Research on a Class of Ordinary Differential Equations and Application in Metallurgy

Abstract

Abstract. This paper mainly discusses the ordinary differential equations of the same class, and when the coefficient is applied by different values or functions, its solutions are also different. Then this equation was solved by analytical method and numerical method, which is applied to solving the practical problems of waste heat in molten slag. The classical four-stage Runge-kutta method and Matlab program were also used for solving this equation, and all proved that this method has practical values.

Differential Equations Models and Their Applications in Metallurgy

“…Let us consider the following delay differential equation: For sake of comparison of our delayed model to the non delayed model introduced in ( [8]), we are using the same coefficients in our numerical study. Also, as in ( [8]), we take the initial temperature of the slag to be T in = T (0) = 1773K. Thus, in our following numerical investigation, we will consider If the delay τ is larger enough than the step of integration, the above fourth Runge-Kutta method can be extended to a fourth Runge Kutta algorithm with delay by expressive constants K1 , K2 , K3 and K4 with the delayed form of the f (t, T (t − τ ), T (t)).…”

“…where T is the temperature of the molten slag, t is the time variable and f is a given function that describes the process of energy transfer. From physical point of view, as stated in [8], models above are based on the fact that convection and radiation are mainly the two manners of heat dissipation, when liquid slag is quenched. The heat transfer between the quenching fluid and the very hot slag is governed by the energy equality cV ρ dT dt (t) = Ah(T (t) − T f ) + Aεσ 0 (T 4 (t) − T 4 f ),…”