The attainable region generated by reaction and mixing

Abstract

The following problem is examined: for a given system of reactions with given kinetics, find all the possible outlet conditions that can be achieved by using any system of steady-flow chemical reactors. The outlet conditions or variables that are considered include concentrations, residence time and temperature. This set of all possible outlet conditions for a given feed was called the Attainable Region by Horn (1964). The boundary of the attainable region is of particular interest as, provided the objective function has open contours over the space of Hie attainable region, the optimum of a system of steady flow reactors will lie in the boundary of the region. More importantly, the optimal reactor structure can be determined from the reactorri that form the boundary of the. attainable region. The prr>-oerties of reaction and mixing are interpreted geometrically and from this a set of necessary conditions for the attainable region is derived. In particular the region must be convex with non-zero reaction vectors on the boundary either pointing into or tangent to the region. A limited, but powerful, sufficiency condition is also derived. The attainable region is constucted for both two and three dimensional examples. It is also shown how the region can be constructed when constraints, such as a specified sequence of reactors, are imposed. The properties of a reactor that lies in the boundary of the attainable region in n-dimensional space are discussed, and in principle the attainable region can be constructed in any number of dimensions. The most important and novel result found is that the method generates the structure of the reactor network that makes up the boundary of the attainable region and hence for many problems the optimal reactor network. This is in contrast to all previous methods where one guessed a network and then optimized it for various parameter values. It was also found that the optimal reactor configuration would in almost also all cases be a series-parallel arrangement of C.S.T.R 's, plug flow reactors and bypasses. Furthermore, the geometry of the boundary of the attainable region gives rise to analytical conditions for optimum reactors structures that are otherwise not readily available. Other interesting results were: - the boundary of the attainable region has very different properties depending on whether the dimension of the space is even or odd, suggesting that the optimization of systems of reactors in even and odd dimensional space could yield rather different results. - the geometric optimization of interstage cooling and coldshot reactors firstly gives insight into the known analytical conditions, but furthermore applies under conditions where the simple analytical optimization breaks down. - the well known properties of plug flow reactors with first order kinetics can be easily explained by the geometric properties of the attainable region.