This paper presents a new model for simultaneous diffusion and precipitation of chemical elements in metallic matrices, a scheme for its numerical solution, and several applications to problems of internal oxidation. The model basically stands as an extension of the classical Wagner model for internal oxidation of steels, but is more much general in that it allows for an arbitrary number of diffusing chemical elements, an arbitrary number of precipitate phases with arbitrary compositions, dependence of diffusion coefficients and solubility products upon (time-dependent) temperature, etc., thus allowing for a much broader range of applications. As a counterpart, it is generally impossible to solve the complex, non-linear equations of the model analytically, but this can be done numerically. The simple but efficient numerical scheme proposed is based on explicit 1D finite differences. Experience has shown that this scheme, in spite of its rusticity and the restrictions it imposes on the time-step, is more efficient than more elaborate strategies based on the finite-element method. The applications presented are concerned with internal oxidation of steels during annealing processes. The model and associated numerical scheme allow for evaluation of the amounts of the various oxide precipitates in the external layer of the sheet. This opens the way, through numerical parametric studies of the influence of the process parameters and the chemical composition, to the improvement of existing treatments and the development of new steel grades. INTRODUCTIONMany processes in the metallurgical industry involve simultaneous diffusion and precipitation of chemical elements: internal oxidation of a steel sheet during some annealing treatment is just one example. The aim of such a treatment is to prevent formation of superficial iron oxides or to eliminate them if they are already present. This is achieved by using a reducing atmosphere consisting mainly of hydrogen and nitrogen. During the process, oxygen, traces of which are present in the annealing atmosphere, diffuses inwards, the oxidizable elements diffuse outwards and various oxide precipitates are formed in the sub-surface alloy region.The aim of this paper is to present a new model for the simultaneous diffusion and precipitation of chemical elements in metallic matrices, its numerical implementation and several applications related to internal oxidation of steel sheets during annealing treatments. Emphasis will be placed on the last topic. The applications envisaged will notably include a study of the influence of the oxygen content of the annealing atmosphere upon the amounts of internal oxides formed, and a systematic investigation, for a given process, of the types of oxides formed for a wide range of steel grades.The first model for internal oxidation was proposed in the seminal work of Wagner. (1) Although it made several severely restricting hypotheses, such as the presence of a single oxidant, consideration of binary alloys only, oxidation of the sole most-react...