Simple Kinetic Equation Research Articles

Thermalization of mini-jets in a quark-gluon plasma

We complete the physical picture for the evolution of a high-energy jet propagating through a weakly-coupled quark-gluon plasma by investigating the thermalization of the soft components of the jet. We argue that the following scenario should hold: the leading particle emits a significant number of mini-jets which promptly evolve via quasi-democratic branchings and thus degrade into a myriad of soft gluons, with energies of the order of the medium temperature T. Via elastic collisions with the medium constituents, these soft gluons relax to local thermal equilibrium with the plasma over a time scale which is considerably shorter than the typical lifetime of the mini-jet. The thermalized gluons form a tail which lags behind the hard components of the jet. We support this scenario, first, via parametric arguments and, next, by studying a simplified kinetic equation, which describes the jet dynamics in longitudinal phase-space. We solve the kinetic equation using both (semi-)analytical and numerical methods. In particular, we obtain the first exact, analytic, solutions to the ultrarelativistic Fokker-Planck equation in one-dimensional phase-space. Our results confirm the physical picture aforementioned and demonstrate the quenching of the jet via multiple branching followed by the thermalization of the soft gluons in the cascades.

Projection of the Dynamics of Electron Transfer Reaction in Dual Space onto the One-Dimensional Slower Reaction Coordinate Axis.

We have derived here for the first time an exact dynamical equation within the domain of classical mechanics for the time dependent density distribution function of one-dimensional reaction coordinate (RC) in the condensed phase for electron transfer reaction by projecting the dynamics of slower modes in multidimensional Liouville space starting with a given set of coordinates of the faster modes. After the faster modes were ensemble averaged, the dynamics of the whole system solely depends on the slower RC. To simplify the complicated equation into a tractable form, benchmark approximations are employed to reduce the formally exact equation into an equation similar to the Smoluchowski equation with a delocalized sink term. As a test case, a Hamiltonian for the solute-solvent system modeled by quadratic functions for fast-relaxing vibrational and slow-relaxing polarization modes, respectively, has been considered. Interestingly, our simplified kinetic equation corresponding to this model Hamiltonian is transformed into the well-known phenomenological Sumi-Marcus equation.

Electron systems out of equilibrium: Nonequilibrium Green's function approach

This review deals with the state of the art and perspectives of description of nonequilibrium many-body systems using the nonequilibrium Green's function (NGF) method. The basic aim is to describe time evolution of the many-body system from its initial state over its transient dynamics to its long time asymptotic evolution. First, we discuss basic aims of transport theories to motivate the introduction of the NGF techniques. Second, this article summarizes the present view on construction of the electron transport equations formulated within the NGF approach to nonequilibrium. We discuss incorporation of complex initial conditions to the NGF formalism, and the NGF reconstruction theorem, which serves as a tool to derive simplified kinetic equations. Three stages of evolution of the nonequilibrium, the first described by the full NGF description, the second by a non-Markovian generalized master equation and the third by a Markovian master equation will be related to each other.

Grain growth kinetics of textured-BaTiO3 ceramics

Textured BaTiO3 (BT) ceramics were fabricated by templated grain growth method. Effects of sintering conditions on the grain growth process of textured-BT ceramics were investigated. Orientation degree increased initially and then decreased with increasing soaking time. The ceramics were composed of equiaxed matrix grains and brick-like template particles. The brick-like particles aligned parallel to the casting direction by observing from SEM images. A (h00)-preferred orientation was confirmed by SAED and XRD patterns. Mechanism of grain growth in textured-BT ceramics was studied. Both consumption of matrix by templates and grain growth of templates determined the orientation degree of ceramics. The kinetic mechanism for grain orientation was also discussed by the simplified phenomenological kinetic equation. The average activation energies were 364 kJ/mol for matrix grain and 918 kJ/mol for template particle, respectively. Finally, a dense ceramic with 85% grain orientation was obtained after sintering at 1400 °C for 2 h.

Collisional relaxation of two-dimensional self-gravitating systems

Systems with long range interactions present generically the formation of quasistationary long-lived nonequilibrium states. These states relax to Boltzmann equilibrium following a dynamics which is not well understood. In this paper we study this process in two-dimensional inhomogeneous self-gravitating systems. Using the Chandrasekhar-or local-approximation we write a simple approximate kinetic equation for the relaxation process, obtaining a Fokker-Planck equation for the velocity distribution with explicit analytical diffusion coefficients. Performing molecular dynamics simulations and comparing them with the evolution predicted by the Fokker-Planck equation, we observe a good agreement with the model for all the duration of the relaxation, from the formation of the quasistationary state to thermal equilibrium. We observe however an overestimate or underestimate of the relaxation rate of the particles with the slower or larger velocities, respectively. It is due to systematic errors in estimating the velocities of the particles at the moment of the collisions, inherent to the Chandrasekhar approximation when applied to inhomogeneous systems. Theory and simulations give a scaling of the relaxation time proportional to the number of particles in the system.

The point kinetic component of neutron noise in an MSR

The point kinetic approximation, and the calculation of the point kinetic component of the neutron noise in Molten Salt Reactors (MSR) is revisited. First, the derivation of the point kinetic equations in an MSR, found in the literature, is discussed. It is shown that to make the equations solvable, some simplifications need to be made whose validity is not justified. Then the point kinetic component of the noise is derived from the full space-frequency dependent solution in an analytical form by a projection to the static adjoint. The solution of the simplified point kinetic equations and the linearly exact point kinetic component obtained by the projection technique are compared quantitatively and it is shown that the solution of the simplified point kinetic equations cannot reconstruct some important features of the true solution.

Unsteady-state kinetic simulation of naphtha reforming and coke combustion processes in the fixed and moving catalyst beds

The work is dedicated to the construction of kinetics models for the naphtha reforming process and the adjacent process of catalyst regeneration by coke combustion. The proposed kinetic model for the reforming process is based on the use of common rate equations for the groups of similar reactions with account of difference in reaction rates for individual homologs within these groups by simple correlations with thermodynamic properties (first of all – with the values of Gibbs free energy) of individual reactions and by other simplification methods. Such approach gives the way to construct the kinetics models optimal from the point of view of compromise between accuracy and simplicity. The proposed naphtha reforming model is characterized with the high level of kinetic scheme detailization (62 individual and group reactants and 146 individual reactions), at the same it is rather simple and provides the accurate description of the experimental data using only 22 kinetic parameters. This model is thermodynamically consistent and provides accurate description of experimental data in a wide range of process parameters. Account of catalyst deactivation by coke deposition in the model gives the way to simulate transient reforming process performance both in fixed and moving catalyst beds. Kinetics of coke combustion for catalysts with moderate coke content (up to 3% mass) may described by simple kinetic equation with apparent reaction rate orders closed to unit for relative coke content and to 1/2 for oxygen. Demonstration simulations of naphtha reforming and coke combustion processes are presented.

Synthesis of magnetic dual-mesoporous carbon spheres and lysozyme release behavior

A simple and novel route for synthesizing magnetic dual-mesoporous carbon spheres was developed. The synthesis of porous silica microspheres and impregnation were conducted in one pot, rendering the presynthesis of silica template and additional impregnation steps unnecessary. The effects of SiO2 loading amount on the morphologies, structural parameters, and magnetic properties of the materials were evaluated by X-ray diffraction, N2 sorption analysis, Raman spectroscopy, scanning electron microscopy, transmission electron microscopy, and physical property measurements. Experimental results show that the specific surface areas and total pore volumes of the carbon spheres increased with SiO2 loading amount. As an antibacterial enzyme, lysozyme was selected as the release protein model. The release behavior of lysozyme was well correlated with time given the use of a simple pseudo second-order kinetic equation. The release rate and maximum release fraction are higher at pH 7 than at pH 11. The maximum release fraction increased with ionic strength.

SCR catalyst coated on low-cost monolith support for flue gas denitration of industrial furnaces

NH3–SCR honeycomb catalyst was prepared by coating powdery active metal components on the surface of blank monolith supports and tested at the temperatures of 250–350°C, and superficial gas velocities of 5.3–6.3m/s. The tests were performed with a simulated gas mixture containing NO, SO2, O2, H2O and N2 or a real flue gas produced on line. The results showed that the coated honeycomb catalyst has an extraordinary high activity for low-temperature deNOx in the presence of 5vol.% steam and 1000ppm SO2. Its activity increased with increasing the coating thickness and reached at 110μm thickness as comparable as that of the 100% active components-molded catalyst, suggesting the deNOx reaction proceeded mainly in the surface layers of the honeycomb cell wall and hence coated honeycomb catalyst should be more cost-efficient than 100% active components-molded one. In a real coal-burned flue gas stream the catalyst maintained essentially its low temperature activity and stability, further verifying its high industrial applicability. Then, the effects of NO concentration and NH3/NO ratio on NO conversion were investigated to seek a suitable kinetic model for prediction of the catalyst’s deNOx rate and reactor design for pilot plant research. The data obtained showed that a simple power-law kinetic equation with first order in NO and zero order in ammonia can be applied to the coated honeycomb catalyst, and was used for determination of the kinetic parameters. Finally, simulation calculation was performed under arbitrary conditions to provide valuable information for design of pilot-plant scale reactors.

Photocatalytic CO2 reduction with H2O vapors using montmorillonite/TiO2 supported microchannel monolith photoreactor

In this study, the performance of a montmorillonite (MMT)/TiO2 coated monolith photoreactor was tested for the photocatalytic CO2 reduction. CH4 and CO were the main products having yield rates of 139 and 52μmolegcatal.−1h−1, respectively. The other adequately significant products were C2H4, C2H6, C3H6 and C3H8. The catalytic reactor performance for CH4 production was in the order of MMT–TiO2–monolith (139μmolegcatal.−1h−1)>TiO2–monolith (82μmolegcatal.−1h−1)>MMT–TiO2–cell (43μmolegcatal.−1h−1)>TiO2–cell (7.7μmolegcatal.−1h−1). The higher yield rates in monolith photoreactor were due to the higher illuminated surface area and efficient light utilization. In addition, the profound hydrocarbon yield rates over MMT/TiO2 nanocatalyst supported microchannels were due to the efficient production and utilization of charges. The reaction rate and the adsorption–desorption phenomenon was postulated according to the Langmuir–Hinshelwood (L–H) model. A simple kinetic equation, derived to model the coupled effect of adsorptive photocatalytic reduction and oxidation, fitted-well with the experimental data.

Sulfite oxidation in seawater flue gas desulfurization by plate falling film corona-streamer discharge

In this paper, sulfite containing effluent from the packed tower of seawater flue gas desulfurization is treated by plate falling film corona-streamer discharge which can solve the scale-up problem of electrical discharge reactors. Effects of pH, electrical discharge intensity, effluent flow rate and temperature on sulfite oxidation are considered. Experimental results indicate that discharge power density is the dominating factor which affects the sulfite oxidation rate by affecting the gaseous radical concentration. Based on experimental results, sulfite oxidation by corona-streamer discharge is proposed as the combination of chain reaction path and radical self-reaction path. In addition, the simplified kinetic equation is expressed in terms of sulfite concentration and discharge power density. Comparing with cylindrical falling film corona-streamer discharge, the plate falling film corona-streamer discharge leads to a fold increase in energy efficiency, besides, scale-up of a plate falling film corona-streamer discharge reactor will affect the energy efficiency insignificantly, all these indicate that the plate falling film discharge has a better potential in large scale utilization than cylindrical falling film discharge.

Application of Fuzzy Fractional Kinetic Equations to Modelling of the Acid Hydrolysis Reaction

In view of the usefulness and a great importance of the kinetic equation in specific chemical engineering problems, we discuss the numerical solution of a simple fuzzy fractional kinetic equation applied for the hemicelluloses hydrolysis reaction. The fuzzy approximate solution is derived based on the Legendre polynomials to the fuzzy fractional equation calculus. Moreover, the complete error analysis is explained based on the application of fuzzy Caputo fractional derivative. The main advantage of the present method is its superior accuracy which is obtained by using a limited number of Legendre polynomials. The method is computationally interesting, and the numerical results demonstrate the effectiveness and validity of the method for solving fuzzy fractional differential equations.

A study on kinetics of NO absorption from flue gas by using UV/Fenton wet scrubbing

The mass transfer-reaction kinetics of NO removal from flue gas by using UV/Fenton wet scrubbing were investigated, and a simple kinetic equation was also established. The results indicate that the NO absorption rates increase with the increase of H2O2 concentration, Fe2+ concentration (below 0.005mol/L), NO concentration and UV radiation intensity, but decrease with the further increase of Fe2+ concentration (above 0.005mol/L). The absorption process of NO by using UV/Fenton wet scrubbing is found to be a pseudo-first-order fast reaction. The NO absorption rate can be effectively increased by raising the chemical reaction rate, the gas–liquid contact area and the NO concentration. The validation results of kinetic equation indicate that the calculated values are in good agreement with the experimental values.

Limiting kinetic equation for describing a first-order chemical reaction with spatially nonuniform diffusion and kinetic coefficients

A spatially nonuniform system in which a first-order chemical reaction occurs is considered. Both the diffusion coefficient and the kinetic coefficient of the reaction are assumed to be dependent on spatial coordinates. An asymptotically valid method is proposed for reducing a diffusion equation (under the described conditions) in the presence of a source (a reaction) to a simple kinetic equation containing a constant kinetic coefficient. A functional is derived that makes it possible to use variational methods for determining the kinetic coefficient.

Kinetics of spin relaxation in quantum wires and channels: Boundary spin echo and formation of a persistent spin helix

In this paper we use a spin kinetic equation to study spin polarization dynamics in 1D wires and 2D channels. This approach is valid in both diffusive and ballistic spin transport regimes and, therefore, more general than the usual spin drift-diffusion equations. In particular, we demonstrate that in infinite 1D wires with Rashba spin-orbit interaction the exponential spin relaxation decay can be modulated by an oscillating function. In the case of spin relaxation in finite length 1D wires, it is shown that an initially homogeneous spin polarization spontaneously transforms into a persistent spin helix. An interesting sound waves echo-like behavior of initially localized spin polarization packet is found in finite length wires. We show that a propagating spin polarization profile reflects from a system boundary and returns back to its initial position similarly to the reflectance of sound waves from an obstacle. Green's function of spin kinetic equation is found for both finite and infinite 1D systems. Moreover, we demonstrate explicitly that the spin relaxation in 2D channels with Rashba and Dresselhaus spin-orbit interactions of equal strength occurs similarly to that in 1D wires of finite length. Finally, a simple transformation mapping 1D spin kinetic equation into the Klein-Gordon equation with an imaginary mass is found thus establishing an interesting connection between semiconductor spintronics and relativistic quantum mechanics.

Hamiltonian formulation of the effective kinetic theory for superfluid Fermi liquids

We present in a local form the time dependent effective description of a superfluid Fermi liquid which includes Landau damping effects at T≠0. This is achieved by the introduction of an additional variable, the quasiparticle distribution function, which obeys a simple kinetic equation. The transport equation is coupled with first order equations for the Goldstone mode and the particle density. We prove that a main feature of this formulation is its Hamiltonian structure relative to a certain Poisson bracket. We construct the Hamiltonian to quadratic order.

Modeling of low-speed rarefied gas flows using a combined ES-BGK/DSMC approach

Modeling of low-speed gas flows at moderate Knudsen numbers, where conventional continuum CFD methods are inapplicable and kinetic methods are either inaccurate of prohibitively expensive, presents a significant challenge for computational modeling. In this work, an approach that combines a finite-volume solution of the ES-BGK model kinetic equation and the statistical DSMC method to accurately predict radiometric forces on a vane in large vacuum chamber filled with rarefied gas is presented. The approach in effect combines the accuracy of a statistical solution of the Boltzmann equation with the numerical efficiency of a deterministic solution of simplified model kinetic equations.

Towards physically realizable and hyperbolic moment closures for kinetic theory

The numerical prediction of continuum and non-equilibrium flows by using fully hyperbolic and realizable mathematical descriptions that follow from moment closures of gas kinetic theory is reviewed. A brief review is first given of some of the current capabilities and limitations of moment closures for predicting a range of continuum and non-equilibrium flow. Next, an extended but hyperbolic Gaussian closure for diatomic gases that does not account for heat-transfer effects, as well as a regularized version of this closure that incorporates anisotropic thermal-diffusion effects via the inclusion of higher order terms having an elliptic nature, are both reviewed and applied to a number of canonical flow problems. The numerical results for the Gaussian closures clearly demonstrate the capabilities and potential of moment closures and purely hyperbolic treatments. Following these reviews, a somewhat novel hierarchy of physically realizable and hyperbolic moment closures is considered and described. This alternative hierarchy is based on modifications to the more common maximum-entropy hierarchies, so as to ensure the validity and integrability of the approximate distribution function for all values of the velocity moments that are physically realizable. The predictive capabilities of this new closure hierarchy are then explored by considering numerical solutions of the closures for a one-dimensional kinetic equation with a relaxation-time collision operator and by comparing the closure solutions to discrete numerical solutions of this simplified kinetic equation. The study concludes with a brief summary of the findings and a discussion of the potential of the physically realizable and hyperbolic moment closures for application to fully three-dimensional physics.

A study on thermal combustion of lean methane–air mixtures: Simplified reaction mechanism and kinetic equations

Homogeneous combustion of lean methane–air mixtures occurs in both thermal and catalytic reverse-flow reactors. The modeling and design of such reactor requires a reliable and relatively simple kinetic model of the combustion. Although a number of detailed descriptions of the kinetics of homogeneous combustion are available in the literature (including those which take into account complex free-radical reactions), their practical usefulness is doubtful. In the present study a simple one- or two-stage model is proposed which directly describes global kinetics and neglects intermediates and radicals. Special attention is paid to the mechanism of the possible formation of carbon monoxide. The studies were done both for the combustion in the free space and for the process taking place over a monolith packing.

Kinetics of devolatilization in the thermal destruction of lignite particles

The first step is kinetic study of the drying and pyrolysis of lignite particles in the range 700–900 ° C, which is characteristic of coal gasification in an autothermal reactor with an inverse thermal wave. Analysis of the experimental data yields a linear kinetic equation for the mass loss of the coal particles in thermal destruction. The simplified kinetic equation is convenient for technological calculations and also for mathematical simulation of the heat and mass transfer in coal gasification.