The attainable region generated by reaction and mixing
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Date
2015-01-27
Authors
Hildebrandt, Diane
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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.