The P-Graph framework was published by professors Ferenc Friedler and L.T. Fan in the early ‘90s.  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.

Figure 2 is the conventional block diagram of the operating unit represented by the P-Graph in Figure 1. Figure 1. P-Graph representation of the operating unit separating mixture ABC into component A and mixture BC. Fig. 2. Block diagram corresponding to the P-Graph in Fig 1.

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.

### Example 1

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).

### Example 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.

A P-Graph can capture not only the syntactic but also the semantic contents of a process structure. For the two examples, three different P-Graphs can be constructed to uniquely represent the four cases. Note that cases (1.2) and (2.1), which are identical, are uniquely represented by the P-Graph in Fig. 5, case (2.2) in Fig. 6, and case (1.1) in Fig. 7. Fig. 5. P-Graph uniquely representing cases (1.2) and (2.1). Fig. 6. P-Graph uniquely representing case (2.2). Fig. 7. P-Graph uniquely representing case (1.1).