Method for the calculation of the species balance in the reaction zone

A class of processes for which the problem of species balance can be solved without using any assumptions concerning the mechanism of the chemical event of the reaction was found in the problem of species balance in the reaction zone. The mathematical statement of the problem in this case was discussed; it was shown that the solution is mainly determined by the system geometry. The method for reducing the problem to determination of the probability of species entering the chemical event zone is substantiated; this can be done by the Green function method.     

Numerical Comparison Between Tikhonov Regularization and Singular Value Decomposition Methods Using the L Curve Criterion

Many problems in chemistry, physics and engineering requires the inversion of a Fredholm integral equation of first order. To find the solution of this problem one has to deal with ill-posed problem and special techniques have to be used. In this paper two of the most common methods, the Tikhonov regularization and the singular value decomposition, are compared when finding the solution of a model integral equation. The regularization parameter in the Tikhonov regularization and the dimension of the subspaces in the singular value decomposition were chosen using the L curve criterion. The analytical solution of the model integral equation was taken as a reference to analyze the results. The advantages of each method, with the presence of errors in the data, is presented and it is argumented the superiority of the singular value decomposition when dealing with this kind of problem.     

Continuous symmetry measures: Finding the closest C2-symmetric object or closest reflection-symmetric object using unit quaternions

A closed-form solution is provided for the problem of finding the closest reflection-symmetric object, and its relation to the problems of finding the closest achiral object, the closest inversion-symmetric object, and the closest projection plane is discussed. The key to the solution is reducing this problem to the problem of finding the best (least squares) c₂ rotation between two sets of points, in addition to solving the problem of finding the closest C₂-symmetric object. The solution is derived using the quaternion representation of rotation. It is shown that calculation of the best c₂ rotation can be reduced to the diagonalization of the outer product matrix of the pair (between the two sets). ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 772–780, 1999     

An analysis of the siegert eigenvalue problem for autoionizing states

We show that the divergent integrals which appear in a direct matrix solution to the Siegert problem for autoionizing (or electron scattering) state energies and widths can be cancelled exactly. When this is done the Siegert problem becomes essentially a bound state problem. We also show that the resulting non-hermitian secular equation which requires several non-hermitian diagonalizations in the iterative solution for the complex energy can be exactly reduced by a partitioning technique to a single hermitian diagonalization (for a single open channel) with the subsequent iterative solution of a simple algebraic equation.     

Validated computing approach for high-pressure chemical and multiphase equilibrium

For the computation of chemical and phase equilibrium at constant temperature and pressure, there have been proposed a wide variety of problem formulations and numerical solution procedures, involving both direct minimization of the Gibbs energy and the solution of equivalent nonlinear equation systems. Still, with very few exceptions, these methodologies may fail to solve the chemical and phase equilibrium problem correctly. Nevertheless, there are many existing solution methods that are extremely reliable in general and fail only occasionally. To take good advantage of this wealth of available techniques, we demonstrate here an approach in which such techniques can be combined with procedures that have the power to validate results that are correct, and to identify results that are incorrect. Furthermore, in the latter case, corrective feedback can be provided until a result that can be validated as correct is found. The validation procedure is deterministic, and provides a mathematical and computational guarantee that the global minimum in the Gibbs energy has been found. To demonstrate this validated computing approach to the chemical and phase equilibrium problem, we present several examples involving reactive and nonreactive components at high pressure, using cubic equation-of-state models.     

Scattering by a hard sphere in a laser field

We consider the scattering of electrons by a hard sphere in the presence of a strong laser field. If the electrons are nonrelativistic and if the radiation field is treated as a monochromatic plane wave in the dipole approximation, the solution of our problem can be expressed in terms of well-known functions. The crux for finding this solution lies in the fact that in the potential-free region (V = 0) a space-translated solution can be written down and expressed in spherical polar coordinates. For this solution it is relatively easy to derive the matching equations on the surface of the scattering sphere of radiusR, whereV = ∞ and Ψ(R) = 0.     

Diffusion-reaction approach to electronic relaxation in solution. Exact solution for delta function sink models

We give a general method for finding the exact solution for the problem of electronic relaxation in solution, modelled by a particle undergoing diffusive motion in a potential in presence of a delta function sink. The diffusive motion is described by the Smoluchowski equation and the sink could be a delta function of arbitrary position and strength. The solution requires the knowledge of the Laplace transform of the Green’s function for the motion in the absence of the sink. We use the method to find the solution of the problem in the case where the diffusive motion is on a parabolic potential. This has been an unsolved problem for some time and is of considerable importance as a model for non-radiative electronic relaxation of a molecule in solution. The solution is analyzed to obtain the viscosity and temperature dependences of the rate constants.     

Applications of Haar basis method for solving some ill-posed inverse problems

In this paper a numerical method consists of combining Haar basis method and Tikhonov regularization method for solving some ill-posed inverse problems using noisy data is presented. By using a sensor located at a point inside the body and measuring the u(x, t) at a point x?=?a, 0?<?a?<?1, and applying Haar basis method to the inverse problem, we determine a stable numerical solution to this problem. Results show that an excellent estimation on the unknown functions of the inverse problem can be obtained within a couple of minutes CPU time at pentium IV-2.4?GHz PC.     

An analysis of nonlinear ion transport problem including arbitrary valences of oxidized and reduced species

The nonlocal identification problem related to nonlinear ion transport model including diffusion and migration is studied. Ion transport is assumed to be superposition of diffusion and migration under the influence of an electric field. Mathematical modeling of the experiment leads to an identification problem for a strongly nonlinear parabolic equation with nonlocal additional condition. It is shown that the nonlocal identification problem can be reduced to the initial-boundary value problem for nonlinear parabolic equation. Iteration method for numerical solution of this problem is proposed. Numerical results and their interpretation are presented for wide class of materials, including various values of valences and diffusivities of oxidized and reduced species.     

The Rate Problem of a Thermally Regenerable Ion-Exchange System

Earlier equilibrium studies have established the thermal dependence of the equilibrium between salt solution and a mixed bed of weakly basic and weakly acidic ion-exchange resins. High resin utilization can be achieved if the resin properties and equilibrium conditions are optimized; the equilibrium characteristics of polyacrylic acid and polyvinylbenzyldiethylamine resins are quite suited for the practical desalination of brackish waters.

However, the adsorption rates exhibited by normal-sized resin beads of this type are much too slow for satisfactory operation of the process because of the low concentration of protons available for transfer between the resins. It is shown that increasing the porosity of the resins improves amine resin kinetics 10-fold and carboxylic acid resin kinetics 6-fold. Nevertheless such improvements are still inadequate for practical purposes, and it is concluded that for satisfactory rates to be achieved systems having much shorter diffusion paths are necessary.

Two further approaches to the rate problem are discussed, both involving the synthesis of novel resin systems. A mixed bed of microbeads (10-20 p) reacts at acceptable rates but presents mechanical problems; the magnetic flocculation of finely divided magnetic resins is reported as one possible solution to this problem. Another avenue is the synthesis of normal-sized beads of the amphoteric and snake-cage variety. Resins of this type that exchange at suitable rates are described.     

On the Dependence of the Light Scattering Intensity on the Averaged Size of Polydisperse Particles: Comments on the Paper by M.S. Dyuzheva et al. (Colloid J., 2002, vol. 64, no. 1, p. 39)

The second stage in the solution of the inverse problem of dynamic light scattering is analyzed; the dependence of the static scattering intensity on the average fraction size is used to calculate the particle size distribution at this stage. The results of the paper by M.S. Dyuzheva et al. (Colloid J., 2002, vol. 64, no. 1, p. 39) concerning this problem are shown to be erroneous. On the basis of calculations within the framework of the Mie theory, the possibility of a power approximation for the particle size dependence of the scattered intensity and the effect of the particle parameters on the power approximation exponents are discussed.     

Effect of large deformation and material nonlinearity on the JKR (Johnson–Kendall–Roberts) test of soft elastic materials

We study in detail the effect of large deformation and material nonlinearity on the JKR (Johnson–Kendall–Roberts) theory of adhesive contact for two systems. The first is a Neo‐Hookean hemisphere in adhesive contact with a smooth rigid substrate. The second is a smooth rigid spherical indenter in adhesive contact with a Neo‐Hookean half space. We show that our results are special cases of a general theory that models large deformation adhesive contact of spherical lenses. This theory shows that the solution of any large deformation JKR (LDJKR) problem can be obtained from the solution of a corresponding large deformation Hertz (LDH) problem. Using this theory, we extend the small strain JKR theory to the large deformation regime, the only restriction being that the materials are nonlinear elastic or hyperelastic. The adhesive contact problem for the two systems is solved using two methods. In method one, the LDJKR theory is obtained using finite element simulation results for a corresponding LDH problem; in method two, we solve the adhesive contact problems directly using a cohesive zone model to quantify adhesive interaction. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2912–2922, 2006     

Wavelet formulation of the polarizable continuum model

The first implementation of a wavelet discretization of the Integral Equation Formalism (IEF) for the Polarizable Continuum Model (PCM) is presented here. The method is based on the application of a general purpose wavelet solver on the cavity boundary to solve the integral equations of the IEF‐PCM problem. Wavelet methods provide attractive properties for the solution of the electrostatic problem at the cavity boundary: the system matrix is highly sparse and iterative solution schemes can be applied efficiently; the accuracy of the solver can be increased systematically and arbitrarily; for a given system, discretization error accuracy is achieved at a computational expense that scales linearly with the number of unknowns. The scaling of the computational time with the number of atoms N is formally quadratic but a N^1.5 scaling has been observed in practice. The current bottleneck is the evaluation of the potential integrals at the cavity boundary which scales linearly with the system size. To reduce this overhead, interpolation of the potential integrals on the cavity surface has been successfully used. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

On the Pace–Datyner theory of diffusion of small molecules through polymers

The Pace–Datyner theory for diffusion of penetrant molecules in polymers has been analyzed. It has been found that the correct solution of the problem they pose is possible only at 0 K, since then the separation of two chains at x = ∞ is equal to the minimum of the DiBenedetto potential for their interaction. Otherwise the energy of symmetrical separation is infinite. By using the linearization method to solve the differential equation, Pace and Datyner neglected the problem of unnatural boundary conditions at x = ∞ for temperatures above 0 K. The exact numerical solutions of differential equations at temperature 0 K were therefore compared with the results of the Pace–Datyner linear approximation. For temperatures different from 0 K the solution of the problem is possible only when the proper cutoff is imposed. The analytical expression for the coeffients in the DiBenedetto potential has been found, and the potential can be written as     

Double potential step chronoamperometry at spherical electrodes and microelectrodes

A general and explicit analytical easy solution for double potential step chronoamperometry applicable to microelectrodes is presented. This solution considers that both electroactive species are initially present.The solution indicates that the smaller the electrode radius, the more notorious the influence of the diffusion coefficient ratio, so the use of microelectrodes is of great utility for determining the diffusion coefficients of the electroactive couple. The analytical solutions given until now can only be considered as partial solutions for this problem, since they lead to absurd results for electrode radii <10⁻² cm.This equation has also been applied to the study of amalgamation processes and it could also be applicable to the study of the ionic transfer at hanging and microspherical electrolyte drop electrodes (i.e., liquid–liquid interfaces). In these last cases, the use of this equation is limited by the validity of Koutecký’s approximation, which neglects the electrode finite volume.     

The use of an algorithmic method for small molecule superimpositions in the design of antiviral agents

Summary The inability to reliably predict relative orientations of drug molecules within our series of antipicornavirus agents has severely limited the usefulness of available structure-activity data in the drug design process. A reported method of overlapping molecules has been evaluated to see if it could provide a solution to this problem. Although it initially succeeded remarkably well with a series of molecules whose bound X-ray structures were known, this success was shown to be only a function of the bound conformation of these molecules. Thus, this method did not provide a general solution to the problem at hand.     

Adaptative finite element simulation of currents at microelectrodes to a guaranteed accuracy. Application to a simple model problem

In this series of papers we consider the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the ‘edge-effect’. Our approach to overcoming this problem involves the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm. This allows us to calculate the current to within a prescribed tolerance. We begin by demonstrating the power of the method for a simple model problem — an E reaction mechanism at a microdisc electrode — for which the analytical solution is known. In this paper we give the background to the problem, and show how an a posteriori error bound can be used to drive an adaptive mesh-generation algorithm. We then use the algorithm to solve our model problem and obtain very accurate results on comparatively coarse meshes in minimal computing time. We give the technical details of the background theory and the derivation of the error bound in the accompanying paper.     

Transient analysis of electrolytically initiated polymerization

Electrochemically initiated polymerization at a planar electrode is treated theoretically and an approximate solution to the full time-dependent problem is found assuming the electrode process to be reversible (so that the Nernst relation holds). The homogeneous kinetics lead to a second-order coupled non-linear initial-boundary value problem, and the method developed for its solution is general and applicable to other problems. The approximate solution is converted to an equation for the transient current-versus-time curve, for electrode initiated polymerization, which can be used for the experimental determination of the polymerization and termination rate constants. An analysis procedure for such determination is outlined.