×

zbMATH — the first resource for mathematics

Two-dimensional fully adaptive solutions of solid-solid alloying reactions. (English) Zbl 0588.65086
Solid-solid alloying reactions occur in a variety of pyrotechnical applications. They arise when a mixture of powders composed of appropriate oxidizing and reducing agents is heated. The large quantity of heat evolved produces a self-propagating reaction front that is often very narrow with sharp changes in both the temperature and the concentrations of the reacting species. Solution of problems of this type with an equispaced or midly nonuniform grid can be extremely inefficient. In this paper we develop a two-dimensional fully adaptive method for solving problems of this class. The method adaptively adjusts the number of grid points needed to equidistribute a positive weight function over a given mesh interval in each direction at each time level. We monitor the solution from one time level to another to ensure that the local error per unit step associated with the time differencing method is below some specified tolerance. The method is applied to several examples involving exothermic, diffusion-controlled, self-propagating reactions in packed bed reactors.

MSC:
65Z05 Applications to the sciences
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
80A25 Combustion
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hardt, A.P.; Phung, P.V., Combust. flame, 21, 77, (1973)
[2] {\scM. D. Smooke and M. L. Koszykowski}, SIAM J. Sci. Statist. Comput.
[3] Birnbaum, M., Determination of palladium/aluminum reaction propagation rates and temperatures, Sandia national laboratories report, SAND78-8503, (1978)
[4] Birnbaum, M., Studies of palladium/aluminum powders, Sandia national laboratoriezs report, SAND78-0486, (1978)
[5] Booth, F., Trans. Faraday soc., 44, 790, (1953)
[6] Margolis, S.B., SIAM J. appl. math., 43, 351, (1983)
[7] Miller, K.; Miller, R., SIAM J. numer. anal., 18, 1019, (1981)
[8] Gelinas, R.J.; Doss, S.K.; Miller, K., J. comput. phys., 40, 202, (1981)
[9] Davis, S.F.; Flaherty, J.E., SIAM J. sci. statist comput., 3, 6, (1982)
[10] White, A.B., SIAM J. numer. anal., 19, 683, (1982)
[11] Bolstad, J.H., An adaptive finite difference method for hyperbolic systems in one space dimension, () · Zbl 0607.76110
[12] Dwyer, H.A.; Kee, R.J.; Sanders, B.R., Aiaa j., 18, 1205, (1980)
[13] Winkler, K.A.; Norman, M.L.; Newman, M.J., Adaptive mesh techniques for fronts in star formation, () · Zbl 0575.76075
[14] Djomehri, J.; Miller, K., A moving finite element code for general systems of PDE’s in 2-D, ()
[15] Berger, M.; Gropp, W.D.; Oliger, J., ()
[16] Berger, M., Adaptive mesh refinement for hyperbolic partial differential equations, () · Zbl 0536.65071
[17] Gropp, W.D., SIAM J. sci. statist. comput., 1, 191, (1980)
[18] Dwyer, H.A.; Smooke, M.D.; Kee, R.J., ()
[19] Herbst, B.M.; Mitchell, A.R.; Schoombie, S.W., Equidistributing principles involved in two moving finite element methods, () · Zbl 0532.65074
[20] Smooke, M.D., J. optim. theory appl., 39, 489, (1983)
[21] Smooke, M.D., J. comput. phys., 48, 72, (1982)
[22] Kautsky, J.; Nichols, N.K., SIAM J. sci. statist. comput., 1, 499, (1980)
[23] Deuflhard, P., Numer. math., 22, 289, (1974)
[24] Curtis, A.R.; Powell, M.J.; Reid, J.K., J. inst. math. appl., 13, 117, (1974)
[25] Newsam, G.N.; Ramsdell, J.D., Estimation of sparse Jacobian matrices, Harvard university report TR-17-81, (1981) · Zbl 0558.65030
[26] Coleman, T.F.; More, J.J., Estimation of sparse Jacobian matrices and graph coloring problems, Argonne national laboratory report ANL-81-39, (1981)
[27] Dletz, P.W., Ind. eng. chem. fundam., 18, 283, (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.