Glueckauf model

The main questions which arise with respect to the Glueckauf model discussed above concern the uniqueness and the physical justification of the postulated local charge variation represented by Eq. (44). With reference to the former question, some model calculations of SN for f(CFL) distributions of different shapes and widths may be noted 122,123). it was shown that the magnitude and form of the deviation from EVM behaviour for a given B/A ratio is determined largely by the shape of the f(CFL) function near the lower limit A. The precise form of f(CFL) in the higher CFL region appears to have a relatively minor effect on the conformity of SN to the power law of Eq. (46) or the value of q. 

Table 4. Recent interpretations of coion sorption and diffusion based on the Glueckauf model U0,120)... 

Using the modified Glueckauf model described above, and employing his simplifying mathematical assumptions (the correctness of which has been confirmed by the authors with the aid of numerical methods), one obtains... 

Figure 4.30 Shape of a self-sharpening boundary as calculated by Helffeiich-Glueckauf model, for several combinations of diffusion resistances and separation factor. A = 10, = 6 cm, and...

Also shown are the corresponding curves calculated for the same system assuming a diffusion model in place of the linear rate expression. For intracrystalline diffusion k = 15Dq/v, whereas for macropore diffusion k = 15e /R ) Cq/q ), in accordance with the Glueckauf approximation (21). 

Fig. 15. Theoretical breakthrough curves for a nonlinear (Langmuir) system showing the comparison between the linear driving force (—), pore diffusion (--------------------), and intracrystalline diffusion (-) models based on the Glueckauf approximation (eqs. 40—45). From Ref. 7.

The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

Membrane deterioration may be merely caused by decrease of acetyl content(C ) in the active surface layer as a result of hydrolysis or oxidation, not by structure change. Analysis was carried out based on solution-diffusion model proposed by Lonsdale etal( ), using their measured values of solute diffusivity and partition coefficient in homogeneous membrnaes of various degree of acetyl content and also using those values of asymmetric membranes heat treated at various temperatures measured by Glueckauf(x) ... 

In order to further substantiate this conclusion, it is of interest to compare it with the prediction obtained from a simple theoretical model. Glueckauf s well-known transport model (19, p. 449-453), supplemented by the more modern concept of hydro-dynamic dispersion, is well suited for this purpose. The model simulates dispersion-affected solute transport with ion exchange for which diffusion processes are rate limiting. In his development, Glueckauf assumes 1) exchange takes place in porous... 

Figure 10 Parameters used in Glueckauf s discontinuous model for electrolytes, and their comparison with static permittivity calculated as a function of distance from the ion centre. (A) cations (B) anions C is the centre of the water dipole A = the range where c = 1...

Table 5 Comparison of dielectric decrements calculated on Glueckauf s discontinuous model with bracketed) experimental values... 

The ideal model should be applied to get information about the thermodynamic behavior of a chromatographic column. Through work by Lapidus and Amundson (1952) and van Deemter et al. (1956) in the case of linear isotherms and by Glueckauf (1947, 1949) for nonlinear isotherms, considerable progress was made in understanding the influences of the isotherm shape on the elution profile. This work was later expanded to a comprehensive theory due to improved mathematics. Major contributions come from the application of nonlinear wave theory and the method of characteristics by Helfferich et al. (1970, 1996) and Rhee et al. (1970, 1986, 1989), who made analytical solutions available for Eqs. 6.41 and 6.42 for multi-component Langmuir isotherms. 

As shown by Glueckauf [27,28], the effects of the different phenomena contributing to band broadening are additive. Since in this simplified model axial dispersion is neglected and the kinetics of adsorption-desorption is also ignored, the... 

The solution to this problem (Eqs. 14.2 to 14.4) has been derived by Glueckauf and Coates [7], and Michaels [8] derived the solution of a similar problem (Eqs. 14.2,14.4, and 14.5) assuming as kinetic model the liquid film linear driving force model ... 

Because of its simple mathematical form and its physical consistence, the Linear Driving Force Model (LDFM) is commonly used to describe intraparticular mass transfer kinetics. Glueckauf and Coates first Introduced LDFM [18], which stated that the uptake rate of a species in the particle is proportional to the difference between the concentration of that species at the outer surface of the particle and its average concentration in the interior of the particle ... 

Several modifications of this model can be found in the literature. One that is frequently used considers the mass transfer resistance in the solid phase to be dominant. As proposed by Glueckauf and Coates (1947), an analogue linear driving force approach for the mass transfer in the solid can then be applied and Equations 6.73 and 6.35 are replaced by Equations 6.74 and 6.75. Mathematically, this linear driving force is modeled as the difference between the overall solid loading of Equation 6.17 and an additional hypothetical loading eq, which is in equilibrium with the liquid phase concentration ... 

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