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ŠUMARSKI LIST 1-2/2018 str. 80     <-- 80 -->        PDF

used ρb ever since (e.g., Thomas 1971; Catchpole et al. 1998; Anderson et al. 2015; Rossa and Fernandes 2017), sometimes arguing that its negative effect on R could be explained based on Rothermel’s formulation. Although ρb depends on fuel load (w) and bed height (δ), the effect of w on R is not as consensual as the effect of ρb on R, being variously reported as non-existent (e.g., Cheney et al. 1993), negligible (McAlpine 1995), or positive (e.g., McArthur 1962; Dupuy 1995). Fuel bed height is also recurrently used in empirical formulations and its positive effect on R is quite consensual.
Empirical models are undeniably useful. However, the fundamental reasons behind the effects of input variables are seldom present, as a theoretical basis to understand the physical mechanisms underlying fire spread is missing. We believe that, possibly because of misinterpretation of previous physical modelling efforts, both in the case of the well-established influence of ρb on R as well as in the disputed effect of w on R, the role of such fuel bed parameters in fire spread is not well understood.
This work discusses the influence of some fuel bed parameters on R. The discussion is based on a simplified thermal energy balance of fire spread derived from an R formulation given by the ratio between Q’’ and the energy necessary for fuel ignition (Thomas 1971; Frandsen 1971; Rothermel 1972).
Analiza ravnoteže toplinske energije tijekom širenja požara
Formulation of fire-spread rate – Definiranje stope širenja požara
Equation (1), like that used as a base for Rothermel’s (1972) model development, gives steady-state R as the ratio between Q’’ originated by the flame (heat source) and the energy absorbed by the fuel until ignition is achieved (heat sink):
where Qi is the heat per unit mass necessary for igniting the fuel. Basically, this relation describes what happens but not how it happens. Concluding that a ρb increase will decrease R based only on this thermal energy balance implies ignoring that an increase on the amount of fuel acting as an heat sink also means an increase on the amount of flaming fuel releasing heat and therefore on Q’’, as will be shown below.
Equation (2) gives ρb, which quantifies the amount of fuel per unit volume of the bed:
and Qi can be calculated using:
where cf is the fuel-specific heat, Ti the ignition temperature, Tf the fuel initial temperature, M moisture content expressed as a percentage of the oven-dry fuel mass, cw water-specific heat, Tv water boiling temperature, and Qw water latent heat of evaporation. Q’’ determination in Rothermel (1972) includes a term obtained by integration over flame depth (D). Nevertheless, the resulting propagating flux refers to the horizontal net heat power transferred per unit vertical cross section of the fuel bed (Frandsen 1971). Thus, Q’’ can be computed as:
where Qfl is the power released per unit fireline length, i.e., Byram’s intensity (Byram 1959), used in a plethora of studies (e.g., Kucuk et al. 2015), and η the fraction of heat transferred from the flame to the unburned fuel. On the other hand, Qfl can be obtained from:
where Qf, ffl, and tr are, respectively, fuel low heat content, the fraction of fuel that burns in flaming combustion, and flame residence time. Substituting equation (2) in (1) and successively substituting equation (5) in (4) and in (1) we obtain:
We can confirm that this R formulation does not depend on ρb because both the numerator and denominator of equation (1) are a function of the amount of fuel per unit volume of the bed, i.e., w/δ, which allowed removing it from the equation. This means that the effect of ρb, established in a great number of empirical studies, cannot be directly inferred from the thermal energy balance expressed in equation (1), which is equivalent to equation (6).
Fraction of heat transferred from the flame to the unburned fuel – Udio topline koja se prenosi od plamena do nesagorijelog goriva
A further analysis of equation (6) allows concluding that, because the 1st term on the right side of the equation (D/tr) yields R by itself (Anderson 1964), the 2nd term must equal unity. Having this in mind, we can obtain:
i.e., η can be deduced from the ratio between the energy that a unit mass of fuel needs for being ignited and the energy it