Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-12T11:13:50.707Z Has data issue: false hasContentIssue false
  • English
  • Français

The multiplicity of steady-state solutions arising from microwave heating. I. Infinite Biot number and small penetration depth

Published online by Cambridge University Press:  17 February 2009

M. I. Nelson ,
G. C. Wake ,
X. D. Chen  and
E. Balakrishnan
M. I. Nelson
Affiliation:
Department of Fuel and Energy, The University of Leeds, Leeds LS2 9JT, England.
G. C. Wake
Affiliation:
Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand.
X. D. Chen
Affiliation:
Department of Chemical and Materials Engineering, The University of Auckland, New Zealand.
E. Balakrishnan
Affiliation:
Department of Mathematics and Statistics, Sultan Qaboos University, Sultanate of Oman.
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.

Microwave heating of porous solid materials has received considerable attention in recent years because of its widespread use in industry. In this study, the microwave power absorption term is modelled as the product of an exponential temperature function with function that decays exponentially with distance. The latter describes the penetration of material by the microwaves.

We investigate the phenomena of multiplicity in class A geometries, paying particular attention to how the penetration function affects the behaviour of the system. We explain why the phase-plane techniques which have been used in the case when the penetration term is constant do not extend to non-constant penetration.

Type
Research Article
Information
The ANZIAM Journal , Volume 43 , Issue 1 , July 2001 , pp. 87 - 103
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Ayappa, K. G., Davis, H. T., Crapiste, G., Davis, E. A. and Gordon, J., “Microwave heating: An evaluation of power formulations”, Chem. Eng. Sci. 46 (1991) 1005–1016.CrossRefGoogle Scholar
[2]Boddington, T., Gray, P. and Wake, G. C., “Criteria for thermal explosion with and without reactant consumption”, Proc. Roy. Soc. A 357 (1977) 403–422.Google Scholar
[3]Bowes, P. C., Self-heating: evaluating and controlling the hazards (Elsevier, Amsterdam, 1984).Google Scholar
[4]Chen, X. D., “Microwave heating in an infinite solid slab and its thermal stability analysis using steady state bifurcation theory”, J. Food Eng. 35 (1998) 339–349.CrossRefGoogle Scholar
[5]Coleman, C. J., “The microwave heating of frozen food substances”, Appl. Math. Modelling 14 (1990) 439–443.CrossRefGoogle Scholar
[6]Coleman, C. J., “On the microwave hot spot problem”, J. Austral. Math. Soc. Series B 33 (1991) 1–8.CrossRefGoogle Scholar
[7]Datta, A. K. and Liu, J., “Thermal time distributions in microwave and conventional heating of food”, Trans. Inst. Chem. Engnrs. 70C (1992) 83–90.Google Scholar
[8]Doedel, E., Wang, X. and Fairgrieve, T., AUTO 94 Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, November 1994, Applied Mathematics Report California Institute of Technology Pasadena, California 91125.Google Scholar
[9]Dolande, J. and Datta, A., “Temperature profiles in microwave heating of solids: systematic study”, J. Microwave Power and Electromagnetic Energy 28 (1993) 58–67.CrossRefGoogle Scholar
[10]Enig, J. W., “Critical parameters in the Poisson-Boltzmann equation of steady-state thermal explosion theory”, Combustion and Flame 10 (1966) 197–199.Google Scholar
[11]Frank-Kamenetskii, D. A., Diffusion and heat transfer in chemical kinetics, 2nd ed. (New York, Plenum Press, 1969).Google Scholar
[12]Fujita, H., “On the nonlinear equation ”, Bull. Amer. Math. Soc. 75 (1969) 132–135.CrossRefGoogle Scholar
[13]Gel'fand, I. M., “Some problems in the theory of quasilinear equations”, Uspehi Mat. Nauk 14 (1959) 87–158, English translation, Amer. Math. Soc. Translations 29, 295–381, 1963.Google Scholar
[14]Herve, A.-G., Tang, J., Luedecke, L. and Feng, H., “Dielectric properties of cottage cheese and surface treatment using microwaves”, J. Food Eng. 37 (1998) 389–410.CrossRefGoogle Scholar
[15]Hill, J. M., “Simple exact solutions applicable to microwave heating”, J. Appl. Math. Phys. (ZAMP) 40 (1989) 872–882.CrossRefGoogle Scholar
[16]Hill, J. M. and Marchant, T. R., “Modelling microwave heating”, Appl. Math. Modelling 20 (1996) 3–15.CrossRefGoogle Scholar
[17]Hill, J. M. and Pincombe, A. H., “Some similarity profiles for the microwave heating of a half-space”, J. Austral. Math. Soc. Series B 33 (1991) 29–30.Google Scholar
[18]Hill, J. M. and Smyth, N. F., “On the mathematical analysis of hot spots arising from microwave heating”, Math. Eng. in Industry 17 (1990) 1823–1834.Google Scholar
[19]Joseph, D. D. and Lundgren, T. S., “Quasilinear Dirichlet problems driven by positive sources”, Archive for Rational Mech. and Anal. 49 (1972–1973) 241–269.CrossRefGoogle Scholar
[20]Lin, Y. E., Anantheswaran, R. C. and Puri, V. M., “Finite element analysis of microwave heating of solid foods”, J. Food Eng. 25 (1995) 85–112.CrossRefGoogle Scholar
[21]Lobo, S. and Datta, A. K., “Characterization of spatial non-uniformity in microwave reheating of high loss foods”, J. Microwave Power and Electromagnetic Energy 33 (1998) 158–166.CrossRefGoogle Scholar
[22]Prabhanjan, D. G., Ramaswamy, H. S. and Raghavan, G. S., “Microwave-assisted convective air drying of thin layer carrots”, J. Food Eng. 25 (1995) 283–293.CrossRefGoogle Scholar
[23]Ryynanen, S., “The electromagnetic properties of food materials: a review of the basic principles”, J. Food Eng. 26 (1995) 409–429.CrossRefGoogle Scholar
[24]Steggerda, J. J., “Thermal stability: An extension of Frank-Kamenetsky's theory”, J. Chem. Phys. 43 (1965) 4446–4448.CrossRefGoogle Scholar
[25]Wake, G. C. and Hood, M. J., “Multiplicity of solutions of a quasilinear elliptic equation in spherical domains”, Math. Comput. Modeling 18 (1993) 157–162.CrossRefGoogle Scholar
[26]Wake, G. C., Scott, S. K. and Brindley, J., “Multiplicity of steady state solutions of combustion in a sphere with arbitrary Biot number”, Phys. Lett. 153A (1991) 16–21.CrossRefGoogle Scholar
[27]Kou Wei, Chen, Davis, H. T., Davis, E. A. and Gordon, J., “Heat and mass transfer in water-laden sandstone: Microwave heating”, Amer. Inst. Chem. Eng. J. 31 (1985) 842–848.Google Scholar
[28]Zhu, S., Zhang, Y. and Marchant, T., “A DRBEM model for microwave heating problems”, Appl. Math. Modelling 19 (1995) 287–297.Google Scholar