Theoretical treatment of first-order reversible reactions occurring in a chromatographic reactor, on the basis of consecutive reactions

Summary Analytical peak-shape equations were derived for first-order reversible reactions occurring in a chromatographic reactor by treating the reversible reactions as consecutive reactions with alternating products. The results of the analytical peak-shape equations were compared with those from a numerical solution of the partial differential equation system modeling the chromatographic reactor. For small to medium conversions the correspondence was found to be sufficiently close to enable substitution of the numerical solution in fitting procedures for the determination of rate constants. Presented at the 21^st ISC held in Stuttgart, Germany, 15^th–20^th September, 1996     

Existence, uniqueness, and calculation of the physically acceptable solution of a particular case of the Sherman equations

When a chemical sample made of N elements is analyzed by using sequential selective excitation by monochromatic X-ray beams and selective measurement of the characteristic X-rays, the production of secondary fluorescence does not interfere with the measurements. This experimental situation leads to a particular simple case of the Sherman equations which can be written in this instance as linear equations. The linear equations thus obtained are shown to be very similar to the equations appearing in the classical models of Beattie and Brissey and of Lachance and Traill. The linear algebra proves the existence of N different sets of solutions, but the Perron Frobenius theorem ensures that there is one and only one physically feasible solution, and also leads to the method for obtaining it. This equation solution method can be extended to the equations appearing when standard samples of pure elements are also measured.The propagation of the errors in the measurements to the errors in the sample concentrations has been calculated and simulated, and the results have shown that the solution is well conditioned.     

Analysis of two-dimensional models for liquid chromatographic reactors of cylindrical geometry

This work presents the analytical solutions of two-dimensional isothermal reactive general rate models for liquid chromatographic reactors of cylindrical geometry. Both irreversible and reversible reactions are considered. The model equations form a linear system of convection-diffusion-reaction partial differential equations coupled with algebraic equations for isotherms. Analytical solutions are derived by integrated implementation of finite Hankel transform, Laplace transform, eigen-decomposition technique, and conventional ordinary differential equations solution technique. To verify the analytical results, a high-resolution finite volume scheme is also applied to numerically approximate the model equations. The current results can be very useful to optimize and upgrade the liquid chromatographic reactors.     

Numerical solution of ordinary differential equations by Fluctuationlessness theorem

This paper presents a numerical method based on Fluctuationlessness Theorem for the solution of Ordinary Differential Equations over appropriately defined Hilbert Spaces. We focus on the linear differential equations in this work. The approximated solution is written in the form of an nth degree polynomial of the independent variable. The unknown coefficients are obtained by setting up a system of linear equations which satisfy the initial or boundary conditions and the differential equation at the grid points, which are constructed as the independent variable’s matrix representation restricted to an n dimensional subspace of the Hilbert Space. An error comparison of the numerical solution and the MacLaurin series with the analytical solution is performed. The results show that the numerical solution obtained here converges to the analytical solution without using too many mesh points.     

A New Singular Matrix Method for Balancing Chemical Equations and Their Stability

In this work is given a new singular matrix method for balancing new classes of chemical equations which reduce to an n × n matrix. The method offered here is founded by virtue of the solution of a homogeneous matrix equation by using of Drazin pseudoinverse matrix. The method has been tested on many typical chemical equations and found to be very successful for the all equations in our extensive balancing research. This method works successfully without any limitations. Chemical equations treated here possess atoms with fractional oxidation numbers. Also, in the present work are analyzed some necessary and sufficient criteria for stability of chemical equations over stability of their reaction matrices.     

An Approximate Kinetic Treatment of Slow‐Initiated Living Polymerization. II. First‐Order Initiation and Half‐Order Propagation

An approximate analytical solution of the set of differential equations modeling the anionic polymerization of styrene is presented. By using this solution, a new method for calculating the initiation rate constant for this polymerization process was developed.     

On numerical solution of the integral Hartree-Fock equations for molecules by the method of MO LCAO in the momentum space

To develop a numerical solution of mentioned equations the method of factorized projection of integral operator kernel is applied. All matrix elements of the method are calculated analytically, being expressed in terms of two types of standard integrals: the overlap integrals and one-electron Coulomb integrals. To calculate the integrals we used the O(4)-symmetry of hydrogen-like atomic orbitals as well as operational technique of differentiation with respect to scalar and vector parameters.     

Dynamics of admixture consumption from a gaseous flow in reaction with a solid reactant: An analytical solution to the problem

Dynamics of admixture consumption from gaseous flow during a reaction with a solid reactant is expressed in terms of a set of two partial differential equations. The analytical solution to the problem is found. The solution is illustrated by the plots of admixture distribution in the gas and absorbent.     

A semi-empirical approach for predicting the performance of mixed matrix membranes containing selective flakes

A successful model for mixed matrix membrane performance must address the complex geometry of the problem and accurately treat the diffusion behavior of the host–guest systems being considered. Detailed calculations based on the Maxwell–Stefan equations provide a widely accepted means of treating the diffusion of gases within zeolites. However, a full numerical solution of these equations for a complex mixed matrix membrane geometry does not offer the convenience and transparency that comes with an analytical treatment. At the same time, existing analytical equations which were formulated specifically to address mixed matrix geometry do so under the assumption of very simplistic models for diffusion. Here, an approach is presented for predicting the permeability and selectivity of mixed matrix membranes containing zeolite flakes that combines well-known analytical expressions for mixed matrix membrane performance with Maxwell–Stefan modeling for zeolite diffusion. The constant permeabilities required by the analytical models are calculated by the Maxwell–Stefan equations as a function of operating conditions, and these calculated effective permeabilities are used to predict mixed matrix membrane performance at corresponding operating conditions. The method is illustrated through two case studies: normal- and iso-butane separation by a membrane containing silicalite-1 flakes and carbon dioxide/methane separation by membranes containing CHA-type zeolites. Predictions are compared to experimental results found in the literature for both cases. Also, the applicability of the Maxwell and Cussler analytical models for mixed matrix membrane performance is explored as a function of flake loading and aspect ratio.     

Mathematical Modeling of Ru(VI)-catalyzed Oxidation of Alcohols by Hexacyanoferrate(III)

The title kinetics reaction has been modeled with a system of ordinary differential equations, for the concentrations of the compounds. In these equations, the velocity constants are unknown. In this work, the four constants had been evaluated by minimizing a mean squares expression comparing the experimental measures of the concentration of hexacyanoferrate(III) with the solution of the system of ordinary differential equations. This problem has not a unique solution and there is an infinite set of constants which minimize the expression. Several sets of possible constants have been analyzed. One of them has been obtained estimating two of the constants with the stationary state approach. For the model to be well posed the constants must fulfill a condition. Information about the order of magnitude of the constants has been reached.     

An Approximate Kinetic Treatment of Slow‐Initiated Living Polymerization. I. First‐Order Initiation and Propagation

A new method for deriving the initiation rate constant for a slowinitiated living polymerization process in which all reactions are first order with respect to all participants is presented. The method is based upon an approximate analytical solution of the set of differential equations modeling this class of processes. The solution is found by asymptotic expansion of the unknown functions, using a dimensionless parameter which characterizes the process.     

Evaluation of adiabatic runaway reaction and vent sizing for emergency relief from DSC

Analytical equations related adiabatic runaway reactions to programmed scanning thermal curves from differential scanning calorimetry (DSC) were proposed. Thermal or pressure hazards can be assessed from the adiabatic trajectories expressed in the analytical equations. These industrially energetic materials include polymerizable monomers, unstable organic peroxides and nitro-compounds. Various emergency relief behaviors, such as tempered vapor, gassy, and hybrid were re-evaluated for calculating vent sizing or mass flow rates from DSC thermal curves and the related physical properties.     

Numerical simulation of natural convection of electrolyte solution with three types of ions in the electrochemical cell with vertical electrodes

The mass transfer in the electrolyte solution with three types of ions in the electrochemical cell of square section with vertical electrodes is studied. The mathematical model of the process involves the Navier-Stokes equations in the Boussinesq approximation, the equations of ionic transfer of electrolyte components, which is caused by diffusion, convection, and migration, and the condition of electroneutrality. It is shown that this problem corresponds to a special case of thermosolutal convection with regard for thermodiffusion (the Soret effect), where the cell boundaries are permeable to an impurity and the flux of impurity through the boundary is proportional to the heat flux. Using the numerical simulation, the distributions of concentration of ions, solution density, local and average mass-transfer rates are studied. The approximate analytical equations for the limiting current are obtained for typical electrochemical systems.     

Theoretical analysis of the enzyme reaction processes within the multiscale porous biocatalytic electrodes

Mathematical model describing the oxidation of glucose in a multiscale porous biocatalytic electrode is discussed. The model considers herein is composed of two nonlinear differential equations accounting for reaction and diffusion within the hydrogel film. In this letter, approximate analytical expressions for the concentration of mediator, substrate and current have been obtained using the Adomian decomposition method (ADM). Furthermore, a comparison confirmed that our analytical result fitted very well with the numerical solution (Matlab). Sensitivity analysis of the parameters is also reported.     

HOA (Heaviside Operational Ansatz) revisited: recent remarks on novel exact solution methodologies in wavefunction analysis

A viable methodology for the exact analytical solution of the multiparticle Schrodinger and Dirac equations has long been considered a holy grail of theoretical chemistry. Since a benchmark work by Torres-Vega and Frederick in the 1990s, the QPSR (Quantum Phase Space Representation) has been explored as an alternate method for solving various physical systems. Recently, the present author has developed an exact analytical symbolic solution scheme for broad classes of differential equations utilizing the HOA (Heaviside Operational Ansatz). An application of the scheme to chemical systems was initially presented in Journal of Mathematical Chemistry (Toward chemical applications of Heaviside Operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise 1/r(I) radial potential in quantum phase space. Journal of Mathematical Chemistry, 2009; 45(1):129–140). It is believed that the coupling of HOA with QPSR represents not only a fundamental breakthrough in theoretical physical chemistry, but it is promising as a basis for exact solution algorithms that would have tremendous impact on the capabilities of computational chemistry/physics. The novel methods allow the exact determination of the momentum [and configuration] space wavefunction from the QPSR wavefunction by way of a Fourier transform. In this note some remarks, examples and further directions, concerning HOA as a tool to solve and provide analytical insight into solutions of dynamical systems occurring in, but not limited to Mathematical Chemistry, are also posited.     

Wavelet algorithm for solving integral equations of molecular liquids. A test for the reference interaction site model

A new efficient method is developed for solving integral equations based on the reference interaction site model (RISM) of molecular liquids. The method proposes the expansion of site-site correlation functions into the wavelet series and further calculations of the approximating coefficients. To solve the integral equations we have applied the hybrid scheme in which the coarse part of the solution is calculated by wavelets with the use of the Newton-Raphson procedure, while the fine part is evaluated by the direct iterations. The Coifman 2 basis set is employed for the wavelet treatment of the coarse solution. This wavelet basis set provides compact and accurate approximation of site-site correlation functions so that the number of basis functions and the amplitude of the fine part of solution decrease sufficiently with respect to those obtained by the conventional scheme. The efficiency of the method is tested by calculations of SPC/E model of water. The results indicated that the total CPU time to obtain solution by the proposed procedure reduces to 20% of that required for the conventional hybrid method.     

An analytical solution for the zero frequency hyperbolic RUM modes of distortion of SiO2-tridymite

The constraint equations controlling the existence of soft, Rigid Unit Mode (RUM) modes of distortion of the ideal tridymite tetrahedral framework structure are derived and used to obtain an analytical solution for the potential soft phonon mode wave-vectors and their associated displacement patterns. One of two types of RUM known to exist are characterized by modulation wave-vectors perpendicular to 〈110〉 and atomic displacement patterns involving tetrahedral edge rotation while a second type of RUM is found to be soft on a hyperbolic reciprocal space distribution and to involve tetrahedral rotation about in-plane rotation axes. The analytical form of this hyperbolic diffuse distribution is derived as is an analytical expression for the associated RUM eigenvector of each point on the diffuse distribution.