Abstract
A prototype chemical reaction is examined in both one and two-dimensional continuous-flow microwave reactors, which are unstirred so the effects of diffusion are important. The reaction rate obeys the Arrhenius law and the thermal absorptivity of the reactor contents is assumed to be both temperature- and concentration-dependent. The governing equations consist of coupled reaction-diffusion equations for the temperature and reactant concentration, plus a Helmholtz equation describing the electric-field amplitude in the reactor. The Galerkin method is used to develop a semi-analytical microwave reactor model, which consists of ordinary differential equations. A stability analysis is performed on the semi-analytical model. This allows the stability of the system to be determined for particular parameter choices and also allows any regions of parameter space in which Hopf bifurcations (and hence periodic solutions called limit-cycles) occur to be obtained. An excellent comparison is obtained between the semi-analytical and numerical solutions, both for the steady-state solution and for time-varying solutions, such as the limit-cycle.
Similar content being viewed by others
References
C. R. Strauss. A combinatorial approach to the development of environmentally benignorganic chemical preparations. Aust. J. Chem. 52 (1999) 83–96.
A. Zlotorzynski. Theapplication of microwave radiation to analytical and environmental chemistry. Crit. Rev. Anal. Chem. 25 (1995) 43–76.
B. Liu and T. R. Marchant. The microwave heating of two-dimensional slabs withsmall Arrhenius absorptivity. IMA J. Appl. Math. 62 (1999) 137–162.
G. A. Kriegsmann.Thermal runaway and its control in microwave heated ceramics. Mat. Res. Soc. Symp. Proc. 269 (1992) 257–264.
B. Liu and T. R. Marchant. On the occurrence of limit-cycles during feedbackcontrol of microwave heating. J. Math. Computer Model. (2002) to appear.
M. R. Booty, J.K. Bechtold and G. A. Kriegsmann. Microwave-induced combustion: a one-dimensional model. Combust. Theory Modelling 2 (1998) 57–80.
B. S. Tilley and G. A. Kriegsmann.Microwave enhancedchemical vapor infiltration: a sharp interface model. J. Engng. Math. 41 (2001) 33–54.
F. Chemat, D. C. Esveld, M. Poux and J. L. DiMartino. The role of selective heating in the microwaveactivation of heterogeneous catalysis reactions using a continuous microwave reactor. J. Microwave Power and Electromagnetic Energy 33 (1998) 88–94.
L. K. Forbes. Limit-cycle behaviour in a modelchemical reaction: the Sal'nikov thermokinetic oscillator. Proc. R. Soc. Lond. A 430 (1990) 641–651.
A. R. Von Hippel, Dielectric materials and applications. Cambridge: MIT press (1954)438 pp.
D. Acierno, V. Fiumara, M. Frigione, D. Napoli, I. M. Pinto and M. Ricciardi. Comparativestudy of microwave and thermal curing of epoxy resins. In: J. M. Catala-Civera, F. L. Penaranda-Foix, D. Sanchez-Hernandez, and E. de los Reyes (eds.): 7th Inter. Conf. on Microwave and High-Frequency Heating. (1999) pp. 209–212.
G. A. Kriegsmann. Cavity effects in microwave heating ofceramics. SIAM J. Appl. Math. 57 (1997) 382–400.
J. H. Merkin, J. H., V. Petrov, S.K. Scott and K. Showalter. Wave-induced chaos in a continuously fed unstirred reactor. J. Chem. Soc., Faraday Trans. 92 (1996) 2911–2918.
J. M. Hill and M. J. Jennings. Formulation of modelequations for heating by microwave radiation. Appl. Math. Modelling 17 (1993) 369–379.
M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory. NewYork: Springer-Verlag (1985) 463 pp.
T. R. Marchant. Cubic autocatalytic reaction-diffusionequations: semi-analytical solutions. Proc. R. Soc. London A 458 (2002) 873–888.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lee, M., Marchant, T. Semi-analytical solutions for continuous-flow microwave reactors. Journal of Engineering Mathematics 44, 125–145 (2002). https://doi.org/10.1023/A:1020820213620
Issue Date:
DOI: https://doi.org/10.1023/A:1020820213620