Computing the Stoichiometric Subspace

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 that a system of reactions with associated kinetics is available, obeying a certain reaction stoichiometry. Since expressions for the rate of formation are known and available, it is possible to compute the AR for the system in concentration space. The particular region computed, defined by the kinetics for the system, will exist as a convex polytope residing in R . 

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. 

Imagine now that a different feed point is specified. Mass balance constraints on the system may be invoked that are consistent with the new feed, which in turn affects that shape of S. Thus, although S does not change with different kinetics, it is affected by a change mfeed and stoichiometry. 

CONCEPT Stoichiometric subspace or mass balance triangle  

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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. 

Initial polytope computation Given a feed point Cf and reaction stoichiometry, compute the stoichiometric subspace S in the H-representation. Let represent the convex polytope at iteration k of the construction process. Initially, Pq is equal to the stoichiometric subspace S. 

In Chapter 8, we showed how the stoichiometric subspace for the methane steam reforming reaction can be computed in concentration space. Since the reaction occurs in the gas phase, it is more appropriate to determine the stoichiometric bounds in mass fraction space. This approach is preferable as the density of the mixture is no longer required to be constant. Compute the stoichiometric subspace for the CH4 steam reforming reaction and compare it to the answer obtained in Chapter 8. Assume that a feed molar vector of Uf = [1,1, l,0,0] kmol/s is available, and that the gas mixture obeys the ideal gas assumption to simplify calculations. Assume a constant pressure and temperature of P = 101 325 Pa and T = 500 K, respectively. 

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. 

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. 

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. 

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... 

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 ... 

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. 

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. 

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... 

We can hence calculate the extreme points of the region defined by Equation 8.2b in extent space by vertex enumeration. The set of extents forming the extreme points of the stoichiometric subspace is found to be [1.0, 0.0], [-0.25, 0.75]T, [0.0, 1.0]T, [0.0, 0.0]. From this set, equivalent extreme points in concentration space may be computed using Equation 8.1. The stoichiometric subspace resides as a two-dimensional subspace in IR . This subspace may be projected onto different component spaces for visualization. A number of example component pairs are shown in Figure 8.4(a). 

In this instance, S is smaller than previously computed. Observe also that whereas in the previous example the feed point is situated on a point on the boundary of the stoichiometric subspace, the feed point lies within the new region. 

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... 

We will begin by discussing a number of important formulae for converting common process variables involving moles to equivalent quantities involving mass fraction. These concepts are not difficult to understand, however, they are fundamental to how the computation of ARs in mass fraction space must be organized. Discussion of how the stoichiometric subspace may be computed and how residence time may be incorporated in mass fraction space is also provided. From this, a number of examples are provided that demonstrate the theory. In particular, isothermal and nonisothermal unbounded gas phase systems shall be investigated. 

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). 

The feasible region is then given by the system of linear inequalities defined by Equation 9.20. In a procedure similar to that given in the steam reforming example, the vertices of the stoichiometric subspace may be computed using a vertex enumeration program. The results of the computation are shown in Figure 9.6. 

In Section 9.2.7, the stoichiometric subspace for the CH4 steam reforming reaction was computed. In reality, the system of equations given involves the CH4 reforming reaction, as well as the water-gas shift reaction. Both of these reactions are important, for instance, in Fischer-Tropsch synthesis reactors (Anderson et al., 1984 Dry, 2002). It follows that it would be useful to understand the limits of achievability for this system. 

Results The results of the computation, using the parallel complement AR construction method, are shown in Figure 9.9(a). Note that the AR computed for this system is smaller than the stoichiometric subspace computed in Section 9.2.7. In particular, although the stoichiometric subspace predicts a maximum CO mass fraction of approximately 0.9, the maximum achievable... 

To compute the equivalent stoichiometric subspace in concentration space, the feed molar flow rates must be converted to a feed concentration vector Cf. The methods described in Chapter 8 for constant density systems may then be applied. The results of the constant density system and the region obtained via mass fractions are shown together in Figure 9.5. 

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