Transport corrected diffusivity

The relationship between the corrected transport diffusivity and the self-diffusivity is more complex. Darken assumed that these quantities should be... 

Fig. 16 Variation of diffusivity with carbon number for linear alkanes in silicalite at 300 K showing comparison between self-diffusivities and corrected transport diffusivities obtained by different techniques, o MD simulation [74] hierarchical simulation [75] + QENS [78] A PFG NMR [76] V single crystal membrane [65] A ZLC [77]. The ZLC values were calculated based on the assumption of isotropic diffusion in an equivalent spherical particle. The present figure has been modified by the addition of further experimental data from a figure originally presented by Jobic [78]...

The concentration profile is steeper for the MacCormack method than for the upstream derivatives, but oscillations can still be present. The flux-corrected transport method can be added to the MacCormack method. A solution is obtained both with the upstream algorithm and the MacCormack method and then they are combinea to add just enough diffusion to ehminate the oscillations without smoothing the solution too much. The algorithm is comphcated and lengthy but well worth the effort (Refs. 37, 107, and 270). 

The effect of this subtle difference in device function can be seen when the measured signal in the presence of biofouling is modeled. As a model patient, we considered the transient response of an individual with basal insulin provided after each of the three daily meals. Blood glucose dynamics predicted by Sorensen was corrected for diffusion to subcutaneous tissue using the mass transport model of Schmidtke et al.24 25 Figure 11.1 shows a model comparison between the sensor response of an electrochemical sensor and an optical sensor with an assumed... 

Figure 9 Use of the Koutecky-Levich equation to correct for diffusive transport effects on the anodic dissolution of Cu in 1 mol dm-3 NaCl recorded on a rotating disk electrode.

In this text, the conversion rate is used in relevant equations to avoid difficulties in applying the correct sign to the reaction rate in material balances. Note that the chemical conversion rate is not identical to the chemical reaction rate. The chemical reaction rate only reflects the chemical kinetics of the system, that is, the conversion rate measured under such conditions that it is not influenced by physical transport (diffusion and convective mass transfer) of reactants toward the reaction site or of product away from it. The reaction rate generally depends only on the composition of the reaction mixture, its temperature and pressure, and the properties of the catalyst. The conversion rate, in addition, can be influenced by the conditions of flow, mixing, and mass and heat transfer in the reaction system. For homogeneous reactions that proceed slowly with respect to potential physical transport, the conversion rate approximates the reaction rate. In contrast, for homogeneous reactions in poorly mixed fluids and for relatively rapid heterogeneous reactions, physical transport phenomena may reduce the conversion rate. In this case, the conversion rate is lower than the reaction rate. 

It should be noted here that Eq. 30D can be used to correct for mass transport only when steady-state measurements are concerned, such as those obtained with the RDE. It is not applicable for polarography or for any other method in which the current varies with time. The reason is rather subtle when such methods are used, the activation controlled current and the diffusion controlled current depend differently on time. As a result, the dependence of measured current on time varies with potential in the region of mixed control, and a simple correction for diffusion limitation, following Eq. 30D is not valid. 

In contrast to all the other techniques considered in this paper, in sorption experiments molecular migration is observed under nonequilibrium sorption conditions. Therefore, instead of self-diffusivities, D, in this case transport diffusivities. A, are derived. It is generally assumed (see, e.g.. Refs. 366) that the corrected diffusivities. Do,... 

In order to determine the activation energy of the difiuaon, the uptake experiments were conducted at temperatures in the range 398 to 473 K. In Table 2, results are compiled. The errors of the transport diffusion coefficient are estimated to be 0.75 10 cmVs. A thermodynamic correction [13] of the transport diffiiavity has not been applied. However, since the... 

One such approach is based on the concepts of non-linear flux limiters introduced by van Leer [193] and Boris and Book [13]. The work of Boris and Book [13] and Zalesak [213] determine the basis for a group of methods called flux correction transport (FCT) schemes. The schemes of Smolarkiewicz [175] is representative for this group. In the FCT schemes a first order accurate monotone scheme is converted to a high resolution scheme by adding limited amounts of an anti-diffusive flux. The work of van Leer [193, 195], on the other hand, represents an extension of the ideas of Gudunov [64] to higher order accuracy. 

Fig. 17.2 Tafel plots for the (normalized, dimensionless) current, yjy, that accompanies hydrogen evolution in a solution containing 3.4 mM HCl + 1.0 M KCl, corrected for diffuse-double-layer effects, mass transport controlled kinetics and ohmic potential drop, measured at three temperatures (5, 45, 75°C all results fall on the same line of this reduced plot) at a dropping mercury electrode. The slope obtained from this plot is 0.52, independent of temperature. (Based on data from E. Kirowa-Eisner, M. Schwarz, M. Rosenblum, and E. Gileadi, J. Electroanal. Chem. 381, 29 (1995) and reproduced by the authors.)...

NMR PFG measurements determine the tracer or self-diffusivity (D ) under equilibrium conditions with no concentration gradient. n any sorption rate measurement it is the transport diffusivity under the influence of a concentration gradient which is measured. In general these two quantities are not the same but the relationship between them can be established from irreversible thermodynamics. (17,18) In the low concentration limit the thermodynamic correction factor vanishes and the transport and self diffusivities should approach the same limit. Since ZLC measurements are made at low concentrations within the Henry s Law region the diffusivity values should be directly comparable with the NMR self-dif fusivities. ... 

The diffusivity (D) defined in this way is not necessarily independent of concentration. It should be noted that for diffusion in a binary fluid phase the flux (/) is defined relative to the plane of no net volumetric flow and the coefficient D is called the mutual diffusivity. The same expression can be used to characterize migration within a porous (or microporous) sohd, but in that case the flux is defined relative to the fixed frame of reference provided by the pore walls. The diffusivity is then more correctly termed the transport diffusivity. Note that the existence of a gradient of concentration (or chemical potential) is implicit in this definition. 

Abstract Neutron scattering was first used to derive the self-diffusivities of hydrocarbons in zeolites, but transport diffusivities of deuterated molecules and of molecules which do not contain hydrogen atoms can now be measured. The technique allows one to probe diffusion over space scales ranging from a few A to hundreds of A. The mechanism of diffusion can, thus, be followed from the elementary jumps between adsorption sites to Lickian diffusion. The neutron spin-echo technique pushes down the lower limit of diffusion coefficients, traditionally accessible by neutron methods, by two orders of magnitude. The neutron scattering results indicate that the corrected diffusivity is rarely constant and that it follows neither the Darken approximation nor the lattice gas model. The clear minimum and maximum in diffusivity observed by neutron spin-echo for n-alkanes in 5A zeolite is reminiscent of the controversial window effect . 

Coherent QENS measurements and MD simulations have been performed for N2 and CO2 in silicalite [30,31]. It has been found that the self-diffusivities of the two gases decrease with increasing occupancy, while the transport diffusivities increase. For a comparison with other systems, it is appropriate to remove the influence of the thermodynamic correction factor and to discuss the collective mobility in terms of the corrected diffusivity (also called Maxwell-Stephan diffusivity). Dq(c) is directly obtained from the Simula-... 

Fig. 9 Transport diffusivities (squares) and corrected diffusivities (circles) obtained for CF4 in silicalite at 200 K, by QENS (filled symbols) and simulations (open symbols)...

Do is generally referred to as the corrected or Maxwell-Stefan diffusivity, and r is called the thermodynamic correction factor, which corrects for the nonlinearity between the pressure and the concentration of the adsorbate. Often, the corrected diffusivity is used in experimental studies where the transport diffusion is measured. Although Do can still depend on the concentration, in systems near the saturation limit or in the low concentration (Henry s law) regime this dependence has been experimentally shown to be quite small, and the use of the corrected diffusivity helps in directly comparing experimental results under different conditions [3]. 

When the isotherm is linear, the thermodynamic correction factor is unity, meaning that the corrected diffusivity is the transport diffusivity at zero loading conditions. 

The above equations (10.2-34 to 10.2-36) are non-linear due to the thermodynamic correction factor in the transport diffusivity term. The method we have been using in solving nonlinear partial differential equations is the orthogonal collocation method. We again apply it here, and to do so we define the following non-dimensional variables and parameters ... 

The intracrystalline diffusivity, D, is a function of concentration in the crystal. It is a constant only when the adsorption isotherm is linear, and unlike the diffusion process in the macropore the intracrystalline diffusion is activated. Intracrystalline diffusion has been dealt with in Section 10.2, where we have shown that the transport diffusivity is related to the corrected diffusivity as follows ... 

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