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.     

Prediction of the bed-load transport by gas–liquid stratified flows in horizontal ducts

Solid particles can be transported as a mobile granular bed, known as bed-load, by pressure-driven flows. A common case in industry is the presence of bed-load in stratified gas–liquid flows in horizontal ducts. In this case, an initially flat granular bed may be unstable, generating ripples and dunes. This three-phase flow, although complex, can be modeled under some simplifying assumptions. This paper presents a model for the estimation of some bed-load characteristics. Based on parameters easily measurable in industry, the model can predict the local bed-load flow rates and the celerity and the wavelength of instabilities appearing on the granular bed.     

On the hyperbolicity of the two-fluid model for gas–liquid bubbly flows

The hyperbolicity condition of the system of partial differential equations (PDEs) of the incompressible two-fluid model, applied to gas–liquid flows, is investigated. It is shown that the addition of a dispersion term, which depends on the drag coefficient and the gradient of the gas volume fraction, ensures the hyperbolicity of the PDEs, and prevents the nonphysical onset of instabilities in the predicted multiphase flows upon grid refinement. A constraint to be satisfied by the coefficient of the dispersion term to ensure hyperbolicity is obtained. The effect of the dispersion term on the numerical solution and on its grid convergence is then illustrated with numerical experiments in a one-dimensional shock tube, in a column with a falling fluid, and in a two-dimensional bubble column.     

The Lyapunov–Movchan method in problems of the stability of flows and deformation processes

The extension of Lyapunov's method to continuous mechanical systems are discussed. An annotated bibliography of papers is given in which, based on the Lyapunov–Movchan method, with the construction of corresponding functionals, a direct analysis is carried out of the stability of motion (deformation) of continuous mechanical systems. The material is divided into sections, devoted to the following: (a) the extension of the mathematical apparatus as a whole to continuous and dynamic systems, (b) the stability of elastic, elastoplastic and viscoelastic deformable solids, (c) stability in aeroelasticity and hydroelasticity theory, (d) the linearized theory of hydrodynamic stability, and (e) the stability with reference to perturbations of material functions in the theory of constitutive relations.     

Convergence of lattice Boltzmann methods for Navier–Stokes flows in periodic and bounded domains

Combining an asymptotic analysis of the lattice Boltzmann method with a stability estimate, we are able to prove some convergence results which establish a strict relation to the incompressible Navier–Stokes equation. The proof applies to the lattice Boltzmann method in the case of periodic domains and for specific bounded domains if the Dirichlet boundary condition is realized with the bounce back rule.     

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.     

Chaotic analysis of pressure fluctuation signal in the gas–liquid–solid slurry column

The pressure signal of a slurry column is easily obtained by using a pressure sensor, and a chaotic analysis method is used to analyze these signals in order to indicate the flow pattern of the slurry column. The slopes of the correlation integral curve indicate the flow pattern of the slurry column in various operating conditions. The flow pattern is dispersed bubble regime when the superficial velocity is low and the correlation integral curve has two slopes. The flow pattern changes into transition regime with increase in the superficial velocity, the correlation integral curve has only one slope. In the case of the flow pattern becoming a slugging regime, there are several slopes to the correlation integral curve. So it is convenient to find out the flow pattern in the slurry column by solving the slopes of the correlation integral of the pressure signal. The maximum Lyapunov exponent represents the chaos in a slurry column with various solid holdups. The maximum Lyapunov exponent is nearly similar at different heights when the flow patterns are dispersed bubble regime and slugging regime, but the maximum Lyapunov exponent at the axial height is quite different when the flow pattern is transition regime.     

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.     

Assessment of turbulence modeling for gas flow in two-dimensional convergent–divergent rocket nozzle

In the present study, the turbulent gas flow dynamics in a two-dimensional convergent–divergent rocket nozzle is numerically predicted and the associated physical phenomena are investigated for various operating conditions. The nozzle is assumed to have impermeable and adiabatic walls with a flow straightener in the upstream side and is connected to a plenum surrounding the nozzle geometry and extended in the downstream direction. In this integrated component model, the inlet flow is assumed a two-dimensional, steady, compressible, turbulent and subsonic. The physics based mathematical model of the considered flow consists of conservation of mass, momentum and energy equations subject to appropriate boundary conditions as defined by the physical problem stated above. The system of the governing equations with turbulent effects is solved numerically using different turbulence models to demonstrate their numerical accuracy in predicting the characteristics of turbulent gas flow in such complex geometry. The performance of the different turbulence models adopted has been assessed by comparing the obtained results of the static wall pressure and the shock position with the available experimental and numerical data. The dimensionless shear stress at the nozzle wall and the separation point are also computed and the flow field is illustrated. The various implemented turbulence models have shown different behavior of the turbulent characteristics. However, the shear-stress transport (SST) k–ω model exhibits the best overall agreement with the experimental measurements. In general, the proposed numerical procedure applied in the present paper shows good capability in predicting the physical phenomena and the flow characteristics encountered in such kinds of complex turbulent flow.     

Unified account of gas pollutants and greenhouse gas emissions: Chinese transportation 1978–2004

To facilitate the aggregation of both quantity and quality of waste emissions, the concept of chemical exergy combining the first and second laws of thermodynamics is introduced for a unified account of gas pollutants and greenhouse gases, by a case study for the Chinese transportation system 1978–2004 with main gas pollutants of NO, SO₂, CO and main greenhouse gases of CO₂ and CH₄. With chemical exergy emission factors concretely estimated, the total emission as well as emission intensity by exergy of the overall transportation system and of its four modes of highways, railways, waterways and civil aviation are accounted in full detail and compared with those by the conventionally prevailing metrics of mass, with essential implications for environmental policy making.     

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.     

Spectral methods for bivariate Markov processes with diffusion and discrete components and a variant of the Wright–Fisher model

The aim of this paper is to study differential and spectral properties of the infinitesimal operator of two dimensional Markov processes with diffusion and discrete components. The infinitesimal operator is now a second-order differential operator with matrix-valued coefficients, from which we can derive backward and forward equations, a spectral representation of the probability density, study recurrence of the process and the corresponding invariant distribution. All these results are applied to an example coming from group representation theory which can be viewed as a variant of the Wright–Fisher model involving only mutation effects.     

Efficient exponential Runge–Kutta methods of high order: construction and implementation

BIT Numerical Mathematics - Exponential Runge–Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly...     

Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

Wavelet method is a recently developed tool in applied mathematics. Investigation of various wavelet methods, for its capability of analyzing various dynamic phenomena through waves gained more and more attention in engineering research. Starting from ‘offering good solution to differential equations’ to capturing the nonlinearity in the data distribution, wavelets are used as appropriate tools at various places to provide good mathematical model for scientific phenomena, which are usually modeled through linear or nonlinear differential equations. Review shows that the wavelet method is efficient and powerful in solving wide class of linear and nonlinear reaction–diffusion equations. This review intends to provide the great utility of wavelets to science and engineering problems which owes its origin to 1919. Also, future scope and directions involved in developing wavelet algorithm for solving reaction–diffusion equations are addressed.     

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–?ojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.     

Well-posedness of Hamilton–Jacobi equations in population dynamics and applications to large deviations

We prove Freidlin–Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth–death processes, Galton–Watson trees, epidemic SI models, and prey–predator models.The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton–Jacobi equations. The notable feature for these Hamilton–Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton–Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.     

Modified Lagrange–Galerkin methods of first and second order in time for convection–diffusion problems

We introduce modified Lagrange–Galerkin (MLG) methods of order one and two with respect to time to integrate convection–diffusion equations. As numerical tests show, the new methods are more efficient, but maintaining the same order of convergence, than the conventional Lagrange–Galerkin (LG) methods when they are used with either P ₁ or P ₂ finite elements. The error analysis reveals that: (1) when the problem is diffusion dominated the convergence of the modified LG methods is of the form O(h ^m+1 + h ² + Δt ^q ), q = 1 or 2 and m being the degree of the polynomials of the finite elements; (2) when the problem is convection dominated and the time step Δt is large enough the convergence is of the form O(\frach^m+1Dt+h²+Dt^q){O(\frac{h^{m+1}}{\Delta t}+h^{2}+\Delta t^{q})} ; (3) as in case (2) but with Δt small, then the order of convergence is now O(h ^m  + h ² + Δt ^q ); (4) when the problem is convection dominated the convergence is uniform with respect to the diffusion parameter ν (x, t), so that when ν → 0 and the forcing term is also equal to zero the error tends to that of the pure convection problem. Our error analysis shows that the conventional LG methods exhibit the same error behavior as the MLG methods but without the term h ². Numerical experiments support these theoretical results.     

Optimal control of diffusion processes and hamilton–jacobi–bellman equations part 2 : viscosity solutions and uniqueness

We consider general optimal stochastic control problems and the associated Hamilton–Jacobi–Bellman equations. We develop a general notion of week solutions – called viscosity solutions – of the amilton–Jocobi–Bellman equations that is stable and we show that the optimal cost functions of the control problems are always solutions in that sense of the Hamilton–Jacobi–Bellman equations. We then prove general uniqueness results for viscosity solutions of the Hamilton–Jacobi–Bellman equations.