Heterogeneous catalytic systems have been widely studied, both experimentally and theoretically. Mathematical modelling of such processes can be classified depending upon the approach taken to address three issues.
Models for heterogeneous catalysis were comprehensively reviewed by Razon and Schmitz . Recently, very detailed models for heat-transfer processes in heterogeneously catalysed reactions have been developed [Hayes & Kolaczkowski 1997].
The existence of oscillatory behaviour has attracted a great detail of experimental and theoretical interest, particular for isothermal mechanisms using Langmuir-Hinshelwood kinetics. Reviews of these and other aspects of heterogeneous catalysis of a general interest include [Sheintuch & Schmitz 1977; Slinko & Slinko 1978, 1982; Razon & Schmitz 1987, Doraiswamy 1991, Scott 1991, Warnatz 1992, Schüth et al 1993, Sheintuch 1997]. Scott  notes that
``many theoreticians have also taken advantage of the `grey areas' of uncertainty in the form of appropriate reaction rate laws, using putative surface reactions as an excuse for almost any otherwise unbelievable but convenient polynomial or transcendental form.''
There have been very few studies of non-isothermal heterogeneously catalysed reactions that incorporate inter-phase heat and mass transfer processes. In fact most models in the mathematics literature treat heterogeneous catalysed reactions as a single-step exothermic reaction, ignoring the adsorption and desorption processes. The porous catalyst particle model developed by Elnashaie and Cresswell [1973, 1974]. appears to be the only simple model that explicitly models adsorption and desorption processes, rather than modelling these as the product of a mass transport coefficient and a difference of concentration levels. They considered the case when the active species is weakly adsorbed onto the catalyst surface. Isothermal heterogeneously catalysed reactions in a CSTR, assuming Langmuir-Hinshelwood kinetics, were comprehensively treated by Chang and Aluko  and Aluko and Chang  who showed that even simple Langmuir-Hinshelwood mechanisms for bi-molecular reactions can have oscillatory behaviour.Subramanian and Balakotaiah  have classified the steady-state solutions and the dynamic behaviour of a well-mixed two-phase heterogeneous catalytic reactor model. In addition to assuming adiabatic conditions they represent adsorption and desorption processes by a simple mass transfer process and assume that the heat of adsorption is negligible. Furthermore they use the Frank-Kamenetskii variables, using the exponential approximation. In a latter paper Christoforatou et al  develop a runaway criterion for adiabatic catalytic reactors, considering both plug flow and CSTR reactors. Although the latter paper uses the model developed in Subramanian and Balakotaiah  it is curious that neither paper refers to the other. Christoforatou et al [1998)] imply that if the residence time is kept below a critical level corresponding to an ignition limit-point, then the temperature of the reactor can not rise to an undesirable level. This is only true if perturbations on the reactor parameters can not occur, as the high-temperature ignition steady-state co-exists with the low-temperature steady-state over a range of residence times between those corresponding to the extinction limit point and the ignition limit point. Furthermore, the results of Subramanian and Balakotaiah  show that there are some operating conditions in which perturbations on the reactor conditions can cause the temperature to reach the ignition state for arbitrary small residence times.