E.CAFARO, L.RANABOLDO and A.SALUZZI Dipartimento di Energetica, Politecnico di Torino, Italy Abstract
The transient fire growth in enclosed spaces is modelled using concepts of non linear dynamical systems theory and the theory of stochastic processes. The mathematical model, derived by a simplified thermodynamic approach to the problem, keeps into account two non linear effects on the burning rate of the fire: the radiation feedback from the hot layer to the fuel and the switch-over between fuel and ventilation control. The zone model used to derive the evolution of the temperature excess of the hot smoke layer is recast in the form of a gradient type dynamical system. A swallowtail catastrophe function is introduced to approximate the fire potential function and to define the boundaries of stable system behaviour. To perform a probabilistic analysis the temperature excess of the smoke layer is considered to be a stochastic variable the time evolution of which is modelled by a Langevin equation.
The numerical solution of the corresponding Fokker-Planck equation allows to determine the probability density function of the stochastic process.
Introduction
Fires in enclosed spaces represent phenomena exhibiting a complex dynamical behaviour at variation of the ratio of air mass flow rate to volatilized fuel mass flow rate. Flashover and extinction jumps as well as hysteresis between the fuel volatilization rate and openings on the compartment have been experimentally observed and recast in mathematical models [1]. Deterministic and stochastic approaches have been used to model the fire growth process: the former tend to fall within the two categories of zone models and field models while the latter views fire spread as a percolation process [1,2].
In a previous paper, by developing a thermodynamic approach introduced in [3], we derived a two layer type dimensionless model, keeping into account the radiation feedback effect and the transition effect from fuel control to ventilation control of the burning rate. Emphasis is placed on jumping phenomena and hysteresis [4]. The evolution of the temperature excess of the smoke layer has been modelled by a first order ordinary differential equation depending on a set of control parameters characterizing the heat and mass transfer processes in the compartment. The numerical integration of the
model, performed by a fourth order Runge-Kutta method for time dependent cases and by a Gauss method for steady state conditions, shows the existence of two stable branches of solution and an unstable one depending on the values of control parameters.
In the present paper the derived model is recast in the form of a dynamical system by introducing a fire potential function recognized as a swallowtail catastrophe function [5,6]. To perform a stochastic analysis of the problem we perturb the system by a white noise term and transform the Langevin equation into a Ito-type stochastic differential equation. Finally we determine the probability density function for the temperature excess stochastic variable by numerical solution of the corresponding monodimensional time dependent Fokker-Planck equation [7,8].
Fire Engineering and Emergency Planning. Edited by R.Barham.
Published in 1996 by E & FN Spon. ISBN 0 419 20180 7.
Deterministic model
The dimensionless mathematical model is defined by the following first order ordinary differential equation [4]:
(1) where
(2)
(3) The quantity θ represent the temperature excess of smoke layer, the dimensionless time, V the ratio of the smoke volume to the compartment volume, µ the dimensionless fuel-ventilation control function, Nv the control parameter defined as the product of the ventilation parameter and the inverse heat release parameter, the ratio of the combustion energy to a reference energy, mf* the dimensionless fuel mass flow rate, α the dimensionless area of the volatilized fuel, γ the radiation heat exchanged by flame and fuel, ε the global emissivity of the smoke layer.
The adopted in the model structure of dimensionless fuel-ventilation control function is recovered from correlation of some experimental results about the combustion efficiency of polymeric materials [9].
The model can be recast in the form of a dynamical system:
(4) where
(5) Fire engineering and emergency planning 92
The shapes of the fire potential function P for different values of control parameters are showed in figs. 1, 2.
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fig.1
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fig.2
Comparing the figs.2 with the following fig.3 we notice that varying the Nv parameter the potential function P shows either three or one noteful points corresponding to the stationary solution branches of the temperature excess evolution equation.
The first point of minimum represent the extinction jump, the point of maximum the unstable solution branch, the second point of minimum the flashover jump. When the solution exhibits one noteful point only it represent the flashover jump.
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fig.3
The fire potential function can be approximated by means of the following swallowtail catastrophe function [5,6]:
(6) The coefficients of the polynomial have been determined by spline interpolation of the potential function.
Stochastic analysis
Let us consider the smoke layer temperature a stochastic variable the time evolution of which Is governed by the Langevin equation [7]:
(7) where
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In the eq. (7) the quantity is a white noise term with zero mean value and variable dependent variance function.
The Langevin equation can be rewritten as a Ito-Stratonovich stochastic differential equation [7]:
(8) where is a standard Wiener process and D is the infinitesimal variance function which characterize the intensity of the noise.
The probability density function of a stochastic differential equation obeys a deterministic partial differential equation called Fokker-Planck equation [7,8]:
(9) As the probability density function converges to a stationary density which is either a generalized function or a proper probability density function depending on the form of the potential function and the infinitesimal variance function.
The stationary solution of the Fokker-Planck equation when the infinitesimal variance function is constant reads:
(10)
where C is a normalization constant.
The eq. (10) implies:
(11) The eq. (11) defines an affine transformation of the potential function. Therefore, the whole apparatus of catastrophe theory, in particular, the classification of the degenerate singularities of the potential function, now applies without change to the logarithm of the stationary probability density function.
In the interior of its domain the stationary probability density function has differentiable relative maxima (modes) and minima (antimodes) which coincide with the relative minima and maxima of the potential function.
The modes and antimodes of the stationary probability density function are non trivially related to the relative minima and maxima of the potential function if the variance function is dependent from the stochastic variable. In this case, namely, the stationary probability density function is given by:
(12)
The eq. (12) can be rewritten as:
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(13)
Let’s define the shape function of F* to be
(14) From the eqs.(13, 14) it can be seen that the modes and the antimodes of the stationary probability density occur at the zeroes of the shape function which do not necessarily coincide with the zeroes of the fire potential function.
The values of the control parameters selected in the numerical integration of the Fokker-Planck equation correspond to those used for determining the shape of the potential function showed in fig.1.a . Some results of the stochastic analysis are showed in figs. 4, 5, 6, 7.
fig. 4
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fig. 5
fig. 6
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fig. 7
Conclusions
The analysis performed show how the catastrophe theory and the stochastic processes theory can be used to understand the main features of the complex dynamical behaviour of the fires in buildings. The reduction of fires growth process in compartments to a dynamical system of polynomial type allows to determine the exact scaling of the control parameters in physical simulation of the interesting process. The most relevant control parameters result to be the ratio of the air mass flow rate to the global heat exchange coefficient and the fuel mass flow rate.
The probabilistic analysis allows to affirm that stochastic systems with multiple stable equilibria may nevertheless exhibit unimodal stationary probability densities.
References
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