Coefficient reducing

Graham-Uranoff They studied multicomponent diffusion of electrolytes in ion exchangers. They found that the Stefan-Maxwell interaction coefficients reduce to limiting ion tracer diffusivities of each ion. 

It can be noted that in general this result predicts that the ratio of the dispersion coefficient to the free-solution diffusion coefficient is different from the ratio of the effective mobility to the free-solution mobility. In the case of gel electrophoresis, where it is expected that the (3 phase is impermeable (i.e., the gel fibers), the medium is isotropic, and the a phase is the space between fibers, the transport coefficients reduce to... 

The standard Rodbard-Ogston-Morris-Killander [326,327] model of electrophoresis which assumes that u alua = D nlDa is obtained only for special circumstances. See also Locke and Trinh [219] for further discussion of this relationship. With low electric fields the effective mobility equals the volume fraction. However, the dispersion coefficient reduces to the effective diffusion coefficient, as determined by Ryan et al. [337], which reduces to the volume fraction at low gel concentration but is not, in general, equal to the porosity for high gel concentrations. If no electrophoresis occurs, i.e., and Mp equal zero, the results reduce to the analysis of Nozad [264]. If the electrophoretic mobility is assumed to be much larger than the diffusion coefficients, the results reduce to that given by Locke and Carbonell [218]. 

If mx = m2, then An = B = 0 and the coefficients (8.2) reduce to those for a homogeneous sphere. We also have ima 0An = lima >0 = 0 therefore, in the limit of zero core radius the coefficients (8.2) reduce to those for a homogeneous sphere of radius b and relative refractive index m2, as required. When m2= 1, the coefficients reduce to those for a sphere of radius a and relative refractive index mx this gives us yet another check on the correctness of our solution. 

The axial Poiseuille flow occurs when 6 = n/2, with x taking the role of z, and y taking the role of De/2 — r. The metric coefficient reduces to h = De/2 — y = r. The expected axisymmetric momentum equations for axial Poiseuille flow can be recovered by substituting f — ur and carrying out the independent variable transformation to exchange r for v. The chain rule for the independent variable transformation provides that... 

The customary techniques used by the formulator to overcome several everyday problems are also discussed below. These techniques include formulation processes and materials to control flow, extend temperature range, improve toughness, match thermal expansion coefficients, reduce shrinkage, increase tack, and modify electrical and thermal conductivity. 

When one is far from resonance, then a further simplification is possible. Kleinman74 showed [15] that energy is simply exchanged among the fields E Ej, and Ek along the three axes i, j, and k, so that the suffixes can be interchanged freely this means that the three fields act independently and can be applied in arbitrary order (at resonance, one field will distort the electronic cloud in such a way that a second field will act on a severely perturbed electron configuration). Thus, when this is not a problem (i.e., when one is far from resonance) for SHG the 27 (or 18) coefficients reduce to 10 ... 

Note Since the model is linear for the special case considered, the same equation is also satisfied by the other three variables.) The following observations may be made from Eq. (98) that expresses the dimensionless dispersion coefficient A (i) The first term describes dispersion effects due to velocity gradients when adsorption equilibrium exists at the interface. We note that this expression was first derived by Golay (1958) for capillary chromatography with a retentive layer, (ii) The second term corresponds to dispersion effects due to finite rate of adsorption (since this term vanishes if we assume that adsorption and desorption are very fast so that equilibrium exists at the interface), (iii) The effective dispersion coefficient reduces to the Taylor limit when the adsorption rate constant or the adsorption capacity is zero, (iv) As is well known (Rhee et al., 1986), the effective solute velocity is reduced by a factor (1 + y). (v) For the case of irreversible adsorption (y — oo and Da —> oo), the dispersion coefficient is equal to 11 times the Taylor value. It is also equal to the reciprocal of the asymptotic Sherwood number for mass transfer in a circular... 

The output solutions contain only enriched products without auxiliary ion impurities. At the same time that the ion separation takes place concentration of the mixture components occurs. The product concentration in column I equals the total concentration of the ion mbmire put into the column and the product concentration in column II, that is, the ion-displacer concentration. This is riiy, other conditions being equal, the use of more concentrated solutions is beneficial. However, it should be taken into account that increase in the total concentration of the mixture of ions separated may result in a decrease in the sorbability differences between the separated and auxiliary ions. Moreover, the resultant decrease in the single-stage separation coefficient reduces the efficiency of the column operation. 

Figure 6.10 The short-time diffusion coefficient reduced by that of the zero-concentration limit, Dsi(f))/Do, as a function of particle concentration

Water dynamics is slowed down by the electric field of the cation, as revealed by diffusion coefficient reduced by a factor of two, compared with pure SPC/E water [132]. A reduction of D of water in ionic solutions is also observed experimentally, with values, determined with the tracer technique, ranging from 1.22 10-5 cm /s for Li+ to 0.52 and 0.53 10 5 cm /s for Fe3+ and Al3+, respectively [206]. 

Table 3.12 The modulating coefficients, reduced to their simplest ratios, for the linear combinations over the uj[here and vj [here from equation 3.21 to form the group orbitals of the F3 triangle displayed in the 3rd and 4th columns of Figure 3.6 using the vector surface harmonics of Table 3.11.

This equation shows that the electric field increases the drift velocity, whereas the drag coefficient reduces it. We define the proportionality constant between the drift speed (the magnitude of the drift velocity) and the field to be the mobility, /t, of the ion (a positive number), giving the following relation ... 

Partial parameter, cubic equation of state 2d virial coefficient, density expansion Partial molar second virial coefficient Reduced second virial coefficient... 

It follows from Eq. (12.30) that as h goes to zero, the exponential term goes to unity and the rejection coefficient reduce to zero. On the contrary as Jy tends to become very large, the exponential term tends towards infinity, and the rejection coefficient approaches a specific limiting value for a given solute. 

The rank of this matrix is again three. With n = 6, the number of independent reactions is now 6 -3 = 3. With the above given elements of the matrix D, the vector equation for the stoichiometric coefficients reduces to... 

Sedimentation The settling of suspended particles or droplets due to gravity or an applied centrifugal field. The sedimentation rate (or velocity) divided by acceleration is termed the sedimentation coefficient. The sedimentation coefficient extrapolated to zero concentration of sedimenting species is termed the limiting sedimentation coefficient. The sedimentation coefficient reduced to standard temperature and solvent is termed the reduced sedimentation coefficient. If the sedimentation coefficient is... 

The Stokes law or hydrodynamic diffusion coefficient should then be observed. Thus at high salt concentration, the effects of ionic shielding discussed in Section 9.2 are unimportant, and the diffusion coefficient reduces to that of the isolated macroion. Physically this occurs because now a fluctuation in the counterions can be neutralized by a concentration fluctuation in ions of opposite charge, so that large electrical forces need not be exerted on the macroion. 

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