Keywords

1 Introduction

Forests cover about 39% of the area of mainland Portugal, being mainly composed of pine trees, eucalyptus, cork, holm oaks, oaks and chestnuts [1]. In particular, pine forests occupy an area of about one million hectares [1]. Official reports prepared in previous years usually refer to more than 20,000 occurrences of forest fires per year [2], which makes Portugal one of the countries in all of Europe with the highest incidence rate of forest fires [3, 4]. The high number of forest fires with extensive burnt areas has been causing more and more negative impacts on “carbon storage, biodiversity conservation, hydrologic processes, and economic and social services” [5]. Thereafter, studies on the behavior of forest fires and their impacts on impairment and tree mortality are increasingly relevant. The mechanism of direct tree death from fire is the cambium necrosis via heat transfer to the crown, stem and root tissue [6]. The heat transfer occurs by convection, conduction, and radiation and all of these processes can cause either damage or death to the tree. The lethal threshold for trees is obtain for temperatures above or equal to 60 °C, although longer exposure at lower temperatures can also cause tissue death [7].

The numerical model of the pine tree trunk developed in this study is based on the numerical model that emulates the human body geometry. Like the numerical model of the human body, the numerical model of the pine tree trunk also uses energy balance equations. The numerical model of the human body was applied in the development of the model that simulates the human body thermal response. For example, the human body thermal response model was used in the works presented in articles by Conceição [8, 9], Conceição and Lúcio [10, 11], and Conceição et al. [12,13,14].

The numerical model uses the generalized mesh to establish the differential energy equations that describe the thermal behavior existing inside the pine tree trunk. In boundary conditions, the numerical model uses energy balance equations. The thermal flow processes defined by these equations are as follows: by conduction with the interior of the pine tree trunk; by convection (forced, natural or mixed) between the surface (bark) of the pine tree trunk and the surrounding environment; by radiation between the surface (bark) of the pine tree trunk and the surrounding environment; and by radiation between the surface (bark) of the pine tree trunk and the fire front. In the calculation of radiative exchanges between the pine tree trunk and the surrounding environment (including the fire front), a concept similar to heat exchanges between surfaces within building compartments is used. The modeling of this heat exchange process between surfaces was implemented in the numerical model of the thermal response of buildings with complex topology or in passenger car cabinets. Its application was effective in the studies developed by Conceição and Lúcio [15,16,17,18], and Conceição et al. [19,20,21,22,23].

In buildings, the temperature of the opaque surfaces is calculated layer by layer. In occupants, the temperature is calculated in different concentric layers. In the case of the pine tree trunk, an adapted 2D mesh is used. The temperature distribution inside the pine tree trunk is calculated using a finite differences implicit model.

The main objective of this numerical work, still in an initial phase, is to allow the definition of a methodology that allows assessing the sustainability of a pine forest in the presence of forest fires with different levels of intensity. In this context, this work begins by studying the temperature distribution in a pine tree trunk in the presence of a progressive fire front. Four scenarios were defined for two flame temperatures, 500 °C and 750 °C, and two wind speeds, 0.1 m/s and 1.0 m/s.

2 Numerical Model

The following thermal phenomena are considered for the pine tree trunk: heat conduction inside the pine tree, heat convection between the pine tree surface and the air environment and heat exchange by radiation between the pine tree trunk surface and the surrounding body surfaces, namely, the fire front, the fuel bed and the sky.

The hypothesis used to establish the energy balance equations are as follows:

  • The heat flux is two-dimensional;

  • The air temperature around the trunk, that is uniform and equal to the environment temperature;

  • Use of heat transfer coefficients by convection developed for isothermal surfaces;

  • The pine tree trunk is composed by bark and cambium;

  • The fire effects around the trunk are not considered.

The type of grid used in the numerical simulation influences the results. In this study a numerical grid generation where the mesh is adapted to the body surface contours was developed using the finite difference method. The mesh used in the pine tree trunk has 30 × 20 points. In this adaptive grid generation, a physical space and a computational space were considered. The idea of this method consists of transforming the physical domain into the computational plan. This grid transformation is done by two elliptic partial differential equations, of Poisson’s type. The adaptive grid generation used in this work can be seen in Fig. 1. The data input of this model are the wind speed, the fire front conditions (dimensions, tilt angle, flame temperature, fire rate spread), the tree dimensions, initial distance of the fire front from the tree and other initial conditions. This model was calibrated and validated in the work of Conceição et al. [24]. In this model, no moisture content was considered in the pine tree trunk. In this model, the following simplifications were also considered: constant dimensions, flame temperature and spread velocity of the fire front; homogeneous trunk constitution.

Fig. 1.
figure 1

Adaptive grid generation used in the pine tree trunk (30 × 20 grid points).

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3 Methodology

The scheme of the forest fire scenery used in the numerical simulation is presented in Fig. 2. This scheme is constituted by an inclined fire front, a pine tree trunk and a fuel bed. Figure 2 also shows the symbology used in the representation of the wind speed (vair), the fire spread rate (R), the dimensions of the fire front and the pine tree trunk. The fuel bed is considered to have finite dimensions a × b.

The simulation analyzes the situation in which the fire front moves at a constant fire spread rate from a distance of 5 m upstream of the pine tree trunk to a distance of 25 m downstream of the tree trunk. The input data of the simulation are present in Table 1. The output data of the simulation are the temperature distribution obtained in a plane that cuts the pine tree at a height of 2 m at 30 points (P) equidistant distributed along the bark of the pine tree trunk, Fig. 3a, and at 20 points (Q) distributed along the radius of the pine tree trunk, Fig. 3b. In Fig. 3b, the line of points chosen is in a plane perpendicular to the direction of propagation of the fire front. The temperature distribution is obtained for the following Cases: a flame temperature of 500 °C and a wind speed of 0.1 m/s (Case A) or 1.0 m/s (Case B); a flame temperature of 750 °C and a wind speed of 0.1 m/s (Case C) or 1.0 m/s (Case D).

Table 1. Input data of the numerical simulation.
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Fig. 2.
figure 2

Scheme of the forest fire scenery used in the numerical simulation: on the left is the fire front and on the right is the tree trunk. The dimensions of the trunk are represented by the diameter d and height zt and those of the fire front by height zf, width b and slope ϕ.

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Fig. 3.
figure 3

Location of the points distributed along the pine tree trunk, where the temperature is numerically calculated, and the fire front: a) P-points equidistant distributed along the bark; b) Q-points distributed along the radius.

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4 Results and Discussion

The evolution of the temperature values obtained at selected points of the pine tree trunk for the wind speeds of 0.1 m/s and 1.0 m/s points is presented for a flame temperature of 500 °C and 750 °C in, respectively, Fig. 4 and Fig. 5. In these Figures, the P1-P15 points are located on the bark of the pine tree trunk facing upstream of the fire front, the points P16-P30 are located on the bark of the pine tree trunk facing downstream of the fire front, and the points Q1-Q20 are located on the line of the tree trunk radius. In Fig. 4 and Fig. 5, the dashed line represents the pine tree trunk lethal threshold (Ttrunk ≥ 60 °C) [7].

Fig. 4.
figure 4

Evolution of the temperature values obtained at selected points of the pine tree trunk for a flame temperature of 500 °C when the wind speed is: a) 0.1 m/s; b) 1.0 m/s. The dashed line represents the pine tree trunk lethal threshold (Ttrunk ≥ 60 °C).

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Regarding to the results, the temperature distribution presents higher values when the flame temperature is 750 °C than when it is 500 °C and, for these flame temperatures the temperature distribution presents higher values when the wind speed is 0.1 m/s than when it is 1.0 m/s. For both flame temperatures, the temperature reaches higher values in the pine tree trunk bark upstream of the fire front than downstream of the fire front. Inside the pine tree trunk, the temperature reaches higher values at its outermost points. The highest temperature is reached at a point in the bark of the pine tree trunk facing upstream of the fire front, this value being about 130 °C for the flame temperature of 500 °C and about 245 °C for the flame temperature of 750 °C. Inside the pine tree trunk, the highest temperature is reached at the point located on the outermost ring, this value being about 46 °C for the flame temperature of 500 °C and about 106 °C for the flame temperature of 750 °C.

Fig. 5.
figure 5

Evolution of the temperature values obtained at selected points of the pine tree trunk for a flame temperature of 750 °C when the wind speed is: a) 0.1 m/s; b) 1.0 m/s. The dashed line represents the pine tree trunk lethal threshold (Ttrunk ≥ 60 °C).

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For the flame temperature of 500 °C, the temperatures in the tree remain below the lethal threshold for a wind speed of 1 m/s, which means that, in this case, the pine tree trunk withstands the passage of the fire front. For a speed of 0.1 m/s, only 11 points of the bark of the pine tree trunk turned upstream of the fire front reach the lethal threshold, which will cause the death of their tissues; in the other points of the bark of the pine tree trunk turned downstream of the fire front and inside the pine tree trunk, the temperature is below the lethal threshold. In this case, it can be stated that the pine tree trunk suffers only minor damage to its bark and that it will also resist fire well.

For the flame temperature of 750 °C, the temperatures in the tree remain below the lethal threshold in the interior of the pine tree trunk for a wind speed of 1 m/s; only for a few points located on the bark of the pine tree trunk reach the lethal threshold. In general, in this case, the pine tree trunk withstands the passage of the fire front. For a speed of 0.1 m/s, all points located on the bark of the pine tree trunk turned reach the lethal threshold, which will cause the death of their tissues; inside the pine tree trunk, only two points located on the two outermost rings reach the lethal threshold. In this case, it can be stated that the pine tree trunk suffers some more extensive damage than for the flame temperature of 500 °C, but only causing the death of more superficial located tissues. However, these injuries are not enough to cause the tree to collapse, so it can be considered that it will resist the action of the fire front.

In short, according to the results, for the assessed flame temperatures and wind speeds considered, the existence of a good cleaning of the land surrounding the tree and an effective pruning up to two meters in height of the tree trunk allows the tree resist a forest fire with only damage to its bark and eventually in some points of its outer ring. Thus, for medium-sized forest fires presented here, the tree can survive, only with the bark burned, therefore it is not necessary to cut it down. In fact, it is possible to achieve a sustainable and fire-resilient forest through effective management that limits the available combustible matter and ensures its regular cleaning.

5 Conclusion

A numerical model of the pine tree trunk thermal behavior, nearby a forest fire environment, was developed and applied. The pine tree thermal behavior numerical model is based on energy equations for the tree elements, which, in turn, are substantiated on heat exchange processes. To assess the resilience of a pine tree to a forest fire, the temperature distribution in its trunk was evaluated for two flame temperatures (500 °C and 750 °C) and two wind speeds (0.1 m/s and 1.0 m/s).

The main conclusions are that the temperature distribution values in the pine are higher for the flame temperature of 750 °C and a wind speed of 0.1 m/s. In general, it appears that only the tissues of the pine tree trunk bark and, eventually, of the first ring located near the surface of the pine tree trunk reach the lethal threshold. The pine tree trunk withstands a flame temperature of 500 °C, for both wind speeds, and a flame temperature of 750 °C, for a wind speed of 1.0 m/s.

Therefore, the results make it evident that a good cleaning of the forest land and that an effective pruning of the trees make them more resilient to low and medium intensity fires.

As future work, it is proposed to study the influence of environmental variables on the thermal behavior of the tree in the presence of a forest fire, to introduce in the thermal behavior tree model the equations that translate the effect of the tree transpiration and, finally, to establish a forest fire prevention index.