Limitations in using Euler's formula in the design of heat exchanger networks with Pinch Technology

Pinch Technology developed by Linnhoff and other workers has been widely adopted and considered to be one of the most successful techniques in process energy integration. The number of heat exchanger units is one of the most important aspects in the problem and Euler's formula has been applied in correlating the number of units and loops in the network. However, planarity of a graph is required for the application of Euler's formula, a fact that has been ignored in previous literature. It is demonstrated in this paper that Euler's formula cannot always be applied in Pinch Technology, by presenting examples of non-planar graphs resulting from thermodynamically feasible heat exchanger networks.