Research Article Mathematical Modeling of Heat and Mass ...

Research ArticleMathematical Modeling of Heat and Mass TransferProcesses with Chemical Reaction at Polymeric MaterialIgnition by Several Small-Size Hot Particles

Dmitrii O. Glushkov and Pavel A. Strizhak

National Research Tomsk Polytechnic University, 30 Lenin Avenue, Tomsk 634050, Russia

Correspondence should be addressed to Dmitrii O. Glushkov; [email protected]

Received 9 May 2014; Revised 7 July 2014; Accepted 13 October 2014

Academic Editor: Christopher Gunaseelan Jesudason

Copyright © 2015 D. O. Glushkov and P. A. Strizhak. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Numerical research of interconnected heat and mass transfer processes in the “two hot particles—polymeric material—air” systemwas executed. The joint effect of several local heat sources on the main integrated characteristic of ignition process (ignition delaytime) was established. Two ignitionmodels characterized by the relative positioning of hot particles on a polymericmaterial surfacewere revealed. Besides, there were established characteristics of local heat sources and the distance between them (700K < 𝑇𝑝 <

1150K and 𝐿 > 1.5 or 𝑇𝑝 > 1150K and 0.25 < 𝐿 < 1.5) when regularities of heat and mass transfer processes in the “twohot particles—polymeric material—air” system are similar to regularities of heat and mass transfer processes in the “single hotparticle—polymeric material—air” system.

1. Introduction

In recent years, polymeric materials (polymethyl methacry-late, polystyrene, polyethylene, etc.) have been widelyadopted in various industries as decorative and constructiveelements. Polymeric material products are very susceptible tothermal effects [1–4] even at rather low outside temperature(𝑇 ≈ 400–600K). Under conditions of some technologicalprocesses (at increased ambient temperature) strength char-acteristics are changed, melting occurs, dangerous carcino-gens and are emitted. Possible temperatures of technologicalprocesses for power production can reach more than 1000K.Under such conditions, the probability of local power sources(metal andnonmetallic particleswarmed to high temperaturewith sizes about several millimeters) formation is high [5–9].

The numerical research results [10, 11] were obtainedfor thermal conduction and thermal convection processesduring a polymeric material ignition by a single metalparticle heated to high temperature. Established theoreticalconsequences can be used for developing guidelines andmethods to reduce the flammability, ignition preventing, and

subsequent stages of polymeric material combustion pro-cesses. However, in practice, several (two, three, etc.) small-size particles heated to high temperature can cause fires.Ignition conditions and heat transfer characteristics may bedifferent 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 timeof solid condensed substance (composite propellant) at highconcentration (large number per unit area surfaces) of hotparticles in the gas stream equals the values of ignition delaytime at condensed substance heating by a massive plate withconstant (during ignition period) high temperature [5, 6].Therefore, more detailed information about characteristics ofphysical and chemical processes at polymeric material heat-ing by several hot particles is necessary for the developmentof relevant precautionary activities.

The purpose of the present study was to develop themathematical model and analyze the characteristics of inter-connected heat and mass transfer processes during inter-action of two small-size metal particles heated to hightemperatures with a polymeric material at the accounting

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 614143, 8 pageshttp://dx.doi.org/10.1155/2015/614143

2 Mathematical Problems in Engineering

of thermal conduction in condensed substance and thermaldecomposition 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 liquidcondensed substances are defined by the distance betweentwo neighboring particles at the various quantities of hotparticles (local heat sources) falling to the flammablematerialsurface.Therefore, previous study results [12] were taken intoaccount at the problem statement.

A scheme with two steel particles heated to high tem-perature in the parallelepiped shape (with the same sizes)situated on the surface of typical polymeric material, poly-methyl methacrylate (PMMA), was chosen to simulate theconditions of hot particle flow interaction with polymericmaterial. We reasonably [12] used the “two hot particles—polymeric material—air” system (Figure 1) instead of the“several hot particles—polymeric material—air” system forheat and mass transfer process investigation. Areas with sizes𝑥𝐿 and 𝑦𝐿 much larger than the sizes of hot particles 𝑥𝑝 and𝑦𝑝 were allocated in polymeric material and air (Figure 1).

The rate of PMMA thermal decomposition processaccelerated at polymeric material near-surface layer heating(at 0 < 𝑡 < 𝑡𝑑) by the heat of hot particles. Gaseous productsof polymericmaterial pyrolysismixedwith an oxidizer (air) atthe diffusion in outside gas area.The gasmixture was warmedby the thermal convection at its movement along lateral sides(𝑥 = 𝑥1, 𝑥 = 𝑥2, 𝑥 = 𝑥3, 𝑥 = 𝑥4, and 𝑦1 < 𝑦 < 𝑦2)

of heat sources (Figure 1). The ignition occurred when thetemperature and concentration of combustible component(PMMA gas) reached the critical values.

The assumptions for the problem statement of heat andmass transfer process include the following.

(1) A gas substance with known kinetic parameters isformed as a result of PMMA thermal decomposition.The realization of only one “effective” oxidation reac-tion where one substance reacts was assumed.

(2) A possible burning out of the polymeric material isnot considered. It was found by the authors [12] thatthe burning out of substance near-surface layer hasan insignificant effect on the ignition characteristicsat local heating during short time period (less than0.5 s).

(3) Surfaces of hot particles and polymericmaterial (𝑥1 <𝑥 < 𝑥2, 𝑥3 < 𝑥 < 𝑥4, 𝑦 = 𝑦1) have ideal thermalcontact. The possibility of gas gap formation is notconsidered.

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 bothhot particles to the heating of polymeric material andgas mixture.

(2) The temperature of gas area in a zone of intenseexothermic reaction exceeds the initial temperature oflocal heat sources.

x

y

1

3

0

22

yL

xL

y1

xp xp

y2 yp

x1 x2 x3 x4

Δx

Figure 1: A scheme of the solution domain at 𝑡 = 0: 1: oxidizer (gasmixture at 0 < 𝑡 ≤ 𝑡𝑑); 2: hot particle; 3: polymeric material.

3. Mathematical Model and Solution Method

The mathematical model describes the interconnected pro-cesses of thermal conduction and thermal decomposition ina condensed substance and thermal convection and diffusionprocesses and oxidation reaction in the gas area are rep-resented by the system of nonstationary partial differentialequations (0 < 𝑡 < 𝑡𝑑).

For gas mixture (0 < 𝑥 < 𝑥1, 𝑥2 < 𝑥 < 𝑥3, 𝑥4 < 𝑥 < 𝑥𝐿,𝑦1 < 𝑦 < 𝑦2; 0 < 𝑥 < 𝑥𝐿, 𝑦2 < 𝑦 < 𝑦𝐿), the followingequations and conditions were implemented.

Poisson’s equation:

𝜕2𝜓

𝜕𝑥2+

𝜕2𝜓

𝜕𝑦2= −𝜔. (1)

The equation of gas mixture movement:

𝜕𝜔

𝜕𝑡

+ 𝑢

𝜕𝜔

𝜕𝑥

+ ]𝜕𝜔

𝜕𝑦

= 𝜐1 (𝜕2𝜔

𝜕𝑥2+

𝜕2𝜔

𝜕𝑦2) + 𝛽𝑔

𝜕𝑇1

𝜕𝑥

. (2)

The thermal convection equation:

𝜌1𝐶1 (𝜕𝑇1

𝜕𝑡

+ 𝑢

𝜕𝑇1

𝜕𝑥

+ ]𝜕𝑇1

𝜕𝑦

) = 𝜆1 (𝜕2𝑇1

𝜕𝑥2+

𝜕2𝑇1

𝜕𝑦2) + 𝑄1𝑊1.

(3)

The diffusion equation:

𝜌1 (

𝜕𝐶𝑓

𝜕𝑡

+ 𝑢

𝜕𝐶𝑓

𝜕𝑥

+ ]𝜕𝐶𝑓

𝜕𝑦

) = 𝜌1𝐷1 (𝜕2𝑇1

𝜕𝑥2+

𝜕2𝑇1

𝜕𝑦2) −𝑊1.

(4)

The balance equation:𝐶𝑓 + 𝐶𝑜 = 1. (5)

The heat balance equation for hot particles (𝑥1 < 𝑥 < 𝑥2,𝑥3 < 𝑥 < 𝑥4, 𝑦1 < 𝑦 < 𝑦2):

𝜌2𝐶2

𝜕𝑇2

𝜕𝑡

= 𝜆2 (𝜕2𝑇2

𝜕𝑥2+

𝜕2𝑇2

𝜕𝑦2) . (6)

Mathematical Problems in Engineering 3

The heat balance equation for the polymericmaterial (0 <𝑥 < 𝑥𝐿, 0 < 𝑦 < 𝑦1):

𝜌3𝐶3

𝜕𝑇3

𝜕𝑡

= 𝜆3 (𝜕2𝑇3

𝜕𝑥2+

𝜕2𝑇3

𝜕𝑦2) − 𝑄3𝑊3. (7)

The initial conditions (𝑡 = 0):

𝑇 = 𝑇0 at 0 < 𝑥 < 𝑥𝐿, 0 < 𝑦 < 𝑦1;𝑇 = 𝑇𝑝 at 𝑥1 < 𝑥 < 𝑥2, 𝑥3 < 𝑥 < 𝑥4, 𝑦1 < 𝑦 < 𝑦2;

𝑇 = 𝑇0, 𝐶𝑓 = 0, 𝜓 = 0, 𝜔 = 0

at 0 < 𝑥 < 𝑥1, 𝑥2 < 𝑥 < 𝑥3, 𝑥4 < 𝑥 < 𝑥𝐿, 𝑦1 < 𝑦 < 𝑦2;0 < 𝑥 < 𝑥𝐿, 𝑦2 < 𝑦 < 𝑦𝐿.

(8)

The boundary conditions (0 ≤ 𝑡 ≤ 𝑡𝑑):

𝑥 = 0, 𝑥 = 𝑥𝐿, 0 < 𝑦 < 𝑦1:𝜕𝑇3

𝜕𝑥

= 0;

𝑥 = 0, 𝑥 = 𝑥𝐿, 𝑦1 < 𝑦 < 𝑦𝐿:

𝜕𝑇1

𝜕𝑥

= 0,

𝜕𝐶𝑓

𝜕𝑥

= 0,

𝜕𝜓

𝜕𝑥

= 0;

𝑥 = 𝑥1, 𝑥 = 𝑥3, 𝑦1 < 𝑦 < 𝑦2:

− 𝜆1

𝜕𝑇1

𝜕𝑥

= −𝜆2

𝜕𝑇2

𝜕𝑥

, 𝑇1 = 𝑇2,

𝜕𝐶𝑓

𝜕𝑥

= 0,

𝜕𝜓

𝜕𝑥

= 0,

𝜓 = 0;

𝑥 = 𝑥2, 𝑥 = 𝑥4, 𝑦1 < 𝑦 < 𝑦2:

− 𝜆2

𝜕𝑇2

𝜕𝑥

= −𝜆1

𝜕𝑇1

𝜕𝑥

, 𝑇2 = 𝑇1,

𝜕𝐶𝑓

𝜕𝑥

= 0,

𝜕𝜓

𝜕𝑥

= 0,

𝜓 = 0;

𝑦 = 0, 0 < 𝑥 < 𝑥𝐿:𝜕𝑇3

𝜕𝑦

= 0;

𝑦 = 𝑦1, 0 < 𝑥 < 𝑥1, 𝑥2 < 𝑥 < 𝑥3, 𝑥4 < 𝑥 < 𝑥𝐿:

− 𝜆3

𝜕𝑇3

𝜕𝑦

= −𝜆1

𝜕𝑇1

𝜕𝑦

+ 𝑄1𝑊1, 𝑇3 = 𝑇1,

𝜌1𝐷1

𝜕𝐶𝑓

𝜕𝑦

= −𝑊3,

𝜕𝜓

𝜕𝑦

= 0;

𝑦 = 𝑦1, 𝑥1 < 𝑥 < 𝑥2, 𝑥3 < 𝑥 < 𝑥4:

− 𝜆3

𝜕𝑇3

𝜕𝑦

= −𝜆2

𝜕𝑇2

𝜕𝑦

, 𝑇3 = 𝑇2;

𝑦 = 𝑦2, 𝑥1 < 𝑥 < 𝑥2, 𝑥3 < 𝑥 < 𝑥4:

− 𝜆2

𝜕𝑇2

𝜕𝑦

= −𝜆1

𝜕𝑇1

𝜕𝑦

, 𝑇2 = 𝑇1,

𝜕𝐶𝑓

𝜕𝑦

= 0,

𝜕𝜓

𝜕𝑦

= 0, 𝜓 = 0;

𝑦 = 𝑦𝐿, 0 < 𝑥 < 𝑥𝐿:𝜕𝑇1

𝜕𝑦

= 0,

𝜕𝐶𝑓

𝜕𝑦

= 0,

𝜕𝜓

𝜕𝑦

= 0.

(9)

Stream function 𝜓 and vortex velocity vector 𝜔:

𝑢 =

𝜕𝜓

𝜕𝑦

, V = −𝜕𝜓

𝜕𝑥

, 𝜔 = rot𝑦V⃗ =𝜕V𝜕𝑥

−

𝜕𝑢

𝜕𝑦

. (10)

The mass rate of combustible gas mixture oxidation [14]:

𝑊1 = 𝜌1𝑘0

1b𝑛𝑜b𝑚𝑓exp(− 𝐸1

𝑅𝑇1

) , (11)

where 𝑛 and𝑚 are constants (𝑛 = 𝑚 = 1).The mass rate of polymeric material pyrolysis:

𝑊3 = ∫

𝑦1

0

𝜌3𝑘0

3exp(−

𝐸3

𝑅𝑇𝑠

)𝑑𝑦. (12)

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

𝐷1 = 𝐷0(𝑇1

273

)

1.7

. (13)

Volume fractions of gas mixture components were deter-mined by its dimensionless mass concentrations:

𝜑𝑓 =

𝐶𝑓/𝜌𝑓

𝐶𝑓/𝜌𝑓 + 𝐶𝑜/𝜌𝑜

, 𝜑𝑜 + 𝜑𝑓 = 1. (14)

Thermophysical properties of gasmixture were defined as

𝜆1 = 𝜆𝑓𝜑𝑓 + 𝜆𝑜𝜑𝑜,

𝐶1 = 𝐶𝑓𝜑𝑓 + 𝐶𝑜𝜑𝑜,

𝜌1 = 𝜌𝑓𝜑𝑓 + 𝜌𝑜𝜑𝑜.

(15)

The system of (1)–(7) with the corresponding initialand boundary conditions was solved by the finite differencemethod. The equations of elliptic type (Poisson’s and gasmixture movement) were solved by the alternating directionmethod. Difference analogues of heat balance and diffu-sion equations were solved by the locally one-dimensionalmethod. A system of differential equations was solved by theiteration method and the sweep method at each iteration (fornonlinear equations), using the implicit four-point differencescheme.

In developing the algorithm for solving numerically theignition problem, we used the elements of the algorithmdeveloped for the numerical simulation of conjugate heattransfer processes at the local heating of an area with limitedsizes [15]. To improve the accuracy of integrated charac-teristics computation, we used not less than 500 knots ofdifference grid for each coordinate and chose a short time stepΔ𝑡 = 10

−6 s.The verification of numerical research results wasexecuted similar to [10, 11] by the test of conservation for theused difference scheme.The error of the energy conservationlaw in the solution area does not exceed 2.7% (Figure 1).


Table 1: Thermophysical properties of substances.

Substance 𝜌, kg/m3𝐶, J/(kg⋅K) 𝜆, W/(m⋅K)

Air 1.161 1190 0.026Steel 7831 470 49PMMA (solid) 1200 1466.5 0.19PMMA (gas) 1.29 1005.6 0.025

4. Results and Discussion

The numerical investigations were carried out for the follow-ing values of parameters [16–21]: the heat effect of oxidationreaction𝑄1 = 2.5×10

7 J/kg; the heat effect of PMMA thermaldecomposition𝑄3 = 10

6 J/kg; activation energy 𝐸1 = 0.125×106 J/mol, 𝐸3 = 0.13 × 10

6 J/mol; preexponential factor 𝑘01=

1010 s−1, 𝑘0

3= 2.82×10

9 s−1; the thermal expansion coefficient𝛽 = 0.0009K−1; the kinematic viscosity coefficient 𝜐1 = 1.4 ×10−5m2/s; the diffusion coefficient𝐷0 = 8.12×10

−6m2/s; theinitial temperature of air and polymeric material 𝑇0 = 300K;temperature of PMMA pyrolysis beginning 𝑇𝑠 = 500K; hotparticles sizes 𝑥𝑝 = 4 × 10

−3m, 𝑦𝑝 = 2 × 10−3m; solution

area sizes 𝑥𝐿 = 20 × 10−3m, 𝑦𝐿 = 20 × 10

−3m. Thermo-physical properties of the substances are presented in Table 1(Figure 1) [16–19].

The purpose of interconnected heat and mass transferprocess research at the ignition of polymeric material con-sisted of the investigation of the joint effect of several localheat sources on the main integrated characteristic: ignitiondelay time 𝑡𝑑 (Figure 1). In previous studies [10, 11], for asingle particle, it was established that the heat content ofa local heat source (characterized by its initial temperature𝑇𝑝) is the main factor determining 𝑡𝑑. It is also reasonablefor two particles to carry out the analysis of the influenceof the distance between heat sources Δ𝑥 on ignition delaytime (Figure 1). Therefore, the numerical investigation wasexecuted by varying the initial temperature of heat sourcesbetween 700K < 𝑇𝑝 < 1500K and varying the parameter of 𝐿(where 𝐿 = Δ𝑥/𝑥𝑃) that characterizes the distance betweentwo particles in the range of 0.25 < 𝐿 < 2.

Thedependence of polymericmaterial ignition delay timeon the value of 𝐿 at the initial temperature of heat sources 𝑇𝑝= 900K is shown in Figure 2. It was found that in the “twohot particles—polymeric material—air” system the value of𝑡𝑑 rises for increasing 𝐿 from 0.25 to 1.5. In case of 𝐿 > 1.5,the ignition delay time does not change (𝑡𝑑 = 0.0936 s). Itcorresponds to the ignition period of PMMAat its interactionwith a single hot particle (curve 1 in Figure 3).Thus, the valueof 𝐿 = 1.5 is the limit (for 𝑇𝑝 = 900K) when the particles stillhave a joint effect on the intensity of heat and mass transferprocesses in the system (Figure 1).

It is seen from Figure 2 that the maximum change ofignition delay time in the “two hot particles—polymericmaterial—air” system is 6.7% at the parameters 0.25 < 𝐿 <

1.5 and 𝑇𝑝 = 900K. The joint effect of several particles at 𝑡𝑑decreases for increasing the heat sources initial temperature.The dependence of PMMA ignition delay time at the initialtemperature of a single particle (curve 1) and two particles

L

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

0.091

0.092

0.093

0.094

0.095

0.087

0.088

0.089

0.09

t d(s

)

Figure 2: PMMA ignition delay time 𝑡𝑑 versus 𝐿 at 𝑇𝑝 = 900K.

700 800 900 1000 1100 1200 1300 1400 1500

0.2

0.25

0.3

0

0.05

0.1

0.15

2

1

t d(s

)

Tp (K)

Figure 3: PMMA ignition delay time 𝑡𝑑 versus initial temperatureof particle 𝑇𝑝: 1: one particle; 2: two particles at 𝐿 = 0.25.

(curve 2) at 𝐿 = 0.25 is shown in Figure 3. The values ofignition delay time at 𝑇𝑝 > 1150K do not differ from thecases of a single hot particle and two hot particles. It can beexplained by the fact that for increasing 𝑇𝑝 the heat contentof particles rises. The influence of neighboring particles onthe warming of polymeric material near-surface layer bythermal conduction and gas mixture by thermal convectiondecreases. In this case, the temperature of PMMA pyrolysisproducts increases and less time is required for warming themixture of combustible gases and oxidizer. Ignition zones asin case of a single particle [10, 11] are formed near the lateralsides of the local heat sources and move to the “polymericmaterial—hot particle” border (𝑥 = 𝑥1, 𝑥 = 𝑥2, 𝑥 = 𝑥3,𝑥 = 𝑥4, and 𝑦 → 𝑦1). Besides, it was determined thatat the conditions of 700K < 𝑇𝑝 < 1150K and 𝐿 > 1.5 or𝑇𝑝 > 1150K and 0.25 < 𝐿 < 1.5 the characteristics of heatand mass transfer processes at the PMMA ignition by severalhot particles are identical to values of 𝑡𝑑 [10, 11] calculated atthe PMMA ignition by a single hot particle.

The isotherms in the “two hot particles—polymericmaterial—air” system are shown in Figure 4 at the ignitionmoment for three different values of 𝐿. Opposite to the“single hot particle—polymeric material—air” system [10, 11]at variation of 700K < 𝑇𝑝 < 1150K and 0.25 < 𝐿 < 1.5, twoignitionmodels are realized. It is characterized by localization


3

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5

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8

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11

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400

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x (m)

y (m

)

×10−3

(c)

Figure 4: Isotherms (𝑇,K) at ignition moment 𝑡𝑑 = 0.087 s, 𝐿 = 0.25 (a), 𝑡𝑑 = 0.092 s, 𝐿 = 0.5 (b), 𝑡𝑑 = 0.096 s, and 𝐿 = 1 (c) at 𝑇𝑝 = 900K:1: gas mixture; 2: hot steel particle; 3: PMMA.

of an oxidation reaction in the gas area relative to the surfacesof polymeric material and local heat sources. It is seen inFigure 4(a) that one local ignition zone is formed in case ofthe small (𝐿 = 0.25) distance between particles. The ignitionzone is located on the symmetry axis 𝑥 = 𝑥𝐿/2 of the solutionarea (Figure 1).Themaximum temperature gradients and thelargest concentration of combustible products (PMMA gas)occurred in comparison with areas 𝑥 < 𝑥1 and 𝑥 > 𝑥4 as aresult of hot particles joint effect. For increasing 𝐿 and other

equal conditions, two ignition areas were formed near theinternal lateral sides (𝑥 = 𝑥2, 𝑥 = 𝑥3, and 𝑦1 < 𝑦 < 𝑦2)of hot particles (Figures 4(b) and 4(c)).

It is seen from Figure 5 that two main convective whirl-winds intensifying heat and mass transfer processes formedin the gas area at various distances between hot particles.

It leads to the decrease of temperature and concentrationof combustible gases near the external lateral sides (𝑥 = 𝑥1,𝑥 = 𝑥4, and 𝑦1 < 𝑦 < 𝑦2) of hot particles. Therefore,


×10−5

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0.0050.005

0.010.01 0.015

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0.015 0.02

0.02

x (m)y (m)

0 01 0.0150.015

(b)

×10−5

−3

−2

−1

1

2

3

00

0

0.0050.005

0.010.01 0.015

0.015 0.02

0.02

x (m)y (m)

0 01 0 0150.015

(c)

Figure 5: Stream function (𝜓, m2/s) at ignition moment 𝑡𝑑 = 0.087 s, 𝐿 = 0.25 (a), 𝑡𝑑 = 0.092 s, 𝐿 = 0.5 (b), 𝑡𝑑 = 0.096 s, and 𝐿 = 1 (c) at𝑇𝑝 = 900K.

in these zones, gases-phase ignition does not occur. At thesmall distance between heat sources diffusion (Figure 5(a)),convection and heat sink from zones 𝑥2 < 𝑥 < 𝑥3 and 𝑦1 <𝑦 < 𝑦2 is absent (Figure 1). Ignition occurs near the symmetryaxis 𝑥 = 𝑥𝐿/2 in the zone with maximum concentrationand temperature of gas mixture (Figure 1). For increasingthe distance between hot particles (Figures 5(b) and 5(c)) inzones 𝑥2 < 𝑥 < 𝑥3 and 𝑦1 < 𝑦 < 𝑦2, the secondary convectivewhirlwinds forms. It leads to the decrease of temperature andconcentration of gas mixture near the symmetry axis 𝑥 =

𝑥𝐿/2. In such conditions, ignition occurs near the internal lat-eral sides (𝑥 = 𝑥2, 𝑥 = 𝑥3, and 𝑦1 < 𝑦 < 𝑦2) of heat sources.

5. Conclusions

(1) The predictivemathematicalmodel of interconnectedheat and mass transfer processes at the ignition of

polymeric material by several small-size hot metallicparticles was developed. It considers thermal con-duction and thermal decomposition in a condensedsubstance, thermal convection, and diffusion andoxidation reaction in the gas area.

(2) Characteristics of a local heat sources and its relativelocation on a polymeric material surface (700K <

𝑇𝑝 < 1150K and 𝐿 > 1.5 or 𝑇𝑝 > 1150K and0.25 < 𝐿 < 1.5) were establishedwhen the regularitiesof heat and mass transfer processes in the “two hotparticles—polymeric material—air” system are simi-lar to regularities of heat and mass transfer processesin the “single hot particle—polymeric material—air”system.

(3) The developed mathematical model can be used inmechanical engineering for a definition of the most


fire danger parts of polymericmaterial constructionalproducts during the process of its interaction withseveral local heat sources which are formed as a resultof technological processes. Besides, the model canbe used in chemistry for the definition of effectivekinetic characteristics of polymeric material thermaldecomposition and oxidation.

Nomenclatures and Units

𝐶: Constant specific heat, J/(kg⋅K)𝐶𝑓: Dimensionless concentration of

combustible gases in the gas mixture𝐶𝑜: Dimensionless concentration of oxidant in

the gas mixture𝐷0: Coefficient of diffusion (at 𝑇 = 293K),

m2/s𝐷1: Coefficient of diffusion, m2/s𝐸: Activation energy, J/mol𝑔: Gravitational acceleration, m/s2𝑘0: Preexponential factor, s−1𝐿: Dimensionless parameter, characterized

the distance between two neighboringparticles (𝐿 = Δ𝑥/𝑥𝑃)

𝑄1: Heat effect of combustible gas mixtureoxidation reaction, J/kg

𝑄3: Heat effect of polymeric material thermaldecomposition reaction, J/kg

𝑅: Ideal gas constant, J/(mol⋅K)𝑡: Time, sΔ𝑡: Time step, s𝑡𝑑: Ignition delay time, s𝑇: Temperature, K𝑇𝑝: Initial temperature of hot particles, K𝑇𝑠: Temperature of the thermal

decomposition beginning for polymericmaterial, K

𝑇0: Initial temperature of air and polymericmaterial, K

𝑢, V: Components of combustible gas velocityat the projection onto the axes 𝑥, 𝑦, m/s

𝑊1: Mass rate of combustible gas mixtureoxidation reaction, kg/(m3⋅s)

𝑊3: Mass rate of polymeric material thermaldecomposition reaction, kg/(m3⋅s)

𝑥, 𝑦: Cartesian coordinates, mΔ𝑥: Distance between two particles, m𝑥𝐿, 𝑦𝐿: Solution domain sizes, m𝑥𝑝, 𝑦𝑝: Hot particle sizes

(𝑥𝑝 = 𝑥2 − 𝑥1, 𝑦𝑝 = 𝑦2 − 𝑦1), m.

Greek Symbols

𝛽: Coefficient of thermal expansion, K−1𝜆: Thermal conductivity, W/(m⋅K)𝜌: Density, kg/m3𝜐: Coefficient of kinematic viscosity, m2/s𝜑𝑓: Volume fractions of PMMA gasification products

𝜑𝑜: Volume fractions of air𝜓: Stream function, m2/s𝜔: Vortex velocity vector, 1/s.

Subscripts

1: Gas mixture (air and PMMA gas)2: Hot particles3: Polymeric material.

Conflict of Interests

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

Acknowledgment

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

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