METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 29B, JUNE 1998—651 Model for Temperature Profile Estimation in the Refractory of a Metallurgical Ladle T.P. FREDMAN and H. SAXE ´ N 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 com- bination 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 cam- paign of 26 casting cycles. Reasonable agreement between the measured and simulated variables could be established, which demonstrates the feasibility of the approach. I. INTRODUCTION IN continuous casting, thermal control of liquid steel plays an important role in product quality and ladle refrac- tory life. A small difference between targeted and achieved temperatures on teeming may have very negative conse- quences for surface quality, cleanliness, tundish flow- through, nozzles, casting schedules, energy economy, etc. Tapping of a furnace, thereby filling the ladle with molten metal, and emptying the ladle during the casting process are, among foundrymen, referred to by the terms tapping and teeming, respectively. The thermal state of the refrac- tory lining of the ladle upon tapping affects the cooling of the heat (the batch of liquid steel) until teeming. The tem- perature drop of the heat is mainly caused by convective heat transfer to the ladle refractory and partly from the free surface of the bath. After teeming, when the hot inner walls of the ladle are exposed to the surroundings, the radiative and convective losses from the ladle lining are large. Thus, the heat stored within the inner parts of the refractory when the ladle is filled will be lost to the environment during the period when the ladle is empty. In order to reliably estimate the energy storage in the working lining, it is necessary to know the temperature field in it before and after tapping and teeming. Under industrial conditions, however, tem- perature measurements are available only occasionally and are often problematic due to high temperatures and a hostile environment. Reliable measurement data on thermal con- ditions within the ladle refractory are also difficult to pro- duce, as process scheduling and geography move the ladle around in the plant during the casting cycle. Therefore, one has to resort to numerical estimation of needed variables. T.P. FREDMAN, Research Scientist, and H. SAXE ´ N, Professor, are with the Heat Engineering Laboratory, A ˚ bo Akademi University, FIN- 20500 A ˚ bo, Finland. Manuscript submitted February 19, 1997. Calculation of transient heat conduction is mostly done by numerical solution of the heat-conduction equation, espe- cially for more complicated shapes and boundary condi- tions, although analytical solutions for a number of simple situations have been derived. [1,2] The problem is compli- cated by the fact that industrial ladle linings consist of sev- eral layers, with different material and thermal properties. In addition, it is difficult to quantify the influence on the boundary conditions for the field computation of factors such as wear, sculling, slag attack, and metal penetration into the refractory. The aforementioned circumstances favor a simple model capable of predicting major trends and changes in the heat balance of the ladle refractory. The simplest approach would be to formulate a statistical model for the energy storage in the lining based on measurements of the steel temperature drop for different process conditions. However, such a model cannot be applied outside the original domain of measurements and generally requires extensive data in order to work reliably. A better approach seems to be to reduce the geometry to one dimension and to calculate the temperature profile in the radial direction within the refrac- tory. [3] This can be done for a number of positions around the ladle to produce an overall estimate of the heat storage in the lining. The one-dimensional solution of the heat-con- duction equation is easily done numerically by standard methods, but these do not separate the dynamics of the field from its geometry, as is the case with certain classical so- lution methods assuming constant thermal properties and boundary heat-transfer coefficients with varying tempera- ture at the hot face, and, thus, do not provide as much insight into the process. The solution of the heat-conduction equation by various analytical techniques involves a Sturm– Liouville eigenvalue problem, [4] which in the case of com- posite media is fairly complicated, due to the discontinuous coefficients of the governing differential equation. Ra- mkrishna and Amundson [5] have given a convenient for-