Modeling of the transient thermal state of metallurgical ladles is motivated by the need for estimating the drop in temperature of the liquid metal in the ladle. On-line estimation of the state is required, since the same ladle is used in a number of casting cycles with rapid changes in boundary conditions for the temperature field, and the conditions in the current as well as previous cycles affect the thermal state. Although a large number of methods for the numerical solution of conduction-diffusion partial differential equations have been developed, there are still advantages to keeping thermal field computations at a relatively simple level, in contrast to the situation in the design process of ladles, where two-dimensional modeling may be required. Extensive computations under nonverifiable boundary and initial parameter values are not especially suited for real-time simulation of industrial processes. This article presents a novel approach to the solution of the one-dimensional transient heat conduction problem applied to ladle linings, relying on classical analytical techniques in combination with numerical methods. The performance of the model was validated by a comparison of predictions to thermocouple measurements from the refractory of a steelmaking ladle during a campaign of 26 casting cycles. Reasonable agreement between the measured and simulated variables could be established, which demonstrates the feasibility of the approach.

A two-dimensional mathematical model of steelmaking ladles is presented. The model can be used as a design tool, by which a number of variables such as holding time, material choice, and refractory layer thicknesses can be studied with regard to their influence on the steel temperature evolution during casting. In addition, the model can act as a decision support and as a basis for automation of temperature control. Temperature measurements from an operational steelmaking ladle are compared to simulation results obtained with the model, demonstrating its feasibility and applicability to steelmaking. I. BACKGROUNDwas done by numerical integration of the heat-conduction equation for the ladle wall. [4,5,[11][12][13][14] Due to the intricate IN modern steelmaking, after the introduction of the conboundary-condition changes for the equation as the ladle tinuous casting process and new refractory materials, steel cycles through various stages of secondary metallurgy, onetemperature control within the narrow bounds called for by dimensional solutions at different height positions in the quality requirements has become increasingly demanding.refractory have been adopted as a sufficient means for steel As holding times for steel in the ladle have increased and temperature prediction. [14,15] Although this approach is process logistics often over-rule thermal-and energyappropriate for on-line simulation, more powerful modeling efficiency aspects, improved strategies for heat-loss estimaand simulation tools are required to achieve a deeper undertion for the steel are necessary. Contemporary refractory standing of thermal phenomena in the ladle cycle, e.g., materials, with improved durability and longer campaign through parametric studies. life, are thermally inferior to traditional brick material, i.e.,Another application of thermal modeling of metallurgical their heat conductivity and specific heat are larger, resulting ladles is the design of refractory configurations. [16,17,18] This in increased heat losses from the steel. Over multiple ladle has been a growing field, with the advent of new materials cycles, this can also lead to unacceptably high shell temperacapable of withstanding longer campaigns and higher temtures for the ladle, introducing problems with shell buckling peratures but, unfortunately, having less-advantageous and lining separation. thermal properties. Traditionally, engineering of ladle refractories has been done by confirming, with steady-state calculations, that the refractory and shell temperature stayed II. INTRODUCTION within permissible temperature bounds. Although this Over the years, a number of thermal models for ladle mostly produced acceptable lining-material assemblies, systems has been presented. Early contributions feature simsince the steady state would give substantially overestimated ulation of heat losses and refractory temperature profiles on temperatures, it was hard to foresee long-term effects over an analog computer. [1,2] More-recent approaches range in multiple ladle cycl...

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