Estimations of the parameters of a thermal explosion in a triaxial ellipsoid
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Estimations of the parameters of a thermal explosion in a triaxial ellipsoid G. N. Kuvyrkin, I. Y. Savelyeva
and V. S. Zarubin
Abstract. The formulation of the nonlinear problem corresponding to the process of stationary heat conduction in homogeneous triaxial ellipsoid with increasing temperature and intensity of volumetric energy release was used to build a variational form of a mathematical model of this process. This form includes a functional defined on a set of continuous and piecewise differentiable functions that approximate the temperature distribution in the volume of an ellipsoid and take a given value of temperature on its surface. An analysis of the stationary points of the functional makes it possible to estimate the combination of determining parameters corresponding to the temperature distribution in the ellipsoid before the occurrence of a thermal explosion. Comparison of the integral error caused by the use of various approximating functions allows to choose the function that most accurately describes the temperature state of the ellipsoid preceding the thermal explosion. Estimations of the parameters of the thermal explosion are obtained under the assumption of an exponential increase in the intensity of volumetric energy release in an ellipsoid with increasing temperature. Mathematics Subject Classification. Primary 80M30; Secondary 97M50. Keywords. Variational form of the model, Stationary functional point, Thermal explosion, Temperature state.
1. Introduction The occurrence of volumetric energy release in select solids and particles caused by exothermic reactions and thermal decomposition processes occurring in them is characteristic for chemical-technological processes and during storage of various substances and products [1–4]. A ball, for which, with an exponential dependence of the volumetric energy release intensity, a critical combination of parameters is known corresponding to the occurrence of the so-called thermal explosion [5,6] is usually taken as the average shape of such bodies and particles. In this case, the thermal energy released in the volume of the ball can no longer be diverted through its surface to the environment, i.e., the existence of steady temperature distribution is impossible. When the shape of a solid body deviates from the ball, the ratio of its area to volume increases, which, ceteris paribus, improves the conditions for the removal of thermal energy, released in the volume, to the environment. However, even for an ellipsoid of revolution, with a semi-axis ratio close to unity, that differs little in shape from the ball, the theoretical determination of the critical combination of parameters characterizing the onset of a thermal explosion requires overcoming significant mathematical difficulties associated with the analysis of the existence and stability of the solution of a nonlinear partial differential equation [6]. With an increase in the deviation of this ratio from unity or a transition to an arbitrary t
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