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Showing papers on "Stefan number published in 2004"


Journal ArticleDOI
TL;DR: In this paper, the spreading and solidification processes of a molten metal droplet impinging on a heated solid surface are studied theoretically. But unlike the original model, the initial conditions are modeled in the present work at an instant away from the start of impact.

9 citations


Proceedings ArticleDOI
01 Jan 2004
TL;DR: In this article, the authors derived a differential equation for the solidification thickness as a function of the internal heat generation (IHG) and the Stefan number, which includes the temperature of the boundary.
Abstract: This paper presents solutions to a one-dimensional solid-liquid phase change problem using the integral method for a semi-infinite material that generates internal heat. The analysis assumed a quadratic temperature profile and a constant temperature boundary condition on the exposed surface. We derived a differential equation for the solidification thickness as a function of the internal heat generation (IHG) and the Stefan number, which includes the temperature of the boundary. Plots of the numerical solutions for various values of the IHG and Stefan number show the time-dependant behavior of both the melting and solidification distances and rates. The IHG of the material opposes solidification and enhances melting. The differential equation shows that in steady-state, the thickness of the solidification band is inversely related to the square root of the IHG. The model also shows that the melting rate initially decreases and reaches a local minimum, then increases to an asymptotic value.Copyright © 2004 by ASME

5 citations


Journal ArticleDOI
TL;DR: In this paper, the effect of basal solidification on viscous gravity currents is analyzed using continuum models, and a simplified version of this model is determined in the lubrication and large-Bond-number limit.
Abstract: The effect of basal solidification on viscous gravity currents is analysed using continuum models. A Stefan condition for basal solidification is incorporated into the Navier-Stokes equations. A simplified version of this model is determined in the lubrication and large-Bond-number limit. Asymptotic solutions are obtained in three parameter regimes. (i) A similarity solution is possible in the following cases: the two-dimensional problem when volume per unit length (V) is proportional to time (t) raised to the power 7/4(V = qt7/4) and the Julian number (v3g2 /q4 ) is large, where v is kinematic viscosity, q is a constant of proportionality and g is the acceleration due to gravity; the axisymmetric problem when volume is proportional to time raised to the power 3 (V = Qt3) and the dimensionless group vg/Q is large, where Q is a constant of proportionality. In both cases, the front is found to depend on time raised to the power 5/4, as it does in the absence of solidification, but the constant of proportionality satisfies a modified system of equations. (ii) In the case of large Stefan number and small modified Peclet number (Peδ2 ≪ 1, where Pe is the Peclet number and δ is the aspect ratio), asymptotic and numerical methods are combined to produce the most revealing results. The temperature of the fluid approaches the melting point over a short time-scale. Over the long time-scale, the solid/liquid interface is determined from the conduction of latent heat into the solid. Strong coupling is observed with the basal solidification modifying the flow at leading order. The solidification may retard and eventually arrest the front motion long before complete phase change has taken place. (iii) In the case of constant volume and large modified Peclet number (Peδ2 ≫ 1), similarity solutions are found for the solidification at the base of the gravity current on the short time-scale. The coupling is weak on this time-scale with the solidification being dependent on the front position but not influencing the fluid motion at leading order. Over the long time-scale, the drop completely solidifies. Analytical solutions are not obtained on this time-scale, but scalings are deduced.

4 citations


Proceedings ArticleDOI
01 Jan 2004-Volume!
TL;DR: In this article, a volume-averaged mass and momentum equation for phase change in metal foams with phase change material (PCM) is proposed, with the Brinkman-Forchheimer extension to the Darcy law to model the porous resistance.
Abstract: Transient solid/liquid phase change occurring in metal foams impregnated with phase change material (PCM) is investigated. Natural convection in the melt is considered. Volume-averaged mass and momentum equations are employed, with the Brinkman-Forchheimer extension to the Darcy law to model the porous resistance. Owing to the difference in the thermal diffusivities between the metal foam and the PCM, local thermal equilibrium between the two is not assured. Assuming equilibrium melting at the pore scale, separate volume-averaged energy equations are written for the solid metal foam and the PCM, and are closed using an interstitial heat transfer coefficient. The enthalpy method is employed to account for phase change. The governing equations are solved implicitly using a finite volume method on a fixed grid. The influence of Rayleigh number, Stefan number, and interstitial Nusselt number on the temporal evolution of the melt front location, temperature differentials between the solid and fluid, and the melting rate is documented and discussed. The merits of incorporating metal foam for improving effective thermal conductivity of thermal storage systems are discussed.Copyright © 2004 by ASME

4 citations


Proceedings ArticleDOI
01 Jan 2004
TL;DR: In this article, an improvement is introduced to the conservation element and solution element (CE/SE) phase change scheme presented previously, which addresses a well known weakness in numerical simulations of the enthalpy method when the Stefan number is small (less than 0.1).
Abstract: An improvement is introduced to the conservation element and solution element (CE/SE) phase change scheme presented previously. The improvement addresses a well known weakness in numerical simulations of the enthalpy method when the Stefan number, (the ratio of sensible to latent heat) is small (less than 0.1). Behavior of the improved scheme, at the limit of small Stefan numbers, is studied and compared with that of the original scheme. It is shown that high dissipative errors, associated with small Stefan numbers, do not occur using the new scheme.Copyright © 2004 by ASME

2 citations


Proceedings ArticleDOI
01 Jan 2004
TL;DR: In this paper, a quasi-steady phase-change heat transfer solution is developed for modeling the laser transformation hardening process, in which the governing equation, boundary, and interface conditions are transformed to coordinates moving with the heat front and expressed in a dimensionless form.
Abstract: A quasi-steady phase-change heat transfer solution is developed for modeling the laser transformation hardening process. The surface of a plane slab is heated by a linear moving heat front. The governing equation, boundary, and interface conditions are transformed to coordinates moving with the heat front and expressed in a dimensionless form. By means of a product solution, the governing equation is changed to a Klein-Gordon equation, which is, in turn, solved for temperature expressed in integral equations. Systematic procedures are developed to solve for the phase-change interface positions and, subsequently, the temperature distribution. A parametric study is conducted to investigate the heat transfer effects of various thermal properties. The numerical results show that the Peclet number has a dominant effect over the Stefan number in determining the depth of the phase-change penetration. Accounting for phase change depresses the temperature on the leading side while elevates the temperature on the trailing side of the heat front, in conformity with the source-and-sink principle developed for the solution of the moving heat front, phase-change problems.A quasi-steady phase-change heat transfer solution is developed for modeling the laser transformation hardening process. The surface of a plane slab is heated by a linear moving heat front. The governing equation, boundary, and interface conditions are transformed to coordinates moving with the heat front and expressed in a dimensionless form. By means of a product solution, the governing equation is changed to a Klein-Gordon equation, which is, in turn, solved for temperature expressed in integral equations. Systematic procedures are developed to solve for the phase-change interface positions and, subsequently, the temperature distribution. A parametric study is conducted to investigate the heat transfer effects of various thermal properties. The numerical results show that the Peclet number has a dominant effect over the Stefan number in determining the depth of the phase-change penetration. Accounting for phase change depresses the temperature on the leading side while elevates the temperature on the t...

1 citations


Proceedings ArticleDOI
05 Jan 2004
TL;DR: In this article, high purity pivalic acid (PVA) dendrites were observed under convection-free conditions on STS-87 as part of the United States Microgravity Payload Mission (USMP4) flown on NASA's space shuttle Columbia in 1997.
Abstract: High-purity pivalic acid (PVA) dendrites were observed under convection-free conditions on STS-87 as part of the United States Microgravity Payload Mission (USMP4) flown on NASA’s space shuttle Columbia in 1997. Our telemetry video data show that PVA dendrites melt without relative motion with respect to the quiescent melt phase. With a small fixed superheat above the melting point, ∆T ≡ Tm − T∞, designated in the theory by a Stefan number, dendritic fragments melt and shrink steadily. Fragmentation of the dendrites is observed at higher initial supercoolings. Individual fragments follow a squareroot time-dependence as predicted using a quasi-static conduction-limited approach [1]. Agreement between the analytic theory and experiments was found when the melting process occurs under shape-preserving conditions, where needle-like crystal fragments may be approximated as prolate spheroids with a constant C/A ratio. In microgravity experiments where C/A ratio is not constant, because of interactions in the mushy zone, a “sectorizing” approach was employed that divides the melting process into a series of steps, each approximated b ya constant average value of the C/A ratio. Sectorization allows prediction of melting kinetics using quasistatic theory. Theoretical Stefan numbers were calculated for each sector of melting independently using the initial and final axial lengths for that interval.