A simple model for vertical annular and horizontal stratified two-phase flows with liquid entrainment and phase transitions: one-dimensional steady state conditions

A simple one-dimensional three-fluid model is presented for the simulation and analyses of vertical annular and stratified horizontal or inclined two-phase flows. The model has been verified for various experimental data: developing annular flow, momentum transfer in an annular flow, plane flow with a hydraulic jump, flooding in a horizontal pipe, and stratified flow with direct steam condensation. Emphasis has been laid upon several mass, momentum and energy interfacial transfer processes. New correlations are proposed for the droplet entrainment intensity in annular flow and for steam direct contact condensation on the liquid film in a stratified flow. The liquid entrainment in the annular flow is correlated with the liquid film thickness. Direct contact condensation is correlated with the turbulent convective heat transfer in the liquid film. It has been shown that the present model is able to predict all dominant processes in both types of flow.     

Macroscopic balance equations for two-phase flow models

The macroscopic, or overall, balance equations of mass, momentum, and energy are derived for a two-fluid model of two-phase flows in complex geometries. These equations provide a base for investigating methods of incorporating improved analysis methods into computer programs, such as RETRAN, which are used for transient and steady-state thermal-hydraulic analyses of nuclear steam supply systems. The equations are derived in a very general manner so that three-dimensional, compressible flows can be analyzed. The equations obtained in this report supplement the various partial differential equation two-fluid models of two-phase flow which have recently appeared in the literature. The primary objective of the investigation is the macroscopic balance equations. The flow-field model equations required by the balance equations are not discussed nor have applications and numerical solution techniques been considered.     

The one-dimensional two-fluid model with momentum flux parameters

A characteristic analysis for the governing differential equations for the compressible one-dimensional two-fluid model is presented. The momentum flux parameters for the gas and liquid phase are introduced to incorporate the effect of void fraction and velocity profiles. A characteristic equation is derived. Two roots of the equation determine the choked flow condition. The other two roots determine the stability of the differential equations in response to the short wave length disturbances. It is shown that the compressible one-dimensional two-fluid model is stable for the whole range of flow regime by use of appropriate momentum flux parameters. The choked flow condition is calculated and is compared with that of the conventional model.     

A self-standing two-fluid CFD model for vertical upward two-phase annular flow

In this paper, a new two-fluid CFD (computational fluid dynamics) model is proposed to simulate the vertical upward two-phase annular flow. This model solves the basic mass and momentum equations for the gas core region flow and the liquid film flow, where the basic governing equations are accounted for by the commercial CFD package Fluent6.3.26^®. The liquid droplet flow and the interfacial inter-phase effects are accounted for by the programmable interface of Fluent, UDF (user defined function). Unlike previous models, the present model includes the effect of liquid roll waves directly determined from the CFD code. It is able to provide more detailed and, the most important, self-standing information for both the gas core flow and the film flow as well as the inner tube wall situations.     

A remedy for the ill-posedness of the one-dimensional two-fluid model

A one-dimensional two-fluid model with explicit momentum flux parameters is proposed as a remedy for the mathematical ill-posedness of the standard one-dimensional two-fluid model. By constructing a simplified two-phase flow using existing correlations for the distribution parameter C_o and velocity profile, the momentum flux parameter is determined as a function of void fraction and density ratio for the whole range of flow regime. By performing a linear stability analysis for a two-phase flow in a channel described by the proposed model, an analytical expression for the growth factor is derived as a function of wave number, void fraction, drag coefficient, and relative velocity. From the analytical expression for the growth factor, the stability criteria are derived as a function of void faction, density ratio, and momentum flux parameters. It is shown that the two-phase flow in a channel described by the proposed model is stable in the whole range of flow regime.     

Development of one-dimensional two-fluid model with consideration of void fraction covariance effect

Accurate evaluation of gas-liquid two-phase flow behavior within rod bundle geometry is crucial for the safety assessment of the nuclear power plants. In safety assessment codes, two-phase flow in rod bundle geometry has been treated as a one-dimensional flow. In order to obtain the reliable one-dimensional two-fluid model, it is essential to utilize proper area-averaged models for governing equations and constitutive relations. The area-averaged interfacial drag term utilized to evaluate two-phase interfacial drag force is typically given by the drift-flux parameters which consider the velocity profile in two-phase flow fields. However, in a rigorous sense, the covariance due to void fraction profile is ignored in traditional formulations. In this paper, the rigorous formulation of one-dimensional momentum equation was derived by taking consideration of void fraction covariance, and a new set of one-dimensional momentum equation and constitutive relations for interfacial drag was proposed. The newly obtained set of formulations was embedded into TRAC-BF1 code and numerical simulation was performed to compare against the traditional model without covariance. It was found that effect of covariance was almost negligible for steady-state adiabatic conditions, but for high void fraction condition with added perturbation, the traditional model underpredicted the damping ratio at around 8%.     

Numerical simulations of basic interfacial instabilities with incompressible two-fluid model

The incompressible two-fluid model for stratified flow was improved. The interface of the stratified two-phase flow was successfully recognized and sharpened within the two-fluid model. After the advection step of volume fraction the numerical diffusion of the interface was reduced in such a way that the thickness of the interface is kept constant during the simulation. The surface tension force was implemented in the system of the two-fluid model equations. The two basic instabilities of stratified flows: Rayleigh-Taylor and Kelvin-Helmholtz instability were used to validate the proposed two-fluid model. The proposed two-fluid model with interface sharpening presents a step towards the simulations of mixed flows, where locally dispersed flow or stratified flow will be simulated with appropriated submodels within the two-fluid model.     

Simulation of SBLOCA based on an Improved Choked Flow Model for RELAP5/MOD3 Code

This paper is a continuation of the present author’s previous publication dealing with a new choked flow model for two-phase flow. The model based on a hyperbolic one-dimensional two-fluid model, where in the momentum equations the terms representing the interfacial pressure difference has been included in lieu of the virtual mass force terms. The new choked flow model is an improvement upon the choked flow model of the current RELAP5/MOD3 code, which itself is based on the Trapp–Ransom method. The author compares the predictions of this improved model with Trapp–Ransom model and Henry–Fauske model, for an assumed flow in a vertical pipe. The author simulates a typical PWR system with a hypothetical SBLOCA as well, and compares the system behaviors predicted by RELAP5/MOD3, based on the aforementioned choked flow models. He shows that the improved choked flow model leads to better predictions.     

New algorithm for the numerical simulation of two-phase stratified gas–liquid flow and its application for analyzing the Kelvin–Helmholtz instability criterion with respect to wavelength effect

In this article, the transient condition of two-phase stratified gas–liquid flow was investigated using numerical simulation. The basis of the method involves the one-space dimensional transient solution of the governing equations using the two-fluid model. In this paper, an analogy between the SIMPLE algorithm in two-space dimensional single-phase flow and one-space dimensional two-phase stratified flow is established through the application of a special algorithm created to solve the conservation equations. After the modeling was established and justified, wave growth was examined in two-phase stratified flow in a horizontal duct. The results were then compared with the results of the previously published articles. The results show that the classical criterion for the Kelvin–Helmholtz (K–H) instability is consistent when the long wavelength with small amplitude is considered. In this case (of the K–H instability criterion), the wavelength effect on this instability and pressure variation on the two phases interface was consistent with prior researchers’ correlations. However, as the wavelengths decreased, the results indicated that the K–H instability criterion is over-predicted and must be modified. The application of the present numerical simulation method improved the results, and the consistency with the analytical solution is higher in comparison with other well-known computer codes.     

On the stability of a one-dimensional two-fluid model

An analysis on the stability of the governing differential equations for area averaged one-dimensional two-fluid model is presented. The momentum flux parameters for gas and liquid are introduced to incorporate the effect of void fraction profiles and velocity profiles. The stability of the governing differential equations is determined in terms of gas and liquid momentum flux parameters. It is shown that the two-fluid model is well posed with certain restrictions on the liquid and gas momentum flux parameters. Simplified flow configurations for bubbly flow, slug flow, and annular flow are constructed to test the validity of proposed stability criteria. The momentum flux parameters are calculated for these flow configurations by assuming a power-law profile for both velocity and void fraction. Existing correlation for volumetric distribution parameter C_o is used. By employing simplified velocity profiles, the void fraction profile is determined from C_o correlation. It is found that the void fraction is wall-peaked at low void fraction and it becomes center-peaked as the void fraction increases. A simplified annular flow is also constructed. With these flow configurations, the momentum flux parameters are determined. It is shown that the calculated momentum flux parameters are located in the stable region above the analytically determined stability boundary. The analyses results indicate that the use of momentum flux parameter is promising, since they reflect flow structure and help to stabilize the governing differential equations.     

The development and validation of a natural circulation analysis code for marine reactors

A code PNCMC (Program for Natural Circulation under Motion Conditions) has been developed for natural circulation simulation of marine reactors. The code is based on one-dimensional two-fluid model in noninertial frame of reference. The body force term in the momentum equation is considered as a time dependent function, which consists of gravity and inertial force induced by three-dimensional ship motion. Staggered mesh, finite volume method, semi-implicit first order upwind scheme and Successive Over Relaxation (SOR) method are used to discretize and solve two-phase mass, momentum and energy equations. Single-phase natural circulation experiments under rolling condition performed in Institute of nuclear and new energy technology of Tsinghua University and two-phase natural circulation experiments under rolling condition performed by Tan and colleagues are used to validate PNCMC. The validation results indicate that PNCMC is capable to investigate the single-phase and two-phase natural circulation under rolling motion.     

Modeling approaches for strongly non-homogeneous two-phase flows

In some situations, two-phase flows exhibit strongly non-homogeneous behaviors like the one where the flow domain is divided into a bubbly region and a droplet region separated by a free surface. In such a situation, the gas bubbles in the bubbly region and the gas sky in the droplet region can exhibit very different fields of velocity and temperature. The same observation can be made for the liquid droplets and the continuous liquid in the bubbly region. The classical ‘two-fluid model’ can be too limited in its capabilities to correctly predict such types of flow, especially in the region around the free surface. In this paper, we analyze three different models more adapted to this kind of situation. The first one is a four-field model where one set of balance equations is written for each of the four fields: continuous liquid, continuous gas, liquid droplets and gas bubbles. This model is particularly heavy since it basically contains 12 balance equations. Therefore, two simplified two-field models have been written by summing the equations of the four-field model two by two, in two different ways. In the first two-field model, the equations are summed by region, giving three balance equations for the bubbly mixture and three balance equations for the droplet mixture. In the second two-field model, the equations are summed by phase, giving a model analogous to the usual two-fluid model, but with additional terms coming from the different fields interactions. The closure problem is discussed for each model and the three models are compared according to several criteria.     

Experimental study on the air/water counter-current flow limitation in a model of the hot leg of a pressurized water reactor

An experimental investigation on the air/water counter-current two-phase flow in a horizontal rectangular channel connected to an inclined riser has been conducted. This test-section representing a model of the hot leg of a pressurized water reactor is mounted between two separators in a pressurized experimental vessel. The cross-section and length of the horizontal part of the test-section are (0.25 m × 0.05 m) and 2.59 m, respectively, whereas the inclination angle of the riser is 50°. The flow was captured by a high-speed camera in the bended region of the hot leg, delivering a detailed view of the stratified interface as well as of dispersed structures like bubbles and droplets. Countercurrent flow limitation (CCFL), or the onset of flooding, was found by analyzing the water levels measured in the separators. The counter-current flow limitation is defined as the maximum air mass flow rate at which the discharged water mass flow rate is equal to the inlet water mass flow rate.From the high-speed observations it was found that the initiation of flooding coincides with the formation of slug flow. Furthermore, a hysteresis was noticed between flooding and deflooding. The CCFL data was compared with similar experiments and empirical correlations available in the literature. Therefore, the Wallis-parameter was calculated for the rectangular cross-sections by using the channel height as length, instead of the diameter. The agreement of the CCFL curve is good, but the zero liquid penetration was found at lower values of the Wallis parameter than in most of the previous work. This deviation can be attributed to the special rectangular geometry of the hot leg model of FZD, since the other investigations were done for pipes.     

A unified model considering force balances for departing vapour bubbles and population balance in subcooled boiling flow

Population balance equations combined with a three-dimensional two-fluid model are employed to predict subcooled boiling flow at low pressure in a vertical annular channel. The MUltiple-SIze-Group (MUSIG) model implemented in CFX4.4 is extended to account for the wall nucleation and condensation in the subcooled boiling regime. A model considering the forces acting on departing bubbles at the heated surface is formulated. This model provides the capacity of complex analyses on the bubble growth and departure for a wide range of wall heat fluxes and flow conditions.Comparison of model predictions against local measurements is made for the void fraction, bubble Sauter mean diameter and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcoolings. Good agreement is achieved with the local radial void fraction, bubble Sauter mean diameter and liquid velocity profiles against measurements. However, significant weakness of the model is evidenced in the prediction of the vapour velocity. Work is in progress to circumvent the deficiency of the MUSIG boiling model by the consideration of additional momentum equations to better represent the momentum forces acting on the range of bubble sizes in the bulk subcooled liquid.     

Two Phase Pressure Drop Multipliers and ICE Technique Applied in a Multibubble Slug Ejection Model

Two-phase flow of coolant is described in subassemblies of liquid metal fast breeder reactors by a one-dimensional two-phase multibubble slug ejection model. Vapor flow of sodium between the slugs is modeled as annular flow with a moving liquid film of variable thickness wetting structural surfaces. The two-phase pressure drop multiplier concept introduced by Lockhart-Martinelli is used in the frame of the slug ejection model with an algorithm that simultaneously computes interfacial friction coefficient and liquid film axial velocity distribution. The implicit continuous Eulerian (ICE) technique developed by Harlow and Amsden for computing pressure distribution in a continuum medium is applied in heterogeneous two-phase flow to the governing equations for the vapor phase. This solution method is more stable than the numerical solution by finite differences of the vapor momentum equation.     

Direct contact condensation induced transition from stratified to slug flow

Selected condensation-induced water hammer experiments performed on PMK-2 device were numerically modelled with three-dimensional two-fluid models of computer codes NEPTUNE_CFD and CFX. Experimental setup consists of the horizontal pipe filled with the hot steam that is being slowly flooded with cold water. In most of the experimental cases, slow flooding of the pipe was abruptly interrupted by a strong slugging and water hammer, while in the selected experimental runs performed at higher initial pressures and temperatures that are analysed in the present work, the transition from the stratified into the slug flow was not accompanied by the water hammer pressure peak. That makes these cases more suitable tests for evaluation of the various condensation models in the horizontally stratified flows and puts them in the range of the available CFD (Computational Fluid Dynamics) codes. The key models for successful simulation appear to be the condensation model of the hot vapour on the cold liquid and the interfacial momentum transfer model. The surface renewal types of condensation correlations, developed for condensation in the stratified flows, were used in the simulations and were applied also in the regions of the slug flow. The “large interface” model for inter-phase momentum transfer model was compared to the bubble drag model. The CFD simulations quantitatively captured the main phenomena of the experiments, while the stochastic nature of the particular condensation-induced water hammer experiments did not allow detailed prediction of the time and position of the slug formation in the pipe. We have clearly shown that even the selected experiments without water hammer present a tough test for the applied CFD codes, while modelling of the water hammer pressure peaks in two-phase flow, being a strongly compressible flow phenomena, is beyond the capability of the current CFD codes.     

A 2D numerical simulation of sub-cooled flow boiling at low-pressure and low-flow rates

The main purpose of this study is to apply a two-fluid mathematical model to numerical simulation of two-phase flow at low-pressure condition. Although models of sub-cooled boiling flow at one-dimension and high-pressure have been studied extensively, there are few equivalent studies for numerical simulation at two-dimension and low-pressure (1-2 bar) conditions. Recent literature studies on sub-cooled boiling flow at low-pressure have shown that empirical models developed for high-pressure situations are not valid at low-pressures. Since the mathematical model used in this study is accomplished at low-pressure, the transport equations for the variables of each phase are substituted in low-pressure. The governing equations of two-phase flow with an allowance to inter-phase transfer of mass, momentum and heat, are solved using a two-fluid; non-equilibrium model. The finite volume discretization scheme is used to create a linearized system of equations that are solved by SIMPLE staggered grid solution technique for a rectangular channel. Improvement of the void fraction prediction of our model for the case of low-pressure sub-cooled flow boiling conditions was achieved. It is found that the heat transfer due to evaporation and surface quenching is higher than that by convection. Good agreement is achieved with the predicted results against the experimental data’s available in the literatures for a number of test cases.     

Verification of the two-phase stratified-flow model in by separate effect tests

The code which is being developed by the Gesellschaft für Anlagen- und Rcaktorsicherheit (GRS) mbH is intended to cover, by means of a single code, the entire spectrum of loss-of-coolant and transient accidents in pressurized and boiling water reactors. The actual version Mod 1.1-Cycle A has a five-equation two-phase model based on the conservation laws for liquid mass, liquid energy, vapor energy and overall momentum. The relative velocity between liquid and vapor is determined by a full-range drift-flux model for two-phase flow in horizontal and vertical pipes. The verification of this drift-flux model is carried out by both large-scale experiments and single-effect tests. The single-effect test ECTHOR investigates stratified flow during the clearance of a water-filled loop seal by a forced air flow through the loop. ECTHOR is a French test for the consideration of two-phase flow regimes in pipes for the development of the codes. The experiments are dedicated to investigating typical two-phase flow during small break loss of coolant accidents (LOCA) in pressurized water reactors (PWR).As a measure, the remaining water level in the loop is determined as a function of the air flow rate. For the verification, a comparison between and computations, on the one hand, and experiments on the other hand is carried out. The results compare very well to each other. Test runs on different numerical grids show convergence to an asymptotic limit with increasing grid refinement.     

Numerical simulation of secondary flow in bubbly turbulent flow in sub-channel

Secondary flow in bubbly turbulent flow in sub-channel was simulated by using an algebraic turbulence stress model. The mass, momentum, turbulence energy and bubble diffusion equations were used as fundamental equation. The basis for these equations was the two-fluid model: the equation of liquid phase was picked up from the equation system theoretically derived for the gas-liquid two-fluid turbulent flow. The fundamental equation was transformed onto a generalized coordinate system fitted to the computational domain in sub-channel. It was discretized for the SIMPLE algorithm using the finite-volume method. The shape of sub-channel causes a distortion of the computational mesh, and orthogonal nature of the mesh is sometimes broken. An iterative method to satisfy a requirement for the contra-variant velocity was introduced to represent accurate symmetric boundary condition. Two-phase flow at a steady state was simulated for different magnitudes of secondary flow and void fraction. The secondary flow enhanced the momentum transport in sub-channel and accelerated the liquid phase in the rod gap. This effect was slightly mitigated when the void fraction increased. The acceleration can contribute to effective cooling in the rod gap. The numerical result implied a phenomenon of industrial interest. This suggested that experimental approach is necessary to validate the numerical model and to identify the phenomenon.     

环形狭缝通道内环状流模型的数值分析

对环形狭缝通道内的环状流建立了分离流模型。应用质量、动量和能量守恒方程 ,加上相应的边界条件和使方程组封闭的经验关系式 ,对环形狭缝通道的内、外液膜厚度、液膜内的速度分布和温度分布 ,以及内、外管的换热系数进行了数值计算求解