Independent reactions problem

In section 11.3 vie showed that the difficult problem of solving the flux relations can be circumvented rather simply when the stoichiometric relations are satisfied by the flux vectors, but the treatment given there was limited to the case of a single Independent chemical reaction, when the stoichiometric relations permit all the flux vectors to be expressed in terms of any one of them. The question then arises whether any comparable simplification is possible v en the reactants participate in more than one independent reaction. 

The maximum set will consist of Equations (14.1) and (14.3) and N versions of Equation (14.2), where N is the number of components in the system. The maximum dimensionality is thus 2- -A. It can always be reduced to 2 plus the number of independent reactions by using the reaction coordinate method of Section 2.8. However, such reductions are unnecessary from a computational viewpoint and they disguise the physics of the problem. 

Analytical treatment of the diffusion-reaction problem in a many-body system composed of Coulombically interacting particles poses a very complex problem. Except for some approximate treatments, most theoretical treatments of the multipair effects have been performed by computer simulations. In the most direct approach, random trajectories and reactions of several ion pairs were followed by a Monte Carlo simulation [18]. In another approach [19], the approximate Independent Reaction Times (IRT) technique was used, in which an actual reaction time in a cluster of ions was assumed to be the smallest one selected from the set of reaction times associated with each independent ion pair. 

The number of independent reactions R can be found simply as the rank of the matrix of stoichiometric coefficients %J with dimension Sx r such that R < r. Different methods can be applied, such as reduction to triangular matrix by Gaussian elimination for small-size matrices, or computer methods for larger problems. 

The lacking special description of the Gibbs phase rule in MEIS that should be met automatically in case of its validity is very important for solution of many problems on the analysis of multiphase, multicomponent systems. Indeed, without information (at least complete enough) on the process mechanism (for coal combustion, for example, it may consist of thousands of stages), it is impossible to specify the number of independent reactions and the number of phases. Prior to calculations it is difficult to evaluate, concentrations of what substances will turn out to be negligibly low, i.e., the dimensionality of the studied system. Besides, note that the MEIS application leads to departure from the Gibbs classical definition of the notion of a system component and its interpretation not as an individual substance, but only as part of this substance that is contained in any one phase. For example, if water in the reactive mixture is in gas and liquid phases, its corresponding phase contents represent different parameters of the considered system. Such an expansion of the space of variables in the problem solved facilitates its reduction to the CP problems. 

If chemical reactions occur, then we must introduce a new variable, the i coordinate e for each independent reaction, in order to formulate the mate balance equations. Furthermore, we are able to write a new equilibrium rela [Eq. (15.8)] for each independent reaction. Therefore, when chemical-rea equilibrium is superimposed on phase equilibrium, r new variables appear r new equations can be written. The difference between the number of va and number of equations therefore is unchanged, and Duhem s theorem originally stated holds for reacting systems as well as for nonreacting syste Most chemical-reaction equilibrium problems are so posed that it is 1 theorem that makes them determinate. The usual problem is to find the corn-tion of a system that reaches equilibrium from an initial state of fixed an of reacting species when the two variables T and P are specified. 

When the reactivity of the centre is determined not only by the last added unit but also by the last but one unit, we speak of the penultimate effect. Merz et al. treated this problem using eight independent reactions [200, 201 ]. 

It will be assumed in discussing the solution of the differential equations relating conditions within a tubular reactor that the calculations are to be carried out with some sort of automatic digital computer. The problem is just manageable with a desk computing machine in the one-dimensional approximation when there is only one independent reaction, but even in this simplest case, this is an expensive way to work, particularly if a fairly large number of solutions is required. 

Merz et al. treated this problem using eight independent reactions [200, 201 ]. 

In the previous approach, a differential equation is required for each component. Therefore Us differential equations are solved simultaneously in this approach. One can instead solve the problem by using one differential equation for each independent reaction for a total of Hi differential equations. Because rii < Us, this second approach produces fewer differential equations. Normally rii is not much less than ris, and the computational expenses of the two approaches are similar. Whether we write differential equations for the species or the reaction extents is largely a matter of taste. 

If equations (A) and (B) are now used to express the rate laws for the three independent reactions given in the problem statement in terms of product concentrations and the initial concentrations of reactants,... 

A simple and direct method of determining the equilibrium composition of a complex reaction is to solve simultaneously all of the equations comprising the complex network. The actual number of equations to be solved is equal to the number of independent reactions of the network. Chapter 5 deals formally with the treatment of complex reactions, and the method outlined therein can be applied to this problem. Where the number of reactions is relatively small, say 2 or 3, simple, less formal methods can be used as illustrated in the following example. 

To solve reaction equilibrium problems, we must combine material balances with the criteria for reaction equilibria. Consequently, such problems bear a superficial resemblance to isothermal flash calculations, though in the case of reaction equilibria the material balances are applied to elements, not species. For H independent reactions involving C species in a single phase at fixed T and P, the criteria for equilibrium were given in 7.6.1,... 

Also note that although this problem involves three independent reactions, we need not identify explicitly any three reactions. Particular reactions and their stoichiometric... 

In a typical problem, multiple reactions are taking place in a multiphase system at fixed T and P, and we are to compute the equilibrium compositions of all phases. At this point, such calculations raise no new thermodynamic issues for example, for (R independent reactions occmrring among C species distributed between phases a and P, the problem is to solve the phase-equilibrium criteria... 

Flexibility. Higher dimensional systems can be broken down into a number of smaller and simpler two-dimensional problems. For systems involving exactly two independent reactions, the resulting AR can still be computed with the method. Two-dimensional constructions provide a minimum working dimension from which candidate regions may be determined. 

In the following sections, we wish to describe an AR construction method that specifically employs information from the complement region. This method is robust in that it is able to handle a wide variety of problem types. The method is also parallel in nature, which allows computation to be split over multiple computing nodes for addressing large problems that involve many independent reactions. 

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