Introduction
What is P-graph?
A process graph or P-graph in short is a unique bipartite graph representing
the structure of a process system. In such a graph, the operating units
are denoted by horizontal bars, and their input and output materials by
solid circles. A P-graph is a directed graph; the direction of the arcs
is the direction of the material flows in the network; it is directed to
an operating unit from its input materials and from an operating unit to
its output materials. The P-graph is illustrated in Fig. 1.
Fig. 1. P-graph representation of the operating unit separating
mixture
ABC into component
A and mixture
BC.
Figure 2 is the conventional block diagram of the operating unit represented
by the P-graph in Fig. 1.
Fig. 2. Block diagram corresponding to the P-graph in Fig 1.
Why do we need P-graph?
P-graphs have been proposed to alleviate difficulties encountered
by approaches based on conventional graphs, e.g. digraph and signal-flow
graph.
In the digraph representation of a process system, the
operating units correspond to the vertices, and the connections to the
arcs of the graph. In the signal-flow graph representation of a process
system, the vertices of the graph are associated with the materials of
the process. While these conventional graphs are suitable for representing
and analyzing a process system (e.g., Mah, 1983, 1990; Dudczak, 1986),
they are not suitable for process synthesis as demonstrated in the following
simple examples.
Cases (1.1) and (1.2) described below can be represented by the
same digraph shown in Fig. 3.
Case (1.1). Two different materials are produced separately,
one by operating unit 02 and the other by operating unit 03. Moreover,
it is necessary to feed both of these materials to operating unit 01 to
generate the final product.
Case (1.2). One material is produced by both operating
units 02 and 03. This material is subsequently fed to operating unit
Ol to generate the final product.
Fig. 3. Digraph: note that it is incapable of uniquely characterizing
a synthesis problem as demonstrated by example 1.
Note that while both operating units 02 and 03 are necessary
to produce the product in case (1.1), either unit is sufficient in
case (1.2).
Cases (2.1) and (2.2) described below are represented by the
same signal-flow graph of Fig. 4.
Case (2.1). Two separate operating units, one receiving
material B as its input and the other receiving material C as its input,
produce the same material which is subsequently fed to another operating
unit where material A (product) is generated.
Case (2.2). A single operating unit, receiving materials
B and C as its inputs, produces a material which is subsequently fed
to another operating unit where material A (product) is generated.
Fig. 4. Signal-flow graph: note that it is incapable of uniquely
characterizing a synthesis problem as demonstrated by example 2.
Note that while case (2.1) requires three operating units,
case (2.2) requires only two units. Obviously, the semantics, i.e. meaning,
of the figure is unclear. As in Fig. 3, Fig. 4 fails to describe the
process system in clear semantics.
Examples 1 and 2 demonstrate that neither of the two most popular
conventional graphs is semantically rich enough to faithfully represent
a process structure. The semantics of a process structure is concerned
with the meaning of individual materials and operating units and the
relationship between them, while the syntax of the process structure is
concerned with the ordered organization of the flow of the materials
and the operating units.
Both a digraph and a signal-flow graph can orderly encode a
process structure into a graph representation. However, as demonstrated
by the examples above, the former is not sufficient to uniquely represent
individual materials and their relationship, and the latter is not sufficient
to uniquely represent individual operating units and their relationship.
Hence, a graph more sophisticated than a conventional one, such as the
digraph or signal-flow graph, is required to uniquely characterize a synthesis
problem.
