Large deviations for Glauber dynamics of continuous gas

This paper is devoted to the large deviation principles of the Glauber-type dynamics of finite or infinite volume continuous particle systems.We prove that the level-2 empirical process satisfies the large deviation principles in the weak convergence topology,while it does not satisfy the large deviation principles in the T-topology.     

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.     

Greenhouse gas abatement — toward Pareto-optimal decisions under uncertainty

This paper examines the economic logic of integrated assessment — balancing the costs against the benefits of greenhouse gas abatement. Stylized facts are employed in a multiregion computable general equilibrium model with a public good. The percentage shares of global emissions are determined outside the model — based upon some form of international agreement — and emission rights are tradeable between regions. The analysis is confined to Pareto-optimal (cooperative) solutions. We focus on the sensitivity of initial decisions to low-probability, high-consequence scenarios associated with cumulative emissions. For simplicity, there are only two regions, two tradeable goods, two time periods, and two states-of-world. With the particular form of public good model adopted here (production rather than utility function impacts), it turns out that a Pareto-optimal hedging strategy is indepedent of the emission shares allocated to each region. Equity issues may be separated from those of economic efficiency. Similar results extend to cases in which there are additional regions, tradeable goods, time periods, and states-of-world.Presented at the Conference on the Economics of Global Environmental Change, Birmingham University, May 9–11, 1994. This research was funded by the Electric Power Research Institute (EPRI). The views presented here are solely those of the individual authors, and do not necessarily represent the views of EPRI or its members.     

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 lattice gas automaton approach to “turbulent diffusion”

A periodic Kolmogorov type flow is implemented in a lattice gas automaton. For given aspect ratios of the automaton universe and within a range of Reynolds number values, the averaged flow evolves towards a stationary two-dimensional ABC type flow. We show the analogy between the streamlines of the flow in the automaton and the phase plane trajectories of a dynamical system. In practice flows are commonly studied by seeding the fluid with suspended particles which play the role of passive tracers. Since an actual flow is time-dependent and has fluctuations, the tracers exhibit interesting intrinsic dynamics. When tracers are implemented in the automaton and their trajectories are followed, we find that the tracers displacements obey a diffusion law, with “super-diffusion” in the direction orthogonal to the direction of the initial forcing.     

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.     

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.     

A fully meshless method for ‘gas – evaporating droplet’ flow modelling

A meshless method for modelling two-phase flows with phase transition is described. The method is based on consideration of three systems: viscous-vortex blobs, thermal-blobs and droplets; and can be applied for numerical simulation of 2D non-isothermal flows of ‘gas-evaporating droplets’ in the framework of the one-way coupled two-fluid approach. The carrier phase is viscous incompressible gas. The dispersed phase is presented by a cloud of identical spherical droplets, and, due to evaporation, the radius and mass of droplets are time dependent. The carrier phase parameters are calculated using the viscous-vortex and thermal-blob method; the dispersed phase parameters are calculated using the Lagrangian approach. Two applications have been considered: (i) a standard benchmark – Lamb vortex; (ii) a cold spray injected into a hot quiescent gas. In the latter problem three cases corresponding to three droplet sizes were investigated. The smallest droplets (of the three cases considered) are more readily entrained by the carrier phase and form ring-like structures; the flow shows better mixing. Larger droplets evaporate less intensively. The medium sized droplets collect into two narrow bands stretched along the jet axis. The largest droplets form a two-phase jet, which remains close to the jet axis. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

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.     

Explicit calculation of the partition function of a one-dimensional δ-function bose gas

We present a method to calculate explicitly the partition function of a one-dimensional δ-function bose gas. The cluster integrals are obtained for up to the four particles and are shown to be the same as those by the thermal Bethe ansatz method (TBA).     

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.     

International environmental cooperation: a new eye on the greenhouse gas emissions’ control

This paper re-examines the formation of International Environmental Agreements (IEAs) modelled as a two-stage non-cooperative game when countries’ strategies to control pollution are complementary. This new assumption relying on empirical and theoretical evidences means that reinforcement effects do exist between countries’ strategies when polluting or abating. From a deliberately conventional model the results established analytically strongly contrast with those in the literature on IEAs. We find that the unique stable agreement can consist in half countries involved in the negotiation; we demonstrate that the environmental impact of such cooperation is almost total: it tends toward the one of the full cooperative solution. Even if the incentives to free-ride are less strong, we do not observe the formation of the “grand” coalition: not all the countries sign the agreement. We also explain why the level of cooperation is decreasing with the perception countries have of the seriousness of the problem.     

Theoretical analysis of Rayleigh–Taylor instability on a spherical droplet in a gas stream

A linear analysis of the Rayleigh–Taylor (R–T) instability on a spherical viscous liquid droplet in a gas stream is presented. Different from the most previous studies in which the external acceleration is usually assumed to be radial, the present study considers a unidirectional acceleration acting on a spherical droplet with arbitrary initial disturbances and therefore can provide insights into the influence of R–T instability on the atomization of spherical droplets. A general recursion relation coupling different spherical modes is derived and two physically prevalent limiting cases are discussed. In the limiting case of inviscid droplet, the critical Bond numbers to excite the instability and the growth rates for a given Bond number are obtained by solving two eigenvalue problems. In the limiting case of large droplet acceleration, different spherical modes are asymptotically decoupled and an explicit dispersion relation is derived. For given Bond number and Ohnesorge numbers, the critical size of stable droplet, the most-unstable mode and its corresponding growth rate are determined theoretically.     

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.     

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.