In order to reduce the production cost of the shape in the foundry industries, we have to test and choose one of several mould materials. This way is not easily or impossible, in some cases, applicable. In fact, the high temperature and complex geometry of each mould make difficult to obtain the real property values of the material during the solidification process. The best way, in our point of view, is to simulate the heat transfer in the solidification process. For this, we propose to solve appropriate governing heat transfer equation with a curvilinear formulation using a convective heat transfer between the mould and the shape. The main objective of this paper is to develop a finite volumes method with the curvilinear formulation in order to discretizate the space (x,y,z). The time dependency will be linearised by the semi-implicit scheme. The boundary conditions used in this approach is the convective term in the solidification phase. We divide this paper in five parts. The first part, the mathematical model that describes a heat transfer between the mould and the piece is exposed. Then we expose the used boundary conditions. The second one, the finite volume method to solve such equation is detailed. After this, we present the test case, in the third part; then we expose the numerical results comparing with other boundary conditions. We conclude this paper with some critics and perspectives of this research work.

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