Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-13T04:19:27.170Z Has data issue: false hasContentIssue false
  • English
  • Français

Some similarity temperature profiles for the microwave heating of a half-space

Published online by Cambridge University Press:  17 February 2009

James M. Hill  and
Adrian H. Pincombe
James M. Hill
Affiliation:
Department of Mathematics, University of Wollongong, Wollongong NSW 2500, Australia.
Adrian H. Pincombe
Affiliation:
Department of Mathematics, University of Wollongong, Wollongong NSW 2500, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

There is presently considerable interest in the utilisation of microwave heating in areas such as cooking, sterilising, melting, smelting, sintering and drying. In general, such problems involve Maxwell's equations coupled with the heat equation, for which all thermal, electrical and magnetic properties of the material are nonlinear. The heat source arising from microwaves is proportional to the square of the modulus of the electric field intensity, and is known to increase with increasing temperature. In an attempt to find a simple model of microwave heating, we examine here simple transient temperature profiles corresponding to a heat source with spatial exponential decay but increasing with temperature, for which we assume either a power-law dependence or an exponential dependence. The spatial exponential decay is known to apply exactly when the electrical and magnetic properties of the material are assumed constant. A number of transient temperature profiles for this model are examined which arise from the invariance of the governing heat equation under simple one-parameter transformation groups. Some closed analytical expressions are obtained, but in general the resulting ordinary differential equations need to be solved numerically, and extensive numerical results are presented. For both models, these results indicate the appearance of moving fronts.

Type
Research Article
Information
The ANZIAM Journal , Volume 33 , Issue 3 , January 1992 , pp. 290 - 320
Copyright
Copyright © Australian Mathematical Society 1992

References

[1] Coleman, C. J., “The microwave heating of frozen substances”, Appl. Math. Modelling 14 (1990) 439–443.CrossRefGoogle Scholar
[2] Coleman, C. J., “On the microwave hotspot problem”, J. Austral. Math. Soc. Ser. B 33 (1991) 1–8.Google Scholar
[3] Hill, J. M., “Simple exact solutions applicable to microwave heating”, J. Appl. Math. Phys. (ZAMP) 40 (1989) 872–882.Google Scholar
[4] Hill, J. M. and Smyth, N. F., “On the mathematical analysis of hot-spots arising from microwave heating”, Math. Engng. Ind. 2 (1990) 267–278.Google Scholar
[5] Metaxas, A. C. and Meredith, R. J., “Industrial microwave heating”, I.E.E. Power Eng. Ser. 4 (Peter Peregrinus, London, 1983).Google Scholar
[6] Murphy, G. M., Ordinary differential equations and their solutions, (Van Nostrand, New York, 1960).Google Scholar
[7] Pincombe, A. H. and Smyth, N. F., “Microwave heating of materials with low conductivity”, Proc. R. Soc. Lond. A, to appear.Google Scholar
[8] Smyth, N. F., “The microwave heating of bodies with temperature dependent properties”, Wave Motion 12 (1990) 171–186.CrossRefGoogle Scholar
[9] Tralli, N., Classical electromagnetic theory, (McGraw-Hill, New York, 1963).Google Scholar