One-dimensional time-dependent modeling of conductive heat transfer during the melting of an initially subcooled semi-infinite PCM
Résumé
The melting problem of a phase change material PCM confined in a semi-infinite medium has been explored numerically. This PCM is completely in a solid state at times T ? 0 and its temperature T0 is uniform and constant which can be different from the melting temperature Tpc (T0 ? Tpc). At the time t > 0, the left side X = 0 of the semi-infinite medium is suddenly subjected to a constant temperature Tw and maintained at this value which is higher than the threshold value Tpc. This involves the development of a liquid layer which progresses within the semi-infinite medium starting from the origin X = 0. The phase change from the solid state to the liquid state is described by a pure conduction model. This problem is called Stefan problem whose analytical solution has been found by Neumann. The problem consists in determining simultaneously the distribution of the temperature in the liquid and solid phases as a function of the spatial and temporal coordinates X and t respectively for t > 0, as well as the melting front position Xf as a function of t, moving front which separates the solid phase from the liquid phase. This study is based on the resolution of the Fourier's conduction equation in the two phases coupled with the Stefan condition at the phase change interface on one side and with the initial and boundary conditions on the other side. At the solid-liquid interface, there is the condition of continuity of temperature, on the one hand, and the one of a discontinuity of heat flux, on the other hand, because of the phase change. The numerical modeling of the melting problem involves finite volume method. The algebraic equations obtained after discretization are solved iteratively by using the line by line method with which the algorithm TDMA is associated. The numerical treatment of this 1D transient problem is ensured by a simulation code conceived for 2D cases while fixing an infinitely small length in the 2nd direction. A fully implicit scheme for the discretization in time and a combination of the forward and backward finite differences with second order accuracy at the node changing phase for the discretization in space have been used. The influence of the number of nodes, the number of iterations, the space step ?X and the time step ?t during the numerical simulations of this problem has been investigated in order to ensure the convergence of the solution. In order to be able to represent graphically the exact Neumann's solution of the present problem, an interpolation by polynomial collocation of the error function and complementary error function has been adopted firstly. Afterwards, the root ? of the transcendental equation is located in the interval [A~B] = [0,01~3,0] then obtained using the iterative dichotomy algorithm. The numerical results obtained of the present investigation have been successfully validated in comparison with the analytical solution resulting from the similarity approach of Neumann. The transient 1D numerical model with two phases has thus allowed an accurate prediction of the temperature evolution in the solid and liquid phases as well as of the melting front position evolution. After validation, the results obtained have been presented in the form of graphs in order to highlight the impact of certain parameters. So, the temporal progression of the melting front for two different initial temperatures and two different thermal conductivities of the PCM as well as the evolution of the PCM temperature for different axial positions, at different times and for two different values of the thermal conductivity of the PCM have been plotted and commented.
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