Schneider rate equations

From the undisturbed nucleation rate it is possible to calculate the undisturbed sum of radii, total surface area, and volume of the caps per unit volume, by applying the Schneider rate equations [3,119]. With the shish growth rate, the undisturbed total length, surface area, and volume of the cylinders per unit volume can be calculated from the Eder rate equations [3,15]. A correction for swallowing of nuclei by growing crystallites and impingement of these crystallites is made to calculate the real volume fraction of semicrystalline material [122-125]. The undisturbed volume fractions of the caps and cylinders are added up in this correction. 

Another method is to independently measure the adsorption equilibrium constants of the components involved, and use them in determining the rate equation. Since the adsorption equilibrium values are very different under actual reaction conditions than under the conditions they are normally determined, care must be taken to ensure their measurement under reaction conditions. A chromatographic method based on the use of central moments appears to be particularly useful (Kubin, 1965 Kucera, 1965 Schneider and Smith, 1968). A combination of this method with statistical analysis is a sound strategy for determining the rate equation for a given reaction. This method has been successfully employed by Raghavan and Doraiswamy (1977) for the isomerization of butenes. 

In the developments already presented, the rate equations used were simple power law expressions. But, as discussed previously, catalytic rate equations are much more complex and often require the use of LHHW models. Many attempts have been made to incorporate these models in the analysis (e.g., Chu and Hougen, 1962 Krasuk and Smith, 1965 Roberts and Satterfield, 1965, 1966 Hutchings and Carberry, 1966 Schneider and Mitschka, I966a,b Kao and Satterfield, 1968 Rajadhyaksha et al., 1976 see in particular Aris, 1975 Luss, 1977). Clearly, graphical representation becomes cumbersome when a large number of adsorbed species is involved. However, the problem is quite tractable where only one species is adsorbed. 

Schneider et al. [66] differentiated Equation (15.8) with respect to time and obtained a system of differential equations, called the rate equations, enabling the creation of auxiliary functions 0i(t) interrelated in the following way ... 

The effect of diffusion in such a reaction can be determined by writing the concentrations of B, R, and S in terms of the concentration of A. This allows us to use the mass balance of Equation 7.14a with the rate term written exclusively in terms of the concentration of A within the pellet (Schneider and Mitschka, 1966b Xu and Chuang, 1997) ... 

Other authors [64,140,141,145] have based their calculations on Ozawa s equation extended by Billon et al. to nonconstant cooling rates (Eq. 15.12). These kinetic laws have been modified to integrate flow effects. To avoid such simplified approaches, some authors used the most general form of Avrami s theory (Eq. 15.7 and Eq. 15.8), either in its initial form [148] or after the mathematical transformations proposed by Schneider et al. [66] or Haudin and Chenot [67]. Schneider et al. and Haudin and Chenot obtained a set of differential equations (see Section 15.3.3) that is more... 

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