Rate laws continued parallel

However, we have to reflect on one of our model assumptions (Table 5.1). It is certainly not justified to assume a completely uniform oxide surface. The dissolution is favored at a few localized (active) sites where the reactions have lower activation energy. The overall reaction rate is the sum of the rates of the various types of sites. The reactions occurring at differently active sites are parallel reaction steps occurring at different rates (Table 5.1). In parallel reactions the fast reaction is rate determining. We can assume that the ratio (mol fraction, %a) of active sites to total (active plus less active) sites remains constant during the dissolution that is the active sites are continuously regenerated after AI(III) detachment and thus steady state conditions are maintained, i.e., a mean field rate law can generalize the dissolution rate. The reaction constant k in Eq. (5.9) includes %a, which is a function of the particular material used (see remark 4 in Table 5.1). In the activated complex theory the surface complex is the precursor of the activated complex (Fig. 5.4) and is in local equilibrium with it. The detachment corresponds to the desorption of the activated surface complex. 

Consider a continuous fiber-reinforced ceramic as a multiphase system where the individual phases are parallel to one another and to the uniaxial loading direction. The fibers (or fiber bundles), matrix, and interface zone are treated as individual phases. In general, each phase undergoes elastic-plastic (creep) deformation. In the present analysis, the creep rate of each phase, e is assumed to obey a general creep law of the following form... 

The continuity of the flow rate over the thickness of the filtration domain dictates that the throughflow velocity be independent from z, i.e. u(z) = U. Integrating Darcy s law over each of the two porous media can be written by a direct adaptation of [14.8] to the notations of Figure 14.5. The force of gravity is oriented downward, parallel to the Oz axis. We obtain ... 

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