Reaction diffusion definition

By definition chirality involves a preferred sense of rotation in a three-dimensional space. Therefore, it can only be affected by a modification of the nonscalar fields appearing in the rate equations. For a reaction-diffusion system [equations (1)] these fields are descriptive of a vector irreversible process, namely, the diffusion flux J of constituent k in the medium. According to irreversible thermodynamics, the driving force conjugate to diffusion is... 

Given these definitions, the steady state reaction diffusion equations for oxygen and oxy-myoglobin in the muscle fiber in radial coordinates are... 

In applying the burning method of diffusivity evaluation, it is important to ascertain that the reaction is definitely diffusion controlled. When this is the case, no increase of rate should result when the temperature of combustion is raised. This can serve as a test for this requirement. As a reference point for orientation, a 3 to 4 mm, diameter particle of... 

Note that, unlike in the definition of effectiveness factor for catalytic reactions (Chapter 7) where the normalizing rate was the rate of reaction, here the normalizing rate is the rate of mass transfer. Thus the reaction is considered the intruder (albeit benevolent or enhancing), whereas for catalytic reactions, diffusion was the intruder (often, but not always, retarding). For a pseudo-mth-order reaction, one can write... 

Besides the simple mathematical approach of combining the rate equation and the diffusion equation, two fundamental approaches exist to derive the reaction-diffusion equation (2.3), namely a phenomenological approach based on the law of conservation and a mesoscopic approach based on a description of the underlying random motion. While it is fairly straightforward to show that the standard reaction-diffusion equation preserves positivity, the problem is much harder, not to say intractable, for other reaction-transport equations. In this context, a mesoscopic approach has definite merit. If that approach is done correctly and accounts for all reaction and transport events that particles can undergo, then by construction the resulting evolution equation preserves positivity and represents a valid reaction-transport equation. For this reason, we prefer equations based on a solid mesoscopic foundation, see Chap. 3. 

A similar experiment may be performed using equal amounts of catalyst in chambers of different diameters. Previous work [3] has shown, however, that the reaction is definitely not isothermal in copper tubes surrounded by liquid nitrogen if the inside diameter of the tubes exceeds about 0.19 in. This would make it difficult to study diffusion effects in tubes of different diameters. 

Before delving into the details of bifurcation theory, I wish to elaborate more fully on the phenomena to be addressed. Figure 1 illustrates some of the variety of spiral states typically found in excitable media. These have been obtained from numerical simulations of a reaction-diffusion model discussed in the next section. Each state is represented by a segment of the path traced out by the spiral tip as it evolves in time. Figures 1(a) and (b) show periodic states the spiral tips trace out circles as the waves rotate. (The definition of the spiral tip is given later in the chapter it is not particularly important here.) Figures l(c-h) show a variety of meandering states for these cases the tip paths are flower patterns of the type first observed by Winfree. 

With these definitions at hand, I turn to the quantitative analysis of spiral dynamics in the reaction-diffusion model, starting with the dynamics as a function of the parameter a, the other parameters being fixed as in the one-parameter cut shown in Figures 5 and 6 6 = 0.05, e = 0.02 and Dy = 0. 

The linear stability of these new supercritical branches is then tested (Ai = Ai s + One finds that if 90 < 9nd, as is usually the case in isotropic reaction-diffusion problems (heterogeneous catalysis may provide counterexamples), the stripes are stable and the squares unstable. The reverse condition leads to the opposite conclusion. Therefore stripes and squares are mutually exclusive [33]. However no definitive conclusion regarding the stability of the winning state should be drawn as its stability must still be tested with respect to perturbations involving higher values of M. As always their stability should also be checked regarding wavenumber changes (Section 4). 

Fig. 16. Comparison between the exact stationary single-front solution (n(a ),t (a )) (full line) and the external approximation of the same solution (U x),V x)) (dashed line) of the reaction-diffusion model (3) with Dirichlet boundary conditions the piecewise linear slow manifold is given by Equation (7) with 6 = 0. (a) The functions u x) and U x) note that u and U differ significantly in the inner region only (b) phase portrait in the (n, v) plane by definition, the points (U x),V[x)) are located on the slow manifold (dashed line) (c) comparison between the exact solution (full line) and fhe function U[x) + i(0 (dashed line) (d) comparison between the exact solution v x) (full line) and the external function V x) (dashed line).

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

The importance of dilfusion in a tubular reactor is determined by a dimensionless parameter, SiAt/S = QIaLKuB ), which is the molecular diffusivity of component A scaled by the tube size and flow rate. If SiAtlB is small, then the elfects of dilfusion will be small, although the definition of small will depend on the specific reaction mechanism. Merrill and Hamrin studied the elfects of dilfusion on first-order reactions and concluded that molecular diffusion can be ignored in reactor design calculations if... 

The concept of surface concentration Cg j requires closer definition. At the surface itself the ionic concentrations will change not only as a result of the reaction but also because of the electric double layer present at the surface. Surface concentration is understood to be the concentration at a distance from the surface small compared to diffusion-layer thickness, yet so large that the effects of the EDL are no fonger felt. This condition usually is met at points about 1 nm from the surface. 

Diffusion in Binary Electrolytes at Nonzero Currents Consider a reaction in which one of the ions of the binary solution is involved. For the sake of definition, we shall assume that its cation is reduced to metal at the cathode. The cation concentration at the surface will decrease when current flows. Because of the electroneutrality condition, the concentration of anions should also decrease under these conditions (i.e., the total electrolyte concentration c. should decrease). 

The trends of behavior described above are found in solutions containing an excess of foreign electrolyte, which by definition is not involved in the electrode reaction. Without this excess of foreign electrolyte, additional effects arise that are most distinct in binary solutions. An appreciable diffusion potential q) arises in the diffusion layer because of the gradient of overall electrolyte concentration that is present there. Moreover, the conductivity of the solution will decrease and an additional ohmic potential drop will arise when an electrolyte ion is the reactant and the overall concentration decreases. Both of these potential differences are associated with the diffusion layer in the solution, and strictly speaking, are not a part of electrode polarization. But in polarization measurements, the potential of the electrode usually is defined relative to a point in the solution which, although not far from the electrode, is outside the diffusion layer. Hence, in addition to the true polarization AE, the overall potential drop across the diffusion layer, 9 = 9 + 9ohm is included in the measured value of polarization, AE. ... 

In order to obtain a definite breakthrough of current across an electrode, a potential in excess of its equilibrium potential must be applied any such excess potential is called an overpotential. If it concerns an ideal polarizable electrode, i.e., an electrode whose surface acts as an ideal catalyst in the electrolytic process, then the overpotential can be considered merely as a diffusion overpotential (nD) and yields (cf., Section 3.1) a real diffusion current. Often, however, the electrode surface is not ideal, which means that the purely chemical reaction concerned has a free enthalpy barrier especially at low current density, where the ion diffusion control of the electrolytic conversion becomes less pronounced, the thermal activation energy (AG°) plays an appreciable role, so that, once the activated complex is reached at the maximum of the enthalpy barrier, only a fraction a (the transfer coefficient) of the electrical energy difference nF(E ml - E ) = nFtjt is used for conversion. 

The limitation of using such a model is the assumption that the diffusional boundary layer, as defined by the effective diffusivity, is the same for both the solute and the micelle [45], This is a good approximation when the diffusivities of all species are similar. However, if the micelle is much larger than the free solute, then the difference between the diffusional boundary layer of the two species, as defined by Eq. (24), is significant since 8 is directly proportional to the diffusion coefficient. If known, the thickness of the diffusional boundary layer for each species can be included directly in the definition of the effective diffusivity. This approach is similar to the reaction plane model which has been used to describe acid-base reactions. 

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