Research Article Numerical and Experimental Research of ...of the established [ ] integrated...

Research ArticleNumerical and Experimental Research of Heat and MassTransfer at the Heterogeneous System Ignition by Local EnergySource with Limited Heat Content

Dmitrii O. Glushkov, Genii V. Kuznetsov, 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 14 March 2014; Accepted 5 June 2014; Published 24 June 2014

Academic Editor: Zhijun Zhang

Copyright © 2014 Dmitrii O. Glushkov et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Numerical and experimental investigations were executed for determination of macroscopic regularities of heat and mass transferprocesses under the conditions of the phase transformations and chemical reaction at the ignition of vapors coming from fabricsimpregnated by combustible liquid into oxidant area at the local power supply. It was established that initial temperature Θ𝑝 > 1of local energy source and volume fraction 𝜑 > 30% of combustible liquid vapors in fabric are necessary for realization of ignitionconditions in a system “fabric—combustible liquid—oxidant.”. Thus three ignition modes are possible for such system. The mostwidespread mode is an arrangement of ignition zone near the lateral side of local energy source. Also we obtained approximatingexpressions of ignition delay time on initial temperature and characteristic size of a local energy source for fabrics impregnated bysome kinds of combustible liquids (gasoline, kerosene, and diesel fuel). Its dependences may be useful at engineering calculationsof fire danger for processes of single hot particles interaction with liquid combustible substances.

1. Introduction

Numerical and experimental investigations of ignition pro-cesses for solid [1–4] and liquid [5–8] condensed substances,polymer materials [9–11], and gas-vapor mixtures [12–14] bythe energy source with limited power consumption (metallicand nonmetallic particles, wires, cores, concentrate radiationflows, etc.) were held in recent years. Investigation resultsallowed defining extreme and optimum ignition conditionsof high-energy materials under the local heating for the mostimportant practical supplements (special equipment, fireand explosion safety of substances and materials, chemicalindustry, and others).

In power engineering and mechanical engineering atvarious technological processes products from fabric ratherand often can be applied to cleaning surfaces, puttinggreasing, removing surplus of greasing, and so forth. As arule, such fabrics after corresponding technological processeshave high fire danger. The numerical investigation results of

regularities for heterogeneous systems (fabrics impregnatedby combustible liquid) ignition by the local energy sourceare reduced in [15].The characteristic fabric thicknesses werechosen essentially bigger than heating source rates. Becauseof the established [15] integrated ignition characteristics(especially ignition delay time 𝑡𝑑) can be considered as highvalues.

Actually realization of other conditions is possible. Moreoften local heating sources (particles, wires, cores, shavingsand etc.) have rates equitable or even exceeding the thicknessof fabrics impregnated by combustion liquid. Numerical andexperimental investigations of ignition conditions for suchstructures are interesting.

The aim of the present work is to research numericallyand experimentally themacroscopic regularities and extremeignition conditions of typical liquid combustible substances(fuels) vapors coming from fabric surfaces impregnated by it(fabric thicknesses equitablewith hot particle rates) under thelocal heating.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 281527, 9 pageshttp://dx.doi.org/10.1155/2014/281527

2 Mathematical Problems in Engineering

2. Problem Statement

The problem statement is similar as in the work [15]. Thesystem “hot particle—fabric impregnated by combustibleliquid—gas mixture” was considered (Figure 1). Unlike theproblem [15] fabric thickness 𝑍1 in present work is compara-ble with characteristic rates of energy source (hot particle)𝑅𝑝(𝑅𝑝 = 𝑅1) and𝑍𝑝 (𝑍𝑝 =𝑍2 –𝑍1). We considered 2D problemstatement for heat and mass transfer processes investigationin decision area (Figure 1). Though, it was established thatthe difference between similar integrated characteristics at 2Dand 3D problem statements makes less than 10% accordingto analysis of numerical research results [16]. Nevertheless at3D statement the mathematical model is more difficult andcalculations are more complex than at 2D statement.

It was shown that at initial moment (𝜏 = 0) fabric isimpregnated by liquid. Its volume fraction (𝜑) was known.Components of combustible liquid start evaporating inten-sively when near-surface layer was heated up to high tem-peratures by energy of particle fallen on the fabric surface.Generated fuel vapors mix with oxidant (air) is heated by theparticle energy. At the critical temperature and concentrationof gas mixture ignition happened.

The investigations were held for liquid fuels (gasoline,kerosene, and diesel fuel) and widespread fabrics (wool,silk, and flax). The steel particle in the shape of disk withsmall rates 𝑅𝑝 and 𝑍𝑝 was chosen as a source of heat.The characteristic rates of domains 𝑅𝐿 and 𝑍𝐿 for ignitionproblem solution were taken far more than 𝑅𝑝 and 𝑍𝑝(Figure 1).

Numerical simulation was held with the followingassumptions.

(1) The substance with known characteristics appearedas a result of evaporation. The “effective” values ofactivation energy 𝐸 and preexponential factor 𝑘0 areusually [17] determined in experimental research ofkinetic parameters for oxidation reaction of liquidvapors. The realization of one “effective” reactionis supposed, in which one substance participates.Appropriately, the theoretical analyses of investigatedprocess are possible when kinetic scheme, for whichvalues 𝐸 and 𝑘0 are known, is used.

(2) The thermophysical characteristics of particle, fabric,liquid, and air substances are constant. It is establishedthat for investigating range of temperatures in system“hot particle—fabric impregnated by combustibleliquid—gas mixture” (Figure 1) thermophysical char-acteristics changing depend on interacting substancestemperature insignificantly.

(3) The contact between the particle and the fabric is per-fect. Numerical and experimental investigations [3, 4]showed that the particle and condensed substancesurfaces do not deform at the small rates (less than1.5m/s) of particle movement.

1

2

3

ZZL

Z2

Z1

Zp

R

R1 RL

Rp

0

Figure 1: A scheme of the solution domain of the problem at 0 < 𝜏 <𝜏𝑑: 1—gas mixture, 2—metal particle, and 3—fabric impregnated byliquid combustible substance.

The ignition conditions for the heat and mass transfermodel were the following [18].

(1) The heat released as a result of chemical reaction ofgas oxidation in air exceeds that transferred from theparticle to the substance and the gas mixture.

(2) The gas mixture temperature is higher than the initialtemperature of the hot steel particle.

At the numerical research the main integrated character-istic ignition delay time was defined as time period from themoment of contact between hot particle and fabric surface tothe moment of formulated ignition conditions performance.

3. Mathematical Model

The mathematical model describes processes of thermalconduction in hot steel particle and fabric at their inter-connection, crystallization of particle material at its cooling,evaporation of combustible liquid at heating near-surfacelayer of fabric impregnated by liquid fuel, thermal convectionand diffusion of evaporation products (combustible gases)at their mixing with air, and oxidation of gas mixture atits heating. The set of nonlinear nonstationary differentialequations [17, 19–22] at 0 < 𝜏 < 𝜏𝑑 is follows:

𝑍1 < 𝑍 < 𝑍2, 𝑅1 < 𝑅 < 𝑅𝐿;

𝑍2 < 𝑍 < 𝑍𝐿, 0 < 𝑅 < 𝑅𝐿;

(1)

the equation of gas mixture (evaporation products and air)movement is

1

Sh𝜕Ω

𝜕𝜏+ 𝑈

𝜕Ω

𝜕𝑅+ 𝑉

𝜕Ω

𝜕𝑍− 𝑈

Ω

𝑅

=1

Re[𝜕2Ω

𝜕𝑅2+1

𝑅

𝜕Ω

𝜕𝑅+𝜕2Ω

𝜕𝑍2−Ω

𝑅2] +

𝜕Θ1

𝜕𝑅;

(2)

Poisson’s equation is

𝜕2Ψ

𝜕𝑅2−1

𝑅

𝜕Ψ

𝜕𝑅+𝜕2Ψ

𝜕𝑍2= −𝑅Ω; (3)

Mathematical Problems in Engineering 3

the thermal conduction equation for gas mixture consideringoxidation reaction is

1

Sh𝜕Θ1

𝜕𝜏+ 𝑈

𝜕Θ1

𝜕𝑅+ 𝑉

𝜕Θ1

𝜕𝑍

=1

RePr[𝜕2Θ1

𝜕𝑅2+1

𝑅

𝜕Θ1

𝜕𝑅+𝜕2Θ1

𝜕𝑍2] +

𝑄𝑜𝑊𝑜𝑧𝐿

𝜌1𝐶1Δ𝑇𝑉𝑚

;

(4)

the thermal diffusion equation for evaporation products is

1

Sh𝜕𝐶𝑓

𝜕𝜏+ 𝑈

𝜕𝐶𝑓

𝜕𝑅+ 𝑉

𝜕𝐶𝑓

𝜕𝑍

=1

ReSc[

𝜕2𝐶𝑓

𝜕𝑅2+1

𝑅

𝜕𝐶𝑓

𝜕𝑅+

𝜕2𝐶𝑓

𝜕𝑍2] −

𝑧𝐿𝑊𝑜

𝜌4𝑉𝑚

;

(5)

the balance equation for mass conservation law is

𝐶𝑓 + 𝐶𝑜 = 1;

𝑍1 < 𝑍 < 𝑍2, 0 < 𝑅 < 𝑅1;

(6)

the thermal conduction equation formetallic particle consid-ering its material crystallization is

1

Fo2𝜕Θ2

𝜕𝜏=𝜕2Θ2

𝜕𝑅2+1

𝑅

𝜕Θ2

𝜕𝑅+𝜕2Θ2

𝜕𝑍2+𝑄𝑐𝑊𝑐𝑧𝐿

𝑧𝑝Δ𝑇𝜆2

;

0 < 𝑍 < 𝑍1, 0 < 𝑅 < 𝑅𝐿;

(7)

the thermal conduction equation for fabric impregnated bycombustible liquid is

1

Fo3𝜕Θ3

𝜕𝜏=𝜕2Θ3

𝜕𝑅2+1

𝑅

𝜕Θ3

𝜕𝑅+𝜕2Θ3

𝜕𝑍2. (8)

Dimensionless complexes are

Sh =𝑉𝑚𝑡𝑚

𝑧𝐿

, Re =𝑉𝑚𝑧𝐿

𝜐,

Pr =𝜐𝜌𝐶

𝜆, Sc = 𝜐

𝐷, Fo =

𝜆𝑡𝑚

𝜌𝐶𝑧2𝐿

.

(9)

We set temperatures of hot particle, fabric impregnatedby combustible liquid and air, fuel vapors concentration ingas mixture, values of stream function, and vortex velocityvector as initial conditions (𝜏 = 0):

(i) 0 < 𝑅 < 𝑅𝐿, 0 < 𝑍 < 𝑍1:

Θ3 = Θ0; (10)

(ii) 0 < 𝑅 < 𝑅1, 𝑍1 < 𝑍 < 𝑍2:

Θ2 = Θ𝑝; (11)

(iii) 𝑅1 < 𝑅 < 𝑅𝐿,𝑍1 < 𝑍 < 𝑍2; 0< 𝑅 < 𝑅𝐿,𝑍2 < 𝑍 < 𝑍𝐿:

Θ1 = Θ0, 𝐶𝑓 = 0, Ψ = 0, Ω = 0. (12)

We used boundary perfect thermal contact conditions,taking into account liquid evaporation and condition ofequality to zero gradients of corresponding functions at thestatement of boundary conditions (0 < 𝜏 ≤ 𝜏𝑑):

(i) 𝑅 = 0, 𝑅 = 𝑅𝐿, 0 < 𝑍 < 𝑍1:

𝜕Θ3

𝜕𝑅= 0; (13)

(ii) 𝑅 = 0, 𝑍1 < 𝑍 < 𝑍2:

𝜕Θ2

𝜕𝑅= 0; (14)

(iii) 𝑅 = 0, 𝑍2 < 𝑍 < 𝑍𝐿; 𝑅 = 𝑅𝐿, 𝑍1 < 𝑍 < 𝑍𝐿:

𝜕Θ1

𝜕𝑅= 0,

𝜕𝐶𝑓

𝜕𝑅= 0,

𝜕Ψ

𝜕𝑅= 0; (15)

(iv) 𝑅 = 𝑅1, 𝑍1 < 𝑍 < 𝑍2:

𝜕Θ2

𝜕𝑅= (

𝜆1

𝜆2

)(𝜕Θ1

𝜕𝑅) , Θ2 = Θ1,

𝜕𝐶𝑓

𝜕𝑅= 0,

𝜕Ψ

𝜕𝑅= 0, Ψ = 0;

(16)

(v) 𝑍 = 0, 0 < 𝑅 < 𝑅𝐿:

𝜕Θ3

𝜕𝑍= 0; (17)

(vi) 𝑍 = 𝑍1, 0 < 𝑅 < 𝑅1:

𝜕Θ3

𝜕𝑍= (

𝜆2

𝜆3

)(𝜕Θ2

𝜕𝑍) , Θ3 = Θ2; (18)

(vii) 𝑍 = 𝑍1, 𝑅1 < 𝑅 < 𝑅𝐿:

𝜕Θ3

𝜕𝑍= (

𝜆1

𝜆3

)(𝜕Θ1

𝜕𝑍) −

𝑄𝑒𝑊𝑒𝑧𝐿

Δ𝑇𝜆3

,

Θ3 = Θ1,

𝜕𝐶𝑓

𝜕𝑍= −

𝑊𝑒𝑧𝐿

𝜌1𝐷1

,𝜕Ψ

𝜕𝑍= 𝑈, −

𝜕Ψ

𝜕𝑋= 𝑉;

(19)

(viii) 𝑍 = 𝑍2, 0 < 𝑅 < 𝑅1:

𝜕Θ2

𝜕𝑍= (

𝜆1

𝜆2

)(𝜕Θ1

𝜕𝑍) , Θ2 = Θ1,

𝜕𝐶𝑓

𝜕𝑍= 0,

𝜕Ψ

𝜕𝑍= 0, Ψ = 0;

(20)

(ix) 𝑍 = 𝑍𝐿, 0 < 𝑅 < 𝑅𝐿:

𝜕Θ1

𝜕𝑍= 0,

𝜕𝐶𝑓

𝜕𝑍= 0,

𝜕Ψ

𝜕𝑍= 0. (21)


The transition to the dimensionless variables (𝑅 = 𝑟/𝑧𝐿,𝑍 = 𝑧/𝑧𝐿, 𝜏 = 𝑡/𝑡𝑚, 𝑈 = 𝑢/𝑉𝑚, 𝑉 = V/𝑉𝑚, 𝑉𝑚 = √𝑔𝛽Δ𝑇𝑧𝐿,Θ = (𝑇 − 𝑇0)/Δ𝑇, Ψ = 𝜓/𝜓0, Ω = 𝜔/𝜔0, 𝜓0 = 𝑉𝑚𝑧

2

𝐿, and

𝜔0 = 𝑉𝑚/𝑧𝐿) was performed for the following scale values:characteristic size of solution area 𝑧𝐿 = 0.02m; time scale 𝑡𝑚 =1 s; temperature scale 𝑇𝑚 = 1000K; gravitational acceleration𝑔 = 9.8m/s2.

The mass rate of gas mixture oxidation [17] is

𝑊𝑜 = 𝜌1𝑘0𝐶𝑓𝐶𝑜 exp [−𝐸

𝑅𝑡𝑇1

] . (22)

The mass rate of combustible liquid evaporation at heat-ing near-surface layer of fabric impregnated by it is

𝑊𝑒 =𝐴 (𝑃𝑛− 𝑃)

√2𝜋𝑅𝑡𝑇dr/𝑀. (23)

Accommodation coefficient for process of combustibleliquid evaporation is

𝐴 =35

(𝑃𝑛)0.56. (24)

The mass rate of particle material crystallization at itscooling is

𝑊𝑐 = 𝑉𝑐𝜌2. (25)

The linear rate of particle material crystallization is

𝑉𝑐 =𝛿 (𝑥, 𝑦, 𝑡 + Δ𝑡) + 𝛿 (𝑥, 𝑦, 𝑡)

Δ𝑡, (26)

where 𝛿(𝑥, 𝑦, 𝑡 + Δ𝑡), 𝛿(𝑥, 𝑦, 𝑡), is distances from the bottomside of a particle to the crystallization front on (𝑡 + Δ𝑡)th and𝑡th time steps, m.

The system of (2)–(8) with initial and boundary con-ditions was solved by the finite difference method. Tosolve difference analogs of differential equations the locallyone-dimensional method was applied. Nonlinear differenceanalogs of differential equations were solved by the iterationmethod. To solve one-dimensional differential equations thedouble sweep method with the implicit four-point schemewas applied. We selected no less than 400 knots of thedifference net for each of the coordinates and used time step10−6 s.

The reliability of the obtained results was verified by thecomparison with the experimental data. Besides check of theenergy conservation law in the solution field was carried outaccording to algorithm given in [5, 6, 8]. The error of theenergy conservation law at change of initial temperature andthe rates of a hot particle did not exceed 2.5%.

4. Experimental Method

At the experimental investigations we used the plant(Figure 2) and methods described in [23].The single metallicdisk with different rates of radius 𝑅𝑝 = 0.15–0.25 and height𝑍𝑝 = 0.15–0.25 (Figure 1) was used as a heat source at various

13

12 5

2 10 11 9 8 2

7

413

6

Figure 2: Schematic diagram of experimental installation: 1—heating device, 2—support, 3—thermocouple, 4—ceramic core, 5—temperature sensor, 6—metal particle, 7—basis of experimentalinstallation, 8—fire-resistant platform, 9 and 10—flame registrationsystem (radiation source and detector), 11—fabric impregnated byliquid combustible substance, 12—analog to digital converter, and13—computer.

series of experiments. The values of disk height (𝑍𝑝) andradius (𝑅𝑝) were chosen for providing good contact with afabric surface. The experiments showed that location of toosmall particles (𝑅𝑝 = 𝑍𝑝 < 0.1) at the contact moment withfabric was not stable even when settling rate was conservative(up to 1-2m/s).

Themetallic particle falling on the fabric surface had solidstructure and did not become deformed because the particlerate did not exceed 1.5m/s when it contacted with fabricsurface. Fabric shape did not change too. The experimentswere carried out in well-repeatable conditions.

The metallic disk is heated to the high temperaturein heating furnace [23] with function of internal volumetemperature stabilization (up to Θ𝑝 = 1.473) during a longtime (Figure 2).

The moment of contact between hot particle and fabricsurface was fixed automatically by flame registration system.It consists of radiation source and detector. A light beambetween radiation source and detector was blocked at theparticle falling. The first signal via the analog to digitalconverter comes to the personal computer. The moment ofliquid fuel evaporation products ignition was registered byphoto cell. It formed a repeated signal at flame appearance.The second signal was fixed by the personal computer too.Time period between two signals characterized ignitiondelay time (𝑡𝑑) in the system “fabric—combustible liquid—oxidant” (Figure 1).

The error of hot particle temperature measurement esti-mated by the methods [24] did not exceed 1-2%. It has beenestablished that the temperature of a particle decreases lessthan 3-4K during falling, because the error of temperaturedetermination was less than ±0.5%. This deviation can beneglected in the analysis because the particle temperaturewasmore than 900K at experiments. Systematic inaccuracy oftime-keeping was ±0.005% and corresponded to computerpossibilities.

Variable error of ignition delay time determination wascalculated according to the results of experiment. Eightexperiments were carried out for each particle rate at thefixed value of Θ𝑝. The average values of 𝜏𝑑 and mean-square


deviations (𝑆) were determined by the methods [24] for eachvalue of Θ𝑝. The values of 𝑆 were differing (from ±0.01 sto ±0.17 s) for each type of studied fabrics and combustibleliquids. Corresponding variation coefficients ranged from±4.5% to ±15%. Calculated mean-square deviations and vari-ation coefficients are acceptable for conducted experimentsdue to realization of complication mechanism for studiedprocess.

Visual observations over the ignition of heterogeneousstructures were insufficient for adequate description of pro-cess mechanism. So the video recording [23] with frequencyat 50 frames per second was used for details studied mech-anisms and allocation rather subtle effects after repeatedanalysis of video frames.

5. Results and Discussion

The numerical simulation was carried out for the followingvalues of thermophysical and thermochemical parameters:

gasoline: 𝐶 = 2060 J/(kg⋅K), 𝜌 = 751 kg/m3, 𝜆 =0.116W/(m⋅K), 𝑄𝑒 = 29.4 ⋅ 10

3 J/kg, and 𝑀 =100 kg/kmol;gasoline vapors: 𝐶 = 2280 J/(kg⋅K), 𝜌 = 2.5 kg/m3,𝜆 = 0.027W/(m⋅K), 𝑄𝑜 = 45 ⋅ 10

6 J/kg, 𝐸 = 130 ⋅103 J/mole, 𝑘0 = 7 ⋅ 10

6 s−1, 𝛽 = 0.0012K−1, and 𝐷 =14.12 ⋅ 10

−6m2/s; 𝜐 = 1.21 ⋅ 109m2/s;kerosene: 𝐶 = 2190 J/(kg⋅K), 𝜌 = 885 kg/m3,𝜆 = 0.117W/(m⋅K), 𝑄𝑒 = 26.1 ⋅ 10

3 J/kg, and 𝑀 =166.2 kg/kmol;kerosene vapors: 𝐶 = 2370 J/(kg⋅K), 𝜌 = 2.8 kg/m3,𝜆 = 0.028W/(m⋅K), 𝑄𝑜 = 43.8 ⋅ 10

6 J/kg, 𝐸 = 190 ⋅103 J/mole, 𝑘0 = 7 ⋅ 10

7 s−1, 𝛽 = 0.00096K−1, and𝐷 =8.07 ⋅ 10

−6m2/s; 𝜐 = 0.66 ⋅ 109m2/s;diesel fuel: 𝐶 = 2980 J/(kg⋅K), 𝜌 = 887.7 kg/m3,𝜆 = 0.1169W/(m⋅K), 𝑄𝑒 = 25 ⋅ 10

3 J/kg, and 𝑀 =150 kg/kmol;diesel fuel vapors: 𝐶 = 3230 J/(kg⋅K), 𝜌 = 3.1 kg/m3,𝜆 = 0.029W/(m⋅K), 𝑄𝑜 = 42 ⋅ 10

6 J/kg, 𝐸 = 250 ⋅103 J/mole, 𝑘0 = 9 ⋅ 10

8 s−1, 𝛽 = 0.0009K−1, and 𝐷 =5.29 ⋅ 10

−6m2/s; 𝜐 = 1.15 ⋅ 109m2/s;wool: 𝐶 = 1721 J/(kg⋅K), 𝜌 = 1320 kg/m3, and 𝜆 =0.052W/(m⋅K);silk: 𝐶 = 1386 J/(kg⋅K), 𝜌 = 1560 kg/m3, and 𝜆 =0.06W/(m⋅K);flax: 𝐶 = 1580 J/(kg⋅K), 𝜌 = 1500 kg/m3, and 𝜆 =0.088W/(m⋅K);steel: 𝐶 = 470 J/(kg⋅K), 𝜌 = 7831 kg/m3, 𝜆 =49W/(m⋅K), and 𝑄𝑐 = 205 ⋅ 10

3 J/kg;oxidant (air): 𝐶 = 1006 J/(kg⋅K), 𝜌 = 1.161 kg/m3,and 𝜆 = 0.026W/(m⋅K).

Particle rates 𝑅𝑝 = 0.15–0.25 and 𝑍𝑝 = 0.15–0.25; fabricthicknesses 𝑍1 = 0.15; domain solution rates 𝑅𝐿 = 𝑍𝐿 = 1;

particle initial temperature Θ𝑝 = 1–1.5; fabric and oxidantinitial temperature Θ0 = 0.3; absolute gas constant 𝑅𝑡 =8.31 J/(mol⋅K).

Executed earlier numerical investigations results of igni-tion processes in the heterogeneous system “hot particle—fabric impregnated by combustible liquid—gas mixture” [15]show that volume fraction of combustible liquid vapors infabric 𝜑 and heat content of ignition source (that dependson the temperature Θ𝑝 and rates 𝑅𝑝, 𝑍𝑝) are the basicparameters, which have bigger effect on the necessary andsufficient ignition conditions performance. Therefore it isinteresting to estimate influence of these parameters onignition integrated characteristics.

At numerical simulation the volume fraction value 𝜑 wasvaried within the range from 5 up to 50% in conditionsof fabric fixed thickness 𝑍1. The value 𝜑 was controlled byfabric weighing before and after impregnation. Numericallyand experimentally was established that the stable ignition ofthe heterogeneous system (Figure 1) takes place under fabrichigh-porous structure (up to 30–40% of the heterogeneoussystemmass is the liquid condensed substance). So regardlessof the local source temperature Θ𝑝 and rates 𝑅𝑝, 𝑍𝑝, it canbe concluded that the requirement 𝜑 > 30% should beaccomplished for ignition realization in the system (Figure 1).

It is significant that the established limit value 𝜑 is lessthan in [15]. It can be explained by the fact that the localenergy supply from particle to fabric with thickness 𝑍1 >0.25 leads to warming up only and sufficiently thin near-surface layer [15]. So small portion of the liquid takes partin the endothermic phase transition. As a result of theexecuted numerical investigations it was established that thin(𝑍1 < 0.15) fabrics can become warm even on all thicknessesof 𝑍1. It depends on temperature and the power sourcerates. Therefore blowing vapor from the fabric surface tooxidizer environment is more intense than it was supposedin statements [15]. As a consequence, a few smaller values ofcombustible liquid concentration in fabric are necessary forignition realization in system (Figure 1) in comparison withresults [15].

The dependencies of ignition integrated characteristics(time 𝜏𝑑) on temperature and rate𝑅𝑝 (with fixed𝑍𝑝) ofmetal-lic particle for fabrics impregnated by different combustibleliquids (gasoline, kerosene, and diesel fuel) are present onFigures 3 and 4. We used mathematical model (2)–(8) forcalculating these characteristics. Earlier it was established [5,6] that correlation of the square of local power source contactwith the condensed substance and square of gas mediumdetermined the possibility of ignition conditions realization.Therefore the particle crosswise size 𝑅𝑝 that determined itscontacting area with fabric was varied.

Figure 3 shows the experimental values 𝜏𝑑 and approx-imate curve obtained by the least square method. There isgood correlation of executed numerical and experimentalinvestigation results. Deviations of experimental values 𝜏𝑑(on the approximate curve) rather corresponding calculatedvalues do not exceed 18%. With temperature increasing Θ𝑝these deviations are reduced to 11% (at Θ𝑝 = 1.45). Thisfeature can be explained by the fact that we used known[25–27] values of oxidation reactions kinetic parameters


1

2

3

𝜏d

0.20

0.16

0.12

0.08

0.04

01.25 1.30 1.35 1.40 1.45

Θp

Figure 3: Dimensionless ignition delay time versus dimensionlessinitial temperature of the energy source: 1—experimental values,2—theoretical values, and 3—approximating curve for experimentalpoints.

1

2

3

Rp

𝜏d

0.10

0.09

0.08

0.070.150 0.175 0.200 0.225

Figure 4: Dimensionless ignition delay time versus rate of a heatingsource 𝑅𝑝 at𝑍𝑝 = 0.4 andΘ𝑝 = 1.35: 1—diesel fuel, 2—kerosene, and3—gasoline.

(𝑘0 = const and 𝐸 = const) at the simulation. For approximat-ing the theoretical models to real applications it is advisableto take into account the dependence of 𝑘0 and 𝐸 on thetemperature according to [28]. It is known [28] that thisfeature is important at the local heating of small droplets,thin films, and large amounts of liquid condensed substancesby sources with limited power consumption. The similartask requires special consideration. Therefore physical andchemical processes features associated with dependencies 𝑘0= 𝑓(Θ𝑝) and 𝐸 = 𝑓(Θ𝑝) are not analyzed in this paper.

Comparison of dependences 𝜏𝑑 = 𝑓(Θ𝑝) shown inFigures 3 and 4 with results of investigations [15] allowsdrawing a conclusion that the ignition delay time for fabricswith small thickness𝑍1 (Figure 1) is less in several times thananalog characteristic for fabric with thicknesses essentiallybigger than heating source rates [15]. It can be explainedby the intensification of heat fabric process with decreasing𝑍1. As a consequence the injection of evaporation productsincreases near the heating source. The smaller quantity ofenergy of a local source is spent for heterogeneous structurewarming up andmore quantity of energy is used on warming

up of oxidizer with fuel vapors mix. It leads to decreasing ofheat time and oxidation reaction acceleration.

Dependences (Figure 4) illustrate that the source rates(especially 𝑅𝑝) have less impact on integrated ignitioncharacteristics compared to Θ𝑝 (Figure 3). Therefore even arelatively small area of particle contact with the impregnatedfabric is sufficient to achieve the high velocities of the phasetransition and the oxidation reaction acceleration (whenΘ𝑝 > 1). At the same time the dependences of evaporationvelocity𝑊𝑒 and oxidation velocity𝑊𝑜 on the temperature arenonlinear.Therefore, even a small (±0.1) change ofΘ𝑝 affects𝜏𝑑 significantly.

For dependencies shown in Figures 3 and 4 the approxi-mation expressions were obtained:

𝜏𝑑 = 7.271−9.719Θ𝑝 +3.264Θ2

𝑝at 1.25 < Θ𝑝 < 1.5, 𝑅𝑝

= 0.15 and 𝑍𝑝 = 0.25;

𝜏𝑑 = 0.147−0.579𝑅𝑝+1.116𝑅2

𝑝at 0.15 < 𝑅𝑝 < 0.25,𝑍𝑝

= 0.25 and Θ𝑝 = 1.35, combustion liquid—gasoline;

𝜏𝑑 = 0.145−0.481𝑅𝑝+0.976𝑅2

𝑝at 0.15 < 𝑅𝑝 < 0.25,𝑍𝑝

= 0.25 and Θ𝑝 = 1.35, combustion liquid—kerosene;

𝜏𝑑 = 0.178−0.749𝑅𝑝+1.541𝑅2

𝑝at 0.15 < 𝑅𝑝 < 0.25,𝑍𝑝

= 0.25 and Θ𝑝 = 1.35, combustion liquid—diesel fuel.

The developed mathematical model allows getting alarge group of approximating expressions by varying thebasic parameters of the process over a wide range (0.15 <𝑅𝑝 < 0.25, 1 < Θ𝑝 < 1.5). However it is impossible toformulate the approximating expressions taking into accountthe dependences of the ignition delay time 𝜏𝑑 on initialsource temperatureΘ𝑝 and its rates𝑅𝑝 and𝑍𝑝 correspondingto experimental and theoretical results. Due to the factthat the more parameters or factors that are taken intoaccount at the approximation expressions making the greaterapproximation error it is expedient to get an approximationexpression of the forms 𝜏𝑑 = 𝑓(Θ𝑝) and 𝜏𝑑 = 𝑓(𝑅𝑝).

The difference between ignition integrated characteris-tics of fabric impregnated by some kinds of combustibleliquids (gasoline, kerosene, and diesel fuel) was establishedas a result of numerical and experimental investigations ofignition processes in system (Figure 1). These characteristicscorrespond well to the results of experimental investigationsof ignition process in homogeneous system (liquid fuels) [23].It was determined that the ignition delay time for systemswith gasoline vapors is minimal and for systems with dieselfuel vapors is maximal (Figure 4) under otherwise equalconditions. Values of 𝜏𝑑 for systems with kerosene vaporshave some intermediate values (Figure 4). It is caused by theevaporation regularities (the phase transition temperature,kinetics, chemical composition, etc.) for different fuels [23].In experiments with the fabrics impregnated by gasolineit was established that the conditions (by concentration𝐶𝑓) sufficient for ignition (in small fabric vicinity) arecarried out even before the particle falling on the surfaceof heterogeneous structure. The particle and fabric contactleaded to temperature increase in reaction zone. The stableignition of fabrics impregnated by kerosene and diesel fuel


00.10.20.30.40.50.60.70.80.9

0.60.50.40.30.20.100.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.11

2

3

Z

R

(a)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.60.50.40.30.20.10

0.01

0.02

0.03

0.04

0.05

0.06

3

2

1

Z

R

(b)

Figure 5: IsothermsΘ (a) and isolines of fuel concentration𝐶𝑓 (b) at the system “steel particle—thewoolen fabric impregnated by kerosene—air” at ignition (𝜏𝑑 = 0.314) for Θ𝑝 = 1.15, 𝑅𝑝 = 0.15, 𝑍𝑝 = 0.25, and 𝜑 = 0.35: 1—gas mixture, 2—metal particle, and 3—fabric impregnated byliquid combustible substance.

took place only under relative long-lived contact with hotparticle (necessary conditions on 𝐶𝑓 and Θ𝑝 were achievedin great times in comparison with gasoline). At the sametime the experimental results showed that ignition happenedconsistently for all types of combustible liquids in spite of theintegrated characteristics difference.

The numerical analysis of ignition modes characterizedby the ignition delay time and location of oxidation reaction(the relative contact border of ignition source and substancesurface) was accomplished for system (Figure 1) similar asin [5, 6, 8]. It was established that realization of three igni-tion modes is possible for the system “fabric—combustibleliquid—oxidant” (Figure 1) during the local heating.Howeverin contrast to the homogeneous liquid fuels [5, 6, 8], themode when zone of exothermal reaction formulates nearthe lateral sides of ignition source (Figure 5) is the mosttypical at variation of Θ𝑝, 𝑅𝑝, 𝑍𝑝, and 𝜑 in wide ranges.It can be explained according to the fact that the contactsquare of the heating source with a substance in the system“fabric—combustible liquid—oxidant” is less than in thesystem “combustible liquid—oxidant” (particle immersionhappens in liquids and square of heat sink into near-surfacelayer increases).Therefore the local energy source cools downmore intensively at interaction with liquid in comparisonwith the system (Figure 1). The temperatures (Figure 5(a))and concentrations (Figure 5(b)) sufficient to ignition arereached near the lateral sides of hot particle.

Videograms illustrated that realization of ignition con-ditions and burning absence are shown in Figures 6 and7 appropriately. It was established that at the particle tem-perature Θ𝑝 < 1 the ignition does not take place in sys-tem (Figure 1). The intensive evaporation (Figure 7) withoutoxidation reaction and subsequent flame appearance areimplemented. The experimental results showed that ignitionhappened consistently at Θ𝑝 > 1 (Figure 6). Thus Θ𝑝 = 1 isthe limit (low) temperature of steel particle (𝑅𝑝 = 0.15 and𝑍𝑝 = 0.25) at which ignition happens. In case of increasing𝑅𝑝 and 𝑍𝑝 the limit value of Θ𝑝 decreases but does notreach 0.95 under any circumstances. The limit value of Θ𝑝

3

2

1

Figure 6: Frame of experiment with ignition (Θ𝑝 = 1.25, 𝑅𝑝 = 0.15,and 𝑍𝑝 = 0.25): 1—woolen fabric impregnated by kerosene, 2—steelparticle, and 3—flame.

3

2

1

Figure 7: Frame of experiment without ignition (Θ𝑝 = 1, 𝑅𝑝 = 0.15,and 𝑍𝑝 = 0.25): 1—woolen fabric impregnated by kerosene, 2—steelparticle, and 3—smoke without flame.

could be reduced at the fabric thickness 𝑍1 decreasing andcombustible liquid volume fraction 𝜑 increasing. However,in this case, it is difficult to provide the conditions for thesystem output on stationary combustion mode (without theflares and subsequent extinction).


6. Conclusions

The necessary (𝜑 > 30%) and sufficient (Θ𝑝 > 1) ignitionconditions of heterogeneous system, widely used fabrics(wool, silk, and flax) impregnated by typical combustibleliquids (gasoline, kerosene, and diesel fuel) under the localheating, were established as a result of numerical and experi-mental investigations.The impact extents of temperature andrates of local energy source and combustible liquid volumefraction in fabric on the ignition integrated characteristicswere determined. It was established that the values of Θ𝑝and 𝜑 have dominant role in combustion process. Besides itwas shown that the main integrated characteristics (ignitiondelay time) for systems with thin high-porous fabrics can besignificantly less than analog characteristic for large thicknessfabrics [15].

Besides the good correspondence of numerical andexperimental results (numerical simulation assumptionswere proved and developed physical and mathematical mod-els were confirmed) was illustrated. Deviations of numericaland experimental values of 𝜏𝑑 did not exceed 18% undercondition of the main processes parameters variation in wideranges. The good correlation of investigation results withregularities [23] (about modes and ignition mechanisms) forhomogeneous structures (liquid fuels) was established.

Investigation results allow making conclusion about thehigh possibility of ignition conditions realization at cooper-ation of typical local heat sources with heterogeneous struc-tures, in particular, fabrics impregnated by liquid condensedsubstances. In the case of a small fabric thickness and itshigh porosity the ignition process is characterized by small(𝜏𝑑 < 0.1) ignition delay time close to similar characteristicfor homogeneous liquid fuels.

Nomenclatures and Units

𝐴: Accommodation coefficient𝐶: Specific heat capacity, J/(kg⋅K)𝐶𝑓: Dimensionless mass fuel vapors

concentration in gas mixture𝐶𝑜: Dimensionless mass oxidizer

concentration in gas mixture𝐷: Coefficient of diffusion, m2/s𝐸: Activation energy, J/molFo: Fourier number𝑔: Gravitational acceleration, m/s2𝑘0: Preexponential factor, s

−1

𝑀: Molecular mass of liquid, kg/kmol𝑃: Pressure of vapors above fabric surface,

N/m2𝑃𝑛: Saturation pressure of vapors, N/m2

Pr: Prandtl number𝑄𝑐: Heat of particle material crystallization,

J/kg𝑄𝑒: Heat of liquid evaporation, J/kg𝑄𝑜: Heat of oxidation reaction of fuel vapors

in air, J/kg

𝑟, 𝑧: Cylindrical coordinates, m𝑟𝐿, 𝑧𝐿: Solution area rates, m𝑟𝑝, 𝑧𝑝: Particle rates, m𝑅, 𝑍: Dimensionless analogues of 𝑟, 𝑧𝑅𝐿, 𝑍𝐿: Dimensionless analogue of 𝑟𝐿, 𝑧𝐿𝑅𝑝, 𝑍𝑝: Dimensionless analogue of 𝑟𝑝, 𝑧𝑝𝑅𝑡: Absolute gas constant, J/(mol⋅K)Re: Reynolds number𝑆: Mean-square deviationsSc: Schmidt numberSh: Strouhal number𝑡: Time, s𝑡𝑑: Ignition delay time, s𝑡𝑚: Time scale, s𝑇: Temperature, KΔ𝑇: Temperature differential (Δ𝑇 = 𝑇𝑚 – 𝑇0), K𝑇dr: Fabric surface temperature, K𝑇𝑚: Temperature scale, K𝑇0: Initial temperature, K𝑢, V: Rates of gas mixture in a projection to axes

𝑟 and 𝑧, m/s𝑈, 𝑉: Dimensionless analogue of 𝑢, V𝑉𝑐: Linear rate of particle material

crystallization, m/s𝑉𝑚: Convection rate scale of fuel vapors, m/s𝑊𝑐: Mass rate of particle material

crystallization, kg/(m2⋅s)𝑊𝑒: Mass rate of liquid evaporation, kg/(m

2⋅s)

𝑊𝑜: Mass rate of fuel vapors oxidation in air,kg/(m3⋅s).

Greek Symbols

Θ: Dimensionless temperatureΘ0: Dimensionless initial temperature of air and fabricΘ𝑝: Dimensionless initial temperature of particleΨ: Dimensionless stream function analogueΩ: Dimensionless analogue of vortex velocity vector𝛽: Coefficient of thermal expansion, K−1𝜆: Thermal conductivity, W/(m⋅K)𝜌: Density, kg/m3𝜏: Dimensionless time𝜏𝑑: Dimensionless ignition delay time𝜐: Coefficient of kinematic viscosity, m2/s𝜑: Volume fraction𝜓: Stream function, m3/s𝜓0: Scale of stream function, m

3/s𝜔: Vortex velocity vector, 1/s𝜔0: Scale of vortex velocity vector, 1/s.

Subscripts

1: Gas mixture2: Metallic particle3: Fabric impregnated by combustible liquid4: Fuel vapor.


Conflict of Interests

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

Acknowledgments

The reported study was partially supported by Grant ofthe President of the Russian Federation (Project no. MK-2391.2014.8) and State task “Nauka” (code of the federal targetscientific and technical Program no. 2.1321.2014).

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