Differential Equations Models and Their Applications in Metallurgy

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

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.

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References

  1. O. Arino, M.L. Hbid, E. Ait Dads, Delay differential equations and applications, Springer, Netherlands, (2006).
  2. N. A. Ayunni Sabri, M. bin Mamat, Solving delay differential equations (DDEs) using Nakashima’s 2 stages 4 th order Pseudo-Runge-Kutta Method, World Appl. Sci. J. 21 (2013), 181-186.
  3. L. Cheng, M. Xu, L. Wang, From Boltzmann transport equation to single-phase-lagging heat conduction, Int. J. Heat Mass Transfer. 51 (2008), 6018–6023. https://doi.org/10.1016/j.ijheatmasstransfer.2008.04.004.
  4. A. S. Eremin, A. R. Humphries, A. A. Lobaskin, Some issues with the numerical treatment of delay differential equations, AIP Conf. Proc. 2293 (2020), 100003. https://doi.org/10.1063/5.0027149.
  5. J. K. Hale, S. M. Verduyn Lunel, Introduction to functional differential equations, Springer Verlag, Berlin, 1993.
  6. S. Hamze, E. Witrant, D. Bresch-Pietri, C. Fauvel, Estimating heat-transport and time-delays in a heat exchanger, in: 2018 IEEE Conference on Control Technology and Applications (CCTA), IEEE, Copenhagen, 2018: pp. 1514–1519. https://doi.org/10.1109/CCTA.2018.8511359.
  7. F. Ismail, R. A. Al-Khasawneh, A. S. Lwin, M. B. Suleiman, Numerical treatment of delay differential equations by Runge–Kutta method using Hermite interpolation, MATEMATIKA: Malaysian J. Ind. Appl. Math. 18 (2002), 79-90
  8. C. Liu, H. Wu, J. Chang, Research on a class of ordinary differential equations and application in metallurgy, in: R. Zhu, Y. Zhang, B. Liu, C. Liu (Eds.), Information Computing and Applications, Springer Berlin Heidelberg, Berlin, Heidelberg, 2010: pp. 391–397. https://doi.org/10.1007/978-3-642-16339-5_52.
  9. M. Massoudi, P. Wang, A brief review of viscosity models for slag in coal gasification, DOE/NETL-2012/1533, National Energy Technology Laboratory, Pittsburgh, PA, 2011.
  10. E. Matinde, G.S. Simate, S. Ndlovu, Mining and metallurgical wastes: a review of recycling and re-use practices, J. South. Afr. Inst. Min. Metall. 118 (2018), 825-844. https://doi.org/10.17159/2411-9717/2018/v118n8a5.
  11. D. Xie, Y. Pan, R. Flann, B. Washington, S. Sanetsis, J. Donnelley et al., Heat recovery from slag through dry granulation, in: 1st CSRP Annual Conference. Melbourne (Australia), vol. CSIRO Minerals, pp. 29–30, 2007.
  12. M. Xu, L. Wang, Dual-phase-lagging heat conduction based on Boltzmann transport equation, Int. J. Heat Mass Transfer 48 (2005), 5616–5624. https://doi.org/10.1016/j.ijheatmasstransfer.2005.05.040.
  13. H. Zhang, H. Wang, X. Zhu, Y.-J. Qiu, K. Li, R. Chen and Q. Liao, A review of waste heat recovery technologies towards molten slag in steel industry, Appl. Energy 112 (2013), 956-966. https://doi.org/10.1016/j.apenergy.2013.02.019.