The interaction of a number of self-heating bodies depositing heat into a common finite heat bath and thereby influencing each other is a problem of great practical importance in many areas including storage and transport of self-heating materials, drums of chemicals, foodstuffs etc. Critical conditions for the complete assembly of interacting heat producers (thermons) are derived under various assumptions and modes of ignition are identified. These include cooperative modes as well as modes which are simply perturbations of ignition for single thermons.

The storage of bagasse, which is principally cellulose, presents many problems for the sugar industry, one of which is bagasse loss due to spontaneous combustion. This is an expensive problem for the industry as bagasse is used as a fuel by sugar mills, and for cogeneration of electricity. Self-heating occurs in the pile through an oxidation mechanism as well as a moisture dependent reaction. The latter reaction is now known to exhibit a local maximum, similar to the heat release curves found in cool-flame problems. Bagasse typically contains 45–55% by weight of water when milling is completed and the question of how to reduce the moisture content is important for two reasons. Firstly, wet bagasse does not burn nearly as efficiently as dry bagasse, and secondly, self-heating is greatly enhanced in the presence of water, for temperatures less than 60–70°C.

An existing mathematical model is used, but modified to take into account the newly observed peak in the moisture dependent reaction. Most of the previously reported complex bifurcation behaviour possible in this model is not realized when physically realistic parameter values are used. The bifurcation diagram describing the long-time steady-state solution is the familiar S-shaped hysteresis curve. In the presence of the new form of the moisture dependent reaction, an intermediate state can be found which is not a true steady-state of the system as, in reality, the characteristics of the pile slowly change as water is lost. This state corresponds to observations of an elevated temperature (around 60–70°C) which persists for long periods of time. Approximate equations can then be defined which predict this intermediate state, and hence a different hysteresis curve is found. A simple explanation for the process by which water is lost from the pile is obtained from these equations and an analytical expression is given for the exponential decay of water levels in the pile.

A model is presented for describing the propagation of a one-dimensional wave of permanent form in a compressible gas in a pipe. Energy is lost to the system through the walls of the pipe, but the combustion wave produces heat through an exothermic chemical reaction. The full set of equations for the model is reduced to a phase-plane system, and it is shown that, for small amplitude waves, a weakly non-linear analysis leads to a temperature profile that is a classical solitary wave. A novel shooting method is developed for the full non-linear problem, and this confirms and extends the solitary-wave solution, up to a value of the temperature amplitude at which the wave begins to develop a shock. The jump conditions across the shock are presented, and numerical integration is used to continue the solution for temperature amplitudes at which a shock is present. An example is given of the extreme situation in which the shock is so strong that all the fuel behind the shock is exhausted.

Many tens of serious incidents involving reactors occur in the developed countries each year. The disaster at the chemical plant in Bhopal, India in 1984 was particularly notable where a thermal runaway process led to more than 3000 tragic fatalities from the cloud of extremely toxic methyl isocyanate that boiled out of a storage tank. This signalled the design of special types of chemical reactors to reduce the risk of thermal runaway by planning (at the design stage) integral safety and thermal stabilization mechanisms. The Endex CSTR (continuously stirred tank reactor) proposed by Gray and Ball [3] involves a reactor in two parts with heat exchange allowed between them. The two parts of the reactor operate side by side in tandem, such that the thermal runaway of one part is offset by an endothermic reaction in the other reactor—hence the term ‘endex’.

It is found that the adiabatic endex system has a large region of parameter space where the operation can be made safe. However adiabatic conditions rely on the continuous supply of reactants to the endothermic side of the reactor, for operation of the system. The risks involved are such that it is always safer to operate batch reactors in a non-adiabatic mode. Thus we consider the limiting case of the approach to adiabatic conditions where although the mathematics produces no oscillatory causes for instability, yet there is a narrow but significant area where the stable solution branch is lost and consequently a persistent and unexpected region of instability in what otherwise appears to be a simple CSTR system.

Steady state solutions for spontaneous thermal ignition in a unit sphere are considered. The multiplicity of unstable, intermediate, steady state, temperature profiles is calculated and shown for selected parameter values. The crossing of the temperature profiles corresponding to the unstable, intermediate, steady states is exhibited in a particular case and is proven in general using elementary ideas from analysis. Estimates of the location of crossing points are given.

Microwave heating of porous solid materials has received considerable attention in recent years because of its widespread use in industry. In this study, the microwave power absorption term is modelled as the product of an exponential temperature function with function that decays exponentially with distance. The latter describes the penetration of material by the microwaves.

We investigate the phenomena of multiplicity in class A geometries, paying particular attention to how the penetration function affects the behaviour of the system. We explain why the phase-plane techniques which have been used in the case when the penetration term is constant do not extend to non-constant penetration.

Oxygen index methods have been widely used to measure the flammability of polymeric materials and to investigate the effectiveness of fire-retardants. Using a dynamical systems framework we show how a limiting oxygen index can be identified with an appropriate bifurcation.

The effectiveness of fire-retardants in changing the limiting oxygen index is calculated by unfolding the bifurcation point with a suitable non-dimensionalised variable, which depends upon the mode of action of the additive. In order to use this procedure it is essential the model is non-dimensionalised so as to retain the variables of interest as distinct continuation parameters.

A simple model for the propagation of a combustion wave is proposed and the speed of propagation is predicted. It is assumed that the reactant ignites at a specified temperature and then burns until depleted with reaction rate dependent on temperature and reactant concentration. The exact solution and linear stability are determined in the case of constant heat generation and a numerical scheme is developed to generate traveling wave solutions in the more general case. This numerical method is applied to the case where the temperature dependence of the reaction rate is modeled by the Arrhenius function.

The steady-state heating of a two-dimensional slab by the TE10 mode in a microwave cavity is considered. The cavity contains an iris with a variable aperture and is closed by a short. Resonance can occur in the cavity, which is dependent on the short position, the aperture width and the temperature of the heated slab.

The governing equations for the slab are steady-state versions of the forced heat equation and Maxwell's equations while fixed-temperature boundary conditions are used. An Arrhenius temperature dependency is assumed for both the electrical conductivity and the thermal absorptivity. Semi-analytical solutions, valid for small thermal absorptivity, are found for the steady-state temperature and the electric-field amplitude in the slab using the Galerkin method.

With no-iris (a semi-infinite waveguide) the usual S-shaped power versus temperature curve occurs. As the aperture width is varied however, the critical power level at which thermal runaway occurs and the temperature response on the upper branch of the S-shaped curve are both changed. This is due to the interaction between the radiation, the cavity and the heated slab. An example is presented to illustrate these aperture effects. Also, it is shown that an optimal aperture setting and short position exists which minimises the input power needed to obtain a given temperature.

In a recent paper Weber et al. [9] examined the propagation of combustion waves in a semi-infinite gaseous or solid medium. Whereas their main concern was the behaviour of waves once they had been initiated, we concentrate here on the initiation of such waves in a solid medium and have not examined in detail the steadiness or otherwise of the waves subsequent to their formation. The investigation includes calculations for finite systems. The results for a slab, cylinder and sphere are compared.

Critical conditions for initiation of ignition by a power source are established. For a slab the energy input is spread uniformly over one boundary surface. In the case of cylindrical or spherical symmetry it originates from a cylindrical core or from a small, central sphere, respectively. The size of source and reactant body is important in the last two cases. With the exception of the initial temperature distribution, the equations investigated are similar in form to those of Weber et al. [5,9] and, as a prelude to the present study, with very simple adaptation, it has been possible to reproduce the results of the earlier work. We then go on to report the result of calculations for the initiation of ignition under different geometries with various initial and boundary conditions.

Self-heating in packed paniculate that is exothermically reactive is a major cause of fire and explosion in the powder industry. This study is focused on part of the Auckland development of a mathematical model dealing with this hazardous process in industry using milk powder as an example. Milk powder is a primary powdered food product around the world.

An update of the detailed mathematical model is given here, and predictions are made using the model to simulate the basket-heating behaviour of a milk powder in the laboratory (so the model can thus be validated). Basket heating in an oven is a standard laboratory technique for measuring the exothermic reactivity of a solid material.

After a favorable comparison with the laboratory results, several aspects of basket-heating were investigated with a view to further improving the technique. Firstly, the model was used to explore the effect of elevated ambient humidity and initial sample water content upon the heating process in the basket. Secondly, the model was used to explore the cross over phenomenon which is related to a novel procedure for measuring activation energy and exothermicity (that is, the Crossing-Point-Temperature (CPT) method, which is a new version of the basket heating technique). The predictions together with the experimental evidence show that the reaction kinetics obtained using the Heat Release (HR) method (another version of the basket heating technique well published in the literature) may not be correct, especially for those measured at elevated oven temperatures and for larger basket sizes. Thirdly, simulations were performed to illustrate that the CPT phenomenon does not just occur at the center of the basket but also occurs everywhere else in the sample. This can become a significant advantage for further development of the CPT method in terms of reducing experimental duration and improving reproducibility.