Arithmetic average model

Other results also confirm the important role of internal diffusion. Experimental activation energies (67—75 kJ mol"1) of the sucrose inversion catalysed by ion exchangers [506—509] were considerably lower than those of a homogeneously catalysed reaction (105—121 kJ mol"1) [505, 506,508] and were close to the arithmetic average of the activation energy for the chemical reaction and for the diffusion in pores. The dependence of the rate coefficient on the concentration in the resin of functional groups in the H+-form was found to be of an order lower than unity. A theoretical analysis based on the Wheeler—Thiele model for a reaction coupled with intraparticle diffusion in a spherical bead revealed [510,511] that the dependence of the experimental rate coefficient on acid group concentration should be close to those found experimentally (orders, 0.65 and 0.53 for neutralisation with Na+ and K+ ions respectively [511] or 0.5 with Na+ ions [510]). 

Simple Mean (SM) amoimts to considering all the available model outcomes in a simple arithmetic average (Polikar 2006) ... 

The effective diameter given by this model is very close to the arithmetic average of the maximum and minimum grain diameter for a narrow grain size range, but for a wider size range it is lower because the smaller particles contribute more surface area. 

Orthogonalized basis states have been used in the calculations by Fritsch et and Stolterfoht. Moreover, Hermitian symmetric matrix elements have been achieved by Pfeiffer and Garcia using the arithmetic averaging procedure. Moreover, Hermitian symmetry is assumed for the analytic model matrix element. ... 

Fig. 19-13. Three-parameter averaging-time model fitted through the arithmetic mean and the second highest 3-hr and 24-hr SOj concentrations measured in 1972 a few miles from a coal-burning power plant. Source From Larsen (21).

Several models have been proposed to estimate the thermal conductivity of hydrate/gas/water or hydrate/gas/water/sediment systems. The most common are the classical mixing law models, which assume that the effective properties of multicomponent systems can be determined as the average value of the properties of the components and their saturation (volumetric fraction) of the bulk sample composition. The parallel (arithmetic), series (harmonic), or random (geometric) mixing law models (Beck and Mesiner, 1960) that can be used to calculate the composite thermal conductivity (kg) of a sample are given in Equations 2.1 through 2.3. 

It is fonnd from Figure 16.3 that, the dielectric constants of the composites are non-linearly dependent on volume % of BNN. This shows that the constituent capacitors formed by dielectrics fillers and polymer in the composites are not in parallel combination. From Figure 16.3, it is clear that the inverse of dielectric constant cnrve is not in a harmonic pattern, constituent capacitors formed by dielectrics fillers and polymer in the composites is not in series combinatiom One can choose to model composites as having capacitance in parallel (upper bound) or in series (lower bound). In practice, the answer will lie somewhere between the two. Physically, in composites with (0-3) structures which generally conform to special logarithmic equation, the relation assumes the form of Lichteneker and Rother s (Lich-teneker, 1956) more appropriate to composite stractures where the two-component dielectrics are neither parallel nor perpendicular to the electric field that is, the vahd averages are neither arithmetic nor harmonic. 

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