Qualitative analysis of thermal explosion of sodium droplet with variable thermophysical properties and thermal radiations

There are many coolants frequently used in the industry for controlling not only heat transfer, but also temperature distribution in a confined domain. However, little is known on the thermal properties of sodium droplets. The qualitative analysis of differential equations that model the thermal explosion, nonlinear dynamic of sodium droplet with variable thermophysical properties when thermal radiations are considered as suggested by Cogley et al, Sohrab et al, and P-1 approximation Sazhin et al is deliberated upon in this study. The governing equations, first-order nonlinear ordinary differential equations, are nondimensionalized using the appropriate similarity variables. The existence and uniqueness of the solutions, concavity, and convexity of the temperature distribution, and positivity nature of the solutions of the dimensionless governing equations are established. It is concluded that there exists a solution for a certain range of the admissible parameters and when the reduced activation energy is negative and temperature distribution fits concavity. More so, the major criteria to obtain a positive solution are outlined in this study.