Kinetic gradient systems

From the Korzukhin theorem follows an important conclusion. Any dynamical systems of the form (4.58) may be regarded as those corresponding to slow dynamics of a standard kinetic system. In other words, the behaviour of dynamical systems can be modelled using chemical reactions. In particular, any of the gradient systems may be modelled in this way. As will be shown in Chapter 5, catastrophes occurring in complex dynamical systems are equivalent to catastrophes appearing in much simpler systems. The latter can be classified — these are so-called standard forms. The standard forms are of the form (4.58) and it follows from the Korzukhin theorem that they can be modelled by the standard equations of chemical kinetics (4.27), corresponding to a realistic mechanism of chemical reactions. 

As an application of the statement on the inverse problem treated in the previous section the question will be investigated here (based upon Toth, 1979) what is the role of gradient systems in chemical reaction kinetics ... 

If the induced kinetic differential equation of a complex chemical reaction with a weakly realistic mechanism is a gradient system, then the mechanism itself is canonically cross-catalytic. 

Toth, J. (1979) Gradient systems are cross-catalytic. React. Kinet. Catal. Lett., 12, 253-1. 

If / / is a Shahshahani gradient system then it is a kinetic system too, because the general form of these equations is... 

Conservative gradient systems among those kinetic equations which have a second order right hand side can only be of the following form ... 

The majority of experimental studies are made in closed systems (without gain or loss of matter), isothermal (exchanging heat with the exterior), and homogeneous. However, in industrial practice, and in the study of rapid reactions and of strongly exothermic processes, one encounters kinetically open systems , which imply the existence of concentration gradients (continuous flow processes) or of thermally adiabatic processes (flames, explosions, etc.). The following definitions apply to closed systems, and their application to open systems will not be detailed until Chapter 3. 

Deviations from the ideal frequentiy occur in order to avoid system complexity, but differences between an experimental system and the commercial unit should always be considered carefully to avoid surprises on scale-up. In the event that fundamental kinetic data are desired, it is usually necessary to choose a reactor design in which reactant and product concentration gradients are minimized (36), such as in the recycle (37) or spinning basket reactor designs (38,39). 

Processes in which solids play a rate-determining role have as their principal kinetic factors the existence of chemical potential gradients, and diffusive mass and heat transfer in materials with rigid structures. The atomic structures of the phases involved in any process and their thermodynamic stabilities have important effects on drese properties, since they result from tire distribution of electrons and ions during tire process. In metallic phases it is the diffusive and thermal capacities of the ion cores which are prevalent, the electrons determining the thermal conduction, whereas it is the ionic charge and the valencies of tire species involved in iron-metallic systems which are important in the diffusive and the electronic behaviour of these solids, especially in the case of variable valency ions, while the ions determine the rate of heat conduction. 

The kinetics of spinodal decomposition is complicated by the fact that the new phases which are formed must have different molar volumes from one another, and so tire interfacial energy plays a role in the rate of decomposition. Anotlrer important consideration is that the transformation must involve the appearance of concenuation gradients in the alloy, and drerefore the analysis above is incorrect if it is assumed that phase separation occurs to yield equilibrium phases of constant composition. An example of a binary alloy which shows this feature is the gold-nickel system, which begins to decompose below 810°C. 

When the mass transfer resistances are eliminated, the various gas-phase concentrations become equal a/(/, r, z) = j(r, z) = a(r, z). The very small particle size means that heat transfer resistances are minimized so that the catalyst particles are isothermal. The recycle reactor of Figure 4.2 is an excellent means for measuring the intrinsic kinetics of a finely ground catalyst. At high recycle rates, the system behaves as a CSTR. It is sometimes called a gradientless reactor since there are no composition and temperature gradients in the catalyst bed or in a catalyst particle. 

As an example, Ji can be a heat flux and X a temperature gradient. The thermodynamic flirxes determine the irreversible time evolution of a system to thermodynamic equilibrium, e.g. a temperature difference can be equalized by a heat flux. In general, the kinetic coefficients A - are non-zero for i implying several so-called... 

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