## Abstract

A generalized Semenov model is proposed to describe the dynamics of compartment fires. It is shown that the transitions to flashover or to extinction can be described in the context of the catastrophe theory (or the theory of dynamical systems) by introducing a suitable potential function of the smoke layer temperature. The effect on the fire dynamics of random perturbations is then studied by introducing a random noise term accounting for internal and external perturbations with an arbitrary degree of correlation. While purely Gaussian perturbations (white noise) do not change the behaviour of the fire with respect to the deterministic model, perturbations depending on the model variable (‘coloured’ noise) may drive the system to different states. This suggests that the compartment fires can be controlled or driven to extinction by introducing appropriate external perturbations.

## 1. Introduction

The study of compartment fires (i.e. fires occurring in spaces delimited by solid walls) received much attention in the past decades, driven by the continuously growing importance of fire safety issues in engineering design, and several deterministic or stochastic models were developed in order to understand the fundamental mechanisms of the fire growth and to assess fire hazard (Birk 1991; Drysdale 1998). Deterministic models can be sorted into zone models, which allow one to find the main parameters of the fire by solving a set of first-order ordinary differential equations derived from global balance equations supplemented by semi-empirical physical models (Mitler 1985), and field models, which simulate the fire evolution within extended geometric domains by solving a set of discretized partial differential equations using general-purpose CFD packages, where specific submodels have been introduced to describe the effects of buoyancy, radiation heat transfer and turbulence (Markatos & Cox 1984). Stochastic approaches view the fire growth as a percolation process, where the transition from non-propagating to propagating fire is described as a phase change phenomenon (Beer 1990).

One of the main objectives of such models is to predict whether a fire will extinguish spontaneously or grow until it reaches a point when the flame spreads almost instantaneously to occupy the whole enclosure, which is known as the flashover (Thomas *et al*. 1980). However, even a qualitative description of the phenomenon is not easy to obtain, both because of the complexity of the physical model, which must take into account the radiative heat exchange between the flame, the fuel and the surroundings, and because most of the physical and environmental parameters (such as the burning surface and the ventilation conditions) are variable in time in a way that is generally not predictable *a priori*. Flashover and extinction jumps (Thomas *et al*. 1980; Bishop *et al*. 1993*a*,*b*; Graham *et al*. 1995), as well as the hysteresis between the fuel-controlled and ventilation-controlled regimes (Holborn *et al*. 1992) have been experimentally observed and recast in mathematical models. More recent studies have aimed at understanding other parametric effects, such as the effect of the thermal inertia of the walls (Graham *et al*. 1999), of the discharge coefficient (Beard 2001) and of the aspect ratio of the compartment (Beard 2003). In particular, mathematical models with one (Beard *et al*. 1992; Graham *et al*. 1995, 1999), two (Bishop *et al*. 1993*a*,*b*) or three state variables (Beard *et al*. 1994–95; Beard 2001, 2003) are available in the literature.

This work aimed to get a deeper understanding of the behaviour of the compartment fires and their stability by studying a thermodynamic model of the fire, conceptually analogous to the Semenov's theory of thermal ignition, in the light of two well-known mathematical tools: the catastrophe theory (Poston & Stewart 1978; Thompson & Stewart 1986) and the theory of random differential equations (Soong 1973).

The catastrophe theory is a set of topological problems useful to classify the degenerate singularities (the central manifold) of the dynamical systems: given the system , where ** x** represents the state of the system and

*p*is a parameter, moving on one of its attractors or in its neighbourhoods, a catastrophe is an abrupt change of state of the system which occurs for a small change of the parameter, for any initial condition. Thus, the catastrophe theory provides a suitable mathematical background to describe the transitions occurring in many combustion problems, and in particular in compartment fires: for example, a simple combustion problem that can be studied in the context of the catastrophe theory is the Semenov theory of thermal ignition (Bobrov & Korobeinikov 1998). The catastrophe theory has been applied to modelling flashover in (Beard

*et al*. 1992), where a swallowtail catastrophe was identified.

The theory of random differential equations (i.e. the differential equations with random initial conditions, random coefficients or random inhomogeneous terms) allows one to investigate the effects of internal and external perturbations (which are random in nature) on the fire dynamics (Cafaro *et al*. 1995). In particular, perturbations can be included in the thermodynamic model as a random inhomogeneous term that yields a Langevin equation for the fire dynamics. The solution is obtained from the associated Fokker–Planck equation, which returns the probability density function of a certain event (the extinction or flashover) with respect to time.

## 2. Generalized Semenov problem for compartment fires

The simplest approach to describe fires in enclosures is the zone model with pool fire combustion, which is schematically represented in figure 1. Within this framework, one can build a thermodynamic model of the fire dynamics where, as in Semenov's theory of thermal explosion, the net rate of increase of the energy of the upper layer (i.e. the combustion products) is given by the difference between the combustion heat transferred to the smoke layer and that rejected to the surroundings (Cafaro *et al*. 1995; Cafaro & Ranaboldo 1997).

The total energy produced by combustion is , where *η*_{c} is the combustion efficiency; is the mass flow rate of fuel; and *H*_{c} is the lower heat of combustion, while the fraction of such energy that is actually transferred to the smoke is given by (Tewarson *et al*. 1993)(2.1)where is the mass flow rate of fresh air and the subscript s denotes the stoichometric conditions. The argument of the exponential in the equation (2.1) is the inverse of the equivalence ratio, and is used to model the effect of ventilation conditions on the combustion efficiency for any kind of fuel. Finally, the energy generation rate can be expressed in the form(2.2)and obviously depends on the smoke temperature, *T*_{g}, and on time.

The mass flow rate of fuel that sustains combustion can be obtained from an energy balance between the vaporization heat of the solid or the liquid fuel, Δ*H*_{vap}, and the radiation heat rate on its surface, *A*_{f}, assuming that the heat source for the fuel vaporization is radiation from the smoke layer and the flame itself (Dorofeev *et al*. 1993)(2.3)where is the radiative heat flux(2.4)The symbols *σ*, *ϕ* and *ϵ* denote the Stefan–Boltzmann constant, the shape factor and the emissivity of the flame (f), the exhaust gas (g) and the combustion surface (b), respectively.

As for the thermal energy transferred to the surrounding environment, this is partly carried by the exhaust gas stream and partly transferred by convection, so that it can be written as(2.5)where *h*_{tot} is the global heat transfer coefficient and *A* is the effective heat transfer surface.

Assuming that radiation between the flame and the compartment walls can be neglected, and that the burning surface does not change in time, the energy equation can be written as(2.6)A further simplification can be obtained if the pressure in the control volume is constant, which allows one to write *ρ*_{g}*T*_{g}=*ρ*_{a}*T*_{a}, and the rate of fuel consumption is small with respect to the mass flow rate of fresh air (i.e. ), so that the equation (2.6) becomes(2.7)Finally, assuming that the reference ambient temperature does not change because of the fire, and that the dimension of the flame is small with respect to the compartment size, *V*_{c}, the model can be reformulated in dimensionless form by introducing the following variables(2.8)(2.9)(2.10)(2.11)(2.12)(2.13)(2.14)(2.15)The quantity *N*_{v} represents the dimensionless mass flow rate of fresh air feeding the fire, and is called ventilation number; *α* is the dimensionless burning surface; *γ* is the dimensionless radiation heat exchange between the flame and fuel; *ϵ* is the global emissivity of the smoke layer; and 1/*λ* represents the dimensionless combustion energy.

Introducing equations (2.8)–(2.15) into equation (2.7) yields the following dimensionless energy balance, where the subscript in *θ*_{g} has been dropped(2.16)Although, this model is extremely simplified, it takes into account at least two important parameters: the radiative thermal exchange between the smoke layer and the flame, which is modelled through *α*, *γ* and *ϵ* and the ventilation conditions that determine the local equivalence ratio of the combustion and are modelled through the ventilation number *N*_{v}.

## 3. Stability analysis

The dimensionless energy equation describing the temperature evolution of the fire (equation (2.16)) can be reformulated as an autonomous dynamical system of gradient type by introducing a suitable potential function (Beard *et al*. 1992)(3.1)where(3.2)According to the catastrophe theory (see, e.g. Poston & Stewart 1978; Thompson & Stewart 1986), the steady-state solutions (equilibrium points) of the dynamical system correspond to the points where the fire potential is stationary (maxima and minima): in particular, maxima represent stable equilibrium points and minima unstable ones.

In practice, the critical points can be found by setting d*θ*/d*τ*=0, i.e.(3.3)In addition to the trivial solution *θ*=−1, which has no physical significance, one can easily show that the equation (3.3) has either one or three positive solutions (*i*=1, 2 and 3). If ∂^{2}*P*/∂*θ*^{2}<0 for (i.e. the function d*θ*/d*τ* has a negative slope with respect to the variable *θ* for ), the equilibrium is stable: increasing the dimensionless temperature *θ* causes d*θ*/d*τ*<0, and vice-versa a temperature reduction is balanced by d*θ*/d*τ*>0. The same arguments allow one to conclude that the steady-state solutions where ∂^{2}*P*/∂*θ*^{2}>0 are unstable.

Steady-state, stable solutions with *θ*<1 correspond to the fires that are not strong enough to increase the temperature significantly above the ambient value (extinction jump), while the solutions with *θ*≫1 correspond to the fires which after an initial transient exhibit the flashover jump.

Figures 2 and 3 show the effects on stability of the ventilation parameter and of the dimensionless burning surface, respectively. For low values of the ventilation parameter, the system shows one stable solution at high temperature, which represents the condition of the fire after flashover. As ventilation increases beyond a critical value, two additional solutions appear: one at low temperature that corresponds to a spontaneous extinction; and the other that is unstable, so that the system will evolve either towards the flashover or towards the extinction depending on the other parameters of the model and on the initial conditions. With a further increase of ventilation, the flashover and the unstable solutions disappear, and the only possible asymptotic condition is extinction. The same qualitative behaviour of the system dynamics can be observed when the dimensionless burning surface is reduced from high values to lower values, an example of which is reported in figure 3.

Figure 4 shows the regions of existence of the steady-state solutions of the dynamical system in the space of the parameters (*α*, *N*_{v}), which can be interpreted as a map or phase diagram, describing the behaviour of the compartment fire with respect to the same parameters. In particular, for high ventilation numbers and small burning surfaces the fire evolves towards extinction, while for large burning surfaces and lower ventilation numbers, the flashover should be expected; in between these regions there is a third one where both extinction and the flashover are theoretically possible. The boundaries between regions correspond to values of the parameters for which the equation (3.3) has a double solution, i.e. when the equation (2.16) is tangent to the line d*θ*/d*τ=*0.

As a concluding remark for this section, one can observe that sometimes the integrand function in equation (3.2) can be approximated by piecewise spline interpolation or by least-squares polynomial best-fit, in order to obtain a simpler formulation of the problem. However, in order to capture the behaviour outlined above (with one or three steady-state solutions for the dynamical system), the polynomial approximating the fire potential function must be at least of fourth order.

## 4. Effect of stochastic perturbations

In the mathematical model of the fire based on energy conservation derived above (equation (2.16)), the only variable is the dimensionless temperature excess, *θ*, and all the physical and environmental parameters are assumed to be constant. This assumption is not realistic, because all the parameters may change in time in a way that is not predictable *a priori*. The simplest way to describe such behaviour is introducing random perturbations into the model (Cafaro *et al*. 1995). Numerical simulations confirm that when deterministic models are perturbed by a random noise term, the output variables exhibit random fluctuations qualitatively similar to those observed in real systems (Hasofer & Beck 1997). Random perturbations can be sorted into internal perturbations, such as changes of the heat transfer parameters and of the burning surface, and external perturbations, such as fluctuations of the ventilation conditions. In general, internal and external perturbations are not independent, but are correlated to a certain extent: for example, a small increase of the ventilation will make oxygen available to burn more fuel, resulting in an increase of the burning surface.

Thus, to account for the parameter fluctuations, one can modify the generalized Semenov model of equation (2.16) by adding two random noise terms, describing the internal and external perturbations, respectively(4.1)where(4.2)The time-dependent variables *w*_{i}(*τ*) and *w*_{e}(*τ*) are Gaussian processes with zero mean and variances 2*D*_{i} and 2*D*_{e}, respectively and an arbitrary degree of correlation *r* (−1≤*r*≤1), which can be written in synthetic form as(4.3)The multiplicative terms *g*_{i}(*θ*) and *g*_{e}(*θ*) account for the dependence of perturbations on the process variable (in this case, the dimensionless temperature excess of the smoke). The two random inhomogeneous terms can be combined together (e.g. Papoulis 1991), so that the equation (4.1) can be reduced to a Langevin equation in an integral form(4.4)where(4.5)and the multiplicative term is given by(4.6)The stochastic differential equation obtained above is associated to a deterministic partial differential equation (the Fokker–Planck equation), which returns the probability density function *F*(*θ*, *τ*) for the dimensionless excess temperature (Soong 1973)(4.7)with the initial condition(4.8)As *τ*→+∞ the probability density function *F*(*θ*, *τ*) tends to a stationary function , which depends both on the fire potential function and on the multiplicative factor of the random noise term, *G*(*θ*). Assuming the obvious boundary conditions and , the stationary solution is(4.9)where *C* is determined by the normalization condition .

The simplest case occurs when the internal and external perturbations are described by two uncorrelated white noise terms, which implies *g*_{i}(*θ*)=*g*_{e}(*θ*)=1 and *r*=0, so that the equation (4.4) reduces to(4.10)and the asymptotic probability density function becomes(4.11)or(4.12)The equation (4.12) defines an affine transformation for the fire potential function introduced above by the equation (3.2). Thus, the whole apparatus of the catastrophe theory and in particular, the classification of degenerate singularities of the fire potential function, applies without changes to the logarithm of the asymptotic probability density function. Hence, has differentiable relative maxima (modes) and minima (anti-modes) that coincide with those of the fire potential function, and indicate the likelihood that one of the stable solutions (the flashover or extinction) appears after a certain time.

Figure 5 shows an example of numerical solution of the equation (4.7), where the parameters of the model are set to the same values of the example reported in figure 2, with *N*_{v}=8. For *τ*→+∞, the probability density converges to an asymptotic curve, indicating that the event with maximum likelihood corresponds to the flashover solution. It is interesting to observe that although, for this choice of the parameters, the deterministic system (equation (2.16)) has multiple stable equilibria, the stochastic system shows a unimodal stationary probability density. In other words, random perturbations drive the system towards one of the two solutions, excluding the other.

As mentioned above, a more realistic situation occurs when the multiplicative factor of the random noise term, equation (4.6), depends on the model variable, *θ*, and eventually on the degree of correlation between the internal and external perturbations, *r*. In this case, the stationary probability density function, equation (4.9), is non-trivially related to the fire potential function, and therefore the modes and anti-modes of do not necessarily correspond to the maxima and minima of *P*(*θ*). In other words, the introduction of a coloured random perturbation can change the behaviour of the fire with respect to the prediction of the deterministic model.

Figure 6 shows the stationary probability density function for a fire with the same parameters of the example reported in figure 2, obtained from the numerical solution of the equation (4.7), in the simple case of linear dependence of the internal perturbation on the temperature of the smoke layer, i.e. . For *N*_{v}=8, the outcome is completely different from the case of purely Gaussian and uncorrelated perturbations: in fact, the system evolves towards extinction instead of the flashover. In the case of *N*_{v}=4, one finds the flashover as in the deterministic model, however, with a smaller most probable temperature of the smoke layer. Finally, for *N*_{v}=12, there is no difference with respect to the deterministic model.

## 5. Conclusions

A generalized Semenov model for compartment fires based on the balance between the energy generated by combustion and that rejected into the surrounding environment through the compartment walls and by the smoke flow, is able to capture some important features of the fire dynamics, including the flashover transition. In particular, the phenomena of the flashover and extinction can be understood in the framework of the catastrophe theory (or alternatively the theory of dynamical systems), by introducing a suitable ‘fire potential’, which is a function of the temperature of the smoke layer. In this context, the flashover and extinction represent the steady-state, stable solutions of the dynamical system, and correspond to the stable critical points (maxima) of the fire potential function.

To account for the effect of stochastic perturbations on the compartment fire dynamics, the deterministic model can be amended by a random noise term, which includes the effects of both internal and external perturbations with an arbitrary degree of correlation, and leads to a stochastic equation of Langevin type. While in the case of purely Gaussian, uncorrelated perturbations (white noise), the stochastic model yields the same solutions as the deterministic model, if perturbations also depend on the model variable (the temperature of the smoke layer) the results can be significantly different. In particular, transitions to the flashover may occur at lower temperatures, or change into transitions to extinction if the deterministic model allows multiple stable solutions.

This result suggests that the dynamics of compartment fires can be controlled and eventually driven towards extinction through the creation of appropriate external perturbations (for example, a forced ventilation), modulated by the temperature of the smoke layer.

## Footnotes

- Received September 22, 2008.
- Accepted November 7, 2008.

- © 2008 The Royal Society