Showing papers on "Stefan number published in 2018"

Heat balance integral methods applied to the one-phase Stefan problem with a convective boundary condition at the fixed face.

Abstract: In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method, assuming a quadratic temperature profile in space. We develop variations of these methods which turn out to be optimal in some cases. Throughout this paper, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to estimate the errors committed by each approach for the corresponding free boundary and temperature profiles.

Studies on the inward spherical solidification of a phase change material dispersed with macro particles

Abstract: This paper investigates numerically the solidification of a phase change material (PCM) dispersed with high conductivity macro particles inside a spherical container. The formulation takes into account the addition of particles invoking an effective thermal conductivity model. A case study has been made comparing the experimental results available in the open literature for the solidification of a pure PCM case (without particles) to investigate the influence of particles. The results show that the addition of particles between 10 and 50% by volume, enhances the heat transfer rate by about 13.5 and 59% respectively. Parametric studies have been carried out to investigate the influence of relevant dimensionless numbers such as Biot number (Bi) and Stefan number (Ste) on the solidification characteristics of the PCM for different particle fractions. The role of particles was found to be significant at lower Ste compared to Bi. It has been concluded that the effect of particle fraction on the solidification is more compared to that of particle-PCM thermal conductivity ratio. It was found that there is no restriction on the choice of particle material for a given PCM, as long as the particle-PCM thermal conductivity ratio considered remain larger than 5.

Basal melting driven by turbulent thermal convection

Abstract: Melting and, conversely, solidification processes in the presence of convection are key to many geophysical problems. An essential question related to these phenomena concerns the estimation of the (time-evolving) melting rate, which is tightly connected to the turbulent convective dynamics in the bulk of the melt fluid and the heat transfer at the liquid-solid interface. In this work, we consider a convective-melting model, constructed as a generalization of the Rayleigh-Benard system, accounting for the basal melting of a solid. As the change of phase proceeds, a fluid layer grows at the heated bottom of the system and eventually reaches a turbulent convection state. By means of extensive Lattice-Boltzmann numerical simulations employing an enthalpy formulation of the governing equations, we explore the model dynamics in two and three-dimensional configurations. The focus of the analysis is on the scaling of global quantities like the heat flux and the kinetic energy with the Rayleigh number, as well as on the interface morphology and the effects of space dimensionality. Independently of dimensionality, we find that the convective-melting system behavior shares strong resemblances with that of the Rayleigh-Benard one, and that the heat flux is only weakly enhanced with respect to that case. Such similarities are understood, at least to some extent, considering the resulting slow motion of the melting front (with respect to the turbulent fluid velocity fluctuations) and its generally little roughness (compared to the height of the fluid layer). Varying the Stefan number, accounting for the thermodynamical properties of the material, also seems to have only a mild effect, which implies the possibility to extrapolate results in numerically delicate low-Stefan setups from more convenient high-Stefan ones.

Numerical investigation and correlations for heat diffusion through planar ablative thermal protection systems

Abstract: In the present study, heat diffusion through a planar ablative Thermal Protection System (TPS) is numerically investigated by modeling the problem as one-dimensional transient heat conduction equation in Cartesian coordinates subject to the adiabatic back wall and aerodynamic heating on the other surface. The surface exposed to aerodynamic heating undergoes sensible heating until the surface temperature reaches an ablative temperature of the material. Further exposure of the material to heat flux results in material getting ablated. Ablation is modeled as Stefan-type wherein layers of material are immediately removed upon melt after reaching ablative temperature. Boundary immobilization method is used to fix the moving boundary and the governing equations are solved using finite difference scheme in space and Crank-Nicolson semi-implicit scheme in time, after expressing them in non-dimensional form. A FORTRAN code is developed to solve the set of equations using Tri-Diagonal Matrix Algorithm (TDMA). Parametric studies are conducted and new correlations are developed for predicting the amount of material ablated as a function of non-dimensional heat flux, Stefan number and non-dimensional time. Correlations are also developed to predict the non-dimensional time when back-wall that protects the vehicle interiors from extreme heat flux environment attains non-dimensional temperature 0.1. Results show that the developed correlations predict the parameter very well and errors are within acceptable limits.

Integral balance methods applied to a non-classical Stefan problem

Abstract: In this paper we consider two different Stefan problems for a semi-infinite material for the non classical heat equation with a source which depends on the heat flux at the fixed face x = 0. One of them (with constant temperature on x = 0) was studied in [4] where it was found a unique exact solution of similarity type and the other (with a convective boundary condition at the fixed face) is presented in this work. Due to the complexity of the exact solution it is of interest to obtain some kind of approximate solution. For the above reason, the exact solution of each problem is compared with approximate solutions obtained by applying the heat balance integral method and the refined heat balance integral method, assuming a quadratic temperature profile in space. In all cases, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to verify the accuracy of the approximate methods