A CORRELATION EQUATION FOR CALCULATING INCLINED JET PENETRATION LENGTH IN A GAS-SOLID FLUIDIZED BED

Numerical simulation of gas-solid flow in a two-dimensional fluidized bed with an inclined jet was performed. The numerical model is based on the two-fluid model of gas and solids phase in which the solids constitutive equations are based on the kinetic theory of granular flow. The improved ICE algorithm, which can be used for both low and high-velocity fluid flow, were used to solve the model equations. The mechanism of jet formation was analyzed using both numerical simulations and experiments. The emergence and movement of gas bubbles were captured numerically and experimentally. The influences of jet velocity, nozzle diameter, nozzle inclination and jet position on jet penetration length were obtained. A semi-empirical expression was derived and the parameters were correlated from experimental data. The correlation equation, which can be easily used to obtain the inclined jet penetration length, was compared with our experimental data and published correlation equations.     

Numerical solution of a two-dimensional fluidized bed model

The numerical solution of a model describing a two-dimensional fluidized bed is considered. The model takes the form of a hyperbolic system of conservation laws with source term, coupled with an elliptic equation for determining a streamfunction. Operator splitting is used to produce homogeneous one-dimensional hyperbolic systems and ordinary differential equations involving the source term. The one-dimensional hyperbolic problems are solved using Roe's method with the addition of an entropy fix. The numerical procedure is second-order in time and first-order in space. Second-order-accuracy in space is obtained using flux limiting techniques. Numerical experiments which show the development of bubbles in the bed are presented. The familiar kidney-shaped bubble, observed experimentally, is found when using the method which is second-order in space. On the same mesh, the first-order method produces bubbles which are no longer kidney-shaped. © 1998 John Wiley & Sons, Ltd.     

A Riemann solver and upwind methods for a two‐phase flow model in non‐conservative form

We present a theoretical solution for the Riemann problem for the five‐equation two‐phase non‐conservative model of Saurel and Abgrall. This solution is then utilized in the construction of upwind non‐conservative methods to solve the general initial‐boundary value problem for the two‐phase flow model in non‐conservative form. The basic upwind scheme constructed is the non‐conservative analogue of the Godunov first‐order upwind method. Second‐order methods in space and time are then constructed via the MUSCL and ADER approaches. The methods are systematically assessed via a series of test problems with theoretical solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

CFD simulation of a gas–solid fluidized bed with two vertical jets

A computational fluid dynamics (CFD) model is used to investigate the hydrodynamics of a gas–solid fluidized bed with two vertical jets. Sand particles with a density of 2660 kg/m³ and a diameter of 5.0 × 10^?4 m are employed as the solid phase. Numerical computation is carried out in a 0.57 m × 1.00 m two-dimensional bed using a commercial CFD code, CFX 4.4, together with user-defined Fortran subroutines. The applicability of the CFD model is validated by predicting the bed pressure drop in a bubbling fluidized bed, and the jet detachment time and equivalent bubble diameter in a fluidized bed with a single jet. Subsequently, the model is used to explore the hydrodynamics of two vertical jets in a fluidized bed. The computational results reveal three flow patterns, isolated, merged and transitional jets, depending on the nozzle separation distance and jet gas velocity and influencing significantly the solid circulation pattern. The jet penetration depth is found to increase with increasing jet gas velocity, and can be predicted reasonably well by the correlations of Hong et al. (2003) for isolated jets and of Yang and Keairns (1979) for interacting jets.     

Three-dimensional multi-phase simulation of the mixing and segregation of binary particle mixtures in a two-jet spout fluidized bed

This study presents a three-dimensional numerical study of the mixing and segregation of binary particle mixtures in a two-jet spout fluidized bed based on an Eulerian–Eulerian three-fluid model. Initially, the particle mixtures were premixed and packed in a rectangular fluidized bed. As the calculation began, the gas stream was injected into the bed from the distributor and jet nozzles. The model was validated by comparing the simulated jet penetration depths with corresponding experimental data. The main features of the complex gas–solid flow behaviors and the mechanism of mixing and segregation of the binary mixtures were analyzed. Moreover, further simulations were carried out to evaluate the effects of operating conditions on the mixing and segregation of binary particle mixtures. The results illustrate that mixing can be enhanced by increasing the jet velocity or enlarging the difference of initial proportions of binary particle mixtures.     

Three-dimensional multi-phase simulation of the mixing and segregation of binary particle mixtures in a two-jet spout fluidized bed

This study presents a three-dimensional numerical study of the mixing and segregation of binary particle mixtures in a two-jet spout fluidized bed based on an Eulerian-Eulerian three-fluid model.Initially,the particle mixtures were premixed and packed in a rectangular fluidized bed.As the calculation began,the gas stream was injected into the bed from the distributor and jet nozzles.The model was validated by comparing the simulated jet penetration depths with corresponding experimental data.The main features of the complex gas-solid flow behaviors and the mechanism of mixing and segregation of the binary mixtures were analyzed.Moreover,further simulations were carried out to evaluate the effects of operating conditions on the mixing and segregation of binary particle mixtures.The results illustrate that mixing can be enhanced by increasing the jet velocity or enlarging the difference of initial proportions of binary particle mixtures.     

A three phase model of a batch fluidized bed

A three phase mathematical model of simultaneous heat and mass transfer of a batch operation for a fluidized bed is presented. The three phases are a solid free bubble, emulsion and solid phases. The model employs an elaborate five equations porosity model. Various correlations for the minimum fluidization parameters are surveyed and compared with the adequate one is being adopted in the model. The governing equations together with the boundary and initial conditions are presented for a cyclic operation of the bed. These are numerically solved for a test case where the bed is charged with silica gel particles to dehumidify a process air stream. Thus the bed works in an air dehumidification mode/bed regeneration mode cyclic operation with matching conditions.Results for the bed operation are presented as the temperature and humidity ratio variations for the test case. The results indicate the ability of the developed model to provide the␣required data for the concerned batch operated fluidized bed. Received on 11 May 1998     

An adaptive numerical scheme for solving incompressible 2‐phase and free‐surface flows

In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.     

Effect of static bed height in the upper fluidized bed on flow behavior in the lower riser section of a coupled reactor

To study olefin reduction by using an auxiliary reactor for FCC naphtha upgrading, a large-scale cold model of a riser-bed coupled to an upper fluidized bed was established. The effect of static bed height in the upper fluidized bed on narticle flow behavior in the lower riser was investigated experimentally. A restriction index of solids holdup was used to evaluate quantitatively the restrictive effect of the upper fluidized bed. Experimental results show that, under the restrictive effect of the upper fluidized bed, the riser could be divided into three regions in the longitudinal direction： accelerating, fully developed and restriction. The axial distribution of solids holdup in the riser is characterized by large solids holdup in the top and bottom sections and small solids holdup in the middle section. Overall solids holdup increased with increasing static bed height in the upper fluidized bed, while particle velocity decreased. Such restrictive effect of the upper fluidized bed could extend from the middle and top sections to the whole riser volume when riser outlet resistance is increased, which increases with increasing static bed height in the upper fluidized bed. The upper bed exerts the strongest restriction on the area close to the riser outlet.     

Comparative analysis of CFD models for jetting fluidized beds: Effect of particle-phase viscosity

Under the Eulerian–Eulerian framework of simulating gas–solid two-phase flow, the accuracy of the hydrodynamic prediction is strongly affected by the selection of rheology of the particulate phase, for which a detailed assessment is still absent. Using a jetting fluidized bed as an example, this work investigates the influence of solid rheology on the hydrodynamic behavior by employing different particle-phase viscosity models. Both constant particle-phase viscosity model (CVM) with different viscosity values and a simple two-fluid model without particle-phase viscosity (NVM) are incorporated into the classical two-fluid model and compared with the experimental measurements. Qualitative and quantitative results show that the jet penetration depth, jet frequency and averaged bed pressure drop are not a strong function of the particle-phase viscosity. Compared to CVM, the NVM exhibits better predictions on the jet behaviors, which is more suitable for investigating the hydrodynamics of gas–solid fluidized bed with a central jet.     

Effect of static bed height in the upper fluidized bed on flow behavior in the lower riser section of a coupled reactor

To study olefin reduction by using an auxiliary reactor for FCC naphtha upgrading, a large-scale cold model of a riser-bed coupled to an upper fluidized bed was established. The effect of static bed height in the upper fluidized bed on particle flow behavior in the lower riser was investigated experimentally. A restriction index of solids holdup was used to evaluate quantitatively the restrictive effect of the upper fluidized bed. Experimental results show that, under the restrictive effect of the upper fluidized bed, the riser could be divided into three regions in the longitudinal direction: accelerating, fully developed and restriction. The axial distribution of solids holdup in the riser is characterized by large solids holdup in the top and bottom sections and small solids holdup in the middle section. Overall solids holdup increased with increasing static bed height in the upper fluidized bed, while particle velocity decreased. Such restrictive effect of the upper fluidized bed could extend from the middle and top sections to the whole riser volume when riser outlet resistance is increased, which increases with increasing static bed height in the upper fluidized bed. The upper bed exerts the strongest restriction on the area close to the riser outlet.     

Evaluation of one‐ and two‐equation low‐Re turbulence models. Part II—Vortex‐generator jet and diffusing S‐duct flows

This second segment of the two‐part paper systematically examines several turbulence models in the context of two flows, namely a vortex flow created by an inclined jet in crossflow, and the flow field in a diffusing S‐shaped duct. The test cases are chosen on the basis of availability of high‐quality and detailed experimental data. The tested turbulence models are integrated to solid surfaces and consist of: Rodi's two‐layer k–ε model, Wilcox's k–ω model, Menter's two‐equation shear–stress‐transport model, and the one‐equation model of Spalart and Allmaras. The objective of the study is to establish the prediction accuracy of these turbulence models with respect to three‐dimensional separated flows with streamline curvature. At the same time, the study establishes the minimum spatial resolution requirements for each of these turbulence closures, and identifies the proper low‐Mach‐number preconditioning and artificial diffusion settings of a Reynolds‐averaged Navier–Stokes algorithm for optimum rate of convergence and minimum adverse impact on prediction accuracy. Copyright © 2003 John Wiley & Sons, Ltd.     

Large eddy simulation of incompressible free round jet with discontinuous Galerkin method

In the paper, discontinuous Galerkin method is applied to simulation of incompressible free round turbulent jet using large eddy simulation with eddy viscosity approach. The solution algorithm is based on the classical projection method, but instead of the solution of the Poisson equation, a parabolic equation is advanced in pseudo‐time, which provides the pressure field ensuring the proper pressure–velocity coupling. For time and pseudo‐time integration, explicit Runge–Kutta method is employed. The computational meshes consist of hexahedral elements with flat faces. Within a given finite element, all flow variables are expressed with modal expansions of the same order (including velocity and pressure). Discretisation of the viscous terms in the Navier–Stokes equations and Laplacian in the Poisson equation is stabilised with mixed finite element approach. The correctness of the solution algorithm is verified in a commonly used test case of laminar flow in 3D lid‐driven cavity. The results of computations of the free jet are compared with experimental and numerical reference data, the latter obtained from the high‐order pseudospectral code. The statistics of centerline flow velocity – mean velocity and its fluctuations – show satisfactory agreement with the reference data. Copyright © 2015 John Wiley & Sons, Ltd.     

HLLC scheme with novel wave‐speed estimators appropriate for two‐dimensional shallow‐water flow on erodible bed

This paper presents a first‐order HLLC (Harten‐Lax‐Van Leer with contact discontinuities) scheme to solve the Saint‐Venant shallow‐water equations, including morphological evolution of the bed by erosion and deposition of sediments. The Exner equation is used to model the morphological evolution of the bed, while a closure equation is needed to evaluate the rate of sediment transport. The system of Saint‐Venant–Exner equations is solved in a fully coupled way using a finite‐volume technique and a HLLC solver for the fluxes, with a novel wave‐speed estimator adapted to the Exner equation. Wave speeds are usually derived by computing the eigenvalues of the full system, which is highly time‐consuming when no analytical expression is available. In this paper, an eigenvalue analysis of the full system is conducted, leading to simple but still accurate wave‐speed estimators. The new numerical scheme is then tested in three different situations: (1) a circular dam‐break flow over movable bed, (2) an one‐dimensional bed aggradation problem simulated on a 2D unstructured mesh and (3) the case of a dam‐break flow in an erodible channel with a sudden enlargement, for which experimental measurements are available. Copyright © 2010 John Wiley & Sons, Ltd.     

Effects of initial static bed height on fractional conversion and bed pressure drop in tapered-in and tapered-out fluidized bed reactors

In this article, a standard 2D Two-Fluid Model (TFM) closed by the kinetic theory of granular flow (KTGF) has been applied to simulate the behavior of tapered-in and tapered-out fluidized bed reactors. In this regard, two types of chemical reactions with gas volume reduction and increase were considered to investigate the effects of initial static bed height on the fractional conversion and bed pressure drop. To validate the CFD model predictions, the results of hydrodynamic simulations concerning bed pressure drop and bed expansion ratio were compared against experimental data reported in the literature and excellent agreement was observed. The obtained simulation results clearly indicate that there is an appropriate static bed height in a tapered-in reactor in which the fractional conversion becomes maximum at this height; whereas variations of static bed height in a tapered-out reactor have insignificant influences on the fractional conversion. Moreover, it was found that the residence time, temperature, and intensity of turbulence of the gas phase are three important factors affecting the fractional conversion in tapered fluidized bed reactors. In addition, it was observed that increasing the static bed height increases the bed pressure drop for both the tapered-in and tapered-out fluidized bed reactors.     

Simulations of vertical jet penetration using a filtered two-fluid model in a gas–solid fluidized bed

The influence of a vertical jet located at the distributor in a cylindrical fluidized bed on the flow behavior of gas and particles was predicted using a filtered two-fluid model proposed by Sundaresan and coworkers. The distributions of volume fraction and the velocity of particles along the lateral direction were investigated for different jet velocities by analyzing the simulated results. The vertical jet penetration lengths at the different gas jet velocities have been obtained and compared with predictions derived from empirical correlations; the predicted air jet penetration length is discussed. Agreement between the numerical simulations and experimental results has been achieved.     

The measurement of bubble flows in fluidized beds by electrical probe

Measurements of bubble velocities, dimensions and flow rates in a two-dimensional fluidized bed by a dual electrical capacitive probe are compared with measurements from cine photography; the cine photographs and electric measurements were taken at the same point in a fluidized bed and at the same time. It was found that both sets of measurements were in agreement only when the conditions of electrical measurement were arranged to exclude spurious signals, while still retaining sensitivity, and when the theory of measurement included the effects of bubble retardation and distortion, and allowed for the stochastic incidence between the bubble front and the probes.     

Multilayer open flow model: A simple pressure correction method for wave problems

This paper describes a pressure correction method for single‐ and multilayer open flow models. The method does not require any complex procedures to solve the discretization of the Poisson equation and is distinguished by a high computational efficiency. The algorithm can easily be adapted to irregular meshes and parallelized. Parabolic interpolation of the pressure profile is used for the free surface. The discretization of the Poisson equation is written in a matrix form, allowing its usage also in the case of basic function expansion of the depth pressure profile. The paper presents the results of algorithm verification where experimental data sensitive to the numerical dissipation of the calculation model was used. Iteration convergence is high including problems with dry‐bed flooding. The complete described technique of pressure correction is implemented in OpenCL on the GPU. Computation time for a test problem solved using CPU and GPU is compared.     

Hydrodynamics of a bubble-induced inverse fluidized bed reactor with a nanobubble tray

This paper reports on the hydrodynamics of a bubble-induced inverse fluidized bed reactor, using a nanobubble tray gas distributor, where solid particles are fluidized only by an upward gas flow. Increasing the gas velocity, the fixed layer of particles initially packed at the top of the liquid starts to move downwards, due to the rise of bubbles in this system, and then gradually expands downwards until fully suspended. The axial local pressure drops and standard deviation were examined to delineate the flow regime comprehensively under different superficial gas velocities. Four flow regimes (fixed bed regime, initial fluidization regime, expanded regime, and post-homogeneous regime) were observed and three transitional gas velocities (the initial fluidization velocity, minimum fluidization velocity, and homogeneous fluidization velocity) were identified to demarcate the flow regime. Three correlations were developed for the three transitional velocities. As the fine bubbles generated from the nanobubble tray gas distributor are well distributed in the entire column, the bed expansion process of the particles is relatively steady.     

A fully coupled numerical technique for 2D bed morphology calculations

A fully coupled two‐dimensional subcritical and/or supercritical, viscous, free‐surface flow numerical model is developed to calculate bed variations in alluvial channels. Vertically averaged free‐surface flow equations in conjunction with sediment transport equation are numerically solved using an explicit finite‐volume scheme using transformed grid in order to handle complex geometry fluvial problems. Convergence is accelerated with use of a multi‐grid technique. Firstly the capabilities of the proposed method are demonstrated by analyzing subcritical and supercritical hydrodynamic flows. Thereafter, an analysis of one‐ and two‐dimensional flows is performed referring to aggradation and scouring. For all reported test cases the computed results compare reasonably well with measurements as well as with other numerical solutions. The method is stable, reliable and accurate handling a variety of sediment transport equations with rapid changes of sediment transport at the boundaries. Copyright © 2002 John Wiley & Sons, Ltd.