Kinetic Formulation Research Articles

A recursive approach for numerically identified DO-BOD interaction and kinetic formulation for water quality probability

In the late 1960s, the mounting public pressure in the developing countries for control of pollution stimulated investment in a host of special studies to find the best alternative ways of protecting the aquatic environment. One of the most important parameters for knowing the water quality is Dissolved Oxygen (DO) because various other parameters are interrelated with it and changes accordingly with DO. Therefore, in the present study, an attempt has been made to summarise the dissolved oxygen-biochemical oxygen demand interaction models to manage the quality of natural water bodies that are subject to pollutant inputs and one must be able to predict the degradation in quality that results from those inputs.

Experimental organic matter maturation at 2kbar: Heat-up effect to low temperatures on vitrinite reflectance

An experimental study was performed to evaluate the effect of heat-up to various low temperatures on vitrinite reflectance (VR) at 2 kbar employing the same previously used heat-up procedures, starting material and apparatus. Heat-up is the isobaric experimental procedure consisting of the increase in temperature of the laboratory vitrinite maturation from room temperature to the desired run temperature Tend of heat-up at which the isobaric-isothermal maturation starts. Experiments employed cold-seal pressure vessels with classical heat-up devices and were carried out on wet (water added) xylite of swamp cypress. Confined system maturation experiments were conducted at 2 kbar and involved temperatures Tend of heat-up of 200, 300 and 400 °C. Heat-up times were ~ 25 min at 200 °C, ~ 35 min at 300 °C and ~ 40 min at 400 °C. Results of this study demonstrate that heat-up has a strong influence on VR at 2 kbar for Tend of heat-up above ~ 250 °C: VR greatly increases with Tend of heat-up during the short heat-up event. This effect of heat-up points out the rapid kinetics of the initial VR increase. At 2 kbar, experimental heat-up to Tend of heat-up < ~ 250 °C does not influence VR. This lack of maturation is regarded as the result of an activation time delaying vitrinite maturation at these very low temperatures. These results are a step toward future isothermal experimental vitrinite maturations and a better kinetic formulation of VR evolution with temperature at 2 kbar. In this respect, the effect of heat-up and the activation time call for performing future series of isothermal laboratory vitrinite maturations at 2 kbar and various temperatures using an extremely immature Type III organic starting material (VR ~ 0.2%) and involving short heating times (a few minutes to a few hours). This aims to investigate the largest possible VR range, to detect the beginning of the isobaric-isothermal vitrinite maturation and to accurately constrain the rapid isobaric-isothermal kinetic evolution of the initial VR increase. Evidently, the activation time and VR gained during heat-up must be considered in the formulation of VR rate equation. Possible directions on how to surmount the effect of heat-up on VR in the kinetic analysis of the isothermal VR evolution are presented. Numerical simulations indicate that the effect of heat-up on the kinetic evolution of the isothermal VR increase diminishes with increasing heating time up to be neutralized for long heating durations. Obviously, VR previously obtained from isothermal experiments performed for long heating durations using classical heat-up procedures can be associated for kinetic studies with further VR resulting from isothermal runs conducted for short heating times in cold-seal pressure vessels equipped with rapid heat-up devices. This avoids the influence of heat-up on VR and thus the kinetic analysis of VR evolution is greatly simplified. Rapid heat-up devices are therefore of interest to carry out future laboratory organic matter maturations for short heating times.

Formulation of Reaction Kinetics for Cyclohexanone Ammoximation Catalyzed by a Clay-Based Titanium Silicalite-1 Composite in a Semibatch Process

This article focuses on the kinetic modeling of cyclohexanone ammoximation catalyzed by the clay-based titanium silicalite-1 (TS-1) composite. With the use of computational tools, modeling was carried out systematically, and a rational interpretation of the reaction mechanism was successfully achieved. By fitting the experimental and calculated data, it was found that cyclohexanone ammoximation over the clay-based TS-1 composite follows the Eley–Rideal mechanism in which the reaction takes place between adsorbed H2O2 and other reactants in the free state. It is concluded that a full understanding of the mechanism and kinetics of the reaction allows one to engineer the overall cyclohexanone ammoximation process by influencing elementary steps. The pre-exponential factor and activation energy of the reaction were also determined as a reference for industrial reactor design.

Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle

We study the comparison principle for anisotropic degenerate parabolic–hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and super-solution, which immediately deduces the L 1 contractivity and therefore, uniqueness of entropy solutions. The method used here is based upon the kinetic formulation and the kinetic techniques developed by Lions, Perthame and Tadmor. By adapting and modifying those methods to the case of Dirichlet boundary problems for degenerate parabolic equations we can establish a comparison property. Moreover, in the quasi-isotropic case the existence of entropy solutions is proved.

A statistical physics perspective on the evolution of ion distributions across low Mach number quasi-perpendicular collisionless shocks

[1] The heating of directly transmitted ions at low Mach number, perpendicular and quasi-perpendicular shocks has been the subject of previous statistical physics studies. In this paper, we use a Hamiltonian formulation of the ion kinetics for a quasi-perpendicular shock model to derive the general solution to Liouville's equation as a function of six invariants, finding forms of these invariants in terms of the upstream parameters. The ion distribution is expressed as a function of one of these invariants, subject to a Maxwellian upstream boundary condition, and the evolution of the distribution through and downstream of the shock is studied. The momentum-space volume within surfaces of constant probability (related to the temperature) is shown to be inversely proportional to an average value of the canonical momentum associated with motion in the direction normal to the shock plane, generalizing a previous result to three-dimensional phase space. We also study the evolution of the distribution properties numerically, in particular noting that the “twisting” of these surfaces in phase space is the result of the unequal guiding center motion of different parts of the distribution (which is not the case for a fully perpendicular shock). This property provides insight into the damping of oscillations in the mean momentum and the temperature for a quasi-perpendicular model (as the distribution is spread about the central value through gyration) and the observed T⊥ > T∥ anisotropy.

Monolithically Patterned Wide–Narrow–Wide All-Graphene Devices

We investigate theoretically the performance advantages of all-graphene nanoribbon field-effect transistors (GNRFETs) whose channel and source/drain (contact) regions are patterned monolithically from a 2-D single sheet of graphene. In our simulated devices, the source/drain and interconnect regions are composed of wide GNR sections that are semimetallic, while the channel regions consist of narrow GNR sections that open semiconducting bandgaps. Our simulation employs a fully atomistic model of the device, contact, and interfacial regions using tight-binding theory. The electronic structures are coupled with a self-consistent 3-D Poisson's equation to capture the nontrivial contact electrostatics, along with a quantum kinetic formulation of transport based on nonequilibrium Green's functions. Although we only consider a specific device geometry, our results establish several general performance advantages of such monolithic devices (besides those related to fabrication and patterning), namely, the improved electrostatics, suppressed short-channel effects, and Ohmic contacts at the narrow-to-wide interfaces.

Kinetic formulation of irreversible evolution of the two-nucleus spin systems

The time irreversible evolution of the two-nucleus spin systems interacting with a magnetic field is analyzed in the framework of the subdynamics theory. The spin systems are determined by the H(1) and C(13) nuclei. The investigation is based on the complex spectral representation of the Liouvillian which gives a rigorous description of irreversibility. The evolution of the density matrix is analyzed for the short and long time regions. Quantum Zeno effect is discussed.

Nonlinear hyperbolic-elliptic systems in the bounded domain

In the article we study a hyperbolic-elliptic system ofPDE. The system can describe two different physical phenomena: 1st one isthe motion of magnetic vortices in the II-type superconductor and 2nd oneis the collective motion of cells. Motivated by real physics, we considerthis system with boundary conditions, describing the flux of vortices (andcells, respectively) through the boundary of the domain. We prove the globalsolvability of this problem. To show the solvability result we use a'viscous' parabolic-elliptic system. Since the viscous solutions do not havea compactness property, we justify the limit transition on a vanishingviscosity, using a kinetic formulation of our problem. As the final resultof all considerations we have solved a very important question related witha so-called 'boundary layer problem', showing the strong convergence of theviscous solutions to the solution of our hyperbolic-elliptic system.

Biogeochemical processes in a clay formation in situ experiment: Part F – Reactive transport modelling

Abstract Reactive transport modelling was used to simulate solute transport, thermodynamic reactions, ion exchange and biodegradation in the Porewater Chemistry (PC) experiment at the Mont Terri Rock Laboratory. Simulations show that the most important chemical processes controlling the fluid composition within the borehole and the surrounding formation during the experiment are ion exchange, biodegradation and dissolution/precipitation reactions involving pyrite and carbonate minerals. In contrast, thermodynamic mineral dissolution/precipitation reactions involving alumo-silicate minerals have little impact on the fluid composition on the time-scale of the experiment. With the accurate description of the initial chemical condition in the formation in combination with kinetic formulations describing the different stages of bacterial activities, it has been possible to reproduce the evolution of important system parameters, such as the pH, redox potential, total organic C, dissolved inorganic C and SO4 concentration. Leaching of glycerol from the pH-electrode may be the primary source of organic material that initiated bacterial growth, which caused the chemical perturbation in the borehole. Results from these simulations are consistent with data from the over-coring and demonstrate that the Opalinus Clay has a high buffering capacity in terms of chemical perturbations caused by bacterial activity. This buffering capacity can be attributed to the carbonate system as well as to the reactivity of clay surfaces.

Modules based on the geochemical model PHREEQC for use in scripting and programming languages

The geochemical model PHREEQC is capable of simulating a wide range of equilibrium reactions between water and minerals, ion exchangers, surface complexes, solid solutions, and gases. It also has a general kinetic formulation that allows modeling of nonequilibrium mineral dissolution and precipitation, microbial reactions, decomposition of organic compounds, and other kinetic reactions. To facilitate use of these reaction capabilities in scripting languages and other models, PHREEQC has been implemented in modules that easily interface with other software. A Microsoft COM (component object model) has been implemented, which allows PHREEQC to be used by any software that can interface with a COM server—for example, Excel ®, Visual Basic ®, Python, or MATLAB ®. PHREEQC has been converted to a C++ class, which can be included in programs written in C++. The class also has been compiled in libraries for Linux and Windows that allow PHREEQC to be called from C++, C, and Fortran. A limited set of methods implements the full reaction capabilities of PHREEQC for each module. Input methods use strings or files to define reaction calculations in exactly the same formats used by PHREEQC. Output methods provide a table of user-selected model results, such as concentrations, activities, saturation indices, and densities. The PHREEQC module can add geochemical reaction capabilities to surface-water, groundwater, and watershed transport models. It is possible to store and manipulate solution compositions and reaction information for many cells within the module. In addition, the object-oriented nature of the PHREEQC modules simplifies implementation of parallel processing for reactive-transport models. The PHREEQC COM module may be used in scripting languages to fit parameters; to plot PHREEQC results for field, laboratory, or theoretical investigations; or to develop new models that include simple or complex geochemical calculations.

Kinetic theory of aggregation in granular flow

AbstractThis article presents a mathematical formulation of the aggregation kinetics in granular flow. The traditional kinetic theory and its generalized application to granular flow does not allow for particle size to change with time thus cannot be used to describe particle flow with aggregation taking place. In this article, a collision success factor, quantifying the completely inelastic collision of particles, is introduced into the evaluation of collision rate. The kinetic transport equations are then transformed to include source terms that account for the effects of particle size and aggregation. The analytical solution of the collision success factor is obtained by integrating the relative velocity distribution function over its velocity domain from 0 to a critical value, which corresponds a balance between the repulsion and attraction forces in a collision. The factor has been found to depend on the mixture granular energy and the critical relative collision energy. © 2011 American Institute of Chemical Engineers AIChE J, 2011

Porphyroblast crystallization kinetics: the role of the nutrient production rate

The mechanisms that govern porphyroblast crystallization are investigated by comparing quantitative textural data with predictions from different crystallization models. Such numerical models use kinetic formulations of the main crystallization mechanism to predict textural characteristics, such as grain size distributions. In turn, data on porphyroblast textures for natural samples are used to infer which mechanism dominated during their formation. Whereas previous models assume that the rate-limiting step for a porphyroblast producing reaction is either transport or growth, the model advanced in this study considers the production of nutrients for porphyroblasts as a potentially rate-limiting factor. This production reflects the breakdown of (metastable) reactants, which at a specific pressure (P) and temperature (T) depends on the bulk composition of the sample. The production of nutrients that potentially contribute to the formation of porphyroblasts is computed based on thermodynamic models. The conceptual model assumes that these nutrients feed into some intergranular medium, and products form by nutrient consumption from that medium, with rates depending on reaction affinity. For any sequence of P-T conditions along a P-T-t path, the numerical model first computes an effective supersaturation (reff) of the product phase(s), then an effective nucleation rate (J), and finally the amount of (porphyroblast) growth. As a result, the model is useful in investigating how the textural characteristics of a sample (of given bulk composition) depend on the P-T-t path followed during porphyroblast crystallization. The numerical model is tested and validated by comparing simulation results with quantitative textural data for garnet porphyroblasts measured in samples from the Swiss Central Alps.

Experimental kinetic study of organic matter maturation: Time and pressure effects on vitrinite reflectance at 400 °C

We carried out a laboratory rate study to elucidate and quantify the effects of time and pressure on vitrinite reflectance ( VR). A series of confined system maturation experiments was conducted at 400 °C and at pressures of 2, 10 and 20 kbar. Experiments were performed on dry (no water added) xylite of swamp cypress and involved run lengths from 0 s to 25 days. At 400 °C, our experimental results demonstrate pressure and heating time to be important variables that promote VR increase and therefore the maturation of Type III organic material. VR increases with time at each investigated pressure. Despite rapid initial kinetics, the increase in VR decelerates with time at each pressure. When VR < 1.44%, increasing pressure reduces the rate of VR increase and hence retards the initial VR enhancement with time. The retarding effect of pressure on VR increase diminishes with enhancing VR. The retardation of VR increase is insignificant for geological maturation at 400 °C because a VR of 1.44% is attained in only a few hours. When VR > 1.44%, increasing pressure counteracts the deceleration of VR increase with time and thus greatly enhances the increase in VR with time. Evidently, vitrinite maturation takes place rapidly in a dry confined system and does not require addition of water to occur. The strong effect of the experimental heat-up on VR is obvious even for very short experiments and must be corrected in kinetic analysis. The evolution of VR with heating time ( t) and pressure ( P) at 400 °C from an initial VR of 0% is well described by our new power law rate equation VR ( P , 400 ° C, t ) = ( k ( P , 400 ° C ) t ) n ( P , 400 ° C ) , where the exponent n( P, 400 °C) and the rate constant k( P, 400 °C) increase with pressure. We regard this kinetic formulation as a step toward a general equation describing VR evolution as a function of time, pressure and temperature for Type III organic matter. The potential of the power law formalism to model VR from any starting VR and for complex metamorphic and heating time histories is shown by making explicit directions on how to use such a kinetic equation.

Approximate Moment Dynamics for Chemically Reacting Systems

In the stochastic formulation of chemical kinetics, the differential equation that describes the time evolution of the lower-order statistical moments for the number of molecules of the different species involved, is generally not closed, in the sense that the right-hand side of this equation depends on higher-order moments. Recent work has proposed a moment closure technique based on derivative-matching, which closes the moment equations by approximating higher-order moments as nonlinear functions of lower-order moments. We here provide a mathematical proof of this moment closure technique, and highlight its performance through comparisons with alternative methods. These comparisons reveal that this moment closure technique based on derivative-matching provides more accurate estimates of the moment dynamics, especially when the population size is small. Finally, we show that the accuracy of the proposed moment closure scheme can be arbitrarily increased by incurring additional computational effort.

A LQ-based kinetic model formulation for exploring dynamics of treatment response of tumours in patients

A kinetic bio-mathematical, linear-quadratic (LQ) based model description for clonogenic survival is presented. In contrast to widely used formulations of models, a dynamic approach based on ordinary differential equations for coupling a repair model with a tumour growth model is used to allow analysis of intercellular process dynamics and submodel interference. The purpose of the model formulation is to find a quantitative framework for investigation of tumour response to radiotherapy in vivo. It is not the intention of the proposed model formulation to give a mechanistic explanation for cellular repair processes. This article addresses bio-mathematical aspects of the simplistic kinetic approach used for description of repair. The model formulation includes processes for cellular death, repopulation and cellular repair. The explicit use of the population size in the model facilitates the coupling of the sub-models including aspects of tissue dynamics (competition, oxygenation). The cellular repair is summarized by using a kinetic model for a dose equivalent Γ describing production and elimination of sublethal lesions. This dose equivalent replaces the absorbed dose used in the common LQ- model. Therefore, this approach is called the Γ- LQ- formulation. A comparison with two kinetic radiobiological models (the LPL model of Curtis and the compartmental model of Carlone) is carried out. The resulting differential equations are solved by numerical integration using a Runge-Kutta algorithm. The comparison reveals a good agreement between the Γ- LQ- formulation and the models of Curtis and Carlone under certain, defined conditions: The proposed formulation leads to results which are identical to the model of Carlone over a wide range of investigated biological parameters and different fractionation schemes when using first order repair kinetics. The comparison with experimental data and the LPL- model of Curtis shows a good agreement of the Γ- LQ- formulation using second order repair kinetics over a wide range of dose rate. Over a limited range, the use of second order repair in the Γ- LQ- formulation approximates the same dose rate dependency of clonogenic survival using only one additional parameter to those of the common LQ model. Within the investigated range of parameters, the presented Γ-LQ- formulation may be used to describe the in-vivo tumour response to radiation. The influence of repopulation, oxygenation and other aspects of tissue dynamics may override the differences between the intrinsic radiosensitivity yielded by each of the models. The proposed model formulation can be extended with additional static and dynamic tissue behaviours. This may be useful for the understanding of the reaction of tissues to heat (hyperthermia) or combined anti-cancer treatments (chemo-radiotherapy).

Formula omitted]-solvability of a full superconductive model

In this article the mean-field vortex model arising from the II-type superconductivity is investigated. The vortex model is reduced to a nonlinear hyperbolic–elliptic system of PDEs in a bounded domain. Motivated by experiments, we consider physical boundary conditions, which describe a flux of superconducting vortices through the boundary of the domain. We prove the global solvability for the system. To show the solvability result we take a vanishing “viscosity” limit in an approximated parabolic–elliptic system. Since the approximated solutions do not have a compactness property, we justify this limit transition, using a kinetic formulation of our problem. The main trick is that instead of the nonlinear system, we have to investigate a linear transport equation.

Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system

This paper deals with the derivation and analysis of the the Hall Magneto-Hydrodynamic equations. We first provide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits. We also propose a kinetic formulation for the Hall-MHD equations which contains as fluid closure different variants of the Hall-MHD model. Then, we prove the existence of global weak solutions for the incompressible viscous resistive Hall-MHD model. We use the particular structure of the Hall term which has zero contribution to the energy identity. Finally, we discuss particular solutions in the form of axisymmetric purely swirling magnetic fields and propose some regularization of the Hall equation.

On the Dirichlet problem for first order quasilinear equations on a manifold

We study the Dirichlet problem for a first order quasilinear equation on a smooth manifold with boundary. The existence and uniqueness of a generalized entropy solution are established. The uniqueness is proved under some additional requirement on the field of coefficients. It is shown that generally the uniqueness fails. The nonuniqueness occurs because of the presence of the characteristics not outgoing from the boundary (including closed ones). The existence is proved in a general case. Moreover, we establish that among generalized entropy solutions laying in the ball‖u‖∞≤R\|u\|_\infty \le Rthere exist unique maximal and minimal solutions. To prove our results, we use the kinetic formulation similar to the one by C. Imbert and J. Vovelle.

Quasilinear theory revisited: general kinetic formulation of wave–particle interactions in plasmas

In laboratory fusion devices radio frequency electromagnetic waves are routinely used for heating plasmas and for controlling current profiles. The evolution of particle distribution function in the presence of electromagnetic waves is derived from fundamental equations using the action-angle variables of the dynamical Hamiltonian. Unlike conventional quasilinear theories (QLTs), the distribution function is evolved concurrently with the particle motion. Since the particle dynamics is time reversal invariant, the master equation for the evolution of the distribution function is also time reversal invariant. A sequential averaging of the master equation over the angles leads to a hierarchy of diffusion equations. The diffusion operator in the equation obtained after averaging over all angles is time dependent, in direct contrast to time independent diffusion operator in QLTs. The evolution of the distribution function with time-dependent diffusion operator is markedly different from quasilinear evolution and is illustrated for current drive by a spectrum of coherent electrostatic waves. A proper description of wave–particle interactions is important for fusion plasmas since the velocity space gradients of the distribution function decisively affect collisional relaxation and the associated transport processes.

A generalized Zel’dovich-Neumann-Döring model of steady gaseous detonation

A model of N-component gas mixture detonation whose combustion zone is formed by n (1 ≤ n ≤ N − 1) independent reactions satisfying the kinetic formulation of the mass action law is studied. In the framework of this model, it is found that there exist two possible types of self-sustaining detonation waves with asymptotically stable combustion zones. It is shown that it is possible to observe only a strong wave propagating in the Chapman-Jouguet regime relative to the flow on the critical surface. It is also shown that, for a perfect gas mixture, this wave appears only when a combustion zone is formed by one or two reactions.