No CrossRef data available.
Article contents
CALCULATION OF CRITICAL PARAMETERS FOR SPONTANEOUS COMBUSTION FOR SOME COMPLEX GEOMETRIES USING AN INDIRECT NUMERICAL METHOD
Part of:
Numerical problems in dynamical systems
Partial differential equations, boundary value problems
Partial differential equations
Published online by Cambridge University Press: 26 February 2018
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.
In the theory of spontaneous combustion, identifying the critical value of the Frank-Kamenetskii parameter corresponds to solving a bifurcation point problem. There are two different numerical methods used to solve this problem—the direct and indirect numerical methods. The latter finds the bifurcation point by solving a partial differential equation (PDE) problem. This is a better method to find the bifurcation point for complex geometries. This paper improves the indirect numerical method by combining the grid-domain extension method with the matrix equation computation method. We calculate the critical parameters of the Frank-Kamenetskii equation for some complex geometries using the indirect numerical method. Our results show that both the curve of the outer boundary and the height of the geometries have an effect on the values of the critical Frank-Kamenetskii parameter, however, they have little effect on the critical dimensionless temperature.
Keywords
MSC classification
Primary:
65P30: Bifurcation problems
- Type
- Research Article
- Information
- Copyright
- © 2018 Australian Mathematical Society
References
Anderson, C. and Zienkiewicz, O., “Spontaneous ignition: finite element solutions for steady and transient conditions”, J. Heat Transfer
96 (1974) 398–404; doi:10.1115/1.3450212.CrossRefGoogle Scholar
Anderson, J. D., Computational fluid dynamics: the basics with applications (McGraw Hill, New York, 1995).Google Scholar
Bartels, R. H. and Stewart, G. W., “Solution of the matrix equation
$ax+xb=c$
[f4]”, Commun. ACM
15 (1972) 820–826; doi:10.1145/361573.361582.Google Scholar
$ax+xb=c$
[f4]”, Commun. ACM
15 (1972) 820–826; doi:10.1145/361573.361582.Google ScholarBoddington, T., Gray, P. and Harvey, D. I., “Thermal theory of spontaneous ignition: criticality in bodies of arbitrary shape”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
270 (1971) 467–506; doi:10.1016/j.amc.2011.06.009.Google Scholar
Bowes, P. C., Self-heating: evaluating and controlling the hazards (Elsevier, Amsterdam, 1984).Google Scholar
Du, X. and Feng, C., “Critical parameters for the thermal explosion of exothermic systems having two-dimensional geometries (in Chinese)”, J. Beijing Inst. Tech.
14 (1994) 37–42.Google Scholar
Frank-Kamenetskii, D. A., Diffusion and heat transfer in chemical kinetics (Plenum Press, New York, 1969).Google Scholar
Golub, G., Nash, S. and Van Loan, C., “A Hessenberg–Schur method for the problem
$ax+xb=c$
”, IEEE Trans. Automatic Control
24 (1979) 909–913; doi:10.1109/TAC.1979.1102170.CrossRefGoogle Scholar
$ax+xb=c$
”, IEEE Trans. Automatic Control
24 (1979) 909–913; doi:10.1109/TAC.1979.1102170.CrossRefGoogle ScholarGray, B., “Spontaneous combustion and self-heating”, in SFPE handbook of fire protection engineering, 5th edn (eds Hurley, M. J.
et al. ), (Springer, New York, 2016) 604–632; doi:10.1007/978-1-4939-2565-0.Google Scholar
Luo, Q., Liang, D. and Mo, S., “Numerical calculation of the critical parameters of Frank-Kamenetskii equation in spontaneous combustion theory”, Numer. Heat Transfer, Part B: Fundamentals
68 (2015) 403–417; doi:10.1080/10407790.2015.1036625.Google Scholar
Moore, G. and Spence, A., “The calculation of turning points of nonlinear equations”, SIAM J. Numer. Anal.
17 (1980) 567–576; doi:10.1137/0717048.Google Scholar
Partridge, P. W. and Wrobel, L. C., “The dual reciprocity boundary element method for spontaneous ignition”, Int. J. Numer. Methods Eng.
30 (1990) 953–963; doi:10.1002/nme.1620300502.Google Scholar
Prata, A. T. and Sparrow, E. M., “Heat transfer and fluid flow characteristics for an annulus of periodically varying cross section”, Numer. Heat Transfer
7 (1984) 285–304; doi:10.1080/01495728408961826.CrossRefGoogle Scholar
Roose, D. and Hlavacek, V., “Numerical computation of Hopf bifurcation points for parabolic diffusion-reaction differential equations”, SIAM J. Appl. Math.
43 (1983) 1075–1085; doi:10.1137/0143070.CrossRefGoogle Scholar
Roose, D. and Hlavacek, V., “A direct method for the computation of Hopf bifurcation points”, SIAM J. Appl. Math.
45 (1985) 879–894; doi:10.1137/0145053.CrossRefGoogle Scholar
Roose, D., Piessens, R., Hlavacek, V. and van Rompay, P., “Direct evaluation of critical conditions for thermal explosion and catalytic reaction”, Combust. Flame
55 (1984) 323–329; doi:10.1016/0010-2180(84)90171-8.Google Scholar
Sexton, M. J., Macaskill, C. and Gray, B. F., “Thermal ignition in rectangular and triangular regions”, ANZIAM J.
42(E) (2000) C1283–C1304; doi:10.0000/anziamj.v42i0.646.Google Scholar
Seydel, R., “Numerical computation of branch points in nonlinear equations”, Numer. Math.
33 (1979) 339–352; doi:10.1007/BF01398649.Google Scholar
Sorensen, D. C. and Zhou, Y., “Direct methods for matrix sylvester and lyapunov equations”, J. Appl. Math.
2003 (2003) 277–303; doi:10.1155/S1110757X03212055.Google Scholar
Tao, W., Numerical heat transfer, 2nd edn (Xi’an Jiaotong University Press, 2001) (in Chinese).Google Scholar
Xue, D. and Chen, Y., Solving applied mathematical problems with MATLAB (Chapman and Hall/CRC, Boca Raton, 2008); doi:10.1201/b17177.Google Scholar
Zhang, Y. P., Su, G. H., Qiu, S. Z., Tian, W. X., Gaus-Liu, X., Kretzschmar, F. and Miassoedov, A., “Numerical study on the heat transfer characteristics of live-l4 melt pool with a partial solidification process”, Prog. Nucl. Energy
74 (2014) 213–221; doi:10.1016/j.pnucene.2014.03.011.Google Scholar
You have
Access