Abstract
In this paper a model for the heterogeneously-catalysed reaction of a
gaseous species undergoing a single-step exothermic reaction in a
well-mixed diabatic continuously stirred tank reactor (CSTR)
is developed and analysed. Low temperature reaction conditions are
assumed, so that the degradation of the inflow species in the
gas-phase is negligible. The adsorption and desorption of the
active species onto a solid catalyst layer is explicitly modelled.
Under the conditions used the classic combustion S-shape steady-state curve is not exhibited. Instead the steady-state diagram consists of two disjoint solution curves: a solution curve containing a steady-state branch corresponding to full coverage of the catalyst with no conversion; and, an isola containing a steady-state branch that corresponds to low coverage of the catalyst with a high conversion of the inflow species. The steady-state curve contains one extinction point and no ignition points. Consequently the initial conditions determine onto which steady-state the system evolves. Large variations in the critical catalytic surface area are found as the initial temperature of the reactor is varied. This is expected to have significant practical implications.
M.I. Nelson, G.C. Wake, and X.D. Chen. Heterogeneously catalysed combustion in a continuously stirred tank reactor - low temperature reactions Combustion Theory and Modelling, 4(1):1-27, March 2000.
Introduction
The continuously stirred tank reactor (CSTR: sometimes referred to as
the open reactor) and the batch reactor (closed system) have been
well documented to be very useful arrangements to investigate a wide
variety of chemical reactions, particularly oscillatory reactions.
Numerous examples of these can be found in [3]. However, there are
still issues that cannot be answered by utilising the closed or
CSTR arrangement as mentioned in [6] and [7]. These include the
effects of the gradual accumulation of certain products and the lack
of outflow have on the dynamical behaviour of a particular system.
Rabai and Epstein [6] suggest that the experimental condition that
``sits in the middle'' of the CSTR and batch conditions, the
semibatch arrangement, is potentially a powerful tool to investigate
chemical systems which have not been fully exploited despite there
being several practical examples where the semibatch mode of
operation is particularly valid ([6] and [7]). These include the
industrial production of oxalic acid by the action of nitrogen
dioxide in hydrolysed starch (which has safety implications due to
high excursions of temperatures), waste treatment plants where
materials accumulate in holding tanks before being treated by
suitable chemicals to convert the waste to less toxic substances,
and also the administration of therapeutic drugs. Furthermore,
Griffiths [5] suggests that studies of semibatch reactors, whereby
one reactant flowing at a controlled rate into another reactant in
a reaction vessel, has potential safety applications since the
semibatch operation is very similar to a combustion hazard scenario
where an inadvertent leakage of a particular chemical through a
valve occurs.
In this paper we consider the gas-phase reaction between hydrogen and
chlorine in the semibatch reactor. Coppersthwaite et al. [1]
were the first to conduct experimental measurements
and numerical simulations for this reaction in the
semibatch mode of operation. These authors showed that this reaction
(under non-adiabatic semibatch operation) can be represented by a
two-dimensional model (based on the Sal'nikov scheme) in which a
reactant concentration (
In this paper we explore this reaction further, firstly by comparing the full system with the reduced model obtained by utilising the quasi stationary-state approximation (OSSA) which was suggested and derived by Coppersthwaite et al [1] (and to a certain extent treated briefly by Warnatz et al [8]. We then investigate the effects of varying the inflow rate of hydrogen and the vessel (or ambient) temperature. Finally we will complete our investigation by first deriving and then examining the results of a reduced scheme based on the Sal'nikov model and compare these results to the more detailed H_{2} + Cl_{2} reaction scheme. We are particularly interested in using the Sal'nikov scheme to predict the onset of oscillations in the semibatch scenario. [1] attempted to do this in their study, however a proper comparison between the full scheme and the Sal'nikov scheme was not undertaken. Furthermore, these authors did not show the derivation of the Sal'nikov scheme. Both of these issues have been undertaken in the present study.
H.S. Sidhu, M.J. Sexton, M.I. Nelson, G.N. Mercer, and R.O. Weber. A Simple Combustion Process in a Semibatch Reactor. In R.L. May, G.F. Fitz-Gerald, and I.H. Grundy, editors, EMAC 2000 Proceedings, pages 251--254. The Institution of Engineers, Australia, 2000. ISBN 085825 705X.
Abstract
Comparison theorems may be used to prove the existence and
uniqueness of solutions to certain types of partial differential
equations. They provide bounds for solutions
and can be used as the basis of numerical techniques for the
computation of solutions. In 1961 Alex McNabb published one of the
first papers extending their use to multi-component systems.
Developments in the theory and applications of such results, through
citations of this original paper, are reviewed.
M.I. Nelson. Comparison Theorems for Multicomponent Diffusion Systems: Developments since 1961. Journals of Applied Mathematics and Decision Sciences, 4(2):151-163, 2000.
Abstract
This paper investigates the bifurcation behaviour of a model oxidation
reaction in a continuously stirred tank reactor (CSTR). We assume that
two gaseous chemical species are pumped separately into the CSTR, at
constant total pressure, reacting to produce an inert product. The
reaction is assumed to be a single step reaction that is described by
Arrhenius kinetics. It is capable of producing oscillatory behaviour as
well as steady state multiplicity in certain parameter regions.
Bifurcation diagrams in various control parameter spaces are presented.
We show that the system always possesses a globally attracting invariant
region. The equivalence of a CSTR having n feed streams and the one
pipe version, by appropriate re-scaling, is also discussed.
Key Words: exothermic reaction, bifurcation, flammability limits, flow reactor
H.S. Sidhu, M.I. Nelson, G.N. Mercer and R.O. Weber. Dynamical analysis of an elementary X+Y-->P reaction in a continuously stirred tank reactor. Journal of Mathematical Chemistry, 28(4):353-374, 2000.