International Journal of Thermal Sciences 48 (2009) 1501–1509 Contents lists available at ScienceDirect International Journal of Thermal Sciences www.elsevier.com/locate/ijts Analytical determination and numerical computation of extinction coefficients for vegetation with given leaf distribution Z. Acem a , A. Lamorlette b , A. Collin b , P. Boulet a,∗ a LEMTA, Nancy-Université, CNRS, Faculté des Sciences et Techniques, BP 239, 54506 Vandœuvre lès Nancy Cedex, France b LEMTA, Nancy-Université, CNRS, 2 Avenue de la Forêt de Haye, BP 160, 54504 Vandœuvre lès Nancy Cedex, France article info abstract Article history: Received 23 May 2008 Received in revised form 30 September 2008 Accepted 14 January 2009 Available online 6 February 2009 Keywords: Extinction Absorption Radiative properties Vegetation Ray tracing method A study has been carried out, with both analytical and numerical methods, on the effective extinction coefficient of a medium featuring a given vegetation. The studied domain contains a heterogeneous medium involving in successive cases: leaves under planophile, erectophile, plagiophile, extremophile orientations, and a mixing of these conditions, all situations assuming a uniform distribution in space. A ray tracing method has been used in order to identify an effective extinction coefficient after computation of the mean free path. For these academical situations, comparisons with some analytical formulations have been carried out in order to achieve the validation of the numerical method. Then, the role of a heterogeneous distribution in space has been studied, introducing morphological properties (orientation and spatial heterogeneities) which produce a set of leaves featuring a sort of virtual vegetation, numerically built in a box. In particular, conditions of leaves mainly located at the periphery of a crown have been considered. Numerical results show that analytical relations for the extinction coefficient may be used for the different leaf orientations, when the spatial distribution is assumed homogeneous. However, some discrepancies from these solutions appear when heterogeneous distributions in space are used, suggesting that such relationships will become less accurate when applied on a true vegetation. The numerical tool is now ready and will be used on a vegetation built with true morphological characteristics.  2009 Elsevier Masson SAS. All rights reserved. Introduction When modeling forest fire propagation, one peculiar difficulty is related to the radiative transfer role. The fact that radiation is a major propagation actor is well known, but the way it has to be taken into account is an open research question. It is obvious that a complete fire propagation model has to involve not only radia- tive transfer, but also numerous combined sub-problems for heat and mass transport, combustion phenomena, turbulence, etc., with supplementary difficulties brought by the heterogeneous structure of the medium under study and the multi scale situation of the problem (see for example [1] for a recent overview and contribu- tion on that topic). Some progress related to each sub-model is then required before gathering all the dedicated tools in a com- bined simulation. In this frame we are studying radiative transfer in order to improve our understanding of the involved mecha- nisms and also in order to provide input data directly useful for fire models. Simulation of radiative transfer is often considered as a hard task alone, in particular due to the complexity of this heat transfer mode and to a lack of knowledge relative to the required * Corresponding author. Tel.: +33 383 684 686. E-mail address: pascal.boulet@lemta.uhp-nancy.fr (P. Boulet). input properties (radiative properties of the vegetation and spec- tral distribution of the radiation emitted by flames for the present domain). The non-gray nature of the radiative transfer is obvious, although often omitted for the sake of computational cost reduc- tion. In a recent contribution present authors have presented a preliminary study of the radiative properties of the vegetation, in which experimentations have been carried out on six species of the Mediterranean region [2]. Spectral transmissivity and reflectiv- ity have been measured and the absorptivity of a single leaf has been deduced. Then a ray tracing method has been used in order to identify an extinction coefficient of a medium built numerically with given leaves uniformly distributed in space and with random orientation. This coefficient can be considered as an effective ex- tinction coefficient corresponding to an equivalent homogeneous medium. The sharp spectral variations of the radiative properties of the studied species have been clearly observed. Discrepancy from the currently used assumption of a vegetation behaving like a black radiative surface has been studied and comparisons with the well known De Mestre’s relationship [3] have been performed. It has been shown that even when the vegetation properties shift from the blackbody ideal case, an easy correction of the relationship can be done multiplying the extinction coefficient by a factor equal to 1 − τ , where τ is the transmissivity of the vegetation species. 1290-0729/$ – see front matter  2009 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2009.01.009