Page 179 - Sustainability in the Process Industry Integration and Optimization

156 Cha p te r S e v e n

have been met largely by employing the well-established mathematics
of graph theory, which can be regarded as a branch of combinatorics.
Thus was developed the graph-theoretic, algorithmic method
described in this section. The method is based on using P-graphs to
extract the universal combinatorial features (properties) inherent in
feasible processes. Such properties can be expressed mathematically
as a set of axioms that characterize the combinatorial feasibility of
processing networks.
A given process network is said to be combinatorially feasible (or to
be a solution structure) if it satisfies the following five structural
axioms:

(S1) Every final product and target is represented in the
structure.
(S2) An entity represented in the structure has no input if and
only if it represents a raw material or precursor.
(S3) Every operating unit represented in the structure is
defined in the problem.
(S4) Any operating unit represented in the structure has at
least one path leading to a final product or a final target.
(S5) An entity belongs to the structure if and only if it is either
an input entity to or an output entity from at least one
operating unit already represented in the structure.
Figure 7.3 illustrates the extreme reduction in the search space
that results from this approach. The universe of all possible networks
is reduced to a much smaller space containing only those networks
that satisfy the axioms—in other words, the combinatorially feasible
(CF) networks. Clearly this reduction will drastically reduce the
required computational effort. Search-space reductions by a factor of
nearly a billion have been reported in some of the real-life process
synthesis tasks performed to date using this axiomatic approach.
Note that each feasible network, including the optimal network, is an
element of the set of combinatorially feasible networks.
Figure 7.4 depicts two process structures that are not combina-
torially feasible. The P-graph in Figure 7.4(a) shows a process structure
in which material F is consumed as an input. Yet because material F is
not a raw material and was never produced, the structure is not
combinatorially feasible according to Axiom (S2). In the P-graph of
Figure 7.4(b), operating unit O produces only by-product B. Here O
3 3
does not output any final product or material that is later used to yield
a final product, so the process structure violates Axiom (S4). In short,
the structural properties expressed by Axioms (S1)–(S5) are necessary
conditions for process structures to be feasible. This means that
reducing the search space to combinatorially feasible structures does
not result in the loss of any practically feasible or optimal processes.