This paper studies two-step exothermic reactions with and without reactants consumption subject to Arrhenius kinetics model. The resulting system of equations modelling the physical problem is coupled non-linear ordinary differential equations. Analytical investigations are possible based on some classical Frank-Kamenetskii criteria. Exact analytical expressions are derived for temperatures and ignition times based on the assumption of without reactant consumption. Thereafter, the entire system of equations is solved numerically to account for effect of various dimensionless parameters with reactants consumption. The numerical solutions, graphs and the statistical measures of skewness and kurtosis for dimensionless concentrations and , and temperature profiles are presented in this work.
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.
The present study addresses the problem of ignition of a single sodium droplet, which is an important issue for the nuclear facilities safety. The study follows the approach of previous works and extends the results of those papers to the case of radiative heat loss. The contribution of the thermal radiation is taken into account based on the P-1 approximation for thermal radiation transfer. An extension of solutions of the existing model is obtained in the presence of radiative heat loss for ignition time and critical temperature by exploiting the sensitivity of the process to large chemical activation energy. Different qualitative effects of varying the dimensionless convective heat loss parameter with ignition time and critical temperature are presented in the graphs. The results show that the inclusion of additional heat sink mechanism, that is, radiative heat loss, causes significant delays in the ignition time and reduces the critical temperature with respect to results of previous studies.
The effect of the radiation on the droplet ignition was investigated. The Rosseland approximation was employed in the modelling of the radiation heat transfer and temperatures of the droplet. The governing equation was expressed in dimensionless form and was solved analytically according to Makino [3]. The MAPLE algebraic computation package was employed to implement the analytical solution to generate the numerical results. The results showed that radiation contributes to the ignition delay and increase the temperature at which the thermal ignition will occur.
This paper is essentially devoted to the property of solutions to a system of ordinary differential equations modelling thermal explosion in combustible dusty gas containing fuel droplet with generalised temperature dependent rate of reaction governed by Arrhenius power-law model. Theorems are stated and proofs provided on the qualitative properties of new system equations governing the physical model. New closed-form solutions are obtained based on quadratic approximations to the Arrhenius terms under realistic conditions. The results show that the delay before ignition depend significantly on interphase heat exchange parameter and energy needed to transfer heat from gas phase to solid phase parameter . It is intended to describe the numerical analysis of the new problem in a later paper.
The problem of thermal explosion in combustible gas mixtures containing fuel droplets is extended to permit a more general temperature dependent rate of reaction for most typical practical reactions based on Arrhenius equation under physically reasonable assumptions. A detailed numerical analysis of the resulting system of coupled non-linear ordinary differential equations is performed to account for numeric exponent effects relating to most typical practical reactions such as sensitized, Arrhenius and bimolecular reactions respectively. The computed results reveal different dynamic delay-type behaviours and are illustrated graphically in this study.
In this paper, we study the properties of solutions to a system of coupled non-linear ordinary differential equations governing the problem of thermal explosion in a combustible gas containing fuel droplets with generalised temperature dependent rate of reaction. Theorems on the properties of solutions to the new physical model problem such as existence and uniqueness, concavity and convexity are formulated based on some criteria and the proofs of the theorems are established accordingly. Analytical investigations are possible based on some reasonable assumptions and approximations. Consequently, these provide analytical solutions to the temperature of the combustible gas and radius of the fuel droplets. The results reveal different dynamic behaviours and are illustrated graphically in this work. Also, different qualitative effects of varying the dimensionless parameters are reported.
We examine steady incompressible flow of viscous liquids between parallel heated walls of plane Couette device. The temperature of the upper and lower walls of the device are maintained at T =Tb and T = T0 respectively. Of a particular interest are exact analytical solutions of the coupled nonlinear differential equations resulting from plane Couette flow obtained for the temperature and velocity distributions respectively. The criterion for which the solutions are valid was determined by the temperature difference, α, between the upper and lower walls. The analysis reveals that the shear stress obtained at the walls exists when the temperature difference α > 0.
This study discusses the influence of heat loss on the critical ignition temperature and Frank-Kamenetskii parameter of a non-linear ordinary differential equation arising in thermal sensitized reaction. The reaction obeys the Arrhenius expression with temperature dependent preexponential factor, taking heat exchange between the reacting material and its surrounding into account. The consequences of heat loss are explored within the framework of one dimensional, steady state model. The numerical estimations based on shooting method techniques show the effect of heat loss parameter on the critical values of ignition temperature and Frank-Kamenetskii parameter.
We invstigate the effect of Frank-Kamenetskii parameter d on the thermal explosion in a flammable gas with fuel droplets. It is shown that the induction period decreases as d increases. The droplets increases the volumes of the gas as they evaporate. The combustion medium is quasi-steady after explosion and the medium could no longer be regarded as homogeneous.