On dimensionality of attainable region construction for isothermal reactor networks

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Abstract

In this paper, a number of novel properties of reactor network synthesis (RNS) problems are presented and proved within the Infinite DimEnsionAl State-space (IDEAS) framework for isothermal reactor networks with and without the constant density assumption. These properties can then be used to reduce the dimension of the composition space (i.e., concentration space for constant density fluid (CDF) reactor networks, mass fraction space for variable density fluid (VDF) reactor networks) within which RNS is conducted and subsequently the attainable region (AR) construction. This procedure may be used to overcome the difficulty preventing the AR methodology and other RNS methods from being used for real industrial size problems.

Section snippets

Introduction and motivation

Horn (1964) proposed the concept of the AR, which can be defined as the set of all points in the concentration space that are attainable by some steady-state reactor network from a given feed. The AR technique has gained intense research attention after a geometric targeting procedure was proposed for the AR construction by Glasser, Hildebrandt, and Crowe (1987). This procedure, based on geometric interpretations of reaction and mixing, constructs the AR by recursive application of the

Preliminaries

Compositions of a stream can be represented by concentrations, molar fractions or mass fractions. As a result the AR construction or other RNS methods can be conducted in any of these three composition spaces. When the constant density assumption is valid, the concentration space is often used. In fact, it is shown that under the constant density assumption, the IDEAS framework and the Shrink–Wrap algorithm are applicable to the volumetric flowrate/concentration CDF CSTR/PFR models in Burri et

Constant density fluid reactor networks

In this section, the additional constant density assumption is made, that is the density changes caused by reaction or mixing are ignored.

Variable density fluid reactor networks

In this section, density changes are considered and isobaric condition is assumed. The mass flowrate/mass fraction VDF CSTR/PFR models can be written as follows:zioutziin=σMiRi(Cout)i=1,,n1Ckout=Ckout(zout,T,P)k=1,,nV=σqdzidσ¯=MiRi(C)i=1,,n1Ck=Ck(z,T,P)k=1,,nσ¯=0zi=ziini=1,,n1σ¯=σzi=ziouti=1,,n1V=σqwhere q is the mass flowrate; ziin and ziout are the ith component’s mass fractions at the reactor inlet and outlet respectively; Mi is the ith component’s molecular weight; σ=V

Conclusions

It is proved within the IDEAS framework that when some component generation rates are linearly related, concentrations (mass fractions) of corresponding components of any stream within CDF reactor networks (VDF reactor networks) are also linearly constrained. This development allows the construction of the AR and other RNS methods being conducted possibly in a lower dimensional composition space. Means of systematically achieving these linear relations are provided and proved. These results are

Acknowledgments

The authors gratefully acknowledge financial support from the National Science Foundation through grant CTS 0301931 and the equipment donated by Intel Corporation through its Higher Education Program.

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