Kinetically attainable region

From the above said, it may be concluded that a detailed kinetic model of coal combustion process that combines all three basic processes can not virtually be constructed, as it is impossible to do for each process separately. Therefore, the empirical models based on separation and experimental study of the limiting stages are extensively used. Such models separately do not reveal general regularities and do not allow the generalized conclusions to be drawn. The thermodynamic model makes it possible to study the whole attainability region and hence to consider states of the considered system as a whole and to keep track of the variation in the amounts of any component as a function of some or other kinetic constraints. The latter are written, as was shown above, easily enough even for such complex processes as coal combustion. 

Pb being product concentration in the permeate. Once steady state conditions are attained, all enzyme is virtually confined within region B at a concentration Eb=N/Vb. Therefore, enzyme kinetics in region B can be experimentally determined through Equation 32. For details on the initial conditions and integration of the set of Equations 30 and 31 readers are referred to Reference 30. 

Figure 6.7 illustrates the attainable region for van de Vusse kinetics (van de Vusse, 1964), based on the reactions ... 

A systematic method for the construction of the attainable region using CSTRs and ITRs, with or without mixing and bypass, for a system of chemical reactions, as presented by Hildebrandt and Biegler (1995), is demonstrated for van de Vusse kinetics ... 

Because of their asymmetry, CDs exhibit chiral effects towards chiral molecules under FAB" and MALDl conditions. The main ambiguity of these studies remains regarding the environment in which chiral recognition occurs, whether in the bulk matrix, in the selvedge vaporization region, or in the gas phase. Besides, neither MALDI nor EAB ensure attainment of purely kinetic or equilibrium conditions so as that quantitative interpretation of the MS patterns in terms of relative stabihty of diastereomeric host/guest intermediates or transition stmctures... 

The writers have found in their laboratory that invariably after a certain burnoff (depending upon the reactor, temperature, and sample), a subsequent extended period of constant reaction rate, expressed in grams of carbon reacting per unit time, is attained. In this bumoff region, there obviously is equilibrium between the rate of formation of the surface-oxygen complex and its removal with a carbon atom. It is felt that this is the reaction rate most characteristic of a given temperature and should be used in kinetic calculations. In principle, Wicke (31) concurs with this reasoning and reports reactivity data only after the sample has attained a total surface area which is virtually constant. 

Therefore, cp = /T. There was an opinion that a2(oJD c [or a(a>a/D c)1/2] is a generally applicable criterion of the attainment of the kinetic region. However, this is not always true. In the case of a reversible first-order reaction ... 

A detailed study of model (16) for CO oxidation on polycrystalline platinum was carried out by Makhotkin et al. [139]. Numerical experiments revealed that the bulk diffusion effect on the character of reaction dynamics is rather different and controlled by the following factors (1) the initial composition of catalyst surface and bulk, (2) the steady state of its surface and bulk, and (3) the position of the region for slow relaxations of kinetic origin (see ref. 139). As a rule, diffusion retards the establishment of steady states, but the case in which the attainment of this state is accelerated by diffusion is possible. 

The objective function (7) in accordance with the general purpose of MEIS that was mentioned in the introduction, i.e., finding the state with extreme value of the system property of interest to a researcher, in this case determines the extreme concentration of the given set of substances. Equality (8) represents a material balance. Expression (9) represents the region of thermodynamic attainability from point y. It is obvious that in Dt(y) the inequalities are satisfied G(xeq) < G(x) < G(y), where xeq—the final equilibrium point. Inequalities (10) are used to set the constraints on macroscopic, including irreversible, kinetics. Presence of this constraint makes up principal difference of the model (7)-( 12) from previous modifications of parametric MEISs. The choice of equations for the calculation of individual terms under the sign of sum in the right-hand side of equality (11) depends on the properties of the considered system. 

The paraffins dehydrogenation on platinum-alumina catalysts proceeds with constant rate up to some time-on-stream after which a slow deactivation of the catalysts takes place Since relative changes of the catalyst activity ( characterized by reaction rate) are proportional to relative amounts of the deposited coke it can suppose that coke formation is the main reason of deactivation. Deactivation can be related with an attainment of a threshold in coke concentration (Co) on catalysts. The threshold amounts are 1.8 wt.% for A-I, 6,8% and 2.2% for A-II and A-IXI catalysts respectively. The isobutane dehydrogenation in non-stationary region (C > Co) is described by the following kinetic equation ... 

For a chemical reaction system, the characteristics of the periodic solutions are uniquely determined by the kinetic constants as well as by the concentrations of the reactants and final products. Starting from the neighborhood of steady state as an initial condition, the system asymptotically attains a closed orbit or limit cycle. Therefore, for long times, the concentrations sustain periodic undamped oscillations. The characteristics of these oscillations are independent of the initial conditions, and the system always approaches the same asymptotic trajectory. Generally, the further a system is in the unstable region, the faster it approaches the limit cycle. 

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