Numerical modeling of unsteady flow in steam turbine stage

This work deals with numerical solution of unsteady flow in turbine stage. We use models of compressible single-phase flow of air and two-phase flow of wet steam. Presented numerical methods are based on different stator-rotor matching algorithms, as well as different numerical schemes. Numerical results achieved by both methods and flow models are discussed.     

Numerical analysis of flow-induced gas entrainment in roll coating

Gas entrainment by plane liquid jets which plunge into a liquid pool is analyzed by numerical simulations. The numerical model is based on the equations of incompressible newtonian fluids flow. The two-phase flow problem is described with the volume-of-fluid method. The dynamic behaviour of the interface is characterized by two similarity parameters, the capillary number Ca = uη/σ and the property number Γ = σ(ρ/η⁴g)^1/3 where u is the velocity, η the dynamic viscosity, σ the interface tension, ρ the density and g the gravitational constant. Numerical simulations are performed with the open source CFD code OpenFOAM. In the simulations the stability of the gas–liquid meniscus is tested for different sets of Ca and Γ. Critical values of Ca which indicate the beginning gas entrainment are deduced from the inspection of the simulation results. The findings of the numerical investigations agree well with corresponding experimental results.     

Two-phase modeling of batch sedimentation

A two-phase model for the simulation of sedimentation processes is presented. The model solves the continuity and momentum equations for the pure-clear liquid and the sludge phases, and it is verified against a well-known benchmark problem, for which analytical solutions exist. Numerical simulations of a typical 1-D batch sedimentation process for mono-dispersed particles are carried out and results are found to be in satisfactory agreement with experimental data and model predictions of other researchers. A further expansion of the model to two-dimensions leads to predictions of the dynamic behavior of settling tanks and the effect of the inclination angle on the sedimentation process.     

Viscous potential flow analysis of interfacial stability with mass transfer through porous media

The interfacial stability with mass transfer, surface tension, and porous media between two rigid planes will be investigated in the view of viscous potential flow analysis. A general dispersion relation is obtained. For Kelvin-Helmholtz instability, it is found that the stability criterion is given by a critical value of the relative velocity. On the other hand, in the absence of gravity the problem reduces to Brinkman model of the stability of two fluid layers between two rigid planes. Vanishing of the critical value of the relative velocity gives rise to a new dispersion relation for Rayleigh-Taylor instability. Formulas for the growth rates and neutral stability curve are also given and applied to air-water flows. The effects of viscosity, porous media, surface tension, and heat transfer are also discussed in relation to whether the system is potentially stable or unstable. The Darcian term, permeability’s and porosity effects are also concluded for Kelvin-Helmholtz and Rayleigh-Taylor instabilities. The relation between porosity and dimensionless relative velocity is also investigated.     

Transient modeling of non-isothermal, dispersed two-phase flow in natural gas pipelines

Unsteady-state or transient two-phase flow, caused by any change in rates, pressures or temperature at any location in a two-phase flow line, may last from a few seconds to several hours. In general, these changes are an order of magnitude longer than the transient encountered during single-phase flow. The primary reason for this phenomenon is that the velocity of wave propagation in a two-phase mixture is significantly slower. Interfacial transfer of mass, momentum and energy further complicate the problem. It is primarily due to the numerical difficulties anticipated in accurately modeling transient two-phase flow that the state of the art in this important area is restricted to a handful of studies with direct applicability to petroleum and gas engineering. A limited amount of information on the subject of two-phase transport phenomena is available in the petroleum engineering literature. Most of the publications for two-phase flow of gas assume that temperature is constant over the entire length of the pipeline.This study is the first effort to simulate the non-isothermal, one-dimensional, transient homogenous two-phase flow gas pipeline system using two-fluid conservation equations. The modified Peng–Robinson equation of state is used to calculate the vapor–liquid equilibrium in multi-component natural gas to find the vapor and liquid compressibility factors. Mass transfer between the gas and the liquid phases is treated rigorously through flash calculation, making the algorithm capable of handling retrograde condensation. The liquid droplets are assumed to be spheres of uniform size, evenly dispersed throughout the gas phase.The method of solution is the fully implicit finite difference method. This method is stable for gas pipeline simulations when using a large time step and therefore minimizes the computation time. The algorithm used to solve the non-linear finite difference thermo-fluid equations for two-phase flow through a pipe is based on the Newton–Raphson method.The results show that the liquid condensate holdup is a strong function of temperature, pressure, mass flow rate, and mixture composition. Also, the fully implicit method has advantages, such as the guaranteed stability for large time step, which is very useful for simulating long-term transients in natural gas pipeline systems.     

Numerical approximation for a Baer-Nunziato model of two-phase flows

We present a well-balanced numerical scheme for approximating the solution of the Baer-Nunziato model of two-phase flows by balancing the source terms and discretizing the compaction dynamics equation. First, the system is transformed into a new one of three subsystems: the first subsystem consists of the balance laws in the gas phase, the second subsystem consists of the conservation law of the mass in the solid phase and the conservation law of the momentum of the mixture, and the compaction dynamic equation is considered as the third subsystem. In the first subsystem, stationary waves are used to build up a well-balanced scheme which can capture equilibrium states. The second subsystem is of conservative form and thus can be numerically treated in a standard way. For the third subsystem, the fact that the solid velocity is constant across the solid contact suggests us to compose the technique of the Engquist-Osher scheme. We show that our scheme is capable of capturing exactly equilibrium states. Moreover, numerical tests show the convergence of approximate solutions to the exact solution.     

On the application of mixture model for two-phase flow induced corrosion in a complex pipeline configuration

This study introduces the application for the mixture model to simulate the liquid–liquid flow through complex pipeline configurations. The model is validated by comparing model predictions with published experimental data and showed reasonable agreement. The model is used to calculate the naphtha–water flow through a complex pipeline configuration with straight pipes and elbow fittings. The selected pipeline suffers from corrosion problems. The effect of different fittings on the pipeline is taken into account. The results obtained here showed that the mixture model is appropriate two-phase flow model and could be used to explain the reasons why specific locations in the pipeline suffer from corrosion problems while other locations do not suffer from these problems. These locations are predicted with good agreement with field measurements of corrosion distribution. It was concluded through this study that the mixture model can predict the two-phase flow features with reasonable accuracy and during relatively short computational time.     

On front solutions of the saturation equation of two-phase flow in porous media

We investigate the existence of “front” solutions of the saturation equation of two-phase flow in porous media. By front solution we mean a monotonic solution connecting two different saturations. The Brooks–Corey and the van Genuchten models are used to describe the relative-permeability – and capillary pressure–saturation relationships. We show that two classes of front solutions exist: self-similar front solutions and travelling-wave front solutions. Self-similar front solutions exist only for horizontal displacements of fluids (without gravity). However, travelling-wave front solutions exist for both horizontal and vertical (including gravity) displacements. The stability of front solutions is confirmed numerically.     

Numerical modeling of baffle location effects on the flow pattern of primary sedimentation tanks

Computational fluid dynamics (CFD) is used extensively by engineers to model and analyze complex issues related to hydraulic design, planning studies for future generating stations, civil maintenance and supply efficiency. In order to find the optimal position of a baffle in a rectangular primary sedimentation tank, computational investigations are performed. Also laboratory experiments are conducted to verify the numerical results and the measured velocity fields which were by Acoustic Doppler Velocimeter (ADV) are used. The GMRES algorithm as a pressure solver was used in the computational modeling. The results of computational investigations performed in the present study indicate that the favorable flow field (uniform in the settling zone) would be enhanced for the case that the baffle position provide small circulation regions volume and dissipate the kinetic energy in the tank. Also results show that the GMRES algorithm can obtain the good agreement between the results of numerical models and experimental tests.     

Modelling and simulation of moving interfaces in gas-assisted injection moulding process

The mathematical models of gas–liquid two-phase flow are introduced, in which the multi-mode eXtended Pom–Pom (XPP) model is selected to predict the viscoelastic behavior of polymer melt. The gas-penetration process is simulated using Level Set/SIMPLEC methods, which can capture the moving interfaces at different time, including the gas–melt interface and the melt front. The physical features such as velocity, temperature and elasticity are described at different time. The influences of gas delay time and injection pressure on gas-penetration time and penetration length are analyzed. The numerical results show that the Level Set/SIMPLEC methods can precisely trace the two moving interfaces in gas-penetration process, the fractional coverage increases at very low Deborah numbers, while at higher Deborah numbers the fractional coverage decreases, and the penetration length is affected significantly by gas delay time and injection pressure.     

An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media

A new model describing immiscible, compressible two-phase flow, such as water-gas, through heterogeneous porous media is considered. The main feature of this model is the introduction of a new global pressure and the full equivalence to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic equation (the global pressure equation) and a nonlinear diffusion-convection one (the saturation equation). Under some realistic assumptions on the data, we show an existence result with the help of appropriate regularizations and a time discretization. We use suitable test functions to get a priori estimates. In order to pass to the limit in nonlinear terms, we also obtain compactness results which are nontrivial due to the degeneracy of the system.     

Numerical investigation of the effect of helix angle and leaf margin on the flow pattern and the performance of the axial flow cyclone separator

A two-stage turbulence model based on the RNG κ–ε model combined with the Reynolds stress model is developed in this paper to analyze the gas flow in an axial flow cyclone separator. Five representative simulation cases are obtained by changing the helix angle and leaf margins of the cyclone. The pressure field and velocity field of the five cases are simulated, and then the effects of helix angle and leaf margins on the internal flow field of the cyclone are analyzed. When the continuum fluid (air) flow is relatively convergent, the discrete particle phase is added into the continuous phase and the gas-solid two-phase flow is simulated. One-way coupling method is used to solve the two-phase flow and a stochastic trajectory model is implemented for simulation of the particle phase. Finally, the pressure drop and separation efficiency of one case are measured and compare quantitatively well with the numerical results, which validates the reliability and accuracy of the simulation method based on the two-stage turbulence model.     

Modelling of two-phase incompressible flows in ducts

An improved level-set method for capturing interfaces combined with a second-order projection method for solving the time-dependent incompressible Navier–Stokes equations is implemented to compute two-phase viscous flows of Newtonian fluids on Cartesian staggered meshes. Various arrays of deformable droplets are suspended in an ambient fluid of different viscosity in ducts and the deformation and migration of droplets as they move along the duct are computed and discussed.     

Nonlinear Kelvin-Helmholtz instability of two miscible ferrofluids in porous media

The nonlinear theory of the Kelvin-Helmholtz instability is employed to analyze the instability phenomenon of two ferrofluids through porous media. The effect of both magnetic field and mass and heat transfer is taken into account. The method of multiple scale expansion is employed in order to obtain a dispersion relation for the first-order problem and a Ginzburg–Landau equation, for the higher-order problem, describing the behavior of the system in a nonlinear approach. The stability criterion is expressed in terms of various competing parameters representing the mass and heat transfer, gravity, surface tension, fluid density, magnetic permeability, streaming, fluid thickness and Darcy coefficient. The stability of the system is discussed in both theoretically and computationally, and stability diagrams are drawn. Received: July 25, 2002; revised: April 16, 2003     

Homogenization of compressible two-phase two-component flow in porous media

The paper is devoted to the homogenization of immiscible compressible two-phase two-component flow in heterogeneous porous media. We consider liquid and gas phases, two-component (water and hydrogen) flow in a porous reservoir with periodic microstructure, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. Phase exchange, capillary effects included by the Darcy–Muskat law and Fickian diffusion are taken into account. The hydrogen in the gas phase is supposed compressible and could be dissolved into the water obeying the Henry law. The flow is then described by the conservation of the mass for each component. The microscopic model is written in terms of the phase formulation, i.e. the liquid saturation phase and the gas pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion–convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem. We rigorously justify this homogenization process for the problem by using the two-scale convergence.     

Analytical estimation of liquid film thickness in two-phase annular flow using electrical resistance measurement

This paper presents an analytical solution to estimate the liquid film thickness in two-phase annular flow through a circular pipe using electrical resistance tomography. Gas–liquid flow with circular gas core surrounded by a liquid film is considered. Conformal mapping is employed to obtain the analytic solution for annular flow with an eccentric circular gas core. The liquid film thickness for an arbitrary annular flow is estimated by comparing the resistance values for concentric and eccentric annular flows. The film thickness estimation has a good performance when the normalized distance between the gas core center and the flow center is less than 0.2 and the void fraction is greater than 0.4, the estimated error of the normalized thickness is less than 0.04.     

Numerical evaluation of two recoloring operators for an immiscible two-phase flow lattice Boltzmann model

The lattice Boltzmann method is applied to the study of immiscible two-phase flows using a Rothman-Keller-type (RK) model. The focus is on the algorithm proposed by Latva-Kokko and Rothman, which has been modified and integrated into the Reis and Phillips model, which belongs to the RK family. A key element of the RK model is the recoloring step applied at the interface of two fluids, at which the fluids are separated and sent to their own region. When convection is weak, the interface in the Reis and Phillips model suffers from “lattice pinning”, which is a problem that may prevent the interface from moving. While the recoloring algorithm proposed by Latva-Kokko and Rothman diminishes this problem, it was not used in the work of Reis and Phillips. This is the framework in which the present study has been conducted. Its scope is twofold: first, to integrate and adapt the Latva-Kokko and Rothman recoloring algorithms for reducing the lattice pinning problem found in the Reis and Phillips model; and second, to conduct a set of numerical tests to show that the combination of the two algorithms leads to an improvement in the quality of the results, along with a better convergence. The context of the work is two-dimensional, with the D2Q9 lattice used as the basic computational element.     

Modelling and simulation of wave transformation in porous structures using VOF based two-phase flow model

This paper represents the results of wave transformation in porous structures and hydraulic performance of a vertical porous seawall. The study was carried out using a VOF based two-phase numerical hydrodynamic model. The model was developed by coupling an ordinary porous flow model based on extended Navier–Stokes equations for porous media, and a two-phase flow model. A unique solution domain was established with proper treatment of the interface boundary between water, air and the structure. The VOF method with an improved fluid advection algorithm was used to trace the interface between water and air. The resistance to flow caused by the presence of structural material was modeled in terms of drag and inertia forces. The parameters that govern resistance to flow in a porous media were calibrated for a typical structural setup and then the computational efficacy of the model was evaluated for several wave and structural conditions other than the calibrated setup. A set of comparisons of wave properties in and around the structure showed that the model reproduced reasonably good agreement between computed results and measured data. The model was then applied to investigate wave transformation in a vertical porous structure. The role of porosity and width of a structure in reducing wave reflection and increasing energy dissipation was investigated. It is confirmed that there exists an optimum value of structure width and porosity that can maximize hydraulic performances of a porous seawall.     

Reduced gradient method combined with augmented Lagrangian and barrier for the optimal power flow problem

A new approach for solving the optimal power flow (OPF) problem is established by combining the reduced gradient method and the augmented Lagrangian method with barriers and exploring specific characteristics of the relations between the variables of the OPF problem. Computer simulations on IEEE 14-bus and IEEE 30-bus test systems illustrate the method.