Non-stationary Diffusion Equation Research Articles

Critical conditions of forced ignition in Ni-Al, Ti-Al powder systems under solid-phase synthesis

A theoretical study of the critical conditions of forced solid-phase ignition in Ti-Al, Ni-Al systems was carried out. Based on the analysis of the non-stationary diffusion equation with moving boundaries, the critical powers of the heating source and the corresponding critical temperatures were determined during the formation of NiAl, TiAl3 intermetallic compounds using the state diagrams of these binary systems. The diffusion activation energies, frequency factors, particle sizes, and heat removal conditions were considered as variable parameters. The diagrams of critical parameters which relate the corresponding specific heating powers with the indicated values were constructed. It has been established that for the same values of the parameters and under identical synthesis conditions, the critical temperatures in these systems differ by more than one hundred degrees. The latter is associated with a significant difference in the width of the solubility region for the corresponding compounds of the composition NiAl, TiAl3. The performed analysis allows separating the modes of isothermal sintering and thermal explosion which makes it possible to control the thermal and diffusion modes of synthesis. The comparison of the calculation results with the experimental data showed their satisfactory agreement.

Theoretical and Numerical Analysis of Nonlinear Processes in Amperometric Enzyme Electrodes with Cyclic Substrate Conversion

A theoretical model of amperometric enzyme electrodes has been developed in which chemical amplification occurs in a single enzyme membrane via cyclic substrate conversion. The system is based on non-stationary diffusion equations with a nonlinear factor related to the Michaelis–Menten kinetics of the enzymatic reaction. By solving the nonlinear equations using the AGM technique, simple analytical expressions of concentration substrate, product, and amperometric current response are derived. Further, biosensor sensitivity, resistance, and gain are obtained from the current. MATLAB programming was used to carry out the digital simulation. The analytical results are validated with the numerical results. The effect of substrate concentration, maximum enzymatic rate, and membrane thickness on biosensor response was evaluated.

Sensitivity and resistance of amperometric biosensors in substrate inhibition processes

A theoretical model of a sensitivity and resistance of amperometry biosensors with substrate inhibition kinetics is discussed. This model is based on the system of non-stationary diffusion equations containing a non-linear term related to non-Michaelis-Menten kinetics of the enzymatic reaction. This paper presents the approximate analytical expression of sensitivity and resistance of biosensor for small values of reaction diffusion parameters. The effect various parameters such as thickness of enzyme layer, bulk substrate concentration, Michaelis-Menten and saturation constant on sensitivity and resistance of biosensor are discussed.

Analytical Expression of Effectiveness Factor for Immobilized Enzymes System with Reversible Michaelis Menten Kinetics

The mathematical model of immobilized enzyme system in porous spherical particle is presented. This model is based on a non-stationary diffusion equation containing a nonlinear term related to Michaelis-Menten kinetics of enzymatic reaction. A general and closed form of an analytical expression pertaining to the substrate concentration profile and effectiveness factor are reported for all possible values of Thiele modules φ andα . However, we have employed New Homotopy Perturbation Method (NHPM) to solve the nonlinear reaction/diffusion equation in immobilized enzymes system. Therefore, analytical results were found to be in an appropriate agreement with simulation result.

Steady-state current in product inhibition kinetics in an amperometric biosensor: Adomian decomposition and Taylor series method

Theoretical models of an amperometric biosensor with product inhibition kinetics are discussed. These models are based on non-stationary diffusion equations containing a non-linear term related to non-Michaelis-Menten kinetics of the enzymatic reaction. The approximate analytical representation of steady-state concentrations is provided for small values Thiele modulus and all other input variables. Here the Adomian decomposition method and Taylor series method are used to find the analytical expressions for the concentration of substrate, product, current and sensitivity. A comparison of our approximate analytical results with numerical simulation is also presented. A satisfactory agreement is noted. The effect of the parameters Michaelis - Menten constant, inhibition constant and bulk concentration of substrate on the biosensor sensitivity and resistance are discussed.

Ring Current Proton Dynamics Driven by Wave-Particle Interactions During a Nonstorm Period

Modeling of pitch angle scattering of ring current protons at interaction with electromagnetic ion cyclotron waves during a nonstorm period was considered very seldom. Therefore it is used correlated observation of enhanced electromagnetic ion cyclotron (EMIC) waves and dynamic evolution of ring current proton flux collected by Cluster satellite near the location L = 4.5 during March 26–27, 2003, a nonstorm period (Dst > –10 nT). Energetic (5–30 keV) proton fluxes are found to drop rapidly (e.g., a half hour) at lower pitch angles, corresponding to intensified EMIC wave activities. As mathematical model is used the non-stationary one-dimensional pitch angle diffusion equation which allows to compute numerically density of phase space or pitch angle distribution of the charged particles in the Earth’s magnetosphere. The model depends on time t, a local pitch angle and several parameters (the mass of a particle, the energy, the McIlwain parameter, the magnetic local time or geomagnetic eastern longitude, the geomagnetic activity index, parameter of the charged particle pitch angle distribution taken for the 90 degrees pitch angle at t = 0, the lifetime due to wave–particle interactions). This model allows numerically to estimate also for different geophysical conditions a lifetime due to wave–particle interactions. It is shown, that EMIC waves can yield decrements in proton flux within 30 minutes, consistent with the observational data. The good consent is received. Comparison of results on full model for the pitch angle range from 0 up to 180 degrees and on the model for the 90 degrees pitch angle is lead. For a perpendicular differential flux of the Earth’s ring current protons very good consent with the maximal relative error approximately 3.23 % is received

Modeling of the Thermal Decomposition of Methane and the Formation of Solid Carbon Particles

Mathematical models of methane decomposition during its pyrolysis, including nonstationary diffusion equations with and without taking into account the convective component of the decomposition process, have been developed and verified. Their exact analytical solutions have been obtained and further used in the study of a mathematical model of the formation of solid carbon particles arising during methane heating to the pyrolysis temperature. By studying the rate and intensity of the formation of solid carbon particles, it has been shown that particles of the smallest size are obtained at higher decomposition temperatures and for a shorter time.

The complete discretization of the dual mixed method for the heat diffusion equation in a polygonal domain

The purpose of this paper is to prove a priori error estimates for the completely discretized problem of the dual mixed method for the non-stationary heat diffusion equation in a polygonal domain of R2. Due to the geometric singularities of the domain, the exact solution is not regular in the context of classical Sobolev spaces. Instead, one must use weighted Sobolev spaces in our analysis. To obtain optimal convergence rates of the discrete solutions, it is necessary to refine adequately the considered meshings near the reentrant corners of the polygonal domain. In a previous work, using the Raviart–Thomas vectorfields of degree 0 for the discretization of the heat flux density vector, we have obtained a priori error estimates of order 1 for the semi-discrete solutions of this problem. In our actual paper, we complete the discretization of the problem in time by using Euler’s implicit scheme, and we obtain optimal error estimates of order 1, in time and space. Numerical results are given to illustrate this improvement of the convergence orders.

Methodical Aspects of Creation and Calculation of the Space Kinetics Benchmarks

The problems arising at creation and calculation of the space kinetics benchmarks are considered by three tasks. The connection between the nonstationary diffusion equation and the inverse solution of the kinetics equation (ISKE) is considered with the different methods of preparation and presentation of kinetic parameters. The following benchmarks are considered: the nondiffusion Small LWR benchmark from the series of Takeda’s tasks, complemented by modeling the motion of control rods; Ferguson’s benchmark, enlarged by calculation of the reactivity change with ISKE; TWIGL benchmark—2D LWR model with insertion of positive reactivity.

An efficient wavelet-based optimization algorithm for the solutions of reaction-diffusion equations in biomedicine

In this paper, a mathematical model of nonlinear reaction-diffusion equation with Michaelis–Menten kinetics in a solid of planar and spherical shape is discussed. The proposed model is based on non-stationary diffusion equation containing a non-linear term related to Michaelis–Menten kinetics of the enzymatic reaction. An efficient wavelet-based spectral method has been developed for the analytical expressions pertaining to substrate concentration for all parameter values. The efficiency of the proposed wavelet method is confirmed by mean of the computational CPU time. The proposed wavelet-based results are compared with Adomian Decomposition Method (ADM). Satisfactory agreement with ADM results is observed. Moreover, the use of the wavelet method is found to be simple, efficient, flexible, and straight forward. Also, it requires less computation costs.

In Ukrainian

The paper presents a system of equations that determines the effective lifetime of minority charge carriers in macroporous silicon. The system of equations was found from the diffusion equation of minority carriers recorded for the macroporous layer and the single-crystal substrate. The solution of nonstationary diffusion equation written for a macroporous layer and a single-crystal substrate is complemented by boundary conditions at the surfaces of a sample of macroporous silicon and at the interface between the macroporous layer and the single-crystal substrate. The effective lifetime of minority charge carriers in macroporous silicon on a single crystal substrate depends on such values as: the minority carrier lifetime in the bulk, the diffusion coefficient of charge carriers, the thickness of the single crystal substrate, the average diameter of the macropores, the average distance between the centers of macropores, the surface recombination rate, the volume fraction macropore. The effective recombination of excess charge carriers in macroporous silicon is determined by the recombination of excess charge carriers on the surface of macropores and limited by the diffusion of charge carriers from the single crystal substrate to the recombination surfaces in the macroporous layer. Using the system of equations, we calculated and shown in the figure the effective lifetime of minority charge carriers in macroporous silicon dependent on the depth of the macropores. To verify the accuracy of calculations performed using a system of analytical equations, which determines the effective lifetime of minority charge carriers in macroporous silicon on a single crystal substrate, we used a numerical method. The numerical method showed the coincidence of the calculations on the effective lifetime of minority carriers. When the depth of macropores is close to the size of the sample of macroporous silicon, a discrepancy of calculations is observed.

OPTIMIZATION OF IGNITION ZONE OF ADVANCED FAST REACTOR, WORKING IN NUCLEAR BURNING WAVE MODE

This article deals with the problem of optimizing composition and structure of the ignition zone of the fast reactor operating in the self-sustaining mode of nuclear burning wave with the purpose of its smooth start-up and reducing fissile material amount in initial assembly. The cylindrical homogeneous reactor with the ignition zone placed in the center or near the cylinder end is considered. The analysis has been performed basing on solving the non-stationary neutron diffusion equation together with the fuel burn-up equations and the equations of nuclear kinetics for precursor nuclei of delayed neutrons, with using the radial buckling approximation. An optimized structure of the ignition zone has been proposed, which ensures a smooth transition of the reactor to the self-sustaining nuclear burning wave mode, avoiding an excessive energy release, which is observed when using a simplified scheme of ignition zone. Comparison of the startup variants with the ignition zone at the cylinder end and at its center shows the benefits of the second one.

К РЕШЕНИЮ ЗАДАЧ НЕСТАЦИОНАРНОЙ ДИФФУЗИИ В СЛОИСТЫХ СРЕДАХ

The issues of nonstationary diffusion in layered structures are considered. When designing the devices for implementing mass transfer processes, it is necessary to take into account the properties of the substance and the nature of the processes. Design time reduces significantly and the efficiency of the devices is higher if a good physical model is built and a mathematical analysis with kinetics of the processes is applied. The difficulties of theoretical analysis and calculation of mass transfer are determined by the complexity of the transfer mechanism to and from the phase boundary. Therefore, simplified models of mass transfer processes are used in which the mass transfer mechanism is characterized by a combination of molecular and convective mass transfer. Many important practical problems involve the calculation of nonstationary diffusion (Fick's second law of diffusion) for a certain volume of substance (substances). For qualitative evaluation of processes, in the case of symmetry, volumetric issues can be considered as one-dimensional tasks, i.e. dependent on one coordinate. The general solution of the non-stationary diffusion equation for layered environments is proposed. The case of non-stationary boundary conditions of the third kind on the external surface and boundary conditions of the fourth kind conjugation for contiguous layers has been considered. The solution is obtained by separating the Fourier variables by the eigenfunctions of the problem using the Duhamel integral. The proposed solution is explicit and due to the recurrent form of the basic relations can be useful in numerical calculations

Kinetic Mechanism for Modelling of Electrochemical Mediatedenzyme Reactions and Determination of Enzyme Kinetics Parameters

The non-steady state current density for reversible electrochemical coupled with a homogeneous enzyme reaction and a constant potential is presented in this manuscript for the first time. The model is based on non-stationary diffusion equations with semi infinite boundary condition containing a nonlinear term related to the kinetics of an enzymatic reaction. The nonlinear differential equation for the mediator is solved for reversible homogeneous enzyme reaction. Approximate analytical expressions for the concentration of the mediator and corresponding current for non-steady state conditions are derived. Kinetic parameters are also determined such as Michaelis–Menten constants for substrate (KMS) and mediator (KMM) as well as catalytic rate constant (kcat). Upon comparison, we found that the analytical results of this work are in excellent agreement with the numerical (Matlab program) and existing limiting case results. The significance of the analytical results has been demonstrated by suggesting two new graphical procedures for estimating the kinetic parameters from the current densities.

Analysis of the Numerical Solution in the One Dimensional Non-Stationary Diffusion Equation using Schemes of Finite Mimetic Differences and Crank Nicolson

espanolSe propone la solucion numerica de la ecuacion de difusion no estatica unidimensional, desarrollando un algoritmo en software Matlab version 7.0, para lo cual se combina el esquema de diferencias finitas mimeticas en la aproximacion de los operadores diferenciales del continuo (gradiente y divergencia) para la variable espacial, sobre una malla uniforme, cuyos operadores diferenciales discretos presentan una aproximacion de segundo orden y el enfoque en diferencias finitas tipo Crank-Nicolson para obtener aproximaciones en la variable temporal. Este algoritmo propuesto para los enfoques mimeticos y Crank-Nicolson presentan mejor aproximacion que el esquema en diferencias finitas tipo Crank- Nicolson. Ademas se calcula el error de aproximacion generado entre la solucion numerica y la solucion analitica usando la norma del maximo para la ecuacion de difusion no estacionaria con condiciones de frontera tipo Robin. EnglishThe numerical solution of the one-dimensional non-static diffusion equation is proposed, developing an algorithm in software Matlab version 7.0, for which the mimetic finite difference scheme is combined in the approximation of the differential operators of the continuum (gradient and divergence) for the spatialvariable, on a uniform grid, whose discrete differential operators have a second order approximation and the finite difference approach type Crank Nicolson for approximations in the temporary variable.This proposed algorithm for the mimetic and Crank Nicolson approaches has a better approximation than the finite-difference Crank Nicolson-type scheme.In addition, the approximation error generated between the numerical solution and the analytical solution is calculated using the maximum standard for the non-stationary diffusion equation with Robin type boundary conditions.

Inhomogeneous dopant distribution in III–V nanowires

We present a theoretical study of the dopant spatial distribution in III–V nanowires grown by molecular beam epitaxy. The evolution of the dopant concentration is obtained by solving the non-stationary diffusion equation. Within the model, it is shown why and how the dopant inhomogeneity appears, as observed experimentally in the case of Be doping of GaAs nanowires and in other material systems.

Stability of Difference Splitting Schemes for Convection Diffusion Equation

We consider numerical modeling of the propagation of pollution in the air on the basis of geometrical splitting method for three-dimensional nonstationary convection diffusion equations. Splitting difference schemes in the form of explicit computing schemes are proposed to solve the obtained one-dimensional problems. The approximation, monotonicity, and stability of the proposed difference schemes are investigated.

The Stefan outflow in a multicomponent vapor–gas atmosphere around a droplet and its role for cloud expansion

A new comprehensive analysis of Stefan׳s flow caused by a free growing droplet in the vapor–gas atmosphere with several condensing components is presented. This analysis, based on the nonstationary heat and material balance and diffusion transport equations, shows the appearance of the Stefan inflow in the vicinity of the growing droplet and the Stefan outflow at large distances from the droplet as a consequence of nonisothermal condensation. For an ensemble of droplets in the atmospheric cloud, this outflow provides an increase of the total volume of the cloud, which can be treated as cloud thermal expansion and leads to the rise of the cloud as a whole due to increasing its buoyancy. We have formulated the self-similar solutions of the nonstationary diffusion and heat conduction equations for a growing multicomponent droplet and have derived analytical expressions for the nonstationary velocity profile of Stefan׳s flow and the expansion volume of the vapor–gas mixture around the growing droplet. To illustrate the approach, we computed these quantities in the case of droplet of stationary composition in air with several specific vapors (C2H5OH/H2O; H2SO4/H2O; H2O).

An efficient Chebyshev wavelet based analytical algorithm to steady state reaction–diffusion models arising in mathematical chemistry

A mathematical model of amperometric biosensors has been developed. The model is based on non-stationary diffusion equation containing a nonlinear term related to non-Michaelis–Menten kinetics of the reaction. In this paper, an efficient Chebyshev wavelet based approximation method is introduced for solving the steady state reaction–diffusion equations. Illustrative examples are given to demonstrate the validity and applicability of the method. The power of the manageable method is confirmed. Moreover the use of Chebyshev wavelet method is found to be simple, flexible, efficient, small computation costs and computationally attractive.

Simulating Plant Metabolic Pathways with Enzyme-Kinetic Models Using a New Approach to Homotopy Perturbation Method

The mathematical models of plant metabolic pathways is discussed. Kinetic modeling is the most detailed mathematical description of a metabolic network and constitutes an important branch in the growing fields of systems biology. This model is based on non-stationary diffusion equations containing a nonlinear term related to the Michaelics-Menten kinetics. An analytical expression of metabolic concentration was obtained for all values of parameters using new approach to Homotopy perturbation method. Our analytical results were compared with simulation results. Satisfactory agreement with simulation data is noted.