Analytical Study of Solidification of Liquid Alloy in a Two-Stage Solidification Front Zygmunt Lipnicki 1 , Adam W. Bydałek 2* , Tomasz Małolepszy 3 1 Institute of Environmental Engineering, University of Zielona Góra, Zielona Góra 65-516, Poland 2 Institute of Materials and Biomedical Engineering, University of Zielona Góra, Zielona Góra 65-516, Poland 3 Institute of Mathematics, University of Zielona Góra, Zielona Góra 65-516, Poland Corresponding Author Email: a.bydalek@iimb.uz.zgora.pl https://doi.org/10.18280/ijht.390302 ABSTRACT Received: 19 April 2021 Accepted: 27 May 2021 In this paper we are investigating theoretically the process of solidification of a liquid alloy in the space. A simplified quasi steady-state analytical model for the solidification process is used. This model describes the phase change phenomenon with the two-phase solidification front, so the two-phase layer between the liquidus and the solidus is analyzed with a solidification liquid being overheated above the solidification temperature. The solution of the problem can be reduced to the solution of a system of differential equations, which has been solved numerically. From this model, an influence of various dimensionless parameters on the solidification process can be clearly seen. The obtained numerical results are presented in graphical figure. In addition, the variable coefficient of a heat transfer on the surface of the solidification front during the solidification process is also calculated. Keywords: development of a simple analytical model, heat transfer, phase change, two-phase layer 1. INTRODUCTION One of the stages during a manufacturing process of an alloy is its solidification [1-12]. A very interesting and important publication on an alloy solidification is the work of Beckermann and Viskanta [1] from 1993, where the mathematical modeling of mass, momentum, heat and transport phenomena that accompany the alloy solidification is reviewed. This article focuses mainly on the role of the two- phase layer on the solidification front in the casting of metal alloys. As the two-phase layer at the front plays a particularly important role in the solidification process, there are many studies in the professional literature related to the modeling of solidification process that includes the two-phase zone [6, 8, 10-12]. However, there are only a few studies in the literature that describe the macroscopic process of alloying in a simple manner [5, 7, 11], which is one of the goals of this work. The important aim of the work is, however, to determine, based on the theoretical foundations, the heat flow between the casting mold and the metal that solidifies in it. A simplified quasi-stationary analytical model of the crystallization process has been proposed. The phenomenon of the phase change with a two-stage solidification front, which takes into account the areas of columnar and equiaxial crystals, has been also described. In this model, the two-phase layer between liquidus and solidus is analyzed. To adjust the model to real conditions, overheating of the metal above the liquidus temperature is included. Furthermore, the variable heat transfer coefficient on the surface of the solidification front during the solidification process is determined. To our best knowledge, there are no studies in the scientific literature that theoretically analyze the solidification process in the two-phase zone. The distribution of heat sources and temperature in this zone in time and space are particularly interesting and important. The mentioned parameters significantly influence the development and the progress of the solidification process in this two-phase space. A novelty in the presented work is the analytical determination of the development of the solidification front: its position and velocity in the solidifying alloy on the basis of Stefan equations, and in particular the determination of the temperature distribution in a two-phase space using the classical Poisson equations. This publication presents an example of the solidification process in the continuous casting of a brass alloy. The article discusses the problem of solidification of alloys with particular emphasis on the two-stage crystallization front. The research part presents the simulation results of the two- stage front of the crystallization setting in continuous casting. Finally, conclusions were built, indicating further research directions. 2. GENERAL STATEMENT OF THE PROBLEM There are many solutions, both analytical and numerical, in the literature relating to the modeling of crystallization processes [1-12]. So far, however, none has fully described these nonlinear and non-stationary phenomena. The process of solidification of the metal alloy in this work is described analytically with quasi-stationary heat flow equations using the Stefan solidification model. In this work, a numerical analytical model of solidification of a metal alloy (brass) was solved. The time and velocity of its solidification were determined. In the two-phase zone, the quasi-stationary solidification process with heat sources was solved using the Poisson differential equation. The calculation results are presented in the form of graphs and tables. The diagram of solidification of the cooled metal with the participation of a crystallizer (referred further as the cold wall) International Journal of Heat and Technology Vol. 39, No. 3, June, 2021, pp. 681-687 Journal homepage: http://iieta.org/journals/ijht 681