Matrix stoichiometric coefficient

Stoichiometric coefficient of compound j in reaction i Stoichiometric coefficient matrix... 

Stoichiometric coefficient matrix for a system of reactions Molar concentration vector... 

N Null space of the stoichiometric coefficient matrix A N = null(AT)... 

The Stoichiometric Coefficient Matrix Consider a general system involving n components participating in d reactions. The first k species are reactants whereas the remaining (n - k) are products. Expressing all reactions in terms of all species then gives ... 

The set of concentrations stoichiometrically compatible with the feed point is hence directly related to the stoichiometric coefficient matrix A and the feed point. The general matrix equation relating this is... 

Two reactions are present involving three components. Letting rows 1-3 correspond to A-C, respectively, the stoichiometric coefficient matrix A is therefore a 3 x2 matrix, given by... 

Since there are two independent reactions participating in the system, we expect the set of points generated by the system to reside in a two-dimensional subspace in IR. The matrix equation describing this space is determined by the stoichiometric coefficient matrix A and the feed point Cf ... 

The dimension of the AR is equal to the number of independent reactions participating in the system. This is because the number of independent reactions defines the dimension of the stoichiometric subspace (the rank of the stoichiometric coefficient matrix A), and the AR must reside in the stoichiometric subspace. 

Hence, the dimension of the AR is governed by the reaction stoichiometry of the system. It is determined by forming the stoichiometric coefficient matrix A and then computing its rank (the number of independent rows or columns in A). 

Consideration of nonnegativity constraints, together with the stoichiometric coefficient matrix, can be used to form a set of linear relations that mathematically describe the stoichiometric subspace. 

To show this, we use Equation 6.4 to express all species concentrations within a mixture in terms of the stoichiometric coefficient matrix A and extent of reaction vector e... 

Null Space For many purposes in AR theory, it is useful to understand the set of concentrations that lie perpendicular (orthogonal) to S, which are spanned by the stoichiometric coefficient matrix A. For instance, the computation of critical DSR solution trajectories and CSTR effluent compositions that form part of the AR boundary require the computation of this space. It is therefore important that we briefly provide details of this topic here. It is simple to show from linear algebra that all points orthogonal to the space spanned by the columns of A are those that obey the following relation ... 

Here, mattix A is the stoichiometric coefficient matrix formed from the reaction stoichiometry, and C is any concentration vector that satisfies Equation 6.7. Determination of the subspace of concenttations orthogonal to the stoichiometric subspace, is in fact equivalent to computing the mil space of A. We shall denote by N the mattix whose columns form a basis for the null space of A. Linear combination of the columns in N hence generates the set of concentrations orthogonal to the stoichiometric subspace. 

The stoichiometric coefficient matrix is formed by storing the stoichiometric coefficients for each reaction participating in the system as a column of A. Thus for a single reaction involving three components, A has size 3x1 ... 

The system under consideration involves two reversible reactions and an irreversible reaction involving component A and intermediate component C. Five reactions are thus present in total. To determine the maximum number of parallel reactor structures, we first must determine the dimension of the stoichiometric subspace. The stoichiometric coefficient matrix is then given by ... 

From Equation 6.13, we have that matrix N must be orthogonal to the stoichiometric subspace S. Recall that the columns of the stoichiometric coefficient matrix A span S. (Each column in A represents a reaction... 

Notice that if we substitute matrix X discussed in this example with the stoichiometric coefficient matrix A, then the columns in X (Xj and X2) represent two reactions participating in n-dimensional concentration space, R". Hence, to compute N, we simply determine the stoichiometric coefficient matrix A (as in Section 6.2.1.3), and then compute the null space of A. From linear algebra, we can show that if A has size nxd (n components participating in d reactions), the size of matrix N will benx(n-d). 

The stoichiometric coefficient matrix A is formed from the reaction stoichiometry of the system. There are three reactions (d = 3) involving five components (n = 5) ... 

Matrix N has columns that form a basis for the nullspace of the stoichiometric subspace S, and hence there must he (n-d) columns in N if there are n components participating in d independent reactions (the stoichiometric coefficient matrix A has size nxd). 

The stoichiometric coefficient matrix A may be formed in the usual manner... 

This is a system of four independent reactions (d=4) in six components (n = 6). The stoichiometric coefficient matrix is... 

Given the system of reactions, it is possible to determine the stoichiometric coefficient matrix for this system. 

The stoichiometric coefficient matrix A is found from the reaction stoichiometry... 

A(C) is found from the controllability matrix E for the CSTR. To construct E, the set of vectors orthogonal to the stoichiometric subspace must be known. This is done by finding a basis for the null space of the vectors spanned by the reaction system given by the stoichiometric coefficient matrix A. 

Recall from Chapter 6 that S is calculated by expressing each independent reaction in terms of the extent of reaction vector, e, and the stoichiometric coefficient matrix A. 

The system involves five components. For simplicity, only forward reactions are considered (although the stoichiometric subspace does not change if reverse reactions are considered as well). The stoichiometric coefficient matrix A may be formed in the usual manner giving... 

Observe that the coefficients of j and 2 belong to the stoichiometric coefficient matrix A. It follows that the system may be written in matrix form as follows ... 

The stoichiometric coefficient matrix A can be formed, given now as follows ... 

The set of concentrations belonging to the overall stoichiometric subspace, S, can be found by first computing Sj and S2, which are the stoichiometric subspaces for Cfi and Cf2, and then calculating the convex hull of Sj and S2. The stoichiometric coefficient matrix A for the set of... 

Similar to the procedure carried out in Chapter 8, computation of the stoichiometric subspace S begins with the stoichiometric coefficient matrix A. The dimension of S in mass fraction space is equivalent to that in concentration space, and it is found by computing the rank of A. This is determined by the number of independent reactions present in the system. For n components in d reactions, the size of A is (nxd). 

Here, e is the molar extent of reaction, A is the stoichiometric coefficient matrix, and % is the feed molar flow rate vectoruf= [njf,n2f,. .., n f]. This can be converted to a system of mass fractions giving... 

The stoichiometric coefficient matrix A for this set of reactions is given by... 

---