Stoichiometric matrix, augmented

Given a set of reactions, determine which of these are key to describing all other reactions as combinations of this set and how to do this. Also determine which additional balances appear and which thermodynamic characteristics (enthalpy change, equilibrium coefficient) are dependent. This will be addressed by the augmented stoichiometric matrix, introduced in Section 2.3. 

Tables 2.1 and 2.2 summarize the eharacteristic features of the RREFs of the augmented moleeular matrix and the augmented stoichiometric matrix.

Table 2.2 Characteristics of the RREF of the augmented stoichiometric matrix...

Remark 7.6. The analysis framework we presented is also applicable if an inert component is used to increase the heat capacity of the reaction mixture. In this case, the model (7.2f) would be augmented by the equations corresponding to the model of the separation unit. However, the stoichiometric matrix S and reaction rates r would remain unchanged, since the inert component does not partake in any reaction. Furthermore, the analysis can be applied if more complex correlations are used for the physical parameters of the system (e.g., temperature dependence of heat capacities and densities), as long as the basic assumptions (7.27), (7.29), and (7.30) apply. 

To summarize, we perform a singular value decomposition of the augmented formula matrix to obtain the matrices U, W, and V. With these, we use (11.2.10) to obtain a particular basis vector N for the range. From V, we form P and then use (11.2.7) to obtain all sets of stoichiometric coefficients Vy. Then we combine N and Vy into (11.2.5) to determine all sets of mole numbers that satisfy the elemental balances. Therefore, a singular value decomposition provides the number of independent reactions 91, all sets of 91 independent stoichiometric coefficients Vy, and all possible combinations of mole numbers N that satisfy the elemental balances. A computer program for performing the decomposition is contained in the book by Press et al. [9] routines for performing the decomposition are also available in MATLAB and in Mathematica . 

For a set of reactions involving intermediates, which typically cannot be measured, determine the overall reactions, that is, the ones not involving such intermediates, and find the numbers by which these can be written as combinations of the given reactions (the so-called Horiuti numbers). This will be addressed by augmenting a specially crafted stoichiometric matrix, as explained in Section 2.4. 

In Section 2.2, we have shown how the molecular matrix, when augmented with a unit matrix and converted to the corresponding RREF, yields a basis for all stoichiometrically acceptable reactions. In reality, however, many of these reactions may be chemically impossible. Therefore, a special stoichiometric matrix can be considered, in which only selected reactions occur. 

In this application, the stoichiometric matrix was not augmented, and we cannot tell by which linear combination of the original reactions the final result was obtained. Sometimes it is useful to keep track of the original reactions, for example when thermodynamic data such as enthalpy changes are required or for the calculation of equilibrium coefficients. 

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