Complement region

Figure 4.34 (a) Three rate vectors residing in the complement region (b) rays extended backward from the three points of interest and (c)... 

The complement region to the AR is the region in space that does not include the AR (it is the remaining space excluding the AR). 

Consider the hypothetical candidate AR given in Figure 4.34(a). Three points numbered 1-3 are shown, lying in the complement region to the candidate AR. Straight lines may be extended backward from the points shown, creating three rays, which are shown in Figure 4.34(b). 

In Chapter 8, we shall discuss an automated AR construction algolthm that uses this property of the complement region to compute candidate regions. This algorithm can then be used to verify this statement... 

In the following sections, we wish to describe an AR construction method that specifically employs information from the complement region. This method is robust in that it is able to handle a wide variety of problem types. The method is also parallel in nature, which allows computation to be split over multiple computing nodes for addressing large problems that involve many independent reactions. 

Assume now that a set of N states Cj, C2,. .., Cj j is generated in the complement region S X. In other words, N concentration vectors that lie in S (but not in P ) are generated so that Cj, C2,. .., Cn S Xj.. Suppose that there is a point C in this set that when the rate evaluated at C, r(C), is extended backward, it intersects the boundary of region P. We shall denote the point of intersection as C. Since C and C lie on a straight line in the direction of r(C), it follows that the vector (C - C ) is collinear with r(C). Furthermore, since the ray has been extended backward from C, there exists a positive scalar t such that... 

If a point C in the complement region does not satisfy the CSTR condition, then it simply means that it is not achievable, by the CSTR condition, at the current iteration. However, C might still be attainable at a later point ... 

Here and are matrices representing region Pj. as a list of hyperplane constraints in state space. C, is hence a feasible point if it satisfies all hyperplane constraints given by Equation 8.9. Intersections with the current region are performed with the same system of inequality constraints. For each potential concentration C, generated in the complement region S X, it is possible to express the intersection point C as a linear combination of the rate vector at C, and a scalar variable t. 

The CSTR equation can be used to find points in the complement region that expand the region. Many points can be checked at once, which means that the AR can be computed from many CSTR evaluations in parallel. 

An Improved LP Formulation The method given in Section 8.6.1.3 does not make use of information regarding the complement region, and the full solution must be... 

Note that these equations are similar to the original mass balance equations for a single feed. The summation over k indicates that there are K feed points from the candidate region boundary and M other CSTRs from the complement region. The total plant feed flow rates may then also be... 

Vj s in this instance refer to the CSTRs operating in the complement region. This is similar in form to the LP given in Section 8.6.1.3, however, now extended to K feed points and with a different objective function. 

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