Solution of one problem of nonlinear adsorption kinetics by the method of lines

The method of lines is constructed and proved for numerical solution of a nonlinear initial-boundary-value problem of parabolic type describing the adsorption of a substance from an aqueous solution of bounded volume by a spherical adsorbent. The method is developed under natural assumptions on the smoothness of the solution of the original problem. The rate of convergence of the method depends on the smoothness of the initial function and is of order O(h) if v₀(x) 0, O(h^1/2) if v₀(x) C¹[0, 1], and 0(|v ₀(x)|W ₂ ¹ (O,h)).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 24–30, 1987.     

Finite-difference solution of a nonlinear initial-boundary-value problem for a nonlinear parabolic equation of second order

An implicit finite-difference scheme is constructed for solving a nonlinear initial-boundary-value problem for a nonlinear homogeneous parabolic equation of second order with a nonlinear boundary condition that contains the time derivative of the sought function. The results are used for numerical solution of the mathematical model of internal-diffusion kinetics of adsorption from a constant bounded volume.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 34–46, 1988.     

Asymptotics for models of non‐stationary diffusion in domains with a surface distribution of obstacles

We consider a time‐dependent model for the diffusion of a substance through an incompressible fluid in a perforated domain Ω_?, with n = 3,4. The fluid flows in a domain containing a periodical set of “obstacles” (Ω\Ω_?) placed along an inner (n ? 1)‐dimensional manifold . The size of the obstacles is much smaller than the size of the characteristic period ?. An advection term appears in the partial differential equation linking the fluid velocity with the concentration, while we assume a nonlinear adsorption law on the boundary of the obstacles. This law involves a monotone nonlinear function σ of the concentration and a large adsorption parameter. The “critical adsorption parameter” depends on the size of the obstacles , and, for different sizes, we derive the time‐dependent homogenized models. These models contain a “strange term” in the transmission conditions on Σ, which is a nonlinear function and inherits the properties of σ. The case in which the fluid velocity and the concentration do not interact is also considered for n ≥ 3.     

Uniqueness of a solution to the stationary problem of kinetics of growth of thin film of vapor phase on solid bases

A mathematical model of many-layer adsorption is considered. Sufficient conditions for the uniqueness of a solution of the class of periodic functions are found. The solvability and the uniqueness theorems for the Dirichlet problem are established for sufficiently small values of a parameter. Bibliography: 1 title. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 193–203.     

Specific surface area of mercuric sulphide

The specific surface area of mercuric sulphide (a) has been determined by three methods: (a) gas adsorption method (N₂ adsorption), (b) air permeability method and (c) microscopic method. The area values are compared. In the gas adsorption method, the effect of degassing temperature on the surface area of solid was studied. In the permeability method the porosity range over which the specific surface area remains constant was also determined.     

A regularizing Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem in chromatography

Competitive adsorption isotherms must be estimated in order to simulate and optimize modern continuous modes of chromatography in situations where experimental trial-and-error approaches are too complex and expensive. The inverse method is a numeric approach for the fast estimation of adsorption isotherms directly from overloaded elution profiles. However, this identification process is usually ill-posed. Moreover, traditional model-based inverse methods are restricted by the need to choose an appropriate adsorption isotherm model prior to estimate, which might be very hard for complicated adsorption behavior. In this study, we develop a Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem. The solvability and convergence for the proposed inverse method are studied. In particular, using a problem-adapted adjoint, we obtain a convergence rate under substantially weaker and more realistic conditions than are required by the general theory. Based on the adjoint technique, a numerical algorithm for solving the proposed optimization problem is developed. Numerical tests for both synthetic and real-world problems are given to show the efficiency of the proposed regularization method.     

Effect of electrophysical activation of components on adhesive interaction in polymer composites

The key problem in designing polymeric composites is ensuring a sufficiently strong adhesive bond between the polymer and the filler (or the reinforcement). One method for strengthening the intermolecular interaction at the phase interface is electrophysical activation of the particle surface. We have investigated the absorption activity and adhesiveness of powdered inorganic fillers and polymeric binders with triboelectric activation and activation in a corona discharge. We consider a broad class of disperse materials. We show that electrophysical activation of powdered materials causes changes in the concentration of donor and acceptor centers on the particle surface and in the adsorption activity of the materials as a whole. The presented results can be used for goal-directed selection of the most effective method of activation for disperse materials; they also are evidence for the important role of adsorption forces in the adhesion phenomenon.Report presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6, pp. 734–741, November–December, 1995.     

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.     

Über den Einfluss der Oberflächeneigenschaften von Halbleitern auf ihre Eiskeimfähigkeit

The factors controlling the ice nucleation efficiency are studied in this paper on suitably chosen model substances, in particular on silicon. It is found that the critical supersaturation for the growth of ice on a (111)-plane of GaAs is 25% at –25°C; this quantity as well as the number of growing ice crystals (cm^–2) were found to be independent of the dislocation density. The critical supersaturation for the growth of ice on the (111)-plane of Si is strongly dependent on the electrical conductivity of the crystals, but independent of the sign of the majority charge carriers. Epitaxial growth is observed on the hexagonal substrate GaSe only, but not on the cubic GaAs and Si.On the basis of the classical nucleation theory the free energy of adsorption of H₂O on GaSe G _ads =0,48 eV and the specific interfacial free energy _SL =23,2 erg · cm^–2 are evaluated. This indicates that the outermost layer of the ice embryo is in a liquid-like state.Water adsorption isotherms were measured gravimetrically down to –20°C and were found to be of type II (BET, [33]). The amount of adsorbed water and the isosteric heat of adsorption at a given relative pressure depend on the doping of the sample.From this it is concluded that low conductivity material had more adsorption sites than high conductivity material, but these fewer sites on the high conductivity samples were more active in collecting water molecules. The larger water clusters are formed on high conductivity samples in agreement with the higher nucleation efficiency observed in mixing cloud chamber experiments.The electrical conductivity and the sign of the thermoelectric effect of a thin silicon on sapphire film have been measured as a function of oxygen and water vapor pressures. It is found that band bending towards ann-type surface occurred during water chemisorption on Si.This indicates that chemisorbed water molecules act as donor surface states. The charge exchanged between adsorbate and adsorbent is larger on high conductivity samples on account of their higher initial surface potential.Chemisorption sites on Si are proportional to the doping concentration and they produce relatively large water clusters. Hence doping results in a higher nucleation efficiency.     

Pore morphological identification of hydrotalcite from nitrogen adsorption

The characteristic energy of nitrogen adsorption in a meso-porous hydrotalcite at 77 K demonstrates the fractality over wider range of pore sizes including meso and macro-pores. More irregular, wider pore-size distribution exhibits higher adsorption potential than that with more regular, narrower pore-size distribution. The changes in differential molar entropy during the adsorption indicate that an irreversibly chaotic process associated with transition from a disordered gaseous state to a more ordered adsorbed state occurs. The differential entropy of adsorption decreases at increasing fractality. The increase in ‘pure’ isosteric enthalpy of adsorption with increasing fractality indicates that the energy is closely related to the pore-size distribution.     

Modelling of displacement washing of pulp fibers using orthogonal collocation on finite elements

Flow of fluid through packed bed of porous particles is modelled with the help of Peclet number (Pe) and Biot number (Bi). Packed bed is divided into three zones, flowing liquor, intrapore solute present in pores of particles and solute adsorbed on particle surface. Langmuir isotherm is used to describe the relationship between intrapore solute concentration and concentration of solute adsorbed on particle surface, whereas the bulk fluid concentration and the intrapore solute concentration are related by linear adsorption isotherm. Model predicted values are also compared with the experimental values. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Matrix transformations of summability domains of generalized matrix methods in Banach spaces

The necessary and sufficient conditions for a matrix M to be a transform from the summability domain of generalized matrix method A into the summability domain of another generalized matrix method B are proved. The elements of Mare continuous linear operators from a Banach space X into another Banach space Y, and the elements of A and B are continuous linear operators from X into X and from Y into Y, respectively. As a special case there are considered the case when A is the generalized Riesz method.     

A Method of Deducing L-Polyhedra for n-Lattices

We suggest a method for selecting an L-simplex in an L-polyhedron of an n-lattice in Euclidean space. By taking into account the specific form of the condition that a simplex in the lattice is an L-simplex and by considering a simplex selected from an L-polyhedron, we present a new method for describing all types of L-polyhedra in lattices of given dimension n. We apply the method to deduce all types of L-polyhedra in n-dimensional lattices for n=2,3,4, which are already known from previous results.     

Crossing numbers of sequences of graphs II: Planar tiles

We describe a method of creating an infinite family of crossing‐critical graphs from a single small planar map, the tile, by gluing together many copies of the tile together in a circular fashion. This method yields all known infinite families of k‐crossing‐critical graphs. Furthermore, the method yields new infinite families, which extend from (4,6) to (3.5,6) the interval of rationals r for which there is, for some k, an infinite sequence of k‐crossing‐critical graphs all having average degree r. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 332–341, 2003     

Contaminant Transport with Adsorption in Dual-Well Flow

Numerical approximation schemes are discussed for the solution of contaminant transport with adsorption in dual-well flow. The method is based on time stepping and operator splitting for the transport with adsorption and diffusion. The nonlinear transport is solved by Godunov's method. The nonlinear diffusion is solved by a finite volume method and by Newton's type of linearization. The efficiency of the method is discussed.     

Inverse Obstacle Problem for the Non-Stationary Wave Equation with an Unknown Background

We consider boundary measurements for the wave equation on a bounded domain M ? ?² or on a compact Riemannian surface, and introduce a method to locate a discontinuity in the wave speed. Assuming that the wave speed consist of an inclusion in a known smooth background, the method can determine the distance from any boundary point to the inclusion. In the case of a known constant background wave speed, the method reconstructs a set contained in the convex hull of the inclusion and containing the inclusion. Even if the background wave speed is unknown, the method can reconstruct the distance from each boundary point to the inclusion assuming that the Riemannian metric tensor determined by the wave speed gives simple geometry in M. The method is based on reconstruction of volumes of domains of influence by solving a sequence of linear equations. For τ ∈C(?M) the domain of influence M(τ) is the set of those points on the manifold from which the distance to some boundary point x is less than τ(x).     

An Algorithm for a Letter-Based Representation of All-Pairwise Comparisons

All-pairwise comparisons among a set of t treatments or groups are one of the most frequent tasks in applied statistics. Users of statistical software are accustomed to the familiar lines display, in which treatments that do not differ significantly, are connected by a common line or letter. Availability of the lines display is restricted mainly to the balanced analysis of variance setup. This limited availability is at stark variance with the diversity of statistical methods and models, which call for multiple comparisons. This article describes a general method for graphically representing any set of t(t?1)/2 all-pairwise significance statements (p values) for t treatments by a familiar letter display, which is applicable regardless of the underlying data structure or the statistical method used for comparisons. The method reproduces the familiar lines display in case of the balanced analysis of variance. Its broad applicability is demonstrated using data from an international multienvironment wheat yield trial and from a fish catching survey.     

Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systems

A variant of balancing domain decomposition method by constraints (BDDC) is proposed for solving a class of indefinite systems of linear equations of the form (K?σ²M)u=f, which arise from solving eigenvalue problems when an inverse shifted method is used and also from the finite element discretization of Helmholtz equations. Here, both K and M are symmetric positive definite. The proposed BDDC method is closely related to the previous dual–primal finite element tearing and interconnecting method (FETI‐DP) for solving this type of problems (Appl. Numer. Math. 2005; 54 :150–166), where a coarse level problem containing certain free‐space solutions of the inherent homogeneous partial differential equation is used in the algorithm to accelerate the convergence. Under the condition that the diameters of the subdomains are small enough, the convergence rate of the proposed algorithm is established, which depends polylogarithmically on the dimension of the individual subdomain problems and which improves with a decrease of the subdomain diameters. These results are supported by numerical experiments of solving a two‐dimensional problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical study of the adsorption of dyes from textile effluents

The textile industry is one of the greatest generators of liquid effluents, due to the high quantities of water used in the dyeing process. Fixed bed adsorption columns using activated carbon have been widely used in industrial processes for the removal of contaminants from textile effluents. In this study we present the modelling of an adsorption process applied to textile dyes, using fixed bed columns. This model permits the prediction of the dye concentration at the adsorption column outlet, considering the influence of various operational parameters. The adsorption isotherms of the dye Basic Green 4, over granular activated carbon at 25 °C in an aqueous solution, were determined experimentally through a comprehensive series of tests. The Langmuir and Radke–Prausnitz models gave the best results for the adsorption isotherms. Tests were carried out in fixed bed columns to determine the breakthrough curves, with variations in feed rate, feed concentration, diameter of the column and mass transfer coefficient. The experimental conditions were simulated using a transient mathematical model. The data obtained numerically and experimentally were compared to validate the mathematical model proposed.     

A Variant of the Inverted Lanczos Method

In this note we study a variant of the inverted Lanczos method which computes eigenvalue approximates of a symmetric matrix A as Ritz values of A from a Krylov space of A ^–1. The method turns out to be slightly faster than the Lanczos method at least as long as reorthogonalization is not required. The method is applied to the problem of determining the smallest eigenvalue of a symmetric Toeplitz matrix. It is accelerated taking advantage of symmetry properties of the correspond ng eigenvector.This revised version was published online in October 2005 with corrections to the Cover Date.