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ISSN Online: 2377-424X

ISBN Print: 0-89116-130-9

International Heat Transfer Conference 6
August, 7-11, 1978, Toronto, Canada

THE TEMPERATURE FIELD IN THE VICINITY OF A SPHERICAL INCLUSION

Get access (open in a dialog) DOI: 10.1615/IHTC6.410
pages 285-290

Resumo

The steady temperature distribution in a semi-infinite solid medium of constant conductivity, with a finite inclusion close to the wall is obtained exactly. This type of problem is encountered in metallurgy, thermography and when noninvasive techniques for establishing the presence of foreign bodies and heat sources are required.

The problem studied in the present paper is that of an isothermal sphere imbedded close to an isothermal or adiabatic free surface of the semi-infinite solid. Existing solutions of such problems are based on the method of reflections, in which the wall is simulated by a mirror-image sphere. Such an approach is inaccurate when the sphere is close to the wall (i.e. the distance between the free surface and the sphere center is of the order of the sphere radius), and various improvements, by continuing the imaging process have been suggested in the past.

An exact solution in series form is presented here. This solution is obtained by means of a transformation to the bispherical coordinate system and is in the form of a series of Legendre polynomials and is rapidly (exponentially) convergent.

The temperature distribution in the solid, as well as the heat transfer are then calculated and exact values for the form-factors are established. Comparison with existing approximate results show that errors of tens of percents occur when the sphere center is less than 1.5 radii from the free surface. Furthermore, surface distributions of the temperature (in the adiabatic wall case) are given.