Anderson criterion

GP 9[ [R 16]The extent of internal transport limits was analysed for the wide fixed-bed reactor, using experimental data on carbon monoxide conversion and matter and process parameter data for the reactants [78]. The analysis was based on the Weisz modulus and the Anderson criterion for judging possible differences between observed and actual reaction rates. As a result, it was found that the small particles eliminate internal transport limitations. 

GP 11] [R 5] A judgement on heat-transfer limitations on the reaction rate according to the Anderson criterion was made [121]. This inequality predicts no such limitations in the boundary layer when the Anderson criterion is smaller than 1. Using process parameter data applied in a number of experiments, the highest value found was 2.2 10 so that no heat-transfer limitations have to be assumed. 

If the Anderson criterion is not satisfied then, as first pointed out by Mott (1966), since states are likely to become localized in the tail of a band, there exists a critical energy Ec (the mobility edge )f separating localized from non-localized states (Fig. 1.21). The simplest definition of Ec in terms of the behaviour of the conductivity a(E) is as follows ... 

The critical value of VJB for complete localization is about three. Since the band widths are of order 5 eV, a very large disorder potential is needed to localize all the electronic states. It was apparent from early studies of amorphous semiconductors that the Anderson criterion for localization is not met. Amorphous semiconductors have a smaller disorder potential because the short range order restricts the distortions of the bonds. However, even when the disorder of an amorphous semiconductor is insufficient to meet the Anderson criterion, some of the states are localized and these lie at the band edges. The center of the band comprises extended states at which there is strong scattering and... 

Criteria have also been developed for evaluating the importance of intraphase and interphase heat transfer on a catalytic reaction. The Anderson criterion for estimating the significance of intraphase temperature gradients is [J. B. Anderson, Chem. Eng. Sci., 18 (1963) 147] ... 

With respect to the heat transfer, the resistance of the channel wall is typically small as compared to the resistance within the porous catalytic layer. Strongly exothermic (or endothermic) reaction may cause a temperature gradient within the catalytic layer. To achieve isothermal behavior, the thickness of the catalytic layer should not exceed the value specified by Equation 9.13, which is a modified form of the Anderson criterion [14] ... 

To conclude this discussion, we point out the close correspondence between the criterion Eq. (5.58) for the existence of surface states and the Anderson criterion for the localization of electronic states in disordered solids [35]. Here, too, the existence of localized states is inferred, if the random variation W of local site potentials is comparable to the bandwidth B. For W < B, localized states are found in the wings of the band, while the band center is composed of extended states. If W > B, even the states at the center of the band become localized. 

In this case, the ML parameter estimates are obtained by minimizing the following quadratic optimality criterion (Anderson et al., 1978 Salazar-Sotelo et al. 1986)... 

To check whether the particle can be considered as isothermal - whether there are important concentration gradients or not - the criterion of Anderson [16] can be used. If... 

The model also assumes that the disorder is uniform. The disorder is introduced at the beginning of the calculations by the Anderson localization criterion, but the effects of disorder on the transfer of the electron from site to site is not considered at a microscopic level. Scaling theory, which is described next, considers the microscopic disorder and reaches some diflerent conclusions. 

Anderson, T.B. Jackson, R.A. Fluid mechanical description of fluidized beds stability of the state of uniform fluidization. I EC Fundam. 1968, 7,12. Verloop, J. Heertjes, P.M. Shock waves as a criterion for the transition from homogeneous to heterogeneous fluidization. Chem. Eng. Sci. 1970, 25, 825. 

Internal heat transport limitations were considered by J.B. Anderson (Chemical Engineering Science 18 (1963) 147) and according to the following criterion such limitations do not occur if... 

Another criterion for metallicity arises from electron localization induced by disorder proposed by Anderson. Electrons diffuse when the disorder is small, but at a critical disorder they do not diffuse (giving rise to zero conductivity). A transition from the metallic to the insulating state occurs as the disorder increases. Mott proposed that there exists a minimum metallic conductivity, a j , or maximum metallic resistivity, pmax. for which the material may still be viewed as being metallic, prior to the localization of electrons due to disorder. The Omin is given by where I is the mean free path of the electrons. Mott s... 

The minimum metallic conductivity criterion of Mott emphasizes the role of disorder. It was first realized by Anderson that electronic states originally extended can become spatially localized if the disorder is strong enough. Mott argued that in a disordered metal, the minimum possible value of the mean free path is the interatomic spacing. For stronger disorder, the picture of electron waves scattered randomly is inappropriate the electron state is localized, and one has a nonmetal. Thus, a minimum electrical conductivity, characterizes the... 

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