Glueckaufs approximation

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) ... 

Glucosyl-fructose, 24 237 D-Glucuronic acid, 4 711 Glucuronic pathway, of ascorbic acid biosynthesis, 25 762-763, 763-764 Glue, kaolin application, 6 688t Glueckauf approximation, adsorption, 1 608, 609... 

This relationship provides an extension of the Glueckauf approximation for systems in which more than one mass transfer resistance is significant,... 

A further generalization of the Glueckauf approximation is suggested by comparison of the moments for the simple linear rate plug flow model (model la) and the general diffusion model with axial dispersion (model 46). One may define an overall effective rate coefficient (k ) which includes both the effects of axial dispersion and mass transfer resistance ... 

The linear driving force (LDF) approximation is obtained when the driving force is expressed as a concentration difference. It was originally developed to describe packed-bed dynamics under linear eqm-librium conditions [Glueckauf, Trans. Far Soc., 51, 1540 (1955)]. This form is exact for a nonlinear isotherm only when external mass transfer is controlling. However, it can also be used for nonlinear sys-... 

The internal mass transfer rate coefficient, kg is difficult to evaluate. It can be approximated as the time-averaged value proposed by Glueckauf [42]. 

As in Eq. (51), Np = 60DvDve/Fdp2. For this case (r = 0), = 4ir2/60. Glueckauf and Coates (G5) have provided the linear-driving-force approximation of Eq. (37), which in the irreversible case becomes... 

The attention of many workers has been given to the equilibrium-limited case (proportionate-pattern) of multiple adsorption (D2, Wl, W3, W7, among others). In the constant-pattern case, Fujita s work has already been discussed (Section III, B, 2b). Also, using the theoretical-plate approximation to a packed column, plate-by-plate calculations can be carried out in the constant-pattern case exactly as for continuous (countercurrent) distillation this treatment is suggested from work on chromatographic and displacement problems by Mair (M2), Spedding (S6), and Glueckauf (G3). Moreover the linear-equilibrium result can be extended, in a nearly trivial fashion, to any number of components. 

The adequacy of the Glueckauf linear rate approximation may be judged by comparing the curves calculated from the diffusion models with the curves calculated fom the linear rate expression, which are also shown in Figure 8.17. The linearized rate approximation provides a good representation of the desorption curves over the entire range of nonlinearity. The approximation is... 

In principle, mass transfer in solid particles is distributed in both time and distance, calling for the use of a PDE (Pick s equation) for a rigorous description of the process. In an elegant study conducted in the 1950s, it was shown by Glueckauf that the results of the formal treatment can be approximated by a volumetric solid-phase mass transfer coefficient, given by... 

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