# Mathematical Modeling of Forest Fire Containment Using a Wet Line Ahead of the Combustion Front

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{fcm}and total span L was considered (Figure 1). Some part of it was in a wetted state (the span of this part was L

_{gr}, zone 1). The forest fuel surface is subjected to the radiant heat flux from a high-temperature source (a flame with a length H

_{rad}and temperature T

_{rad}). As the forest fuel part wetted with water heats up, the moisture at the wet/dry interface evaporates. Then, the water vapors pass through the drained forest fuel layer (3) and are injected into the near-surface gas region (zone 4).

_{rad}).

_{fcm}< y < H):

_{0}is the initial temperature of the region; D is the water vapor diffusion coefficient in the air.

_{a}= 1005 J/(kg·K), c

_{H}

_{2O}= 4200 J/(kg·K)—heat capacity of air and water vapors; λa = 0.0259 W/(m·K), λ

_{H}

_{2O}= 0.0237 W/(m·K)—thermal conductivity of air and water vapors; ρ

_{a}= 1.205 kg/m

^{3}, ρ

_{p}= 0.598 kg/m

^{3}—density of air and water vapors.

_{gr}, 0 < y < H

_{fcm}, Figure 1), the heat transfer is given by the thermal conductivity equation in the generalized statement by Stefan [35,36]:

_{w}—phase transition heat; ε—smoothing interval, K; c

_{w}, λ

_{w}, ρ

_{w}—specific isobaric heat capacity, thermal conductivity and density of wetted forest fuel; c

_{pd}, λ

_{d}, ρ

_{d}—specific isobaric heat capacity, thermal conductivity and density of dry forest fuel.

_{gr}< x < L, 0 < y < H

_{fcm}) is given by a two-dimensional unsteady equation of thermal conductivity [34,37,38]:

_{gr}] (average draining depth) at the point t was calculated from the following ratio:

_{water}and ρ

_{p}are the densities of water and vapor.

_{f}is the rate of the water vapor filtration through the porous forest fuel frame; ρ

_{p}is the water vapor density; c

_{p}is the heat capacity of water vapors.

_{fcm}, x ϵ [0; L

_{gr}] (Figure 1), the concentration of water vapors and the vertical component of their velocity were set:

_{ri}comes to the gas/forest fuel interface (y = H

_{fcm}, 0 ≤ x ≤ L) from the source with a length H

_{rad}(flame) with a constant temperature T

_{rad}. The value of q

_{ri}on the forest fuel surface was determined using the zonal method [37,39]. In the case of an isothermal surface with a length H

_{rad}, the resultant heat flux equals [37]:

_{0i}and E

_{0k}are the total radiant fluxes per unit area of a black body of the i-th and k-th surfaces, W/m

^{2}; A

_{i}= 1 and A

_{k}= 0.97 are the emissivities of the i-th and k-th surfaces; φ

_{i}

_{,k}are local angular coefficients (depending only on the body system geometry).

_{x}is a step on the forest fuel surface in the X-axis direction.

## 3. Results and Discussion

^{2}and flame length of 0.15 m. The radiation source temperature was taken to be 900 K. This ensured that the average heat flux to the forest fuel surface was close to the one obtained experimentally [3]. Moreover, based on the experimental research findings obtained under laboratory conditions [27], as assumption was made that the time of the complete burnout of forest fuels when using a wet line ahead of the combustion front was 120 s. The typical wet line width in the experiments [3] was no more than 0.5 m with a material sample thickness up to 0.05 m. Therefore, the mathematical modeling results in Figure 3 are shown at a point t = 120 s at a control line span L

_{gr}of 0.3 m. The thickness of a coniferous forest fuel sample H

_{fcm}was 0.05 m. These conditions can be considered general for different values of the radiant heat flux and flame length when analyzing the typical fire suppression conditions. By analyzing the mathematical modeling results under these conditions and taking account of the experimental data on fire types, characteristics and sizes [3,4,27], it is possible to predict how far the flame length affects the containment time and liquid consumption volume when using different component compositions. The experiments [3,4,27] compare the integral characteristics of forest fuel combustion when using liquids without additives, slurries, emulsions and solutions.

_{gr}= 0.2; 0.3; 0.4 m), other things being equal. It is clear that an increase in the control line span leads to a lower maximum temperature of the forest fuel surface (its unwetted part). It is apparent from Figure 4 that the T(x) curve is quite complex.

_{gr}= 0.2 m)—corresponds to the span of the control line from which all the moisture has already evaporated. A great temperature variation over the surface of this region illustrates the distributed flows from the combustion front, on the one hand, and the effect of water and water vapors, on the other hand. Farther away from the combustion front, the forest fuel surface temperature quickly decreases due to the moisture evaporating in the control line structure. The local minima and maxima on the three curves T(x) at different spans of control lines illustrate the second characteristic region that includes a small area to the left of the right boundary of the control line and to the right of this boundary. The local maxima of the surface temperature on the curves in Figure 4 correspond to a small area behind the control line on the right, where water and its vapors have a modest effect. The third area of T(x) is characterized by a slight rise in the temperature of the forest fuel surface due to radiant heating from the flame front that is quite far from the forest fuel.

_{rad}of 0.15 m, the minimum span of the control line L

_{gr}, at which combustion is contained, is 0.3 m.

_{gr}can be attributed to the fact that the flame length during the numerical experiment was taken to be constant. In a real fire, the flame length ahead of the wet line decreases over time (which was recorded in the experiments [27]). This process reduces the heat flux to the forest fuel surface, and the fire suppression condition is satisfied at lower L

_{gr}.

_{e}, characteristic width) of the wet line, which should be placed ahead of the combustion front using hand tools or an aircraft, calculated for varying flame lengths. The effective wet line span increases with an increase in the flame length. In a real fire, the higher the flame, the greater the heat flux supplied to the forest fuel surface and the higher the vaporization rates. This process results in a buffer vapor layer emerging between the forest fuel and the flame zone. As a result, the rates of the physical and chemical processes decrease, which stabilizes the forest fuel surface temperature to some degree.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Scheme of solution domain 1—wetted forest fuel; 2—dry part of forest fuel; 3—drained part of forest fuel; 4—mixture of water vapors and air; 5—forest fire front.

**Figure 3.**Temperature distribution (

**a**) (K) in the forest fuel layer and in gas; contour lines of the stream function (

**b**) (m

^{2}/s) of the mixture of air and water vapors near the forest fuel surface; contour lines of the velocity vector module (

**c**) (m/s) near the forest fuel surface; mass concentration of water vapors in the air near the forest fuel surface (

**d**).

**Figure 5.**Maximum forest fuel surface temperature depending on the control line span and the effective control line span (typical size of the wetted forest material strip ahead of the combustion front) at different flame lengths.

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**MDPI and ACS Style**

Kuznetsov, G.; Kondakov, A.; Zhdanova, A. Mathematical Modeling of Forest Fire Containment Using a Wet Line Ahead of the Combustion Front. *Fire* **2023**, *6*, 136.
https://doi.org/10.3390/fire6040136

**AMA Style**

Kuznetsov G, Kondakov A, Zhdanova A. Mathematical Modeling of Forest Fire Containment Using a Wet Line Ahead of the Combustion Front. *Fire*. 2023; 6(4):136.
https://doi.org/10.3390/fire6040136

**Chicago/Turabian Style**

Kuznetsov, Geniy, Aleksandr Kondakov, and Alena Zhdanova. 2023. "Mathematical Modeling of Forest Fire Containment Using a Wet Line Ahead of the Combustion Front" *Fire* 6, no. 4: 136.
https://doi.org/10.3390/fire6040136