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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 80
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 4

A Numerical Algorithm for Solving a Multivalued Equation arising in Fire Spread Modelling

L. Ferragut, I. Asensio and S. Monedero

Departement of Applied Mathematics, University of Salamanca, Spain

Full Bibliographic Reference for this paper
L. Ferragut, I. Asensio, S. Monedero, "A Numerical Algorithm for Solving a Multivalued Equation arising in Fire Spread Modelling", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Fourth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 4, 2004. doi:10.4203/ccp.80.4
Keywords: fire, radiation, moisture, pyrolysis, wind, slope.

Summary
Many existing physical models for fire spread in porous fuel bed use the principle of energy conservation applied to the preheated fuel. Generally, radiation is considered as the dominant mechanism of the fuel preheating. On the other hand slope and wind effects as well as the initial vegetation moisture have to be taken into account in order to obtain reliable rates of fire spread. Physical models from fundamental conservation equations and complex physics have been developed [1]. These valuable approaches are computationally expensive and too slow to be used in real time mode, even with fast and parallel processing. Besides, several works have appeared recently where one or two dimensional physical models are considered in order to simulate fire spread in small computers, with moderate simulation times, see for example [2].

This paper is a contribution to generally applicable models of fire spread through fuel beds, by means of simple models, but taking into account local radiation, moisture content and wind and slope effects. Particularly the influence of the moisture content and eventually heat absorption by pyrolysis, can be represented as two free boundaries, and are treated in this paper using a multivalued operator representing the enthalpy. The maximal monotone property of this operator allows the implementation of a numerical algorithm with well-known convergence properties.

The non dimensional equations governing the fire spread in a region with boundary are:

 (1)

 (2)

 (3)

with the corresponding boundary conditions. The unknowns , and are the enthalpy, temperature and mass fraction of solid fuel, respectively. The enthalpy is an element of a multivalued operator given by:

where and represent the water evaporation temperature and the pyrolysis solid fuel temperature, respectively. The quantities and are the evaporation heat and pyrolysis heat respectively. The right hand side of equation (1) describes the source term due to fuel combustion. This model is a variant of the models in [3], where we have introduce the influence of moisture content, and local radiation effects as in [4].

The numerical algorithm is a semi-implicit scheme by discretizing the total derivative. The basic idea is to treat implicitly the positive terms. At each time step the resulting nonlinear equations are solved by means of the Yosida aproximation of the multivalued operator and the Bermúdez-Moreno algorithm. [5].

We consider the series of experiments that were carried out in a low speed wind tunnel by [6]. They were performed in order to observe wind driven effects, slope and moisture content effects in fire across pine needles beds. As in the experimental results, qualitatively, numerical results are similar for low and high fuel moisture content, but the rate of spread is always lower for the later. In absolute terms the difference increase with increasing wind velocity. The higher the fuel moisture content, the lower the combustion temperatures reached, which lowers the rate of spread.

References
1
R.R. Linn, "Transport model for Prediction of Wildland Behaviour", Los Alamos National Laboratory, Scientific Report, LA1334-T, 1997
2
J.L. Dupuy, M. Larini,"Fire spread through a porous forest fuel bed: a radiation and convective model including fire-induced flow effects", Int. J. of Wildland Fire, 9, 155-172, 1999. doi:10.1071/WF00006
3
G. Cox G, "Combustion Fundamentals of Fire", Academic Press, London, UK, 1995.
4
M.I. Asensio, L. Ferragut, "On a wildland fire model with radiation", Int. J. Numer. Meth. Engng, 54, 137-157, 2002. doi:10.1002/nme.420
5
A. Bermúdez, C. Moreno, "Duality methods for solving variational inequalities", Comp. and Math. Appl.,7, 43-58, 1981. doi:10.1016/0898-1221(81)90006-7
6
J.M.C. Mendes-Lopes, J.M.P. Ventura, J.M.P. Amaral, "Rate of spread and flame characteristics in a bed of pine needles", III International Confer. on Forest Fire Research. 14th Conference on Fire and Forest Meteorology, Vol. I, 497-511, Luso, 1988.

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