One-dimensional Two-phase Flow Research Articles

1次元二相流コードRELAP5コードと3次元二相流コードACE-3Dの熱的結合(原子炉伝熱流動解析の革新(3),原子炉伝熱流動解析の革新)

RELAP5 one-dimensional two-phase flow code is thermally coupled with ACE-3D multi-dimensional two-phase flow code for analyses of thermal-hydraulic behavior of a heat exchanger. RELAP5 code is suitable for analysis of one-dimensional flow in heat exchanger condenser tubes of interest, but is inappropriate for multi-dimensional flow analysis required to simulate flows in a large volume secondary water pool. Therefore, ACE-3D code is coupled with RELAP5 code as if it works as a subroutine of RELAP5 code. RELAP5 code gives the thermal boundary condition to ACE-3D code and takes the ACE-3D calculation results as the boundary conditions. This paper introduces the coupled code with some calculation results. Key Words: PCCS, RELAP5, ACE-3D, thermal-coupling, ROSA-V, LSTF

Limit boundary conditions for finite volume approximations of some physical problems

In the industrial context, finite volume schemes are used to compute an approximation of the solution of a system of equations set on a certain domain. When this domain is bounded, some numerical boundary conditions have to be implemented in order to complete the computation of the finite volume scheme. This is a tricky step in the elaboration of the scheme, which is still not mastered. In fact, at a closer sight, it appears that there is a deep interaction between the understanding of the physical phenomena at the boundary of the domain and the implementation of the numerical boundary conditions. Unfortunately, this link is not always completely intelligible and a reason for this lack of clarity is the fact that, whereas the continuous equation satisfied by the limit of the numerical solution is known, the boundary conditions satisfied by this very limit are not well-understood. The purpose of this paper is to clarify this point in three industrial situations of one-dimensional two-phase flows.

A COMPARATIVE ASSESSMENT OF THE PERFORMANCE OF MASS CONSERVATION-BASED ALGORITHMS FOR INCOMPRESSIBLE MULTIPHASE FLOWS

This work is concerned with the implementation and testing, within a structured collocated finite-volume framework, of seven incompressible-segregated multiphase flow algorithms that belong to the mass conservation-based algorithms (MCBA) group in which the pressure-correction equation is derived from overall mass conservation. The pressure-correction schemes in these algorithms are based on SIMPLE, SIMPLEC, SIMPLEX, SIMPLEM, SIMPLEST, PISO, and PRIME. The performance and accuracy of the multiphase algorithms are assessed by solving eight one-dimensional two-phase flow problems spanning the spectrum from dilute bubbly to dense gas-solid flows. The main outcome of this study is a clear demonstration of the capability of all MCBA algorithms to deal with multiphase flow situations. Moreover, results displayed in terms of convergence history plots and CPU times indicate that the performance of the MCBA versions of SIMPLE, SIMPLEC, and SIMPLEX are very close. In general, the performance of SIMPLEST approaches that of SIMPLE for diffusion-dominated flows. As expected, the PRIME algorithm is found to be the most expensive, due to its explicit treatment of the phasic momentum equations. The PISO algorithm is generally more expensive than SIMPLE, and its performance depends on the type of flow and solution method used. The behavior of SIMPLEM is consistent, and in terms of CPU effort it stands between PRIME and SIMPLE.

On convergence of finite volume schemes for one-dimensional two-phase flow in porous media

This paper deals with development and analysis of finite volume schemes for a one-dimensional nonlinear, degenerate, convection-diffusion equation having application in petroleum reservoir and groundwater aquifer simulation. The main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term. We analyze and compare three families of numerical schemes corresponding to explicit, semi-implicit, and implicit discretization of the diffusion term and a Godunov scheme for the advection term. L ∞ stability under appropriate CFL conditions and BV estimates are obtained. It is shown that the schemes satisfy a discrete maximum principle. Then we prove convergence of the approximate solution to the weak solution of the problem. Results of numerical experiments using the present approach are reported.

Thermal hydraulic modeling of a natural circulation loop

The experiment was carried out on the test loop HRTL-5, which simulates the geometry and system design of a 5 MW nuclear heating reactor. The analysis was based on a one-dimensional two-phase flow drift model with conservation equations for mass, steam, energy and momentum. Clausius–Clapeyron equation was used for the calculation of flashing front in the riser. A set of ordinary equations, which describes the behavior of two-phase flow in the natural circulation system, was derived through integration of the above conservation equations for the subcooled boiling region, bulk boiling region in the heated section and for the riser. The method of time-domain was used for the calculation. Both static and dynamic results are presented. System pressure, inlet subcooling and heat flux are varied as input parameters. The results show that subcooled boiling in the heated section and void flashing in the riser have significant influence on the distribution of the void fraction, mass flow rate and flow instability of the system, especially at low pressure. The response of mass flow rate, after a small disturbance in the heat flux is shown, and based on it the instability map of the system is given through experiment and calculation. There exists three regions in the instability map of the investigated natural circulation system, namely, the stable two-phase flow region, the unstable bulk and subcooled boiling flow region and the stable subcooled boiling and single phase flow region. The mechanism of two-phase flow oscillation is interpreted.

Homogeneous hydraulic and pneumatic conveying of solid particles

A discussion is provided of the difficulties and limitations involved in the application of the one-dimensional two-phase flow model to estimate pressure gradients associated with the transport of particles. Two particular cases are investigated, namely, hydraulic conveying and pneumatic transport. The validity of the model is checked by comparing experimental data with predicted values of pressure gradients and void fractions. New approaches for estimating the interaction forces in hydraulic and pneumatic transport are proposed. It is demonstrated that, despite its limitations, the two-phase flow model may provide good predictions for the pressure gradients.

Steady-state critical two-phase flashing flow with possible multiple choking phenomenon

In this paper, the modelling of pure fluid steady-state adiabatic flashing flow through a pipe involving an abrupt enlargement is presented. The thermal non-equilibrium two-phase flow model DEM has been adjusted for subcooled inlet conditions close to saturation and/or for inlet two-phase mixture state. The multichoking flow phenomenon occurs when two basic criteria are simultaneously fulfilled. A general procedure taking into account this possible occurrence is developed on the basis of two iterative algorithms. The first algorithm is applied to the mass flow rate upstream from the enlargement and the second algorithm is based on the length of the pipe downstream from the enlargement. The proposed methodology involves the improved physical one-dimensional two-phase flow model DEM and the global non-equilibrium model through the abrupt enlargement. (C) 1999 Elsevier Science Ltd. All rights reserved.

99/01812 Pressurised steam drying of Australian low-rank coals. Part 1. Equilibrium moisture contents: Bongers, G. D. et al. Fuel Process. Technol., 1998, 57, (1), 41–54

A mathematical model for mass, momentum and heat transfer in a one-dimensional two-phase flow is presented. This model was applied to the drying process of wet coal particles in a gas flow. The coal particles were assumed to have a wet core and a dry oute crust. The evaporation process of liquid from a particle was assumed to be governed by diffusion through the particle crust and convection into the gas medium. As evaporation proceeds, the wet core shrinks as the particle dries. The drying process is assumed to stop when: the moisture content of a particle falls to a predefined value; or the crust temperature reaches the ignition temperature: or break-up of the particle, caused by a pressure rise in the wet core, has occurred. The model was based on the one-dimensional balance equations for mass, momentum and energy of the gas and the dispersed phases. The system of the governing equations was represented by first-order differential equations and solved simultaneously by a numerical method. The model permitted calculation of the mass transfer ratio, change of the core diameter, and change of the average temperatures of the core and the crust. Four operating conditions were simulated using the model: isothermal: adiabatic: fixed wall temperature: and known heat flux. The model is also capable of simulating a dispersed gas-solids flow, without mass or heat transfer, in a one-dimensional transport system. The prediction of the numerical simulation, for the last case, was compared with experimental results of coal-nitrogen conveying in high pressure systems.

Classification of one-dimensional steady-state two-phase geothermal flows including permeability variations—II. The general case

In Part II of this paper the analysis is extended to include both conduction and non-zero net mass flow. We are able to distinguish between low and high enthalpy two-phase reservoirs, where the former must terminate in single-phase liquid conditions at depth, while the latter may end in single-phase liquid (for low permeabilities), single-phase vapour (for intermediate permeabilities), or can be unstable when extended (for high permeabilities). If, and only if, both conduction and net mass flow are non-zero a class of models can be identified with the property that a permeability increase (downwards) is associated with a liquid saturation decrease. This provides a prototype for a vapour-dominated geothermal reservoir.

Classification of one-dimensional steady-state two-phase geothermal flows including permeability variations—I. Theory and special cases

This paper describes a graphical method for classifying reservoir models of the above type, and for clarifying the connection between permeability (and permeability gradients) and saturation. The theory behind the method is reviewed, and then the technique is applied to the example of a bottom-heated geothermal heatpipe, which is shown to have a maximum length, beyond which the only possible stable continuation (downwards) is into single-phase vapour. A permeability increase (downwards) always acts to induce a liquid saturation increase. A similar conclusion holds for non-zero net mass flow but with no conduction. Our theoretical results agree with those obtained by numerical simulation.

Mechanistic Modeling of Critical Flow of Initially Subcooled Liquid Containing Dissolved Noncondensables Through Cracks and Slits Based on the Homogeneous Equilibrium Mixture Method

A model for critical flow in capillaries and cracks of an initially subcooled liquid containing a dissolved noncondensable gas is presented. The model is based on the iterative numerical solution of, and the imposition of critical flow conditions on, one-dimensional two-phase flow conservation equations, everywhere assuming homogeneous equilibrium two-phase flow, and equilibrium between liquid and vapor-noncondensable mixture phases with respect to the concentration of the noncondensable.Model predictions are compared with data from two different sources with good agreement, indicating that the assumption of complete equilibrium between the two phases is adequate for estimating the critical flow in microchannels and cracks. The effect of dissolved noncondensables is examined, and it is shown that the desorption of dissolved noncondensables from water can lead to a slight (up to several percent) reduction in the critical flow rate.

Discharge and transport of nearly buoyant granular solids in liquids: Part II: An investigation of the effects of particle properties on mixture-discharge rates and pore-pressure profiles

Three different solid-liquid food analogues and one model food were used in a series of flow experiments in a model conical hopper and vertical stand-pipe system. The mixture discharge rates, the liquid fraction in the discharge and the differential pore pressures set up during flow along the hopper and stand-pipe walls were monitored. The liquid fraction in the discharge decreases from an asymptotic limit set at high mixture rates to values considerably less than 0.5 as the discharge rate is reduced. Individual solid particle properties affect both the value of the critical discharge rate, below which the discharge has a higher solid content and the limiting value of the discharge rate, above which the liquid content remains unchanged. The results obtained with near-spherical particles couple well with the tomographic observations of the corresponding flow fields presented in Part I (Faderani et al., 1998, Chem. Engng Sci. 53, 553), which indicate the onset of a settling suspension well within the conical hopper at low mixture-discharge rates but a packed-bed flow extending almost to the plane of the hopper orifice as the mixture rate is increased. With cylindrical particles, the packed-bed to settling suspension flow transition is postponed to occur within the vertical stand-pipe resulting in a high liquid content in the discharge even at small-mixture discharge rates. The profiles of pore pressure measured along the hopper and stand-pipe walls during steady discharge agree well with the observed flow regime transitions and the changes in the liquid content of the discharge. The values of pore pressures corresponding to the packed-bed flow regime are compared with the predictions based on a modified form of the Ergun equation first proposed by Mills Lamptey and Thorpe (1990, Chem. Engng Sci. 46, 2197). The interstitial fluid pressures measured in the suspension regime are also compared with the theoretical predictions corresponding to the incipient settling condition. Good agreement with experiment is reported in both regimes when the transition from packed-bed to settling suspension occurs at the vicinity of the hopper orifice. A simple, first-order theoretical treatment based on Wallis' Drift-Flux Model (1969, One-dimensional Two-phase Flow, McGraw-Hill, London) explains well the observed relationships between the mixture discharge rates, the voidage profiles within the flow field and the liquid content of the discharge.

Drying of wet solid particles in a steady-state one-dimensional flow

Abstract A mathematical model for mass, momentum and heat transfer in a one-dimensional two-phase flow is presented. This model was applied to the drying process of wet coal particles in a gas flow. The coal particles were assumed to have a wet core and a dry oute crust. The evaporation process of liquid from a particle was assumed to be governed by diffusion through the particle crust and convection into the gas medium. As evaporation proceeds, the wet core shrinks as the particle dries. The drying process is assumed to stop when: the moisture content of a particle falls to a predefined value; or the crust temperature reaches the ignition temperature: or break-up of the particle, caused by a pressure rise in the wet core, has occurred. The model was based on the one-dimensional balance equations for mass, momentum and energy of the gas and the dispersed phases. The system of the governing equations was represented by first-order differential equations and solved simultaneously by a numerical method. The model permitted calculation of the mass transfer ratio, change of the core diameter, and change of the average temperatures of the core and the crust. Four operating conditions were simulated using the model: isothermal: adiabatic: fixed wall temperature: and known heat flux. The model is also capable of simulating a dispersed gas-solids flow, without mass or heat transfer, in a one-dimensional transport system. The prediction of the numerical simulation, for the last case, was compared with experimental results of coal-nitrogen conveying in high pressure systems.

Parametric analysis of two-phase flow instability in a channel with inlet and outlet hydraulic resistances

The mechanism of low-frequency self-oscillating instability of a one-dimensional two-phase flow in a channel with inlet and outlet hydraulic resistances is considered. The mechanism is based on the sensitivity of the inlet flow rate of the liquid to the pressure variation inside the channel and the sensitivity of the pressure to the variation of the outlet gas flow rate (with a constant mass rate of the liquid-gas phase transition per unit volume). A spectral analysis of the stability of the steady solution of the boundary-value problem for a hyperbolic-type nonlinear system of equations is performed within the framework of a two-velocity model of a gas-liquid flow. Parametric boundaries of the region of instability are obtained. The existence of self-oscillations in this range of parameters is supported by a numerical solution of the unsteady boundary-value problem.

Applicability of the ZND Detonation Model to Vapor Explosions

Thermal detonation theory, has been increasingly applied to vapor explosions. However, there are many questions and interrogations in previous works due to the lack of detailed explanation of the basis of the calculations and due to the incomplete interpretations or internally inconsistent discussions. Therefore, the present study calculates the Hugoniot curves for water and CFCl2 systems to discuss again the applicability of the ZND detonation model to vapor explosion. The results reveal that a solution of detonation may be obtained in a wide range of initial void fraction if steady, one-dimensional and homogeneous two-phase flow is assumed. Three types of solution of C-J point can be obtained, depending on the initial void fraction: the two-phase condition, the kink-point condition and the condensed-phase condition. However, the assumptions involved in the two-phase condition and the condensed phase condition solutions are not always satisfied. On the other hand, the kink-point condition, which has been often excluded from the discussion in previous works, provides a solution that may be actualized.

New considerations on the mass and energy balances in one-dimensional two-phase flow at steady state

A new equation to be added to the classical mass balance expressions for two-phase flow is presented. It is based on the definition of new differential control volumes of variable length which are proportional to the gas velocity in a compressible flow. The new equation is equivalent to the gas—solids velocity ratio being constant throughout the duct, and it is used to derive a new expression of the energy balance for a twophase, non-reacting flow. Through this energy balance, new correlations for the pressure drop in pneumatic conveying lines are obtained, showing an excellent agreement with experimental data from the high-pressure research facility of the Institute of Gas Technology, Chicago, IL. Finally, a more general equation, which is also valid for the mass balance of reacting flows, is supplied.

Applicability of the ZND Detonation Model to Vapor Explosions.

Thermal detonation theory, has been increasingly applied to vapor explosions. However, there are many questions and interrogations in previous works due to the lack of detailed explanation of the basis of the calculations and due to the incomplete interpretations or internally inconsistent discussions. Therefore, the present study calculates the Hugoniot curves for water and CFCl2 systems to discuss again the applicability of the ZND detonation model to vapor explosion. The results reveal that a solution of detonation may be obtained in a wide range of initial void fraction if steady, one-dimensional and homogeneous two-phase flow is assumed. Three types of solution of C-J point can be obtained, depending on the initial void fraction: the two-phase condition, the kink-point condition and the condensed-phase condition. However, the assumptions involved in the two-phase condition and the condensed phase condition solutions are not always satisfied. On the other hand, the kink-point condition, which has been often excluded from the discussion in previous works, provides a solution that may be actualized.

WAVE PHENOMENA AND ONE-DIMENSIONAL TWO-PHASE FLOW MODELS. PART II: A REFERENCE CASE: FLOW, WITHOUT MASS TRANSFER, OF TWO FLUIDS OF CONSTANT PROPERTIES

This paper describes analytical work on one-dimensional modeling of two-phase flows. The main focus is on discussing modeling strategies, in particular with relation to propagation phenomena. The "drift flux" approach, with suitable generalization, and the "two-fluid" approach are put in perspective.

Interfacial wave stability of concurrent two-phase flow in a horizontal channel

A wave front perturbation method is applied to a one-dimensional two-phase flow in a horizontal duct to analyze the interfacial wave stability. The governing equations are thus reduced to a first-order ordinary differential equation (ODE). The linear solution of this ODE agrees with the conclusion from the small amplitude wave perturbation approach. The non-linear theory suggests that the linear stability result is sufficient for predicting instability in wavy flow, and may over-predict the critical gas velocity for large disturbances. Accordingly, a weak non-linear stability criterion is obtained. The type of unstable wave, under-cut wave or roll wave, also can be predicted analytically.

Onset of slugging criterion based on characteristics and stability analyses of transient one-dimensional two-phase flow equations of two-fluid model

A two-step approach has been used to obtain a new criterion for the onset of slug formation: (1) In the first step, a more general expression than the existing models for the onset of slug flow criterion has been derived from the analysis of singular points and neutral stability conditions of the transient one-dimensional two-phase flow equations of two-fluid model. (2) In the second step, introducing simplifications and incorporating a parameter into the general expression obtained in the first step to satisfy a number of physical conditions a priori specified, a new simple criterion for the onset of slug flow has been derived. Comparisons of the present model with existing models and experimental data show that the present model agrees very closely with Taitel & Dukler's model and experimental data in horizontal pipes. In an inclined pipe ( θ = 50°), however, the difference between the predictions of the present model and those of existing models is appreciably large and the present model gives the best agreement with Ohnuki et al.'s data.