Mathematical modeling of heat and mass transfer processes at the ignition of a liquid condensed substance by an immersed hot particle

A numerical investigation of heat and mass transfer processes at the heating of combustible liquid was carried out at the interaction of hot small-size steel particle with gasoline. Developed mathematical model considers at two-dimensional statement thermal conduction, thermal convection, transfer of energy by phase change (evaporation of liquid fuel and crystallization of particle material), partial immersion of hot particle in liquid fuel, forming of vapor gap between hot particle and liquid fuel, diffusion of fuel vapors in oxidizer, dependence of thermophysical characteristics of interactive substances on temperature. It was established that the highest rates of heat and mass transfer processes in a system “hot particle – gasoline – air” are possible at temperature of hot particle higher than melting temperature of it material due to the additional heat released at the crystallization of material.


Introduction
Heating, evaporation and ignition of liquid condensed substances drops at contact with hot plate are strongly investigated [1][2][3][4][5][6].Such interest can be explained by absence of information about main regularities of interconnected heat and mass transfer processes.Heating, evaporation and ignition of liquid condensed substances by local energy sources with small sizes (for example, heated till high temperatures metallic and nonmetallic particles with limited heat content) is known less.
Interaction processes between heated particles formed at cutting, welding and friction of metals with solid and liquid combustible substances are significant processes from the point of industrial safety.These processes are characterized by high rates of chemical reactions, heat and mass transfer, small sizes and high temperatures of energy sources, consequently high fire density.
The general ignition theory for liquid combustible substances at different mechanisms of energy supply (thermal conduction, thermal convection, and thermal radiation) is not developed.However similar theory for solids combustible substances exists [7][8][9].This result can be explained by more difficult heat and mass transfer mechanisms for liquids in comparison with solids, and absence of experimental data.Nowadays results of theoretical investigations of ignition processes of typical liquid fuels by single heated till high temperatures steel particle are known [10][11][12].A few factors influencing on heat transfer characteristics at the ignition were not considered at these papers.For example, the dependences of thermophysical characteristics on temperature for substances and possible processes transfer of energy by phase changes at the crystallization of hot particle material.

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The purpose of present work is the numerical investigation of basis regularities for heat and mass transfer processes at immersion of a metallic hot particle in a liquid fuel.Mathematical model of process considers two-dimensional heat and mass transfer, kinetics of evaporation and ignition processes, crystallization of hot particle material, partial immersion of hot particle in liquid fuel, forming of vapor gap between hot particle and liquid fuel, and dependence of thermophysical characteristics of substances on temperature.

Problem statement
The problem of liquid fuel (gasoline) ignition by single hot steel particle was examined.It was supposed that local energy source (small sizes particle) fell into gasoline surface.Particle immersed in liquid (Fig. 1) and near-surface layer of condensed substance was heated.The evaporation began at the achievement of phase change conditions.The fuel vapor gap was formed between the energy source and the liquid fuel (Fig. 1).The rate of vapor gap was defined by the particle heat content, thermophysical and kinetic characteristics of liquid condensed substance.Fuel vapor got up and diffused at air.As the result flammable gas-vapor mixture was formed nearby contact boundary between the particle and the liquid.Ignition conditions for mixture realized at critical values of fuel concentration and it temperature.
Theoretical investigations of interconnected heat and mass transfer process in the system (Fig. 1) were carried out for parallelepiped-shape particle with sizes L p (L p = X 1 ) and H p (H p = Y 4 -Y 2 ).Sizes of solution area L and H were changed greatly more than L p and H p .
Follow ignition conditions were assumed [7]: 1. Heat released due to chemical reaction of fuel vapors oxidation is more than heat transferred from a hot particle to liquid condensed substance and air.2. Temperature of gas-vapor mixture exceeds the initial temperature of a particle.

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Thermophysical Basis of Energy Technologies

Mathematical model
System of transient differential equations [13,14] for concerned gas-phase ignition process (Fig. 1) includes thermal convection equation, Poisson's equation, thermal conduction equation, diffusion equation and balance equation for fuel vapors in the air; thermal conduction equations for hot particle and liquid fuel.At dimensionless variables (0 )); t is time, s; t d is ignition delay time, s; t m is time scale, s; X and Y are dimensionless coordinates of Cartesian display correspond to x and y; Sh is Strouhal number; is dimensionless vortex velocity vector; is dimensionless time; d is dimensionless ignition delay time ( d = t d /t m ); U and V are dimensionless rates of fuel vapors at projection on x and y; Re is Reynolds number; Gr is Grashof number; is dimensionless temperature; is dimensionless stream function; Pr is Prandtl number; Q o is heat of oxidation reaction of fuel vapors in air, J/kg; W o is mass rate of fuel vapors oxidation in air, kg/m 3 s; is density, kg/m 3 ; C is specific heat, J/(kg • K); T is temperature, K; T is differential temperature ( T = T m -T 0 ), K; T m is temperature scale, K; T p is initial particle temperature, K; T 0 is initial liquid and air temperature, K; V m is rate scale of convection (V m = g T L), m/s; g is gravitational acceleration, m/s 2 ; is solid coefficient of expansion, K −1 ; C f is mass concentration of fuel vapors in gas-vapor mixture; C o is mass concentration of oxidant in gas-vapor mixture; Sc is Schmidt number; Fo is Fourier number; Q cr is heat of energy source crystallization, J/kg; W cr is mass rate of particle crystallization, kg/(m 2 • s); is thermal conductivity, W/(m.K); subscripts 1, 2, 3, 4, 5 correspond to air, hot particle, liquid fuel, fuel vapor, gas-vapor mixture.
Initial ( = 0) conditions: boundary perfect thermal contact conditions were accepted for thermal conduction equations, besides condition of equality to zero gradients of corresponding functions for thermal convection, diffusion and Poisson equations were assumed; on boundaries "liquid -fuel vapor" conditions of equal to zero gradients of corresponding functions were accepted for all equations.
The scale magnitudes were used to transfer dimensionless variables: L is characteristic size of solution area, m; V m is scale of convection rate of fuel vapors nearby liquid surface, m/s; t m is time scale, s (t m =5 s); T m is temperature scale, K (T m = 1000 K).Formulas for V m , W cr , W o and dimensionless complexes are presented in papers [10][11][12].
The volume fractions of vapor and gas components in mixture were calculated from it mass concentrations according to the following expressions: Thermophysical characteristics of gas-vapor mixture as heterogeneous system were calculated according to the formulas: Size of fuel vapor gap between particle and liquid were determined at algorithm given in the paper [11].

Solution method
The solution algorithm of Eqs. ( 1)-( 7) was similar to one used in papers [10][11][12].This system of equations with initial and boundary conditions was solved by the finite difference method.The equations of elliptic type (1) and ( 2) were solved by the alternating directions method.The locallyone-dimensional method was applied to the solution of difference analogues for Eqs. ( 3)- (7).Nonlinear difference analogues of differential equations were solved by iteration method.The sweep method using the implicit four-dot difference scheme was applied to solution of one-dimensional difference equations.The reliability of numerical research results was determined by the verification of difference scheme conservation.The error of the energy conservation law in the field of the solution area did not exceed 2.5%.

Thermophysical Basis of Energy Technologies
Numerical values of dimensionless ignition delay time * * d versus dimensionless initial particle temperature p are presented in the Table 1.The analysis of ignition delay time shows that heat and mass transfer processes at system (Fig. 1) are implemented with less d at increase of initial particle temperature.It can be explained by the increase of heat transfer rates from hot particle to liquid fuel and air.At this conditions temperature of gas-vapor mixture near particle increases.Accordingly probability of mixture ignition also increases.
It is significant that maximum deviation of * * d and * d [11] is not more than 11%.The relatively small numerical values (Table 1) can be explained by no significant change of thermophysical characteristics for substances at temperature interval ( p = 1-2).This result confirms relevancy of using for analysis of investigated process of mathematical model with assumption = const, C = const, = const for substances [11].
Welding and cutting of metal construction are often accompanied by particle formation at molten state.It can be supposed that energy transfer by phase changes (crystallization of hot particle material) is significant when particle interconnected with combustible liquid.Values of dimensionless ignition delay time * * d versus dimensionless initial temperature p of hot particle are presented in the Table 2.The initial temperature of energy source is varied over the range including particle melting temperature.For steel particle dimensionless melting temperature is cr = 1.5.
It is obvious (Table 2) that ignition delay time * * d is significantly decreased in comparison with * * * d at initial particle temperature more than cr .It result can be explained by the additional heat release due to the crystallization of metal and consequently the acceleration of physical and chemical processes within the system (Fig. 1).According to the Table 2 values of are significantly increased at p that exceed particle melting temperature ( p > 1.9).However at p less than 1.6 the crystallization process of the particle negligently influence on numerical results of researches (Table 2).By this factor can be neglected at these temperatures.01025-p.5 Numerical research results allow drawing a conclusion that model [7] of solid condensed substance ignition by massive energy source (hot plate) can not be used at analysis of ignition by single hot particle.It is due to the fact that small-size particle have limited heat content.Such particle is strongly cooled at immersion in liquid.Consequently temperature of particle surfaces and heat flow in evaporation zone strongly decrease.The ignition process at these conditions is often transient.Therefore ignition delay time of liquid by heated single particle is significantly limited.It can not increase 1 s in contrast to values of analogue characteristic (more than 10 s) for solid condensed substances [7] by heated massive energy source (hot plate).
The main peculiarities of investigated process are endothermic phase change at evaporation of liquid fuel and exothermic phase change at crystallization of hot particle material unlike to ignition process of solid fuel by single particle.Therefore methods, algorithms and models used at numerical investigation of powders and solid fuels ignition [7] can not be effective for research of liquid fuels ignition.
Distributions of and C f at the system "hot particle -gasoline -air" at p =1, H * p = 0.2, L * p = 0.05 (at ignition moment) graphically illustrated in Figs. 2 and 3. Isotherms ( ) are presented in Fig. 2. It is shown that the cooling of the particle occurs from all its borders.Due to the fact the ignition of gas-vapor mixture realizes at small sizes gas area over particle (X = 0, 0.5 < Y < 0.55).Obtained result can be explained by specific character of investigated process.Endothermic effect of phase exchange of combustible liquid evaporation is the reason of vapor temperature on the boundary "liquid -air" is not enough to ignite gas-vapor mixture.Ignition begins possible only after the additional heating fuel vapors while its moving nearby the side edges 01025-p.6 Thermophysical Basis of Energy Technologies particle.This is the reason of significant heterogeneousness of temperatures (Fig. 2) and fuel vapors concentration (Fig. 3) which is observed within system (Fig. 1).
It was established (Figure 3) that local maximum of fuel concentration C f locates nearby the evaporation boundary (0.1 < X < 0.4, Y = 0.45).Fuel concentration decreases at vapor moving from liquid surface to gas area.It is caused by the most part of fuel vapor enter the air nearby hot particle.The fluxes of heated vapors unite while reaching the upper side of the particle.Therefore C f is increased in this gas area.At the ignition moment the rate of combustible vapors oxidation raises rapidly and it concentration decreases near energy source vicinity (Fig. 3).
Analysis of Figs. 2 and 3 shows that areas of significant changes of main parameters for investigated process have small sizes relative to energy source sizes.It factor and other factors (high temperature and small sizes of energy source, high rates of physical and chemical processes) complicate experiment investigation of liquid ignition processes at concerned statement (Fig. 1).
It is obvious that determination of kinetics for ignition of liquid by single hot particle realizes with difficult factors which are no characteristic for ignition of combustible liquid drops on hot surface [15].Ignition delay time at concerned system depends on sizes and heat content of particle.Correspondingly values of activation energy E and preexponential factor k 0 also change.

Conclusions
Numerical research results of interconnected heat and mass transfer processes at the ignition of liquid fuel by heated till high temperatures particle allow drawing a conclusion about obvious expressed 01025-p.7 mechanism of gas-phase ignition for combustible liquids.Oxidation reaction for fuel vapor begins at the significant increase of mixture temperature over the fuel evaporation temperature.Endothermic phase change at evaporation of liquid fuel reduces rates of heat and mass transfer in a system "hot particlegasoline -air".The highest rates of heat and mass transfer processes are possible at temperature of hot particle higher than melting temperature of it material due to exothermic phase change at crystallization of hot particle material.
Mathematical model and investigated regularities can be useful at development of power effective ignition system due to heat transfer enhancement at local energy sources interaction with various combustible liquid substances.
The reported study was supported by State task "Nauka" (code of the federal target scientific and technical program No. 2.1321.2014).

Figure 3 .
Figure 3. Distribution of fuel concentration C f at the system "hot particle -gasoline -air" at ignition moment ( d = 0.286).
boundary perfect thermal contact conditions taking into account liquid evaporation were accepted for thermal conduction equation, besides second-type boundary conditions taking into account vapors blow-in on border were assumed for thermal convection, diffusion and Poisson equations;on axis of symmetry and boundaries (

Table 1 .
Dimensionless ignition delay time at the system "hot particle -gasoline -air" versus dimensionless initial particle temperature p .

Table 2 .
Dimensionless ignition delay time at the system "hot particle -gasoline -air" versus dimensionless initial particle temperature p .
d is dimensionless ignition delay time without taking into account the particle material crystallization; * * * d is dimensionless ignition delay time at the taking into account the particle material crystallization; is deviation ( = ( * * d - * * * d )/ * * d • 100), %.