Equal diffusion coefficients

The physics of the problem under study is assumed to be governed by the compressible form of the Favre-filtered Navier-Stokes energy and species equations for an ideal gas mixture with constant specific heats, temperature-dependent transport properties, and equal diffusion coefficients. The molecular Schmidt, Prandtl, and Lewis numbers are set equal to 1.0, 0.7, and 1.43, respectively [17]. 

Pearson, J. E. and Horsthemke, W. (1989). Turing instabilities with nearly equal diffusion coefficients. J. Chem. Phys., 90, 1588-99. 

A study was also made of release with equal diffusion coefficients in the coating and the kernel. The description of delay time is much the same as in the previous case. Figure 13 illustrates this similarity in delay time, as well as later release behavior. In this case also, the t1/2 plot shows a long linear region. The reason is probably that depletion effects are reasonably followed by this function and not that there is kernel... 

In fact, the earlier treatments from the Czechoslovakian school [11] did not involve simplifying assumptions as described above. However, in most cases, only expressions for the d.c. limiting current were derived. Extensions to this work, describing the complete d.c. polarogram, are known from several workers [168—170]. Usually, the mathematics are kept simple by adopting the Nemst equation instead of some rate equation and by assuming equal diffusion coefficients, e.g. DA — DQ — Dqa > etc ... 

The problem was considered at the rotating disc electrode, making the assumptions of equal diffusion coefficients, that nx = n2 — n and that the second electron transfer is easier than the first to give [209, 210]... 

To avoid bulky calculations, we restrict ourselves by the following problem statement particles A and B have equal diffusion coefficients, DA = Db = D, fluctuating particle sources in Poisson statistical properties ... 

Fig. 5.14. The time dependence of the survival probability for one-dimensional diffusion with mobile, noninteracting traps for various values of k. Curve 1 is exact for static particles (k = 0) and identical to the Smoluchowski result curve 4 is exact for static traps (n = 1), curve 2 for equal diffusion coefficients (k = 1/2), and curve 3 for k = 1 are obtained from...

To simplify mathematical manipulations, let us consider now the case of equal diffusion coefficients, Da = D, in which case the similar correlation functions just coincide, Xv r),T) = X(t),t). Taking into account the definition of correlation length Id = VDt, where D = Da + D = 2D a, as well as time-dependence of new variables r) and r, one gets from (5.1.2) to (5.1.4) a set of equations... 

The addition of microspheres lowers the glass transition temperature of the epoxy binder (Fig. 13). This seems to be because the filler causes defects in the matrix network. Equal diffusion coefficients of filled and unfilled epoxy binder indicates, therefore, that the diffusion processes are insensitive to binder changes. The sorption of water by epoxy resins is in fact known to depend mainly on their polarity and only slightly on the three-dimensional compactness of the network. 

For the sake of simplicity only electrode processes in which the oxidized and reduced species are soluble in the electrolytic solution and have equal diffusion coefficients will be considered. 

In this section, microdisc electrodes will be discussed since the disc is the most important geometry for microelectrodes (see Sect. 2.7). Note that discs are not uniformly accessible electrodes so the mass flux is not the same at different points of the electrode surface. For non-reversible processes, the applied potential controls the rate constant but not the surface concentrations, since these are defined by the local balance of electron transfer rates and mass transport rates at each point of the surface. This local balance is characteristic of a particular electrode geometry and will evolve along the voltammetric response. For this reason, it is difficult (if not impossible) to find analytical rigorous expressions for the current analogous to that presented above for spherical electrodes. To deal with this complex situation, different numerical or semi-analytical approaches have been followed [19-25]. The expression most employed for analyzing stationary responses at disc microelectrodes was derived by Oldham [20], and takes the following form when equal diffusion coefficients are assumed ... 

Equation (4.50) for the particular case of equal diffusion coefficients and rK 0 was deduced by Kambara [22] by applying the Superposition Principle. 

Using the expressions of the currents corresponding to the first and second potentials applied given by Eqs. (4.28) and (4.29), the expression of the current in ADDPV at any electrode geometry assuming equal diffusion coefficients for species O and R and cR = 0 is... 

Equation (5.64) is equivalent to the following expression deduced by Reinmuth by using Laplace transform and assuming equal diffusion coefficients of species O and R and = 0 [27],... 

According to Scheme 7.1 and Eq. (7.1), by subtracting the currents corresponding to the consecutive potential pulses Ep and Ep (see Eq. (5.24)), the DSCVC response for any electrode geometry when equal diffusion coefficients for species O and R are assumed is... 

This expression determines the kinetics of the excitation accumulation from IV (0) = 0 up to the stationary value N, after instantaneous switching the permanent illumination at t 0. This is a particular case of the more general convolution recipe derived in Ref. 201 for pulses of arbitrary shape. Hence, for equal diffusion coefficients, the convolution formula follows from IET as well as from the many-particle theory employed in Ref. 201. The generality of the latter allows us to use in the convolution formula the system response P(t), calculated with any available theory. The same is valid for the stationary equation (3.458), used above. Although P(t) obtained with different theories is different, as well as N (t), the relationship between N and P remains the same. 

The assumption of equal diffusion coefficients (Do = Dr = D) allows the problem to be described in terms of a single species (R). The boundary conditions are of the form ... 

These equations are called the Navier-Stokes equations, and when supplemented by the state equation for fluid pressure and species transport equations, they form the basis for any computational model describing the flows in fires. For simplicity, several approximations are inherent (see Equation 20.3) (no Soret/Dufour effects, no viscous dissipation, Fickian diffusion, equal diffusion coefficients of all species, unit Lewis number). 

Modification to the totally irreversible case (kb = 0) is trivial, as is the simplification to equal diffusion coefficients (d = 1). 

There are two unknowns, so we need one more equation. This comes from the fact that the flux of substance A at the electrode must be equal and opposite to that of substance B. [f we assume equal diffusion coefficients for the moment, this means... 

One of these specific cases would be to consider a system with equal diffusion coefficients... 

Equations (13.110) yield acceptable values if (1/2) vpproduct inhibition for equal diffusion coefficients of. S, and S2 (Glansdorff and Prigogine, 1971). 

In the following we shall restrict ourselves to binary symmetrical electrolytes of which the Ions have equal diffusion coefficients (D = D = D). Then the concentration profiles are identical for all Ions. For that situation [4.8.8] reduces to... 

Returning to the case of equal diffusion coefficients, the task Is now to solve [4.8.10 and 11], subject to the outer boundary conditions... 

To simplify mathematical manipulations, let us consider now the case of equal diffusion coefficients. Da = Db, in which case the similar correlation functions just coincide, = X t],t). Taking into account the defi-... 

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