Potential transients for an electrochemical corrosion reaction under constant current conditions

Due to the electrochemical nature of almost all corrosion reactions, electrochemical methods are commonly used to measure the corrosion rate of a metal in the laboratory or in the field. In particular, steady state methods are the most widely used for corrosion rate measurements. Transient methods, which can be much more efficient, traditionally rely on an equivalent linear circuit representing the surface kinetics, with negligible mass transport effects. This has been reported to predict transients which are not observed experimentally in many practical situations. In this paper, we consider the galvanostatic method, wherein a constant current is applied across a corroding metal surface and the transient potential response is recorded. The resulting boundary value problems incorporating mixed kinetic and diffusion control involve highly nonlinear, coupled boundary conditions. We present numerical and approximate analytical solutions which can be incorporated into corrosion analysis routines in order to calculate corrosion parameters. The analytical expressions open the possibility of measuring corrosion parameters by merely fitting a class of elementary functions to experimental potential transients. This leads to a significant reduction in the number of computations required for the curve fitting, and hence increasing the overall efficiency of the measurement process compared to the conventional steady state methods.     

Analytical approximate solution of the cooling problem by Adomian decomposition method

The Adomian decomposition method (ADM) can provide analytical approximation or approximated solution to a rather wide class of nonlinear (and stochastic) equations without linearization, perturbation, closure approximation, or discretization methods. In the present work, ADM is employed to solve the momentum and energy equations for laminar boundary layer flow over flat plate at zero incidences with neglecting the frictional heating. A trial and error strategy has been used to obtain the constant coefficient in the approximated solution. ADM provides an analytical solution in the form of an infinite power series. The effect of Adomian polynomial terms is considered and shows that the accuracy of results is increased with the increasing of Adomian polynomial terms. The velocity and thermal profiles on the boundary layer are calculated. Also the effect of the Prandtl number on the thermal boundary layer is obtained. Results show ADM can solve the nonlinear differential equations with negligible error compared to the exact solution.     

Exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term

The present work derives the exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term. The work also emphasizes the role of reaction–diffusion models as important particular cases of much more general equations in the kinetic theory of active particles. The analytical expression derived shows the structure of the solution and the contributions of different terms of the model to it. The result obtained enables one to solve the Cauchy problem indicated by using the exact analytical representation rather than numerical methods, which are usually time-consuming, especially when the number of spatial dimensions is greater than 2.     

The decomposition method for Cauchy reaction–diffusion problems

In this paper, the solution of Cauchy problems for the reaction–diffusion equation is obtained using the decomposition method. In the case when the reaction parameter is time-dependent only, an analytical solution in series form can be derived, otherwise symbolic numerical computations may need to be performed.     

Polarography of potassium ethylxanthate,a titrant in amperometry

Polarograpbic studies of potassium ethylxanthate (RSH or xantbate) at the dropping mercury electrode (DME) reveal that the product of the anodic reaction is strongly adsorbed at the mercury drop as indicated by a prewave. The adsorbed film (0·15 mM to 1·0 mM xanthate solution) greatly affects the characteristics of the anodic wave of xanthate in aqueous medium. The current of main wave is proportional to the concentration of xanthate up to 2·5 mM in aqueous medium at DME. This electrode is used as an indicator electrode for amperometric titrations of metal ions with xanthate. Proper buffer composition and pH are developed for the accurate determination of metal ions. The combined use of the cathodic current of metal ions and the anodic current of xanthate under controlled pH conditions makes possible the amperometric titration of two metal ions in a mixture.     

Time/depth dependent diffusion and chemical reaction model of chloride transportation in concrete

This study focuses on the physical and chemical processes that control the transport of chloride ions into concrete structures. An analytical solution of a diffusion reaction model is presented for determining the time/depth dependent chloride diffusivities considering both diffusion process and binding mechanism of chloride occur simultaneously. The diffusion-reaction model, which is based on the Fick’s second law of diffusion and a mathematical formulation for an irreversible first-order chemical reaction, is used to precisely describe the diffusion mechanism of chloride diffusion process. When the chemical reaction is considered, the free chloride concentration is slowly reduced since some of the free chloride ions have reacted with cement paste such that the diffusion coefficient is also reduced simultaneously. The diffusion-reaction model predicts a longer service life than the total and free chloride diffusion models that do not consider the effect of the chemical reaction during the chloride diffusion process.     

A REACTION–DIFFUSION EQUATION MODELING THE INVASION OF THE ARGENTINE ANT POPULATION,LINEPITHEMA HUMILE,AT JASPER RIDGE BIOLOGICAL PRESERVE

Abstract A continuous reaction–diffusion model is developed for the invasive Argentine ant population within a preserve in northern California. The model is a second‐order partial differential equation incorporating a logistic growth term. The dispersal distance traveled during the reproductive process of budding is used to estimate the diffusion coefficient. The model has two homogeneous steady states, one occurring at the propagation front where the Argentine ant population does not yet exist and one occurring where the population has reached carrying capacity. The traveling wave solutions of the model depict the population density for a given time and location. Using current research, parameter values for the model are estimated and a traveling wave solution for the average parameter values is numerically demonstrated.     

Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach

In this paper, we propose and analyse a mathematical model to study the mathematical aspect of reaction diffusion pattern formation mechanism in a predator-prey system. An attempt is made to provide an analytical explanation for understanding plankton patchiness in a minimal model of aquatic ecosystem consisting of phytoplankton, zooplankton, fish and nutrient. The reaction diffusion model system exhibits spatiotemporal chaos causing plankton patchiness in marine system. Our analytical findings, supported by the results of numerical experiments, suggest that an unstable diffusive system can be made stable by increasing diffusivity constant to a sufficiently large value. It is also observed that the solution of the system converges to its equilibrium faster in the case of two-dimensional diffusion in comparison to the one-dimensional diffusion. The ideas contained in the present paper may provide a better understanding of the pattern formation in marine ecosystem.     

Diffusion of methylene blue in glass fibers – Application of the shrinking core model

Diffusion of mass in a solid cylinder with concentration dependent diffusivity (or temperature-dependent thermal conductivity in case of heat diffusion) does not admit of an analytical solution except in special cases. The ‘shrinking core model’ has been used to develop an approximate analytical solution in certain circumstances. The model, generally useful to describe heterogeneous solid–fluid reactions, is applied to theoretically analyze the adsorption–diffusion phenomena of methylene blue dye in a glass fiber in the present work. Theoretical equations have been derived for the case of diffusivity as an exponential function of concentration. The diffusivity parameters are evaluated by global minimization of the error between the experimental and the theoretical concentration history. Other forms of diffusivity, namely constant diffusivity and diffusivity varying linearly with concentration are found to involve larger errors. A parametric sensitivity analysis of the error has been done. The shrinking core model could satisfactorily interpret the experimental dye concentration profile in the substrate.     

Gas absorption with first order chemical reaction in a laminar falling film over a reacting solid wall

In the present work, a general case of gas absorption with first order irreversible chemical reaction in a liquid film, for laminar flow over a solid wall, has been analyzed theoretically. First order chemical reaction between the diffused solute and the wall is also considered. Laplace transform followed by power series method has been applied to solve the governing equations. Thereafter, the obtained analytical solution of the developed general model has been successfully verified by an explicit numerical scheme. The general model has also been reduced to six simplified cases, tackled by previous workers and an excellent agreement in the solutions is observed. Moreover, the results are validated by the experimental data available in the literature. The obtained concentration profiles in both the phases have been used to find the absorption rates and enhancement factor.     

Unsteady state mathematical model of chemically reactive pollutants from an instantaneous line source into a stable atmospheric boundary layer

A time dependent atmospheric model represented for chemically reactive primary pollutants emitted from an elevated line source into a stable atmospheric boundary layer over a surface terrain. The model obtained from an analytical solution of the atmospheric diffusion equation with the quadratic diffusion coefficient (exchange coefficient) and the variable wind velocity taken to be of three different types’ viz. constant, constant shear and parabolic functions of vertical height. The pollutants considered to be of chemically reactive primary pollutants emitted from a time-dependent line source of Instantaneous type. In order to facilitate the application of the model the results for the general situation that includes chemical reaction rate & time dependent source incorporated in the model.     

Approximate analytical solution of two-dimensional space-time fractional diffusion equation

This work presents an iterative scheme for the numerical solution of the space-time fractional two-dimensional advection–reaction–diffusion equation applying homotopy perturbation with Laplace transform using Caputo fractional-order derivatives. The solution obtained is beneficial and significant to analyze the modeling of superdiffusive systems and subdiffusive system, anomalous diffusion, transport process in porous media. This iterative technique presents the combination of homotopy perturbation technique, and Laplace transforms with He's polynomials, which can further be applied to numerous linear/nonlinear two-dimensional fractional models to computes the approximate analytical solution. In the present method, the nonlinearity can be tackle by He's polynomials. The salient features of the present scientific work are the pictorial presentations of the approximate numerical solution of the two-dimensional fractional advection–reaction–diffusion equation for different particular cases of fractional order and showcasing of the damping effect of reaction terms on the nature of probability density function of the considered two-dimensional nonlinear mathematical models for various situations.     

A time integration procedure for plastic deformation in elastic-viscoplastic metals

A simple unconditionally stable numerical procedure for time integration of the flow rule for large plastic deformation of an elastic-viscoplastic metal is developed. Specific attention is focused on a unified set of constitutive equations which represents a generalization (for large deformation and thermomechanical response) of the Bodner-Partom model [6, 7]. An analytical solution is obtained for large deformation simple shear at constant shear rate. Numerical examples of simple shear, a corner test exhibiting the transition from uniaxial compression to shear, and simple tension are considered which demonstrate the stability and accuracy of the procedure. It is shown that the same procedure can be used for a rate insensitive metal characterized by a yield function as well as for a rate sensitive metal characterized by an overstress model. Finally, an appendix is provided which records the basic equations associated with the small deformation theory.     

Convergence of an operator splitting method on a bounded domain for a convection-diffusion-reaction system

We solve a convection-diffusion-sorption (reaction) system on a bounded domain with dominant convection using an operator splitting method. The model arises in contaminant transport in groundwater induced by a dual-well, or in controlled laboratory experiments. The operator splitting transforms the original problem to three subproblems: nonlinear convection, nonlinear diffusion, and a reaction problem, each with its own boundary conditions. The transport equation is solved by a Riemann solver, the diffusion one by a finite volume method, and the reaction equation by an approximation of an integral equation. This approach has proved to be very successful in solving the problem, but the convergence properties where not fully known. We show how the boundary conditions must be taken into account, and prove convergence in L1,loc of the fully discrete splitting procedure to the very weak solution of the original system based on compactness arguments via total variation estimates. Generally, this is the best convergence obtained for this type of approximation. The derivation indicates limitations of the approach, being able to consider only some types of boundary conditions. A sample numerical experiment of a problem with an analytical solution is given, showing the stated efficiency of the method.     

Second-order kinetics for EC' reactions at a spherical microelectrode

A second-order (nonlinear) model is derived for steady-statekinetics of an EC' (catalytic electrochemical) reaction at aspherical microelectrode in the case where the electron transferprocess is followed by a homogeneous chemical reaction regeneratingthe electroactive species. An asymptotic analysis of the modelis performed, and the asymptotic results are compared with thosefrom a numerical solution of the full nonlinear model. It isshown that in the fast reaction limit, where the current atthe electrode takes its maximum possible value, the concentrationsof the reactants are controlled by diffusion both close to andfar from the electrode, with significant chemical activity occurringonly in a narrow zone standing off the electrode. Also, it isshown that an equation obtained from a different asymptoticlimit may be used to predict the limiting current at the microelectrodein all circumstances. The reasons for the surprising measureof agreement at the surface of the electrode are discussed,the predictions from the model of the limiting current are compared(favourably) with experimental results, and the model is comparedwith the standard pseudo-first-order model, which, althoughalso based on a linearization of the governing equations, hasa restricted range of validity.     

Modeling particle deposition from fully developed turbulent flow

An unsteady state transfer of immersed particles within the interval between the arrival of eddies is solved by use of the Laplace transform schemes. The mean particle flux and the mean particle transport mechanisms are automatically considered on the average sublayer growth period by formulating the mean distributions as a stochastic process with the aid of exponentially distributed density function. The proposed relationship for the particle deposition velocity of average time domain obtained by this analysis is expressed as the form of analytical equation, with the inclusion of the effects of Brownian diffusion, turbulent eddy diffusivity, turbophoresis, and thermophoresis. The solution of this equation is in reasonable agreement with the measured deposition velocities for three distinct categories. This mathematical framework offers a simple computation tool of practical use to aerosol engineers and can further extend by including appropriate forces in the analytical formulation through the equilibrium among acceleration terms.     

On the use of homotopy analysis method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheets

In the present work, unsteady MHD flow of a Maxwellian fluid above an impulsively stretched sheet is studied under the assumption that boundary layer approximation is applicable. The objective is to find an analytical solution which can be used to check the performance of computational codes in cases where such an analytical solution does not exist. A convenient similarity transformation has been found to reduce the equations into a single highly nonlinear PDE. Homotopy analysis method (HAM) will be used to find an explicit analytical solution for the PDE so obtained. The effects of magnetic parameter, elasticity number, and the time elapsed are studied on the flow characteristics.     

Analytical solution for laterally loaded long piles based on Fourier–Laplace integral

Piles are frequently used to support lateral loads. Elastic solutions based on the Winkler foundation model are widely used to design laterally loaded piles at working load. This paper reports a simplified analytical solution for laterally loaded long piles in a soil with stiffness linearly increasing with depth. Based on a Fourier–Laplace integral, a power series solution for small depth and a Wentzel–Kramers–Brillouin (WKB) asymptotic solution for large depth are derived. By using this analytical solution, the deflection and bending moment profiles of a laterally loaded pile can be obtained through simple calculation. The proposed power series solution is exact for infinitely long piles. Numerical examples show that this solution agrees well with other existing methods on predicting the deflection and bending moment of laterally loaded piles. The WKB asymptotic solution developed in this study has never been introduced before. The simplified analytical solution obtained in this study provides a better approach for engineers to analyze the responses and design of laterally loaded long piles.     

A study about the solution of convection–diffusion–reaction equation with Danckwerts boundary conditions by analytical,method of lines and Crank–Nicholson techniques

We compare different solutions of the convection–diffusion–reaction problem with Danckwerts boundary conditions. Analytical solution is found, and method of lines and Crank–Nicholson method are described, applied, and compared for different values of Péclet and Damköhler numbers. The eigenvalues and eigenfunctions have been obtained for all the possible values of the dimensionless parameters. And the analytical expression of the concentration has been calculated with the optimum number of terms in the Fourier expansion.     

Modeling the reactive processes within a catalytic porous medium

A one-dimensional modelling approach to the reactive processes within a heated homogeneously premixed fuel–air mixture in its passage through a non-adiabatic catalytically reactive porous medium is described. The main focus of this contribution was comparison of the results obtained while using different modeling approaches that include mass diffusion to solid pores versus neglecting it; single step reaction versus detailed kinetic simulation; adiabatic versus non-adiabatic reactor operation; two different approaches accounting for radiation heat transfer. This model was tailored to our experimental results so as to obtain original kinetic data for corresponding global reactions for different types of catalysts and validate at the same time the predictive approaches.