The Method of Bounding Hyperplanes

The central approach to constmction is to iterate over each comer of the current polytope P introduce new hyperplanes that eliminate unattainable space. Sharp corners of the polytope are slowly smoothed out by the introduction of additional bounding planes, and the level of accuracy obtained is hence a strong function of the number of unique hyperplanes that are introduced. Curvature of a region, such as that generated by a PFR manifold on the AR boundary, may be approximated by the use of many bounding hyperplanes. 

A space that contains only unachievable states. 

The remaining space, containing the trae AR and other unachievable states that have yet to be eliminated. 

Point 1 is feasible due to a condition, developed by Abraham and Feinberg (2004), which guarantees the denial of 

In particular, this elimination condition states that if a rate vector r(C ), evaluated at a point C, on a hyperplane points into the hyperplane, then C is not achievable. By moving hyperplanes inward starting from the extreme points of the polytope, we ensure that only unachievable states are removed from the space. 

---

Figure 8.21 Graphical representation of the method of bounding hyperplanes. Hyperplanes are introduced at the comers of the polytope.

AR constructions of systems involving five independent reactions have been investigated using the method of bounding hyperplanes (Abraham, 2005). Systems involving a larger number of independent reactions are difficult to construct due to the increased computational complexity at present, which is primarily due to the evaluation of points obtained from higher dimensional Cartesian products. 

Figure 8.23 Summary of construction results for the three-dimensional Van de Vusse kinetics, using the method of bounding hyperplanes.

S.2.2 The Algorithm The actual method of bounding hyperplanes may now be described. The following algorithm describes a general overview of the method ... 

A candidate AR construction method that utilizes hyperplanes to carve away unachievable space shall be discussed in Section 8.5.2. Linear constraints, such as non-negativity constraints on component concentrations and flow rates, may also be expressed in the form of a hyperplane equation. Hyperplanes therefore also arise in establishing bounds in state space. In Section 8.6, superstructure methods shall be described for the computation of candidate ARs. These methods, at their core, rely on the solution of a large... 

Enantiomeric control is more difficult if the excited molecular potential energy surfaces do not possess an appropriate minimum at the o hyperplane configurations (see Figs. 1 and 2). In this case the method introduced in this section is not applicable. One may however be able to apply the laser distillation procedure by adding a molecule B to the initial L, D mixture to form weakly bound L — B and B - D, which are themselves right- and left-handed enantiomeric pairs [83]. The molecule B is chosen so that electronic excitation of B — D and L — B forms an excited species G, which has stationary rovibrational states that are either symmetric or antisymmetric with respect to reflection through t7>,. The species L - B and B - D now serve as the L and D enantiomers in the general scenario above, and the laser distillation procedure described above then applies. Further, the molecule B serves as a catalyst that may be removed from the final product by traditional chemical means. 

An alternative method for computing S exists that relies on computing a basis for the null space of the stoichiometric matrix A. Vectors belonging to null (A ) are normal to S, and these vectors may be related to bounding hyperplane constraints that define S. This method is described in Feinberg (1987,2000a, 2000b). 

Candidate AR construction via bounding hyperplanes (either via translation or rotation) may be extended to allow for constructions involving a parameter that may affect the direction of the rate vector. A system involving temperature-dependent kinetics is a well-known example of this. In Figure 8.26(b), the rotated hyperplanes method is used to compute an AR with temperature-dependent kinetics. At each point of evaluation, a temperature range between 300 and 1000 K is generated and rate vectors are checked for tangency with the hyperplane. This allows for... 

---