The general theory of polarographic kinetic currents

A unified mathematical formulation of the transport problem to the dropping electrode, accounting for diffusion, chemical reactions in solution and electrode reactions, is given. The method for the reduction of the problem to a diffusion problem where the effect of chemical reactions is restricted to boundary conditions is described. Conditions are defined for appearance of a kinetic current and, on the contrary, of a diffusion current dependent on equilibrium constants of chemical reactions. The general results are exemplified by the case of a single chemical and a single electrode reaction.     

One-dimensional diffusion model in an inhomogeneous region

A one-dimensional model is developed to describe atomic diffusion in a graphite tube atomizer for electrothermal atomic adsorption spectrometry. The underlying idea of the model is the solution of an inhomogeneous one-dimensional diffusion equation, with the diffusion coefficient being a function of temperature over the entire inhomogeneous region. An analytical solution of the problem is obtained in the form of a Green’s function.     

An analytical solution for the transient response in a diffusion cell of the Wicke-Kallenbach type

An analytical solution is developed for the problem of transient diffusion and first order reaction in a solid flanked by two well mixed fluid compartments. This solution is necessary and useful in describing unsteady-state measurements in a Wicke-Kallenbach diffusion cell or reaction studies in a single pellet reactor. It is shown that a proper choice of an inner product vector space, following the methodology developed by Ramkrishna and Amundson [14, 15], leads readily to the desired solution and guarantees its completeness. This solution is valid for the equivalent heat transfer problem also.     

Inequalities constraining the value of the averaged diffusion coefficient in a heterogeneous layer

A problem describing the steady-state diffusion in a heterogeneous medium bounded by parallel planes with fixed concentrations at the boundaries is studied. An analytical solution in terms of a small-parameter power series taking into account a quadratic expansion term is obtained. It is shown that the effective diffusion coefficient in this approximation is constrained by two nontrivial inequalities. Reasons suggesting that these inequalities will also be valid for the exact solution to the problem are given.     

A SIMPLIFIED DESCRIPTION OF CHAR COMBUSTION

A simplified analysis of carbonaceous particle combustion is presented that includes the effects of pore diffusion and growth as well as gas-phase heat and mass transfer. The combustion dynamics are described by time-dependent equations for particle temperature, radius and a number of intraparticle conversion variables. These are coupled to pseudosteady equations for gas-phase transport and internal reaction and diffusion. The differential equations for gas-phase transport are reduced by quadrature to a nonlinear boundary condition to the intraparticle boundary value problem. Numerical calculations are performed for conditions relevant to pulverized coal combustion. An analytical solution of the intraparticle problem, pertinent to the regime of strong diffusional limitations, reduces the intraparticle solution into a set of two quadratures which drastically simplifies the numerical calculations. The simplified intraparticle solution is in excellent agreement with the full solution at 1800 K free stream temperature and fair agreement at 1500 K.     

Transient diffusional interactions between solid bodies and isolated fluids

A generalized Sturm-Liouville approach is used to provide an efficient solution procedure for the problem of one-dimensional diffusion accompanied by an irreversible first-order reaction in a body suddenly immersed in an isolated volume of moderately stirred fluid. Specific results are presented for the slab, solid and hollow cylinders, and the sphere. The non-reactive, non-adiabatic calorimeter problem is solved by reduction to a closely related adiabatic problem with first-order heat generation in the body, the solution to which is contained in the earlier developments of the paper.     

Automodel problem of the growth of the hydrate particle in aqueous solution of gas

The problem of the growth of hydrate particle in an aqueous solution of gas, the intensity of which is determined by the diffusion of gas to its surface, is considered. An analytical solution that enables one to determine the law of growth depending on the concentration of the hydrate-forming gas in water and hydrate, as well as the diffusion coefficient, has been obtained.     

Unsteady mass transfer around axisymmetric drops of revolution in variable diffusion coefficient liquids

Unsteady mass transfer in the continuous phase around axisymmetric drops of revolution at high Peclet numbers has been theoretically studied. The liquid is a binary system, having a variable diffusion coefficient, which depends on the solute concentration. The solution to the problem was obtained by extending the theory of Favelukis and Mudunuri, developed for a constant diffusion coefficient liquid. The procedure consists of transforming the differential mass balance, for a binary system, into a partial differential equation which has an analytical solution, and an ordinary differential equation that needs to be solved numerically. Solutions to a large number of problems can be immediately obtained with the only requirements being the shape of the drop, the tangential velocity at the surface of the drop and an expression for the variable diffusion coefficient liquid. An approximate analytical solution is also suggested which is in excellent agreement with the numerical results.     

On the Applicability of a Quasi-Stationary Solution of the Diffusion Equation for the Hydrate Layer Formed at the Gas–Ice (Water) Interface

The problem of methane hydrate formation when the process is controlled by gas diffusion in the hydrate layer formed at the gas–ice (or water) interface is solved. It is shown that an approximate quasi-stationary solution of the diffusion equation is in good agreement with its numerical solution over a wide range of the solubility of the gas in the hydrate, which is dependent on pressure. It is found that the time for the complete transition of the water (or ice) phase into the hydrate state decreases with an increase in the saturation concentration of the mobile gas in the hydrate. The kinetic equations derived based on a quasi-stationary solution of the diffusion equation, which are relationships for the intensity of hydrate formation in snow-containing (or water-containing) formations during the filtration of hydrate-forming gases, are used to describe the concentration fields of the diffusing gas and the dynamics of hydrate layer growth.     

Characterization of water penetration inside organic coatings by capacitance measurements

Knowledge of the solution of transport equations allows one to determine parameters which are needed to evaluate the protective properties of organic coatings, particularly the penetration of water inside the coatings. The form of equations representing the processes of transport and their usefulness should be verified by experimental results. The mathematical aspect of the problem is studied in the framework of the theory of partial differential equations. Various methods of solving the equations, the problem of their univocal character, their regularity and properties are considered. The form of a particular solution depends on the imposed problem. The method of determination of the depth of water penetration inside the organic coatings is based upon the solutions of the transport equation. The use of Boltzmann transformation allows the concentration profiles dependent on additional parameters (e.g., temperature) to be represented. The solutions of the diffusion equation for various limiting conditions as well as the methods of determination of diffusion coefficient are presented. The method of evaluation of the depth of medium penetration inside the protective coating in the case of a non-stationary process is described. The use of Boltzmann transformation made it easier to analyse the solution of the diffusion equation. The dependence of the water diffusion coefficient in epoxide-phenol lacquer coatings on temperature was determined and the applicability of Arrhenius' law was found in the temperature range from 303 to 363 K. Knowledge of the diffusion coefficients enabled the time of water penetration inside organic coatings to be determined.     

A Sharp Drying Front Model

The drying of materials is often described by nonlinear diffusion equations. Up to now the only way to solve these equations is by numerical simulations. Recently an analytic solution has been proposed for the drying problem. Based on this solution a sharp drying front model is presented. Measured moisture profiles during drying and the drying curve of gypsum are compared with approximate models.     

Solution of nonlinear boundary value problems—X: The false transient method

The solution of a nonlinear boundary value problem may be found by solving a pertinent transient equation until the solution ceases to change significantly (the false transient method). The method is used to get a solution of a strongly nonlinear diffusion problem as well as towards solution of equations arising in the boundary layer theory. If multiple solutions occur the false transient method is not capable of calculating all profiles. For the boundary layer problems it is difficult to construct a reliable false transient equation if it is not based on a physically “sound” transient model.     

Investigation and calculation of diffusion processes in thin fibre materials and fibre-forming polymers

A solution to the boundary-value problem for diffusion processes in thin fibre materials and fibre-forming polymers is presented. Quantitative estimations for the diffusion coefficients applicable to washing and drying of thin fibre materials and fibre-forming polymers were obtained. __________ Translated from Khimicheskie Volokna, No. 3, pp. 7–8, May–June, 2007.     

AN INTEGRAL-SPECTRAL FORMULATION FOR CONVECTIVE-DIFFUSIVE TRANSPORT IN A PACKED-BED WITH ADSORPTION AT THE WALL AND BULK REACTION

A solution for a convective-diffusive transport problem coupled with bulk reaction and adsorption at the wall in packed-bed reactor has been obtained. Such solution has been derived in terms of a particular solution that satisfies the non-homogeneous boundary conditions, at the wall, and the solution to the differential model with homogeneous boundary conditions. The latter is given in terms of integral equations whose kernel is the Green function of the parabolic differential operator of the transport problem. The Green function is expanded in terms of the eigenvalues and the eigenfunctions of the radial diffusion operator of the differential problems. This methodology allows a decoupling of the linear aspects of the problem (i.e., transport part) of the nonlinear source (i.e., kinetic of the reaction) and facilitates the convergence of the Fourier series involved in the spectral expansion in the case of a need for numerical solutions. Details about the solution methodology and the identification and solution of the eigenvalue problem, are given.     

Estimation of concentration-dependent diffusion coefficient in drying process from the space-averaged concentration versus time with experimental data

The estimation of a concentration-dependent diffusion coefficient in a drying process is known as an inverse coefficient problem. The solution is sought wherein the space-average concentration is known as function of time (mass loss monitoring). The problem is stated as the minimization of a functional and gradient-based algorithms are used to solve it. Many numerical and experimental examples that demonstrate the effectiveness of the proposed approach are presented. Thin slab drying was carried out in an isothermal drying chamber built in our laboratory. The diffusion coefficients of fructose obtained with the present method are compared with existing literature results.     

Mechanism of bubble formation in the drying of polymer films

A common problem in making thin polymer films by solution processing is the undesirable formation of bubbles during the drying process. These bubbles appear well below the boiling point of the solvent. Experience indicates, however, that the degassing of the polymer solutions reduces bubble formation, indicating a relationship with the presence of air. This work is based on a hypothesis that if the solubility of air in the polymer solution increases with solvent concentration, then the solution can become supersaturated with air as the concentration of the solvent is reduced during the drying process. To test this hypothesis the system poly(vinyl acetate)‐toluene‐nitrogen was chosen. Previously published data on the solubility and diffusion of nitrogen in the polymer‐solvent system were used. Different diffusion models based on the friction coefficients and free‐volume model were then used to correlate the diffusivity data so that the diffusion behavior of the ternary system can be predicted over a broad range of conditions. Finally, the thermodynamic and diffusivity correlations were incorporated into a multicomponent drying model which included main and cross‐diffusion terms to predict saturation behavior in the polymer solution during the drying process. The model without the cross‐diffusion terms represents the ideal system in which the diffusion of one component does not affect the diffusion of others. The drying model did not predict supersaturation of nitrogen when cross‐diffusion terms were neglected. Supersaturation of nitrogen was predicted, however, when the cross‐diffusion terms are included. Therefore, the cross‐diffusion terms in the mass transfer model are essential for the development of nitrogen supersaturation. Also different diffusion models based on the friction coefficients led to qualitatively similar predictions for the supersaturation of nitrogen. The simulation's results supported our experimental observations regarding bubble formation. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

THE LAMINAR FLOW TUBULAR REACTOR WITH HOMOGENEOUS AND HETEROGENEOUS REACTIONS I. Integral Equations for Diverse Reaction Rate Regimes

Design equations for non-isothermal Laminar Flow Tubular Reactors (LFTRs) with homogeneous and heterogeneous - at the reactor wall - reactions with arbitrary kinetic equations have been satisfactorily treated transforming the original P.D.E. problem into a system of integral equations. The kernels of the integral operators are related to an eigenvalue problem which does not depend on the kinetic equations; this avoids repetitive computational effort in the treating of different reaction kinetics.

To render a more efficient numerical treatment and according to the governing reaction rate regime, modified expressions of the general solution were obtained was follows: (i) a solution with kernels depending only on the diffusion and convective times was obtained for a low reaction regime; (ii) another solution with kernels including the reaction lime, besides the diffusion and convective ones, was necessary for a fast reaction regime and (iii) the local quasi-steady-state approximation was obtained as limiting case of solution (ii) for a instantaneous reaction regime.     

THE LAMINAR FLOW TUBULAR REACTOR WITH HOMOGENEOUS AND HETEROGENEOUS REACTIONS I. Integral Equations for Diverse Reaction Rate Regimes

Design equations for non-isothermal Laminar Flow Tubular Reactors (LFTRs) with homogeneous and heterogeneous - at the reactor wall - reactions with arbitrary kinetic equations have been satisfactorily treated transforming the original P.D.E. problem into a system of integral equations. The kernels of the integral operators are related to an eigenvalue problem which does not depend on the kinetic equations; this avoids repetitive computational effort in the treating of different reaction kinetics.

To render a more efficient numerical treatment and according to the governing reaction rate regime, modified expressions of the general solution were obtained was follows: (i) a solution with kernels depending only on the diffusion and convective times was obtained for a low reaction regime; (ii) another solution with kernels including the reaction lime, besides the diffusion and convective ones, was necessary for a fast reaction regime and (iii) the local quasi-steady-state approximation was obtained as limiting case of solution (ii) for a instantaneous reaction regime.     

A Sharp Drying Front Model

Abstract

The drying of materials is often described by nonlinear diffusion equations. Up to now the only way to solve these equations is by numerical simulations. Recently an analytic solution has been proposed for the drying problem. Based on this solution a sharp drying front model is presented. Measured moisture profiles during drying and the drying curve of gypsum are compared with approximate models.     

Geometrical analysis of dynamic problem on the membrane transport using spectral solution

The problem analyzed in this paper is a specific application of the composite membrane. General diffusion and convection formulation is presented for the dynamic problem. The spectral analysis considers convective transport of a single solute species across a one dimensional membrane system. The solution is obtained using operator theoretic methods. The geometrical structure of the spectrum of the operator is determined for the complete range of the various parameters including the distribution coefficient, the convective velocity and the diffusion coefficient. The structure of the spectrum allows a complete characterization of all the eigenvalues of the system in terms of all of these physical parameters. Calculation of the first eigenvalue for a number of cases shows its variation with the convective velocity for various medium porosities and allows a priori estimates of the profiles.