Combinatorial aspects of sharp split separation systems synthesis

Thompson and King (1 972) presented a closed form expression for the number of different possible separation sequences arising when an n-component mixture is separated into pure products using sharp component separators with one input and two outputs. Subsequently, Shoaei and Sommerfeld (1986) pointed out that the number of sequences is the series of Catalan numbers (Alter, 1971), but they did not show how to derive the closed form formula. This paper will demonstrate that this may be done using a general mathematical technique-generating functions. The analysis is extended to combinatorics of sequences of sharp separators with more than two outputs. Finally expressions for the number of distinct separators will be derived. An underlying assumption throughout the paper is that the components in any stream are “sorted”; components appearing together will always appear in the same order. Sequences of Two-Output Sharp Separators A sharp two-output-component separator is a device where a subset of the feed components leave entirely in the separator’s top stream and the rest leave entirely in the other, the bottom stream. Thus, the remaining separation problem originating from the top stream will be totally independent of the one originating from the bottom stream. The different separation sequences may be represented as paths in a tree, as illustrated in Figure 1 for a four-component example. It is seen from the figure that this example involves five different paths; five alternative separation sequences are possible. The number of separation sequences may be defined recursively for the general case: a stream consisting of n components may be split in n - 1 different ways in one sharp two-output separator. For each of these alternative splits, the number of different separation sequences is equal to the number of separation sequences originating from the separator’s top stream multiplied by the number of separation sequences originating from the separator’s bottom stream. A stream with only one