Optimization of Batch Reactions in Series with Uncertainty

Certain types of chemical reactions, such as the global deprotection of a polypeptide, are extremely complex. As a result, it may be very difficult or expensive to develop accurate models of these chemical reactions. Without a satisfactory kinetic model for the reaction, it is difficult to develop an optimum operating policy that will maximize the profit. Stochastic optimization is applied in this work to an example process step to obtain the optimum reaction temperature and reaction time. In the case of the “here and now” problem, the optimal conditions are a lower reaction temperature and a longer reaction time than obtained from the deterministic problem. The average reaction time for the “wait and see” problem is also longer than the deterministic case, but the average reaction temperature is very close to that of the deterministic problem. Both normally and uniformly – distributed uncertain parameters are considered.     

Studies in oxidation

The kinetics of the oxidation of toluene and its ring-substituted derivatives as well as of diphenyl methane by chromic acid have been studied in acetic acid-water systems. The reaction had to be in unbuffered solutions to get a measureable rate. Product analysis showed a step-wise reaction. Rate constants have been evaluated, using a graphical method, for a three-stage consecutive and concurrent reaction, the values being obtained for the first and last stages alone. Electron attracting substituents retard the reaction while those facilitating electron release accelerate the reaction. Diphenylmethane also undergoes only a step-wise reaction with the second step much slower than the first. A comparison of experimental activation energies indicates clearly that the entropy term of the rate equation exerts possibly an even greater influence on the reaction.     

Bogoliubov averaging principle of stochastic reaction–diffusion equation

The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation.     

EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

In this paper, we consider a nonlinear system of reaction diffusion equations arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing pulse solutions) and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equations. The Evans functions for the standing pulse solutions are constructed explicitly.     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

Numerical exploration of a system of reaction–diffusion equations with internal and transient layers

A reaction pathway for a classical two-species reaction is considered with one reaction that is several orders of magnitudes faster than the other. To sustain the fast reaction, the transport and reaction effects must balance in such a way as to give an internal layer in space. For the steady-state problem, existing singular perturbation analysis rigorously proves the correct scaling of the internal layer. This work reports the results of exploratory numerical simulations that are designed to provide guidance for the analysis to be performed for the transient problem. The full model is comprised of a system of time-dependent reaction–diffusion equations coupled through the non-linear reaction terms with mixed Dirichlet and Neumann boundary conditions. In addition to internal layers in space, the time-dependent problem possesses an initial transient layer in time. To resolve both types of layers as accurately as possible, we design a finite element method with analytic evaluation of all integrals. This avoids all errors associated with the evaluation of the non-linearities and allows us to provide an analytic Jacobian matrix to the implicit time stepping method. The numerical results show that the method resolves the localized sharp gradients accurately and can predict the scaling of the internal layers for the time-dependent problem.     

Reaction of hydrazobenzene with schiff bases,aldehydes and acetonylacetone in acetic acid

The reaction of hydrazobenzene with benzylideneaniline in acetic acid gives dibenzylidenebenzidine. It has been shown that in this reaction, acetic acid functions as an acid catalyst causing benzidine rearrangement, while benzylideneaniline serves as a potential source of aldehyde. The driving force for the observed transarylidenation reaction appears to be the enhanced conjugation in the resulting product. The reaction between dibenzylidenebenzidine and cinnamalaniline (cinnamaldehyde) has yielded the dicinnamylidene benzidine, supporting the above conclusion.     

我国上市公司首次披露股权激励计划的市场反应分析

本文首次以《上市公司股权激励管理办法》颁布以来沪深两市首次披露股权激励计划的上市公司作为研究对象,运用事件研究法计算超额收益的方式,检验了股权激励计划披露前后超额收益率的变动情况.研究发现,在股权激励计划披露后,证券市场有显著的正面反应;在股权激励计划披露前34天的短时窗内证券市场有不显著的正面反应,在披露前60天到90天的长时窗内,证券市场有负面反应.     

Transient diffusion-controlled smoulder propagation: a similarity solution approach

One-, two- and three-dimensional time-dependent smoulder propagationthrough solid reactants with plane bounding surfaces is considered.Propagation is maintained by the diffusion of oxidizer fromthe boundaries to the smouldering reaction zone. The resultingburnt solid is assumed to be porous and the unreacted solidis taken to be sufficiently dense for no oxidizer to be present.The activation energy of the reaction is taken to be so largethat the exothermic reaction term has a delta-function behaviour.This enables the reaction zone to be approximated by a narrowreaction front and results in the equations of heat and masstransfer being decoupled away from the front. The assumption,based on experimental observations, that the reaction frontspropagate with a speed proportional to t^–1/2, where tis the time, permits the introduction of similarity variables.The resulting intermediate asymptotic equations, lying betweenthose for very small and very large times, are solved and theequation of the smouldering fronts determined for each geometryis considered.     

Stability and attractivity of periodic solutions of parabolic systems with time delays

This paper is concerned with the existence, stability, and global attractivity of time-periodic solutions for a class of coupled parabolic equations in a bounded domain. The problem under consideration includes coupled system of parabolic and ordinary differential equations, and time delays may appear in the nonlinear reaction functions. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement using Schauder fixed point theorem, while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme. The results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system. Applications are also given to three Lotka-Volterra reaction diffusion model problems, and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions.     

Pinning and mode-locking of reaction fronts by vortices

We present results of experiments on the behavior of reaction fronts in the presence of vortex-dominated flows. The flow is either a single vortex or a chain of vortices in an annular configuration, and the reaction is the excitable Belousov–Zhabotinsky chemical reaction. If the vortex chain oscillates periodically in the lateral direction, the reaction front often mode-locks to the oscillations, propagating an integer number of wavelengths of the flow (two vortices) in an integer number of drive periods. In the presence of a uniform “wind”, the front often freezes, remaining pinned to the leading vortex and neither propagating forward against the wind nor being blown backward by it. Studies with an individual vortex verify the ability of a moving vortex to pin and drag a reaction front. We use this pinning behavior to explain the mode-locking for the oscillating case.     

Analytic Representations of Thermonuclear Reaction Rates

The basic physical principles common to all nonresonant and resonant thermonuclear reaction rates are taken as a basis to derive the standard forms of both types of reaction rates. Results are given for the closed-form representations of the standard nonresonant and resonant thermonuclear reaction rates in terms of a certain type of Meijer's G-function. Then, for physical reasons the nonresonant reaction rate with modified Maxwell-Boltzmann distribution is considered. The mathematical method for the derivation of closed-form representations for the modified nonresonant reaction-rate integral, which is also suitable for numerical computations, is given in detail. Some new techniques developed by the authors are used for deriving the results. The reaction-rate systematics thus obtained in terms of Meijer's G-function is a contribution to Fowler's methods for the numerical computation of thermonuclear reaction rates.     

Exact solutions of potentiostatic current transients for a corrosion reaction under mixed charge transfer and diffusion control

An analytical solution has been obtained for the current response transients to a potentiostatic step for a corrosion reaction where a planar metal electrode is immersed in an acidic electrolyte. This solution can be used to provide values of important kinetic parameters of the reaction such as corrosion current and Tafel slopes. The model is derived for conditions of mixed charge transfer and diffusion control and includes the metal ion deposition partial reaction. A Nernstian diffusion model is used, where anodic and cathodic species diffuse in a finite diffusive layer. A numerical solution has previously been published [1], but it is believed that this is the first time an analytical solution has been presented. Transients obtained by including both the metal dissolution and metal-ion deposition (back) reactions, are compared to those obtained by neglecting the metalion deposition reaction. It is shown that neglecting the back reaction can cause significant errors especially when large cathodic potential steps are applied and as the corrosion current approaches the limiting current density of the cathodic reaction.     

Auto-ignition of diesel spray using the PDF-Eddy Break-Up model

This work presents the development and implementation of auto-ignition modelling for DI diesel engines by using the PDF-Eddy Break-Up (PDF-EBU) model. The key concept of this approach is to combine the chemical reaction rate dealing with low-temperature mode, and the turbulence reaction rate governing the high-temperature part by a reaction progress variable coupling function which represents the level of reaction. The average reaction rate here is evaluated by a probability density function (PDF) averaging approach. In order to assess the potential of this developed model, the well-known Shell ignition model is chosen to compare in auto-ignition analysis. In comparison, the PDF-EBU ignition model yields the ignition delay time in good agreement with the Shell ignition model prediction. However, the ignition kernel location predicted by the Shell model is slightly nearer injector than that by the PDF-EBU model leading to shorter lift-off length. As a result, the PDF-EBU ignition model developed here are fairly satisfactory in predicting the auto-ignition of diesel engines with the Shell ignition model.     

PERIODIC REACTION AND TURBULENCE OF CHEMICAL BRUSSELATOR

In this paper,we discuss the reaction model of tri-molecules and the exi- stence of periodic reaction under certain original density conditions of two ob- jects without diffusion phenomena.We prove that the periodic reaction exhibit some chemical phenomena that violate the second law of thermo-mechanics. Finally,we research similar properties and phenomena concerning Belousov- Zhabotinsky reaction model.     

Entropy functionals for finding requirements in hierarchical reaction-diffusion models for inflammations

Based on an established model for liver infections, we open the discussion on the used reaction terms in the reaction-diffusion system. The mechanisms behind the chronification of liver infections are widely unknown, therefore we discuss a variety of reaction functions. By using theorems about existence, uniqueness, and nonnegativity, we identify properties of reaction functions which are indispensable to modelling liver infections. We introduce an entropy functional for reaction-diffusion models of this type, which allows predictions of the longtime behavior of the solutions. As a result, we find more conditions on the reaction functions to derive a model covering different inflammation courses. Finally, we discuss the models in the frame of a hierarchical model family.     

Stability estimates of solutions to extremal problems for a nonlinear convection-diffusion-reaction equation

We consider an identification problem for a stationary nonlinear convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of the substance. This problem is reduced to an inverse extremal problem by an optimization method. The solvability of the boundary value problem and the extremal problem is proved. In the case that the reaction coefficient is quadratic, when the equation acquires cubic nonlinearity, we deduce an optimality system. Analyzing it, we establish some estimates of the local stability of solutions to the extremal problem under small perturbations both of the quality functional and the given velocity vector which occurs multiplicatively in the convection–diffusion–reaction equation.     

Exact multiplicity of solutions and S-shaped bifurcation curve for a class of semilinear elliptic equations

The set of steady state solutions to a reaction-diffusion equation modeling an autocatalytic chemical reaction is completely determined, when the reactor has spherical geometry, and the spatial dimension is n=1 or 2 for any reaction order, or n?3 for subcritical reaction order. Bifurcation approach and analysis of linearized problems are used to establish exact multiplicity and precise global bifurcation diagram of positive steady states.     

Periodic homogenization of strongly nonlinear reaction–diffusion equations with large reaction terms

We study in this article the periodic homogenization problem related to a strongly nonlinear reaction–diffusion equation. Owing to the large reaction term, the homogenized equation has a rather quite different form which puts together both the reaction and convection effects. We show in a special case that the homogenized equation is exactly of a convection-diffusion type. This study relies on a suitable version of the well-known two-scale convergence method.