Influence of density change during melting inside a cavity: Theoretical scaling laws and numerical analysis

Abstract

An exact prediction of melting processes is important for the design of latent heat thermal energy storages. However, in phase change models based on continuum theory, the effect of density change during melting is often not considered, although there is no clear estimate of the error caused by this assumption. To overcome this problem, we use scaling analysis to predict the difference between two macroscopic models that handle the density change during melting differently. As a test case, we choose the melting in a two-dimensional cavity heated from one side. Numerical simulations are used to confirm the scaling analysis. Hence, two different models are built and implemented into OpenFoam. In the first model, the density is constant and equal in both phases and the natural convection is taken into account by the Boussinesq approximation. In the second model, the density is different for both phases and varies with the temperature in the liquid phase. The scaling analysis reveals that for small Stefan numbers, the theoretical maximum deviation in the average phase front position at the same time between these two models is solely controlled by the liquid/solid density ratio. Furthermore, this maximum deviation is different for the different regimes of the melting process. The results of the numerical simulations confirm these results and show that the maximum deviation occurs at the end of the convection dominated regime. Our analysis further reveals that while the results strongly differ in time (≈11% for Octadecane), the flow field and the shape of the phase front is nearly the same if the time is shifted or the latent heat is scaled accordingly. Taking into account the uncertainty prevailing in the literature regarding the latent heat of fusion and other material properties of PCM, this may explain why many researchers found excellent agreement between experiments and their simulations despite neglecting the density change.