On mathematical models and numerical simulation of the fluidization of polydisperse suspensions

The well-known Masliyah–Lockett–Bassoon (MLB) model for sedimentation of small particles is extended to fluidization of polydisperse suspensions. For N particle species that differ in size and density, this model leads to a first-order system of N conservation laws, which in general is of mixed (in the case N = 2, hyperbolic–elliptic) type. By a simple algebraic steady-state analysis, we derive necessary compatibility conditions on the size and density parameters that admit the formation of stationary fluidized beds. We then proceed to determine the composition of polydisperse fluidized beds of given compatible species by varying the fluidization velocity and the initial composition of the suspensions, and prove that, within the framework of the MLB model combined with the Richardson–Zaki formula, the constructed bidisperse beds always cause the equations to be hyperbolic. This means that these states are always predicted to be stable. The transient behaviour of the MLB model applied to fluidization is illustrated by three numerical examples, in which the system of conservation laws is solved for N = 2, N = 3 and N = 5, respectively. These examples illustrate the effects of bed expansion and layer inversion caused by successively increasing the applied fluidization velocity and show that the predicted fluidized states are indeed attained.     

On the accuracy of two-fluid model predictions for a particular gas–solid riser flow

Literature presents a huge number of different simulations of gas–solid flows in risers applying two-fluid modeling. In spite of that, the related quantitative accuracy issue remains mostly untouched. This state of affairs seems to be mainly a consequence of modeling shortcomings, notably regarding the lack of realistic closures. In this article predictions from a two-fluid model are compared to other published two-fluid model predictions applying the same closures, and to experimental data. A particular matter of concern is whether the predictions are generated or not inside the statistical steady state regime that characterizes the riser flows. The present simulation was performed inside the statistical steady state regime. Time-averaged results are presented for different time-averaging intervals of 5, 10, 15 and 20 s inside the statistical steady state regime. The independence of the averaged results regarding the time-averaging interval is addressed and the results averaged over the intervals of 10 and 20 s are compared to both experiment and other two-fluid predictions. It is concluded that the two-fluid model used is still very crude, and cannot provide quantitative accurate results, at least for the particular case that was considered.     

Numerical Computation of Heat and Mass Transfer in Fluidized Beds with Liquid Injection

The goal of this work is the simulation of concentration and temperature distributions insides fluidized bed with a uniform liquid distribution. Further, a physically based 2D model is developed for the heat and mass transfer processes in fluidized beds with a spray nozzle. The model is a coupled and semi-linear system of convection-reaction-diffusion equations. We considered the numerical solution of these semi-linear partial differential equations with discrete boundary conditions using linear finite elements on an adaptive triangular grid in space and implicit methods in time. We present calculations using, semi implicit and implicit methods in time, and different solvers for solving the linear systems. The complex correlations of mass and liquid flow rates, mass transfer, heat transfer, drying, and transient two dimensional air humidity, air temperature, degree of wetness, liquid film temperature and particle temperature were simulated. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

河床流体模型方程的稳定化有限差分法及非线性稳定性分析

1 引言本文考虑如下问题: μ(x+2π,t)=μ(x,t), x∈R,t∈[0,τ], (1.2) μ(x,0) =μ_0(x) β,ε,σ∈R,ε,σ>0. (1.3) 该模型描述河床流体流动,其中μ(x,t)为实值函数,它代表河床流体中微粒沉淀(concen—tration)在空间方向上的周期小扰动。G.H.Ganser和D.A.Drew用摄动法对该问题进行了分析,认为该问题是非线性不稳定的。 数值研究表明,对该问题,采用通常的差分方法和Galerkin有限元是不稳定的。文     

Detailed computational and experimental fluid dynamics of fluidized beds

To describe the hydrodynamic phenomena prevailing in large industrial scale fluidized beds continuum models are required. The flow in these systems depends strongly on particle–particle interaction and gas–particle interaction. For this reason, proper closure relations for these two interactions are vital for reliable predictions on the basis of continuum models. Gas–particle interaction can be studied with the use of the lattice Boltzmann model (LBM), while the particle–particle interaction can suitably be studied with a discrete particle model. In this work it is shown that the discrete particle model, utilizing a LBM based drag model, has the capability to generate insight and eventually closure relations in processes such as mixing, segregation and homogeneous fluidization.     

Simulation study on the effect of gas permeation on the hydrodynamic characteristics of membrane-assisted micro fluidized beds

Recent research has shown the potential of membrane-assisted fluidized bed reactors for various applications, and for ultra-pure hydrogen production in particular. Due to the excellent mass transfer characteristics of fluidized beds, concentration polarization (i.e. mass transfer limitation) can be overcome and the production capacity of membrane-assisted fluidized bed reactors could be further improved by maximizing the installed membrane area per unit volume, leading to the concept of a micro-structured membrane-assisted fluidized bed reactor. In this study, numerical simulations have been systematically carried out with a discrete particle model to investigate in detail the effects of gas addition and extraction through the confining porous membrane walls on the hydrodynamic characteristics of a single membrane-assisted micro fluidized bed compartment. In particular, the effect of the permeation ratio (amount of gas permeated through the membrane relative to the amount fed) and the installed membrane area on the hydrodynamics was investigated. Gas addition or extraction via the porous membrane walls confining the emulsion phase was simulated via inward or outward directed fluxes of the gas phase, which was found to have a very pronounced influence on the bed hydrodynamics. The effects of gas permeation on the solids circulation pattern, solids holdup distribution and porosity probability density function in membrane-assisted micro fluidized beds have been discussed in great detail. It has been found that gas permeation can have an adverse effect on the bed expansion caused by gas by-passing either through the bed center for the case of gas extraction or close to the membrane walls for the case of gas addition. In addition, the formation of densified zones (increased solids holdup) close to the membrane wall that was observed in case of gas extraction may increase the bed-to-membrane mass transfer resistance. These effects may strongly decrease the gas–solid contacting and the gas residence time, which may deteriorate the reactor performance. On the other hand, it is shown that these problems caused by gas permeation may be avoided by properly tuning the gas velocity through the membrane via membrane area and other design parameters and operating conditions.     

Characteristic development of hyperbolic two-dimensional two-fluid model for gas–liquid flows with surface tension

A new hyperbolic, two-dimensional two-fluid model is developed to properly solve two-phase gas–liquid flows. Adopting the interfacial pressure jump terms in the momentum equations, the numerical stability is confirmed owing to the improvement in the mathematical property of the equation system. The derivation of the interfacial pressure jump terms is based on the infinitesimal surface-tension effect incorporated in the pressure difference at the gas–liquid interface. Through the characteristic analysis on the equation system, the eight eigenvalues are obtained analytically and they are proved real values representing phasic convective velocities and phasic sound speeds. Furthermore, the characteristic sound speeds are comparable with the earlier experimental data in excellent agreements. In addition, the eigenvectors are obtained analytically and they are shown to be linearly independent. Consequently, the governing equation system is mathematically hyperbolic with reasonable characteristic speeds by which the upwind numerical method avails. Advantage and possibility of the present model are discussed in some detail.     

流态化非均匀结构的概率模型

通过对实验数据的直方图的分析和统计检验的手段,找到了定量描述一类气固流化订系统非均匀结构动态行淡的概率分布－七参娄威布尔分布,提示了这种复杂系统内非均匀性的一些规律,模拟结果是理想的。     

On a Moving Boundary Problem Aristing in Fluidized Bed Combustion

We consider a model describing the combustion of a coal particlein a fluidized bed, in which attrition plays a dominant role.The model consists of (1) a quasi-linear elliptic equation forthe oxygen concentration, supplemented by boundary conditionson the moving surface representing the burning particle interface,(2) an evolution equation for the carbon consumption, and (3)an equation governing the motion of the interface in terms ofa specified function of the carbon consumption at the interface.We prove a global existence and uniqueness result, togetherwith a priori bounds for the solution; the existence of travellingwaves will also be established.     

Towards the problem of hydrodynamic stability of plane-parallel dusty-gas flows

Several modifications of the classical Saffman formulation of the dusty-gas flow linear stability problem are considered. Dispersed flows are described by a two-fluid model. Linear stability problems are reduced to the solution of modified Orr-Sommerfeld equations which are solved by the orthogonalization method. It is shown, that the additional factors taken into account in the problem formulation affect significantly the flow stability limits. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hydrodynamical instability initiation in two-phase stratified flow using Spectral Method

In this article, the hydrodynamical instability initiation criterion in two-phase stratified flow in a horizontal duct is examined. The nonlinear two mass and two momentum conservation equations are used for numerical simulation using the two-phase two-fluid model. The model is solved using the Finite Volume and Spectral Methods, respectively. This paper is the first to utilize the Spectral Method for the simulation of two-phase flow problems. Using the Spectral Method, we show that the numerical error and CPU time decreases noticeably relative to the Finite Volume Method. The well established Kelvin–Helmholtz (K–H) instability is selected for the test case and comparison. The results taken from each set of computer codes developed in this paper are highly compatible with the theoretical and experimental results of previous researchers who used alternative numerical methods. The results obtained from the Spectral Method in comparison with the results of other well known codes exhibit greater consistency with prior analytical results, but with much smaller computer calculation time. The step taken in the present study shows a positive progress in two-phase two-fluid model numerical solution with hydrostatic assumption. It is recommended the research to be continued with two-phase two-fluid model but with hydrodynamical assumption.     

Two-dimensional numerical modelling of shallow water flows over multilayer movable beds

The two-dimensional modelling of shallow water flows over multi-sediment erodible beds is presented. A novel approach is developed for the treatment of multiple sediment types in morphodynamics. The governing equations include the two-dimensional shallow water equations for hydrodynamics, an Exner-type equation for morphodynamics, a two-dimensional transport equation for the suspended sediments, and a set of empirical equations for entrainment and deposition. Multilayer sedimentary beds are formed of different erodible soils with sediment properties and new exchange conditions between the bed layers are developed for the model. The coupled equations yield a hyperbolic system of balance laws with source terms. As a numerical solver for the system, we implement a fast finite volume characteristics method. The numerical fluxes are reconstructed using the method of characteristics which employs projection techniques. The proposed finite volume solver is simple to implement, satisfies the conservation property and can be used for two-dimensional sediment transport problems in non-homogeneous isotropic beds without need of complicated three-dimensional equations. To assess the performance of the proposed models, we present numerical results for a wide variety of shallow water flows over sedimentary layers. Comparisons to experimental data for dam-break problems over movable beds are also included in this study.     

Introduction to nonclassical shock solutions of the modified Burgers‐Korteweg‐de Vries equation

A general property of nonlinear hyperbolic equations is the eventual formation of discontinuities in the propagating signal. These discontinuities are not uniquely defined by the initial data for the problem and a central issue is the identification of acceptable weak solutions. Particular difficulties arise when the hyperbolic system ceases to be genuinely nonlinear in some of its characteristic fields. This equates in the case of a scalar law to the lack of convexity in the flux function. Here a representative example is provided by the modified Korteweg‐de Vries‐Burgers equation which exhibits a quadratic as well as a cubic nonlinear term and arises in a variety of engineering applications including weakly nonlinear waves in fluidized beds and two‐layer fluid flows. Its solutions have the distinguishing feature to generate undercompressive or nonclassical shocks in the hyperbolic limit with dispersion and dissipation balanced. The resulting rich variety of wave phenomena: shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations, which have no counterpart in classical shock theories is discussed.     

Exact solutions of a two-fluid model of two-phase compressible flows with gravity

We aim at determining and computing a class of exact solutions of a two-fluid model of two-phase flows with/without gravity. The model is described by a non-hyperbolic system of balance laws whose characteristic fields may not be given explicitly, making it perhaps impossible to solve the Riemann problem. First, we investigate Riemann invariants in the linearly degenerate characteristic fields and obtain a surprising result on the corresponding contact waves of the model without gravity. Second, even when gravity is allowed, we show that smooth stationary solutions can be governed by a system of differential equations in divergence form, which determines jump relations for any stationary discontinuity wave. Using these relations, we establish a nonlinear equation for the pressure and propose a method to compute the pressure and then the equilibria resulted by a stationary wave.     

Error Analysis of a Spectral Projection of the Regularized Benjamin–Ono Equation

The regularized Benjamin–Ono equation appears in the modeling of long-crested interfacial waves in two-fluid systems. For this equation, Fourier–Galerkin and collocation semi-discretizations are proved to be spectrally convergent. A new exact solution is found and used for the experimental validation of the numerical algorithm. The scheme is then used to study the interaction of two solitary waves.AMS subject classification (2000) 35Q53, 65M12, 65M70.Received September 2004. Revised January 2005. Communicated by Uri Ascher.Henrik Kalisch: This work was supported in part by the BeMatA program of the Research Council of Norway.     

CFD simulation of gas–solid flow with a cluster structure-dependent drag coefficient model in circulating fluidized beds

To model the effect of clusters on hydrodynamics of gas and particles phases in risers, the interfacial drag coefficient is taken into account in computational fluid dynamic simulations by means of a two-fluid model. The momentum and energy balances that characterize the clusters in the dense phase and dispersed particles in the dilute phase are described by the multi-scale resolution approach. The model of cluster structure-dependent (CSD) drag coefficient is proposed on the basis of the minimization of energy dissipation by heterogeneous drag (MEDHD) in the full range of Reynolds number. The model of CSD drag coefficient is then incorporated into the two-fluid model to simulate flow behavior of gas and particles in a riser. The distributions of volume fraction and velocity of particles are predicted. Simulated results are in agreement with experimental data published in the literature.     

Iterative methods for finding the boundaries of unrecovered viscoplastic oil in multilayer beds

We suggest a generalized statement of the problem of finding the boundaries of limit-equilibrium unrecovered viscoplastic oil in multilayer beds in the form of a mixed variational inequality. We analyze the solvability of the problem. For its solution, we suggest an iterative method whose each step can essentially be reduced to the solution of the Dirichlet problem for the Poisson equation.     

Convergence of the Euler–Maxwell two-fluid system to compressible Euler equations

The combined non-relativistic and quasi-neutral limit of two-fluid Euler–Maxwell equations for plasmas is rigorously justified in this paper. For well-prepared initial data, the convergence of the two-fluid Euler–Maxwell system to the compressible Euler equations is proved in the time interval where a smooth solution of the limit problem exists.