Effective retardation factor equation

Thirty-six undisturbed soil columns were taken on a 6 x 6 sampling grid immediately adjacent to the field core locations, and were brought to the laboratory. The columns were leached at 2 cm/d until steady state was reached, at which time a pulse of KC1 and napropamide was added to the inlet end. Affluent breakthrough curves for each chemical were fitted to the convection-dispersion equation by the method of moments (11). The effective retardation factor R, which may be calculated from the ratio of the chloride and napropamide vnap velocity parameters obtained by fitting the convection-dispersion equation, is equal to... 

In this discussion equation (8) is used to model the transport of TCE during the test. The parameters needed for the model include the effective dispersion/diffusion coefficients, retardation factor, and the pore fluid velocity. 

Finally, the effects of retardation and sorption/desorption were included in the model. Kaolinite has a low CEC (about 0.01 mg/g) and low buffering capacity (Mitchell, 1993), hence retardation for ff+ transport may not be significant. However, due to the change in pH environment, the tortuosity and the presence of other species, there may be a significant influence of retardation to the transport of H+ (Acar and Alshawabkeh, 1993). An experimentally obtained retardation factor of Jid was applied to the rate of concentration change term for the mass transport equation of H+. 

Adsorption of material, for example polymer, will always have the same gross effect on the position of the effluent compared with that of an inert tracer. The effluent will be retarded relative to the tracer (i.e. it will appear later) because of the retention process, although this also depends on the magnitude of the IPV/excluded volume effect—the product of quantities / F—as discussed in Section 7.2.1. Considering the case of a linear adsorption isotherm (i.e. adsorption, T, is linear with c) when there is no IPV/excluded volume effect (/= ), the retardation factor, Fr, as defined in Equation 7.12, is greater than unity. Hence, the component velocity of the adsorbed/retained species will be less than that of an inert tracer by the factor 1/Fr whereas the tracer travels at the same average velocity as the fluid. 

Petersen [12] points out that this criterion is invalid for more complex chemical reactions whose rate is retarded by products. In such cases, the observed kinetic rate expression should be substituted into the material balance equation for the particular geometry of particle concerned. An asymptotic solution to the material balance equation then gives the correct form of the effectiveness factor. The results indicate that the inequality (23) is applicable only at high partial pressures of product. For low partial pressures of product (often the condition in an experimental differential tubular reactor), the criterion will depend on the magnitude of the constants in the kinetic rate equation. 

Until now we have ignored an important factor. The electric field affects not only the surface charges of the particle, but also the ions in the electrical double layer. The counterions in the double layer move in a direction opposite to the motion of the particle. The liquid transported by them inhibits the particle motion. This effect is called electrophoretic retardation. Therefore the equation is only valid for D [Pg.77]

Generally, the rate of alkaline hydrolysis of a series of substituted phenyl benzoates was decreased in the presence of 0.5 m BiuNBr, the retardation being larger for esters with electron-donating substituents. The data from 22 esters were fitted to a multiparameter equation, the results showing that solvent electrophilicity was the main factor responsible for changes in the ortho, para and meta polar substituent effects with medium.15... 

The effects of micelles on the rates of nucleophilic aromatic substitution reactions (equations 42a-42e) follow a similar pattern. The rate constant for reaction (42a) was unaffected by the presence of micellar sodium dodecyl sulfate, even though 2,4-dinitrofiuorobenzene partitions strongly in favor of the micelles and the free base of glycineamide is not appreciably solubilized. On the other hand, the rate constant for the reaction with glycylglycine (equation 42b) decreased by a factor of 3-6 in micellar sodium dodecyl sulfate solution and increased by a factor of ca. 15 in the presence of CTAB (Herries et al., 1964). Since the rate retardation in NaLS solutions was observed to parallel the partitioning of 2,4-dinitrofluorobenzene in favor of the micellar phase and the... 

Hydrodynamic Retardation. Smoluchowski assumed in the derivation of his equations that )pair = Z)1+Z)2, but this is not true if the diffusing particles are relatively close to each other. When two particles come close, the liquid between them has to flow out of the gap, and this means that (a) the local velocity gradient is increased and (b) the flow type becomes biaxial elongation rather than simple shear (see Section 5.1). Both factors cause the effective viscosity to be increased, which means in turn that the mutual diffusion coefficient of the particles is decreased, the more so as the particle separation (h) is smaller. The phenomenon is called the Spielman-Honig effect. 

The diffusion equations just used are simplifications of more complex processes. The F factor was empirically derived and must take into account those matrix pore geometric factors contributing to decreases in diffusion rates. Such factors may include pore tortuosity, dead-end pores, and pore constrictions. Initial modeling studies suggest that constrictions, in particular, have large effects in retarding release (8,9). 

Therefore, the net effect of a homogeneous, rapid, reversible reaction is to retard the rate of diffusion of solute through the tissue. Solutions to this equation are identical to solutions of the pure diffusion equation (compare Equation 3-31 with Equation 3-52), except that the diffusion coefficient is reduced by a factor equal to the binding constant plus unity. These same equations can be used to evaluate penetration into tissues when more complicated equilibrium expressions are appropriate, by substituting the non-linear equilibrium expression into Equation 3-50 and solving the resulting equation (see [7]). 

When the particle size becomes comparable to the wavelength of light, a retardation effect occurs and has to be taken into account in analysis [44], Radiation damping becomes significant as the particle volume increases. These factors can be taken into account in the following equation. 

Here, we have also assumed that the particles do not interact before they are in contact and all collisions lead to doublet formation. Moreover, hydrodynamic interactions have been neglected. An experimental verification of this formula showed, not unexpectedly, deviations. The coagulation was slower than that predicted by the rate constant given in equation (1.31). Derjaguin, in 1966, proposed the reason for this was that the particles interact hydrodynamically when they were sufficiently close to each other. The dispersion medium has to be removed from the space between the particles when they approach one another and the motion of the particles is retarded. The effect is in many cases quite large, i.e. about a factor of 2. This can be expressed as a reduction in the diffusion coefficient. Honig and co-workers have derived an approximate equation for how the diffusion coefficient D(H) varies with the interparticle surface-to-surface distance H. The expression is as follows ... 

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