Critical times in multilayer diffusion. II: approximate solutions. (English) Zbl 1177.80023

Summary: Traditional averaging methods for multilayer diffusion give inaccurate approximations of critical time behaviour, such as the heating time of a material. In particular, they fail to capture the importance of layer order. We use a perturbation expansion of an exact solution to find a simple approximate solution which accurately describes the critical time for transport across multiple layers. This approximate solution is then used to find a correction for the averaging method which captures the key critical time behaviour.


80A20 Heat and mass transfer, heat flow (MSC2010)
80M25 Other numerical methods (thermodynamics) (MSC2010)


Full Text: DOI


[1] Barry, S. I.; Sweatman, W. L.: Modelling heat transfer in steel coils, Anziam j. (E) 50, C668 (2009) · Zbl 1359.80005
[2] M. McGuinness, W.L. Sweatman, D.Y. Baowan, S.I. Barry, Annealing steel coils, in: T. Marchant, M. Edwards, G.N. Mercer (Ed.), MISG 2008 Proceedings, 2009. · Zbl 1246.80003
[3] Yuen, W. Y. D.: Transient temperature distribution in a multilayer medium subject to radiative surface cooling, Appl. math. Model. 18, 93-100 (1994) · Zbl 0796.73076
[4] Pontrelli, G.; De Monte, F.: Mass diffusion through two-layer porous media: an application to the drug-eluting stent, Int. J. Heat mass transfer 50, 3658-3669 (2007) · Zbl 1113.80013
[5] Landman, K.; Mcguinness, M.: Mean action time for diffusive processes, J. appl. Math. decis. Sci. 4, No. 2, 125-141 (2000) · Zbl 0998.76083
[6] Mcnabb, A.; Wake, G. C.: Heat conduction and finite measures for transition times between steady states, IMA J. Appl. math. 47, No. 2, 192-206 (1991) · Zbl 0767.35018
[7] Mcnabb, A.: Mean action times, time lags and mean first passage times for some diffusion problems, Math. comput. Model. 18, No. 10, 123-129 (1993) · Zbl 0802.35122
[8] Hickson, R. I.; Barry, S. I.; Mercer, G. N.: Critical times in multilayer diffusion. Part 1: exact solutions, Int. J. Heat mass transfer (accepted) (2009) · Zbl 1177.80022
[9] Hickson, R. I.; Barry, S. I.; Mercer, G. N.: Exact and numerical solutions for effective diffusivity and time lag through multiple layers, Anziam j. (E) 50, C682-C695 (2009) · Zbl 1359.80006
[10] Azeez, M. F. A.; Vakakis, A. F.: Axisymmetric transient solutions of the heat diffusion problem in layered composite media, Int. J. Heat mass transfer 43, 3883-3895 (2000) · Zbl 1073.80508
[11] Ash, R.; Barrer, R. M.; Palmer, D. G.: Diffusion in multiple laminates, Brit. J. Appl. phys. 16, 873-884 (1965)
[12] Ash, R.; Barrer, R. M.; Petropoulos, J. H.: Diffusion in heterogeneous media: properties of a laminated slab, Brit. J. Appl. phys. 14, 854-862 (1963)
[13] Barrer, R. M.: Diffusion and permeation in heterogenous media, Diffusion in polymers, 165-215 (1968)
[14] Maple, Maplesoft, Waterloo, Ontario. Available from: <http://www.maplesoft.com/>.
[15] Crank, J.: The mathematics of diffusion, (1957) · Zbl 0077.32604
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.