## Abstract

We present numerical investigations of a spherically symmetric model for the ignition and subsequent combustion of low-exothermicity porous materials exposed to a constant, continuous heat source. We account simultaneously for oxidant, the gas-dynamic processes, including a gaseous product of reaction, and a solid product that is allowed to assume different physical properties from the solid reactant. For external conditions that are typical of natural convection, the model exhibits striking novel behaviour, including the possibility of a potentially dangerous high-temperature ‘burnout’ at the external surface of the material, which triggers a reverse combustion wave propagating from the outer surface of the solid towards the heat source. This phenomenology is controlled largely by the diffusion of oxygen entering the system. We identify the effects that convection and product properties have on combustion of the solid, particularly on the formation of a reverse wave. Applications of the approach to specific problems are discussed and future work is outlined.

## 1. Background and introduction

Ignition and self-sustained combustion of low-exothermicity materials are issues of great practical importance, particularly for avoiding or controlling fire and explosion hazards in industries that handle combustible solids in particulate or connected (e.g. fibrous or foam-like) form. One of the more prevalent problems is that of smouldering, a low-temperature, flameless form of combustion, controlled to a large degree by the rate of oxygen transport to the combustion zone (Ohlemiller 1995, 2002). The reaction continues to consume oxygen as fast as it is able to reach the reaction zone, causing a total depletion of oxygen locally.

Understanding this type of combustion is important for a number of reasons. Apart from yielding toxic products, it is a common precursor of fire (flaming combustion) based on secondary (often much more exothermic) oxidation of the initial char product, or of explosion. The smouldering can start deep within the solid fuel and can go unnoticed for long periods of time, taking hours or even days to develop. In a series of experiments, Palmer (1957) demonstrated that a layer of sawdust 1 m in depth can take up to two weeks to smoulder through entirely. The potential for a sudden flare-up when the smoulder zone reaches the surface of the solid with a rich oxygen supply is then an extra hazard (see figure 5 of this paper and the related discussion), potentially catastrophic if it leads to a dust explosion. Settings for such events come in the form of grain elevators, bins and silos, flour mills and other environments where foodstuffs, plastics, wood, metals or other exothermic materials are found in powder form. According to several surveys (e.g. Abbott 1988; Jeske & Beck 1989), smouldering ‘nests’ are the cause of a sizeable fraction of these and other major accidents in powder and dust handling plants.

A common initiator of smouldering, or indeed any form of combustion, is a localized heat source or ‘hotspot’, possibly caused by mechanical friction or an electrical fault in processing equipment, or even by localized spontaneous exothermic reaction in the material itself. A hierarchy of models for the combustion of reactive solids subject to external heating, in the form of a constant heat flux have been described earlier (Brindley *et al.* 1999, 2001*a*,*b*; Mcintosh *et al.* 2003; Shah *et al.* 2003, 2004, 2006), motivated primarily by the need for a fundamental understanding of industrial ignition hazards. These investigations have been concerned with the conditions required for self-sustained combustion to be initiated and for the most part have assumed that oxygen is in plentiful supply. Less attention has been paid to the oxygen-limited case that is the subject of this investigation, and which can lead to new behaviour (demonstrated in later sections) such as total internal burnout and, when accompanied by a finite rate of heat loss to the external surface of the material, to the formation of a reverse wave.

The well-established theory of spontaneous ignition in solids does not entirely lend itself to the case in which the solid has low exothermicity. The classic Frank-Kamenetskii theory for predicting the conditions under which ignition (or thermal runaway) in a spatially distributed system will occur (Frank-Kamenetskii 1955), idealizes the rapid acceleration to high temperatures by its identification with the development of a mathematical singularity in the temperature solution. The great majority of related models neglect any reactant consumption, fail to account for oxygen availability in the reaction zone and assume that the reaction products are the final products. They are valid only for systems that do not depart far from their initial state prior to the onset of ignition (Burnell *et al.* 1989; Gray & Wake 1993; Watt *et al.* 1999) since their large heat of reaction (high exothermicity) provides a sufficient trigger for ignition while the reactant consumption is still (mathematically) negligible. However, this is not true for low-exothermicity materials, in which ignition is preceded by significant reaction consumption. In this case, the outcome of the competing effects of reaction consumption and of temperature increase is that both the time and location of an ignition event, if indeed it occurs at all, are *a priori* unknown (Brindley *et al.* 1999, 2001*a*,*b*; Mcintosh *et al.* 2003; Shah *et al.* 2003, 2004, 2006). In such cases, a full Frank-Kamenetskii or asymptotic analysis (as in Linãn & Williams (1971), Kapila (1981) and Telengator *et al.* (1999)) is generally restricted to highly simplified mathematical models (Boddington *et al.* 1977; Burnell *et al.* 1983; Bebernes & Lacey 1992; Shah & Wake 2004; Shah *et al.* in press). Numerical modelling, however, reveals that, in contrast to high-exothermicity materials, reactant depletion in the materials of interest here can inhibit the birth of a self-sustaining combustion wave even when the ignition temperature (as defined in the Frank-Kamenetskii sense) is attained. The effects of gas convection, heat exchange between the solid and gas phases, and limited oxygen further complicate the problem.

A closely related issue is that of filtration combustion, in which an exothermic reaction front propagates through a porous solid that reacts with an oxidizing gas flowing through its pores (usually forced by imposed pressure gradients). Important applications include self-propagating high-temperature synthesis (SHS). Modelling has been mainly confined to the propagation and stability of the established combustion front, rather than its initiation (Aldushin & Matkowsky 1998; Margolis 1998; Leach *et al.* 2000; Wahle *et al.* 2003). In most practical cases, ignition is stimulated artificially, and, provided it is successful, its detailed phenomenology has been of less concern in that context. In natural smouldering (i.e. no forced oxidizer), the speed of the combustion front, if one exists at all, will be slower. The pressure gradients that develop in the gas provide the only driving force for convection (Darcy's law), which is usually opposed to the diffusive flux of oxygen. This scenario has received far less attention, except in specialized cases such as cigarette burning (e.g. Rostami *et al.* 2003), where again the authors are not concerned with the initiation of the wave.

Throughout this introduction, the word ‘ignition’ has been used fairly loosely, reflecting its historic range of use in the literature. In fact, it is not easy (perhaps not even possible) to provide a unique definition of ignition. In a mathematical sense, it is often associated with the appearance of a singularity at a finite time or with a jump from one branch to another in a bifurcation diagram; physically, it is usually taken to mean the onset of some form of self-sustaining combustion. Throughout this paper, comprising as it does the results of extensive numerical modelling, we shall adopt the latter, if somewhat imprecise, descriptive use.

The model we explore represents a small spherical ‘hotspot’ supplying a constant heat flux, embedded in a much larger body of porous solid reactive material, powder or matrix. The solid reacts exothermically with oxygen to form both solid and gaseous products, which are allowed to have separate temperatures.

The outline of the paper is as follows: in §2, a mathematical model is presented, together with a detailed discussion of the assumptions and approximations involved; the numerical results are presented in §3 and are discussed in §4, where we also indicate future avenues of investigation.

## 2. Model assumptions and equations

We consider the combustion of a porous solid, X, with oxygen, O, producing both gas and solid products, S and Z, respectively. For simplicity, we assume single-step, irreversible kinetics(2.1)with unit stoichiometric coefficients. A similar scheme is found in Staggs (2001), and more complex kinetics are discussed in DiBlasi (2000). We briefly outline the most important features and assumptions. A more detailed account can be found in Shah *et al.* (2004, 2006).

Separate energy balances are expressed for the gas and solid phases, with a linear rate of heat exchange between the two. The density and thermal capacity of the solid far exceed those of the gas, thus the energy released in the reaction is primarily spent on heating the solid. Accordingly, we account for heat production by reaction only in the energy equation representing the solid, with the assumption that the gas is heated through the heat exchange and by the impact of the gaseous product at the temperature of the solid.

We assume a single, well-mixed gas phase with a single density, specific heat capacity and thermal conductivity.

We use the ideal gas and Darcy's law for the gas phase, and make a dilute-mixture assumption, with nitrogen as the dominating component. The full Chapman–Enskog expression is used for the molecular diffusion coefficients, with a Bruggeman correction to account for the tortuosity of the diffusion path.

The assumptions above enable us to write the equations for energy (temperature), oxygen concentration, solid reactant volume fraction, gas continuity, the ideal-gas law and Darcy's law as follows:(2.2a)(2.2b)(2.2c)(2.2d)(2.2e)(2.2f)(2.2g)Here, is the temperature of the gas (solid) phase; is the gas pressure; is the gas (solid reactant/product) density; *Y* is the mass fraction of oxygen, taken with respect to the mass of the gas phase; *ϕ* is the porosity, *ψ* is the volume fraction of solid reactant and *ψ*_{S} the volume fraction of solid product.

The various parameters are defined as follows: *Q*′ is the heat release per kmol of X; *D*′ is the binary diffusion coefficient for oxygen; *A*′ is the pre-exponential constant and *E*′ is the Arrhenius activation energy for the reaction rate; *W*_{X} and *W*_{O} are the molar masses of oxygen and Z, and is an average for the gas phase; *κ*′ is the permeability of the solid phase and *ν*′ is the dynamic viscosity of the gas phase; *μ*′ is a volumetric heat-exchange coefficient between the solid and gas; are the specific heat capacity and thermal conductivity for the gas phase (solid reactant/product); and *ϵ* is the fraction of the mass of reactants, X+O, that forms gas.

The relative permeability in equation (2.2*f*) is approximated by the Kozeny–Carman law (Bear 1972, p. 166),(2.3)where *K*′ is a shape factor and is a mean pore diameter. The constant *k*′ is a modified permeability, which we shall refer to as the Kozeny–Carman constant. Note that the value of *κ*′ depends entirely on the solid, whereas the dynamic viscosity is a property of the gas.

The molecular diffusion coefficient is derived from a Bruggeman correction (in place of a tortuosity) and the Chapman–Enskog theory applied to a nitrogen–oxygen mixture (Bird *et al.* 2002, p. 526).(2.4)where *Ω*_{D,ON} is the collision integral and *σ*_{ON} is the collision diameter (in Å), calculated from the molecular diameters of oxygen and nitrogen (these are taken from Bird *et al.* (2002) to yield the value in table 1).

The heat-exchange coefficient *μ*′ is the interfacial heat transfer coefficient between the solid and gas, , multiplied by the specific interface surface area (total surface area/total volume). The coefficient is related to the local Nusselt number by , where is an average pore diameter. The Nusselt number is typically correlated experimentally as a function of the Reynolds, Prandlt and Grashof numbers (Incropera & DeWitt 2002, ch. 6).

The temperature of the solid is higher than that of the gas phase and gas product leaves the reaction zones at the temperature of the solid. We account for this temperature difference by placing the term (arising from the solid energy equation), which represents the rate of heat-content change as a result of the gaseous product of reaction, in the gas energy equation. Additionally, is replaced with to ensure consistency in the heat capacitance of the gas phase. The resulting extra source term in the energy equation for the gas phase can be seen naturally as the change in the heat capacity of the gas. Radiative losses are neglected since the temperature at the external surface of the material is considered relatively low. In cases where there is substantial reactant depletion at this surface, these losses may have an effect.

The first of equations (2.2*g*) relates the solid reactant volume fraction, *ψ*, to the porosity, *ϕ*. It is arrived at through the following mass balance: let *V*′ be a control volume with porosity (subscript referring to solid or gas) and solid reactant volume fraction *ψ*. The volume fraction vacated by any reacted X, from the initial time, is given by (1−*ϕ*_{0})−*ψ*, where *ϕ*_{0} is the initial porosity. Thus, the mass of reacted X is given by . The corresponding mass of reacted oxygen is given by . Therefore, the mass of solid product is , where *ϵ* is the fraction of the mass of X+O that forms S. Since the solid product has a constant density , the volume occupied by *S* is given by , yielding equations (2.2*g*).

Note finally that on the right-hand side of equation (2.2*a*), *Γ*=(1−*ϵ*)(1+*W*_{X}/*W*_{O})−1 is the net mass of gas produced (or consumed) per unit mass of oxygen (reacting in stoichiometric proportion with X).

### (a) Boundary and initial conditions

We assume a spherical hotspot of radius supplying a constant heat flux, embedded in a much larger concentric spherical volume of porous solid, radius . Other, more complicated boundary shapes will have some effects but we expect them to be quantitative rather than qualitative. The temperature at the hotspot surface is uniform, equalizing between solid contact and gas contact very rapidly, for a typical average pore size. Thus, it is realistic to assign equal gas and solid phase temperatures at the hotspot surface and to average the flux with respect to volume (to take account of the difference in thermal conductivity between the two phases). The boundary conditions for the temperatures are therefore(2.5)corresponding to a heat flux *p*′ (and total power ) applied at the surface of the hotspot, , and heat transfer with coefficient *h*_{a} at the outer surface . The latter coefficient can be predicted from the relationship , with conditions pertaining to the environment (see below). A wide variation in arises from that in the Nusselt number, *Nu*.

A particularly convenient, and often adopted, idealization of the conditions at is obtained by setting , i.e. imposing a condition of instantaneous relaxation to the conditions in the environment, in which case . In the results section, we provide a comparison of this with the more general form given by equation (2.5), demonstrating that the value of is a key parameter to the problem posed.

The hotspot is assumed to be impenetrable to gas so that the gas velocity, and therefore diffusive flux of oxygen, is zero at its surface. The boundary conditions on the gas density and oxygen mass fraction are(2.6)The conditions at the outer edge are consistent with an imposed atmospheric pressure . The condition on the mass fraction at the outer surface is idealized; we assume a constant mass fraction (but not a constant concentration). The molar mass of gas, *W*, is determined from a mass average of the molar masses of the constituents, assumed to be oxygen and nitrogen(2.7)where *Y*_{a} and *Y*_{N} are the mass fractions of oxygen and nitrogen at the outer surface, respectively.

Finally, the initial conditions are(2.8)

### (b) Non-dimensionalization

For the computations, it is convenient to non-dimensionalize the variables as follows:(2.9)noting that *Y*, *ϕ*, *ψ* and *ψ*_{S} are already dimensionless, and to introduce the following non-dimensional parameters:(2.10)This yields (from equations (2.2*a*–*g*) to (2.8))(2.11a)(2.11b)(2.11c)(2.11d)(2.11e)(2.11f)The initial-boundary conditions (2.5)–(2.8) become(2.12a)

(2.12b)

Problem (2.11*a*–*f*), (2.12*a*,*b*) was solved using the finite-element method implemented in COMSOL, with the general form option. All calculations were performed with a quadratic Lagrange basis using between 240 and 1920 elements.

Unless otherwise stated, the physical parameters assume the values given in table 1. It is worth noting the small value of solid conductivity. This value is typical of many woods (including sawdust), in the range 0.05–0.5 W m^{−1} K^{−1}, and is roughly equal to the value for carbon monoxide, carbon dioxide and air at high temperature.

## 3. Results and discussion

As a starting point, we use the constant temperature (*T*_{g}=*T*_{s}=1) boundary condition at the outer boundary *r*=*r*_{1}, i.e. we set *h*_{a}=∞. The significant effect of replacing this condition with a finite rate of heat loss (a Robin condition) is also examined.

For the case of abundant oxygen supply, there is a critical hotspot power above which a self-sustained combustion wave is formed, causing complete burnout of the solid except for a small layer near the outer boundary owing to the cold boundary conditions. Figure 1 shows the profiles of solid reactant volume fraction and temperature in this (supercritical) regime. In this example, one dimensionless time unit corresponds to 0.49 min, so that combustion is complete inside 13 min. For lower hotspot powers, no such wave is formed and the burning proceeds not only much more slowly but also incompletely (in finite time).

### (a) Oxygen-limited combustion: general characteristics

When the reaction is oxygen limited, an entirely different picture emerges, as demonstrated in figure 2. For *h*_{a}=∞, there is again a critical power below which no significant combustion occurs. Above this critical value, a zone emerges, at an interior location separate from the hotspot, in which combustion proceeds until all reactant is consumed in a relatively short period of time and beyond which combustion occurs at an extremely slow rate. The combustion is controlled by the delivery of oxygen to the reaction zone. In the early stages of reaction, there is a partial burning of the solid, limited by the amount of oxygen initially in the pores.

Beyond this initial period, a sensitive balance between oxygen concentration and temperature (favourable to burning) is achieved, and at this stage two combustion fronts develop, one propagating ‘backward’ from the fully burnt zone in the direction of the hotspot and the other ‘forward’ in the direction of the outer boundary (see the profiles of reaction rate). As the hotspot power increases, both the rate and the degree of degradation of the solid increase. In this example, *h*_{a}=∞, so that a large portion of the solid material in contact with the surface *r*=*r*_{1} remains unreacted.

### (b) Sensitivity to boundary conditions with the environment: variations of

With a finite heat transfer coefficient at the outer surface (), a qualitative change in behaviour from that discussed above (and illustrated in figure 2) is observed for hotspot powers greater than a critical value. The following possibilities exist.

For small (subcritical) hotspot power, no significant combustion is observed.

For hotspot power greater than a critical value (supercritical), burning of the type seen in figure 2 emerges, an example of which is illustrated in figure 3

*a*. The heat transfer coefficient in this example takes the value . A dimensional time unit corresponds to 9.05 s, so that the profile at*t*=2200 is reached within approximately 5.5 h.At a second (greater) critical hotspot power, the development initially mirrors that of the previous example, i.e. figure 3

*a*, but, crucially, a ‘reverse’ combustion wave eventually forms at the*outer*boundary*r*=*r*_{1}, as demonstrated in figure 3*b*. In this example, for and a hotspot power of 10 W, the temperature rise at the outer boundary is enough to initiate a (reverse) combustion wave that propagates back into the material, totally consuming any remaining reactant. The wave forms at*t*≈3000, or in dimensional terms*t*′≈6.1 h.As the hotspot power is increased, the temperature rise at the outer boundary occurs more rapidly, so that a reverse wave forms with increasingly less reactant consumption prior to its birth. An example is given in figure 4, which clearly demonstrates the rapid temperature rise at the outer boundary at around

*t*=2800 (or*t*′=3.96 h). Within a short period of time centred around the birth of the reverse wave, the solid temperature rises substantially ahead of the front, preheating the virgin solid to a temperature well above that required for self-sustained combustion. The rate at which combustion proceeds is therefore limited entirely by the (low) oxygen concentration at the combustion front. The enhanced preheating of the solid arises from the simultaneous supply of heat from the reaction zone and the hotspot ahead of it (and therefore the simultaneous conduction of heat to a region ahead of the front from both directions). This is demonstrated by the evolving solid temperature profile in figure 4*e*, with the development of a large (negative) gradient at*r*=*r*_{1}. In the region between the front and the outer boundary, the temperature eventually falls below that at the front, a consequence of the convection of heat in the gaseous phase directed towards*r*=*r*_{1}. It is worth noting that the gas and solid temperatures in this example (figure 4) equilibrate relatively rapidly because of the large heat-exchange rate (*μ*′) assumed. The effect of variations*μ*′ will be presented in a separate publication.At fixed hotspot power (and heat flux), the extent of material degradation increases with decreasing

*h*_{a}, as demonstrated in figure 3*c*,*d*(or by comparing figure 2 with figure 4). In this sense, at a fixed hotspot power or flux, there is a critical value of below which the reverse wave phenomenon may exist. As another indicator of the difference in burning for values of on either side of the critical value (equivalent to the difference on either side of the critical values of for fixed ), we refer to figure 5. This figure shows the evolution of the reaction rate with , for and , the latter value being small enough to trigger a reverse wave. For , the temperature rise at the boundary is greater. Even though the concentrations of oxygen and solid reactant are lower, the exponential dependence of the reaction on temperature dominates, ensuring that a reverse wave is formed.As is lowered, the critical hotspot power required for a reverse combustion wave decreases.

The relatively small values of for which the reverse wave is apparent in these examples are typical of natural convection under conditions of laminar flow in the environment: ; much higher values may be achieved by forced convection of cool air. Unsurprisingly, the magnitude of below which the reverse wave may form is highly dependent on the physical and chemical properties of the solid (particularly the activation energy) and the ambient temperature.

### (c) Solid product properties

One of the features of the present model is the freedom to assume different properties for the reactant and product solids. We examine differences in the specific heat capacity, density and thermal conductivity (each in isolation).

Figure 6*a* shows the evolution of *ψ* when the product thermal conductivity takes the value 0.2 W m^{−1} K^{−1}, with all other parameters as in table 1. As in the example of figure 4, a reverse wave is formed, but the time to initiation is shorter, and the wave that eventually forms travels at a greater speed (an approximately 25% increase). This latter significant aspect is probably due to enhanced preheating through the increase in solid-phase conductivity.

The effect of a decrease in the specific heat capacity of the product solid to *C*^{′}_{S}=750 J kg^{−1} K^{−1}, with all other parameters as in table 1, is demonstrated in figure 6*b*. Comparison with figure 4 again reveals a shorter initiation time to the reverse wave. In this case, the lower heat capacitance of the solid phase is the likely cause.

Differences in density have a mixed effect on the results. Figure 7 compares the base case of figure 4, in which the two solids possess the same density, to the case in which the product density takes the value 200 kg m^{−3} (all other parameters as in table 1). In the latter case we see from the corresponding profiles of *ψ* that a reverse wave does not form. Here, the lighter product density leads to a greater occupation of volume at the state of complete combustion, compared with the case presented in figure 4 (see the profiles of *ψ*_{s} in figure 7).

It is not an easy matter to pinpoint the causes for this trend; there are several competing processes. In figure 8, the evolutions of the differences in several quantities between the examples of figures 4 and 7 are depicted (value for minus the value for ). For , the greater volume occupied by the product leads to a greater effective thermal conductivity of the solid phase ( in equation (2.2*b*)), which raises the temperature more rapidly in a region surrounding the hotspot. However, the convective heat flux is lower in this case as a consequence of the much lower porosity (see the profiles of the difference) and this inhibits the transport of heat to the outer surface to the extent seen in the case of . This is reflected in the profiles of reaction rate difference, which form a positive peak corresponding to that in the temperature difference and are negative in a region between this peak and the outer surface.

It appears then that the cause for the lack of a reverse wave in figure 7 is due to restricted transport of heat by convection directed towards the outer surface and not, as one might expect, a reduced oxygen supply to the reaction zone as a consequence of the contracting pore volume. As a result of the confinement of heat to a region close to the hotspot, the solid degrades preferentially in that region, which results in the porosity-difference profiles in figure 8. At the same time, insufficient heat is sent to the outer surface to trigger a reverse wave.

## 4. Summary and future direction

The class of problems that forms the subject of this paper is relevant to a wide range of industrial processes of a potentially hazardous nature. The model used is an extension and generalization of those introduced in Brindley *et al.* (1999, 2001*a*, *b*), Mcintosh *et al.* (2003) and Shah *et al.* (2003, 2004, 2006), particularly in relation to the boundary conditions and product properties. Complete description of all possible phenomenology is impracticable in a single paper, but the exploration of effects of variations in several influences and properties has displayed qualitative behaviour that, in addition to its own scientific interest, will prove valuable in informing more detailed modelling of specific situations.

Two features of particular interest stand out: firstly, the widely occurring behaviour in which complete burnout of the reactant occurs initially at an interior point of the material rather than at the hotspot surface and secondly, the crucial importance of interactions with the ambient environment, expressed through conditions at the outer boundary, *r*=*r*_{1}.

The first of these may be readily understood in terms of the competing effects of oxygen limitation (itself the consequence of further competition between inward diffusion of oxygen and outward mass flow of gaseous product) and heat input arising from exothermic reaction as well as conduction from the hotspot. The reaction is most vigorous where the combination of these effects, mediated by local reactant depletion, is maximized. This is expressed through the term *Q*′Ω′ in equation (2.2*b*); near the hotspot, the oxygen deficiency effectively removes this term, far from the hotspot the temperature is too low. In contrast to the simpler case considered, for example in Brindley *et al.* (2001), in which a concept of ‘criticality’, expressed there as a critical value of hotspot power, separates two distinct qualitative behaviours, we now have at least three qualitative scenarios each occurring in extended regions of parameter space: slow reaction (figure 2); initial interior burnout followed by a self-propagating front travelling ‘backwards’ towards the hotspot (figure 4); and a forward self-propagating combustion front (figure 1). The parameter space in question is high dimensional, reflecting the many physical properties and external influences that can be important in this sensitive phenomenology. In the current model, table 1 lists about 20, each of which takes a value specific to a particular physical configuration.

All the phenomena described above are seen when the outer boundary, *r*=*r*_{1}, is maintained at a constant (low) ambient temperature. When this condition is relaxed, a new kind of behaviour is revealed, as demonstrated in figure 4. For sufficiently small values of the power, *p*′, the qualitative behaviour is similar to that for the cold boundary case (figure 3*a*). However, for larger values of *p*′, the temperature at the outer boundary rises sufficiently to trigger a combustion wave that propagates backwards into the material, fully consuming any remaining reactant. A similar criticality in behaviour may be achieved through variation in the heat transfer coefficient at the boundary *r*=*r*_{1}, (figure 3*c*,*d*). The physics is vividly exhibited in figure 5, which shows that, in appropriate circumstances, the temperature at this boundary leaps to a very high value during a rapid (and complete) burnout of the reactant. This second ‘hotspot’ is able to trigger the reverse combustion wave that, fed by oxygen diffusing from the rear, is able to self-propagate, driven by the heat from reaction (it is worth noting that this phenomenon is not restricted to the small spherical geometry considered here, but is possible in, for example, the one-dimensional case of a planar hot surface).

This phenomenology implies yet another ‘criticality’ for complete burnout in a relatively short time. Perhaps more importantly, the high temperature attained at the boundary could pose a major hazard for further fire or explosion in the ambient surroundings.

Differences in the physical properties of the materials, involving porosity, permeability, thermal conductivity, exothermicity, etc., can all influence these several criticalities, but the distinct qualitative phenomena are remarkably robust to substantial variations in parameter values; figure 9 shows, schematically, the domains in, for example, *p*′, space for which the various phenomena may occur. Knowledge of their existence could be exploited to identify and reduce or eliminate hazards in particular cases. Further results of our own numerical experiments will be reported elsewhere. In particular, these include a more systematic exploration of the consequences of variations in the material and hotspot radii, *r*_{1} and *r*_{0}, respectively, the mass fraction of the product that is in solid form, *ϵ*, initial porosity and pore/particle size (which influence the diffusion and permeability constants, heat exchange and possibly reaction rate), the gas properties and, in more detail, the density of the product solid, . With variations in *ϵ* and , there is a possibility of reducing the porosity from its initial value (as indicated in figure 7); such ‘tumescent’ behaviour may be valuable in inhibiting oxygen supply and hence acting as a fire retardant.

## Footnotes

- Received November 19, 2006.
- Accepted January 12, 2007.

- © 2007 The Royal Society