MEIS with variable parameters

Currently the authors are developing three classes of models of extreme intermediate states (MEIS) (1) with variable parameters (2) with variable flows, and (3) those describing spatially inhomogeneous systems. All these classes of the models are formulated and analyzed in terms of MP, which, in the authors opinion, can be defined as a mathematical theory of equilibrium states. It is natural to start the analysis of the created modifications with the MEIS with variable parameters, which is the closest in character to the traditional equilibrium thermodynamics models. 

With fixed T, P, and initial composition of the components y of physicochemical system this model will take the form find 

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. 

Writing model (7)-( 12) we use two main assumptions following from the previous text  

each of these points can be attained from y along the equilibrium, in a general case irreversible, trajectory along which the condition of monotonous change in the characteristic thermodynamic function is met (as applied to the described case—the Gibbs energy). 

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The examples of specifying the representation of individual expressions in system (7)-(12) and application of MEIS with variable parameters are presented in Section 5. 

Decrease of the initial information volume directly and largely depends on the fact that there is no need to know a complete detailed mechanism of the studied process. The use of MEIS with variable parameters (7)-(12) calls for the information on individual limiting stages only. Substitution of the assignment of the list of reactions by the assignment of... 

At fixed T, P and initial composition of components y of the physicochemical system the MEIS with variable parameters and subjective extremality criterion (the maximum concentration of a set of useful or harmful products of the process) has the form ... 

As opposed to the described MEIS with variable parameters and the mechanisms of physicochemical processes in this case we will try to determine the objective function of applied model for a dissipative system based on the equilibrium principle of conservative systems, i.e. the Lagrange principle of virtual works. Derivation will be given on the example of the closed (not exchanging the fluid flows with the environment) active (with sources of motive pressures) circuit. The simplest scheme of such a circuit is presented in Fig. 3,a. A common character of the chosen example is explained by the easiness of passing to other possible schemes. For example, if at the modeled network nodes there are external... 

There is an appreciable groundwork for development of methods for construction and analysis of trajectories on the MEIS base. It resulted from extensive studies on the search for results of processes possible in the region of thermodynamic attainability D ( /). A key component of these studies is an extension of the experience gained in the convex analysis of ideal systems (Van der Waals and Konstamm, 1936 Zeldovich, 1938 Gorban, 1984) to modeling of real ones (Antsiferov, 1991 Kaganovich et al., 1989, 1993, 1995, 2007, 2010). They may include an ideal (or real) gas phase, plasma, condensed substances, solutions and other components. The components of this analysis are determination of particularities of the objective function and study on the properties of set Dj(y). For MEIS with variable parameters solution of the first mentioned problem proves to be non-tiivial, when the sum in expression (8) with the constant value of y is replaced by the... 

MEIS with variable parameters and subjective extremality criteria of type (8)-(13) may turn out even more convenient objects for DP applications than MEIS represented by schemes in the form of graphs. This is due to the fact that the stepwise optimization on the basis of these models (e.g. determination of concentrations for the specified set of substances) can be performed without dividing the model into interacting parts. This fact ensures additivity of the objective function. Certainly it does not mean that construction of DP algorithms for MEIS (8)-(13) is an easy process. Dividing the computation process into individual steps and taking an account of specific features of these steps are still rather difficult due to a lack of formalized techniques to perform those processes. 

Prior to discussion of modeling the complete mechanisms of chemical systems on the basis of MEIS (29)-(34), we will present capabilities of the approximate analysis of the efficiency of such mechanisms by MEIS (7)-(12) with variable parameters. This possibility was studied in Kaganovich and Filippov (1995 Kaganovich et al., 1993). Let us consider a process during which some initial composition a should be used to get the maximum quantity of products b. Let the point y in Figure 7 denote an initial state of the reactive system. The point m corresponds to the maximum thermodynamically admissible concentration of b. The points l and k represent states of a chemical system with the use of catalysts that provide... 

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