Studies on the Effects of Interphase Heat Exchange during Thermal Explosion in a Combustible Dusty Gas with General Arrhenius Reaction-Rate Laws

A mathematical model for thermal explosion in a combustible dusty gas containing fuel droplets with general Arrhenius reaction-rate laws, convective and radiative heat losses, and interphase heat exchange between gas and inert solid particles is investigated. The objective of the study is to examine the effects of interphase heat exchange between the gas and solid particles on i ignition of reacting gas, ii accumulation of heat by the solid particles during combustion process iii evaporation of the liquid fuel droplets, and iv consumption of reacting gas concentration. The equations governing the physical model with realistic assumptions are stated and nondimensionalised leading to an intractable system of first-order coupled nonlinear differential equations, which is not amenable to exact methods of solution. Therefore, we present numerical solutions as well as different qualitative effects of varying interphase heat exchange parameter. Graphs and Table feature prominently to explain the results obtained.


Introduction
The concept of thermal explosion in combustible dusty gas containing fuel droplets is of great importance in safety aspect of nuclear facilities, furnaces, gas turbines and internal combustion engines, coal mine, and so forth.The essence of the inert solid particles in the combustible gas is to delay the ignition or explosion which could cause catastrophe.The phase of research on combustion and explosions that began at the end of the nineteenth century and continues to this day is associated with the invention of the internal combustion engine, with the development of explosive technology and of internal ballistics for artillery, and, in recent decade, with the extensive introduction of jet and diesel engines.In many to see that the critical and maximum points in the gas temperature-concentration domain could coincide.It was found that the presence of a solid produces more than one critical temperature.The limiting cases of the problem were also offered.The numerical solution showed that the supercritical trajectory shows a thermal runaway for the gas over the solid at the end of reaction.
In this paper, an attempt has been made to extend the problem of thermal explosion in a combustible gas mixture with addition of inert solid particles and to generalise the problem based on temperature dependence of the reaction rate i.e., Arrhenius Power-law model equation 22-24 given as K T g A 0 T g /T g0 n exp −E/R u T g while taking into account convective and radiative heat losses, temperature dependence of density, and thermal conductivity of the gas.Therefore, a generalised physical model for thermal explosion in combustible dusty gas mixture containing fuel droplets is developed in the present paper.The main interest is focused on numerical solutions of the system of coupled nonlinear ordinary differential equations governing the physical model.Parametric analysis is performed, and the results show that the delay before ignition of the reacting gas, accumulation of heat by dusty particles, evaporation of liquid fuel droplets, and consumption of reacting gas concentration depend significantly on interphase heat exchange between the gas and solid particles.

Mathematical Model
In this study, we consider combustible dusty gas with addition of fuel droplets.Dusty gas is a combustible gas with addition of solid particles.It is assumed that the solid phase is inert, monosized, and uniformly heated, and the dusty gas is optically thick.The dependence of the convective heat transfer coefficient on both gas temperature and droplet radii is taken into account.Thermal conductivity of both monoatomic and polyatomic gases is proportional to T g .Therefore, we assume λ g T g λ g0 T g /T g0 .The rate of reaction is based on generalised temperature-dependent Arrhenius's equation.Temperature dependence of the gas density and preexponential factor is given as ρ g T g ρ g0 T g0 /T g and A T g A 0 T g /T g0 n , respectively.Following the realistic assumptions above, the system of governing equations has the form:

2.4
The system 2.1 -2.4 includes energy equations for the reacting gas 2.1 and solid particles 2.2 , a mass equation for the liquid droplet 2.3 and a concentration equation for the reacting gas mixture 2.4 .Initial conditions for the system:

Nondimensional Analysis
In this study, we introduce the following dimensionless variables: We assume that at initial stage the temperature of gas and solid particles are the same, that is, T g0 T s0 .Therefore, using 2.6 in 2.1 -2.5 , the dimensionless system of governing equations has the following form: with the initial conditions:

2.11
In 2.7 -2.11 , the following dimensionless parameters have been introduced:

2.12
Some special cases of the system of coupled nonlinear ordinary differential equations 2.7 -2.10 with initial conditions 2.11 and related problems have been studied for n / 0 and α 3 0 see, e.g., 22, 23, 25 and the references therein .In the events that n 0 and parameters α 1 , α 2 , and α 3 are varied, analyses have been performed both analytically and numerically in 2, 8-17, 26, 27 , and references cited therein .In this study, attention is focused on the generalization of the problem and effect of the interphase heat exchange between the reacting gas and solid particles on the ignition of reacting gas, accumulation of heat by dusty particles, rate of evaporation of liquid fuel droplets, and depletion of reacting gas concentration.

Numerical Computation
In this section, we present numerical solutions of the system of governing equations 2.7 -2.10 with initial conditions 2.11 .Meanwhile, before actual computation, we begin by deriving the exact integral for the system of ordinary differential equations as follows: Multiplying 2.7 by α 4 , we get

3.1
Adding 3.1 and 2.10 together gives

6
Journal of Applied Mathematics Rearranging, we get

3.3
From 2.8 and 2.9 , we obtain where N is constant of integration.Applying the initial condition 2.11 in 3.6 , we get Using 3.7 in 3.6 gives exact integral of the system of equations 2.7 -2.10 as Now, by substituting η in 3.8 into 2.7 , it is possible to obtain the following equation: 3.9 The resulting systems of 2.8 , 2.9 , and 3.9 are coupled nonlinear differential equations and do not possess closed form solutions.Therefore, a numerical technique based on finite difference approximation is used to solve the problems.The effect of the interphase heat exchange parameter is adequately examined on the ignition time, combustible gas temperature, solid particle temperature, droplet radius, and combustible gas concentration as shown in the figures.

Results
The general characteristic features of Figures 1-4 show time histories of dimensionless gas temperature θ g , solid particle temperature θ s , gas concentration η and liquid-fuel droplet radius r for typical practical rate of reactions with various values of heat exchange parameter α 3 when δ 0.3, β 0.05, B 0.008, μ * 5, The qualitative impacts of interphase heat exchange between gas and solid particles α 3 in this study are ascertained in the figures.We notice in Figure 1 the occurrence of convectional thermal explosions in sensitized, Arrhenius, and bimolecular rate of reactions, respectively.Meanwhile in bimolecular rate of reaction, when α 3 1.0, it is quite interesting to note that the problem has a quasi-steady solution after explosion.In this case, the medium is no longer homogeneous and diffusion term has to be taken into consideration.The medium could not be regarded as well stirred.Of a particular interest is the notable delay in the ignition time as the interphase heat exchange between gas and solid particles increases.
In Figure 2, it is observed that the heat exchange between the gas and solid particles raises the temperature of the solid particles at different time for sensitized, Arrhenius, and bimolecular rate of reactions.This implies that the solid particles accumulate heat over a period of time from reacting gas, since the solid particles are inert.
Figure 3 shows that the size of the liquid-fuel droplet radius significantly reduces as heat exchange between the gas and solid particles increases for sensitized, Arrhenius, and bimolecular rate of reactions.This reveals that the rate of evaporation of the liquid-fuel droplet substantially depends on heat exchange between the reacting gas and solid particles.We note that when α 3 0.5 and α 3 1.0, the liquid-fuel droplet completely evaporated just before ignition.
Figure 4 reveals a remarkable reduction in gas concentration as heat exchange between the gas and solid particles increases for sensitized, Arrhenius, and bimolecular rate of reactions.This depicts depletion of gas concentration during the combustion process.
In Figure 5, we present variation of θ g , θ s , r, and η with numeric exponent for various values of α 3 at ignition point.Important facts to note from these results are the following.i As can be seen from Figures 5 a , 5 c , and 5 d , an increase in interphase heat exchange between the gas and solid particles causes significant reduction in the values of gas temperature, droplet radius, and gas concentration whereas in Figure 5 b it raises the value of solid particle temperature at ignition point for sensitized, Arrhenius, and bimolecular reaction rates, respectively.
ii It is also obvious from Figures 5 a and Table 1 shows the variation of dimensionless ignition times τ ig with interphase heat exchange parameter α 3 for typical sensitized, Arrhenius, and bimolecular rate of reactions at  ignition point.It is significant to note from the table that increase in heat exchange between the gas and solid particles causes a notable delay in the ignition time of the combustible gas mixture for sensitized, Arrhenius, and bimolecular rate of reactions.The results also reveal that responsiveness to delay in ignition time is characterized by numeric exponent, which means that delay is more enunciate in sensitized reaction compared to Arrhenius and bimolecular reactions, respectively.

Conclusion
The problem of thermal explosion in combustible dusty gas mixtures containing fuel droplets have been extended to permit a more general temperature dependent rate of reaction for most typical practical reactions under physically reasonable assumptions.The mathematical formulation involves a system of four highly nonlinear ordinary differential equations, which was solved numerically by finite difference approximation scheme.In multiple phase processes, interphase heat exchange plays the role of heat losses in homogeneous combustible gas mixtures.The model with heat losses provides more detailed information about the effects of interphase heat exchange and numeric exponent characterising sensitized, Arrhenius, and bimolecular rate of reactions.The study revealed that interphase heat exchange   between gas and solid particles, and numeric exponent play important roles in determining the following: 1 occurrence of convectional thermal explosions and quasi-steady solution after explosion.
2 significant delay in ignition time.
3 reduction in inert solid particles temperature.
4 remarkable reduction in the size of the liquid-fuel droplet leading to complete evaporation.
The results highlighted above do guarantee safety in practical explosion and combustion devices.

Greek Symbols
ρ: Density of the combustible gaseous mixture kg/m 3 λ: Thermal conductivity Wm −1 K −1 ϕ: Volumetric phase content dimensionless σ: Stefan-Boltzmann's constant β: Dimensionless activation energy τ: Dimensionless time δ: Dimensionless parameter expressing reciprocal of the characteristic time for adiabatic temperature rise α 1 : Dimensionless parameter expressing heat loss via convection from gas phase α 2 : Dimensionless parameter expressing heat loss via radiation from gas phase α 3 : Dimensionless parameter expressing interphase heat exchange between gas and solid particle μ: Dimensionless parameter expressing energy needed to transfer heat from gas phase to solid phase μ * : Dimensionless parameter expressing energy needed to evaporate all fuel droplets ψ: Dimensionless parameter expressing energy needed to consume all gas concentration.
5 b that an increase in numeric exponent n ascertains reduction in the values of gas and solid particles temperature at ignition point, whereas in Figures5 d and 5c we observe increment in the values of gas concentration and insignificant reduction in the values of droplet radius at ignition point, respectively.

3 Figure 5 :
Figure 5: Variation of θ g , θ s , r, and η with numeric exponent for various values of α 3 at ignition point.

A:
Preexponential factor 1/s B: Dimensionless parameter expressing ratio of latent heat of evaporation and specific combustion energy c f : Molar concentration of combustible gaseous mixture kmol/m 3 C pg : Specific heat capacity of the gas phase at constant pressure JK −1 kg −1 C ps : Specific heat capacity of the solid phase at constant pressure JK −1 kg −1 E: Activation energy J/kmol L: Latent heat of evaporation J/kg m f : Molar mass kg/kmol n: Numerical exponent n d : Number of droplets per unit volume m −3 n s : Number of solid particles per unit volume m −3 Q: Specific combustion energy J/kg R d : Droplet radius m R s : Solid particle radius m R u : Universal gas constant Jkmol −1 K −1 t: Time s T g0 : Combustible gas initial temperature K T s0 : Dusty particle initial temperature K T g : Combustible gas temperature K T s : Dusty particle temperature K .

Table 1 :
Variation of ignition times with interphase heat exchange parameter at ignition point when n