1. Introduction

The mathematical analysis of heat transfer in a billet during the continuous casting process has been investigated from the early stages of the development of continuous casting machines. In this study, some fundamental works related to the numerical solution of heat transfer in solidifying billets are presented. A pioneer work is mentioned in [1] in which a mathematical model of heat transfer in continuously cast steel slabs is described. A computer program was developed to model the problem using a 1D transient conduction equation with the appropriate boundary conditions. Some years later, the design of the mold and spray sections of a continuous casting machine were examined in detail with the aid of a 2D heat-flow mathematical model [2]. Experimental data obtained from commercial casters were used for the validation of the deduced results. In that monumental work, the various aspects of heat transfer in the various sections of a casting machine were

analyzed in detail. Later on, surface and subsurface defects in the cast billets became apparent, and research work included mechanical phenomena (like stress and strain) together with the heat transfer analysis; an early typical work on the subject was presented in [3]. It is worth mentioning that the caster automation includes most of the heat transfer formulation and analysis, albeit not published. Consequently, the present literature survey mostly presents indicative published works upon heat transfer analysis. In the 1990s, the continuing casting of steel started maturing, and this was reflected by two studies with practical implementations [4, 5]. In a review paper [6], the evolution of heat transfer and mechanical studies for the continuous casting were presented. Furthermore, an analysis on the ideal taper prediction for billet casting and a thorough analysis of thermomechanical behavior in billet casting were also presented [7, 8]. The salient features of heat transfer in the secondary cooling zone during the continuous casting of steel were examined in [9]. Primary cooling is considered the start-up of steel solidification process in a water-cooled copper mold, and secondary cooling zone is considered the region just after the mold until the end of the caster in which the billet is cooled by spray nozzles or air-mist systems. Radiation heat exchange between the billet surface and the environment plays also an important role in the secondary cooling zone. Fluid-flow phenomena in the mold captured the interest, and as of that, heat transfer studies were also reported on the subject later on; a typical work is given in [10]. In most similar works, general-purpose heat-transfer software has been deployed either as a direct prediction tool or as a verification one. A typical work is given in [11] in which a 1D heat-transfer simulation program (CON1D) was successfully verified against a 3D finite element model developed in Abaqus. Near-netshape cast sections appeared in industrial practice in the early 2000, and therefore, their thermal-mechanical modeling during casting was developed as well [12]; in this type of analysis, the use of a package like Abaqus has been indispensable. In a relatively recent work [13], the 1D differential equation for heat transfer was solved in order to study the evolved microstructures during the solidification of round billets (so, the radial direction for heat transfer prevails). In the cases under study, the billet diameters were in the range from 210 up to 410 mm; the validation of their model was performed by deploying a semi-analytical model [14] for the prediction of the surface temperature of a cast round billet. The numerical model was used to calculate the local solidification time, the local gradient, and the local solidification time as a function of the distance from the round billet surface; furthermore, a simplified relation for the prediction of the columnar to equaxial transition was proposed. Deploying the finite volume method, a real-time 2D heat transfer and solidification model for continuous casting [15] were developed and tested online. The behavior was satisfactory with less than 10°C deviations on surface central temperatures. A dynamic heat transfer model was developed [16] in order to study the effect of the casting speed on the temperature field of continuously cast billets; various steel grades were taken under consideration. A real-time heat-transfer model and a heat-transfer coefficient identification method [17] were developed. The model validation was achieved by the measurement of surface temperatures by a CCD system that appeared to eliminate the impact of scales on the temperature measurement and keep the measured surface temperature fluctuation within the range of ˜10°C. A finite volume method and the alternating direction implicit algorithm (ADI) were selected in order to develop a real-time heat transfer model [18] for the dynamic control of continuous cast billets. The system was applied online to control the electric current of the final electromagnetic stirring (FEMS) system. A simulation model of solidification and heat transfer of 150-mm-square billets was developed [19] for the continuous casting of grade 40 Cr. Nailing tests and temperature measurements were applied in order to fine-tune

A Numerical Solution Model for the Heat Transfer in Octagonal Billets DOI: http://dx.doi.org/10.5772/intechopen.84305

the model with maximum errors of less than 2%. In another study [20], different micro-segregation models coupled with fluid flow and heat transfer were run to study macrosegregation phenomena in a round billet with a 165-mm diameter. It was predicted that heavy centerline segregation occurs at the end of solidification when the central solid fraction reaches the value of 0.7. The need for high-speed casting under very controllable conditions has led the researchers to seek billet sections that can succeed in the demanding steelmaking environment. Recently, one of the leading manufacturers in the steelmaking sector, Danieli, has announced [21] the octagonal billet as a proven potential shape for successful high-speed casting. Consequently, the present author has taken the opportunity to study the heat transfer in a solidifying billet based on fundamental principles. The work has been carried out for a larger billet size than the one tested in [21] and for two special steel grades, the peritectic grade S355 J2 and the medium-high-grade 42CrMo4, that are both important for many end-user applications.