ratio of the mass velocity to the heat flux was lower than 2.16 kg/(kWs). As the pressure increased far from the critical pressure, the amount of deterioration decreased. Correlations of heat transfer coefficients of the forced-convection heat transfer on the top and bottom of the tube have been provided, and can be used to predict heat transfer coefficient of spirally water wall in supercritical boilers.     

The Flow Resistance and Heat Transfer Characteristics of Micro Pin-Fins with Different Cross-Sectional Shapes

The flow resistance and heat transfer characteristics of deionized water flowing through a rectangular channel (60 mm × 5.2 mm × 0.5 mm) with staggered array micro pin-fin circular, diamond, and elliptical groups are experimentally investigated over Reynolds numbers ranging from 8 to 1,000, and the investigation shows that the flow resistance increases due to the endwall effect and large pin-fins density at low Re. With the increase in Re, the endwall effect is weakened, but the flow resistance still increases due to the appearance of vortex resistance, and the heat transfer is enhanced due to the flow disturbance or transition from laminar flow to turbulent flow. The experimental results are also compared with predictions of the theoretical correlations for the staggered array micro pin-fin groups, and the comparisons indicate that only the correlation related to the diamond shaped micro pin-fin groups approximately agrees with experimental data, and the other correlations do not describe well the flow and heat transfer characteristics covering laminar, transitional, and turbulent states in circular and elliptical test sections.     

Deterioration of heat transfer to supercritical helium at 2·5 atmospheres

Heat transfer has been investigated for supercritical helium at 2·5 atm flowing inside a vertical tube with inlet bulk fluid temperatures less than the transposed critical temperature. Results indicate that for high heat flux conditions, the heat-transfer coefficient passes through a maximum and then deteriorates as the fluid temperature approaches the transposed critical temperature. This is contrary to the predictions of a correlation developed in an earlier study of supercritical helium heat transfer under low heat flux conditions, which only predicts enhancement in heat transfer as the transposed critical temperature is approached.The experimental data are presented and conditions under which heat-transfer deterioration was observed are discussed. The probable limitations to the validity of the above mentioned heat-transfer coefficient correlation, developed for a different range of experimental data, are also discussed.     

Analysis of forced convection heat transfer to supercritical carbon dioxide inside tubes using neural networks

The modeling of forced convection heat transfer for carbon dioxide flowing inside a heated tube at supercritical conditions was studied. The conventional models in the literature tend to modify a constant property correlation by including thermodynamic property terms that follow the heat flux trends. An innovative heuristic method is assumed here for the first time to draw the case-specific heat transfer coefficient correlation from the experimental data on said quantity alone.Neural networks were used since they constitute a general, powerful function-approximator tool proving able to represent a conventional heat transfer surface precisely in the present case. Four different correlation architectures were considered for the neural network function, alternatively based on dimensionless groups and on directly accessible physical quantities as independent variables. In all these architectures, the optimal functional form of the correlation was obtained using a completely heuristic procedure based exclusively on experimental data, reaching an accuracy comparable with the experimental uncertainties declared.An improved performance of the present model was found with respect to conventional correlations. On all the data sets, the third architecture reaches an AAD of 3.98% against 4.09% for the conventional equation and the fourth architecture an AAD of 2.67% against 4.30% for the conventional equation. Besides both these NN architectures present Bias values very close to 0, whereas the conventional equation has a Bias considerably greater.     

Heat transfer characteristics of a temperature-dependent-property fluid in shell and coiled tube heat exchangers

An experimental investigation was performed to study the heat transfer characteristics of temperature-dependent-property engine-oil inside shell and coiled tube heat exchangers. For this purpose, a well-instrumented set-up was designed and constructed. Three heat exchangers with different coil pitches were selected as the test section for counter-flow configuration. Engine-oil was circulated inside the inner coiled tube, while coolant water flowed in the shell. All the required parameters like inlet and outlet temperatures of tube-side and shell-side fluids, flow rate of fluids, etc were measured using appropriate instruments. An empirical correlation existed in the previous literature for evaluating the shell-side Nusselt number was invoked to calculate the heat transfer coefficients of the temperature-dependent-property fluid flowing in the tube-side of the heat exchangers. Using the data of the present study, an empirical correlation was developed to predict the heat transfer coefficients of the temperature-dependent-property fluid flowing inside the shell and coiled tube heat exchangers.     

Experimental investigation of heat transfer of supercritical carbon dioxide flowing in a cooled vertical tube

This work presents an experimental investigation on the heat transfer characteristics of a cooled vertical turbulent flow of supercritical carbon dioxide. The test section consists of two sectors (vertical tube-in-tube) connected in series by means of a U-bend. An integral method is used to treat the experimental data. Experimental results illustrate the influence of buoyancy forces on the heat transfer process. Results are presented in dimensionless form commonly used in the studies of mixed convection in heating. Correlations have been developed for upward and downward flows. The present results complete the literature on turbulent vertical mixed convection under cooling conditions.     

Development of a new forced convection heat transfer correlation for CO2 in both heating and cooling modes at supercritical pressures

Experimental and numerical investigations on forced convection heat transfer of carbon dioxide at supercritical pressures in a prototypic printed circuit heat exchanger under both cooling and heating conditions have been performed in this present study. The experiment test section has nine semi-circular channels with a hydraulic diameter of 1.16 mm and a length of 0.5 m. Primary operational parameters include inlet pressure of 7.5–10 MPa, mass fluxes of 326 kg/m² s and 762 kg/m² s, inlet temperatures from 10 °C to 90 °C and the average heat flux was 30 kW/m². Beyond reproducing the regular experimental cases, numerical modeling also implemented higher heat fluxes of 60 kW/m² and 90 kW/m² in order to investigate the effect of heat flux. Good agreement was found between the experiments and FLUENT simulations using an SST k–w model with the near-wall region being completely resolved. The distinctive behavior of convection heat transfer at supercritical pressures between heating and cooling modes was systematically analyzed. A more physically reasonable property-averaging technique, Probability Density Function (PDF)-based time-averaged property, was developed to account for the effect of nonlinear dependency of properties on instantaneous local temperature. Furthermore, experimental and computational data were compared to empirical predictions by the Dittus–Boelter and Jackson correlations. The results showed that Dittus–Boelter correlation has better precision for the average value of the predicted heat transfer coefficient but cannot take account of the effect of heat flux. In contrast, the Jackson correlation, with property ratio correction terms to account for the distribution of the properties in the radial direction, could predict the distinction of heat transfer characteristics under heating and cooling conditions. However, it overestimates the average value of heat transfer coefficient in the whole range of the experiment conditions. Finally, a new correlation evaluated by PDF-based time-averaged properties for forced convection heat transfer of CO₂ in both heating and cooling mode at supercritical pressures was developed. Comparison of experimental and computational data with the prediction results by the new developed correlation reveals that it works quite well; i.e., more than 90% data in either heating or cooling mode with various heat fluxes are predicted within an accuracy of ±25%.