Mathematical modeling of spatial-temporal structures in a heterogeneous catalytic system

The article focuses on mathematical modeling of the spatial-temporal structures that appear in a heterogeneous catalytic reaction on the surface of a catalyst. A system of consistent mathematical models has been developed for a three-component reaction, describing self-organization phenomena on macro, meso, and micro levels. Qualitative analysis of the solutions of the ODE system (macro level) produces the existence regions of spatial-temporal structures of various types in distributed meso- and micro-level models. A point model is applied to predict the shape of traveling impulses and fronts; the switching direction in a bistable medium is determined analytically. Solutions constructed and investigated for a PDE multicomponent reaction-diffusion system describe trigger waves, single traveling impulses, phase and spiral waves. Spatial-temporal structures on the atomic level are investigated by the Monte Carlo method. Direct and inverse trigger waves are implemented, as well as single traveling impulses and spiral waves. The effect of internal fluctuations in the reaction system is investigated.     

Dynamics of a system of nonlinear differential equations

This is a qualitative analysis of a system of two nonlinear ordinary differential equations which arises in modeling the self-oscillations of the rate of heterogeneous catalytic reaction. The kinetic model under study accounts for the influence of the reaction environment on the catalyst; namely, we consider the reaction rate constant to be an exponential function of the surface concentration of oxygen with an exponent μ. We study the necessary and sufficient conditions for the existence of periodic solutions of differential equations as depending on μ. We formulate some sufficient conditions for all trajectories to converge to a steady state and study global behavior of the stable manifolds of singular saddle points.     

A New Model of Thermal Desorption of Nitrogen from a Nonhomogeneous Catalyst Surface

A new mathematical model is proposed for temperature recombination of atomic nitrogen on the (111) face of the iridium single crystal and on iridium foil, allowing for geometrical nonhomogeneity of the catalyst surface. The model describes the main qualitative features of laboratory experiments: the effect of the catalyst surface structure and the adsorbed oxygen on nitrogen thermal desorption spectra. Conditions are determined when an additional low-temperature peak appears in the N₂ thermal desorption spectrum. The results are compared with calculations using traditional models of associative thermal desorption from homogeneous surfaces, including models that allow for lateral interactions in the adsorption layer. The numerical analysis is carried out using deterministic and stochastic approaches.     

Synthesis and micro-structural study of one-dimensional nano-materials

Silicon nano-wires (SiNWs) and boron nitride nano-tubules (BN-NTs) were successfully synthesized by excimer laser ablation at high temperature. These one-dimensional nano-materials synthesized by this method have a very high yield, a uniform diameter distribution, and a high purity. Micro-structures of these nano-materials were investigated by transmission electron microscopy (TEM). The SiNWs have a high density of structural defects of microtwin, stacking faults, and low-angle boundary, which are closely related to the formation of SiNWs and the determination of morphology of the nano-wires. BN-NTs are mainly single atomic-layered and the outer surface of tubules is clean without any attachment. The formation of single atomic-layered tubule is attributed to the catalyst effect which makes the axial rate of BN-NTs much higher than the radial growth     

Existence of weak positive solutions to a nonlinear PDE system around a triple phase boundary, coupling domain and boundary variables

We consider a 2D nonlinear system of PDEs representing a simplified model of processes near a triple-phase boundary (TPB) in cathode catalyst layer of hydrogen fuel cells. The particularity of this system is the coupling of a variable satisfying a PDE in the interior of the domain with another variable satisfying a differential equation (DE) defined only on the boundary, through an adsorption-desorption equilibrium mechanism. The system includes also an isolated singular boundary condition which models the flux continuity at the contact of the TPB with a subdomain. By freezing certain terms we transform the nonlinear PDE system to an equation, which has a variational formulation. We prove several L^∞ and W1,p a priori estimates and then by using Schauder fixed point theorem we prove the existence of a weak positive bounded solution.     

On the growth of a polymer layer around a catalytic particle: a free boundary problem

The problem studied is a substantial part of a larger mathematical model describing the heterogeneous Ziegler-Natta polymerization process of a gaseous monomer. Here we concentrate on the growth of a polymer layer around a spherical catalytic particle (the whole process taking place in an agglomerate of such particles). The velocity field of the growing layer is determined the absorption rate of the monomer at the catalyst surface. The monomer diffuses through the expanding layer. Its concentration at the outer surface (the free boundary) is supposed to be a known function of time. Global in time existence and uniqueness of a regular solution are proved. Received May 12, 1996     

Chemical waves and localized chaos in a model of heterogeneous catalytic reaction in a porous particles

A mathematical model of an oscillatory chemical reaction in a porous catalyst particle is considered. The model describes an oscillatory medium uniformly distributed throughout the volume of a spherical particle. The dynamical interaction of the reaction with the diffusive flow of the gaseous reagent inside the pores generates nonstationary dissipative structures in the oscillatory medium on the surface of the catalyst. Depending on the pressure in the gaseous phase, the model produces specific chemical waves and localized spatio-temporal chaos. The study was partially supported by the Russian Foundation for Basic Research (grant No. 96-03-34427a). Translated from Chislennye Metody i Vychislitel'nyi Eksperiment, Moscow State University, pp. 31–43, 1998.     

On the stagnation-point flow towards a stretching sheet with homogeneous–heterogeneous reactions effects

The effects of homogeneous–heterogeneous reactions on the steady boundary layer flow near the stagnation point on a stretching surface is studied. The possible steady-states of this system are analyzed in the case when the diffusion coefficients of both reactant and auto catalyst are equal. The strength of this effect is represented by the dimensionless parameter K and K_s. It is shown that for a fluid of small kinematic viscosity, a boundary layer is formed when the stretching velocity is less than the free stream velocity and an inverted boundary layer is formed when the stretching velocity exceeds the free stream velocity. The uniqueness of this problem lies on the fact that the solutions are possible for all values of λ > 0 (stretching surface), while for λ < 0 (shrinking surface), solutions are possible only for its limited range.     

Modeling the Oscillations of the CO + O2 Reaction Rate in a Granular Catalyst Layer

A new point model is proposed for the oxidation reaction of CO on the surface of a Pd cluster. The model reflects the oxidation–reduction mechanism on the palladium surface and incorporates all the stages of the kinetic scheme from the previous three-component model. A new assumption based on a series of experimental facts claims that the adsorbed oxygen atoms may diffuse into deeper-lying layers of the crystal lattice and thus influence the processes in the adsorption layer due to the micro size of the palladium clusters. The new point system has been built into the distributed general model for a granular catalyst developed previously. As a result, for parameter values corresponding to experimental conditions we have managed for the first time to obtain a wide region of chaos and complex mixed modes, close to those observed in real experiments.     

Simulation of fluid catalytic cracking operation

Since its introduction in 1942 the fluid catalytic cracking (FCC) has been the most important and widely used process for the production of gasoline from heavy distillates. In most refineries the capacity of the FCC unit is second only to that of the crude distillation unit. Often an FCC unit is referred to as the heart of a modern refinery oriented toward maximum production of gasoline.The basic step in the FCC process is the recirculation of the catalyst through the reactor, stripping and regenerator. In the reactor system the hydrocarbon feed is heated and cracked. Coke (or carbon) may be produced and may deposit on the catalyst reducing its activity and selectivity. When the catalyst is circulated to the regenerator carbon is burned off causing the heating of the catalyst before its return to the reactor part. The products from the reactor are separated in a main fractionator into gas and liquid streams normally including a recycle feed to the reactor.The operation of an FCC unit requires the manipulation of a large number of controlled variables affecting its performance. Major process variables such as reactor temperature, catalyst circulation rate, catalyst inventory and recycle feed rate can be varied to influence the product yields and to accomodate widely different feedstocks. Unpredictable variation can occur in feed stock, catalyst quality and equipment performance. Most normal variation can be accomodated by a small change in operating conditions.For a new plant, comparison of actual versus predicted performance provides a valuable check on the validity of the design correlations and a guide for future laboratory and engineering research.The objectives of the present work are to simplify the complicated FCC process variables and to develop a computer model to simulate the operation of an FCC at different conditions. This includes the prediction of the effects of the operating variables on the reactor product yields. These products include fuel gas, C₃, C₄ gasoline, light gas oil and coke. The model provides a good base for troubleshooting and debottlenecking and may be useful in optimal control of the FCC.     

Optimization of bifunctional catalysts in tubular reactors

A number of chemical reactions of industrial importance, especially reforming reactions, are most effectively promoted by bifunctional catalysts in tubular reactors. For the determination of the optimal catalyst blend, a mathematical model for a general first-order kinetic reaction scheme is developed. The introduction of state-space definitions leads to a model described by a bilinear system. Based on that system and a suitably defined performance index, necessary conditions for the optimal catalyst blend are derived by means of the maximum principle. Special attention is directed to the singular solution of the optimization problem.This paper was presented at the 14th Joint Automatic Control Conference, Columbus, Ohio, 1973.     

On stability in terms of two measures of a hybrid system having partly invisible solutions

We consider three different dynamic systems. The first runs “smoothly” during a certain finite time interval, undergoes an abrupt change in the dynamics during the next (finite) time interval and is governed by the second system. The solution of this second system lies on a surface for a finite amount of time and becomes invisible. At the beginning of the third phase, the system is subjected to an impulse which causes the solution to leave the surface and we have the new hybrid impulsive system. In this paper, we employ two measures to find suitable conditions so that the new system again runs as smoothly as the first.     

Instability Paths in the Kirchhoff–Plateau Problem

The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a soap film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. Adopting a variational approach, we define an energy associated with shape deformations of the system and then derive general equilibrium and (linear) stability conditions by considering the first and second variations of the energy functional. We analyze in detail the transition to instability of flat circular configurations, which are ground states for the system in the absence of surface tension, when the latter is progressively increased. Such a theoretical study is particularly useful here, since the many different perturbations that can lead to instability make it challenging to perform an exhaustive experimental investigation. We generalize previous results, since we allow the filament to possess a curved intrinsic shape and also to display anisotropic flexural properties (as happens when the cross section of the filament is noncircular). This is accomplished by using a rod energy which is familiar from the modeling of DNA filaments. We find that the presence of intrinsic curvature is necessary to obtain a first buckling mode which is not purely tangent to the spanning surface. We also elucidate the role of twisting buckling modes, which become relevant in the presence of flexural anisotropy.     

Optimal System of Loops on an Orientable Surface

Every compact orientable boundaryless surface M can be cut along simple loops with a common point v₀, pairwise disjoint except at v₀, so that the resulting surface is a topological disk; such a set of loops is called a {\it system of loops} for M. The resulting disk may be viewed as a polygon in which the sides are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is a combinatorial surface, and that each edge has a given length, we are interested in a shortest (or optimal) system of loops homotopic to a given one, drawn on the vertex-edge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its homotopy class. We give an algorithm to build such a system, which has polynomial running time if the lengths of the edges are uniform. As a byproduct, we get an algorithm with the same running time to compute a shortest simple loop homotopic to a given simple loop.     

The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces

We introduce a way to represent pairs (E,?), where E is a bundle on a Riemann surface and ? is a logarithmic connection in E, based on a representation of the surface as the quotient of the exterior of the unit disc. In this representation, we write the local isomonodromic deformation conditions for the pairs (E,?). These conditions are written as a modified Schlesinger system on a Riemann sphere (reduced to the ordinary Schlesinger system in the typical case) supplemented by a certain system of linear equations.     

Star Product for Second-Class Constraint Systems from a BRST Theory

We propose an explicit construction of the deformation quantization of a general second-class constraint system that is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class constraint system and can also be understood as a far-reaching generalization of the Fedosov quantization. The effective gauge system is quantized by the BFV–BRST procedure. The star product for the Dirac bracket is explicitly constructed as the quantum multiplication of BRST observables. We introduce and explicitly construct a Dirac bracket counterpart of the symplectic connection, called the Dirac connection. We identify a particular star product associated with the Dirac connection for which the constraints are in the center of the respective star-commutator algebra. It is shown that when reduced to the constraint surface, this star product is a Fedosov star product on the constraint surface considered as a symplectic manifold.     

A synopsis of elliptic PDE models for grid generation

This paper is devoted to an analytical comparison of the various elliptic partial-differential-equation (PDE) models which are in current use for grid generation. These comparisons, particularly between the equations from the Laplace-Poisson system and the equations from a Gaussian approach, have yielded useful expressions connecting the 3D Laplacians and the surface Beltramians. This effort specifically been successful when the transverse coordinate leaving the surface is orthogonal to the surface. Equations which are derivable from Cartesian-type Poisson and those obtained by using the variational priciple in surface coordinates have also been considered.     

A Stochastic Model for Assessing Synchronization among Spike Trains in a Population of Motor Neurons

Synchronization among discharges in a population of motor neurons is of interest because of its potential to characterize physiological changes related to the neuromuscular system. Milner-Brown et al. (1973) developed a method to quantify synchronization in a population of motor neurons, in which synchronization is measured by averaging the spike-triggered surface electromyograms (EMG) waveforms. The surface EMG method opened a way to assess motor neuron synchrony in a large population of motor neurons instead of only a few, allowed investigators to track the same or similar groups of motor neurons longitudinally, and overcame the limit of examining only a few motor neurons using cross correlation. However, experimental results have suggested that the surface EMG method does not accurately and consistently detect motor neuron synchrony under some experimental conditions (Yue et al., 1995). This article reports our attempts to improve this method by establishing a new mathematical framework for the surface EMG procedure and to propose a general model based on this framework. The proposed model includes existing methods such as that of Hamm et al. (1985) as special cases. Based on the proposed model, we proposed a new synchronization index and performed computer simulation that indicated that the new index detects synchronization consistently with relatively high accuracy. Though based on the neuromuscular system, the proposed model should be extendable for detecting synchronization in other nervous systems.     

Elastic-plastic halfspace simulation

Due to the roughness of technical surfaces only the surface peaks are in contact for moderate contact pressures. Thus, the real contact area is smaller than the apparent contact area. Contact forces can only occur in the real contact area. Consequently it is necessary to determine the deformation of surface asperities in order to analyse the tribological properties of surfaces. The real contact area is usually small in initial contact. This leads to large contact pressures which in turn lead to the plastic deformation of surface roughness peaks. Therefore an elastic-plastic model is necessary. The halfspace model seems to be beneficial because there is only a system of equations on a surface mesh to be solved and not on a volume mesh like in the Finite-Element-Method. This leads to a much smaller system of equations which should allow reasonable calculation times even for large contact surfaces. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)