Normalisation diffusion coefficient

We are still dealing with two species as in the uncoupled case and the same boundary conditions apply they are reformulated in the present matrix-vector form here. As noted above, there is a common condition for all experiments, the flux condition (6.30), generalised to include the normalised diffusion coefficients, to the gradient condition (6.32), and we now write out its discrete form fully, pairing the two species terms for each spatial index ... 

The effect of slip coefficient on concentration polarisation (CP) was mathematically modeled for flat membrane and tubular membrane systems [12,13,15,16]. Lowering of CP due to slip coefficient as a function of product water recovery ( ) for different normalised diffusion coefficients (a) is shown in Figure 6.8. The data show that CP decreases both with and a. Since a is a measure of particle diffusion from the membrane surface to the bulk solution, slip-flow possibly augments diffusive back-transport of particles from the membrane surface to the bulk solution. Thus, the slip-flow velocity model possibly accounts for higher or actual UF/MF flux, which is under-predicted by the gel polarisation model discussed in Chapter 1. 

The influence of polydispersity is exemplarily shown in Fig. 4.24 for DLCA aggregates. Variation in primary particle size (monodisperse or log-normal distributed with = 0.4) as well as in aggregation number (log-normal distributed and a self-preserving distribution function, Eq. (4.10)) were considered. As in the previous section, a reciprocal correlation between N and jCp v was assumed for the calculation. The normalised diffusion coefficients are plotted versus a -axis that is scaled with the effective radius of gyration Rg etc (cf- Elq- (4.59)) ... 

In this relation a(r, t) is the experimentally observed signal, s represents random noise, axi r) represents the time invariant systematic noise and aRi(f) the radial invariant systematic noise Schuck [42] and Dam and Schuck [43] describe how this systematic noise is ehminated. x is the normahsed concentration at r and t for a given sedimenting species of sedimentation coefficient 5 and translational diffusion coefficient D it is normalised to the initial loading concentration so it is dimensionless. 

There are not a great number of studies on the viscoelastic behaviour of quasi-hard spheres. The studies of Mellema and coworkers13 shown in Figure 5.5 indicate the real and imaginary parts of the viscosity in a high-frequency oscillation experiment. Their data can be normalised to a characteristic time based on the diffusion coefficient given above. 

The reasons for the great difference in values of reaction- and self-diffusion coefficients of the components of a chemical compound are analysed. For example, in the case of Fe3 s04 the reaction-diffusion coefficient is two orders of magnitude greater than the self-diffusion coefficient of iron ions. For other compounds (A1203, Fe2Al5, Pd2Si, AlSb, etc.) this difference varies from five to ten orders of magnitude. After the normalisation to the same vacancy concentration the values of reaction- and self-diffusion coefficients of the same component become close, if not identical, as it should be from a physical viewpoint. 

When two species are involved, they may have different diffusion coefficients. Here it will be assumed that the two species might be two out of more than two species in a given mechanism, and that normalisation is referred to some species other than these two. Therefore both their diffusion coefficients need to be normalised. Let the two species be called O and R, and the reference species be called A. Then the normalisations are... 

Figure 12 Diffusion coefficients (Ds) of fast diffusion component (A) and slow diffusion component ( ) normalised by diffusion coefficients (D/D0) of PSs in deuterated toluene solution as a function of the PS concentration (CPS).

From the response of the injecting charge normalised to the saturated value based on Equation 8.1, the diffusion coefficient of dopant ions can be estimated. The diffusion constants in polyaniline [13] and poly(o-methoxyaniline) film [23] in various electrolytes at pH = 0 have been summarised. The diffusion constants of poly (o-methoxyaniline) are always larger than that of polyaniline, as the oxidation occurs by ejection of protons in poly (o-methoxyaniline), while in the case of polyaniline the oxidation occurs by injection of bulky anions as discussed previously. 

Fig. 4.25 Effective diffusion coefficients Deg from MA-DLS normalised with the DeffiqRg.eg = 2) versus the scattering vector q scaled with the effective radii of gyration as determined by SLS, for 7 grades of pyrogenic silica and different dispersion procedures (Babick et al. 2012c)...

Each reaction j has its own electrochemical parameters n, the number of transferred electrons p, the standard potential (normalised) k, the standard heterogeneous rate constant a, the transfer coefficient diffusion coefficient D and d, the ratio of D to D, the reference substance s, whose bulk concentration also normalises all the other concentrations. We shall use the subscript j here to avoid confusion with the symbol i, used below to denote current. In the following, the different species concentration samples (at, respectively, 0, h, 2h, etc or 0, h/2, 3h/2 etc) are written in the form, for example, c q, meaning species 1 at sample point 0, etc. 

F ure 12.15 Diffusion coefficient of polystyrene tracer in polyvinyl methyl ether gels as a function of tracer molecular weight. Diffusion coefficients normalised by ratio of molecular weight between crosslinks of gels. Reprinted with permission from [52]. Copyright 1992 American Chemical Society... 

Let the species Ox and Red with concentrations Cq and Cr diffuse in the solution in the direction perpendicular to the electrode. Dq and Dr are the diffusion coefficients. The initial conditions demand that the solution is homogeneous and that the concentrations are equal to c and c for t = 0. Outside the Nernst layer the concentrations are equal to c, the concentrations in the bulk. At the electrode surface, the fluxes of Ox and Red are identical and equal to the normalised faradaic current I /nFA (A electrode area). The charge-transfer resistance is defined as ... 

Figure 8.10 The normalised effective diffusion coefficients as functions of volume fractions for different values of D ID. ...

Another two DFE simulations are also carried out with different distributions of the fast diffusion phase in order to discover the upper bound and lower bound of the relation between the effective coefficient and the volume fraction. Figure 8.11 describes the two cross-sections of the two cases in the first case, the heterogeneous system is composed of two phases in parallel, while in the second case the system is composed of a centrally located fast diffusion phase and a surrounding slow diffusion phase. Figure 8.12 illustrates three relations between the normalised effective diffusion coefficients obtained from the randomly distributed, the parallel distributed and the centtal distributed fast diffusion phase. It can be seen clearly in Figure 8.12 that the Dj, - Vf relation calculated from the first case yields the upper bound while that from the second case yields the lower bound. 

Figure 8.12 The normalised effective diffusion coefficients of the random case shown in Figure 8.8, the parallel case and the central case shown in Figure 8.12 versus the volume fraction when Df/D = 1000.

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