Kinetic Differential Equations Research Articles

Rigorously proven chaos in chemical kinetics.

This study addresses a longstanding question regarding the mathematical proof of chaotic behavior in kinetic differential equations. Following the numerous numerical and experimental results in the past 50 years, we introduce two formal chemical reactions that rigorously demonstrate this behavior. Our approach involves transforming chaotic equations into kinetic differential equations and then realizing them through formal chemical reactions. The findings present a novel perspective on chaotic dynamics within chemical kinetics, thereby resolving a longstanding open problem.

NEW METHODOLOGIES FOR ASSESSING CAVITATION LOAD AND RESISTANCE OF STRUCTURAL MATERIALS AND PROTECTIVE COATINGS IN NUCLEAR POWER INDUSTRY. PART II. POLYFRACTIONAL APPROACH

A fractional approach to cavitation erosion process as proposed previously by one of the authors is explained. The polyfractional erosion process is considered as a superposition of a series of monofractional ones and described by a differential kinetic equation with a set of parameters constituting “the erosion resistance matrix”. The funda-mental methodological considerations are supplemented by describing the most typical applications including pre-diction of erosion of structural materials under given cavitation load, derivation of cavitation erosion fatigue charac-teristics, retrieval of effective erosive load basing on performance of reference materials. Finally, extending the methodology onto thin and thick protective coatings is proposed.

Relaxation and asymptotic expansion of controlled stiff differential equations

The control of relaxation-type systems of ordinary differential equations is investigated using the Hamilton–Jacobi–Bellman equation. Firstly, we recast the model as a singularly perturbed dynamics which we embed in a family of controlled systems. Then we study this dynamics together with the value function of the associated optimal control problem. We provide an asymptotic expansion in the relaxation parameter of the value function. We also show that its solution converges toward the solution of a Hamilton–Jacobi–Bellman equation for a reduced control problem. Such systems are motivated by semi-discretisation of kinetic and hyperbolic partial differential equations. Several examples are presented including Jin–Xin relaxation.

A Precision Benchmark Suite for Nuclear Reactor Point Kinetics Equations via Converged Accelerated Taylor Series (CATS)

Extreme benchmarks of 10 or more places for the point kinetics equations for time-dependent nuclear reactor power transients are rare. Therefore, to establish an extreme benchmark, we employ a Taylor series (TS) with continuous analytical continuation to solve the ordinary differential equations of point kinetics including feedback. Nonlinear Wynn-epsilon convergence acceleration confirms the highly precise solutions for neutron and precursor densities. Through adaptive partitioning of time intervals, the proposed Converged Accelerated Taylor Series, or CATS algorithm in double precision, automatically performs successive mesh refinement to obtain high-precision initial conditions for each subinterval, with the intent to reduce propagation error. Confirmation of 10 to 12 places comes from comparison to the BEFD (Backward Euler Finite Difference) algorithm in quadruple precision also developed by the author. We report benchmark results for common cases found in the literature including step, ramp, zigzag, and sinusoidal prescribed reactivity insertions and insertions with nonlinear adiabatic Doppler feedback. We also establish a suite of new prescribed reactivity insertions and insertions with feedback, based on reactivities with Taylor series representations as suggested by the CATS algorithm.

Optimal Temperature Calculation for Multicriteria Optimization of the Hydrogenation of Polycyclic Aromatic Hydrocarbons by NSGA-II Method

Introduction. Multicriteria optimization, taking into account contradicting criteria, is used to improve production efficiency, reduce costs, improve product quality and environmental safety of processes. The literature describes the application of multicriteria optimization for production purposes, including the selection of reaction conditions and improvement of technological processes. In the presented paper, the object of research is the process of hydrogenation of polycyclic aromatic hydrocarbons (PAH) in the production of high-density fuels. To determine the optimal conditions of the process, the problem of multicriteria optimization based on the kinetic model is solved. The criteria include maximizing the yield of targeted naphthenes and conversion of feedstock. The research objective is to create a program implementing the multicriteria optimization non-dominated sorting genetic algorithm-II (NSGA-II). Due to this, it is possible to calculate the optimal temperature for the PAH hydrogenation process on the basis of the kinetic model.Materials and Methods. The NSGA-II genetic algorithm was used to solve the multicriteria optimization problem. Modified parental and survival selection within the Pareto front was also used. If it was necessary to divide the front, solutions based on the Manhattan distance between them were selected. The program was implemented in Python.Results. In the system of ordinary nonlinear differential equations of chemical kinetics, the concentration was designated yi, the conditional contact time of the reaction mixture with the catalyst — τ. The system was solved for the hydrogenation reaction of polycyclic aromatic hydrocarbons. The calculations showed that at τ = 0 y1(0) = 0.025; y2(0) = 0.9; y6(0) = 0.067; y9(0) = 0.008; yi(0) = 0, i = 3–5, 7, 8, 10–20; Q(0) = 1. The process temperature was considered as a control parameter according to two optimality criteria: maximizing the yield of target naphthenes (f1) at the end of the reaction, and maximizing the conversion of feedstock (f2). Values f1 were in the range of 0.43–0.79; conversion — 0.01–0.03; temperature — 200–300 K. The growth of temperature was accompanied by an increase in the yield of target naphthenes and a decrease in the conversion of feedstock. Each solution obtained was not an unimprovable one. When modeling the process of hydrogenation of PAH, an algorithm was launched with a population size of 100 and a number of generations of 100. A program implementing the NSGA-II algorithm was developed. The optimal set of values of the PAH hydrogenation reaction temperature was calculated, which made it possible to obtain unimprovable values of the optimality criteria — maximizing the yield of target naphthenes and conversion of feedstock.Discussion and Conclusion. The NSGA-II algorithm is effective for solving the problem of non-dominance, and deriving the optimal solution for all criteria. Future research should be devoted to the selection of optimal algorithm parameters to increase the speed of the solution. Based on the obtained theoretical optimal conditions of the PAH hydrogenation reaction, it is possible to implement the process in industry

Learning systems of ordinary differential equations with Physics-Informed Neural Networks: the case study of enzyme kinetics

Physics Informed Neural Networks (PINNs) are a type of function approximators that use both data-driven supervised neural networks to learn the model of the dynamics of a physical system, and mathematical equations of the physical laws governing that system. PINNs have the benefit of being data-driven to train a model, but also of being able to assure consistency with the physics, and to extrapolate accurately beyond the range of data that currently accessible. As a result, PINNs can provide models that are more reliable while using less data. Specifically, the PINNs objective is to learn the solutions of a systems of equations using supervised learning on the available data and incorporating the knowledge of physical laws and constraints into the training process. However, solving single differential equations with a PINN may be relatively simple, solving systems of coupled differential equations may not be so simple. In this study, I present a neural network model specialized in solving differential equations of enzyme kinetics that has the main characteristic of being a demonstrative simple case of coupled equations system. The study focuses mainly on the theoretical aspects of the definition of a physics-informed loss function and shows a case study that highlights the challenges still to be overcome in solving systems of coupled differential equations.

Local Stability, Period-Doubling and 1:2 Resonance Bifurcation for a Discretized Chemical Reaction System

We investigate a discrete counterpart of planar dynamical system of nonlinear differential equations induced by kinetic differential equations for a two-species chemical reaction. Chemical reactions exhibit a wide range of dynamical behavior. We show how the theoretical analysis provides insight into the potential behavior of chemical reaction systems, determining the areas of parametric space which indicate scenarios for local stability, then for one type of bifurcation co-dimension one and one type of bifurcation co-dimension two. Precisely, we prove the existence of period-doubling bifurcation and 1:2 resonance bifurcation also, by using the center manifold theorem and the technique of normal forms. All mathematical investigations are illustrated with numerical examples, bifurcation diagrams, Lyapunov exponents and phase portraits.

Kinetic Modeling of Cornstalk Cellulose Hydrolysis in Supercritical Water: A Comparative Study of the Effects of Temperature and Residence Time on Derivative Production

Kinetic modeling is essential in understanding and controlling the process of cellulose hydrolysis for producing value-added cellulose derivatives. This study aims to adopt a set of dominate kinetic ordinary differential equations of cornstalk cellulose hydrolysis in supercritical water for mechanism-based prediction of the production of cellulose, glucose, fructose, glyceraldehyde, erythrose, 5-hydroxymethyl furfural, glycolaldehyde, threose, aldose, and other cellulose derivatives from cornstalks under processing conditions with a pressure of 89 MPa and a temperature of 378 °C, as considered in a recent experimental study in the literature. The yield rates of several cellulose derivatives, e.g., glucose, fructose, 5-HMF, and erythrose as predicted by the present model, are close to those of experimental measurements. The model is further used to predict the yield rates of a few new cellulose derivatives, e.g., glycolaldehyde, threose, and aldose, that are potentially generated in cornstalk cellulose hydrolysis in supercritical water. The present model and computational simulations can be utilized as a rational tool to predict, control, and optimize the derivative yields in cellulose hydrolysis in supercritical water via tuning the process parameters, and, therefore, are useful for the optimal production of targeted bio-based fuels and chemicals from cornstalks and other agricultural and municipal wastes.

Free-damped vibration tangential wave responses of FG-sandwich merged hemispherical-cylindrical shells under effects of artificial springs at merging and boundary conditions

This article is formed to extract the frequencies with free-damped vibration characteristics of the tangential waves related to the Functionally Graded (FG) sandwich Merged Hemispherical-Cylindrical Shells (MHCS) under elastic springs. The wall of the MHCS contains three layers of embodied Functionally Graded Material (FGM). The material characteristics of the structure change functionally along the thickness of each layer. Equivalent Single Layer (ESL) and Rule of Mixture (RM) are applied to determine the mechanical features of the materials. In addition, eight sandwich types are examined here. Moreover, both merging and support conditions are assumed to be flexible. Artificial transitional and rotational springs are modeled to apply the effect of a semi-rigid merging between hemispherical and cylindrical parts and the support’s flexibility. Further, Donnell's hypothesis and Hamilton's scheme are utilized to acquire the kinetic differential equations related to the MCHS. A recognized semi-analytical method, labeled the Generalized Differential Quadrature Method (GDQM), is being implemented to discretize the kinetic equations. In addition, the tangential wave frequencies are determined by solving the eigenvalue problem. The available results in the submitted framework are compared with the outputs of the FE-based software to validate the proposed procedure. Several innovative parametric studies will be implemented to illustrate the impacts of different general boundary conditions, semi-rigid connections, and material features on the vibrational behavior of MHCS.

The role of thermal noise on key parameters governing the improvement of tokamak fusion reactor efficiency using aneutronic fuel

This article examines the time evolution of energy, particle density, synchrotron radiation loss, plasma frequency, electron particle density, and the relationship between these parameters and blackbody radiation in the presence of thermal noise for the tokamak fusion reactor with aneutronic fuel. To achieve this goal, a suitable thermal noise term is added to the set of energy and particle balance equations. These nonlinear point kinetic coupled differential equations are simultaneously solved by numerical methods. The results of our calculations related to fusion power, particle density, fusion energy density, and quality factor in cases: (i) without and (ii) with noise show that in the conditions where thermal noise is considered, the mentioned parameters and quality factor are increased and in fact system performance improves. Also, our numerical results related to synchrotron radiation loss, plasma frequency, electron particle density, and energy intensity in cases (i) and (ii) show that in the conditions when blackbody radiation occurs (3.75 and 7.225 s respectively for (i) and (ii) cases), synchrotron radiation reaches the maximum values of and for (i) and (ii) cases, respectively.

An Adaptive Auto‐Reduction Solver for Speeding Up Integration of Chemical Kinetics in Atmospheric Chemistry Models: Implementation and Evaluation in the Kinetic Pre‐Processor (KPP) Version 3.0.0

AbstractKinetic integration of large and stiff chemical mechanisms is a computational bottleneck in models of atmospheric chemistry. It requires implicit solution of the coupled system of kinetic differential equations with time‐consuming construction and inversion of the Jacobian matrix. We present here a new version of the Kinetic Pre‐Processor (KPP 3.0.0) for fast integration of chemical kinetics featuring a range of improvements over previous versions in performance, diagnostics, versatility, and community openness. KPP 3.0.0 includes a new adaptive auto‐reduction solver to decrease the size of any mechanism locally and on the fly under conditions where full complexity is not needed, by partitioning species as “fast” or “slow” based on their local production and loss rates. Previous implementations of this adaptive solver suffered from excessive overhead in the repeated construction of the local Jacobian matrix or were hard‐wired to specific mechanisms. Here we retain the general applicability of the method to any mechanism and avoid overhead by using pre‐computed Jacobian matrix terms for the full mechanism and cropping the matrix locally to remove the slow species with no change in memory allocation. We apply this adaptive solver within KPP 3.0.0 to the GEOS‐Chem global 3‐D model of atmospheric chemistry and demonstrate a 32% reduction in solver time while maintaining a mean error lower than 1% for key species in the troposphere.

Heat kernel and gradient estimates for kinetic SDEs with low regularity coefficients

We establish heat kernel and gradient estimates for the density of kinetic degenerate Kolmogorov stochastic differential equations. Our results are established under somehow minimal assumptions that guarantee the SDE is weakly well posed.

Method for calculating the pyrolysis zone in the production unit activated carbon

Activated charcoal can be obtained in many ways. The most promising in terms of resource saving and economic benefits is the method of obtaining activated carbon from organic waste by slow conductive pyrolysis. As a result of such thermochemical conversion, biochar and volatile products - condensable and non-condensable are formed, which are used as fuel for the operation of pyrolysis reactors. Modeling of slow thermal decomposition is carried out using finite difference schemes for heat and mass processes described by differential equations and equations of chemical kinetics. The pyrolysis process equipment is a complex and expensive mechanism, therefore, when developing equipment for pyrolysis, it is necessary to carry out calculations related to the optimization of the equipment design. In this work, as applied to an energy- and resource-saving plant for the production of activated carbon, analytical calculations were carried out and a universal calculation method was developed for the zone of conductive pyrolysis of crushed waste. A mathematical model of the process of conductive slow pyrolysis is presented, which consists of heat and mass transfer equations and equations of chemical kinetics. An algorithm for calculating the pyrolysis zone has been developed. A kinetic dependence of the specific gravity of the solid residue of wood particles during pyrolysis is obtained, which characterizes mass transfer during the thermochemical transformation of waste into a carbonaceous residue, this dependence shows the dynamics of the change in the mass of the solid fraction over time during the decomposition of waste, the kinetic curves show: the rate of chemical reactions occurring during pyrolysis, as well as the final specific gravity of coal.

Stability, Discretization, and Bifurcation Analysis for a Chemical Reaction System

Chemical reactions reveal all types of exotic behavior, that is, multistability, oscillation, chaos, or multistationarity. The mathematical framework of rate equations enables us to discuss steadystates, stability and oscillatory behavior of a chemical reaction. A planar cubic dynamical system governed by nonlinear differential equations induced by kinetic differential equations for a two-species chemical reaction is studied. It is investigated that system has unique positive steady state. Moreover, local dynamics of system is studied around its positive steady state. Existence and direction of Hopf bifurcation about positive equilibrium are carried out. In order to modify the bifurcating behavior, bifurcation control is investigated. Keeping in mind, a consistency preserving discretization for continuous chemical reaction system, a discrete counterpart is proposed, and its qualitative behavior is investigated. Numerical simulation along with bifurcation diagrams are provided to illustrate the mathematical investigations.

PolyODENet: Deriving mass-action rate equations from incomplete transient kinetics data.

Kinetics of a reaction network that follows mass-action rate laws can be described with a system of ordinary differential equations (ODEs) with polynomial right-hand side. However, it is challenging to derive such kinetic differential equations from transient kinetic data without knowing the reaction network, especially when the data are incomplete due to experimental limitations. We introduce a program, PolyODENet, toward this goal. Based on the machine-learning method Neural ODE, PolyODENet defines a generative model and predicts concentrations at arbitrary time. As such, it is possible to include unmeasurable intermediate species in the kinetic equations. Importantly, we have implemented various measures to apply physical constraints and chemical knowledge in the training to regularize the solution space. Using simple catalytic reaction models, we demonstrate that PolyODENet can predict reaction profiles of unknown species and doing so even reveal hidden parts of reaction mechanisms.

Accelerating reactive-flow simulations using vectorized chemistry integration

The high cost of chemistry integration is a significant computational bottleneck for realistic reactive-flow simulations using operator splitting. Here we present a methodology to accelerate the solution of the chemical kinetic ordinary differential equations using single-instruction, multiple-data vector processing on CPUs using the OpenCL framework. First, we compared several vectorized integration algorithms using chemical kinetic source terms and analytical Jacobians from the pyJac software against a widely used integration code, CVODEs. Next, we extended the OpenFOAM computational fluid dynamics library to incorporate the vectorized solvers, and we compared the accuracy of a fourth-order linearly implicit integrator–both in vectorized form and a corresponding method native to OpenFOAM—with the community standard chemical kinetics library Cantera. We then applied our methodology to a variety of chemical kinetic models, turbulent intensities, and simulation scales to examine a range of engineering and scientific scale problems, including (pseudo) steady-state as well as time-dependent Reynolds-averaged Navier–Stokes simulations of the Sandia flame D and the Volvo Flygmotor bluff-body stabilized, premixed flame. Subsequently, we compared the performance of the vectorized and native OpenFOAM integrators over the studied models and simulations and found that our vectorized approach performs up to 33–35× faster than the native OpenFOAM solver with high accuracy.

Multiscale bio-chemo-mechanical model of intimal hyperplasia.

We consider a computational multiscale framework of a bio-chemo-mechanical model for intimal hyperplasia. With respect to existing models, we investigate the interactions between hemodynamics, cellular dynamics and biochemistry on the development of the pathology. Within the arterial wall, we propose a mathematical model consisting of kinetic differential equations for key vascular cell types, collagen and growth factors. The luminal hemodynamics is modeled with the Navier-Stokes equations. Coupling hypothesis among time and space scales are proposed to build a tractable modeling of such a complex multifactorial and multiscale pathology. A one-dimensional numerical test-case is presented for validation by comparing the results of the framework with experiments at short and long timescales. Our model permits to capture many cellular phenomena which have a central role in the physiopathology of intimal hyperplasia. Results are quantitatively and qualitatively consistent with experimental findings at both short and long timescales.

Modeling of the Particle Build-Up Evolution on a Single-Wire Magnetic Capture from Axial Stream Flow

The kinetic equation of the accumulation of magnetic particles from axial flow on a magnetized ferromagnetic wire in an external homogeneous magnetic field has been developed in this study. A new differential equation of the evolution of the accumulation radius over time, which considers both the capture and the detachment of the particles in the accumulation profile on the wire, has been formulated. The evolution of the radius of the accumulation profile over time was obtained from both the differential kinetic equation based on population theory and from the stochastic Fokker–Planck equation. In the limit approach (t→∞), it was observed that the expressions of the saturation radius of the accumulation radius on the magnetized wire of the particles obtained from both models were the same. It is emphasized that the obtained results are valid for both the initial and steady-state build-up of the particle capture process. These results were compared with the experimental results from the literature, and it was observed that the theoretical and experimental results were in good agreement. The effects of both capture and detachment events on the accumulation of particles on the magnetized wire were evaluated.

Mathematical and Analytical Evaluation of Paste Products Quality

Topicality. In modern conditions, the society feels a growing demand in products made from natural ingredients of high nutritional value. Such products include meat pastes, the quality of which depends on their recipe, technological processes, equipment, and modes of its operation. With the implementation of innovative technological regimes and new equipment, there is a necessity to determine and prognosticate the quality of food products at all stages of their production and sale. According to the mentioned above, as well as economic feasibility of using automated systems for technological processes management, there is a need to elaborate a new mathematical and analytical approach to assessing and prognosticating changes in the quality of meat paste with various additives. Aim and research methods. The aim of this research is to elaborate a method for modeling the material system state, based on differential equations of kinetics of biochemical processes, assessment, and prognostication of food quality. Research methods. The method of mathematical and analytical evaluation of the paste products quality is grounded on modified mathematical models, differential equations, visualisation of research results in the form of 3D graphs, obtained by using symbolic computer mathematics. The quality of new meat paste products is compared with the quality of the paste, which is assumed to be relatively optimal according to the main organoleptic parameters of sensory quality assessment and control sample. The control sample of the paste is cooked according to the classical technology, and the recipe (GSTU 4424:2005). The prototype samples are cooked according to the innovative technology, which involves adding mechanically deboned poultry meat to the main raw material. Results. New results of analytical and experimental studies of the quality of meat paste products, cooked according to the classic recipe with the use of mechanically deboned poultry meat, are offered. The method for determining the coefficients, included in the mathematical model for assessing the meat paste products quality, their analytical relation with the optimization parameter, is elaborated. The expediency of using modern methods of symbolic computer mathematics for solving and analysing differential equations, presenting results in 3D graphs, is proved. The conducted research makes it possible to prognosticate the quality of food products, to control possible changes in their recipe while using various additives, to carry out elaborating new paste products. Conclusions and discussion. Based on the results of theoretical and experimental studies, a new mathematical model for estimating the quality of meat paste products in the form of the first-order differential equation, is offered. Its analogue is the equations, recommended for modeling the processes of biochemical kinetics. It is proved that the computer programme of mathematical and analytical research and prognostication the foodstuff quality (Goots et al., 2018) is universal. The offered mathematical model makes it possible to envision the quality of meat paste products, based on organoleptic evaluation. With its help, it becomes possible to determine the vector of possible changes in product quality and its optimisation, while elaborating the 3D graph. Mathematical and analytical assessment of the new paste products quality highlights that the partial replacement of the main meat raw by mechanically deboned poultry meat, and in pastes, made according to classical technology and the GSTU recipe (4424:2005), does not really reduce the parameters of organoleptic evaluation. In some cases, they are even higher than in the control paste samples, and very close to optimal ones. This new mathematical and analytical approach to assessing the paste products quality is promising in new culinary products elaboration in the restaurant business.

Mathematical Modelling of Turbulent Combustion of Two-Phase Mixtures of Gas and Solid Particles with a Eulerian–Eulerian Approach: The Case of Hydrogen Combustion in the Presence of Graphite Particles

The numerical modelling of turbulent combustion of H2–air mixtures with solid graphite particles is a challenging and key issue in many industrial problems including nuclear safety. This study presents a Eulerian–Eulerian model based on the resolution of the Navier–Stokes equations via large eddy simulation (LES) coupled with a system of ordinary differential equations (ODEs) of the detailed chemical kinetics to simulate the combustion of mixtures of gases and particles. The model was applied to predict the transient evolution of turbulent combustion sequences of mixtures of hydrogen, air and graphite particles under low concentration conditions. When applied to simulate lab-scale combustion experiments, the results showed a good agreement between experimental and numerical data using a detailed chemical kinetic model. Moreover, the model was able to predict some key experimental tendencies and revealed that the presence of a low concentration of graphite particles (~96 g/m3) in the scenario influenced the hydrogen combustion dynamics for mixtures of 20% (in volume) of hydrogen in air. Under these conditions, pressure levels reached at the walls of the sphere were increased and the combustion time was shortened. The results also showed the viability of using this kind of a model for obtaining global combustion parameters such as wall pressure evolution with time.