Reaction-diffusion equation general form

Remark 2.4 In the derivation of the generalized reaction-diffusion equation (2.82) we do not explicitly refer to the particular form of the waiting time PDF. Equation (2.82) is valid for arbitrary waiting time PDFs < (t) and has much wider applicability than subdiffusive transport. 

The generalized reaction-diffusion equation (2.82) can be written in a form using fractional derivatives for subdiffusive transport, where the waiting PDF of species i is given in Laplace space by (2.52), (i) 1 —. In that case... 

The evolution equations of such reactions schemes, has a general form called "Reaction-Diffusion Equations" ... 

In the kinetics of formation of carbides by reaction of the metal widr CH4, the diffusion equation is solved for the general case where carbon is dissolved into tire metal forming a solid solution, until the concentration at the surface reaches saturation, when a solid carbide phase begins to develop on the free surface. If tire carbide has a tirickness at a given instant and the diffusion coefficient of carbon is D in the metal and D in the carbide. Pick s 2nd law may be written in the form (Figure 8.1)... 

Pollard and Newman" have also studied CVD near an infinite rotating disk, and the equations we solve are essentially the ones stated in their paper. Since predicting details of the chemical kinetic behavior is a main objective here, the system now includes a species conservation equation for each species that occurs in the gas phase. These equations account for convective and diffusive transport of species as well as their production and consumption by chemical reaction. The equations stated below are given in dimensional form since there is little generalization that can be achieved once large chemical reaction mechanisms are incorporated. 

In electrochemical cells we often find convective transport of reaction components toward (or away from) the electrode surface. In this case the balance equation describing the supply and escape of the components should be written in the general form (1.38). However, this equation needs further explanation. At any current density during current flow, the migration and diffusion fluxes (or field strength and concentration gradients) will spontaneously settle at values such that condition (4.14) is satisfied. The convective flux, on the other hand, depends on the arbitrary values selected for the flow velocity v and for the component concentrations (i.e., is determined by factors independent of the values selected for the current density). Hence, in the balance equation (1.38), it is not the total convective flux that should appear, only the part that corresponds to the true consumption of reactants from the flux or true product release into the flux. This fraction is defined as tfie difference between the fluxes away from and to the electrode ... 

A fundamental fuel cell model consists of five principles of conservation mass, momentum, species, charge, and thermal energy. These transport equations are then coupled with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte. Such convection—diffusion—source equations can be summarized in the following general form... 

It does not appear possible to consider in general form, discarding the assumptions of steady propagation of the regime with constant velocity, the differential equations of heat conduction and diffusion in a medium in which a chemical reaction is also running. 

General Considerations. The continuity equations for a column vector of species taking place in a reaction-diffusion wave phenomenon take the form... 

Mass transport processes - diffusion, migration, and - convection are the three possible mass transport processes accompanying an - electrode reaction. Diffusion should always be considered because, as the reagent is consumed or the product is formed at the electrode, concentration gradients between the vicinity of the electrode and the bulk solution arise, which will induce diffusion processes. Reactant species move in the direction of the electrode surface and product molecules leave the interfacial region (- interface, -> interphase) [i-v]. The - Nernst-Planck equation provides a general description of the mass transport processes. Mass transport is frequently called mass transfer however, it is better to reserve that term for the case that mass is transferred from one phase to another phase. 

POMs exhibit two dominant modes of reactivity in redox processes. There are well-documented variations for each mode. Mode 1, which is the most frequent one when 02 is used as the oxidant, involves initial substrate oxidation (Equation (26)) followed by reoxidation of the reduced POM (Equation (27)). The net reaction is Equation (28). Mode 2, which is most frequent with oxygen donor oxidants including peroxides, involves initial activation of the oxidant, OX, by the POM with formation of a POM-OX complex (Equation (29)). There are three general fates of these complexes. They can directly react with substrate to form product (Equation (30)), transform to another complex, [POM-OX] (Equation (31)) which then oxidizes the substrate (Equation (32)), or form freely diffusing oxidizing intermediates that are not bound to the POM catalyst... 

The simulation model includes the coupled reaction-diffusion-migration equations for interstitial atoms, interstitial clusters and related defects. These equations have the general form for species i ... 

It is well known that when mass transport is purely controlled by the reaction diffusion in the electrode or electrolyte, the electrochemical responses at the rough interface obey the generalized form of the Cottrell equation. For certain cases, however, classical fractal behavior under diffusion-controlled constraint is replaced by mixed kinetic/diffusive control (de Levie and Vogt, 1990 Kant and Rangarajan, 1995 Jung and Pyun, 2006a,b). Taking oc = (dp - l)/2, the chronoamperometric current becomes (Dassas and Duby, 1995 Stromme et al., 1995 Andrieux and Audebert, 2001) ... 

The next step in the development of the thermodynamics of multicomponent systems is the formulation of the equations of change. These equations can, in completely general form, be considerably more complicated than the analogous pure component equations since (1) the mass or number of moles of each species may not be conserved due to chemical reactions, and (2) the diffusion of one species relative to the others may occur if concentration gradients are present. Furthermore, there is the computational difficulty that each thermodynamic property depends, in a complicated fashion, on the temperature, pressure, and composition of the mixture. 

Using artificial neural networks (ANN) the reaction system, including intrinsic reaction kinetics but also internal mass transfer resistances, is considered as a black-box and only input-output signals are analysed. With this approach the conversion rate of the i-th reactant into the j-th species can be expressed in a general form as a complex function, being a mathematical superposition of all above mentioned functional dependencies. This function includes also a contribution of the internal diffusion resistances. So each of the rate equations of Eq. 5 can be described with the following function based on the vanables which uniquely define the state of the system ... 

Paul et al. (25) observed that for polymer volume fractions less than 0.8, the functional dependence of the diffusion coefficients on the polymer volume fraction was, generally, in accordance with Equation 40. Muhr and Blanshard (26) provide additional supporting data on different polymers than those reported by Paul et al, Roucls and Ekerdt (27) measured the diffusion of cyclic hydrocarbons in benzene-swollen polystyrene beads their diffusion coefficients satisfy the general form of Equation 40. The effective dlffuslvltles of organic substrates in crossllnked polystyrene reported by Marconi and Ford (17) also follow trends predicted in Equation 40. In the absence of experimental data, it appears that Equation 40 provides a reasonable, and the simplest, means to estimate D for use in detailed modeling or in estimation methods such as Equation 38. Equation 40 was used by Dooley et al. (11) in their study of substrate diffusion and reaction in a macroreticular sulfonic acid resin which involved vapor phase reactants. 

In Chapter 14, we formulated a number of regimes with corresponding conditions and governing rate equations. In the present chapter, we recast the rate equations in a general form that indicates the relative roles of reaction, liquid film diffusion, and gas film diffusion. Then we briefly discuss the design principles of the more common classes of fluid-fluid reactors. Detailed treatments of design may be found in the books of Astarita (1967), Danckwerts (1970), Shah (1979), Levenspiel (1972, 1993), Doraiswamy and Sharma (1984b), Bisio et al. (1985), and Kastanek et al. (1993). 

The Ginzburg-Landau equation possesses a family of plane wave solutions. They are considered to be a special form of the plane waves whose existence was proved by Kopell and Howard (1973 a) for oscillatory reaction-diffusion systems in general. In view of the physical situation where the Ginzburg-Landau equation arises, the plane waves of Kopell and Howard are expected to reduce to this special form as the point of Hopf bifurcation (of the supercritical type) is approached from above. One of the important conclusions to be drawn below is that all the family of plane waves (including uniform oscillation as a special plane wave) can happen to be unstable, which is a property not shared by the A - co system with a diagonal diffusion matrix, see Sect. 2.4. 

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