Numerical solution of an inverse problem of nonisothermal sorption dynamics

We consider the inverse problem for a mathematical model of sorption dynamics that incorporates diffusion, intradiffusion kinetics, and the heat balance. Two numerical methods are proposed. Their efficiency is investigated in a computer experiment. Translated from Prikladnaya Matematika i Informatika, No. 29, pp. 56–63, 2008.     

Mathematical modeling numerical determination of coefficients in some mathematical models of nonisothermal sorption dynamics

We consider inverse problems for mathematical models of sorption dynamics that allow for diffusion, heat balance, and two types of kinetics assuming temperature-dependent kinetic coefficients. Two methods for numerical solution of inverse problems are proposed. Their efficiency is investigated by computer experiments.     

脑内各向异性扩散传质与吸附反应过程数值分析

对脑组织内传质过程的机理及其影响因素进行了分析,建立了综合考虑脑内物质各向异性扩散、吸附和反应过程的数学模型,模型方程采用隐式控制容积法进行数值求解.计算结果表明:组织迂曲度越大,物质的扩散越慢,当某一方向迂曲度较小时,物质浓度明显增大,物质扩散变快,由于脑组织的非均质性,脑内物质的扩散传递存在着竞争现象;吸附与反应作用会抑制脑内物质传递,吸附速率越大,抑制现象越明显,对于脑内非线性的米氏反应过程,当反应速率常数增大时,稳定浓度会显著减小,同时米氏常数的增大则会使得稳定浓度值增大.相较于吸附过程,米氏过程的抑制性作用更为明显.     

Hopf bifurcation in time-delayed Lotka–Volterra competition systems with advection

In this paper, we study time-delayed reaction–diffusion systems with advection subject to Lotka–Volterra competition dynamics over one-dimensional domains. These systems model the population dynamics of two groups of competing species, with one dispersing randomly and the other a combination of random and biased dispersal (to avoid competition). We show that time-delay(s) in the interspecific competition mechanism can induce instability of the homogeneous equilibrium to the reaction–advection–diffusion systems, and further promote the appearance of time-oscillating spatially inhomogeneous distributions of the species. Our results indicate that these time-delayed systems (both single and double time-delays) can be used to model the well-observed time-periodic distributions of interacting species in natural fields, compared to the systems without time-delay(s).     

Modeling the Oscillations of the CO + O2 Reaction Rate in a Granular Catalyst Layer

A new point model is proposed for the oxidation reaction of CO on the surface of a Pd cluster. The model reflects the oxidation–reduction mechanism on the palladium surface and incorporates all the stages of the kinetic scheme from the previous three-component model. A new assumption based on a series of experimental facts claims that the adsorbed oxygen atoms may diffuse into deeper-lying layers of the crystal lattice and thus influence the processes in the adsorption layer due to the micro size of the palladium clusters. The new point system has been built into the distributed general model for a granular catalyst developed previously. As a result, for parameter values corresponding to experimental conditions we have managed for the first time to obtain a wide region of chaos and complex mixed modes, close to those observed in real experiments.     

Numerical determination of the sorption isotherm for nonisothermal sorption

Two mathematical models of sorption dynamics are considered, one allowing for external diffusion kinetics and the other for internal diffusion kinetics with heat balance. Two inverse problems are investigated for these models and numerical solutions are proposed. The efficiency of the numerical methods is studied in a computer experiment. __________ Translated from Prikladnaya Matematika i Informatika, No. 28, pp. 37–43, 2008.     

Oscillatory modes of adsorption in the seepage of multicomponent systems

A system of equations describing the motion of multicomponent mixtures in a porous medium, taking into account the phenomena of competing adsorption is formulated within the framework of a kinetic approach to the study of sorption processes. An equation of sorption kinetics, which describes a series of oscillatory modes for certain values of the original parameters, is suggested. A numerical analysis of the stability is carried out for the corresponding dynamical system. A qualitative comparison of the results of calculations and experimental data is given.     

Inverse problems for a mathematical model of redox sorption

We consider a mathematical model of sorption that allows for external diffusion kinetics and a redox reaction. Two inverse problems are considered for this model, uniqueness is proved, and numerical solution methods are proposed. The efficiency of the numerical methods is investigated by computer experiments. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 15–23, 2006.     

Unique Solvability of the Inverse Problem with a Time-Dependent Boundary Condition for a Sorption Model

A mixed initial boundary-value problem is considered for nonequilibrium sorption dynamics with inner-diffusion kinetics. The problem allows for convection and longitudinal diffusion and has a time-dependent boundary condition. This condition contains the time derivative of a solution component and constitutes the balance equation for the absorbed mixture near the output boundary of the sorption region—inside the diffusion barrier. Bounds on the solution of the direct problem are obtained: nonnegativity of the solution and its first time derivatives, and boundedness of the solution by known functions. The inverse problem of estimating the nonlinear system parameter—the sorption isotherm—is considered and a uniqueness theorem is proved.     

Moisture permeability of a polymer nanocomposite containing unmodified clay

Results of an investigation into the kinetics of moisture sorption by an unfilled styrene-acrylate copolymer and a nanocomposite containing unmodified montmorillonite clay are reported. The sorption tests were performed at 25°C and different levels of the relative humidity of air up to 80%. The moisture diffusivity was calculated using Fick's equation for one-dimensional diffusion. Data on the influence of montmorillonite concentration on the coefficients of diffusion, solubility, and permeability are obtained. The experimental values of the permeability coefficient are compared with those calculated by using a model taking into account the increased path of a diffusing water molecule caused by the shielding effect of plate like filler particles. Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 5, pp. 723–736, September–October, 2008.     

Iterative Method of Solving the Inverse Problem for a Quasilinear Hyperbolic Equation

We consider the inverse problem for the quasilinear hyperbolic equation connected with the mathematical model of adsorption dynamics with a concentration-dependent kinetic coefficient. An iterative method is proposed that reduces the inverse problem to a nonlinear operator equation.__________Translated from Prikladnaya Matematika i Informatika, No. 17, pp. 5–12, 2004.     

A transfer function model to describe odor causing VOCs transport in a ventilated airspace with mixing/adsorption heterogeneity

The ability of a transfer function modeling technique is evaluated to explain the odor causing VOCs (VOC-odor) transport processes influenced by heterogeneity of adsorption surface of ambient aerosol and air mixing pattern in a ventilated airspace. An advection–reaction impulse/step response function is used to generalize the dynamic transport of VOC-odor in heterogeneous mixing/adsorption ventilated airspace. The system process presented by an ensemble transfer function is solved analytically in the Laplace domain. The model requires the specification of probability density function (pdf) for residence time of airflow and for both equilibrium linear partitioning and first-order mass transfer rate parameters of gas/solid phase to quantify the specific air mixing pattern and transport processes. The model predicts the ensemble mean VOC-odor concentrations for a variety of adsorption kinetics and mixing pattern combinations as a function of the boundary impulse/step response inputs as well as residence time and adsorption rate statistics. The general behavior of output VOC-odor profiles is analyzed through the effects of mean adsorption rate coefficient, mean linear partitioning constant, mixing efficiency, mean residence time and coefficient of variations of both linear partitioning and rate coefficients. This study indicates that when mixing/adsorption heterogeneity exists, simple complete mixing assumption and simple distribution of rate constant are inherently not sufficient to represent a more generally distributed mixing/adsorption process of VOC-odor transport in a ventilated airspace.     

A transfer function technique to describe odor causing VOCs transport in a ventilated airspace with mixing/adsorption heterogeneity

A model describing odor causing volatile organic compounds (VOC-odor) transport in a ventilated airspace influenced by heterogeneity of adsorption surface of ambient aerosol and air mixing pattern is proposed and analyzed based on a transfer function modeling technique. In this study an advection–reaction impulse/step response function for VOC-odor is assumed. The system process presented by an ensemble transfer function is solved analytically in the Laplace domain. The analytical results are then numerically inverted using a modified fast Fourier transform algorithm. The model requires the specification of probability density function for residence time of airflow and for both equilibrium linear partitioning and first-order mass transfer rate parameters to quantify the specific air mixing pattern and transport processes. The model predicts the ensemble mean VOC-odor concentrations for a variety of adsorption kinetics and mixing pattern combinations as a function of the boundary impulse/step response inputs as well as residence time and adsorption rate statistics. The general behavior of output VOC-odor profiles is analyzed through the effects of mean adsorption rate coefficient, mean linear partitioning constant, mixing efficiency, mean residence time and coefficient of variations of both linear partitioning and rate coefficients. It indicates that when mixing/adsorption heterogeneity exists, simple complete mixing assumption and simple distribution of rate constant is inherently not sufficient to represent a more generally distributed mixing/adsorption process of VOC-odor transport in a ventilated airspace.     

Numerical solution of an inverse problem for a two-dimensional mathematical model of sorption dynamics

We consider a two-dimensional mathematical model of sorption that allows for inner-diffusion kinetics as well as longitudinal and transverse diffusion. The inverse problem of determining the sorption isotherm from an experimental dynamic output curve is investigated for this model and stable solution methods are proposed for the inverse and the direct problem. The efficiency of the solution methods is explored in computer experiments.     

Waves analysis and spatiotemporal pattern formation of an ecosystem model

In this paper, we consider a predator–prey model given by a reaction–diffusion system. This model incorporates Holling-type-II (Michaelis–Menten) and modified Leslie-Gower functional responses. We show the existence of qualitatively different types of system behaviors realized for various parameter values. Our model is investigated with methods of the qualitative theory and the theory of bifurcations. We generalize the traveling waves existence method for populations dynamics with positive derivative densities, to the predator–prey system in which growth densities may change sign. Parallel to this is a discussion and an analysis of alternative model outcomes such as complex pattern formation and spatio-temporal chaos behavior.     

A comparative analysis of moisture transport models as applied to an epoxy binder

The results of experimental and theoretical investigations into the kinetics of moisture sorption by a neat epoxy resin obtained from RAE Industries (Reapox 520, D523) are reported. The sorption process was realized in atmospheres with a constant relative humidity of 33, 53, 75, 84, and 97% and a temperature of 50°C. The results obtained showed that the diffusion behavior of epoxy resin did not obey Fick’s law under the experimental conditions considered. Consequently, the application of a non-Fickian diffusion model was necessary. For this purpose, two-phase moisture sorption models, a model with a time-dependent diffusivity, a two-phase material model, as well as relaxation and convection models of anomalous diffusion, were considered. The model parameters were obtained from the approximation of experimental sorption data. A comparative analysis of the sorption models was performed, and the specific features of their applications were estimated. The two-phase material model and the model with varying diffusivity were found to be the most suitable ones due to a good agreement between calculation results and experimental data and the rather small (three or four) number of parameters, which make them more flexible and physically more justified than the classical Fick’s model with its two parameters. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 555–570, July–August, 2007.     

Shadow systems and attractors in reaction-diffusion equations

For a pair of reaction diffusion equations with one diffusion coefficient very large, there is associated a reaction diffusion equation coupled with an ordinary differential equation (the shadow system) with nonlocal effects which has the property that it contains all of the essential dynamics of the original equations. Key words: Attractors, shadow systems, reaction-diffusion equations     

Verification of model for adsorption and reduction of chromium(VI) by Escherichia coli 33456 using chitosan bead as a supporting medium

A non-steady-state mathematical model system for the kinetics of adsorption and reduction of Cr(VI) by Escherichia coli (E. coli) biofilm on chitosan bead (EBCB) process was derived. The mechanisms in the model system included Cr(VI) adsorption by chitosan beads, Cr(VI) bioreduction by E. coli cells and Cr(VI) mass transport diffusion. Batch kinetic tests were performed to determine surface diffusivity of Cr(VI), adsorption parameters for Cr(VI) and biokinetic parameters of E. coli 33456. Experiments were conducted using an EBCB reactor system with high recycled rate to approximate a completely-mixed flow reactor for model verification. The experimental results indicated that E. coli biofilm bioregenerated the chitosan beads after E. coli biofilm has grown significantly. Cr(VI) reducing efficiency by E. coli was about 84% when Cr(VI) concentration in the influent was 5 mg/L at a steady-state condition. The concentration of suspended E. coli cells reached up to 10 mg/L while the thickness of attached E. coli cells was estimated to be 150 μm at a steady-state condition by model prediction. The comparisons of experimental data and model simulation show that EBCB model system for Cr(VI) adsorption and reduction can predict the experimental results well.