Simulation Modeling of the Process of Danger Zone Formation in Case of Fire at an Industrial Facility

Abstract

Proactive prevention and fighting fire at industrial facilities, often located in urbanized clusters, should include the use of modern methods for modeling danger zones that appear during the spread of the harmful combustion products of various chemicals. Simulation modeling is a method that allows predicting the parameters of a danger zone, taking into account a number of technological, landscape, and natural-climatic factors that have a certain variability. The purpose of this research is to develop a mathematical simulation model of the formation process of a danger zone during an emergency at an industrial facility, including an explosion of a container with chemicals and fire, with the spread of an aerosol and smoke cloud near residential areas. The subject of this study was the development of a simulation model of a danger zone of combustion gases and its graphical interpretation as a starting point for timely decision making on evacuation by an official. The mathematical model of the process of danger zone formation during an explosion and fire at an industrial facility presented in this article is based on the creation of a GSL library from data on the mass of explosion and combustion products, verification using the Wald test, and the use of algorithms for calculating the starting and ending points of the danger zone for various factor values’ variables, constructing ellipses of the boundaries of the distribution of pollution spots. The developed model makes it possible to calculate the linear dimensions and area of the danger zone under optimistic and pessimistic scenarios, constructing a graphical diagram of the zones of toxic doses from the source of explosion and combustion. The results obtained from the modeling can serve as the basis for making quick decisions about evacuating residents from nearby areas.

1. Introduction

Many industrial enterprises located in urban areas are characterized by chemical hazards—toxic, fire, and explosive ones. Combined accidents are possible: fire in combination with a toxic accident, when a flammable substance is also a toxic substance, or when non-toxic materials release toxic substances when burned. All this creates a real or potential threat to the health and life of residents of populated areas (often cities with a population of more than a million people), which is realized by fire and the spread of a cloud of harmful substances (combustion gases) to adjacent densely populated areas. It is also appropriate to talk about the importance of modeling danger areas in the context of sustainable development and ensuring safe industrial development to reduce damage to the environment [1,2]. Therefore, the area of distribution of combustion products of various chemicals used in modern industrial enterprises located in warehouses and logistics hubs (plastics, petroleum products, fertilizers, explosives, etc.) is the main qualitative attribute of fire hazards for residents of nearby areas, along with the composition of combustion products [3,4]. The modeled maximum boundaries of hazardous zones are in kilometers, which corresponds to the size of small towns and rural areas and makes this study relevant for them.

At the same time, when predicting the distribution area, materials that emit harmful volatile substances during combustion are a constant value, while the parameters of the danger zone are variable. Therefore, research into the characteristics of the harmful factors of fires at industrial enterprises and the development of effective measures to mitigate or eliminate them is possible only through the use of mathematical simulation modeling of such objects. This is because an emergency cannot be organized or repeated. The simulation model must adequately describe the emergence and development of a danger zone during the combustion of chemicals at an enterprise and allow for calculation of the parameters for the emission of hazardous substances into the atmosphere near residential areas (Figure 1).

The purpose of simulation modeling is to reproduce the behavior of the system under study—the formation of a danger zone from fire with the combustion of chemicals at an industrial enterprise—based on the results of the analysis of the most significant relationships between its elements (statistical experiment). Unlike classical mathematical models, the simulation results of which reflect the time-stable behavior of the system, the results obtained in the simulation model are observations subject to experimental errors. This means that any statement about the magnitude of the parameters of the modeled system must be based on the results of appropriate statistical tests.

Research in the field of simulation modeling of danger areas during emergency situations at industrial enterprises was carried out by H. Wen et al. [9], Huseinov R. [10], T. Drozdova et al. [11], S. Gwynne et al. [12], N. Zaurbekova et al. [13], F. Di Giuseppe et al. [14], etc. In the works of these authors, basic provisions have been developed, and industry-specific features of the danger zone formations in emergency situations of a man-made nature have been highlighted.

In relation to the spread of fire and combustion products, studies of danger areas were carried out by A. Tsvirkun et al. [15], E. Chuvieco et al. [16], I. Teslenko [17], A. Bykov et al. [18], C. Sirca et al. [19], etc. Methods for computer modeling of fire danger zones have been studied by Z. Xu et al. [20], M. Pratt [21], S. Yemelyanenko et al. [22], etc.

The studies of J. Beringer [23], L. Redwood-Campbell et al. [24], S. Ahmed et al. [25], etc., are devoted to the issues of protecting the life and health of workers at industrial enterprises and residents of urban areas during fires. A significant contribution to the study of hazards and risks in emergency situations in urban areas was made by T.M. Ferreira [26,27].

The variability of factors that determine the size of danger zones during fires at industrial enterprises requires high flexibility from the models used; therefore, from the point of view of detailing the behavior of complex systems, simulation modeling (compared to “classical” mathematical modeling) has certain advantages. At the same time, the creation of simulation models of danger zones during fires at industrial facilities is associated with a significant amount of time when trying to optimize the behavior of the simulated system [28].

We consider the opinion of a number of authors that the initial results of simulation modeling of danger zones during fires are unstable [29] and therefore cannot form an absolute idea of their true parameters but give their assessment in a certain approximation that is sufficient for predicting the consequences of fires and planning evacuation measures to save lives and protect the health of residents of settlements within the territory in which industrial enterprises are located.

Thus, the previously performed studies presented important provisions for determining the risks of occurrence and spread of fires with technogenic nature, their industry specificity, and the general principles of calculating the parameters of hazardous zones. At the same time, there is a certain gap in the studies of simulation modeling of the size of hazardous zones in case of fires at an industrial enterprise, associated with mathematical models and the algorithmization of determining their main parameters.

2. Materials and Methods

The first step in creating a simulation model of a danger zone during a fire at an industrial facility is to develop a mathematical description of the formation and distribution of toxic combustion products in the atmosphere—a real system—using the characteristics of the main events. An event—the fire with the release of combustion products of chemicals into the atmosphere—is defined as a point in time at which a change in the characteristics of the system occurs (the distribution of combustion products in the atmosphere) [30].

All toxic combustion products are volatile substances that eventually settle on the surface of the earth. So, when polystyrene burns, styrene is released, which is a toxic sub-stance. Polyvinyl chloride releases dioxins. All synthetic materials emit chlorine, hydrogen chloride, hydrogen cyanide, sulfur, and nitrogen derivatives. To take into account the intensity of the release of these harmful substances into the atmosphere, we used data from previously conducted studies.

To obtain the required modeling results, it is enough to observe the system at those moments in time when events occur. Sharp transitions (“jumps”) made by the model when moving from one event to another indicate that the process occurs in discrete time. It should be noted that in discrete simulation there is nothing in common between real time and model running time. For example, the operating time of a model implemented on a computer is usually significantly less than the operating time of a real system [31,32,33], which is an undeniable advantage of simulation modeling in obtaining the same information about the system, which is especially important for studying fires at industrial facilities, where full-scale experiments are impossible, and each fire is unique. However, obtaining independent observations in a simulation is much more difficult than in conventional laboratory experiment.

Therefore, a simulation model of the formation of a dangerous spread zone of toxic substances in the atmosphere during the combustion of chemicals is adopted in the form of a mathematical and geometric model, as well as an algorithm. The dominant factors in the process of formation of a dangerous zone are the mass of toxic combustion products spreading in the form of a cloud, as well as the speed of its spread.

2.1. Mathematical Model for Calculating the Danger Zone during a Fire at an Industrial Facility

We consider the spread of combustion products with modeling of hazardous areas, which we must take into account to the maximum possible dimensions. Therefore, we use experimental data obtained in other previously performed studies on the maximum burning rate of various chemicals [34], which were obtained by modeling at the stage of volumetric development of a fire, to determine the size of dangerous zones as conservatively as possible.

In Figure 2, the structure of a mathematical model for calculating the main parameters of the danger zone area (DZA) (area and linear dimensions of the ellipse) formed during fires at industrial facilities (with the combustion of chemicals) is presented.

In turn, in the danger zone, the concentrations of harmful substances in the atmosphere—combustion products (carbon monoxide, nitrogen oxides, acrolein, etc.)—exceed, including many times, the maximum permissible values (MPC). All these factors are inputs to the mathematical model.

The output variables of the model are the following: ${\mathrm{S}}_{\mathrm{D}\mathrm{Z}\mathrm{A}}=\frac{\mathsf{\pi }}{4}\mathrm{a}\mathrm{b}$—area of the ellipse of the danger zone during fire at an industrial enterprise, where a and b are the lengths of the axes of the ellipse of the danger zone, respectively.

With the adopted approach, ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$ (expert estimate of the mass of a flammable chemical) should be a random variable taking values in a certain numerical interval. For simulation modeling, it is necessary to submit to the input of the simulation model a sequence of random variable M_E within the boundaries of the expert assessment interval (0.9 ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$, 1.1 ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$).

To obtain disjoint sets of random sequences in the optimistic (“opt”) and pessimistic (“pess”) scenarios, it is proposed to set Condition (1):

$$E_{opt}\left[{\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}\right]={0.95\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}},E_{pess}=1.05{\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}},{\sigma }_{opt}\left[{\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}\right]={\sigma }_{pess}\left[{\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}\right]=0.01{\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$$

(1)

Thus, the maximum deviation (error) in the absolute value of the expert estimate of the mass of a flammable chemical substance will not exceed $\left|\delta \right|=3\sigma =0.03{\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$.

The diagram of the simulation model for the formation of a dangerous zone during fire is presented in Figure 3.

$\mathrm{E}\left[{\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}\right]$—assembly average of an expert estimate of the mass of a flammable chemical substance ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$.

$\mathsf{\sigma }\left[{\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}\right]$—root-mean-square expectation of expert assessment of ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$.

n—sample size of generated random variables of ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$.

${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}={\tilde{\mathrm{m}}}_{E_1}^{\mathrm{*}},{\tilde{\mathrm{m}}}_{E_2}^{\mathrm{*}},\dots ,{\tilde{\mathrm{m}}}_{E_n}^{\mathrm{*}}$—sampling options.

The output variables of the model are the following:

${\tilde{\mathrm{s}}}_{\mathrm{D}\mathrm{Z}\mathrm{A}}$—random value of the area of the danger zone during a fire at an industrial enterprise.

$\tilde{\mathrm{a}},\tilde{\mathrm{b}}$—random values of axes a and b of the danger zone ellipse.

In order for the sample of random numbers to be representative, its size is usually chosen as n = 30. By generating random variables ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$ and substituting them into the simulation model, we obtain a sample of random variables ${\tilde{\mathrm{S}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}{,\tilde{\mathrm{a}}}_{\mathrm{o}\mathrm{p}\mathrm{t},}{\tilde{\mathrm{b}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}$ for the optimistic and ${\tilde{\mathrm{S}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}{,\tilde{\mathrm{a}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s},}{\tilde{\mathrm{b}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}$ for the pessimistic scenarios of the release of harmful substances during a fire, each with size of n = 30.

When creating a mathematical model of a dangerous zone, it is necessary to statistically test the hypothesis that the distribution law of random variables ${\tilde{\mathrm{S}}}_{\mathrm{D}\mathrm{Z}\mathrm{A}},\tilde{\mathrm{a}},\tilde{\mathrm{b}}$ is normal, using the goodness-of-fit criterion ${\chi }^2$ (Pearson chi-square). Then, if the hypothesis about the normality of the law of distribution of random variables ${\tilde{\mathrm{S}}}_{\mathrm{D}\mathrm{Z}\mathrm{A}},\tilde{\mathrm{a}},\tilde{\mathrm{b}}$ is confirmed, then the mathematical expectations $\mathrm{E}\left[{\tilde{\mathrm{S}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right],\mathrm{E}\left[{\tilde{\mathrm{a}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right],\mathrm{E}\left[{\tilde{b}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right],\mathrm{E}\left[{\tilde{\mathrm{s}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right],\mathrm{E}\left[{\tilde{\mathrm{a}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right],\mathrm{E}\left[{\tilde{\mathrm{b}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]$ and the corresponding standard deviations $\mathsf{\sigma }\left[{\tilde{\mathrm{S}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right],\mathsf{\sigma }\left[{\tilde{\mathrm{a}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right],\mathsf{\sigma }\left[{\tilde{b}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right],\mathsf{\sigma }\left[{\tilde{\mathrm{S}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right],\mathsf{\sigma }\left[{\tilde{\mathrm{a}}}_{\Pi \mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right],\mathsf{\sigma }\left[{\tilde{\mathrm{b}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]$ are calculated.

If the hypothesis about the normality of the law of distribution of random variables $\tilde{\mathrm{S}},\tilde{\mathrm{a}},\tilde{\mathrm{b}}$ for the pessimistic and optimistic options is not confirmed, the sample size of the random variable is doubled (n = 60) and sequences of random variables are generated again (${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$). To test the hypothesis about the homogeneity of the parameters’ variance in the danger zone (the area of the ellipse, its length and width), the Cochran criterion is used for samples of random variables of equal size $\tilde{\mathrm{S}},\tilde{\mathrm{a}},\tilde{\mathrm{b}}$.

If the general variances of the populations under consideration are equal to each other, this means that the corrected sample variances of random variables differ slightly from each other. Consequently, the value of the mathematical expectation of the area of the danger zone ellipse $\mathrm{E}\left[{\tilde{\mathrm{S}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right]$ corresponds to the value a₀, the value $\mathsf{\sigma }\left[{\tilde{\mathrm{S}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right]$ to the value ${\sigma }_x$ [8], the value a₁ corresponds to the value $\mathrm{E}\left[{\tilde{\mathrm{S}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]$, and the value $\mathsf{\sigma }\left[{\tilde{\mathrm{S}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]$ to the value ${\sigma }_x$ [8], because according to the Cochran criterion: $\mathsf{\sigma }\left[{\tilde{\mathrm{S}}}_{opt}\right],=\mathsf{\sigma }\left[{\tilde{\mathrm{S}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]$.

Then, the sequential Wald analysis procedure is carried out, which allows either acceptance of the hypothesis H₀, or the alternative H₁, or requires continuing the statistical experiment, increasing the model run counter N by one.

Figure 4 shows an illustration of the proposed procedure for obtaining options for solving the selection problem for the parameter ${\tilde{\mathrm{S}}}_{\mathrm{D}\mathrm{Z}\mathrm{A}}$.

Finally, the expert assessment of ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$ is refined. This number ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$ is fed to the input of the simulation model, the output parameters are calculated, and either hypothesis H₀ or alternative H₁ is accepted. If H₀ or H₁ are not accepted, the quantitative value of the new expert assessment is assigned to a new reference point ${\tilde{\mathrm{m}}}_{E_1}^{\mathrm{*}}$ and new areas of acceptance of the hypothesis and alternative are determined.

Thus, the mathematical model of the spread of a cloud of hazardous components released during a fire and explosion on the surface of the surrounding area is proposed to be considered as a geometric figure on a plane in the form of ellipses with axes numerically equal to the universal means $\mathrm{E}\left[{\tilde{\mathrm{a}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right],\mathrm{E}\left[{\tilde{b}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right],\mathrm{E}\left[{\tilde{\mathrm{a}}}_{pess}\right],\mathrm{E}\left[{\tilde{\mathrm{b}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]$ and the corresponding standard deviations $\mathsf{\sigma }\left[{\tilde{\mathrm{a}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right],\mathsf{\sigma }\left[{\tilde{b}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right],\mathsf{\sigma }\left[{\tilde{\mathrm{a}}}_{pess}\right],\mathsf{\sigma }\left[{\tilde{\mathrm{b}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]$.

2.2. Geometric Model of the Danger Zone of a Fire in Industrial Enterprises

The article examines the process of release of toxic substances from the convective column (a mixture of combustion products with air rising above the fire) and their spread as a result of the action of wind. To determine the geometric characteristics of the danger zone, it is necessary to construct, as indicated above, an envelope line that limits the entire set of pollution spots. A family of such spots fills a certain area called the pollution danger zone.

The left and right boundaries of the danger zone represent a spot of pollution with a radius equal to zero (Figure 4). The size of the circle $a\left(1.5,{\mathrm{t}}^{\mathrm{*}}\right)$ depends on the moment in time t^∗. The position of the center of the circle on the x₁ axis also depends on t^∗. Here, 1.5 is a recommended value of the height of the breathing layer, m.

The radius of the contamination spot becomes zero under the condition ${\mathrm{a}}_1\left(1.5,{\mathrm{t}}^{\mathrm{*}}\right)=1$(2):

$$1=\left\{exp\left[-\frac{{(1.5-x_{30})}^2}{2{\sigma }_2^2\left(u_1{t}^{*}\right)}\right]+exp\left[-\frac{{(1.5+x_{30})}^2}{2{\sigma }_2^2\left(u_1{t}^{*}\right)}\right]\right\}d\left(t^{*}\right)$$

(2)

where x₃₀ is the actual height of the breathing layer, m.

${\sigma }_2^2\left(u_1{t}^{*}\right)$—conditional standard deviation of the hazardous area along the x₂ axis, m.

The distribution function itself is the integral over the area of definition of a random variable from the density functions of the distribution multiplied by the increment of the argument. It is not a standard distribution function (such as exponent, hyperexponent, etc.), so it allowed us to estimate the density of the distribution. One as a relative value means that we cover the entire area of the ellipse of hazardous zone borders that was generated from the two continuous density functions. It aims to more accurately describe the boundaries of the hazardous zone.

The dimensions of the contamination spot are determined by the lengths of the semi-axes of the ellipses ${\mathrm{a}}_1\left(1.5,{\mathrm{t}}^{\mathrm{*}}\right)$ and ${\mathrm{a}}_2\left(1.5,{\mathrm{t}}^{\mathrm{*}}\right)$.

The dynamics of the movement of a pollution spot when its size changes are shown in Figure 4. Starting from the moment ${\mathrm{t}}_{\mathrm{H}}^{\mathrm{*}}$ to the moment ${\mathrm{t}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{*}}$, the radius of the pollution spot begins to decrease and at the moment ${\mathrm{t}}_{\mathrm{K}}^{\mathrm{*}}$ degenerates into a point. The driving force for the movement of the pollution spot is the wind speed U₁. We took into account wind speed as the average for the time of day at which the fire occurred, according to meteorologists.

Each moment in time in the interval (${\mathrm{t}}_{\mathrm{H}}^{\mathrm{*}},{\mathrm{t}}_{\mathrm{K}}^{\mathrm{*}}$) corresponds to a specific pollution spot. To determine the geometric characteristics of a danger zone, it is necessary to construct an envelope line, an isopleth, limiting the entire set of pollution spots. The envelope contour of a family of pollution spots can be constructed using both analytical equations of the envelope and numerically using computer modeling.

The boundaries of the envelope contour on the x₁ axis show the greatest length of the danger zone. At time ${\mathrm{t}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{*}}$, the width of the danger zone is maximum (Figure 5).

If a pollution spot exists (radius is greater than zero), Equation (2) can have two real positive roots ${\mathrm{t}}_{\mathrm{H}}^{\mathrm{*}}$ and ${\mathrm{t}}_{\mathrm{K}}^{\mathrm{*}}\left({\mathrm{t}}_{\mathrm{H}}^{\mathrm{*}}<{\mathrm{t}}_{\mathrm{K}}^{\mathrm{*}}\right)$. This means that the pollution spot exists for a certain interval $\left({\mathrm{t}}_{\mathrm{H}}^{\mathrm{*}},{\mathrm{t}}_{\mathrm{K}}^{\mathrm{*}}\right)$. Outside this interval, the pollution spot does not exist.

The moments of time ${\mathrm{t}}_{\mathrm{H}}^{\mathrm{*}}$ and ${\mathrm{t}}_{\mathrm{K}}^{\mathrm{*}}$ on the x₁ axis correspond to the following coordinates:

$${\mathrm{x}}_{1\mathrm{H}}^{\mathrm{*}}=\mathrm{u}1{\mathrm{t}}_{\mathrm{H}}^{\mathrm{*}},{\mathrm{x}}_{1\mathrm{K}}^{\mathrm{*}}=\mathrm{u}1{\mathrm{t}}_{\mathrm{K}}^{\mathrm{*}}$$

(3)

Linear size (length) of the hazardous pollution zone is as follows:

$$L={\mathrm{x}}_{1\mathrm{K}}^{\mathrm{*}}-{\mathrm{x}}_{1\mathrm{H}}^{\mathrm{*}}$$

(4)

We presented the condition that allows us to find the dependencies for constructing the envelope line as follows:

$$-2u1\left(x1-u1t^{*}\right)-\frac{\partial \left[a^2\left(1.5,t^{*}\right)\right]}{\partial t^{*}}=0$$

(5)

We obtained Equations (6) and (7), which together define the equations of two branches of the isopleth—the pollution spot (quadratic equation for x₂):

$$x2=\mp \sqrt{a^2\left(1.5,t^{*}\right)-\frac{1}{4u_1^2}{\left\{\frac{\partial \left[a^2\left(1.5,t^{*}\right)\right]}{\partial t^{*}}\right\}}^2}$$

(6)

The solution to Equation (7) for x₁ has the following form:

$$x_1=u_1{\mathrm{t}}^{\mathrm{*}}-\frac{1}{2u_1}·\frac{\mathrm{d}\left[{\mathrm{a}}^2\left(1.5,{\mathrm{t}}^{\mathrm{*}}\right)\right]}{\mathrm{d}{\mathrm{t}}^{\mathrm{*}}}$$

(7)

As a result, the contours of spot pollution are determined by the parametric equations of two branches of the envelope line, with the definition of x₂ and x₁, and have the character of unimodal curves. The envelope line has the following property: a tangent at any point of the envelope line is simultaneously a tangent to one of the covered curves in the form of circles (Figure 4). The horizontal tangent to the envelope line at the point of its maximum (for the upper branch of the envelope line) is simultaneously a tangent to the osculating circle having the largest radius a (1.5, t*).

Due to this, the moment in time ${\mathrm{t}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{*}}$, at which the dangerous contamination zone has a maximum width, can be found from the following conditions:

$$\frac{\partial \left[a^2\left(1.5,t^{*}\right)\right]}{\partial t^{*}}=0;\frac{{\partial }^2\left[a^2\left(1.5,t^{*}\right)\right]}{\partial t^{{*}^2}}<0$$

(8)

Using Expression (8), we can write down the necessary condition for the maximum:

$$\frac{d\left[{\sigma }_2^2\left(t^{*}\right)\right]}{d{t}^{*}}·\ln \left[d\left(1.5,t^{*}\right)\right]+{\sigma }_2^2\left(u_1{t}^{*}\right)\frac{1}{d\left(1.5,t^{*}\right)}·\frac{\mathrm{d}\left[d\left(1.5,t^{*}\right)\right]}{\mathrm{d}{t}^{*}}=0$$

(9)

The root of this equation, satisfying the inequality, i.e., sufficient condition for a maximum, is denoted as ${\mathrm{t}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{*}}$.

Thus, the maximum width of the dangerous zone of air pollution during a fire at an industrial enterprise is expressed by Formula (10):

$${\mathrm{I}}_{2\mathrm{m}\mathrm{a}\mathrm{x}}=2\left|\mathrm{a}\left(1.5,{\mathrm{t}}^{\mathrm{*}}\right)\right|$$

(10)

2.3. Algorithms for Determining Air Pollution Danger Zones during a Fire at an Industrial Enterprise

To determine the geometric dimensions of dangerous zones using computer modeling, which is a priori as accurate as possible and allows taking into account a large number of factors and input parameters, it is necessary to take into account the height of the fire zone and the emission of toxic substances. This is due to the fact that in industrial enterprises chemical substances can be stored and entered into production at different levels of workshops and other premises, and their combustion can occur at a certain height, which certainly affects the parameters of the danger zone.

The distance from the source of initial contamination with toxic substances and combustion products ${\mathrm{x}}_{1\mathrm{H}}^{\mathrm{*}}$, at which the pollutants first touch the plane $x_{3h}=1.5$ m—is determined based on the height above the ground.

If the source is terrestrial, i.e., x₃₀ = H = 0, it is accepted that ${\mathrm{x}}_{1\mathrm{H}}^{\mathrm{*}}\cong 0$. If the source is located at a certain height x₃₀ = H, then ${\mathrm{x}}_{1\mathrm{H}}^{\mathrm{*}}$, and then it is determined approximately by the algorithm shown in the form of a block diagram in Figure 6 (starting point of the hazardous pollution zone).

The coordinate ${\mathrm{x}}_{1\mathrm{K}}^{\mathrm{*}}$ (end point of the hazardous pollution zone) is determined by the algorithm shown in Figure 7. In this case, the value of ${\mathrm{x}}_{1\mathrm{K}}^{\mathrm{*}}$ determines the depth of spread of the danger zone during the fire at an industrial enterprise.

The moment in time at which, at the height of the breathing layer, the value of the concentration of toxic substances and combustion products in the air exceeding ${Ca}^{*}$ (for example, ${Ca}^{*}=\mathrm{M}\mathrm{P}\mathrm{C}$) is first recorded is calculated by Formula (11), and when it is not higher than $Ca$ it is calculated by Formula (12):

$${\mathrm{t}}_{\mathrm{H}}^{\mathrm{*}}=\frac{{\mathrm{x}}_{1,\mathrm{H}}^{\mathrm{*}}}{{\mathrm{u}}_1}$$

(11)

$${\mathrm{t}}_{\mathrm{K}}^{\mathrm{*}}=\frac{{\mathrm{x}}_{1,\mathrm{K}}^{\mathrm{*}}}{{\mathrm{u}}_1}$$

(12)

The value ${\mathrm{t}}_{\mathrm{K}}^{\mathrm{*}}$ determines the point in time when the impact of a fire on the population of nearby areas at an industrial enterprise ceases.

The maximum width of the danger contamination zone during a fire at an industrial enterprise is determined in the following sequence:

Step 1. Find the coordinate of the center of the contamination spot with the largest radius:

$${\mathrm{x}}_{1\mathrm{c}}^{\mathrm{m}\mathrm{a}\mathrm{x}}\approx {\mathrm{x}}_{1,\mathrm{H}}^{\mathrm{*}}+\raisebox{1ex}{$\mathrm{L}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$$

(13)

Step 2. Determining the moment in time when the center of the pollution spot moves to the point with coordinates ${\overline{\mathrm{x}}}_{\mathrm{c}}^{\mathrm{m}\mathrm{a}\mathrm{x}}={\left({\mathrm{x}}_{1\mathrm{c}}^{\mathrm{m}\mathrm{a}\mathrm{x}},0,1.5\right)}^{\mathrm{T}}$ is equal to the following:

$${\mathrm{t}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{*}}=\frac{{\mathrm{x}}_{1\mathrm{c}}^{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{u}}_1}$$

(14)

Step 3. Calculation of the largest width of the pollution danger zone:

$${\mathrm{I}}_{2\mathrm{m}\mathrm{a}\mathrm{x}}=2{\mathrm{y}}_{\mathrm{g}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(15)

Step 4. Determination of the area of the pollution danger zone during a fire at an industrial enterprise:

$$\mathrm{S}=2\underset{{\mathrm{t}}_{\mathrm{H}}^{\mathrm{*}}}{\overset{{\mathrm{t}}_{\mathrm{K}}^{\mathrm{*}}}{\int }}\left|{\mathrm{x}}_2\left({\mathrm{t}}^{\mathrm{*}}\right)\right|\mathrm{*}\frac{\mathrm{d}{\mathrm{x}}_1\left({\mathrm{t}}^{\mathrm{*}}\right)}{\mathrm{d}{\mathrm{t}}^{\mathrm{*}}}\mathrm{d}{\mathrm{t}}^{\mathrm{*}}$$

(16)

2.4. Plotting the Ellipses of Air Pollution Danger Zones during a Fire at Industrial Enterprises for Making Evacuation Decisions

In the Russian Federation, the decision-making process for evacuating the population from nearby houses in the event of a fire at an industrial enterprise is regulated by the relevant Government Resolution in general terms, since the formation of a hazardous zone is probabilistic in nature, and uncertainty in determining its exact parameters is taken into account using the following principles:

−

Non-reproducibility of the experiment (a fire cannot be exactly repeated in a series of experiments).

−

Expert assessments introduced into the model also introduce uncertainty.

−

The actions of the population may contain opportunism, and complete evacuation may be difficult.

Therefore, this Decree of the Government of the Russian Federation [35] establishes the following sequence for the implementation of the Evacuation Action Plan, which is created at each industrial enterprise, based on which the following decisions are made:

• Enterprise services—turn on an emergency warning signal (sound—in adjacent areas, via radio and television broadcasts).
• Enterprise services and representatives of the Ministry of Emergency Situations—a primary assessment of the area where the fire occurred, the maximum possible area of the combustion source, and the volume of flammable chemicals.
• Services of the enterprise and representatives of the Ministry of Emergency Situations—determination of the maximum possible limit of the spread of the danger zone for fire and toxic substances in the atmosphere, forecasting the excess of the maximum permissible concentration to levels dangerous to the life and health of the population and farm animals (in rural areas).
• Representatives of the Ministry of Emergency Situations—the actual decision making on the evacuation of the population from residential areas in the probable maximum hazardous pollution zone.
• Representatives of the Ministry of Emergency Situations—informing the population of evacuation orders, accompanying rescue teams, and providing transport if necessary and if possible.

Options for the calculated areas of air pollution danger zones according to Formula (16) may differ in the configuration of the ellipses ${\mathrm{S}}_{\mathrm{D}\mathrm{Z}\mathrm{A}}=\frac{\mathsf{\pi }}{4}\mathrm{a}\mathrm{b}$, that is, areas of equal size can be wider and shorter and narrower and longer. Figure 8 illustrates this statement.

When localizing the consequences of a fire at an industrial enterprise, this circumstance introduces an additional element of uncertainty when making a decision to finish the evacuation (the end of the spread of pollution spots exceeding the MPC for toxic substances and combustion products).

Therefore, the determination of optimistic and pessimistic scenarios for emissions of toxic combustion products into the atmosphere makes it possible to justify three independent options for the decision to evacuate people from residential areas located in the danger zone and adjacent to it. Given that sequential execution of the Wald test requires that ${\mathrm{a}}_1>{\mathrm{a}}_0$, we have $\mathrm{E}\left[{\tilde{\mathrm{S}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]>\mathrm{E}\left[{\tilde{\mathrm{S}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right]$. The Wald test is a statistical test used to test limitations on the parameters of statistical models estimated from sample data [36].

Since the method used in this article is a heuristic type of programming (presumptive), and many data are stochastic, the limitations for our model are related to the limitations of these methods—reliance on elementary information processes organized in a complex hierarchical structure and mutual strengthening and weakening of the action of factors that cause uncertainty and unpredictability of the behavior of the control object.

An official making a decision to evacuate the people, based on data on the most likely geometric parameters of the danger zone, can rely on the hypothesis of an optimistic scenario:

$${\mathrm{H}}_0:{\mathrm{s}}_{\mathrm{D}\mathrm{Z}\mathrm{A}}=\mathrm{E}\left[{\tilde{\mathrm{S}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right]$$

(17)

which means to reject the alternative (accept the pessimistic scenario):

$${\mathrm{H}}_1:{\mathrm{s}}_{\mathrm{D}\mathrm{Z}\mathrm{A}}=\mathrm{E}\left[{\tilde{\mathrm{S}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]$$

(18)

or accept alternative H₁, rejecting hypothesis H₀ (accept the optimistic scenario).

However, it seems appropriate to recommend to the person making the decision to consider a third option for a possible solution, namely ${\mathrm{S}}_{\mathrm{D}\mathrm{Z}\mathrm{A}}=\tilde{\mathrm{S}}$, provided that the input of the model is the value of the expert assessment of the mass of chemical substances emitting toxic pollutants during combustion that passed through the gas-smoke cloud ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$.

The person making the decision to evacuate the population must be able to quickly visually evaluate these three options. Figure 9 shows the ellipses of the areas of the hazardous pollution zone for these three options (optimistic, pessimistic, and neutral values of the area of the danger zone and the radius of its ellipse).

It should be taken into account that decision making is a procedure that cannot be fully formalized. That is why the right to make the final decision belongs to the individual. For example, in Figure 8, the area $\mathrm{E}\left[{\tilde{\mathrm{s}}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\right]$ is the smallest, but the extent of the danger zone is the greatest. One can only assume that when making a decision, the responsible person will take into account in his heuristic (informal) model other factors that are not used in the simulation model.

In relation to the situation under consideration, according to the Wald criterion, it is necessary to decide that the dangerous contamination zone should be represented by an ellipse with axes equal to the following:

$$a_{max}^{*}=\mathrm{E}\left[{\tilde{\mathrm{a}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]+3\sigma \left[{\tilde{\mathrm{a}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right],b_{max}^{*}=\mathrm{E}\left[{\tilde{\mathrm{b}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]+3\sigma \left[{\tilde{\mathrm{b}}}_{\mathrm{p}\mathrm{e}\mathrm{s}\mathrm{s}}\right]$$

(19)

The area of the ellipse, equal to the area of the danger zone—the area of pollution with a concentration of toxic substances above the MPC—will be calculated using Formula (20):

$${\overline{\mathrm{S}}}_{\mathrm{D}\mathrm{Z}\mathrm{A}}=\frac{\pi }{4}{a}_{max}^{*}·b_{max}^{*}$$

(20)

The value ${\overline{\mathrm{S}}}_{\mathrm{D}\mathrm{Z}\mathrm{A}}$ is the most reliable estimation of the contamination zone by toxic combustion products, according to the “three sigma” rule [37].

Outside this zone, the probability of injury to risk receptors is minimal for each specific fire at an industrial enterprise.

Figure 10 presents a family of ellipses—the contours of dangerous zones—for making the final decision on evacuation of the population in case of fire in industrial enterprises.

Using ${\overline{\mathrm{S}}}_{\mathrm{D}\mathrm{Z}\mathrm{A}}$, it is possible to calculate control actions to localize the consequences of fire at an industrial enterprise, which will minimize the risk of damage to public health from nearby residential areas when a cloud of toxic products spreads.

3. Results

3.1. The Final Expression of the Parameters of the Danger Zone When the Pollution Spot Moves

We accept two ways of spreading fire at an industrial enterprise, in which toxic combustion products are emitted into the atmosphere and spread:

(1)

Burning of a collapsed roof, liquid, or solid chemicals only within the boundaries of their storage facilities.

(2)

Burning of a collapsed roof together with the released product and the spread of fire over the entire area of the spilled liquid.

In both cases, the remaining part of the capacitive equipment will be engulfed in flames, which will lead to its destruction no more than 20 min after the fire occurs.

To model the process of propagation of a dangerous zone, it is advisable to consider the process for two time periods:

−

From the moment of occurrence of the initial source of emissions of toxic substances, i.e., time period 0 ≤ t ≤ td (by time t = td the fire source is eliminated).

−

From the moment of elimination of the initial source of fire t = td, when the flow of toxic substances into the atmosphere is stopped (post-emission period) and the movement of the gas-smoke cloud continues in the direction of the wind. In this case, the danger zone is “blurred”, and the concentration of toxic substances decreases.

The isopleth equation has the following form:

$${\left({\mathrm{x}}_{1\mathrm{p}}-{\mathrm{u}}_1{\mathrm{t}}^{\mathrm{*}}\right)}^2+{\mathrm{x}}_{2\mathrm{p}}^2={\mathrm{a}}^2\left(1.5,{\mathrm{t}}_{\mathrm{p}}^{\mathrm{*}}\right)$$

(21)

The roots of this quadratic equation, relative to t^*, are determined from Formula (22):

$${\mathrm{t}}_{\mathrm{1,2}}^{\mathrm{*}}=\frac{{\mathrm{x}}_{1\mathrm{p}}}{\mathrm{u}1}\mp \frac{\mathrm{a}\left(1.5,{\mathrm{t}}_{\mathrm{p}}^{\mathrm{*}}\right)}{\mathrm{u}1}\sqrt{1-\frac{{\mathrm{x}}_{2\mathrm{p}}^2}{{\mathrm{a}}^2\left(1.5,{\mathrm{t}}_{\mathrm{p}}^{\mathrm{*}}\right)}}$$

(22)

An illustration of the movement of a pollution spot in the vicinity of a given point ${\overline{\mathrm{x}}}_{\mathrm{p}}$ (times of approach of the spot of pollution to a given point and departure from it) is shown in Figure 11.

Having the values of a specific time from the beginning of the emission of toxic substances and combustion products $\left({\mathrm{t}}_1^{\mathrm{*}} \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{t}}_2^{\mathrm{*}}\right)$ into the atmosphere, it is possible to determine the average concentration in the interval $∆{\mathrm{t}}^{\mathrm{*}}$.

3.2. An Example of Calculating the Parameters of a Dangerous Zone—Conditional Case of an Explosion and Fire in Nylon Production

There are a certain number of cases of emergency situations at industrial facilities—chemical plants in which there is an explosion of stationary containers with toxic substances and fire in the storage facility, in which both steam and fine aerosol (in an explosion) and combustion products are distributed.

Examples include the explosion at the Shandong Runxing New Material plant producing nylon in 2015, where damage to adiponitrile containers resulted in the release of this toxic substance and combustion products into the atmosphere [38]. Also of note is the 2016 gas tank explosion at a nylon plant in Cantonment, Florida [39].

Combustion products of chemicals used in the production of nylon include, along with carbon monoxide (CO) and nitrogen oxides (NOx), acrolein (H2C=CH-CHO), formaldehyde (HCHO), hydrogen chloride (HCl), and hydrogen cyanide (hydrocyanic acid—HCN). These, together with the possible release of adiponitrile vapors (autoignition at 550 degrees C), are capable of forming a cloud of aerosol, smoke, and gases that can spread over a considerable distance.

We accept the following conditions for the formation of a dangerous pollution zone in the event of a fire in nylon production with the release into the atmosphere of the above gases in the form of an aerosol (adiponitrile) and combustion products:

(A)

The amount of vapor and aerosol generated is about 7–10% of the total mass of adiponitrile in the container. We assume that the remaining amount of the substance will settle on the floor of the storage facility in the form of large drops and then decompose under the influence of high temperatures during explosion and combustion (with the formation of combustion products). The resulting primary aerosol cloud will enter the atmosphere through partial destruction of the walls and roof after the explosion and can spread in the direction of the wind. The maximum height of the storage roof is 10 m. The source of adiponitrile release can be considered a point source, operating at a height of 8 m.

(B)

The period of emergency is summer; the air temperature at the time of explosion and subsequent fire was 250 °C; wind speed at the height of the weather vane is 3.6 m/s, the height of the weather vane is 10 m; time of day—diurnal; cloudless; atmospheric stability class: “convection”; the roughness of the underlying surface is assumed to be 0.3 m (heterogeneous surface with grass and shrubs).

(C)

There are 10 containers in the storage facility in which adiponitrile is stored. We assume that one container was destroyed by the explosion, and the rest of the containers are intact. Part of the generated steam will be released into the atmosphere through areas of damage to the roof and walls of the room; the resulting cloud of steam and fine aerosol will spread in the direction of the wind. Then, various combustion products will spread in the direction of the wind, resulting in a significant depletion of the initial cloud of toxic raw materials and an increase in the flow of combustion products into the atmosphere.

(D)

Let us assume that when the container explodes, the mass of pure adiponitrile, instantly transformed into steam and fine aerosol, will be 1400 kg of the substance (determined on the basis of the mathematical expectation of an expert estimate of the mass of a flammable chemical substance ${\tilde{\mathrm{m}}}_{\mathrm{E}}^{\mathrm{*}}$ in the scheme of a simulation model for obtaining random values of the parameters of the dangerous zone (Figure 2). We also assume that as a result of condensation, approximately 30% of the mass of the substance from the initial release settled in the room. Therefore, the mass of adiponitrile in the cloud, which will spread in the atmosphere before the distribution of combustion products, will be no more than 840 kg.

Figure 12 shows a dangerous pollution zone for the case under consideration—the spread of adiponitrile as a result of an explosion.

Figure 12 reflects the spread coordinates of the cloud from the explosive release and the spread of the toxic substance within the boundaries of the danger zone, beyond which the concentration drops below the MPC level. The dashed line shows the boundaries of the hazardous pollution zone.

To determine the optimistic and pessimistic options for the size of the danger zone during the spread of both an aerosol cloud of a chemical substance as a result of an explosion and combustion products, for this case we used algorithms for calculating the starting and ending points of the danger zone (Figure 5 and Figure 6), equations of two branches of the isopleth (Formulas 7 and 8), a system of equations, and a graphical representation of the movement of a pollution spot, presented in Section 2.2. “Geometric model of the danger zone during the combustion of chemicals during the fire at industrial enterprises” and Section 2.4. “Construction of ellipses of dangerous zones of air pollution during fires at industrial enterprises for making a decision on evacuation”, Figure 8 and Figure 9.

The results of simulation modeling and calculation of danger zone area, obtained using the criteria of optimistic and pessimistic scenarios, in contrast to the only one option, are presented in Figure 13.

Table 1. The table of deviations of explosion and fire parameters used to make evacuation decisions.

Scenario	Deviation, %
	By Mass of Toxic Substance Released	By Area of the Danger Zone	Axis Length of X1	Axis Length of X2	Along the Length of the Isopleth L
Optimistic	−4.7	−8.0	+15.0	−20.0	+10.0
Pessimistic	+5.1	+44.7	+23.0	+17.4	+22.4

shows the numerical values of the deviations of the parameters for calculating the danger zone during an explosion and fire at a nylon production plant, calculated for optimistic and pessimistic emergency scenarios in comparison with the neutral scenario in the considered conditional case. These deviations must be taken into account by the person making the decision to evacuate residents of nearby areas.

The numerical values of the criteria for assessing the danger zone obtained for this particular case indicate that the person making the decision to evacuate must inevitably use intuitive heuristic procedures when making the final decision on the operational management of emergencies.

3.3. Case Studies of Fires at Industrial Enterprises in 2022–2024

The authors are unknown when the press publishes data on accidents, and their exact data are closed. Therefore, this article gives a forecast model. For its complete presentation, an example is presented in Section 3.2. “An Example of Calculating the Parameters of a Dangerous Zone—Conditional Case of An Explosion and Fire in Nylon Production”, which is taken as probable for the types of fires considered in this article.

Further, in order to compare the calculated hazard zone dimensions obtained from the simulation with the actual ones, three cases were analyzed of press-reported fires at industrial enterprises, the data on which were available. The hazardous area dimensions obtained in the simulation for the optimistic and pessimistic scenarios were compared with the reported (used as actual) data in order to validate the reliability of the modeling results.

The following parameters (

Table 2. Parameters used to compare modeling results with actual data of the case—fire at an industrial enterprise.

Parameter	Symbols	Units	Equation
Initial parameters
Wind speed (averaged over the time of day during which the fire occurred)	U	m/s
Mass of burning material evaporated in the cloud	${\tilde{m}}_E^{*}$	kg
Maximum height of the storage	$x_{3h}$	m
Duration of emissions of toxic combustion products into the atmosphere	t*	s
Calculation parameters
Minimum boundary of the danger zone	$x_{1H}^{*}$	m	3
Maximum boundary of the danger zone	$x_{1K}^{*}$	m	3
Danger zone length	L	m	4
Largest width of the pollution danger zone	I2max	m	10
Coordinate of the center of the contamination spot with the largest radius	$x_{1c}^{max}$	m	13
The moment in time when the center of the pollution spot moves to the point with coordinate ${\mathrm{x}}_{1\mathrm{c}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$	$t_{max}^{*}$	s	14
Maximum lengths of the ellipse of the danger zone	$a_{max}^{*}$	m	19
Maximum width of the ellipse of the danger zone	$b_{max}^{*}$	m	19
Pessimistic area of the danger zone	Spess	m2	20
Optimistic area of the danger zone	Sopt	m2	16

) were used to compare modeled and actual (reported) dimensions of hazardous zones.

We used the data of three cases from 2022 to 2024 (Figure 14, Figure 15 and Figure 16):

Case 1. Fire in the warehouse of a plastic processing plant, May 2024, Tver, Russia [40].

Figure 14. Photo of the fire in Tver, 2024 [40].

Figure 14. Photo of the fire in Tver, 2024 [40].

The fire area was about 1000 square meters, duration of intensive burning was 3.5 h, the average wind speed at the fire site was 4.1 m/s (according to meteorological data), the total mass of burning materials (mainly polystyrene and polyvinyl chloride) was 4.8 t, the mass of harmful burning material evaporated (according to copyright ratings based on known methods [34]) was 240 kg (to the greatest extent—toluene, styrene, and dioxins (up to 90% in weight in the air) and to a lesser extent—chlorine, hydrochloride, hydrogen cyanide, sulfur, and nitrogen derivatives (up to 10%)). The height of the fire hearth is 1.5 m.

Case 2. Fire at a plastic plant, Moscow region, Russia, December 2022 [41].

Figure 15. Photo of the fire in Moscow region, 2022 [41].

Figure 15. Photo of the fire in Moscow region, 2022 [41].

The fire area was about 650 square meters, duration of intensive burning was 2.5 h, average wind speed at the fire site was 2.4 m/s, total mass of burning materials (mainly polystyrene and polyvinyl chloride) was 6.5 t, and mass of harmful burning materials evaporated (according to author’s estimates) was 290 kg (toluene, styrene, and dioxins (up to 90% by mass in the air)) and to a lesser extent chlorine, hydrogen chloride, hydrogen cyanide, sulfur, and nitrogen derivatives (up to 10%). The height of the fire hearth was 5 m.

Case 3. Fire at a factory producing polyethylene products, St. Petersburg, Russia, May 2024 [42].

Figure 16. Photo of the fire in St. Petersburg, 2024 [42].

Figure 16. Photo of the fire in St. Petersburg, 2024 [42].

The fire area was about 1200 square meters, duration of intensive burning was 3.2 h, average wind speed in the fire area was 3.6 m/s, total mass of burning materials (polyethylene) was 19 tons, and mass of harmful burning material evaporated (according to author’s estimates) was 140 kg (formaldehyde, benzpyrene). The height of the fire hearth was 3 m.

The actual (reported) data characterizing the cases of these fires and used for validation of the proposed model are presented in

Table 3. Initial data of the cases.

Case	U, m/s	${\tilde{\mathit{m}}}_{\mathit{E}}^{\mathit{*}}$, kg	${\mathit{x}}_{3\mathit{h}}$, m	t*, s
Case 1 (Tver, 2024 [40])	4.1	240	1.5	12,600
Case 2 (Moscow Region, 2022 [41])	2.4	290	5	9000
Case 3 (Saint Petersburg, 2024 [42])	3.6	140	3	11,520

.

The calculations of hazardous area parameters obtained from the modeling and deviations from actual data obtained from reports are presented in

Table 4. Case 1—Tver, 2024.

Parameters	Symbols	Calculated Value	Actual (Reported) Value	Excess of Calculated Parameters over Actual Values, %
Minimum boundary of the danger zone	$x_{1H}^{*},$ m	4162	3600	13.5
Maximum boundary of the danger zone	$x_{1K}^{*},$ m	1334	1100	17.5
Danger zone length	L, m	2831	2300	18.8
Largest width of the pollution danger zone	I2max, m	343	300	12.5
Coordinate of the center of the contamination spot with the largest radius	$x_{1c}^{max},$ m	1397	1250	10.5
The moment in time when the center of the pollution spot moves to the point with coordinate ${\mathrm{x}}_{1\mathrm{c}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$	$t_{max}^{*},$ s	5543	6600	−19.1
Maximum lengths of the ellipse of the danger zone	$a_{max}^{*},$ m	5246	4400	16.1
Maximum width of the ellipse of the danger zone	$b_{max}^{*},$ m	416	360	13.5
Pessimistic area of the danger zone	Spess, m2	1,713,133	1,243,000 (reported)	27.4
Optimistic area of the danger zone	Sopt, m2	1,317,795	1,243,000 (reported)	5.7

, Table 5 and Table 6.

As can be seen from the comparison of the estimated hazardous area parameters obtained from the Case 1 modeling, in all linear parameters they exceed the reported data (which we accept as actual) by 10.5–17.5%, and the maximum estimated hazardous area propagation time is 19.1% longer than the reported one. The estimated hazardous area values are also larger than reported—for the pessimistic scenario by 27.4% and for the optimistic one by 5.7%. Consequently, these data are quite valid for evacuation decision making.

Table 5. Case 2—Moscow Region, 2022.

Parameters	Symbols	Calculated Value	Actual (Reported) Value	Excess of Calculated Parameters over Actual Values, %
Minimum boundary of the danger zone	$x_{1H}^{*}$, m	6193	5400	12.8
Maximum boundary of the danger zone	$x_{1K}^{*},$ m	1972	1600	18.9
Danger zone length	L, m	4224	3500	17.1
Largest width of the pollution danger zone	I2max, m	4918	4400	10.5
Coordinate of the center of the contamination spot with the largest radius	$x_{1c}^{max}$, m	1965	1700	13.5
The moment in time when the center of the pollution spot moves to the point with coordinate ${\mathrm{x}}_{1\mathrm{c}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$	$t_{max}^{*}$, s	3422	3800	−11.0
Maximum lengths of the ellipse of the danger zone	$a_{max}^{*}$, m	6921	5800	16.2
Maximum width of the ellipse of the danger zone	$b_{max}^{*}$, m	543	470	13.4
Pessimistic area of the danger zone	Spess, m2	2,618,320	1,940,000 (reported)	25.9
Optimistic area of the danger zone	Sopt, m2	2,014,092	1,940,000 (reported)	3.7

Table 5. Case 2—Moscow Region, 2022.

Parameters	Symbols	Calculated Value	Actual (Reported) Value	Excess of Calculated Parameters over Actual Values, %
Minimum boundary of the danger zone	$x_{1H}^{*}$, m	6193	5400	12.8
Maximum boundary of the danger zone	$x_{1K}^{*},$ m	1972	1600	18.9
Danger zone length	L, m	4224	3500	17.1
Largest width of the pollution danger zone	I2max, m	4918	4400	10.5
Coordinate of the center of the contamination spot with the largest radius	$x_{1c}^{max}$, m	1965	1700	13.5
The moment in time when the center of the pollution spot moves to the point with coordinate ${\mathrm{x}}_{1\mathrm{c}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$	$t_{max}^{*}$, s	3422	3800	−11.0
Maximum lengths of the ellipse of the danger zone	$a_{max}^{*}$, m	6921	5800	16.2
Maximum width of the ellipse of the danger zone	$b_{max}^{*}$, m	543	470	13.4
Pessimistic area of the danger zone	Spess, m2	2,618,320	1,940,000 (reported)	25.9
Optimistic area of the danger zone	Sopt, m2	2,014,092	1,940,000 (reported)	3.7

For Case 2, the modeled linear parameters of the hazardous area are also 12.8–18.9% larger than the reported ones, and even its optimistic area is 3.7% larger than the actual one. The estimated propagation time of the hazardous area is 11.0% less than the actual (reported), which gives additional time for evacuation planning and implementation.

Table 6. Case 3—Saint Petersburg, 2024.

Parameters	Symbols	Calculated Value	Actual (Reported) Value	Excess of Calculated Parameters over Actual Values, %
Minimum boundary of the danger zone	$x_{1H}^{*},$ m	3311	2800	15.4
Maximum boundary of the danger zone	$x_{1K}^{*},$ m	1055	850	19.4
Danger zone length	L, m	2263	1900	16.0
Largest width of the pollution danger zone	I2max, m	296	250	15.5
Coordinate of the center of the contamination spot with the largest radius	$x_{1c}^{max},$ m	1122	1000	10.9
The moment in time when the center of the pollution spot moves to the point with coordinate ${\mathrm{x}}_{1\mathrm{c}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$	$t_{max}^{*}$, s	4283	4800	−12.1
Maximum lengths of the ellipse of the danger zone	$a_{max}^{*}$, m	4235	3500	17.4
Maximum width of the ellipse of the danger zone	$b_{max}^{*}$, m	365	300	17.8
Pessimistic area of the danger zone	Spess, m2	1,210,108	940,000 (reported)	22.3
Optimistic area of the danger zone	Sopt, m2	980,853	940,000 (reported)	1.1

Table 6. Case 3—Saint Petersburg, 2024.

Parameters	Symbols	Calculated Value	Actual (Reported) Value	Excess of Calculated Parameters over Actual Values, %
Minimum boundary of the danger zone	$x_{1H}^{*},$ m	3311	2800	15.4
Maximum boundary of the danger zone	$x_{1K}^{*},$ m	1055	850	19.4
Danger zone length	L, m	2263	1900	16.0
Largest width of the pollution danger zone	I2max, m	296	250	15.5
Coordinate of the center of the contamination spot with the largest radius	$x_{1c}^{max},$ m	1122	1000	10.9
The moment in time when the center of the pollution spot moves to the point with coordinate ${\mathrm{x}}_{1\mathrm{c}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$	$t_{max}^{*}$, s	4283	4800	−12.1
Maximum lengths of the ellipse of the danger zone	$a_{max}^{*}$, m	4235	3500	17.4
Maximum width of the ellipse of the danger zone	$b_{max}^{*}$, m	365	300	17.8
Pessimistic area of the danger zone	Spess, m2	1,210,108	940,000 (reported)	22.3
Optimistic area of the danger zone	Sopt, m2	980,853	940,000 (reported)	1.1

The hazard zone size modeling data for Case 3 confirm the validity of the model by exceeding the reported linear data by 10.9–17.8%, by 1.1% in area in the optimistic case (22.3% in the pessimistic case); the estimated hazard zone propagation time is 12.1% less than the actual.

4. Discussion

The approach presented in this article to determine the parameters of dangerous zones during a fire at an industrial facility (which may be preceded by the explosion of containers with toxic substances) is based on simulation modeling of the spread of pollutants. The presented mathematical model and method of geometric determination of the linear dimensions and area of dangerous zones allows their calculations within the framework of optimistic and pessimistic scenarios, which, when compared with the neutral scenario, provide justification for a quick decision by the responsible person to evacuate residents from a certain territory. This is the main advantage of the proposed simulation model for calculating the danger zone of contamination with toxic chemicals and combustion products, along with the use of differential equations of two branches of the isopleth for constructing ellipses—the trajectories of movement of the pollution spot.

Reliance on expert assessments of a toxic substance released during a container explosion, followed by the combustion of various chemicals at a nylon producing plant (in the given case), showed that in the algorithm for calculating the area of a hazardous pollution zone there are random factors (speed wind, roughness of the underlying surface, air temperature, etc.). Therefore, the developed method for correcting the parameters of the danger zone when moving in the decision-making procedure from an optimistic to a pessimistic scenario should be applied for each specific emergency.

The limitations of the application of the proposed simulation model are, first, the need to perform complex iterative calculations to accurately determine the linear parameters and area of pollution danger zones and construct ellipses of their boundaries, as well as the need to use expert estimates of the mass of toxic substances entering the atmosphere during an explosion. The need to create a library with the maximum possible number of dangerous zone templates for various parameters and scenarios for the deployment of emergencies with an explosion and fire at a specific industrial facility will reduce the time for the responsible person to make a decision on evacuation.

Overcoming these limitations is associated, first of all, with the development of specialized software for simulating danger zones in real time and creating libraries of their templates, the processing of which can be quickly and accurately carried out by artificial intelligence [43]. This is the subject of subsequent research by the team of authors.

5. Conclusions

To conduct simulation studies of the formation of dangerous zones during fires, which are often preceded by explosions of containers with toxic substances, at industrial enterprises located in urban areas, the authors developed a mathematical model and proposed a geometric method for calculating linear parameters and the area of distribution of spots pollution. The development of the model required a multi-scenario approach, which takes into account both expert assessments of the release of toxic substances into the atmosphere during an explosion and calculations of the parameters of danger areas. Simulation modeling of the spread of a dangerous zone during fire at an industrial enterprise and the application of the sequential Wald analysis procedure, in the opinion of the authors, will allow the person making the decision to manage the elimination of the fatal consequences of an accident to obtain additional comprehensive information on the optimistic and pessimistic scenarios for the formation of this zone.

The main methodologically new result of this study is the method proposed by the authors for calculating the linear parameters of dangerous zones, based on elliptical forecast boundaries determined using the radii of inscribed circles—from the initial to the final linear boundary. In turn, algorithms were proposed to calculate the initial and final values of dangerous zones, which also constitute the novelty of the research.

Testing the simulation model using the example of calculating a conditional case based on real experience of explosions and fires at enterprises producing nylon confirmed its correctness as a working model. Thus, it can be used to conduct simulation studies. During the creation of the simulation model, it was established that, depending on such random variables as wind speed, roughness of the underlying surface, height of the source of explosion of containers with toxic substances, and fire, the linear parameters of the danger zone could vary, while its area can remain constant. This predetermined the need to construct elliptical boundaries of the pollution danger zone, relying on a system of isopleth differential equations and determining the moments in time at which the concentration of toxic substances will exceed the maximum permissible concentration (MPC). The difficulty and complexity of such calculations will undoubtedly require the next step of research to develop software for calculating the parameters of dangerous zones in real time for quickly making evacuation decisions, including mobile applications that can be adapted for decision making on emergency management using artificial intelligence.

A comparative analysis of the data obtained from hazardous area modeling with reported data on three specific cases of industrial fires with emission of toxic substances into the atmosphere allowed us to confirm the validity of the model. The calculated linear dimensions of the hazardous area were 10–20% larger than the actual (as stated in the reports that were available), and the dimensions of the hazardous area obtained from the modeling were 22–27% larger than the actual for the pessimistic scenario and 1–5% larger for the pessimistic scenario. This implies a high validity of the model, allowing a more conservative estimate of the size of the hazardous area.