Numerical modelling of bubbly flows with and without heat and mass transfer

In this paper, modelling gas–liquid bubbly flows is achieved by the introduction of a population balance equation combined with the three-dimensional two-fluid model. For gas–liquid bubbly flows without heat and mass transfer, an average bubble number density transport equation has been incorporated in the commercial code CFX5.7 to better describe the temporal and spatial evolution of the geometrical structure of the gas bubbles. The coalescence and breakage effects of the gas bubbles are modelled according to the coalescence by the random collisions driven by turbulence and wake entrainment while for bubble breakage by the impact of turbulent eddies. Local radial distributions of the void fraction, interfacial area concentration, bubble Sauter mean diameter, and gas and liquid velocities, are compared against experimental data in a vertical pipe flow. Satisfactory agreements for the local distributions are achieved between the predictions and measurements. For gas–liquid bubbly flows with heat and mass transfer, boiling flows at subcooled conditions are considered. Based on the formulation of the MUSIG (multiple-size-group) boiling model and a model considering the forces acting on departing bubbles at the heated surface implemented in the computer code CFX4.4, comparison of model predictions against local measurements is made for the void fraction, bubble Sauter mean diameter, interfacial area concentration, and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Good agreement is achieved with the local radial void fraction, bubble Sauter mean diameter, interfacial area concentration 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 through the consideration of additional momentum equations or developing an algebraic slip model to account for the effects of bubble separation.     

A Locally Conservative Eulerian-Lagrangian Finite Difference Method for a Parabolic Equation

The object of this paper is to define a finite difference analogue of a locally conservative Eulerian—Lagrangian method based on mixed finite elements and to prove its convergence. The method is appropriate for convection-dominated diffusive processes; here, it will be considered in the case of a semilinear parabolic equation in a single space variable.This revised version was published online in October 2005 with corrections to the Cover Date.     

Developments in modelling of gas injection and slag foaming

Gas injection into metallurgical ladles has been an active area of CFD modelling for many years. Recent work with both Eulerian and Lagrangian frameworks is presented for bottom stirring in ladle and steelmaking electric furnace configurations. Comparison with water and liquid metal results shows that the Lagrangian models provide a better representation of the systems. Slag foaming is an important phenomenon in smelting–reduction processes and electric furnace steelmaking. The void fraction in the foam is generally greater than 0.9, a regime that has received considerably less attention than bottom stirring where the local void fraction is less than 0.1. Again, it was found, by comparison with experimental data, that Lagrangian models were generally preferable over Eulerian models.     

Computational fluid dynamics modelling of gas jets impinging onto liquid pools

Gas jets impinging onto a gas–liquid interface of a liquid pool are studied using computational fluid dynamics modelling, which aims to obtain a better understanding of the behaviour of the gas jets used metallurgical engineering industry. The gas and liquid flows are modelled using the volume of fluid technique. The governing equations are formulated using the density and viscosity of the “gas–liquid mixture”, which are described in terms of the phase volume fraction. Reynolds averaging is applied to yield a set of Reynolds-averaged conservation equations for the mass and momentum, and the k–ε turbulence model. The deformation of the gas–liquid interface is modelled by the pressure jump across the interface via the Young–Laplace equation. The governing equations in the axisymmetric cylindrical coordinates are solved using the commercial CFD code, FLUENT. The computed results are compared with experimental and theoretical data reported in the literature. The CFD modelling allows the simultaneous evaluation of the gas flow field, the free liquid surface and the bulk liquid flow, and provides useful insight to the highly complex, and industrially significant flows in the jetting system.     

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.     

VLES turbulence model for an Eulerian–Lagrangian modeling concept for bubble plumes

Traditional Reynolds-averaged Navier–Stokes (RANS) approaches to turbulence modeling, such as the k-ϵ model, have some well-known shortcomings when modeling transient flow phenomena. To mitigate this, a filtered URANS model has been derived where turbulent structures larger than a given filter size (typically grid size) is captured by the flow equations and smaller structures are modeled according to a modified k-ϵ model. This modeling approach is also known as a VLES model (Very Large Eddy Scale model), and provides more details of the transient turbulence than the k-ϵ model at little extra computational cost.In this study a two-phase extension to the VLES model is described. A modeling concept for bubble plumes has been developed in which the bubbles are tracked as particles and the flow of liquid is solved by the Navier–Stokes equations in a traditional mesh based approach. The flow of bubbles and liquid is coupled in an Eulerian–Lagrangian model. Turbulent dispersion of the bubbles is treated by a random walk model. The random walk model depends on an estimation of the eddy life time. The eddy life time for the VLES model differs from a k-ϵ model, and its mathematical expression is derived.The model is applied to ocean plumes emanating from discharge of gas at the ocean floor. Validation with experiments and comparison with k-ϵ model are shown.     

Modelling gas injection of a Peirce–Smith-converter

A commercial CFD-code PHOENICS was used to solve isothermal flow field of gas and liquid in a Peirce–Smith-converter. An Euler–Euler based algorithm was chosen for modelling fluid dynamics and evaluating controlling forces of a submerged gas injection. Predictions were made with a k–ε turbulence model in the body fitted coordinate system. The model has been verified with a 1/4 scale water model, and a parametric study with the mathematical model of submerged gas injection was made for the PS-process and the ladle injection processes. Limits of the modelling technique used were recognised, but calculated results indicate that the present model predicts the general flow field with reasonable accuracy. Predicted bubble distribution, pattern of the flow field and magnitude of flow velocities were used to evaluate scaling factors of physical models and general flow conditions of an industrial PS-converter.     

CFD modeling of hydrodynamic characteristics of a gas–liquid two-phase stirred tank

A three-dimensional CFD model was developed in this work to simulate hydrodynamic characteristics of a gas–liquid two-phase stirred tank with two six-bladed turbines and four baffles, coupling of the Multiple Size Group model to determine bubble size distribution. Important hydrodynamic parameters of the multi-phase system such as volume-averaged overall and time-averaged local gas holdups and axial liquid velocities along time and transversal courses were simulated and analyzed in detail, under varied operating conditions (inlet air flow rate and impeller rotation speed). Model predictions of local transient gas holdup and liquid velocity distributions on vertical and horizontal sections of the tank were also carried out. The overall flow patterns were discussed in detail to assess the mixing. Bubble size distributions were further predicted to reveal the unique properties of gas phase. Experimental measurements of overall gas holdups and local axial liquid velocities were used to validate the developed model.     

CFD modelling of the flow and reactions in the Olympic Dam flash furnace smelter reaction shaft

The performance of flash furnace burners can be evaluated quickly and efficiently using CFD modelling. Gas flows are modelled using the conventional Eulerian approach, while Lagrangian particle tracking is used to model the flow of solid feed through the burner and into the reaction shaft. A composite particle model has been developed that considers the solid feed to be made up of single particles containing appropriate quantities of concentrate, flux and dust. Solid fuels (such as coal) can also be included in the composite particle. Reactions between the solids and gas are then modelled using standard heat and mass transfer relationships. Results from the modelling process are shown for BHP-Billiton’s Olympic Dam copper flash smelter with the burner that was used from 1998–2003. Flow patterns, temperature and gas composition distributions, particle dispersion and residence time, and overall extent of sulphur removal are predicted and used to evaluate furnace performance. However, results are sensitive to the assumed size of the composite particles, and plant measurements are required to determine the appropriate composite particle size to predict quantitative data.     

Numerical study of dynamics of single bubbles and bubble swarms

A systematic computational study of the dynamics of gas bubbles rising in a viscous liquid is presented. Two-dimensional simulations are carried out. Both the dynamics of single bubbles and small groups of bubbles (bubble swarms) are considered. This is a continuation of our previous studies on the two-bubble coalescence and vortex shedding [A. Smolianski, H. Haario, P. Luukka, Vortex shedding behind a rising bubble and two-bubble coalescence: a numerical approach, Appl. Math. Model. 29 (2005) 615–632]. The proposed numerical method allows us to simulate a wide range of flow regimes, accurately capturing the shape of the deforming interface of the bubble and the surface tension effect, while maintaining the mass conservation. The computed time-evolution of bubble’s position and rise velocity shows a good agreement with the available experimental data. At the same time, the results on the dynamics of bubble interface area, which are, up to our knowledge, presented for the first time, show how much the overall mass transfer would be affected by the interface deformation in the case of the bubble dissolution. Another set of experiments that are of interest for chemical engineers modelling bubbly flows concerns the bubble swarms and their behavior in different bubble-shape regimes. The ellipsoidal and spherical shape regimes are considered to represent, respectively, the coalescing and non-coalescing bubble swarms. The average rise velocities of the bubble swarms are computed and analyzed for both regimes.     

Improved Chen–Ricci inequality for curvature-like tensors and its applications

We present Chen–Ricci inequality and improved Chen–Ricci inequality for curvature like tensors. Applying our improved Chen–Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms.     

Numerical simulation two phase flows of casting filling process using SOLA particle level set method

Mould filling process is a typical gas–liquid metal two phase flow phenomenon. Numerical simulation of the two phase flows of mould filling process can be used to properly predicate the back pressure effect, the gas entrapment defects, and better understand the complex motions of the gas phase and the liquid phase. In this paper, a novel sharp interface incompressible two phase numerical model for mould filling process is presented. A simple ghost fluid method like discretization method and a density evaluation method at face centers of finite difference staggered grid are proposed to overcome the difficulties when solving two phase Navier–Stokes equations with large-density ratio and large-viscosity ratio. A new mass conservation particle level set method is developed to capture the gas–liquid metal phase interface. The classical pressure-correction based SOLA algorithm is modified to solve the two phase Navier–Stokes equations. Two numerical tests including the Zalesak disk problem and the broken dam problem are used to demonstrate the accuracy of the present method. The numerical method is then adopted to simulate three mould filling examples including two high speed CCD camera imaging water filling experiments and an in situ X-ray imaging experiment of pure aluminum filling. The simulation results are in good agreement with the experiments.     

Recent applications of CFD modelling in the power generation and combustion industries

Computational fluid dynamics (CFD) modelling is now widely applied as an industrial plant development and process optimisation tool. The steady increase in computer power over recent years has enabled process engineers to model reacting multi-phase flows in a realistic geometry with good mesh resolution. As a result, the number of applications of CFD to industrial processes is also growing rapidly and increasing in sophistication. This paper reviews some of the recent applications of the CFX-4 code [CFX-4.3: Solver Manual, AEA Technology Engineering Software, 1999] to the power generation and combustion industries. The aim is to illustrate what can be done and also to identify trends and those areas where further work is needed. Examples include coal-fired low-NO_x burner design, furnace optimisation, over-fire air, gas reburn, and laminar flames. It is argued that the trend is for CFD models to become more comprehensive and accessible by being coupled to other process models and embedded in automated information and process control systems.     

A 3-dimensional Eulerian–Eulerian CFD simulation of a hydrocyclone

Hydrocyclones are used in mineral industries for classification and separation of solid particles of different sizes and densities suspended in water medium. In the present study an Eulerian–Eulerian CFD simulation of a solid–liquid hydrocyclone has been carried out taking into account two solid phases and one liquid phase. The average size of the larger particle was 0.6117 and that of the smaller particle was 0.09875 mm. Three separate momentum balance equations for the three phases have been considered unlike that in the mixture model where a single momentum equation is solved for the three phases. Two turbulent models i.e. the Reynolds stress model (RSM) and the standard k–ε model were studied. Comparison of the two turbulence models showed slight variation in prediction of the velocity profile and the separation efficiency. The maximum deviation between the two models was observed near the wall where the stress was maximum for larger size particles.     

The effect of modeling of velocity fluctuations on prediction of collection efficiency of cyclone separators

The effect of modeling of velocity fluctuations on the prediction of collection efficiency of cyclone separators has been numerically investigated using the Reynolds stress turbulence model (RSTM) and large eddy simulation (LES). The Eulerian–Lagrangian modeling approach of CFD code Fluent 6.3.26 has been employed to simulate the three dimensional, unsteady turbulent gas–solid flows in a Stairmand high efficiency cyclone. The simulated results have been compared with experimental observations available in the literature. The analysis of results shows that the RSTM and the LES have adequately predicted the mean flow field. Results of the present study demonstrate that the LES has good performance on prediction of fluctuating flow field and collection efficiency for each and every particle size. However, the performance of the RSTM is found poor in terms of prediction of velocity fluctuations and collection efficiency, especially for small particles. This relates to the precessing of the vortex core phenomenon, which is resolved more accurately by LES as compared to the RSTM simulation. The results suggest that the prediction of collection efficiency, especially for small particles is greatly influenced by the simulation of velocity fluctuations in cyclones.     

A computational determination of the Cowper–Symonds parameters from a single Taylor test

A novel technique for the dynamic characterization of metals from a single Taylor impact test is proposed. This computational characterization procedure is based on the formulation and solution of a first class inverse problem, in which the silhouette of the Taylor specimen’s final shape is expressed as a vector of its geometrical moments and used as input parameter. The inverse characterization problem is reduced to an optimization problem where the optimum material parameters for the Cowper–Symonds material model are determined. The optimization process is performed by a range adaptation real-coded genetic algorithm. Numerical example for the characterisation of 1018 steel is implemented and presented to validate the methodology presented in this paper. The effectiveness and simplicity of the proposed characterization procedure makes it an appropriate tool for the characterization of metals at high strain rates.     

Vibrational resonances of nonrigid vehicles: Polygonization and ripple patterns

The well-known phenomenon of ripples on roads has its modern counterpart in ripple patterns on railroads and polygonization of wheels on state-of-the-art lightrail streetcars. Here we study an idealized mechanical suspension model for the vibrational frequency response of a buggy with a nonrigid body (typically, an aluminium chassis and coach). The finite flexural rigidity of the body is an important novel feature. Since the essential physics is described by only one extra material parameter (viz. the stiffness coefficient), the model retains its basic simplicity and can still be analysed exactly. The dynamics (i.e., the Lagrangian equations of motion) are solved in the frequency domain. The motion on a distorted surface is treated as a nonholonomic constraint. Thus we analytically calculate spectra, e.g., the wheel spectrum. This reveals a new, significant wheel resonance (typically near 30–35 Hz), which is confirmed by means of a novel analysis of the wheel’s lift force (taking care of traction forces). At moderate city speeds this resonance agrees with recently observed characteristic ripple patterns on lightrail tracks, with wavelengths of approximately 10–20 cm (amplitudes of the order of a millimeter), and correspondingly polygonized wheels.     

Global weak solutions for a viscous liquid–gas model with transition to single-phase gas flow and vacuum

This work deals with a viscous two-phase liquid–gas model relevant to the flow in wells and pipelines. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. The model is rewritten in terms of Lagrangian coordinates and is studied in a free boundary setting where the liquid and gas masses are of compact support initially, and continuous at the boundary. Consequently, the initial masses involve a transition to single-phase gas flow and vacuum at the boundary. An appropriate balance between pressure and viscous forces is identified which allows obtaining pointwise upper and lower estimates of masses. These estimates rely on the assumption of a certain relation between the rate of degeneracy of the viscosity coefficient and the rate that determines how fast the initial masses are vanishing at the boundary. By combining these estimates with basic energy type of estimates, higher order regularity estimates are obtained. The existence of global weak solutions is then proved by showing compactness for a class of semi-discrete approximations.     

隔水管注气双梯度钻井井筒参数计算研究北大核心CSCD

基于隔水管注气双梯度钻井过程中隔水管环空多相流特性,建立了隔水管注气双梯度钻井环空多相流模型,采用有限差分法对模型进行求解,结合墨西哥湾某口深水井现场数据,分析了钻井参数对井底压力和环空压力的影响,并对注气流量的影响因素进行讨论.研究结果表明:隔水管注气双梯度钻井井底压力比常规钻井更低,更适用于海底窄密度窗口钻井;隔水管注气双梯度钻井在钻井过程中注气流量的大小对井底压力和环空压力影响较大;水深和钻井液密度是影响注气流量的两个重要因素.在隔水管注气双梯度钻井参数设计时,应选择合适的注气流量,且钻井液密度不宜过大,以确保隔水管注气双梯度钻井安全.该研究对隔水管注气双梯度钻井设计及现场作业具有指导意义.