Heat loss analysis in a radiating slab of variable thermal conductivity

Abstract

This article investigates the transfer of heat in a stockpile of reactive materials, that is assumed to lose heat to the environment by radiation. The study is modeled in a rectangular slab whose materials are of variable thermal conductivity. The stockpile's reactive material in this context is one that readily reacts with the oxygen trapped within the stockpile due to exothermic chemical reaction. The study of the combustion process in this case is conducted theoretically by using the Mathematical approach. The differential equation governing the problem is tackled numerically by applying the Runge-Kutta Fehlberg (RKF45) method coupled with the Shooting technique. To investigate the heat transfer phenomena, some kinetic parameters embedded in the governing differential equation, are varied to observe the behavior of the temperature profiles during the combustion process. The results obtained from the temperature profiles, are depicted graphically and discussed accordingly. It was discovered that kinetic phenomena such as the reaction rate parameter, accelerates the exothermic chemical reaction. However, the radiation parameter decelerates the exothermic chemical reaction by lowering the temperature profiles.

Reactant Consumption and Thermal Decomposition Analysis in a Two-Step Combustible Slab

“…(1) and (2). The reactive slab is also assumed to lose heat to the surrounding environment by radiation, and the heat loss is expressed as 𝑞𝑞 = 𝜇𝜇𝜇𝜇(𝑇𝑇 4 − 𝑇𝑇 𝑤𝑤 4 ), according to Stefan-Boltzmann's law. In this case 𝜇𝜇 is the solid slab's surface emissivity, where 𝜇𝜇 is within [0,1], 𝜇𝜇 is the Stefan-Boltzmann constant whose approximate value is 5.6703× 10 −8 𝑊𝑊/𝑚𝑚 2 𝐾𝐾 4 , 𝑇𝑇 is the slab's absolute temperature and 𝑇𝑇 𝑤𝑤 is the ambient temperature.…”

“…The Figure 1 below illustrates the geometry of the problem. Following Lebelo and Makinde [3], Lebelo and Moloi [4], and Lebelo and Makinde [12] the nonlinear ordinary differential equations governing the problem, for the energy and reactant consumption, respectively, are expressed as…”

“…𝑄𝑄 1 and 𝑄𝑄 2 represent the heat of reaction for the first and second steps respectively, 𝐴𝐴 1 and 𝐴𝐴 2 are the rate constants for both steps accordingly, 𝐸𝐸 1 and 𝐸𝐸 2 are the respective activation energies, 𝑅𝑅 the universal gas constant, 𝑙𝑙 the Planck number, 𝑣𝑣 the vibration frequency, 𝐾𝐾 the Boltzmann constant, and 𝑛𝑛 the order of exothermic chemical reaction. The reaction kinetics type is expressed in terms of the symbol 𝑚𝑚, where 𝑚𝑚 = −2 represents the Sensitized (laser induced) kinetics, 𝑚𝑚 = 0 the Arrhenius kinetics, and 𝑚𝑚 = 0.5 the Bimolecular kinetics, following [3,4,12]. The following dimensionless parameters are introduced to Eqs.…”

“…The spontaneous reaction in the picture, causes the accumulation of heat within the stockpile to raise the temperature and should the heat loss to the surrounding environment be exceeded by heat generation due to continued exothermic chemical reaction within the stockpile, thermal runaway that leads to self-ignition may take place [1,2,3]. A one-step combustion process in a rectangular slab was studied extensively in [4,5,6], but in this study, a two-step combustion process, like the one taking place in an automobile exhaust system, is considered. An example of a two-step combustion mechanism is that of combustion of methane with the elimination of the toxic gas carbon monoxide (CO).…”