Overall parabolic equation

Engineering design is usually associated with the exact economic optimum leading to a global minimum. On the other hand, only simple problems, such as determining the minimum value of a parabolic equation, have single optimum solutions. The level of irreversibility introduces thermodynamic imperfections that eventually decrease the overall performance of the system. Some optimization procedures are ... 

Fig. 6. Plotting of Cd desorption by chloride or citrate from Fe oxides formed at initial citrate/Fe(II) MRs of (a) 0 and (b) 0.1 based on overall parabolic diffusion equation.

Develop non-linear equations (i.e., parabolic equations) to relate overall liking/disliking of odor to the concentrations of the components. The equation is ... 

In certain circumstances even the parabolic rate law may be observed under conditions in which the oxide is porous and permeated by the oxidising environment". In these cases it has been shown that it is diffusion of one or other of the reactants through the fluid phase which is rate controlling. More usually however the porous oxide is thought to grow on the surface of a lower oxide which is itself growing at a parabolic rate. The overall rate of growth is then said to be paralinear - and may be described by the sum of linear and parabolic relationships (see equations 1.197 and 1.198). 

The characteristic time scale for the motion of the particle in the parabolic top barrier is the inverse barrier frequency, the sharper is the barrier, the faster is the motion. Typically, atom transfer barrier are quite sharp therefore the key time scale is very short, and the short-time solvent response becomes relevant instead of the long-time overall response given by the ( used in Kramers theory (see eq.(20)). To account for this critical feature of reaction problems, Grote and Hynes (1980) introduce the generalized Langevin equation (GLE) ... 

Incorporation of a diffusion term in nucleation and growth reaction models has been proposed by Hulbert [68]. Interface advance is assiuned to fit the parabolic law and is proportional to but the nucleation step is uninhibited. The overall rate expressions have the same form as the Avrami-Erofeev equation ... 

The authors analysis (38) uses film theory on the gas-side and penetration theory on the liquid side. The penetration theory was adopted as offering a more realistic basis to describe the diffusion and reaction. The set of parabolic partial differential equations which describe this diffusion and reaction were solved in conjunction with the customary boundary conditions, plus a number of subsidiary relationships which ensure electrical neutrality and are compatible with the initial loadings of reagent amine. Equilibrium was assumed to be established in the bulk, but otherwise the fluxes were not jointed to the overall material balances on the gas and liquid phases. Even so, the computation appears quite formidable. 

All symbols have their usual meaning and only more important ones are defined here. Cj is the concentration of component j in the aqueous phase (e.g. polymer, tracer, etc.). The viscosity of the aqueous phase, rj, may depend on polymer or ionic concentrations, temperature, etc. Dj is the dispersion of component j in the aqueous phase Rj and qj are the source/sink terms for component j through chemical reaction and injection/production respectively. Polymer adsorption, as described by the Vj term in Equation 8.34, may feed back onto the mobility term in Equation 8.37 through permeability reduction as discussed above. In addition to the polymer/tracer transport equation above, a pressure equation must be solved (Bondor etai, 1972 Vela etai, 1974 Naiki, 1979 Scott etal, 1987), in order to find the velocity fields for each of the phases present, i.e. aqueous, oleic and micellar (if there is a surfactant present). This pressure equation will be rather more complex than that described earlier in this chapter (Equation 8.12). However, the overall idea is very similar except that when compressibility is included the pressure equation becomes parabolic rather than elliptic (as it is in Equation 8.12). This is discussed in detail elsewhere (Aziz and Settari, 1979 Peaceman, 1977). Various forms of the pressure equation for polymer and more general chemical flood simulators are presented in a number of references (Zeito, 1968 Bondor etal, 1972 Vela etal, 1974 Todd and Chase, 1979 Scott etal, 1987). 

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