Balance polyhedron

The laws of conservation for the catalyst amount c3 + c4 + c5 = 6, = const, and the gas pressure cx + c2 = b2 = const, along with the natural conditions of non-negativity for c account for a convex polyhedron. This polyhedron determined by fixed values of the balances, in this case catalyst and pressure balances, is a balance polyhedron D0. Unlike the polyhedron D, the structure of the balance polyhedron D0 is, as a rule, rather simple (formally D0 is a particular case of reaction polyhedra). If there exists only one type of active site for the catalyst and accordingly one law of conservation with the participation of concentrations of intermediates, then D0 is a product of two simplexes D0(gas) and D0(surf). The dimensions of 2)0(gas) and D0 (surf) is a unit lower than the number of the corresponding substances, gaseous or those on the catalyst surface. Thus in the case under consideration, B0 consists of the vectors... 

For homogeneous (completely flowing) open systems a steady-state point becomes unique and stable at a very high constant velocity of the flow [35]. In this case the concentrations of gas-phase components rapidly become almost constant and their ratios are close to those for the input mixture. This fact is independent of a concrete type of the w(c) function. To confirm this postulate, let us consider eqns. (125) for a balance polyhedron Da. Since vm is very high, the inequality (127) is fulfilled automatically and we can write... 

Thus if the flow velocity in a completely flowing (homogeneous) system is higher than a certain value, the balance polyhedron contains a unique steady-state point that is globally stable, i.e. every solution for the kinetic equations (139) lying in Da tends to it at t - oo. Note that a critical value for the flow velocity at which this effect is obtained can depend on the choice of balance polyhedron (gas pressure). 

If the reaction graph is orientally connected, the phase space of a linear system (a balance polyhedron) has a metric (154) in which all trajectories of the system monotonically converge and the distance between them tends to zero at t - oo. This holds true for both constant and variable coefficients (rate constants), if in the latter case it is demanded that all rate constants have upper and positive lower limits (0 < a < k(t) < / < oo, a, / = const). 

Each internal point z of the balance polyhedron has a set of constants qtj corresponding to the orientally connected graph of the mechanism. Steady-state points (and, more extensively, positive semi-trajectories) on the balance polyhedron boundary are absent since it would contradict the oriented connectivity of the graph for the initial mechanism (a reader can prove this as an exercise). Therefore for any z > 0 there exist such <5 > 0 that, for any solution of eqn. (158) lying in a given balance polyhedron at t = 0, we obtain zM > (5 at t > i and all values of i. Let us consider two solutions for eqn. (158), zm(t) and z(2)(i), lying in the same balance polyhedron t)0. 

As in the case ml = 1, in accordance with the above properties of Jacobian matrix (160), it follows that, under the assumption of the oriented connectivity for the reaction mechanism involving no intermediate interactions, the time shift is the phase space (or balance polyhedron) compression in the metric... 

Any two solutions lying in the same balance polyhedron converge in the metric (161) and p(z(1)(t), z(2)(t)) -> 0 at t -> co. It results, in particular, in the existence, uniqueness and asymptotic stability (in the large) of the steady state in the balance polyhedron. This was confirmed by Vol pert et al. [45] and partly and simultaneously by Bykov et al. [46-48], (Note that all the considerations given also hold for the n A -> LmiBi- type reaction systems.)... 

There is no doubt that studies for the establishment of new classes of mechanisms possessing an unique and stable steady state are essential and promising. On the other hand, it is of interest to construct a criterion for uniqueness and multiplicity that would permit us to analyze any reaction mechanism. An important contribution here has been made by Ivanova [5]. Using the Clark approach [59], she has formulated sufficiently general conditions for the uniqueness of steady states in a balance polyhedron in terms of the graph theory. In accordance with ref. 5 we will present a brief summary of these results. As before, we proceed from the validity of the law of mass action and its analog, the law of acting surfaces. Let us also assume that a linear law of conservation is unique (the law of conservation of the amount of catalyst). 

We know that a PDE is stable as a linear approximation (see Sect. 2). Whence from eqns. (137) and (138) we establish that, at sufficiently low um and vout and t - oo, a solution of the kinetic equations for homogeneous systems tends to a unique steady-state point localized inside the reaction polyhedron with balance relationships (138) in a small vicinity of a positive PDE. If b(c(0)) = 6(c,n) vinjvout, then at low v,n and eout the function c(t) is close to the time dependence of concentrations for a corresponding closed system. To be more precise, if vm -> 0, uout -> 0, vmjvOM, c(0), cin are constant and c(O) is not a boundary PDE, then we obtain max c(t) — cc](t) -> 0, where ccl(t) is the solution of the kinetic equations for closed systems, ccl(0) = c(0),and is the Euclidian norm in the concentration space. 

Figure 6 Polyhedron of material balance (a) and thermodynamic tree (b) of hexane isomerization reaction. T = 600 K, P — 0.1 MPa.

Equation 2 defines a spinodal surface in ((/ -f 1 )-dimensional space based on the (/-dimensional composition polyhedron (providing for the material balance condition, Elqua-tion 1.1.1-40) plus the temperature axis, F = const. Equations 2 and 3 define a i/-dimensional critical surface. 

Due to the natural polymolecularity of any polymer sample, the system P-I-LMWL should be regarded as a polynary one, containing (stability boundary of the single-pheise system state is a spinodal surface in the (i/-t-1 )-dimensional space based on the j/ -dimensional polyhedron of composition (with the material balance taken into account) and the temperature axis, the critical state being defined by a i -dimensional surface. 

Both difficulties were successfully overcome with the help of the affine scaling method suggested by LI. Dikin (Dikin, 1967, 2010 Dikin and Zorcaltsev, 1980). It became a key method for using models (8)-(13) and (14). The method is convenient, as it employs only interior points of the polyhedron of material balance, at which the objective function... 

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