A three-dimensional mathematical model of gaseous hydrogen combustion in a confined space has been developed. The process is described by a system of diferential equations of gas dynamics. The thermodynamic parameters of the mixture and its components are determined as functions of the local temperature and composition of the mixture. Changes in the concentration of fuel and combustion products are determined using conservation laws. The calculation takes into account the rates of decay and formation of components, as well as turbulent difusion. It is assumed that the chemical reaction occurs only in the volume where the fuel concentration is within the flammability limits. The mathematical model has been tested in comparative tests to predict the combustion of a large-scale hydrogen-air mixture in the open atmosphere. A numerical solution algorithm based on the Godunov method has been developed. A software tool has been developed for the engineering analysis of gas-dynamic processes of hydrogen-air mixture formation and combustion in a closed space with natural ventilation. It enables the prediction of excess pressure, temperature, hydrogen and combustion product concentrations, and other thermo-gas-dynamic parameters of the mixture. This prediction can be used to assess hazardous areas of destruction and recommend safety measures.
Assessing the fire safety level of industrial premises is an extremely important task with significant practical significance [ 1, 2, 3, 4]. A comprehensive investigation of combustion dynamics through mathematical and computational modeling (computational fluid dynamics methods) represents one of the most promising directions of research in this field [5, 6, 7, 8].
A three-dimensional mathematical model of gaseous hydrogen combustion in air within a closed space with natural ventilation is presented, with explicit consideration of combustion product formation.The results of solving some problems obtained using the developed Expert-2 software are provided. First, the results of a comparative experiment on predicting the combustion of a large-scale cloud of hydrogenair mixture in the open atmosphere, the formation of a spherical shock wave, and the dynamics of temperature and concentration changes are given. Then, a scenario of instantaneous release of highly compressed hydrogen, followed by the formation and combustion of a hydrogen-air mixture in a naturally ventilated industrial facility, is modeled. Calculations were performed to assess the efect of the combustible load on deflagration indicators (total heat release, average excess pressure, and temperature) in a production facility under various scenarios of deflagration occurrence and development. The peculiarities of hydrogen spread and combustion in air were identified. They are attributed to its low density, high difusion coeficient, rapid chemical reaction with oxygen, wide ignition concentration limits, and high flame propagation rate.
The most accurate modeling of the physical processes of mixing reactive gases with air [9, 10, 11], their combustion (taking into account the chemical interaction of the mixture components) [11, 12], and the subsequent dispersion of the mixture in open [13, 14, 15] or closed [12, 16, 17] spaces with ventilation is only possible using the system of unsteady Navier-Stokes equations for compressible gas [18, 19, 20]. Efective numerical modeling of turbulent flows is carried out by solving the Reynolds-averaged NavierStokes equations, supplemented by a turbulence model [21, 22]. However, most turbulence models do not describe diferent types of flows with the same degree of adequacy. This is especially true for flows with intense separation and/or large pressure and temperature gradients [23, 24, 25]. Therefore, there is a need to develop innovative mathematical models and computational algorithms for the numerical simulation of such flows.
The main objective of this study is to develop a simplified mathematical model that adequately describes the unsteady processes of mixing reactive gases with air, their combustion, and the subsequent dispersion of the mixture in open or closed spaces with forced or natural ventilation. In addition, the study aims to create an efective numerical solution algorithm and implement it in a computer system that can serve as a tool for modeling these complex gas-dynamic processes.
As a result of the structural analysis of the described flow and the decomposition of the complete gas-dynamic mathematical model, it was assumed that the main influence on the process is exerted by convective exchange of mass, momentum, and energy. Thus, to simulate the mixing processes of a three-component gas, it is suficient to use the reduced Navier-Stokes equations (Euler approach using source terms). However, the interaction between turbulence and combustion is a powerful feedback mechanism that ensures the exchange of component concentrations, and this phenomenon can be taken into account using source terms in the conservation laws of the mixture components.
The calculation space Ω is a parallelepiped located in the right coordinate system (Figure 1) and divided into spatial cells, the scale of which depends on the characteristic dimensions of the calculation space (roughness of the surfaces being flowed around, dimensions of the objects being flowed around).
The complete system of non-stationary equations describing the three-dimensional flow of a
threecomponent gas mixture is as follows [26, 27, 28, 29, 30, 31, 32, 33, 34, 35] :
⃗
+
⃗
+
⃗
+
⃗
= ⃗ ,
(
⃗ = [, · , · , · , ]⊤, ⃗ = [︀ · , + · 2, · · , · · , ( + )]︀ ⊤, ⃗ = [︀ · , · · , + · 2, · · , ( + )]︀ ⊤, ⃗ = [︀ · , · · , · · , + 2, ( + )]︀ ⊤, ⃗ = [0, 0, − , 0, − · + / ]⊤, = ( + 1 (2 + 2 + 2)) 2 where is time, , , are components of the velocity vector ⃗, is pressure, and density, is the total energy per unit volume of the gas mixture, is internal energy per unit mass of gas; components of vector ⃗ are projections of distributed volume sources; is acceleration due to gravity; is heat release intensity per unit volume of gas caused by a chemical reaction.
The conservation laws for the components of the mixture (combustible gas, air, and combustion products), taking into account difusion and chemical reaction rates, are as follows [36, 37, 38]: ( · ) + ( · · ) + ( · · ) + ( · · ) = + , where is the relative mass density of the impurity (the ratio of the density of the gaseous impurity to the density of the mixture); = 1, 3, where = 1 corresponds to fuel, = 2 corresponds to air, = 3 corresponds to combustion products; is the rate of change in impurity density as a result of turbulent difusion (according to Fick’s law, = ( · grad), and the difusion coeficient is determined according to Berland [39]; is intensity of change in the density of mixture components caused by a chemical reaction.
The system of equations (
=
1 ∑3︀ =1 ; = ∑3︁ ( ); = ∑3︁ (); ∑3︁ = 1, = =1 =1 =1 .
For an ideal polytropic gas, the value of is related to the values of and of the mixture by the following dependence: = (− 1) .
The rate of change in fuel density as a result of a chemical reaction is determined as the product of the molecular mass of the fuel 1 and its molar rate of change 1. It is believed that the chemical "gross" reaction leads to:
2
∑︁ → 3 3,
=1
where is a stoichiometric coeficient, is a chemical agent ( = 1, 3). The molar rate of change 1 is
defined as:
where is the universal gas constant, 1, 1, 1, ′1, ′2 are parameters obtained based on the
generalization of experimental data [40, 41, 42], [ ] is the molar concentration of the -th component of
the mixture: [ ] = . The Arrhenius reaction constant 1 was adjusted for use with large control
(
= − · 1 · 1,
The heat release intensity per unit volume of gas caused by a chemical reaction is defined as: is equal to 1 ≤ components is set. where is the combustion completeness coeficient, 1 is the lower heat of combustion of fuel.
It is assumed that any component of the air flow velocity is subsonic. The oncoming flow is determined by the values of the total enthalpy 0* = − 1
+ 12 (2 + 2 + 2), the entropy function 0 = , the flow velocity vector (angles , , ) and relative mass concentration of impurity 1 (for gas impurity inflow, 1 ≤
1). The flow parameters at the inlet are determined using the "left" Riemann invariant relation [43]. On the impermeable surfaces of the calculation cells, the "no leakage" condition is satisfied: = 0 (wher→e− is the normal to the cell surface under consideration). The boundary conditions at the outlet of the mixture from the computational domain are specified on the surfaces of the computational cells using the "right" Riemann invariant relation [44].
At the initial moment in time, the parameters of the environment are set in all "gas" cells of the calculation space. In cells occupied by a cloud of impurities, the relative mass concentration of impurities 1. In cells with gas evaporation, the law of change in the flow rate of mixture
Vector equation (
· · , ∫︁ ∫︁ ∫︁ ∫︁ ∫︁ ∫︁ external normal ⃗ ⃗( are vectors ⃗, ⃗, ⃗, respectively. for each computational cell: where is the volume of the elementary calculation cell⃗; is the boundary surface of the cell with the = ⃗ ); ^ is the flux density tensor of conservative variables ⃗, whose columns The conservation law for each component of the mixture (8) can also be represented in integral form ∫︁ ∫︁ ∫︁ ∫︁∫︁ · · + ○ · · · = ( + ) .
The numerical solution of the fundamental equations of gas dynamics (9), (10) was obtained using the Godunov method [44]. The explicit Godunov method is used to solve Euler’s equations, supplemented by the conservation law for the gas mixture concentration in integral form. A first-order finite diference scheme is used to approximate the Euler equations. Second-order central diferences are used for the difusion source term of the gas mixture concentration conservation law. Simple vertical pressure interpolation is applied. The Godunov method has a robust algorithm that is resistant to large-scale disturbances in flow parameters and allows flow parameters to be obtained when modeling the combustion of large-scale gas mixtures. It is assumed that deflagration can occur in the calculation 1 min and 1 max are set based on a generalization of experimental data [45, 46]. cells where the fuel concentration is within the flammability limits 1 min ≤ 1 ≤ 1 max. The values
Based on the mathematical model, a software has been developed for engineering analysis of gas mixture formation, combustion, and dispersion in the atmosphere. It is used in the Expert-2 research software complex of the Rizikon Risk Research Center (Severodonetsk, Ukraine) and the National Aerospace University "Kharkiv Aviation Institute" (Kharkiv, Ukraine). The software allows various hazardous scenarios to be modeled using standalone computers within a reasonable time frame. (9) (10)
The instantaneous release of combustible gas under high pressure into the atmosphere followed by ignition can generate high-temperature zones and pressure waves. These efects may cause injuries to personnel and damage to vital facilities [47]. Temperature and excess pressure values are typically used to assess pressure and thermal loads on building surfaces [48, 49].
The accepted simplified combustion model is suitable for describing the processes under consideration in both open and closed spaces. Turbulent exchange is taken into account in the laws of conservation of mass of the mixture components (8), (10) in the form of source terms. Experimental data have validated the mathematical model. The deflagration of a hemispherical cloud of a homogeneous stoichiometric hydrogen-air mixture (experiments at the Fraunhofer ICT [50, 51, 52]) is modeled under the following conditions: total cloud volume – 2094m3; initial pressure – 98.9 kPa; initial temperature – 283 K; radius of the hemispherical cloud – 10 m. During the calculation, the dynamics of temperature, fuel and combustion product concentration, and pressure are investigated at distances of 5 m (control point B in Figure 2), 8 m (control point C), and 18 m (control point D) from the epicenter of the deflagration (point A).
The calculation space has the following dimensions: length 200 m; width 100 m; height 30 m. The calculation grid has dimensions of 200 × 100 × 30 cells.
Computer specifications: 1 PC with Intel ® Celeron® processor (2.4 GHz), 0.75 GB RAM, Windows XP. Processor operating time is 4 hours.
The validation of the mathematical model of hydrogen deflagration against experimental results of measuring excess pressure in the shock wave front was performed by evaluating statistical indicators recommended in the works [53] (Fractional Bias ( ), Geometric Mean Variance ( ), Geometric Mean Bias ( ), Normalized Mean Square Error ( ), Factor of 2 ( 2), and Correlation Coeficient ()). Only the initial phase of wavefront compression was studied, as its characteristics are used for the probabilistic assessment of the negative impact of the wave on the environment. For the wave at control point B, the following model performance indicators are obtained: = 0.23, = 1.88, = 1.56, = 0.29, 2 = 71%, and = 0.78. At control point C, these indicators have the following values: = − 0.18; = 2.97; = 1.07; = 1.51; 2 = 33%; = 0.01. For point C, they are as follows: = − 0.43; = 4.00; = 2.39; = 0.57; 2 = 42%; = 0.63. The values of the statistical indicators , , and 2 for point B, the values of , , and 2 for point C, and the values of , , , and 2 for point D slightly exceed the limits recommended in the work [53], which is explained by some desynchronization in the time of passage of the shock wave through the control points compared to the real process.
However, the overall qualitative and quantitative analysis of the compression phase of the shock wave at the control points, taking into account the parameters of the wave’s impact on the environment, gives significantly better results: for the maximum excess pressure, we obtained the following indicators’ values: = − 0.02; = 1.04; = 0.95; = 0.04; 2 = 100%; = 0.72.
For the compression phase impulse, = 0.14; = 1.10; = 1.17; = 0.07; 2 = 100%; = 0.73. All relevant model performance indicators are within the recommended limits. Therefore, it can be assumed that the mathematical model of deflagration combustion of a gas-air mixture can be used for the purposes set in the study: to simulate large-scale explosions of a gas-air mixture in the surface layer of the atmosphere; to assess and predict the possible consequences of the baric and thermal efects of combustion products on people and building structures in the epicenter zone of a man-made accident.
The dynamics of excess pressure at control points B and D are shown in Figures 3 and 4 in comparison with experimental data and calculation results obtained using other software [51]. The curves at points B and C are similar. The peculiarities of the combustion model used explain the steeper slope of the calculated curve. The more intense decrease in excess pressure as the shock wave propagates from point C to point D can be attributed to the first-order scheme of the Godunov method.
The temperature dynamics at control points B and D are shown in Figure 5. The flame temperature of the stoichiometric hydrogen-air mixture (Figure 5a) is slightly higher than usual in the experiment, which can be explained by the peculiarities of the model.
In general, the calculation results agree quite well with the experimental data. This allows the developed mathematical model and software to be used to simulate large-scale combustion of a hydrogenair mixture in the atmosphere and to predict the barometric and thermal consequences of hydrogen combustion.
The layout of the enclosed production facility (where hydrogen combustion takes place) is shown in Figure 6. The load-bearing structures of the building are brick walls with a thickness of 0.25 m. The height of the room is 3 m. The room consists of two production subspaces, partially separated by an internal wall 0.25 m thick. The floor thickness is 0.25 m. The room is equipped with two windows (1.5m2 each) and one doorway (2.5m2). There are two entrances (4× 2.5m2 each) for hydrogen-powered vehicles. A high-pressure hydrogen cylinder is stored at point A. The production room is located in a three-dimensional space with environmental conditions.
Factors such as the location of the deflagration, the position and condition (open or closed) of windows and doorways, and the power and nature of hydrogen release can significantly afect the dynamics of deflagration in a production facility. To determine their impact on the dynamics of deflagration in a production facility, two possible scenarios are modeled with diferent parameters of the hydrogen cloud (in particular, the mass concentration of hydrogen) (Figure 7).
It is assumed that hydrogen is released as a result of the destruction (or leakage) of a refuelling cylinder stored in the production facility. The production building’s doors, windows, and airlocks are open during the deflagration. In the first case, the leak from the cylinder leads to the formation of a stoichiometric hydrogen-air cloud (with a radius of 2 m) with the parameters of the surrounding atmosphere (scenario 1). In the second case, the instantaneous destruction of a refuelling set of 12 cylinders (each with a volume of 0.51m3) creates a hydrogen-air cloud (with a radius of 2 m) with an ambient temperature, a pressure of 134.213 kPa, and a mass concentration of hydrogen 1 = 0.111 (scenario 2). It is assumed that for both scenarios, the center of the cloud (and the point of ignition of the flame) is located at point A (Figure 6). The dimensions of the building are 10.5 × 3.25 × 12.75m3. The dimensions of the calculation cells are 0.25 × 0.25 × 0.25m3. The dimensions of the calculation area are 22.5 × 4.25 × 15.75m3. The height of all control points B, C, D, and the deflagration ignition point A (center of the cloud) is 1.125 m.
To analyze the dynamics of hydrogen deflagration in both scenarios, the following calculated data are considered at control points B, C, and D (Figure 6): dynamics of changes in hydrogen mass concentration over time; dynamics of changes in combustion product mass concentration over time; dynamics of excess pressure and temperature over time.
The distribution of the mass concentration of hydrogen and combustion products in the working area during hydrogen deflagration ( 0.15, 0.25, and 0.35 s after ignition) for scenario 1 is shown in Figure 7. The pocket of unburned hydrogen (Figure 7b, Figure 7c) can be explained by the influence of the wall. The dynamics of changes in excess pressure and temperature at control point B for scenario 1 are shown in Figure 8. The excess pressure at point C behaves similarly, and the temperature changes are insignificant. The flame temperature of the hydrogen-air mixture (Figure 8b) is again slightly higher than usual in the experiment, which can be explained by the peculiarities of the model. The dynamics of the mass concentration distribution of hydrogen and combustion products in the working zone during hydrogen deflagration ( 0.10, 0.15, and 0.20 s after the start of deflagration) for scenario 2 is shown in Figure 9. The change in excess pressure and temperature over time at control point B for scenario 2 is shown in Figure 10 (the excess pressure at points C and D behaves similarly to point B, and the temperature changes insignificantly, as in scenario 1).
A comparative analysis of the results of calculations for both scenarios of hydrogen deflagration in a closed zone shows that the control flow parameters difer significantly. In particular, the maximum concentration of combustion products in scenario 2 (Figure 9) is greater than in scenario 1 (Figure 7). The maximum values of excess pressure and temperature in scenario 2 are significantly higher than in scenario 1. These diferences are due to the greater mass of hydrogen fuel involved in combustion and the more intense dispersion of the mixture in scenario 2.
The identified features of hydrogen dispersion and combustion in air during deflagration development are due to the low density of hydrogen, high difusion coeficient, high chemical reaction rate with oxygen, wide range of ignition concentration limits, and high flame propagation rate.
To prevent or minimize the efects of pressure and temperature caused by hydrogen deflagration in a closed production space, some measures are recommended (in the event of high hydrogen concentrations being recorded by sensors): water dispersion (reduces the temperature of the mixture and, as a result, the chemical reaction rate); injection of neutral gas (carbon dioxide) or chemically active additives (increases the minimum ignition concentration limit and, as a result, reduces the temperature of the mixture).
The results of this study can be used in real-time safety management with the help of artificial intelligence tools, namely reinforcement learning (RL) can be applied to ventilation control. For instance, an RL agent trained on the "digital training ground" of our model can make decisions about opening flaps/extraction speed/water spray, while minimizing max and the area of hazardous zones. This directly links RL to the recommendations for reducing concentrations and heat release proposed in the article, thus enabling the improvement of the efect of implementing these recommendations in real-world situations.
A three-dimensional mathematical model of gaseous hydrogen combustion in a confined space has been developed. An algorithm for the numerical solution of the system of basic diferential equations of gas dynamics based on the Godunov method has been developed. A software has been created for the engineering analysis of gas dynamics processes involved in the formation and combustion of hydrogen-air mixtures in a closed space with natural ventilation, allowing the prediction of changes in excess pressure, temperature, hydrogen and combustion product concentrations, as well as other thermo-gas dynamic parameters of the mixture in the space. This prediction can be used to assess hazardous areas of destruction and recommend safety measures.
The results of modeling hydrogen deflagration under various initial parameters of a hydrogen-air cloud released in a closed industrial facility are considered. The features of hydrogen propagation and combustion in air during deflagration development are identified due to the low density of hydrogen, high difusion constant, high chemical reaction rate with oxygen, wide range of ignition concentration limits, and high flame propagation rate. The results of the calculations show that the scale of the deflagration depends on the concentration of hydrogen in the released mixture. Thus, all proposed safety measures that minimize hydrogen concentration can prevent or reduce the barometric and thermal efects caused by hydrogen deflagration in a confined space.
All input decks, meshes, and configuration files used in this study are publicly available at [54].
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