Non-equilibrium Two-phase Flow Research Articles

A note on the relaxation process in a class of non-equilibrium two-phase flow models

A note on the relaxation process in a class of non-equilibrium two-phase flow models

High-order relaxation methods for nonequilibrium two-phase flow equations

PurposeThe purpose of this paper is to investigate the two-phase flow problems involving gas–liquid mixture.Design/methodology/approachThe governed equations consist of a range of conservation laws modeling a classification of two-phase flow phenomena subjected to a velocity nonequilibrium for the gas–liquid mixture. Effects of the relative velocity are accounted for in the present model by a kinetic constitutive relation coupled to a collection of specific equations governing mass and volume fractions for the gas phase. Unlike many two-phase models, the considered system is fully hyperbolic and fully conservative. The suggested relaxation approach switches a nonlinear hyperbolic system into a semilinear model that includes a source relaxation term and characteristic linear properties. Notably, this model can be solved numerically without the use of Riemann solvers or linear iterations. For accurate time integration, a high-resolution spatial reconstruction and a Runge–Kutta scheme with decreasing total variation are used to discretize the relaxation system.FindingsThe method is used in addressing various nonequilibrium two-phase flow problems, accompanied by a comparative study of different reconstructions. The numerical results demonstrate the suggested relaxation method’s high-resolution capabilities, affirming its proficiency in delivering accurate simulations for flow regimes characterized by strong shocks.Originality/valueWhile relaxation methods exhibit notable performance and competitive features, as far as we are aware, there has been no endeavor to address nonequilibrium two-phase flow problems using these methods.

Depressurization of CO2 in a pipe: Effect of initial state on non-equilibrium two-phase flow

It is key in several industrial applications to accurately describe rapid depressurization of liquid and dense phase states. Examples include refrigeration systems, nuclear reactor cooling and CO2 capture and storage (CCS). It is expected that large-scale CO2 pipeline transportation must be deployed as a vital part of reaching net zero emissions by 2050. During rapid depressurization of liquid-like CO2, boiling will in many cases occur out of equilibrium, at a lower pressure than the local saturation pressure. Capturing the non-equilibrium effects is necessary to predict outflow rates and the resulting pressure and temperature inside the pipe. In the present work, we quantify the non-equilibrium effects by studying a series of CO2 pipe depressurization experiments from liquid-like states at initial temperatures from 10°C to 40°C. We compare the experimental results to predictions of the homogeneous equilibrium model (HEM) and a homogeneous relaxation-type non-equilibrium model (HRM*) where the mass-transfer rate from liquid to gas is tuned by a relaxation time. The relaxation time was found to decrease for increasing temperatures, and it was observed to be approximately 60 times longer for the coldest experiment than for the warmest one.

Incorporating bubble evolution and transport in constitutive relationships for quasi- and non-equilibrium two-phase flows in porous media

There is a need to better understand the presence and transport of bubbles in multi-phase subsurface porous media so that these processes can be accurately described, and more efficient engineered solutions can be developed. To this end, constitutive relationships between geometric state variables (fluid-fluid curvature, Jnw; non-wetting phase volume, Vn; fluid-fluid interfacial area, anw; and Euler characteristic, χn) have become increasingly more common in efforts to uniquely predict the state of a two-fluid flow system. Both lattice Boltzmann simulations and fast X-ray microtomography (μCT) imaging experiments have shown that a geometric state function using the non-dimensionalized invariant properties of saturation, specific interfacial area, and Euler characteristic can uniquely predict the mean curvature of the system for both quasi- and non-equilibrium conditions, however, the presence of bubble evolution and the subsequent transport phenomena have not been explored. This study investigates whether the geometric state function remains unique with the inclusion of bubble generation and transport under quasi- and non-equilibrium two-fluid flow. The data presented here suggests that bubble formation and entrapment occur in a manner that cannot be predicted by the more traditional capillary pressure-saturation-interfacial area, Pc(Sw, anw), relationship, and further extensions to the constitutive relationship are needed to fully capture these mechanisms.

A 1-D/3-D coupling approach for compressible non-equilibrium two-phase flows using the Baer-Nunziato model based on the Finite-Volume framework

An efficient bi-directional 1-D/3-D coupling is here proposed for the computation of compressible two-phase flows with the use of the Baer-Nunziato model. The objective is here to couple the 3-D Baer-Nunziato model with its 1-D counterpart involving spatially varying pipe cross-sections. The present coupling acts at the common interface between three-dimensional and one-dimensional computational domains. The proposed approach is built upon the framework of the Finite-Volume method with, in particular, the exchange of the numerical fluxes coming from the approximation of both conservative and non-conservative terms of the convective part of the Baer-Nunziato model. For this purpose, local Riemann problems at the 1-D/3-D common interface are considered for the computation of the corresponding numerical conservative and non-conservative fluxes. These local Riemann problems are based on the variables associated with the 1-D domain on one side and the variables coming from the 3-D domain on the other side of the interface. The potentially non-null tangential phasic velocity components coming from the 3-D domain at the coupling interface are also considered in the local Riemann problems. In addition, adequate approximations of the non-conservative terms associated with changes of volume fraction as well as changes of cross-section are considered at the 1-D/3-D interface. As a consequence, conservation properties such as the uniform pressure and velocity profiles preservation are ensured by the present 1-D/3-D coupling. The pipe cross-section variations are also taken into account in the present method. What is more, no iterative procedure is required in the proposed approach. Several numerical shock-tubes involving both subsonic and supersonic configurations of the Baer-Nunziato model and pipe cross-section variations are then considered to assess the present 1-D/3-D coupling. The influence of the angle between the 1-D and the 3-D domains at the coupling interface is also studied. Finally, the propagation of a shock-wave in a circular tube with a sudden change in cross-section is simulated using a hybrid 1-D/3-D computation showing the potential of the proposed geometrical multi-scale strategy.

Parameter study of the grid spacer position on subcooled boiling in an upward vertical tube at low pressures

One of the principal issues in the phase change heat transfer at the vertical channel that changes the vapour fraction in the two-phase system is the presence of a grid spacer which prompts the temperature and velocity jump. In this study, the subcooled fluid stream rising inside the annular tube with grid spacer has been concentrated to investigate the effect of its location on changes in the gas volume fraction ratio, grid spacer temperature, and fluid flow jumping. The system of non-equilibrium two-phase flow of liquid-vapour exposed to an external heat transfer is modelled by mass, Euler-Euler model momentum, and energy equation of each phase. Validation of developed code is done by experimental results. Results show that extending the gas volume part as a result of a grid spacer presence decreases the Nusselt number, in conclusion, a temperature jump will occur. In the same manner, the presence of the grid spacer changes fluid streamlines. In this assessment, by contemplating the grid spacer in the vertical annular tube, the rising of fluid velocity in the vertical direction is demonstrated. As well the results show that the presence of a grid spacer moves to the liquid stage is less redirected conversely with vapour stage stream because of having more power; along these lines, around the completion of the grid spacer in the channel, the proportion of vapour volume fraction near the divider rises strongly. It has moreover been observed that if there are grid spacer inside the vertical annular tube, there will be no gigantic change in the liquid stream and gas volume division in various parts of the vertical annular tube when stood out from the customary vertical annular tube. The results demonstrate that the grid spacer before the heating zone will not change the vapour volume fraction inside the tube. The worst case happens for the grid spacer near the middle of the heated zone. The local peaks happen after the grid spacer when it is located after the heated zone.

Experimental study of pressure losses in non-equilibrium two-phase flow

Experimental study of pressure losses in non-equilibrium two-phase flow

Development of a special thermal-hydraulic component model for the core makeup tank

We have assessed the applicability of the thermal-hydraulic system analysis code, SPACE, to a small modular reactor called SMART. For the assessment, the experimental data from a scale-down integral-test facility, SMART-ITL, were used. It was conformed that the SPACE code unrealistically calculates the safety injection flow rate through the CMT and SIT during a small-break loss-of-coolant experiment. This unrealistic behavior was due to the overprediction of interfacial heat transfer at the steam-water interface in a vertically stratified flow in the tanks. In this study, a special thermal-hydraulic component model has been developed to realistically calculate the interfacial heat transfer when a strong non-equilibrium two-phase flow is formed in the CMT or SIT. Additionally, we developed a special heat structure model, which analytically calculates the heat transfer from the hot steam to the cold tank wall. The combination of two models for the tank are called the special component model. We assessed it using the SMART-ITL passive safety injection system (PSIS) test data. The results showed that the special component model well predicts the transient behaviors of the CMT and SIT.

Mass flow rate of non-equilibrium subcooled two-phase flow of R600a in a household refrigerator-freezer

A few researchers recently reported the presence of vapor bubbles of R600a at the inlet of the capillary tube of refrigerators under subcooled conditions. However, the effect of a non-equilibrium two-phase state on the refrigerant mass flow rate has not yet been addressed. The aim of this study was firstly to investigate the effects of ambient temperature and R600a charge amount on the refrigerant state at the inlet of the capillary tubes and, secondly, to examine the ability of empirical correlations for prediction of the refrigerant mass flow rate under a non-equilibrium two-phase state. A transparent tube was installed at the inlet of the capillary tubes of a parallel two-circuit cycle refrigerator-freezer for visualization of the refrigerant flow. The visualization results demonstrated the presence of a non-equilibrium state under subcooled conditions. The experimental mass flow rates were compared with predictions by empirical correlations of Rasti et al. and Wolf and Pate. Both correlations predicted the mass flow rate with a large deviation under a non-equilibrium state and acceptable deviation under a fully liquid state. Results showed that the mass flow rate under a non-equilibrium two-phase state significantly decreased compared to the fully liquid condition. Finally, a new empirical parameter was introduced to consider the non-equilibrium two-phase flow. The new parameter was applied to previous empirical correlations and the modified version of Rasti et al.’s correlation showed good agreement with the experimental data under a non-equilibrium two-phase state.

Runge–Kutta discontinuous Galerkin method for Baer–Nunziato model with ‘simple WENO’ limiting of conservative variables

Abstract The work is devoted to the application of Runge–Kutta discontinuous Galerkin (RKDG) method for solving Baer–Nunziato hyperbolic model for nonequilibrium two-phase flows. The approach is based on the application of the simple WENO limiter directly to the conservative variables. Mathematical model and the corresponding numerical algorithm are described. The results of numerical simulations for 1D and 2D tests are presented and discussed.

Riemann solver with internal reconstruction (RSIR) for compressible single-phase and non-equilibrium two-phase flows

A new Riemann solver is built to address numerical resolution of complex flow models. The research direction is closely linked to a variant of the Baer and Nunziato [2] model developed in Saurel et al. [31]. This recent model provides a link between the Marble [19] model for two-phase dilute suspensions and dense mixtures. As the Marble model, the Saurel et al. one is weakly hyperbolic with the same 4 characteristic waves, while the system involves 7 partial differential equations. It implies serious theoretical and practical issues rendering the building of a simple and accurate flow solver difficult. To overcome related difficulties the Riemann solver of Linde [18] is revisited. The method is first examined in the simplified context of the compressible Euler equations. Physical considerations are introduced in the solver improving robustness and accuracy of the Linde method. With these modifications, accuracy is shown to be comparable to the one of the HLLC solver of Toro et al. [35]. Second the two-phase flow model is considered. A locally conservative formulation is built and validated, removing issues related to the non-conservative terms. However, two extra major issues arise from numerical experiments. The solution appears not self-similar and multiple contact waves emerge in the dispersed phase. The design of a Riemann solver is consequently challenging. The modified Linde method is thus examined for the considered flow model. Some basic properties of the equations are used, such as shock relations of the dispersed phase and jump conditions across the contact wave. Thanks to these ingredients the new Riemann solver with internal reconstruction (RSIR), modification of the Linde method, handles stationary volume fraction discontinuities, presents low dissipation for transport waves and handles shocks and expansion waves accurately. It is validated on various test problems showing method's accuracy and versatility for complex flow models. Its capabilities are illustrated on a difficult two-phase flow instability problem, unresolved before.

Numerical Modeling of Wave Processes Accompanying Combustion of Inhomogeneously Distributed Composite Propellant

A mathematical model and a numerical method are proposed for studying the interior ballistic process related to combustion of inhomogeneously distributed propellant in the two-dimensional axisymmetric approximation. The gas–propellant mixture is modeled by a two-phase nonequilibrium heterogeneous medium consisting of a multicomponent gas phase of combustion products and a polydisperse solid phase of propellant granules. The mathematical model of nonequilibrium two-phase flow is based on the nonconservative Euler equations. A Godunov-type scheme with an approximate Riemann solver is developed for their solution. The propellant combustion is considered taking into account the motion of the projectile, which is modelled using the free-boundary method. Results are represented concerning the origin and evolution of the interior ballistic wave process proceeding during the combustion of an inhomogeneously distributed propellant charge and the motion of the projectile. A comparative analysis with the case of static (nonmoving) propellant is carried out.

A Finite-Volume method for compressible non-equilibrium two-phase flows in networks of elastic pipelines using the Baer–Nunziato model

A novel Finite-Volume scheme for the numerical computations of compressible two-phase flows in pipelines is proposed for the fully non-equilibrium Baer–Nunziato model. The present FV approach is the extension of the method proposed in Daude and Galon (2018) in the context of the Euler equations to the Baer–Nunziato model. In addition, proper approximations of the non-conservative terms are proposed to consider jumps of volume fraction as well as jumps of cross-section in order to respect uniform pressure and velocity profiles preservation. In particular, focus is given to the numerical treatment of abrupt changes in area and to networks wherein several pipelines are connected at junctions. The proposed method makes it possible to avoid the use of an iterative procedure for the solution of the junction problem. The present approach can also deal with general Equations Of State. In addition, the fluid–structure interaction of compressible fluid flowing in flexible pipes is also considered. The proposed scheme is then assessed on a variety of shock-tubes and other transient flow problems and experiments demonstrating its capability to resolve such problems efficiently, accurately and robustly.

How to achieve full liquid conditions at the capillary tube inlet of a household refrigerator

The capillary tube with a liquid-to-suction heat exchanger (CT-LSHX) is a component that is widely used in household refrigerators. Recent works have indicated that even when measuring subcooled conditions at the condenser outlet, the condition at the capillary tube inlet is a two-phase flow. The present work was dedicated to analyzing the actual refrigerant conditions at the capillary tube inlet and to investigating how full liquid conditions could be achieved.The research was performed using a typical household refrigerator with corresponding fresh food and freezer compartments, replacing the original refrigerant-to-air condenser with a refrigerant-to-water condenser. This allowed, first, the condensation conditions to be controlled and, second, the estimation of the refrigerant conditions at the condenser outlet from the heat exchanger balance. The obtained results indicated the presence of a non-equilibrium two-phase flow, composed of subcooled vapor and subcooled liquid, at the capillary tube inlet, with both liquid and vapor entering the capillary tube as a vortex with small, fast fluctuations of the liquid level. This non-equilibrium indicated that the subcooling, evaluated from the pressure and temperature of the refrigerant at the condenser outlet, was only apparent and did not allow the evaluation of the actual enthalpy.Finally, by using a smaller capillary tube diameter and increasing the compressor speed, full liquid conditions at the capillary tube inlet were achieved. Furthermore, a performance comparison between the original and the new design revealed that the COP was higher with full liquid conditions.

Numerical Simulation of Non-Equilibrium Two-Phase Wet Steam Flow through an Asymmetric Nozzle

The present study reported of the numerical investigation of a high-speed wet steam flow through an asymmetric nozzle. The spontaneous non-equilibrium homogeneous condensation of wet steam was numerically modeled based on the classical nucleation theory and droplet growth rate equation combined with the field conservations within the computational fluid dynamics (CFD) code of ANSYS Fluent 13.0. The equations describing droplet formations and interphase change were solved sequentially after solving the main flow conservation equations. The calculations were carried out assuming the flow two-dimensional, compressible, turbulent, and viscous. The SST k-ω model was used for modeling the turbulence within an unstructured mesh solver. The validation of numerical model was accomplished, and the results showed a good agreement between the numerical simulation and experimental data. The effect of spontaneous non-equilibrium condensation on the jet and shock structures was revealed, and the condensation shown a great influence on the jet structure.

Effects of a non-equilibrium two-phase refrigerant flow at subcooled temperatures on the performance of an R-600a refrigeration system

Previous researches reported the observation of a thermodynamic non-equilibrium two-phase refrigerant between the condenser outlet and the expansion device inlet under a subcooled condition, where the fluid temperature is below the corresponding saturation temperature. Experiments are conducted to examine the influence of the non-equilibrium subcooled two-phase state of R-600a on the performance of a refrigeration system. The enthalpy of the non-equilibrium state of the refrigerant is calculated using three different methods, and the heat rejection from the condenser and the coefficient of performance (COP) of the refrigeration system are evaluated. Compared to a case in which the refrigerant flows in an equilibrium subcooled state, the condenser heat rejection and the COP both decrease when the refrigerant is in a non-equilibrium subcooled two-phase state.

Thermodynamic Performance Simulation and Modeling of Saturated Steam Turbines

The thermal load performance optimization and control is an important means for performance analysis, control research and operation optimization of steam turbine. It has great significance for operator training, parameter adjustment and decision support. In saturated steam turbine, the rapid expansion of steam will lead to non-equilibrium two-phase condensation flow, and the model establishment is more complicated. In order to study the thermal characteristics of saturated steam turbine during load changing process, the non-equilibrium box model of two phase flow was used to simulate the process of vapor-liquid two phase flow in the saturated steam turbine cascade, analysed the energy and mass exchange process in cascade passage of saturated steam turbine, on the basis of these theories, a segemented lumped parameter model that could reflect parameter distribution between cascade passage of saturated steam turbine was established.

Simulation of high-speed nonequilibrium heterogeneous turbulent flows with phase transition

In this paper the method of modelling of high-speed nonequilibrium two-phase turbulent flows with phase transition in disperse phase is presented. The method proposed in this paper has the following features: 1) the mathematical model includes conservation equations for gas and solid particles, as well as the equations for gas vibration molecular energy, conservation of species and phase transition equation for particles; 2) the equations for the particles are solved mixed Euler-Lagrange formulation; 3) the solver algorithm is two-way coupled; 4) a special turbulence model taking into account the effect of flow compressibility is implemented. The main case considered in this paper is crystallization process of aluminium oxide Al2O3 particles in underexpanded jet flow with cross-flow at different angles of attack. Simulation results for jets of different configuration are presented.

Non-equilibrium two-phase refrigerant flow at subcooled temperatures in an R600a refrigeration system

A refrigeration system is typically designed such that the refrigerant reaches a subcooled state at the condenser outlet. A series of experiments are conducted to investigate the refrigerant state at the condenser outlet and the capillary tube inlet. The refrigeration system used in the present experiment is operated with R600a. The visual observations, as well as temperature and pressure measurements, demonstrate that R600a flows in a non-equilibrium two-phase state at highly subcooled temperatures. Furthermore, a set of equations is proposed for calculating the specific enthalpy of the refrigerant in a non-equilibrium two-phase state, and this calculation method is verified using experimental measurements. It is found that the thermodynamic property table does not provide an appropriate value of specific enthalpy in non-equilibrium conditions, while the enthalpy calculation method proposed in this work is in good agreement with the experimentally measured values.

Viscous Regularization for the Non-equilibrium Seven-Equation Two-Phase Flow Model

In this paper, a viscous regularization is derived for the non-equilibrium seven-equation two-phase flow model (SEM). This regularization, based on an entropy condition, is an artificial viscosity stabilization technique that selects a weak solution satisfying an entropy-minimum principle. The viscous regularization ensures nonnegativity of the entropy residual, is consistent with the viscous regularization for Euler equations when one phase disappears, does not depend on the spatial discretization scheme chosen, and is compatible with the generalized Harten entropies. We investigate the behavior of the proposed viscous regularization for two important limit-cases. First, a Chapman---Enskog expansion is performed for the regularized SEM and we show that the five-equation flow model of Kapila is recovered with a well-scaled viscous regularization. Second, a low-Mach asymptotic limit of the regularized seven-equation flow model is carried out whereby the scaling of the non-dimensional numbers associated with the viscous terms is determined such that an incompressible two-phase flow model, with a properly scaled regularization, is recovered. Both limit-cases are illustrated with one-dimensional numerical results, including two-phase flow shock tube tests and steady-state two-phase flows in converging-diverging nozzles. A continuous finite element discretization is employed for all numerical simulations.