Modified Quasi-Physical Grassland Fire Spread Model: Sensitivity Analysis

Abstract

Developing models for predicting the rate of fire spread (ROS) in nature and analyzing the sensitivity of these models to environmental parameters are of great importance for fire study and management activities. A comprehensive sensitivity analysis of a general and modified quasi-physical model is provided in the current study to predict parameters that affect grassland fire propagation patterns. The model considers radiative heat transfer from the flame and fuel body and convective heat transfer to predict the fire’s rate of spread and the grassland fire patterns. The model’s sensitivity to ten main parameters that affect fire propagation, including temperature, humidity, wind speed, specifications of vegetable fuel, etc., is studied, and the results are discussed and analyzed. The model’s capability is validated with experimental studies and a comprehensive physical model WFDS. The model’s capability, as quasi-physical, faster than the real-time model, shows high consistency in fire propagation parameters compared with experimental real data from the Australian grassland fire Cases C064 and F19. The comprehensive sensitivity analysis provided in this study resulted in a modified equation for the corrected rate of fire spread which shows quite an improvement in ROS prediction from 5% to 65% compared with the experimental results. The study could be a base model for future studies, especially for those researchers who aim to design experiments and numerical studies for grassland fire spread behavior.

1. Introduction

Wildfires are a growing concern for sustainable development [1]. Records indicate that in recent times, the number of wildfires has increased due to population growth, land use change, and climate change. Part of this increase is likely due to improved detection and reporting of fires [2,3,4]. As wildfires increase, integrated strategies for forests, climate, and sustainability modeling become more essential. With this concern, studies should be conducted to link the wildfire effects to sustainable development goals [5,6]. Unfortunately, climate change in the Middle East has affected the growing trend of forests fires and grassland fires which consequently destroy the vegetal cover for a couple of years, affect plant diversity, and promote soil erosion in the forest ecosystems [7,8,9,10,11]. In recent years, there have been frequent fires in the Hur-al-Aazim Wetland in the southern part of the Iran–Iraq border and the Anzali wetland located on the Caspian Sea.

Analyzing forest fire behavior is very important because of all the environmental repercussions [12]. The development and application of fire models has become a priority for many countries as well as international unions of firefighters and environmental organizations. The purpose of modeling fire behavior in grasslands and pastures is to understand and predict possible fire behaviors [13,14,15]. Wildland fire behavior modeling has been active since the 1920s when Hawley presented theoretical considerations regarding factors that influence forest fires [16], and Gisborne presented the objectives of forest fire–weather research and complicated controls of fire behavior [17,18].

Fire behavior models fall into three broad categories based on the way they are constructed: physical and quasi-physical models, empirical and quasi-empirical models, and simulation and mathematical analogue models [19]. Physical models represent both the physics and chemistry of fire spread, such as the WFDS [2], FIRETEC [20], FIRESTAR [21], Weber [22], and Grishin [23]; but combustion chemistry is not directly considered in quasi-physical models, such as ADFA I and II [24], Albini [25], and UoC-B [26]. The physics of fire dynamics is not considered at all in empirical models, such as CSIRO Grass [27], CSIRO Forest [28], and Helsinki [29], while quasi-empirical models, such as TRW [30], NBRU [31], and Nelson [32], do use some form of physical concepts. Simulation studies have also been conducted on the fire spread in wildland–urban interface fires [33,34]. In these studies, various physical, computational, or statistical methods have been used [35]. The WFDS model, known as the 3D physical model, is used for various purposes, including numerical modeling of fire spread through grasslands and shrubs [36] or modeling fire behavior in areas with multiple tree arrangements [37]. A newer version of the WFDS, called WFDS svn9977, based on the FDS6 has been used frequently in recent studies [38,39,40].

Simulation and mathematical analogue models implement existing fire behavior models, such as Firemap [41], SiroFire [42], FARSITE [43], and FlamMap [44], or utilize a mathematical precept rather than a physical one to model the spread of wildland fires [45,46]. In contrast, physical models use mathematical relationships generated from the governing fluid dynamics equations, combustion chemistry, and heat transfer mechanisms including convection and radiation using fuel characteristics implemented on terrain topography and weather conditions [47]. The main features that differentiate an operational model from a physical model is the lack of combustion chemistry and a reliance on the heat flame geometry and flame temperature [19].

Knowing the different categories of wildfire behavior prediction models and the features, advantages, and limitations of each of these categories is the first step in fire behavior modeling and studying towards the improvement of these models. The primary heat transfer mode used in wildfire behavior modeling significantly impacts the model approach [48]. Wildfire propagation is generally driven by radiation and convection; in some studies, radiation is considered the dominant heat transfer mechanism and the convection heat transfer mechanism is less discussed [25]. Radiative models were initially proposed to describe the uniform propagation of a planar fire front. These models assume there is no ambient wind, and expressions are obtained for the forward rate of the flame front [24]. Despite full-radiative wildfire models methods, some authors suggest that convection terms of heat transfer should be considered when calculating fire spread based on flame dynamics in specific fuel beds [26,47,49]. Since convective heat transfer has the greatest impact on fire propagation under windy conditions, high slope, and low fuel loads [30,50], some studies suggest considering both convective heating and cooling in calculations to improve the accuracy of results [51]. For low fuel load, fire can be in boundary layer mode even when the driving wind is not that strong [52]. Predicting the thermal radiation emitted by a flame also plays an essential role in wildfire modeling [53]. The Solid Flame Model (SFM) assumes that the flame absorbs and emits radiation as a surface. The Volume Flame Models (VFMs) refer to models where radiation is emitted from inside the actual flame volume. Studies have shown that both these models provide reasonable results in wildfire modeling, which is a wide-ranging study category [54]. In solid flame models, each fuel element’s emissivity, absorptivity, and configuration factor are parameterized to calculate thermal radiation [55]. Two-dimensional quasi-physical fire propagation models are suitable for faster than real-time operational use [50,56,57].

Considering the listed importance of physical and quasi-physical models of fire behavior prediction, a series of quasi-physical wildfire modeling studies have been mentioned in this section. Koo et al. proposed a modified simple physical model and tested it in a series of laboratory and field experiments at the University of California, Berkeley [26]. Morvan et al. proposed a 3D physical model to study the behavior of vegetation fires at a laboratory scale, concentrating on a detailed physical fire model of forest fuels [58]. Studies also showed that increasing ambient wind speed significantly increases the forward spread of the simulated fires [59,60]. A comparison with the wind tunnel experiment showed that the physical characteristics of the model successfully evaluate the influence of wind and slope on the fire spread rate. Fuel bed slope is another parameter that must be considered in fire spread studies [47,61]. A summary of the main quasi-physical models published in the literature between 1990 and 2020 is presented in

Table 1. Quasi-physical models published in the literature between 1990 and 2020.

Authors (Year)	Origin	Heat Transfer Mechanism	Flame Radiation Type	Flame Equivalent Temperature	Dimension	Plane
de Mestre et al. [24]	Australia	Radiation	SFM	795	1	x
Carrier et al. (1991) [30]	USA	Convection/Diffusion	VFM	-	2	xy
Albini et al. (1996) [25]	USA	Radiation	SFM	1050	2	xz
Santoni and Balbi (1998) [56]	France	Radiation/Convection	SFM	970	2	xy
Catchpole et al. (2002) [51]	Australia and USA	Radiation/Convection	SFM	-	2	xz
Vaz et al. (2004) [62]	Portugal	Mix	SFM	1100	2	xy
Koo et al. (2005) [26]	USA	Radiation/Convection	SFM	1083	2	xz
Mohamed Drissi (2015) [57]	France	Radiation/Convection	SFM	1083	2	xy
J.K. Adou et al. (2015) [50]	Ivory Coast	Radiation/Convection	SFM	1083	2	xy
Balbi et al. (2020) [47]	France	Radiation/Convection	SFM	-	1	x

SFM: Surface Flame Model; VFM: Volume Flame Model.

.

To evaluate and improve fire behavior modeling, there is a need for new methods such as genetic algorithms, which have been considered in recent years [63,64]. Machine learning has also helped experts to increase the accuracy of the wildfire simulation while reducing the speed of calculations [65]. Faster prediction gives the fire management systems a better tool to prevent fire damage [66]. Recent models try to reduce the calculation time for faster fire prediction by reducing the number of model parameters [66,67]. For this purpose, studies have focused on the sensitivity analysis of parameters in recent years [68,69,70].

The current study proposes a modified quasi-physical fire spread model to predict fire propagation patterns in grasslands that considers radiative heat transfer from the flame and fuel body and convective heat transfer. This model considers the characteristics of the vegetative fuel bed in three geometric dimensions. The proposed modified quasi-physical model objectives are model simplicity, adaptability to the Earth’s topographic conditions, and designed to be faster than real-time for operational purposes. The model can also be generalized to various plant fuels. By carrying out a comprehensive sensitivity analysis on all the parameters affecting the fire spread rate and improving the physics of the quasi-physical models developed, especially in the convective heat transfer section, the current research has improved the prediction results of the fire spread rate and the shape and pattern of the fire propagation. To the authors’ knowledge, such a comprehensive sensitivity analysis has not been provided on wildfire quasi-physical modeling in previous studies.

2. Materials and Methods

The proposed model consists of a typical two-dimensional network with circular cells of equal size. Figure 1 presents the schematic of the model network and the relationship between cells. The network of combustible cells, or in other words, the pasture bed that is being modeled, can be created using a schematic of the model network to present a vegetation map. It should be noted that burning cells represent the vegetation pasture in this model. The model was applied to a set of reference fires conducted in controlled conditions (Australian grassland experiments) and cases modeled by WFDS [2].

2.1. Hypotheses

The first step to develop a two-dimensional quasi-physical fire spread model is to determine the correct and pre-proven principles and assumptions. These physical or geometric assumptions can include the characteristics of plant fuels or environmental conditions. A description of these hypotheses is summarized in the following paragraphs:

• Each combustible cell (j) is defined as a cylinder of height (Hj) and diameter (Dj) (Figure 2).
• The combustible cell is considered a healthy cell as long as the cell temperature (Tj) is equal to the ambient temperature (T∞).
• The energy absorbed by the flammable cell when exposed to the fire front is used to increase the temperature of the wet fine fuel cells to the boiling point of water (373 K).
• By evaporating the cell moisture (humidity as fuel) temperature of the fine fuel cells, the temperature of the dry cell is raised to combustion temperature.
• The combustible cell then enters the flame combustion phase for the duration of tc, the flame residence time, while simultaneously transferring heat to adjacent combustible cells by radiation and convection mechanisms.
• In the Solid Flame Model, the visible flame is considered a solid body with uniform radiation in the form of a cylinder from the surface of the emitted thermal radiation.

2.2. Equations

Combustion cell temperature is calculated using the energy conservation equation. The energy received by the flammable cell moves toward the target cell through the two mechanisms of displacement and radiation. The most important heat dissipation mechanism in a flammable cell is radiation [57]. The total heat transfer, q_t, can be calculated as follows:

$$q_t={q_{conv}}^{+}+{q_{rad}}^{+}+{q_{rad}}^{-}$$

(1)

where q_t is the total heat transfer, q_conv is the convection heat transfer, q_rad⁺ is the radiation to cell j, and q_rad⁻ is the radiation losses of cell j. Equation (1) is extended in Equation (2) to obtain the total heat transfer emitted from the burning cell. Then, the radiation and convection parameters are added in Equations (3) and (4).

$$q_{ij}=q_{ij}^{sr}+q_{ij}^{ir}+q_{ij}^{rl}+q_{ij}^{sc}+q_{ij}^{ic}$$

(2)

$${q_{rad}}^{+}+{q_{rad}}^{-}=q_{ij}^{sr}+q_{ij}^{ir}+q_{ij}^{rl}$$

(3)

$${q_{conv}}^{+}=q_{ij}^{sc}+q_{ij}^{ic}$$

(4)

where $q_{ij}^{sr}$ is the surface radiation, $q_{ij}^{ir}$ is the internal radiation, $q_{ij}^{rl}$ is the radiative loss from the combustible cell (the radiation parameters), and $q_{ij}^{sc}$ is the surface convection and $q_{ij}^{ic}$ is the internal convection (the convection parameters). Figure 3 is a schematic of how heat is transferred from the flammable cell to the target cell through the two mechanisms of displacement and radiation [26,47].

2.2.1. Radiation

In the process of fire spread, radiant heat transfer plays a major role in heat transfer mechanisms and fire spread [71]. It is assumed that flammable cell (j) is at the d_ij distance from the burning cell (i). The total heat transfer emitted from the burning cell (i) received by the flammable cell (j) is denoted by q_ij, which is calculated in (2). The right-hand side of (2) is a summation of all possible heat transfer mechanisms, including radiation from the upper surface of cell j, internal radiation from the ember zone of cell j, convection above the cell j, convection above the head of cell j, internal convection inside the fuel bed, and radiation losses at the upper surface of cell j.

Since the air in grasslands is often unstable and the wind speed is non-zero, other preheating heat transfer mechanisms, i.e., turbulent diffusion and phase conduction, convective cooling, and energy absorbed during pyrolysis before combustible cell ignition point, can be ignored [50]. In modeling, only thin, thermally delicate plants burn (such as grasses, thin branches, and the leaves of trees and shrubs with a thickness of less than 6 mm), and only this type of burning determines the fire spread rate [72]. Other fuels, such as tree trunks, burn more slowly and therefore play no role in spreading the fire [72]. All the parameters of the above equations are explained in order below. Radiation to the upper surface of the fuel bed received from the flame to cell j is calculated from (5).

$$q_{ij}^{sr}=\frac{a_{fb}{\epsilon }_{fl}\sigma T_{fl}^4}{H_j}{F}_{ij}$$

(5)

$${\epsilon }_{fl}=1-\mathrm{e}\mathrm{x}\mathrm{p}(-0.6L_{fl})$$

(6)

where ε_fl is the flame emissivity, a_fb is the fuel bed absorptivity, σ is the Stefan–Boltzmann constant (5.67 × 10⁻⁸ W/m² K), and F_ij is the view factor or visibility [26].

The visible flame is considered to have a uniform surface of radiation with cylindrical geometry (Figure 2). F_ij visibility is the fraction of radiation absorbed by cell j due to radiation from cell i which is explained in Appendix A. The View Factor or Configuration Factor is comprehensively described in Henkinson’s study [73,74]. This method is similar to the Monte Carlo method used in Ansys Fluent software version 14.0 or upper. The surface-to-surface (SOS) radiation model calculates the net heat flows for given temperatures of the plane, gray, diffuse surfaces [75]. Analytical expressions to calculate this view factor were introduced in various studies. In a porous fuel bed, unburned cells receive radiant heat flux from the fuel bed environment and the flame front. This internal radiation from cell i to cell j at the distance d_ij is obtained from (7) [26,50].

$$q_{ij}^{ir}=0.25 A_{fb}{\epsilon }_b\sigma T_b^4\exp (-0.25A_{fb}{d}_{ij})$$

(7)

where ε_b is the ember emissivity equal to 1, T_b is the ember temperature equivalent to ignition temperature, and A_fb is the total fuel particle surface area per fuel bed volume. Correspondingly, the heat loss of unburned cells due to radiation to the environment above the fuel bed surface is calculated from (8). The expression $q_{ij}^{rl}$ is equivalent to qrad⁻ in (1).

$$q_{ij}^{rl}=-\frac{{\epsilon }_{fb}\sigma (T_j^4-T_{\infty }^4)}{H_j}$$

(8)

where ${\epsilon }_{fb}$, the diffusion coefficient of the (8) is related to the fuel bed. Further explanation about flame height based on Heskestad equations is presented in Appendix B [76].

2.2.2. Convection

In fire spread modeling, the heat transfer calculations only consider the forced heat transfer due to ambient wind. The heat transfer over the surface of cell j is calculated from the following equation [26,77]:

$$q_{ij}^{sc}=\frac{0.565k_{fl}{Re}_{d_{ij}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{Pr}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{{d_{ij}H}_j}(T_{fl}-T_j)\mathrm{e}\mathrm{x}\mathrm{p}(-0.3d_{ij}/L_{fl}){\beta }_{ij}$$

(9)

$$q_{ij}^{ic}=\frac{0.911k_b{A}_{fb}{Re}_D^{0.385}{Pr}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{D_j}(T_b-T_j)\mathrm{e}\mathrm{x}\mathrm{p}(-0.25A_{fb}{d}_{ij}){\beta }_{ij}$$

(10)

where Pr is the Prandtl number, Re_dij stands for the Reynolds number, U_w is the ambient wind speed, and K_fl is the thermal conductivity. The heat transfer coefficient for a cylinder in a cross-flow used for the interior of the fuel bed is derived from (9) [26].

${Re}_{D_j}=\frac{U_{fb}{D}_j}{{\vartheta }_g}$

$U_{fb}=(1-{\mathrm{\varnothing }}_j)U_w$

${\beta }_{ij}=\left\{\begin{array}{c}1 \overrightarrow{ij} \parallel \overrightarrow{U} \\ 0 \overrightarrow{ij} \nparallel \overrightarrow{U}\end{array}\right.$

The β_ij coefficient shows that the internal convection will be considered in calculations only if the connecting vector from cell burning cell i to combustible cell j is aligned with the wind vector.

2.3. Methodology

The ordinary differential equation system must be solved before the ignition point. The system of equations is as follows:

$$for 0\le t\le t_c\Rightarrow \left\{\begin{array}{c}\frac{{dT}_j}{dt}=\frac{1}{{\rho }_j{C}_{Pj}{\mathrm{\varnothing }}_j} {\displaystyle \sum _{l=sr o rl}}{q}_{ij}^l for T_j\ne 373 K\\ \frac{{dW}_j}{dt}=-\frac{1}{{\rho }_j{h}_{vap}{\mathrm{\varnothing }}_j} {\displaystyle \sum _{l=sr o rl}}{q}_{ij}^l for T_j=373 K\end{array}\right.$$

(11)

where T_j(0) is equal to the ambient temperature (T∞) and W_j(0) is the initial water mass fraction. Each cell has its interactive range using the current method and considering the preheating mechanisms [26].

All the coefficients used in the above equations in each simulation and in accordance with the physical conditions of the environment to be simulated are presented in separate tables. Simulation results of the WFDS model conducted by Mell et al., the developers of WFDS, are used to compare the results of a two-dimensional simulation of the proposed model [1]. The conditions used for the simulation using the WFDS model are listed in

Table 2. Simulation parameters used in the WFDS model (Case F19).

Parameter	Symbol	Value
Domain Size	xyz	200 m × 200 m × 25 m
Minimum Grid Resolution	-	0.1 m
Enthalpy of Combustion	Δhc	15,600 KJ·Kg−1
Wind Speed	U∞	uniform—5 m·s−1
Ambient Temperature	T∞	307 K
Solid Phase Density	ρs	512 kg·m−3
Vegetable Surface to Volume Ratio	σs	12,240·m−1
Height of the Vegetative (Cell j)	Hj	0.51
Maximum Simulation Period	-	138 s

.

3. Results and Discussion

In the current study, the fire spread model was validated with two scenarios: grassland fire experiments conducted in the Northern Territory of Australia, F19, and a complete 3D physical wildland fire dynamics simulator called WFDS. A summary of the main parameters used in the modeling of the current study is listed in

Table 3. Model parameters.

Item	Parameter	Specification	Case F19	Case F19
1	σ	Stefan–Boltzmann constant (W/m2 K)	5.67 × 10−8	5.67 × 10−8
2	αfb	fuel bed absorptivity	0.6	0.6
3	Lfl	flame length (m)	5.0	5.0
4	Dj	length scale inside the fuel bed (m)	2.5	2.5
5	εfl	flame emissivity	-	-
6	εfb	fuel bed emissivity	0.6	0.6
7	εb	ember emissivity	1.0	1.0
8	Hfl	flame height (m)	5.0	5.0
9	Hj	height of the vegetative cell j (m)	0.51	0.21
10	RSFM	Solid Flame Model radius (m)	1.25	1.25
11	Tfl	flame temperature (K)	1083	1083
12	Tb	ignition temperature (K)	500	500
13	Tpyr	pyrolysis temperature (K)	500	500
14	Tinf	ambient temperature (K)	307	305
15	UW	ambient wind (m/s)	4.8	4.6
16	Ufb	velocity of wind inside the fuel bed (m/s)	1	1
17	Kfl	thermal conductivity of air at flame temperature (W/m·K)	70 × 10−3	70 × 10−3
18	Kb	thermal conductivity of air at bed ambient (W/m·K)	40 × 10−3	40 × 10−3
19	ϑg	kinematic viscosity of air at the flame temperature	13 × 10−5	13 × 10−5
20	Pr.	Prandtl number	0.7	0.7
21	Afb	total fuel particle surface area per fuel bed volume	12,240	9770
22	W0	initial vegetable moisture	0.058	0.063
23	Cpwet	fuel particle wet fuel particle wet specific heat capacity	1110 + 3.7 × T(K)	1110 + 3.7 × T(K)
24	∅	packing ratio	0.0012	0.0026

.

3.1. Model Validation

A photograph of the fire perimeter taken during the Australian experiment F19 is shown in Figure 4. The time step had a significant effect on the accuracy of the results of the numerical calculations in the present model. Although reducing the time step increases the accuracy of the calculations, it leads to an increase in computational time and cost, which contradicts the initial goal of developing a simple model. A series of calculations were performed under the same physical conditions for six time steps ranging from 2.5 s to 0.1 s to determine the optimal time step, see Figure 5. The number of burned cells and the average temperature of 3600 cells in the simulation domain were chosen as a base for proper time step selection. The fire spreads in a 200 × 200 m grassland plot with the fuel properties of experiment F19 shown in Figure 6. One of the most important reasons for the oscillations in the fire head location is the interactions of the fire plume and the local wind field, which is visible in Figure 6.

According to calculation results, a time step of 0.25 s is generally used in simulations based on network conditions. Results show a reasonable agreement between the experimental fire propagation line and the 3D physical simulations of the WFDS Model (Figure 6). Comparative results of fire spread rate and fire spread perimeter are included in

Table 4. Simulation results in comparison with simulations on the WFDS model; Case Study F19.

Item	Ignition Line (m)	Time (s)	ROS (WFDS) (m/s)	ROS (Current Study) (m/s)	ROS Difference (%)	Fire Perimeter (WFDS) (m)	Fire Perimeter (Current Study) (m)	Fire Perimeter Difference (%)
1	8.0	60.0	0.85	0.92	7.8	121.7	118.7	2.5
2	8.0	120.0	0.97	1.17	20.7	265.8	316.2	18.9
3	25.0	60.0	1.35	1.14	15.9	195.0	156.6	19.7
4	25.0	120.0	1.32	1.19	9.7	376.4	362.1	3.8

. A 2.5% to 19.7% difference in fire perimeter prediction and a 7.8% to 20.7% difference in the spread prediction rate seems reasonable for a faster-than-real-time prediction model. It can significantly help fire management and firefighting organizations with any preventative measures. Therefore, the results show the current model’s remarkable accuracy compared to the WFDS model’s validated results.

Another excellent indicator of fire behavior is the average width of the fire front is the fire front head width. The head fire width can be directly determined from the model results. The product of the rate of spread and the flame residence time can also calculate this characteristic length.

Where d_head is the head fire front average width, ROS is the rate of fire spread, and t_c is the fire residence time. The value of head fire front average width is calculated by the product of ROS at fire front by t_c in a certain period.

To strengthen the validity of the model results, achieve more accurate evaluation, and improve the quality of the physical model, validation of the model with experimental data was carried out in the case of studies C064 and F19. For case C 064, the rate of fire spread and the shape of the fire front is drawn for three time periods of 56, 86, and 138 s. Similarly, for the case of F 19, the mentioned parameters are fixed for the period of 27, 53, and 100 s (Figure 7).

3.2. Modified Model

It is necessary to search for the existing shortcomings regarding the prediction of heat transfer parameters to improve the results of the general quasi-physical model of fire spread in wildfires. Recent studies on the role of buoyant turbulent flow in the spread of wildfires show that buoyant instabilities lead to the formation of transverse waves toward the center of the fire [78]. As shown in experimental studies, especially those mentioned by Finney et al., streamwise vortex flow instabilities push convective heat transfer through the firing domain [78]. Therefore, unlike previous wildfire spread quasi-physical models, in addition to the cells in the main wind direction, the convective heat transfer term is considered for the induced wind gradient created towards the center of the fire front.

As shown in Figure 7, the model modification leads to a significant improvement in predicting the fire spread contour and fire spread rate.

Table 5. Results of the general model and modified model in comparison with experimental cases F19 and C064.

Item	Case	Time (s)	Interval (s)	ROS (Experiment) (m/s)	ROS (General Model) (m/s)	ROS Difference (%)	ROS (Modified Model) (m/s)	ROS Difference (%)
1	F19	56	56	1.34	1.03	30%	1.29	3%
2	F19	86	30	1.42	1.17	21%	1.58	11%
3	F19	138	52	1.59	1.20	32%	1.59	0%
4	C064	27	27	0.70	0.74	5%	0.93	24%
5	C064	53	26	1.12	0.67	66%	1.06	5%
6	C064	100	47	1.19	0.69	72%	1.12	7%

shows that the model is significantly improved in predicting the fire front line and fire rate of spread by modifying the model’s behavior in predicting convective heat transfer. The considerable reduction of the model error in predicting fire spread for two real and famous case studies, C064 and F19, is listed in Table 5.

Figure 8 compares the results of the developed semi-physical model and the results of other models and extracted relationships. As shown in Figure 8, the result of the prediction of the fire spread rate according to the wind speed for the present model compared to the mathematical model of the Rothermel Model [79], McArthurs’ Model [80], and other prediction equations [27,81] for two cases, C064 and F19, are shown.

Results show that the predicted rate of fire spread in the modified model improves from 5% to 65% versus the previous general model, both in comparison with experimental results.

3.3. Sensitivity Analysis

This section presents the sensitivity analysis of the model to each of the parameters affecting the fire spread rate (ROS). Difficulties in measuring these parameters produce uncertainties in the model results. Therefore, the model’s predictive capability depends on the accuracy of determining these parameters. The range of parameter variations was chosen to cover typical vegetable and meteorological conditions, see

Table 6. Variation range and sensitivity of parameters effect on the rate of spread (Case F19).

Item	Parameter	Reference Value	Variation Range	1st Level	ROS Change @ 1st Level	2nd Level	ROS Change @ 2nd Level
Hf	Vegetative height (m)	0.51	20%	0.408	+0.125	0.612	−0.125
Lf	Flame length	5	20%	4	−0.291	6	+0.291
Tfl	Flame temperature (K)	1083	10%	975	−0.333	1191	+0.375
Tign	Ignition temperature (K)	500	10%	450	+0.167	550	−0.125
T∞	Ambient temperature (K)	307	5%	292	−0.125	322	+0.167
Uw	Wind speed (m/s)	5	20%	4	−0.083	6	+0.083
W0	Initial vegetable moisture	0.058	38%	0.036	+0.208	0.08	−0.125
cp	Fuel specific heat (kJ/kg·K)	1.11 + 3.7 × T	22%	6.10 + 3.7 × T	+0.125	1.610 + 3.7 × T	−0.125
ρs	Solid phase density (kg/m3)	512	25%	384	+0.458	640	−0.291
∅	Packing ratio	0.0012	25%	0.0009	+0.458	0.0015	−0.291

.

The sensitivity analysis makes it possible to estimate responses as functions of these parameters and to make proper correlations between the parameter effects and the system response. The range of changes selected for the parameters of environmental conditions and the physical characteristics of vegetable fuel mentioned in Table 6 are chosen based on the typical range of changes of these parameters in nature. The rate of spread can then be corrected with the following equation:

$${ROS}_c={ROS}_0+\sum _{i=1}^{10}\alpha X_i+{\sum }_{i,j,\dots }(interactions from order 2 to order 10)$$

(12)

where ROS_c and ROS₀ are the corrected and predicted rate of spread, respectively, α is the parameter sensitivity, and X_i is the result of each of the parameters analyzed. High orders of parameter interaction are not noticeable; therefore, only low order parameter interactions are considered [70,84]. By omitting the interaction of parameters in Equation (12), Equation (13) is obtained based on the sensitivity analysis of parameters in Table 6:

$$\begin{array}{ll}{ROS}_c={ROS}_0& -1.225\left(H_f-5\right)+0.291\left(L_f-5\right)+0.0033\left(T_{fl}-1083\right)\\ & \hspace{1em}-0.292\left(T_{ign}-500\right)+0.0097\left(T_{\mathrm{\infty }}-307\right)+0.083\left(U_w-5\right)\\ & \hspace{1em}-7.57\left(W_0-0.058\right)-0.00025\left(c_p-1110-3.7T\right)\\ & \hspace{1em}-0.0029\left({\rho }_s-512\right)-1248(\mathrm{\varnothing }-0.0012)\end{array}$$

(13)

Figure 9 shows the fire propagation contour for case F19 with 8 m and 25 m fire ignition lines at two moments (t = 60 s and t = 120 s). The comparative chart of the relative sensitivity analysis of 10 main parameters to the rate of fire spread (ROS) is also presented in Figure 10.

As shown in Figure 10, the flame temperature has the most significant effect on the rate of fire spread, and increasing the ignition temperature has the most effective relative impact on reducing the rate of fire spread.

Another essential component in evaluating fire propagation models is studying the heat transfer rate for each radiative heat transfer and displacement mechanism and its effect on fire spread. In this regard, Figure 11 depicts a diagram of heat transfer mechanisms from the ignition moment to the ignition moment across with a temperature diagram for the same point on the simulation environment.

As shown in Figure 11, the contribution of internal radiation and internal convection to the fire moment is minimal compared to other heat transfer mechanisms. Therefore, to better separate the charts, these two mechanisms have been removed in Figure 12, and the time interval is limited to 10 s when the fire front approaches the desired point.

The effect of wind speed on the rate of fire spread in three fuel bed packing ratios was investigated, and the results are shown in Figure 13. The ratio of initial moisture content of vegetable fuel on the rate of fire spread at three wind speeds of 2, 5, and 10 m/s was also investigated. Figure 14 shows that the ROS is essentially much slower than dry vegetable fuels for very high fuel moisture content. The range of plant fuel moisture in pasture fires is usually below 0.45.

4. Conclusions

In this study, a comprehensive modified quasi-physical model was developed to predict fire propagation patterns on gentle terrain. Preheating energy transfer mechanisms, including surface radiation from the flame, internal radiation from the ember part, radiative loss, surface convection from the top of the flame, internal convection, and cooling convection from the swamp or wet soil, are considered in the model to predict the fire contour and rate of fire spread on grassland and pastures. The model also includes the effects of vegetation specification, topography and terrain characteristics, and climatic parameters. The proposed model was validated with the WFDS model, and the results showed that it makes the model valuable compared to other physical models. In this model, vegetation is considered a network of combustible cells on a macroscopic fuel map. To make it suitable for fire management and support systems, the model development pattern is considered such that it provides considered results faster than real-time computation period.

In the sensitivity analysis, the most important section of this study, the model’s sensitivity to ambient conditions and vegetative fuel properties that affect fire propagation was explained for academic considerations. Ambient temperature, ambient humidity, and wind speed were considered the most effective ambient conditions. Moreover, the sensitivity of vegetable fuel properties, including specific heat, vegetable height, packing ratio, initial moisture of vegetative fuel, flame temperature, and flame length, were studied separately. Results show that the flame temperature has the most positive, and the ignition temperature has the most negative effect on the rate of fire spread (ROS).

In summary, a significant share of heat transfer occurs around the surface radiation, surface convection, and radiation losses. Therefore, internal radiation and convection could be ignored to reduce the volume of the simulations and increase the computational speed. Based on the analytical results obtained, it can be concluded that before 373 K (the boiling point of water and evaporation of all the moisture in the vegetative fuel), the contribution of the surface convection, and after this temperature, the contribution of the surface radiation to the total heat transfer is the largest among all heat transfer mechanisms.

This sensitivity analysis leads to a general equation for the rate of fire spread which is based on 10 main effecting parameters. Results show that the predicting rate of fire spread in modified model improved from 5% to 65% versus the previous general model, both in comparison with experimental results.