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Research Papers: Melting and Solidification

Heat Conduction Equation Solution in the Presence of a Change of State in a Bounded Axisymmetric Cylindrical Domain

[+] Author and Article Information
Danillo Silva de Oliveira

Departamento de Geofísica, Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, Rua do Matão, 1226, Butantã, São Paulo 05508-090, SP, Brazil

Fernando Brenha Ribeiro1

Departamento de Geofísica, Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, Rua do Matão, 1226, Butantã, São Paulo 05508-090, SP, Brazilbrenha@iag.usp.br

1

Corresponding author.

J. Heat Transfer 133(6), 062301 (Mar 08, 2011) (7 pages) doi:10.1115/1.4003542 History: Received May 07, 2010; Revised January 27, 2011; Published March 08, 2011; Online March 08, 2011

The heat conduction problem, in the presence of a change of state, was solved for the case of an indefinitely long cylindrical layer cavity. As boundary conditions, it is imposed that the internal surface of the cavity is maintained below the fusion temperature of the infilling substance and the external surface is kept above it. The solution, obtained in nondimensional variables, consists in two closed form heat conduction equation solutions for the solidified and liquid regions, which formally depend of the, at first, unknown position of the phase change front. The energy balance through the phase change front furnishes the equation for time dependence of the front position, which is numerically solved. Substitution of the front position for a particular instant in the heat conduction equation solutions gives the temperature distribution inside the cavity at that moment. The solution is illustrated with numerical examples.

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Figures

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Figure 9

Temperature variation in the cylindrical cavity filled with a material characterized by a Stefan number of 0.8393 as function of distance for the different nondimensional times

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Figure 8

Temperature variation inside the cylindrical cavity as function of nondimensional distance of the cavity axis for the different nondimensional times

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Figure 7

Solidification front position Φ(τ) as function of the nondimensional time varying the infilling material

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Figure 6

Freezing front position Φ(τ) as function of the nondimensional time varying only the inner cavity surface temperature

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Figure 5

Φ(τ) as function of the nondimensional time varying the radius of the inner cavity surface

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Figure 4

Freezing front position Φ(τ) as function of the nondimensional time for different values of γ

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Figure 3

Flow diagram of the process adopted to find the initial solution of Eq. 47

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Figure 2

Graphical solution for Φ(τ0) and dΦ(τ0)/dτ with τ0⪡1

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Figure 1

Cylindrical cavity scheme

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