Development of boundary transfer method in simulation of gas–solid turbulent flow of a riser

Computational fluid dynamics (CFD) is used to simulate the behavior of two phase gas solid in a fluid catalytic cracking (FCC) riser. Gas and particle phases are considered as separate fully interpenetrating continuous media within each control volume. Each phase described in terms of its own separate mass and momentum conservation equations. Simple k–epsilon (k_g–?_g) turbulence model is used for the gas phase and the solid phase is handled with the kinetic theory of granular flows. Source terms are used to account for the influence of hydrodynamic drag on the production, dissipation and exchange of turbulent kinetic energy between the phases. For the particles partial slip condition is considered at the wall.     

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.     

Simulation of a reacting gas–liquid bubbly flow with CFD and PBM: Validation with experiments

In this work we use computational fluid dynamics (CFD) to simulate a reactive gas–liquid bubbly system in a rectangular bubble column, operating at low superficial velocities (i.e. homogeneous regime). The gas bubbles, injected in the column through a sparger, contain one of the reactants, namely CO₂, that via mass transfer moves to the continuous liquid phase, where it reacts with NaOH. A key role is played by the bubble size distribution (BSD) and the specific surface area that define the overall mass transfer rate in the CFD model. In order to correctly predict the BSD and the polydispersity of the bubbly system the population balance equation is solved by the quadrature method of moments (QMOM), within the OpenFOAM (v. 2.2.x) two-fluid solver compressibleTwoPhaseEulerFoam. To reduce the computational time and increase stability, a second-order operator-splitting technique for the solution of the chemically reactive species is also implemented, allowing to solve the different processes involved with their own time-scale. To our knowledge this is the first time that QMOM is employed for the simulation of a real reactive bubbly system and predictions are validated against experiments.     

Mathematical simulation of gas–liquid mixture flow in a reservoir and a wellbore with allowance for the dynamical interactions in the reservoir–well system

Fluid dynamic processes related to mature oil field development are simulated by applying a numerical algorithm based on the gas–liquid mixture flow equations in a reservoir and a wellbore with allowance for the dynamical interaction in the reservoir–well system. Numerical experiments are performed in which well production characteristics are determined from wellhead parameters.     

Front tracking for a model of immiscible gas flow with large data

In this paper we study front tracking for a model of one dimensional, immiscible flow of several isentropic gases, each governed by a gamma-law. The model consists of the p-system with variable gamma representing the different gases. The main result is the convergence of a front tracking algorithm to a weak solution, thereby giving existence as well. This convergence holds for general initial data with a total variation satisfying a specific bound. The result is illustrated by numerical examples.     

Computational fluid dynamics and experimental study of the hydrodynamics of a gas–solid tapered fluidized bed

In the present work, experimental and numerical studies for the hydrodynamics in a gas–solid tapered fluidized bed have been carried out. The experimental results obtained by carrying out experiments in a tapered fluidized bed for glass bead (spherical) of 2.0 mm and dolomite (non-spherical particles) of 2.215 mm in diameter, were compared with the computational fluid dynamics (CFD) simulation results, using a commercial CFD software package, Fluent. The gas–solid flow was simulated using the Eulerian–Eulerian model and applying the kinetic theory of granular flow for solid particles. The Gidaspow drag model was used to calculate the gas–solid momentum exchange coefficients. Pressure drops predicted by the CFD simulations agreed reasonably well with experimental measurements for both types (spherical and non-spherical) of particles. Good agreement was also obtained between experimental and CFD predicted bed expansion ratios for both types of particles. Present study provides a useful basis for further works on the CFD of tapered fluidized bed.     

Predicting gas–liquid flow in a mechanically stirred tank

Computational fluid dynamics (CFD) provides a method for investigating the highly complex fluid flow in mechanically stirred tanks. Although there are quite a number of papers in the literature describing CFD methods for modelling stirred tanks, most only consider single-phase flow. However, multiphase mixtures occur very frequently in the process industries, and these are more complex situations for which modelling is not as well developed. This paper reports on progress in developing CFD simulations of gas–liquid mixing in a baffled stirred tank. The model is three-dimensional and the impeller region is explicitly included using a Multiple Frames of Reference method to account for the relative movement between impeller and baffles. Fluid flow is calculated with a turbulent two-fluid model using a finite-volume method. Several alternative treatments of the multiphase equations are possible, including various expressions for drag and dispersion forces, and a number of these have been tested. Variation in bubble size due to coalescence and break-up is also modelled. The CFD simulation method has been used to model a gas-sparged tank equipped with a Rushton turbine, and simulation results are compared with experimental data. Results to date show the correct pattern of gas distribution and the correct trends in local bubble size in the tank. Further work is needed to improve the quantitative agreement with experimental data.     

A Lagrangian-based SPH-DEM model for fluid–solid interaction with free surface flow in two dimensions

A Lagrangian-based SPH-DEM coupling model is proposed to study fluid–solid interaction (FSI) problems with free-surface flow. In this model, SPH uses an incompressible divergence-free scheme for simulating complex flow problems. Based on the Mohr–Coulomb criterion with tension cut, the DEM describes the characteristics of solid deformation and failure by means of contact models between particles. The coupling mechanism between SPH and DEM is realised by the decoupling of the force field during the process of fluid–solid interaction. That is, the motions of fluid and solid particles are reflected by the Navier–Stokes equations and interactions among solid particles are determined by Newton's second law in the DEM. To demonstrate the applicability of the SPH-DEM model, three case studies are used to verify the different fluid interaction situations with rigid bodies, deformable objects, and granular assemblies, respectively. The results of the proposed model shows good agreement with experimental data and indicates that it is capable of capturing the features of solid movement, deformation and failure under complex flow conditions with convincing accuracy and high efficiency.     

Decay of the 3D inviscid liquid–gas two-phase flow model

We establish the optimal \({L^{p}-L^{2}(1 \leq p < 6/5)}\) time decay rates of the solution to the Cauchy problem for the 3D inviscid liquid–gas two-phase flow model and analyze the influences of the damping on the qualitative behaviors of solution. Compared with the viscous liquid–gas two-phase flow model (Zhang and Zhu in J Differ Equ 258:2315–2338, 2015), our results imply that the friction effect of the damping is stronger than the dissipation effect of the viscosities and enhances the decay rate of the velocity. Our proof is based on Hodge decomposition technique, the \({L^{p}-L^{2}}\) estimates for the linearized equations and an elaborate energy method.     

Efficient simulation of gas–liquid pipe flows using a generalized population balance equation coupled with the algebraic slip model

The inhomogeneous generalized population balance equation, which is discretized with the direct quadrature method of moment (DQMOM), is solved to predict the bubble size distribution (BSD) in a vertical pipe flow. The proposed model is compared with a more classical approach where bubbles are characterized with a constant mean size. The turbulent two-phase flow field, which is modeled using a Reynolds-Averaged Navier–Stokes equation approach, is assumed to be in local equilibrium, thus the relative gas and liquid (slip) velocities can be calculated with the algebraic slip model, thereby accounting for the drag, lift, and lubrication forces. The complex relationship between the bubble size distribution and the resulting forces is described accurately by the DQMOM. Each quadrature node and weight represents a class of bubbles with characteristic size and number density, which change dynamically in time and space to preserve the first moments of the BSD. The predictions obtained are validated against previously published experimental data, thereby demonstrating the advantages of this approach for large-scale systems as well as suggesting future extensions to long piping systems and more complex geometries.     

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.     

Predictions of the gas–liquid flow in wet electrostatic precipitators

Based on Computational Fluid Dynamics (CFD), the present paper aims to simulate several important phenomena in a wet type ESP from the liquid spray generation to gas-droplet flow in electric field. A single passage between the adjacent plates is considered for the simulation domain. Firstly, the electric field intensity and ion charge density are solved locally around a corona emitter of a barbed wire electrode, which are applied to the entire ESP using periodic conditions. Next, the Euler–Lagrange method is used to simulate the gas-droplet flow. Water droplets are tracked statistically along their trajectories, together with evaporation and particle charging. Finally, the deposition density on the plate is taken as the input for the liquid film model. The liquid film is simulated separately using the homogenous Eulerian approach in ANSYS-CFX. In the current case, since the free surface of the thin water film is difficult to resolve, a special method is devised to determine the film thickness.As parametric study, the variables considered include the nozzle pressure, initial spray spreading patterns (solid versus hollow spray) and plate wettability. The droplet emission rate and film thickness distribution are the results of interest. Main findings: electric field has strong effect on the droplet trajectories. Hollow spray is preferred to solid spray for its lower droplet emission. The liquid film uniformity is sensitive to the plate wettability.     

Optimal boundary control with critical penalization for a PDE model of fluid–solid interactions

We study the finite-horizon optimal control problem with quadratic functionals for an established fluid-structure interaction model. The coupled PDE system under investigation comprises a parabolic (the fluid) and a hyperbolic (the solid) dynamics; the coupling occurs at the interface between the regions occupied by the fluid and the solid. We establish several trace regularity results for the fluid component of the system, which are then applied to show well-posedness of the Differential Riccati Equations arising in the optimization problem. This yields the feedback synthesis of the unique optimal control, under a very weak constraint on the observation operator; in particular, the present analysis allows general functionals, such as the integral of the natural energy of the physical system. Furthermore, this work confirms that the theory developed in Acquistapace et al. (Adv Diff Eq, [2])—crucially utilized here—encompasses widely differing PDE problems, from thermoelastic systems to models of acoustic-structure and, now, fluid-structure interactions.     

Transient flow in natural gas pipeline – The effect of pipeline thermal model

One-dimensional, nonisothermal gas flow model was solved to simulate the slow and fast fluid transients, such as those typically found in high-pressure gas transmission pipelines. The results of the simulation were used to understand the effect of different pipeline thermal models on the flow rate, pressure and temperature in the pipeline. It was found that simplified flow model with steady-state heat transfer term overestimates the amplitude of the temperature fluctuations in the pipeline. This result indicated that unsteady heat transfer model with the effect of heat accumulation in the surroundings of the pipeline should be used to calculate the gas parameters at locations of interest within high-pressure gas transmission pipelines.     

Impulsive perturbations in a predator–prey model with dormancy of predators

In this paper, a predator–prey model consisting of active and dormant states of predators with impulsive control strategy is established. Using Floquet theories, the small amplitude perturbation technique and the piecewise Lyapunov function method, the conditions of local and global asymptotical orbital stability of the prey-eradication periodic solution are obtained. The boundness and permanence of the impulsive system are proved by the comparison principle. Through numerical simulations, the effects of the impulsive perturbation on the inherent oscillation are investigated, which implies that the impulsive perturbation can lead to period-doubling bifurcation, chaos, and period-halving bifurcation. Moreover, the effects of the impulsive perturbation and hatching rate on the chaos of the system are comparatively studied by numerical simulation. These obtained results can be useful for ecosystem management and for explaining complex phenomena of ecosystems.     

A note on viscous liquid–gas two-phase flow model with mass-dependent viscosity and vacuum

In this paper, we consider a free boundary value problem for two-phase liquid–gas model with mass-dependent viscosity coefficient; the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid, and the fluid velocities are unequal, i.e., $u_g\ne u_l$. The local existence of a weak solution is established when the initial gas mass connects to vacuum continuously.