A Spatial Stefan Problem Modified by Natural Convection: Melting Around a Horizontal Cylinder

Many important problems in engineering science, when formulated in mathematical terms, involve very irregularly shaped boundaries which move with time. Such problems require that the position of the boundaries be determined as part of the solution. Obvious examples can be found in the growth of crystals, in phase-change, bubble dynamics, and in cavitation problems. Irregular, moving boundaries cause enormous complications by adding extra dependent variables to a given analysis and by making it more difficult to apply boundary conditions. This study solves a typical engineering problem--the melting of a solid about a heated horizontal cylinder. A radial transformation is used which transforms highly irregular boundaries into simple geometric shapes and immobilizes moving boundaries. The transformation provides the basis for numerical and perturbation methods used to study the melting problem. Isothermal and constant flux thermal boundary conditions are considered along the heated cylinder. Natural convection heat transfer between vertically eccentric cylinders is examined as a test case. The melting process can be divided up into three distinct stages: conduction, transition, and convection. During the conduction stage, the rates of melting and heat transfer are determined by the Stefan number, St. St is the ratio of thermal energy available in the melt to the thermal energy needed to melt the solid. During the transition stage, convection heat transfer becomes important. In the final convection stage, natural convection is the dominant mode of heat transfer. The strength of the circulation induced by buoyancy force is measured by the Rayleigh number, Ra. For the case of isothermal boundary condition, the average heat transfer rate increases considerably with Ra. For the constant flux case, Ra has only a very minor effect on the average heat transfer rate. Detailed results for the growth of the melt region, local and average heat transfer rates, average shear stresses, and temperature and flow fields are presented. The present results are compared, where possible, with earlier experimental, numerical, and analytical works. The excellent comparisons demonstrate the validity of the radial transformation method in problems with highly irregular, moving boundaries.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Prusa, Joseph Michael

• Engineering, Mechanical

Identifier

(UMI)AAI8310001

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/70113