Two-phase Critical Flow Research Articles

A new paradigm in critical flow analysis: Combining Buckingham Pi theorem with neural network for improved predictions in microchannels

Critical flow phenomena are prevalent in many chemical engineering processes, such as microchannel heat exchangers, nuclear safety analysis, jet pump modeling, and industrial systems utilizing subcooled or two-phase pressurized fluids. Understanding the impact of various physical parameters on critical flow and accurately predicting critical flow mass flux is essential in these processes. This article collects extensive experimental data on subcooled water two-phase critical flow and supplements it with additional experimental data from actual micro-cracked channels, constructing a comprehensive benchmark database. The database covers an inlet pressure range of 1.82–16.29 MPa, an inlet subcooling range of 0.3–119.35 K, and a microchannel hydraulic diameter range of 0.09–2.2 mm. By integrating the classical Buckingham Pi theorem with a data-driven active subspace method, the dominant dimensionless numbers in subcooled water two-phase critical flow were identified, addressing some limitations of the Buckingham Pi theorem by ranking the influences of these dimensionless numbers. Based on these dominant dimensionless numbers, the impact of physical quantities on two-phase critical flow was accurately identified, leading to a concise predictive correlation for critical flow mass flux with a mean absolute percentage error of 20.3 % across 1071 experimental data points, outperforming commonly used critical flow models. Furthermore, leveraging the dominant dimensionless numbers improved existing model results, enhancing the Henry-Fauske model’s overall prediction accuracy from a mean absolute percentage error of 47.6 % to 17.8 %. This study outlines a promising path for discovering physical characteristics and mechanisms in data-driven critical flow, facilitating a deeper understanding of critical flow phenomena.

Study of the effect of virtual mass force on two-phase critical flow

Abstract Critical (choked) flow is a highly concerning phenomenon in safety analysis for nuclear energy. The discharge mass flow rate prediction is crucial for engineering design and emergency response in case of nuclear accidents. Unfortunately, the critical flow is difficult to predict especially when the two-phase flow exists. The accuracy is based on a deeper understanding of the complex phenomenon of critical flow. The influence of virtual mass force on the two-phase critical flow was seldom concentrated on owing to the lack of suitable critical flow models for studies in detail. This study is based on a developed 6-equation two-phase critical flow model. It is confirmed that the virtual mass force contributes to the stability and convergence of the critical flow simulation and it will impact not only the critical mass flux but also the thermal hydraulic parameters along the discharge duct. The magnitude depends on the geometry of the discharge duct and the upstream condition. It is larger when the duct is longer and the pressure is lower. Furthermore, the virtual mass force for each flow regime was studied in detail with a sensitivity study. The results show that the most sensible condition for the virtual mass force is annular flow along a long tube under relatively low pressure. The future work is to develop a correlation of virtual mass force for critical flow specifically since the correlations in the literature were developed under general two-phase flow process conditions.

A novel strategy of thermal management system for battery energy storage system based on supercritical CO[formula omitted

Supercritical CO2 (sCO2) is examined as a working fluid for the first time in a unique thermal management strategy that aims to control undesirable thermal behavior in battery cooling applications. In the pseudocritical region, sCO2 has a high volumetric thermal capacity with low critical pressure and temperature, which not only reduces system complexity and costs but also eliminates the possibility of two-phase critical heat flux and flow instabilities. A pack of 20×5 Li-ion batteries for battery energy storage system (BESS) applications was designed and employed in a structurally optimized thermal management system. Further, the effects of different dielectric fluid media on the number of flow inlets, flow rates, and discharge rates were numerically investigated. Compared with conventional coolants, sCO2 exhibited superior temperature suppression and temperature differences. The rise in both mass flow rate and the number of flow inlets significantly suppressed the maximum temperature and temperature difference. Besides, the pressure drop was slightly increased with the increased mass flow rate. Moreover, sCO2 controls the thermal behavior of a large battery pack at high discharge rates within an optimal range. The finding offers various attributes for cooling large battery packs, particularly for high-power grid stations and BESS. In the future, the practicality of high working pressures and power requirements for the pump should be carefully studied.

Experimental Assessment of the Efficiency of Two-Phase Ejector Components for Isobutane

Two-phase ejectors as well as single phase ejectors can be applied in many branches of industry: refrigeration and heat pump systems, chemical engineering, food processing, and others. Due to the complicated nature of the process of momentum transfer in two-phase ejectors, their design procedure based on the accurate theoretical prediction of the ejector performance is still an open issue. The paper provides its own experimental results of the velocity coefficients of the components of the two-phase ejector, i.e., the motive nozzle, suction chamber, mixing chamber, and diffuser. The results were obtained in the case of isobutane as the working fluid. It was demonstrated that the velocity coefficients may not be treated as constant quantities. Therefore, our own proposed dimensionless relationships describe the velocity coefficients of the components of the ejector that may be applied in the design procedure of the ejector. The two physical parameters, the wet vapour quality and the volumetric entrainment ratio, were selected as the key parameters. In addition, the aspects of the prediction of the critical mass flow rate of the motive nozzles was considered on the basis of the Henry–Fauske model. It was demonstrated that the model accurately predicts the two-phase critical flow under the conditions of a higher range of wet vapour quality.

Two-phase modeling of micro-channel critical flows with inlet sub-cooling: A review and benchmark study

Micro-channel two-phase critical flow is encountered in the chemical industry and power engineering, such as critical flow leakage/release and two-phase micro-channel heat sinks. Realistic modelling of micro-channel two-phase critical flow deserves enough attention. The paper aims to review the existing two-phase critical flow models and to evaluate the application to simulate the micro-channel critical flow. Choking and critical flow criteria of the single-phase flow and two-phase flow are discussed, and the pressure gradient near the critical cross-section is compared. The development as well as the key characteristic of the critical flow model is drawn on. A benchmark of critical flow models against the experimental data for the micro-channel critical flow, especially for the pressure profile, has been carried out to check the ability of the models to accurately reproduce the micro-channel critical flow, as well to reasonably describe the fluid flow and interfacial transfer. The constitutive relations of existing separated flow models are further compared and evaluated depending on the analysis of the fluid flow and interfacial transfer for two-phase flow to assess constitutive relations. Additionally, the bubble nucleation and metastable liquid property are included in separated flow models to explore the boiling delay in the micro-channel critical flow.

Physical Phenomena in Nuclear Thermal Hydraulics and Current Status

116 nuclear Thermal-Hydraulic Phenomena T-HP are identified in the present paper, following documents issued during the last three decades by the Committee on the Safety of Nuclear Installations of Nuclear Energy Agency of the Organization for Economic Cooperation and Development (OECD/NEA/CSNI) and by the International Atomic Energy Agency (IAEA). The derived T-HP list includes consideration of experiments performed in Separate Effect Test (SET) and Integral Effect Test (IET) facilities relevant to reactor coolant system and containment of Water Cooled Nuclear Reactors (WCNR). We consider a dozen WCNR types: Pressurized Water Reactors (PWR), Boiling Water Reactors (BWR), Russian reactors (VVER-440, VVER-1000 and RBMK), pressure tube heavy water reactors by Canada (CANDU) and India (PHWR) and so-called ‘advanced’ reactors (e.g. AP-1000 and APR-1400 designed in US and Korea, respectively). We envisage a variety of applications for the T-HP list. Four of the phenomena are helpful to characterize the current state of art in nuclear thermal-hydraulics: Counter Current Flow Limitation (CCFL), Critical Heat Flux (CHF), reflood and Two-Phase Critical Flow (TPCF). Furthermore, the T-HP identification contributes to addressing the scaling issue, performing uncertainty evaluations, developing constitutive equations and ‘special models’ in codes and prioritizing the research.

Fundamental Study on Injector Flow Characteristics of Self-Pressurizing Fluid for Small Rocket Engines

Abstract Nitrous oxide is a suitable propellant for rocket engines and has been widely used in various countries, given its high saturated vapor pressure, which enables self-pressurization. Because nitrous oxide is in a state of vapor–liquid equilibrium in tanks, it is easy to form a gas–liquid two-phase flow by cavitation in feed line. Since accurately estimating the performance of rocket engines requires evaluating the characteristics of propellant flows, tests reported in this paper were conducted using hybrid rocket engines under three conditions: cold flow test, hot firing test at low back pressure, and hot firing test at high back pressure. With consideration to the subcooling degrees, nitrous oxide may be in an unsteady superheated state in the upstream flow of the injector. In a comparison of the pressure ratios between the injector in each test condition, it is observed that a critical two-phase flow was formed in the injector in the cold flow test and in the low backpressure firing test. In the high backpressure hot firing test, the injector flow may be choked, but the large oscillations were observed in chamber pressure and thrust. According to the FFT analysis results, these oscillations were caused by chugging and acoustic oscillation. In light of these experimental results, it is suggested that when the chamber pressure fluctuates due to combustion instability such as chugging and acoustic oscillation, it may affect the injector flow characteristics and the critical two-phase flow.

A New Homogeneous Non-Equilibrium Model to Compute Vapor-Liquid Two-Phase Critical Pressure Ratios of Multicomponent Hydrocarbon Mixtures

The critical pressure ratio (ηc) is an essential parameter for computing the vapor-liquid two-phase critical pressure and mass flow rate of multicomponent hydrocarbon mixtures flowing through valves and leakage orifices. The Homogeneous Non-Equilibrium Diener-Schmidt (HNE-DS) model widely used to calculate ηc assumes that the fluid's volume linearly changes with the pressure (using the Clausius-Clapeyron equation), which is not suitable for multicomponent gas mixtures. In this paper, a new Homogeneous Non-Equilibrium (new-HNE) model is proposed to calculate ηc of gas mixtures. Firstly, a new critical flow compressibility factor (ωc) is developed from its thermodynamic definition and the Peng-Robinson equation of state (EOS), overcoming the inherent limitations of the Clausius-Clapeyron equation. Then, ηc is correlated to the newly derived ωc by fitting experimental data at various pressures and gas mass fractions of both single-component and multicomponent gas mixtures, yielding the new HNE-DS model. Results show that, for the water-steam and air-water two-phase flow, the average relative deviations (ARD) between the calculated critical pressure ratios and experimental values are equal to 2.8% and 4.93%, respectively, which represents a significant improvement in comparison with the original HNE-DS model. Moreover, this new model is extended to the applications of Liquefied natural gas (LNG)/liquefied petroleum gas (LPG) fluids, and will further contribute to the calculation of the leakage mass flow rate of fluid flowing through the orifices/valves.

Critical flow prediction by system codes – Recent analyses made within the FONESYS network

A benchmark activity on Two-Phase Critical Flow (TPCF) prediction was conducted in the framework of the Forum & Network of System Thermal-Hydraulics Nuclear Reactor Thermal-Hydraulics (FONESYS). FONESYS is a network among code developers who share the common objective to strengthen current technology. The aim of the FONESYS Network is to highlight the capabilities and the robustness as well as the limitations of current SYS-TH codes to predict the main phenomena during transient scenarios in nuclear reactors for safety issues.Six separate effect test facilities, more than 90 tests, both in steady and transient conditions, were considered for the activity. Moreover, two ideal tests were designed for code to code comparison in clearly defined conditions. Overall eight System Thermal-Hydraulic (SYS-TH) codes were adopted, mostly by the developers themselves, ensuring the minimization of the user effect. Results from selected tests were also compared against Delayed Equilibrium Model, not yet implemented in industrial version of SYS-TH codes.Generally, the results of the benchmark show an improvement of the capability of SYS-TH codes to predict TPCF in the last three decades. However, predicting break flowrate remains a major source of uncertainty in accidental transient simulations of Water-Cooled Nuclear Reactors (WCNR). A set of possible actions is proposed to go beyond the current limitations of choked flow models. More detailed guidelines for using 0-D choked flow models is possible by using the experience gained by the benchmark results as well as all available validation results. Progress in understanding and 1-D modelling of flashing and choked flow might be achieved by a deeper physical analysis leading to more mechanistic models based on specific flow regime maps for high speed flow. Also the use of advanced 3-D numerical tools may help to understand and predict the complex 3-D geometrical effects.

Application of stability solver for severe slugging considering a choking device and a tubing connected to a Wet Christmas Tree

This work is an improvement of a previous work (Andreolli et al. 2018) by including a tubing as a subsystem of the pipeline–riser system, a realistic choke valve model to predict the two-phase critical flow and pressure drop before the separator and boundary conditions between stretches of the geometry, in order to improve the determination of the stability boundary. The improved stability solver is applied for a three phase flow (gas, oil and water). Stability for operational points obtained from actual measurements are successfully predicted with the stability maps generated by the solver considering the tubing connected to the Wet Christmas Tree (WCT). A parametric analysis is made in order to evaluate a choke valve influence over stability maps. The stability maps are built by using field conditions at Santos and Campos basins (Brazil). The numerical results present a good agreement with numerical data and real operational conditions. The analysis to evaluate and verify the choke valve influence over stabilization shows a good agreement with the expected behavior.

Critical flow leakage of a vapour-liquid mixture from sub-cooled water: Nucleation boiling study

Two-phase critical flow has been studied extensively by many researchers over the past seven decades, due to its importance in the fields of nuclear engineering and other industries that deal with high energy storage. The paper aims to develop a mathematical model for the critical flow of a two-phase mixture in a narrow channel, considering the irreversible non-equilibrium interphase mass transfer of metastable liquids. Calculations are based on the model to describe the boiling delay and the non-equilibrium vapor generation process during a rapid depressurization. Moreover, the paper provides an explicit algorithm, using the computational pressure step, to seek efficiency and precision as well as the stability of the computation. The possibility of this model to predict the critical mass flux, pressure profiles, as well as the mass transfer between phases, has been validated, by comparison with three mathematical models with respect to experimental results of different authors. The developed non-equilibrium model gives an excellent approach to experimental measures and explains the complex behavior shown by critical blowdowns of pressurized vessels.

Comparison of Two-Phase Critical Flow Models for Estimation of Leak Flow Rate through Cracks

Comparison of Two-Phase Critical Flow Models for Estimation of Leak Flow Rate through Cracks

Critical Mass Flux of Subcooled Liquid Leakage in a Nuclear Plant

Safety analyses of pressurized water reactors and boiling water reactors in the event of small-break loss-of-coolant accidents strongly depend on leakage rate predictions using two-phase critical flow models. The paper aims to revise the critical flow criterion and consider the nonequilibrium phenomena of critical flows in constructing a modified two-phase critical flow model. The model predictions exhibit strong similarities with the experimental values, with prediction deviations of 14.4% for mass fluxes and 19.3% for outlet pressure. The compiled code, according to the proposed model, can be exploited in pressure pipeline designs, providing the theoretical basis for leak-before-break analyses.

V&V&C in nuclear reactor thermal-hydraulics

V&V constitutes a powerful framework to demonstrate the capability of computational tools in several technological areas. Passing V&V requirements is a needed step before applications. Let's focus hereafter to the area of (transient) Nuclear Thermal-hydraulic (NTH) and let's identify V1 and V2 as acronyms for Verification and Validation, respectively. V1 is performed within NTH according to the best available techniques and may not suffer of in-principle deficiencies. This is not the case of V2; understood limits are: 1) Validation implies comparison with experimental data: available experimental data cover a (very) small fraction of the parameter range space expected in applications of the codes; this can be easily seen if one considers data in large diameter pipe, high velocity and high pressure or high power and power density; scaling issue must be addressed which may result in controversial findings. 2) Water is at the center of the attention: the physical properties of water are known to a reasonable extent as well as large variations in values of quantities like density or various derivatives are expected within the range of variation of pressure inside application fields. Although not needed for current validation purposes (e.g. validation ranges may not include a situation of critical pressure and large heat flux) physically inconsistent values predicted by empirical correlations outside validation ranges, shall not be tolerated. 3) Occurrence of complex situations like transition from two-phase critical flow to ‘Bernoulli-flow’ (e.g. towards the end of blow-down) and from film boiling to nucleate boiling, possibly crossing the minimum film boiling temperature (e.g. during reflood). Therefore, whatever can be mentioned as classical V2 is not or cannot be performed in NTH. So the idea of the present paper is to add a component to the V&V. This component, or step in the process, is called ‘Consistency with Reality’, or with the expected phenomenological evidence. The new component may need to be characterized in some cases and is indicated by the letter ‘C’. Then, the V&V becomes V&V&C. The motivations at the bases of the V&V&C are clarified in the paper.

Experimental study on the flow and thermal characteristics of two-phase leakage through micro crack

Better understanding of the leakage characteristic through thin-walled pipes is of great importance to maintain the integrity of reactor pressure boundary. This experimental study focuses on the characteristics of two-phase critical flow through short slits (L/d = 4–21), and the test pressure covers the range of 0.1–10.0 MPa. Cracks with small passage opening (as low as 46.5 μm) were adopted to better simulate the real crack, where both axial and circumferential crack were considered. Therefore the test database was enlarged. Influences of stagnation conditions and crack morphologies were analyzed. The flow mechanism was explained on how flash decides the flow rate by affecting flow pattern and choking: circumferential crack shows a higher leakage than axial crack, due to a weaker choking phenomenon; the flow pattern transition is discovered at low subcooling degree. A dimensionless number β, indicating the fraction of flashing, was proposed and two different flow phenomenon stages were identified and distinguished by β = 15. Besides, correlations were developed for the flow rate through short slits, showing a reasonable accuracy in predicting the test data both in this study and from literature.

Mass transfer characteristics of pipeline leak-before-break in a nuclear power station

This paper aims to develop a correlation for pressure undershoot and revises the hypotheses of nucleation inception in the two-phase critical flow model. The theories of rapid depressurization and isothermal bubble growth were used to construct a modified two-phase critical flow model for leak-before-break analysis. The modified model predictions exhibit strong similarities with the experimental values compared with the Moody and Henry–Fauske models, with prediction deviations of less than 8.9% in the inter-granular stress corrosion crack. The effects of the inlet conditions on the nucleation inception point, as well as the distribution characteristics of the mass quality and void fraction along the channel, were investigated to explore the mass transfer process.

Numerical and Analytical Approaches to Modeling Critical Two-Phase Flow with Granular Layer

Based on data obtained in the previous experimental study conducted by the authors, two approaches are proposed for analytical and numerical modeling of a critical two-phase flow in a pipe with a granular layer. An analytical approach is based on a polytrophic model, while a numerical approach was developed using a smoothed particle hydrodynamics method. A model of isenthalpic flow of vapor–water mixture in a fixed bed of solid particles is considered is this study. The mixture expansion process is considered to be polytropic. Similarly to the known problem of gas dynamics of a granular bed, an analytical relationship for calculation of a critical mass velocity was obtained. The results of the calculation based on the analytical and numerical models were compared with the experimental data and agreement between analytical and numerical data and the experiment was observed.

A code development for leak before break (LBB) leakage from supercritical to subcritical conditions

The leak before break (LBB) concept plays an important role in reactor safety. A code (LEABLE) for leak before break (LBB) leakage estimation was developed in this paper. Complicated crack morphologies and different conditions were taken into account, covering subcritical and supercritical fluids, single phase and multi-phases, choked and unchoked flow. Precise and reliable criteria were proposed for proper model selection. A modified two-phase critical flow model was presented and a single-phase critical flow model for superheat gas was established. Besides, modifications on flashing inception, flow resistance and particulate plugging were conducted in the LEABLE code. Moreover, the code was validated and verified by experimental data for both artificial and natural cracks. Reasonable agreements between experiment values and code predictions are shown for all the conditions. And in the present study the LEABLE shows a higher precision than existing models and commercial softwares (PICEP and SQUIRT). The parameter sensitivity study of LBB leakage was performed as well. The mass flux increases with crack opening crack opening displacement (COD), crack length and the entrance resistance coefficient, and decreases with global and local roughness, number and angle of turns. However, the entrance resistance coefficient and local roughness have limited effects on LBB leakage.

Benchmark of Delayed Equilibrium Model (DEM) and classic two-phase critical flow models against experimental data

The safety analysis of Pressurized Water Reactors, in the event of LOCA, strongly depends on the ability to evaluate the discharge rate of coolant inventory through the breach. Due to the huge pressure difference between the primary system and the reactor containment, the mass flow rate is choked at the break. Under such conditions, both mechanical and thermal equilibrium between phases are not ensured.A general theory to evaluate the two-phase critical mass flow rate is not yet available. However, some models are capable of providing accurate evaluations of either critical mass flux or critical pressure and such a model is the Delayed Equilibrium Model (DEM), which is examined in this article. Here we show how to integrate the DEM system of equations coupling a standard Runge-Kutta method with the Possible-Impossible Flow algorithm. Hence a simple procedure which does not require sophisticate computational schemes.The main objective of this work is to compare DEM, Homogeneous Equilibrium Model, Moody (1965) and Henry and Fauske (1971) models to experimental data. The four models were tested and the results from experimental data containing a sample range in excess of 450 conditions compared in determining an appropriate benchmark. Each of the chosen models is representative of a particular category of critical flow models. Furthermore, two-phase critical models provide good estimations depending on the configuration or set of conditions. Consequently, the models have been individually tested incorporating long tubes, short tubes and slits.This analysis has been carried out for both critical mass flux and critical pressure evaluations.

A polytropic model of a critical two-phase flow in a bed of spherical particles

The paper is concerned with a model of isenthalpic flow of vapor-water mixture in a fixed bed of solid particles. The mixture expansion process is considered to be polytropic. Similarly to the known problem of gas dynamics of a granular bed we obtained the relationships for calculation of a critical mass velocity. The results of the calculation based on a theoretical model are compared with the experimental data obtained in the packed beds of steel balls, 2 mm and 4 mm in diameter.