The Stoichiometric Subspace

1 Introduction Given a system of reactions, it is possible to compute bounds in concentration space wherein all feasible concentrations must lie. This space is typically much larger than the space of achievable concentrations (the AR). Consequently, we can use this space as an upper bound on the set of feasible concentrations that the AR must reside in. We call this space the stoichiometric subspace and denote it by the set S. Determining S is also useful for AR construction algorithms, which are discussed in Chapter 8. 

Relative proportions of each component in the reaction are dictated by the reaction stoichiometry—one mole of nitrogen and three moles of hydrogen combine to form two moles of ammonia by the following reaction. Given a feed composition, species concentrations of all components can be expressed in terms of the extent of reaction e and reaction stoichiometry. Assuming constant density, a mass balance for the ammonia reaction may be written, giving expressions for 

Observe that the left hand side of the equation is simply the concentration vector = [0, Cjj, c Hjl mol/L, and the right hand side is a vector sum, consisting of the feed vector Cf = [c jf, Cn f, CNHjfl mol/L and a vector holding the stoichiometric coefficients of the reaction, a = [-l, -3, 2]T mol/L. 

a number of feed vectors have been specified together with the corresponding set of concentrations satisfying Equation 6.1. Irrespective of the feed vector used. 

If kinetics for the ammonia system is available, computation of the set of achievable concentrations for this system (the AR) would result in concentrations constrained on the line defined by Equation 6.1. The AR for this system would therefore lie in a one-dimensional subspace of R. Thus, the set of points achievable by the ammonia reaction, and thus the AR itself, is inherently one-dimensional for the ammonia reaction, even though there are three components present in the system. 

---

Equation (1.76) expresses the fundamental relation between the atom matrix and the reaction matrix of a closed system. The matrices A and , however, result in the same stoichiometric subspace if and only if the subspace defined by (1.73) and the one defined by (1.74) are of the same dimension, in addition to the relation (1.76). I4e denote the dimension of the stoichiometric subspace by f also called the stoichiometric number of freedom. If the reaction matrix is known, then f = rank(B),... 

Extents can take on negative values Extent of reaction is positive for products and negative for reactants. Concentrations, by comparison, may never take on negative values. This property provides a convenient set of bounds for all species present in a mixture. If concentration is used over extent for a given reactive system, the associated AR for the system must then also always lie in the positive orthant of concentration space." Additionally, using reaction stoichiometry and mass balance constraints, it is possible to compute a definite bound (called the stoichiometric subspace) that the AR must reside in. Similar bounds are rather less well-defined when extent is employed, and thus greater care must be taken. 

In general, A has sizenxd. Each column in A represents a distinct reaction of the system. From the definition of A, the stoichiometric subspace must reside in the space spanned by the columns in A, which correspond to the a vectors for reactions taking part in the system... 

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. 

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. 

We shall discuss in greafer defail how fo numerically compute S for a given reaction and feed point in Chapter 8, but for now it is sufficient for us to have a qualitative understanding of the stoichiometric subspace for use in later sections of this chapter. 

The stoichiometric subspace can be defined by the following system of inequalities... 

We can extend the concept of the stoichiometric subspace, S, to also include multiple feeds. Consider the case when two feeds, Cjj and Cf2, are available for the same reaction stoichiometry. We may compute the stoichiomettic subspacebelonging to Cfi and Cj2 individually, giving Sj and S2, respectively. But since both feeds are achievable, it is also possible to mix between feeds and thus also between points in Sj and S2. Thus, we can compute the convex hull... 

The overall stoichiometric subspace belonging to multiple feeds may therefore be determined by first eom-puting the stoichiometric subspace S,- for each feed Cj, and then computing the convex hull of the set of points for all Sj s, which, for N feeds gives ... 

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. 

There are three components in this system (n = 3). This describes a line in (c -Cb-Cc space) for which all possible concentrations must lie in order to be stoichiometri-cally feasible with the feed. The stoichiometric subspace in this instance is thus a line in passing through the feed point. Note that computation of the null space does not require us to specify a feed point, and so this has been omitted here. 

Since there is only one column in A (corresponding to a single reaction), rank(A) = 1. To compute the set of concentrations orthogonal to the stoichiometric subspace, we compute the null space of A . Hence, since the rank of A is one, we expect the rank of the null space to be (3 - 1) = 2. We may compute the null space using standard methods such as elementary row operations. It... 

Matrix N describes a two-dimensional subspace in that is perpendicular to the stoichiometric subspace. Note that A iii =0, A ii2 = 0 demonstrating that iii and 03 are orthogonal to the columns of A (the stoichiometric subspace). Linear combination of iii and 112 (such as -111 + 2) would therefore also be orthogonal to this space. 

Finally, one may compute the set of vectors orthogonal to the stoichiometric subspace. Again computation gives a set of two vectors that form a basis for the null space. Two representative vectors for this space are given below ... 

Linear combinations of iij and n2 also describe a two-dimensional subspace in and in this instance the dimension of the stoichiometric subspace is the same as the dimension of the null space. 

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 ... 

This analysis is not influenced by specification of the feed point Cf. Computation of the dimension of the stoichiometric subspace involves the reaction stoichiometry alone, and thus the maximum number of parallel structures is the same irrespective of the feed point employed— the actual number of parallel structures required to achieve the AR will, however, be dependent on the reaction kinetics and feed point. 

EXAMPLE 8 Explaining the role of matrix N in Equation 6.13 with relation to the stoichiometric subspace... 

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... 

The columns in A span the stoichiometric subspace S in (Ca-Cb-Cc-Cd-Ce space). To find N—the columns of N that form a basis for the nullspace of S—we simply... 

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). 

Observe that this involves two iterated Lie brackets (zf (C) and zP (C)) and two normal vectors that are orthogonal to the stoichiometric subspace (N = [dj, 02]). 

The stoichiometric subspace is described mathematically as a system of linear inequalities that represent a feasible region in R". These inequalities are formed from mass balance and nonnegativity constraints in terms of the stoichiometric coefficient matrix A and extent of reaction vector e. 

A in this instance is a 4 x 3 matrix (four components and three reactions), with row 1 corresponding to component A, and row 4 corresponding to component D. Computation of the rank of A gives a value of 3, which validates that there are three independent reactions occurring in the system, and that the stoichiometric subspace (and hence also the AR) is a three-dimensional subspace in... 

From Section 7.2.1.1, the dimension of the AR is three (d=3) and there are four components (n = 4). It is expected that the dimension of the subspace orthogonal to the stoichiometric subspace is (4 - 3) = 1. Therefore, for the three-dimensional Van de Vusse system, the null space is given by a one-dimensional subspace (in other words, the basis for the null space is composed of a single, linearly independent vector). This is confirmed when the null space of A is computed, giving... 

Linear combinations of n hence form the set of vectors that are orthogonal to the stoichiometric subspace. From Chapter 6, it is known that the condition for a CSTR to lie on the AR boundary occurs when the controllability matrix E does not contain full rank. An expression may be determined for this by computing the determinant of E and setting it equal to zero. 

To determine whether a CSTR effluent concentration is a critical CSTR point, A(C) = 0, which depends on the controllability matrix E. To find E, vectors forming the subspace orthogonal to the stoichiometric subspace... 

Each row in A represents a component in the BTX system in the following order (i) benzene, (ii) ethylene, (iii) toluene, (iv) xylene, (v) diphenyl, and (vi) hydrogen. Computing rank(A) results in an answer of three, which validates that all three reactions participating in the BTX system are independent. The stoichiometric subspace is hence a three-dimensional subspace residing in re . Matrix N may be computed from null(A ) (the null space to the stoichiometric subspace). We expect the size of N to be 6 X 3, since the total number of species is 6 and rank(A) = 3. Performing the null space computation, we find that the following three vectors... 

Hereafter, it is assumed that the reader is comfortable with the idea of the stoichiometric subspace S, which is discussed in Chapter 6. In this section, we shall describe how to numerically calculate the bounds of S (which is a convex polytope that is expressed as a collection of hyperplane constraints). 

Suppose now that a different set of kinetic expressions for the same system of reactions and feed point is supplied. Since the size and shape of the AR is defined by the kinetics, the corresponding AR for the new kinetics may differ to the original region computed. Although the particular form of the kinetics may change the shape of the AR, all kinetics irrespective of individual form must still obey mass balance constraints defined by the reaction stoichiometry. Hence, the stoichiometric subspace S does not change by the introduction of new kinetics. 

Geometrically, Equation 8.2b describes a set of hyperplane constraints (a set of inequality constraints) that define the stoichiometric subspace in R . 

The following sections will describe how it is possible to compute the stoichiometric subspace by identifying the bounding constraints in extent space that form the feasible region, which is a function of the reaction stoichiometry and feed point. From this information, it is possible to compute the vertices of the region via vertex enumeration, which is described in Section 8.2.2.2. 

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... 

For a reactive system involving n components, n inequality constraints may be formed. The system of inequalities, described by Equation 8.2b, defines the feasible region of all stoichiometrically compatible concentrations in extent space. An extent vector e,- that satisfies the system of inequalities, given by Equation 8.2b, is one that satisfies mass balance constraints defined by the reaction stoichiometry. The corresponding concentration, C, is then said to be feasible—as it lies within the stoichiometric subspace—although it may still not be attainable if it lies outside of the AR. 

---