#### Abstract

Numerical research of interconnected heat and mass transfer processes in the “two hot particles—polymeric material—air” system was executed. The joint effect of several local heat sources on the main integrated characteristic of ignition process (ignition delay time) was established. Two ignition models characterized by the relative positioning of hot particles on a polymeric material surface were revealed. Besides, there were established characteristics of local heat sources and the distance between them (700 K and or K and when regularities of heat and mass transfer processes in the “two hot particles—polymeric material—air” system are similar to regularities of heat and mass transfer processes in the “single hot particle—polymeric material—air” system.

#### 1. Introduction

In recent years, polymeric materials (polymethyl methacrylate, polystyrene, polyethylene, etc.) have been widely adopted in various industries as decorative and constructive elements. Polymeric material products are very susceptible to thermal effects [1–4] even at rather low outside temperature . Under conditions of some technological processes (at increased ambient temperature) strength characteristics are changed, melting occurs, dangerous carcinogens and are emitted. Possible temperatures of technological processes for power production can reach more than 1000 K. Under such conditions, the probability of local power sources (metal and nonmetallic particles warmed to high temperature with sizes about several millimeters) formation is high [5–9].

The numerical research results [10, 11] were obtained for thermal conduction and thermal convection processes during a polymeric material ignition by a single metal particle heated to high temperature. Established theoretical consequences can be used for developing guidelines and methods to reduce the flammability, ignition preventing, and subsequent stages of polymeric material combustion processes. However, in practice, several (two, three, etc.) small-size particles heated to high temperature can cause fires. Ignition conditions and heat transfer characteristics may be different for the “single hot particle—polymeric material—air” system and the “two hot particles—polymeric material—air” system. For example, it is known that ignition delay time of solid condensed substance (composite propellant) at high concentration (large number per unit area surfaces) of hot particles in the gas stream equals the values of ignition delay time at condensed substance heating by a massive plate with constant (during ignition period) high temperature [5, 6]. Therefore, more detailed information about characteristics of physical and chemical processes at polymeric material heating by several hot particles is necessary for the development of relevant precautionary activities.

The purpose of the present study was to develop the mathematical model and analyze the characteristics of interconnected heat and mass transfer processes during interaction of two small-size metal particles heated to high temperatures with a polymeric material at the accounting of thermal conduction in condensed substance and thermal decomposition of polymeric material, thermal convection, and diffusion and oxidation reaction in outside gas area.

#### 2. Problem Statement

It was determined [12] that ignition conditions for liquid condensed substances are defined by the distance between two neighboring particles at the various quantities of hot particles (local heat sources) falling to the flammable material surface. Therefore, previous study results [12] were taken into account at the problem statement.

A scheme with two steel particles heated to high temperature in the parallelepiped shape (with the same sizes) situated on the surface of typical polymeric material, polymethyl methacrylate (PMMA), was chosen to simulate the conditions of hot particle flow interaction with polymeric material. We reasonably [12] used the “two hot particles—polymeric material—air” system (Figure 1) instead of the “several hot particles—polymeric material—air” system for heat and mass transfer process investigation. Areas with sizes and much larger than the sizes of hot particles and were allocated in polymeric material and air (Figure 1).

The rate of PMMA thermal decomposition process accelerated at polymeric material near-surface layer heating by the heat of hot particles. Gaseous products of polymeric material pyrolysis mixed with an oxidizer (air) at the diffusion in outside gas area. The gas mixture was warmed by the thermal convection at its movement along lateral sides (, , , , and of heat sources (Figure 1). The ignition occurred when the temperature and concentration of combustible component (PMMA gas) reached the critical values.

The assumptions for the problem statement of heat and mass transfer process include the following.(1)A gas substance with known kinetic parameters is formed as a result of PMMA thermal decomposition. The realization of only one “effective” oxidation reaction where one substance reacts was assumed.(2)A possible burning out of the polymeric material is not considered. It was found by the authors [12] that the burning out of substance near-surface layer has an insignificant effect on the ignition characteristics at local heating during short time period (less than 0.5 s).(3)Surfaces of hot particles and polymeric material (, , ) have ideal thermal contact. The possibility of gas gap formation is not considered.

Ignition conditions were taken into account [13].(1)The heat release from the oxidation reaction of PMMA gas is more than the heat consumed by both hot particles to the heating of polymeric material and gas mixture.(2)The temperature of gas area in a zone of intense exothermic reaction exceeds the initial temperature of local heat sources.

#### 3. Mathematical Model and Solution Method

The mathematical model describes the interconnected processes of thermal conduction and thermal decomposition in a condensed substance and thermal convection and diffusion processes and oxidation reaction in the gas area are represented by the system of nonstationary partial differential equations .

For gas mixture (, , , ; , ), the following equations and conditions were implemented.

Poisson’s equation:

The equation of gas mixture movement:

The thermal convection equation:

The diffusion equation:

The balance equation:

The heat balance equation for hot particles (, , ):

The heat balance equation for the polymeric material (, ):

The initial conditions :

The boundary conditions :

Stream function and vortex velocity vector :

The mass rate of combustible gas mixture oxidation [14]:where and are constants .

The mass rate of polymeric material pyrolysis:

The diffusion coefficient of polymeric material thermal decomposition products in a gas mixture:

Volume fractions of gas mixture components were determined by its dimensionless mass concentrations:

Thermophysical properties of gas mixture were defined as

The system of (1)–(7) with the corresponding initial and boundary conditions was solved by the finite difference method. The equations of elliptic type (Poisson’s and gas mixture movement) were solved by the alternating direction method. Difference analogues of heat balance and diffusion equations were solved by the locally one-dimensional method. A system of differential equations was solved by the iteration method and the sweep method at each iteration (for nonlinear equations), using the implicit four-point difference scheme.

In developing the algorithm for solving numerically the ignition problem, we used the elements of the algorithm developed for the numerical simulation of conjugate heat transfer processes at the local heating of an area with limited sizes [15]. To improve the accuracy of integrated characteristics computation, we used not less than 500 knots of difference grid for each coordinate and chose a short time step s. The verification of numerical research results was executed similar to [10, 11] by the test of conservation for the used difference scheme. The error of the energy conservation law in the solution area does not exceed 2.7% (Figure 1).

#### 4. Results and Discussion

The numerical investigations were carried out for the following values of parameters [16–21]: the heat effect of oxidation reaction J/kg; the heat effect of PMMA thermal decomposition J/kg; activation energy J/mol, J/mol; preexponential factor , ; the thermal expansion coefficient ; the kinematic viscosity coefficient m^{2}/s; the diffusion coefficient m^{2}/s; the initial temperature of air and polymeric material K; temperature of PMMA pyrolysis beginning K; hot particles sizes m, m; solution area sizes m, m. Thermophysical properties of the substances are presented in Table 1 (Figure 1) [16–19].

The purpose of interconnected heat and mass transfer process research at the ignition of polymeric material consisted of the investigation of the joint effect of several local heat sources on the main integrated characteristic: ignition delay time (Figure 1). In previous studies [10, 11], for a single particle, it was established that the heat content of a local heat source (characterized by its initial temperature ) is the main factor determining . It is also reasonable for two particles to carry out the analysis of the influence of the distance between heat sources on ignition delay time (Figure 1). Therefore, the numerical investigation was executed by varying the initial temperature of heat sources between 700 K < < 1500 K and varying the parameter of (where ) that characterizes the distance between two particles in the range of .

The dependence of polymeric material ignition delay time on the value of at the initial temperature of heat sources = 900 K is shown in Figure 2. It was found that in the “two hot particles—polymeric material—air” system the value of rises for increasing from 0.25 to 1.5. In case of , the ignition delay time does not change ( s). It corresponds to the ignition period of PMMA at its interaction with a single hot particle (curve 1 in Figure 3). Thus, the value of is the limit (for = 900 K) when the particles still have a joint effect on the intensity of heat and mass transfer processes in the system (Figure 1).

It is seen from Figure 2 that the maximum change of ignition delay time in the “two hot particles—polymeric material—air” system is 6.7% at the parameters and = 900 K. The joint effect of several particles at decreases for increasing the heat sources initial temperature. The dependence of PMMA ignition delay time at the initial temperature of a single particle (curve 1) and two particles (curve 2) at is shown in Figure 3. The values of ignition delay time at > 1150 K do not differ from the cases of a single hot particle and two hot particles. It can be explained by the fact that for increasing the heat content of particles rises. The influence of neighboring particles on the warming of polymeric material near-surface layer by thermal conduction and gas mixture by thermal convection decreases. In this case, the temperature of PMMA pyrolysis products increases and less time is required for warming the mixture of combustible gases and oxidizer. Ignition zones as in case of a single particle [10, 11] are formed near the lateral sides of the local heat sources and move to the “polymeric material—hot particle” border (, , , , and ). Besides, it was determined that at the conditions of 700 K < < 1150 K and or K and the characteristics of heat and mass transfer processes at the PMMA ignition by several hot particles are identical to values of [10, 11] calculated at the PMMA ignition by a single hot particle.

The isotherms in the “two hot particles—polymeric material—air” system are shown in Figure 4 at the ignition moment for three different values of . Opposite to the “single hot particle—polymeric material—air” system [10, 11] at variation of 700 K < < 1150 K and , two ignition models are realized. It is characterized by localization of an oxidation reaction in the gas area relative to the surfaces of polymeric material and local heat sources. It is seen in Figure 4(a) that one local ignition zone is formed in case of the small distance between particles. The ignition zone is located on the symmetry axis of the solution area (Figure 1). The maximum temperature gradients and the largest concentration of combustible products (PMMA gas) occurred in comparison with areas and as a result of hot particles joint effect. For increasing and other equal conditions, two ignition areas were formed near the internal lateral sides (, , and ) of hot particles (Figures 4(b) and 4(c)).

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**(b)**

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It is seen from Figure 5 that two main convective whirlwinds intensifying heat and mass transfer processes formed in the gas area at various distances between hot particles.

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**(b)**

**(c)**

It leads to the decrease of temperature and concentration of combustible gases near the external lateral sides (, , and ) of hot particles. Therefore, in these zones, gases-phase ignition does not occur. At the small distance between heat sources diffusion (Figure 5(a)), convection and heat sink from zones and is absent (Figure 1). Ignition occurs near the symmetry axis in the zone with maximum concentration and temperature of gas mixture (Figure 1). For increasing the distance between hot particles (Figures 5(b) and 5(c)) in zones and , the secondary convective whirlwinds forms. It leads to the decrease of temperature and concentration of gas mixture near the symmetry axis . In such conditions, ignition occurs near the internal lateral sides (, , and ) of heat sources.

#### 5. Conclusions

(1)The predictive mathematical model of interconnected heat and mass transfer processes at the ignition of polymeric material by several small-size hot metallic particles was developed. It considers thermal conduction and thermal decomposition in a condensed substance, thermal convection, and diffusion and oxidation reaction in the gas area.(2)Characteristics of a local heat sources and its relative location on a polymeric material surface (700 K < < 1150 K and or K and ) were established when the regularities of heat and mass transfer processes in the “two hot particles—polymeric material—air” system are similar to regularities of heat and mass transfer processes in the “single hot particle—polymeric material—air” system.(3)The developed mathematical model can be used in mechanical engineering for a definition of the most fire danger parts of polymeric material constructional products during the process of its interaction with several local heat sources which are formed as a result of technological processes. Besides, the model can be used in chemistry for the definition of effective kinetic characteristics of polymeric material thermal decomposition and oxidation.

#### Nomenclatures and Units

: | Constant specific heat, J/(kg·K) |

: | Dimensionless concentration of combustible gases in the gas mixture |

: | Dimensionless concentration of oxidant in the gas mixture |

: | Coefficient of diffusion (at K), m^{2}/s |

: | Coefficient of diffusion, m^{2}/s |

: | Activation energy, J/mol |

: | Gravitational acceleration, m/s^{2} |

: | Preexponential factor, s^{−1} |

: | Dimensionless parameter, characterized the distance between two neighboring particles |

: | Heat effect of combustible gas mixture oxidation reaction, J/kg |

: | Heat effect of polymeric material thermal decomposition reaction, J/kg |

: | Ideal gas constant, J/(mol·K) |

: | Time, s |

: | Time step, s |

: | Ignition delay time, s |

: | Temperature, K |

: | Initial temperature of hot particles, K |

: | Temperature of the thermal decomposition beginning for polymeric material, K |

: | Initial temperature of air and polymeric material, K |

, : | Components of combustible gas velocity at the projection onto the axes , , m/s |

: | Mass rate of combustible gas mixture oxidation reaction, kg/(m^{3}·s) |

: | Mass rate of polymeric material thermal decomposition reaction, kg/(m^{3}·s) |

, : | Cartesian coordinates, m |

: | Distance between two particles, m |

, : | Solution domain sizes, m |

, : | Hot particle sizes , m. |

*Greek Symbols*

: | Coefficient of thermal expansion, K^{−1} |

: | Thermal conductivity, W/(m·K) |

: | Density, kg/m^{3} |

: | Coefficient of kinematic viscosity, m^{2}/s |

: | Volume fractions of PMMA gasification products |

: | Volume fractions of air |

: | Stream function, m^{2}/s |

: | Vortex velocity vector, 1/s. |

*Subscripts*

1: | Gas mixture (air and PMMA gas) |

2: | Hot particles |

3: | Polymeric material. |

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The reported study was partially supported by the Russian Science Foundation (no. 14-39-00003).