Differential Equations Models and Their Applications in Metallurgy

Ridha Selmi,Amel Touati,Abir Sboui,Muflih Alhazmi,Hechmi Hattab,Hanen Louati

doi:10.28924/2291-8639-20-2022-7

Abstract

This paper focus on the heat recovery from the metallurgical and mining wastes. We propose and study a new and more realistic mathematical model for heat recovery from molten slag. Our model is based on time delay differential equations. In the theoretical part, we prove that a unique solution exists to the mathematical problem. In the numerical part, we establish an algorithm based on explicit fourth order Runge-Kutta method with delay; the new feature is that the delay must be larger enough than the step of integration. Compared to the classical model (without time delay), the numerical test proves that our model is more efficient and industrially more profitable.

Highlights

Heat recovery from molten slag is nowadays one of the relevant manners to valorize the huge amount of thermal energy provided by such mining waste
From mathematical point of view, models dealing with heat recovery from molten slag are formulated using ordinary differential equations that are taking in account only instantaneous properties of the molten slag
Memory effects caused by the viscoelasticity of the molten slag: As for such property, based on physical experiences and chemical analysis the detailed review paper [9] and references therein putted in evidence the viscoelastic response of molten slag, where according to the authors "at high temperatures, the time dependent constitutive relations are needed for the stress-stress relationship, and for the heat flux vector"(page 26)

Summary

Introduction

Heat recovery from molten slag is nowadays one of the relevant manners to valorize the huge amount of thermal energy provided by such mining waste. Heat transfer; heat recovery from molten slag; delayed convection; mathematical model; time delay-differential equations; Runge-Kutta method; numerical solution. From mathematical point of view, models dealing with heat recovery from molten slag are formulated using ordinary differential equations that are taking in account only instantaneous properties of the molten slag They naturally consider the ordinary time derivative. Memory effects caused by the viscoelasticity of the molten slag: As for such property, based on physical experiences and chemical analysis the detailed review paper [9] and references therein putted in evidence the viscoelastic response of molten slag, where according to the authors "at high temperatures (which is our case), the time dependent constitutive relations are needed for the stress-stress relationship, and for the heat flux vector"(page 26).

Functional setting and theoretical results

Numerical investigation

Conclusion