Melting in a vertical cylindrical tube: Numerical investigation and comparison with experiments

A review of analytical, numerical and experimental investigations of melting and ensuing convection of phase change materials within enclosures with different shapes commonly used for thermal energy storage is presented. The common shapes of the containers being rectangular cavities, spherical capsules, tubes or cylinders (vertical and horizontal depending on orientation of gravity) and annular cavities are covered. Studies focusing on melting within rectangular cavities are categorized into two groups. The first one is melting due to isothermal heating on one or more boundaries, whereas the second is the constant heat flux-assisted melting. Moreover, constrained and unconstrained melting in both spherical and horizontal cylindrical containers were discussed. The effects of the concentric geometry and location of the heating source on melting in horizontal annular spaces are presented. The review concentrated on elucidating the heat transfer mechanisms (conduction and convection) during the multiple stages of the melting process and the effects of these mechanisms on the liquid–solid interface shape and its progress, melting rate, charging time of the storage system, etc. The strength of buoyancy driven-convection varies greatly with the dimensionless Rayleigh or Stefan numbers and depends somewhat on the location of heat source and imposed boundary condition. High dimensionless numbers and/or side position of the heat source ensure the dominant role of natural convection melting, otherwise conduction will be responsible for major melting within the container. Furthermore, the geometrical parameters such as the aspect ratio in rectangular containers and vertical cylindrical ones, diameter or radius in spherical capsules and horizontal cylindrical vessels, and eccentricity in annular cavities are reviewed. In addition, the parameters affecting the thermal behavior of the melting process in various enclosures, i.e. the Nusselt, Rayleigh, Stefan, Prandtl and Fourier numbers and are reviewed.

/pdf/melting-and-convection-of-phase-change-materials-in-3gncexod3b.pdf

Melting and convection of phase change materials in different shape containers: A review

We present a bibliography on moving and free boundary problems for the heatdiffusion equation, particularly regarding the Stefan and related problems. It contains 5869 titles referring to 588 scientific Journals, 122 books, 88 symposia (having at least 3 contributions on the subject), 30 collections, 59 thesis and 247 technical reports. It tries to give a comprehensive account of the western existing mathematicalphysical-engineering literature on this research field. RESUMEN Se presenta una bibliografía sobre problemas de frontera móvil y libre para la ecuación del calor-difusión, en particular sobre el problema de Stefan y problemas relacionados. Contiene 5869 títulos distribuidos en 588 revistas científicas, 122 libros, 88 simposios (teniendo al menos 3 contribuciones en el tema), 30 colecciones, 59 tesis y 247 informes técnicos o prepublicaciones. Se da un informe amplio de la bibliografía matemática, física y de las ingenierías existente en occidente sobre este tema de investigación. Primary Mathematics Subject Classification Number (*): 35R35, 80A22 Secondary Mathematics Subject Classification Number (*): 35B40, 35C05, 35C15, 35Kxx, 35R30, 46N20, 49J20, 65Mxx, 65Nxx, 76R50, 76S05, 76T05, 93C20. (*) Following the 1991 Mathematics Subject Classification compiled by Mathematical Reviews and Zentralblatt fur Mathematik. Primary key words: Enthalpy formulation or method, Filtration, Free boundary problems, Freezing, Melting, Moving boundary problems, Mushy region, Phase-change problem, Solidification, Stefan problem. Secondary key words: Continuous mechanics, Diffusion process, Functional analysis, Heat conduction, Mathematical methods, Numerical methods, Partial differential equations, Variational inequalities, Weak solutions. Palabras claves primarias: Método o formulación en entalpía, Filtración, Problemas de frontera libre, Congelación, Derretimiento, Problemas de frontera móvil, Región pastosa, Problema de cambio de fase, Solidificación, Problema de Stefan. Palabras claves secundarias: Mecánica del continuo, Procesos difusivos, Análisis funcional, Conducción del calor, Métodos matemáticos, Métodos numéricos, Ecuaciones diferenciales a derivadas parciales, Inecuaciones variacionales, Soluciones débiles. El manuscrito fue recibido y aceptado en octubre de 1999. D.A. Tarzia, A bibliography on FBP. The Stefan problem, MAT Serie A, # 2 (2000). 3

/pdf/a-bibliography-on-moving-free-boundary-problems-for-the-heat-zk5g33r15v.pdf

A bibliography on moving-free boundary problems for the heat-diffusion equation. The stefan and related problems

https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1085&context=coolingpubs

Experimental and numerical study of melting in a cylinder

A detailed experimental and analytical study has been performed to evaluate how copper porous foam (CPF) enhances the heat transfer performance in a cylindrical solid/liquid phase change thermal energy storage system. The CPF used in this study had a 95% porosity and the phase change material (PCM) was 99% pure eicosane. The PCM and CPF were contained in a vertical cylinder where the temperature at its radial boundary was held constant, allowing both inward freezing and melting of the PCM. Detailed quantitative time-dependent volumetric temperature distributions and melt/freeze front motion and shape data were obtained. As the material changed phase, a thermal resistance layer built up, resulting in a reduced heat transfer rate between the surface of the container and the phase change front. In the freezing analysis, we analytically determined the effective thermal conductivity of the combined PCM/CPF system and the results compared well to the experimental values. The CPF increased the effective thermal conductivity from 0.423 W/m K to 3.06 W/mK. For the melting studies, we employed a heat transfer scaling analysis to model the system and develop heat transfer correlations. The scaling analysis predictions closely matched the experimental data of the solid/liquid interface position and Nusselt number.

Phase Change Heat Transfer Enhancement Using Copper Porous Foam

Experiments on the melting of a phase-change medium in a vertical tube yielded quantitative results both for the heat transfer and the timewise evolution of the melting front. The upper surface of the phase-change medium was bounded by an insulated air space, which accommodated the volume changes which accompany the melting process. Numerical solutions based on a pure conduction model were also performed for comparison purposes. It was found that the rate of melting and the heat transfer are significantly affected by fluid motions in the liquid melt induced by the volume change and by natural convection, with the former being significant only at early times. For melting initiated with the solid at the phase-change temperatures, the experimentally determined values of the energy transfer associated with the melting process were about 50 percent higher than those predicted by the conduction model. Furthermore, the measured values of the energy stored in the liquid metal were about twice the conduction prediction. A compact dimensionless correlation of the experimental results was achieved using the Fourier, Stefan, and Grashof numbers. Initial subcooling of the solid substantially decreased the rate of melting, with corresponding decreases in the energy transfers for melting and sensible heat storage.

Inward Melting in a Vertical Tube Which Allows Free Expansion of the Phase-Change Medium

Fundamental heat transfer experiments were performed for freezing of an initially superheated or nonsuperheated liquid in a cooled vertical tube. Measurements were made which yielded information about the freezing front and the frozen mass, about the various energy components extracted from the tube, and about the decay of the initial liquid superheat. Four component energies were identified and evaluated from the experimental data, including the latent energy released by the phase change and sensibly energies released from the subcooled frozen solid and the superheated liquid. Initial superheating of the liquid tended to moderately diminish the frozen mass and latent energy extraction at short freezing times but had little effect on these quantitites at longer times. The extracted sensible energies associated with the superheating more than compensated for the aforementioned decrease in the latent energy. Although the latent energy is the largest contributor to the total extracted energy, the aggregate sensible energies can make a significant contribution, especially at large tube wall subcooling, large initial liquid superheating, and short freezing time. Natural convection effects in the superheated liquid were modest and were confined to short freezing times.

J. A. Broadbent

Papers

Inward Melting in a Vertical Tube Which Allows Free Expansion of the Phase-Change Medium

Freezing in a Vertical Tube