# 3. Theory of Reactor Design

In the 1990s I got interested in optimal reactor design, mainly because there was a resurgence of interest in old *attainable region* ideas of Fritz Horn, my late collaborator in chemical reaction network theory. I am very proud of the attainable region work I did — more on that later. However, I now think that the most important work to emerge from my temporary foray into design came from a different line of thought entirely, a line that (strangely) traces back to my seemingly unrelated paper [2.2] on Gibbs’s phase rule!

This work in design theory is summarized in [3.1] below, and its mathematical foundations are described in [3.2]. In rough terms, the central idea is this: Given a network of chemical reactions (with kinetics) and given a speciﬁed commitment of resources, there is an *absolute* and *computable* limit to what can be achieved in ANY steady-state reactor-separator design — a limit that is different from that afforded by stoichiometry or thermodynamics. Knowledge of such a limit is of practical importance, for it gives a benchmark against which actual designs can be judged. (Connections to the mathematics underlying the Gibbs phase rule are discussed in Remark A.1 of paper [3.2].)

As I said earlier, I am very proud of my attainable region work. There are some surprising and beautiful conclusions, and very different parts of mathematics come together in interesting ways. (For me it remains stunning that, associated with a given reaction network with kinetics and a given feed composition, there are certain exceptional numbers — something like eigenvalues — having special significance for reactor synthesis: A classical steady-state CFSTR design can have an optimal conversion relative to all other steady-state designs only if the CFSTR residence time assumes one of those exceptional values.) The review [3.3] was written after I had moved away from attainable region work, and it gives an indication of why I came to prefer ideas in [3.1] and [3.2].

The conference paper [3.4] is also friendly review, this time written for control theorists — I’m not one of those — to indicate why ideas in modern geometric control theory have somewhat surprising connections with Horn’s attainable region approach to optimal reactor design. Despite its intended control-theorist target, I think this article is suited to a much wider chemical engineering audience.

Mathematical underpinnings of [3.3] and [3.4] are given in the three-article series [3.5]-[3.7]. The paper [3.8], based on Thomas Abraham’s PhD thesis, is an extension of these, viewed from a different perspective.