Showing papers on "Stefan number published in 1989"

Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary

Abstract: We study the large time behaviour of the free boundary for a one-phase Stefan problem with supercooling and a kinetic condition u = −e|⋅ṡ| at the free boundary x = s(t). The problem is posed on the semi-infinite strip [0,∞) with unit Stefan number and bounded initial temperature ϕ(x) ≤ 0, such that ϕ → −1 − δ as x → ∞, where δ is constant. Special solutions and the asymptotic behaviour of the free boundary are considered for the cases e ≥ 0 with δ negative, positive and zero, respectively. We show that, for e > 0, the free boundary is asymptotic to , δt/e if 0 respectively, and that when δ = 0 the large time behaviour of the free boundary depends more sensitively on the initial temperature. We also give a brief summary of the corresponding results for a radially symmetric spherical crystal with kinetic undercooling and Gibbs-Thomson conditions at the free boundary.

Dendritic growth of an elliptical paraboloid with forced convection in the melt

Abstract: All experimental observations of the growth of fully developed dendritic ice crystals indicate that the shape of the tip region is an elliptical paraboloid. Therefore, moving-boundary solutions of the three-dimensional Navier-Stokes and energy equations are obtained here for the shape-preserving growth of isothermal elliptical paraboloids by using the Oseen approximation which is valid for the low-Reynolds-number viscous flows which prevail in dendritic growth. Explicit expressions for the flow and the temperature fields are derived in a simple way using Ivantsov's method. It is shown that the growth Peclet number, PG, is a function of the aspect ratio A, the Stefan number St, the Reynolds number Re, and the Prandtl number Pr. As the Reynolds number increases PG becomes linear in St, less dependent on A and ultimately varies roughly as Re½.A comparison between the exact solutions given here and the experiments of Kallungal (1974) indicate that A decreases as Re increases. This result agrees qualitatively with the experiments of Kallungal (1974) and Chang (1985). The differences between theory and experiments for Re > 10−3 may be due to attachment kinetic resistance to growth along the c-axis and capillary effects at the tip which make ice dendrites non-isothermal and create conduction in the solid phase. However, more accurate simultaneous measurements of R1 and R2 are needed to determine definitively the mechanisms responsible for these deviations between theory and experiment.

A Heat Transfer Analysis for Solidification of Slabs, Cylinders, and Spheres

Abstract: The method of strained coordinates is applied to the inward solidification problem. Constant thermal properties are assumed throughout the analysis for the liquid, which is initially at the fusion temperature. A unified approach is adopted that allows the simultaneous treatment of the problem in plane, cylindrical, and spherical geometries for three different types of boundary condition. A general recurrence formula is derived for the determination of the series solutions up to any desired order of the Stefan number. A comparison is made with numerical and regular perturbation solutions in the plane case to illustrate the usefulness and the validity of the method.

Theory of melting with natural convection in an enclosed porous medium

Abstract: This paper reports a complete theory of the melting that occurs in a confined porous medium saturated with phase-change material and heated from the side. Darcy flow characteristics are assumed for the liquid phase. The solid phase is isothermal and at the melting point. The first part of the paper reports a matched boundary layer solution for natural convection dominated melting in the quasi-steady regime. The second part reports a solution for the heat transfer during the two earlier regimes, pure conduction followed by mixed conduction and convection. Together, the two solutions cover the entire history of transient heating administered from the side. This theory shows that the liquid-side Stefan number has a profound effect on the heat transfer and melting rates.

Effects of anisotropic heat conduction on solidification

Abstract: Two-dimensional solidification influenced by anisotropic heat conduction has been considered. The interfacial energy balance was derived to account for the heat transfer in one direction (x or y) depending on the temperature gradient in both the x and y directions. A parametric study was made to determine the effects of the Stefan number, aspect ratio, initial superheat, and thermal conductivity ratios on the solidification rate. Because of the imposed boundary conditions, the interface became skewed and sometimes was not a straight line between the interface position at the upper and lower adiabatic walls (spatially nonlinear along the height). This skewness depends on the thermal conductivity ratio k(yy)/k(yx). The nonlinearity of the interface is influenced by the solidification rate, aspect ratio, and k(yy/k(yx).

On the steady Melting of a Phase-Change Material placed on a Hot Wall

Abstract: The problem of the thermofluid mechanics of the steady squeezing of a viscous fluid between a hot wall and a surface of the phase-change material is studied theoretically. The partial differential system consisting of the Navier-Stokes equation, the energy equation and the boundary conditions is found to be reduced to an ordinary differential system depending on only one coordinate in the direction normal to the surface of the hot wall. This ordinary differential system is solved by using the approximate and numerical methods. The problem is governed by two nondimensional parameters, that is, the Stefan number and the Prandtl number. It is found that the surface of the phase-change material melted by a flat wall with a higher temperature than the melting temperature is always left flat. The relationship between the thickness of the liquid layer, the melting rate and the force acting on the interface is obtained, as well as the distributions of the velocity, the pressure and the temperature in the liquid layer. If the inertia term in the Navier-Stokes equation is disregarded, the ordinary differential system is solved analytically. Its solution is obtained in the closed form. It is found to be very useful in many practical cases.

低融点相変化物質充てん傾斜く形蓄熱槽の動特性に関する研究 : 第1報,蓄熱過程の実験

Abstract: Melting characteristics of a solid phase change material in a inclined rectangular heat storage enclosure heated from one side is investigated experimentally. According to the inclination angle of the enclosure, the natural convection pattern in s liquid phase shifts from three-dimensional flow to two-dimensional one. The interface morphology is found to be dependent on the angle of inclination and heating wall temperature. The time-wise variation of the melt fraction is obtained for various inclination angled. Correlation equations of natural convection in a liquid phase are presented in terms of Rayleigh number, inclination angle and Stefan number.

Melting in a spherical enclosure: an experimental study

Abstract: The Latent Heat Thermal Energy Storage Systems will play an important role in the solution of the energy problems in future. The main advantageous of these systems seems to be storing the large amount of energy in relatively small volumes. But many problems arise at the design stages of these systems. One of them is the selection of the container which includes the phase change material (PCM) and identifies the heat transfer between working fluid and PCM. The container geometry can be chosen as a slab, a cylinder or a sphere.

Study of instantaneous bioheat transfer in human tissue during cryogenic surgical operation

Abstract: An approximate analytical solution presented by Cooper for predicting the radius of the frozen phase, S, of human tissue during cryosurgical operations has been widely used. Based on consideration of the Stefan number, Ste, another approximate analytical solution for S has been developed in this paper. The new technique has the advantages of simple form, reasonable structure, good accuracy, and quantitative agreement with experimental data.