The model problem of a single, irreversible, exothermic reaction taking place in a perfectly mixed continuously stirred tank reactor, and its extensions, has been extensively studied by many authors. The early literature was reviewed by Ray [1977], more recent review articles are those of Razon and Schmitz [1987], detailing the multiplicity and dynamic behaviour of chemically reacting systems, and [Bequette 1991], dealing with the nonlinear control of chemical processes. The classic problem has, at most, seven static steady-state diagrams as control parameters are varied [Razon & Schmitz 1987], including the standard S-shaped combustion curve. We outline some recent extensions of the standard CSTR model which retain a single, one-step, exothermic reaction.

One of the motivations for studying the dynamic behaviour of a
CSTR is to understand the mechanisms by which thermal runaway
occurs, so as to eliminate this phenomena in industry.
The risk of thermal runaway may be reduced
by using a control that changes one of the
inflow parameters, e.g. the inflow temperature, inflow velocity, or
inflow concentration, as a function of the outflow variables,
usually the outflow temperature.
Possio & Pellegrini (1996)
have conducted a detailed bifurcation
analysis of a non-ideal CSTR, in which flow bypass and
dead volume are taken into account,
with a Proportional-Integral (PI) controller acting
on the inlet temperature. They showed that the introduction of the
control can cause unexpected patterns of dynamic behaviour via the
promotion of degenerate, type 1, Hopf bifurcations.
Furthermore the PI controlled CSTR
can destabilise a stable equilibrium point to a limit cycle for any
value of the control constants.
Sexton `et al` (1997)
considered the effect of control on limit cycle behaviour where
the feed flow rate is continuously changed in
response to the outflow temperature. They showed that
oscillatory behaviour can be eliminated by linear proportional
control, although the choice of the gain parameter is not
simple. Furthermore they presented several nonlinear
proportional control functions which ensure that oscillatory behaviour
is eliminated.

Recently an ingenious way of controlling the standard reaction by placing the CSTR within a second CSTR in which an endothermic reaction occurs has been investigated [Gray and Ball 1999; Ball and Gray 1999]. The basic idea is to carefully choose the endothermic reaction so that it `kicks in' when the exothermic reaction is on the point of thermal runaway, cooling it to the desired operating temperature. Although simple in principle, this concept translates into a differential dynamical system of formidable complexity.

The standard CSTR model implicitly assumes that the cooling jacket dynamics are negligible and that the coolant temperature, often referred to as `ambient temperature', is the experimentally manipulated variable. Russo & Bequette [1995, 1995] have included the cooling jacket energy balance, so that the coolant temperature becomes a third state variable. The principal control parameter then becomes the coolant flow rate.

Although research into CSTR models goes back over forty years, no universally accepted non-dimensionalisation scheme has emerged. This lack of consistency has resulted in many different dimensionless forms of CSTR models. The non-dimensionalisation chosen is unimportant in the cataloguing of the distinct types of steady-state diagrams exhibited by a system. However, it is not possible to relate the order in which these steady-state diagrams occurs in the non-dimensionalised model to the order in which they would be encountered experimentally unless the distinguished bifurcation parameters, which depend upon the non-dimensionalisation, correspond to experimentally distinct quantities.

This point has not been appreciated in most of the CSTR studies to-date and it is only recently that consistent use has been made of non-dimensionalised forms in which physically distinct quantities are retained as independent non-dimensionalised parameters [Ball 1999].

**References**

- R. Ball.
The origins and limits of thermal steady-state multiplicity in
continuous stirred tank reactor.
`Proceedings of the Royal Society of London A`, 455:141--161, 1999. - R. Ball and B.F. Gray.
Thermal stabilization of chemical reactors. II Bifurcation analysis
of the Endex CSTR.
`Proceedings of the Royal Society A`, 455:4223-4243, 1999. - R.W. Bequette.
Nonlinear control of chemical proccesses: A review.
`Ind. Eng. Chem. Res`, 30:1391-1413, 1991. - B.F. Gray and R. Ball. Thermal stabilization of chemical reactors. I The mathematical description of the endex reactor.
- C.T. Possio and L. Pellegrini.
The influence of Hopf bifurcation degeneracies on the
dynamic behaviour of a non-ideal PI controlled CSTR.
`International Journal of Bifurcation and Chaos`, 6(7):1255--1266, 1996. - W.H. Ray.
Bifurcation phenomena in chemically reacting systems.
In Paul H. Rabinowitz, editor,
`Applications of Bifurcation Theory`, pages 285-315. Academic Press, Inc, 1977. - L.F. Razon and R.A. Schmitz.
Multiplicities and instabilities in chemically reacting
systems --- A Review.
`Chemical Engineering Science`, 42(5), 1005-1047, 1987. - L.P. Russo and B.W. Bequette.
Impact of process design on the multiplicity behaviour of
a jacketed exothermic CSTR.
`AIChe Journal`, 41(1):135--147, January 1995. - L.P. Russo and B.W. Bequette.
Impact of process design on the open-loop behaviour of
a jacketed exothermic CSTR.
`Computers Chem. Engng`, 20(4):417-426, 1996. - M.J. Sexton, L.K. Forbes, and B.F. Gray.
A study of the dynamics of a continuously stirred tank reactor with
feedback control of the temperature.
`Dynamics and Stability of Systems`, 12(2):89--107, 1997.

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