Chemical kinetics with diffusion

Guzhev,D.S., Kalitkin,N.N., Seifert,P., Shirkov,P.D. (1992) Numerical methods for problems of chemical kinetics with diffusion. Mathematical Modelling 4, 98-110 (Russian). 

Instabilities arise in combustion processes in many different ways a thorough classification is difficult to present because so many different phenomena may be involved. In one approach [1], a classification is based on the components of a system (such as a motor or an industrial boiler) that participate in the instability in an essential fashion. Three major categories are identified intrinsic instabilities, which may develop irrespective of whether the combustion occurs within a combustion chamber, chamber instabilities, which are specifically associated with the occurrence of combustion within a chamber, and system instabilities, which involve an interaction of processes occurring within a combustion chamber with processes operative in at least one other part of the system. Within each of the three major categories are several subcategories selected according to the nature of the physical processes that participate in the instability. Thus intrinsic instabilities may involve chemical-kinetic instabilities, diffusive-thermal instabilities, or hydrodynamic instabilities, for example. Chamber instabilities may be caused by acoustic instabilities, shock instabilities, or fiuid-dynamic instabilities within chambers, and system instabilities may be associated with feed-system interactions or exhaust-system interactions, for example, and have been assigned different specific names in different contexts. 

TABLE 19-1 Numerical Solution of the Mass Transfer Equation for One-Dimensional Diffusion and Hougen-Watson Chemical Kinetics with Dissociative Adsorption of Reactant A2 in Porous Catalysts with Rectangular Symmetry"... 

Let us consider here the effect of segmental mobility on curing rate. The Eiring theory of reaction rates, which considers formation and decomposition of an activated complex, can deal with the simultaneously occuring chemical kinetics and diffusion control characterized, respectively, by some rate constants k and kd. Under certain conditions (prevailing chemical kinetics or diffusion control), the apparent reaction rate constant, kapp, expressed via the Rabinowitch equation... 

When we think of diffusion acting on a system in which there are concentration inhomogeneities, our intuition suggests that diffusion should act to lessen, and eventually eliminate, the inhomogeneities, leaving a stable pattern with concentrations that are equal everywhere in space. As in the case of temporal oscillation, for a closed system the laws of thermodynamics require that this intuition be valid and that the eventual concentration distribution of the system be constant, both in time and in space. In an open system, however, just as appropriate nonlinear rate laws can lead to temporal structure, like oscillations and chaos, the interaction of nonlinear chemical kinetics and diffusion can produce nonuniform spatial structure, as suggested schematically in Figure 14.1. 

The theoretical prediction is supported by the experiments. Patterns that spontaneously form from the uniform state have multiple domains with different characteristic angles. The root-mean-square width A rms of the observed angular distribution function changes with the bifurcation parameter in qualitative accord with theory compare Figure 8b with Figure 8a [13]. A quantitative comparison of experiment and theory would require an evaluation of the coefficients in the Landau-Ginzburg equation from the chemical kinetics and diffusion coefficients of the reactants [47]. 

Lam, S.H. Reduced chemistry-diffusion coupling. Combust. Sci. Technol. 179, 767-786 (2006) Lam, S.H. Model reductions with special CSP data. Combust. Flame 160, 2707-2711 (2013) Lam, S.H., Goussis, D.A. Understanding complex chemical kinetics with computational singular perturbation. Proc. Combust. Inst. 22, 931-941 (1988)... 

In this chapter, we begin to focus on chemical changes and their interaction with diffusion. We are particularly interested in cases in which diffusion and chemical reaction occur at roughly the same speed. When diffusion is much faster than chemical reaction, then only chemical factors influence the reaction rate these cases are detailed in books on chemical kinetics. When diffusion is not much faster than reaction, then diffusion and kinetics interact to produce very different effects. 

There are many potential advantages to kinetic methods of analysis, perhaps the most important of which is the ability to use chemical reactions that are slow to reach equilibrium. In this chapter we examine three techniques that rely on measurements made while the analytical system is under kinetic rather than thermodynamic control chemical kinetic techniques, in which the rate of a chemical reaction is measured radiochemical techniques, in which a radioactive element s rate of nuclear decay is measured and flow injection analysis, in which the analyte is injected into a continuously flowing carrier stream, where its mixing and reaction with reagents in the stream are controlled by the kinetic processes of convection and diffusion. 

When a relatively slow catalytic reaction takes place in a stirred solution, the reactants are suppHed to the catalyst from the immediately neighboring solution so readily that virtually no concentration gradients exist. The intrinsic chemical kinetics determines the rate of the reaction. However, when the intrinsic rate of the reaction is very high and/or the transport of the reactant slow, as in a viscous polymer solution, the concentration gradients become significant, and the transport of reactants to the catalyst cannot keep the catalyst suppHed sufficientiy for the rate of the reaction to be that corresponding to the intrinsic chemical kinetics. Assume that the transport of the reactant in solution is described by Fick s law of diffusion with a diffusion coefficient D, and the intrinsic chemical kinetics is of the foUowing form... 

GASFLOW models geometrically complex containments, buildings, and ventilation systems with multiple compartments and internal structures. It calculates gas and aerosol behavior of low-speed buoyancy driven flows, diffusion-dominated flows, and turbulent flows dunng deflagrations. It models condensation in the bulk fluid regions heat transfer to wall and internal stmetures by convection, radiation, and condensation chemical kinetics of combustion of hydrogen or hydrocarbon.s fluid turbulence and the transport, deposition, and entrainment of discrete particles. 

Chemistry can be divided (somewhat arbitrarily) into the study of structures, equilibria, and rates. Chemical structure is ultimately described by the methods of quantum mechanics equilibrium phenomena are studied by statistical mechanics and thermodynamics and the study of rates constitutes the subject of kinetics. Kinetics can be subdivided into physical kinetics, dealing with physical phenomena such as diffusion and viscosity, and chemical kinetics, which deals with the rates of chemical reactions (including both covalent and noncovalent bond changes). Students of thermodynamics learn that quantities such as changes in enthalpy and entropy depend only upon the initial and hnal states of a system consequently thermodynamics cannot yield any information about intervening states of the system. It is precisely these intermediate states that constitute the subject matter of chemical kinetics. A thorough study of any chemical reaction must therefore include structural, equilibrium, and kinetic investigations. 

It should be mentioned that the predicted curve at highest benzene level in Figure 13 agrees with classical kinetics (no diffusion-control). It is not clear therefore why measured data at even higher benzene concentrations do not agree with classical kinetics. There may be some subtle chemical interactions at these high solvent levels. Duerksen(lT) fomd similar effects with styrene polymerization in benzene and had to correct kp for solvent. 

The Chemkin package deals with problems that can be stated in terms of equation of state, thermodynamic properties, and chemical kinetics, but it does not consider the effects of fluid transport. Once fluid transport is introduced it is usually necessary to model diffusive fluxes of mass, momentum, and energy, which requires knowledge of transport coefficients such as viscosity, thermal conductivity, species diffusion coefficients, and thermal diffusion coefficients. Therefore, in a software package analogous to Chemkin, we provide the capabilities for evaluating these coefficients. ... 

Consider the case when the equilibrium concentration of substance Red, and hence its limiting CD due to diffusion from the bulk solution, is low. In this case the reactant species Red can be supplied to the reaction zone only as a result of the chemical step. When the electrochemical step is sufficiently fast and activation polarization is low, the overall behavior of the reaction will be determined precisely by the special features of the chemical step concentration polarization will be observed for the reaction at the electrode, not because of slow diffusion of the substance but because of a slow chemical step. We shall assume that the concentrations of substance A and of the reaction components are high enough so that they will remain practically unchanged when the chemical reaction proceeds. We shall assume, moreover, that reaction (13.37) follows first-order kinetics with respect to Red and A. We shall write Cg for the equilibrium (bulk) concentration of substance Red, and we shall write Cg and c for the surface concentration and the instantaneous concentration (to simplify the equations, we shall not use the subscript red ). 

The reaction was studied for all coinage metal nanoparticles. In the case of GMEs the rate follows zero-order kinetics with IT for all the coinage metal cases. The observed IT for the Cu catalyzed reaction was maximum but its rate of reduction was found to be minimum. Just the reverse was the case for Au and an intermediate value was obtained for the Ag catalyzed reaction (Figure 7). The adsorption of substrates is driven by chemical interaction between the particle surface and the substrates. Here phe-nolate ions get adsorbed onto the particle surface when present in the aqueous medium. This caused a blue shift of the plasmon band. A strong nucleophile such as NaBH4, because of its diffusive nature and high electron injection capability, transfers electrons to the substrate via metal particles. This helps to overcome the kinetic barrier of the reaction. 

The mass transfer factors with heterogeneous reactions the reaction rate may be controlled by the rates of diffusion of the reacting species rather than the chemical kinetics. 

In order to estimate the region of this approximation applicability, it is necessary to examine macrokinetics of a polymeranalogous reaction with explicit allowance for the diffusion of a reagent Z into a globule. In this case, the profile of its constituent monomeric units will be fuzzy rather than stepwise (see Fig. 1). This brings up two questions. The first one is how this profile depends on kinetic and diffusion parameters of a reaction system. The second question is concerned with the effect of the profile shape on the statistical characteristics of the chemical structure of the products of a polymeranalogous reaction. A rigorous theory has been developed [22,23] which enables us to answer these questions. The main concepts of this theory are outlined in the subsequent Sections. 

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