Exact and numerical solutions for effective diffusivity and time lag through multiple layers

We consider diffusion through multiple layers, with application to heat transport. An exact solution is derived and the time lag for heat conduction across the layers is studied. We show the limitations of traditional methods of averaging the diffusivity, which are only applicable in the steady state or for numerous layers. References J. Absi, D. S. Smith, B. Nait-Ali, S. Granjean, and J. Berjonnaux. Thermal response of two-layer systems: Numerical simulation and experimental verification. J. European Ceramic Soc., 25:367 -- 37, 2005. doi:10.1016/j.jeurceramsoc.2004.02.005. N. M. Aduirre, G. G. De La Cruz, Y. G. Gurevich, G. N. Logninov, and M. N. Kasyanchuk. Heat diffusion in two-layer structures: photoacoustic experiments. Phys. Stat. Sol., 220:781--787, 2000. doi:10.1002/1521-3951(200007)220:1<781::AID-PSSB781>3.0.CO;2-D. R. Ash, R. M, Barrer, and D. G. Palmer. Diffusion in multiple laminates. Brit. J. Appl. Phys., 16(6):873--884, 1965. M. F. Abdul Azeez and A. F. Vakakis. Axisymmetric transient solutions for the neat diffusion problem in layered composite media. Int. J. Heat. Mass Transf., 43:3883--3895, 2000. doi:10.1016/S0017-9310(99)00386-5. R. M. Barrer. Diffusion and permeation in heterogenous media., pages 165--215. Academic Press, London, 1968. S. I. Barry and W. L. Sweatman. Modelling heat transfer in steel coils. ANZIAM J. (E), 2008. (accepted). H. S. Carslaw and J. C. Jaeger. Conduction of heat in solids. Clarendon Press, Oxford, 1959. F. de Monte. Transient heat conduction in one-dimensional composite slab: A `natural' analytic approach. Int. J. Heat Mass Transfer, 43:3607--3619, 2000. doi:10.1016/S0017-9310(00)00008-9. F. de Monte. Multi-layer transient heat conduction using transition time scales. Int. J. Thermal Sci., 45:882--892, 2006. doi:10.1016/j.ijthermalsci.2005.11.006. M. Fukuda and H. Kawai. Diffusion of low molecular weight substances into a fiber with skin-core structure -- rigorous solution of the diffusion equations in a coaxial cylinder of multiple components. Polymer Engineering and Science, 34(4):330--340, 1994. doi:10.1002/pen.760340415. M. Fukuda and H. Kawai. Diffusion of low molecular weight substances into a laminar film. 1: Rigorous solution of the diffusion equations in a composite film of multiple layers. Polymer Engineering and Science, 35(8):709--721, 1995. doi:10.1002/pen.760350811. S. H. Gilbert and R. T. Mathias. Analysis of diffusion delay in a layered medium. Biophys J., 54:603--610, 1988. M. McGuinness, W. L. Sweatman, D. Y. Baowan, and S. I. Barry. Cold point determination in heat treated steel coils. In T. Marchant, M. Edwards, and G. N. Mercer, editors, MISG Proceedings, 2008. J. R. Miller and P. M. Weaver. Temperature profiles in composite plates subject to time-dependent complex boundary conditions. Composite Structures, 59:267--278, 2003. doi:10.1016/S0263-8223(02)00054-5. G. Otouranc and A.Z. Sahin. Eigenvalue analysis of temperature distribution in composite walls. Int J. Energy Res., 25:1189--1196, 2001. doi:10.1002/er.747. G. Pontrelli and F. de Monte. Mass diffusion through two-layer porous media: an application to the drug-eluting stent. Int. J. Heat Mass Transfer, 50:3658--3669, 2007. doi:10.1016/j.ijheatmasstransfer.2006.11.003. N. Shigesada, K. Kawasaki, and E. Teramoto. Travelling periodic waves in heteregeneous environments. Theor. Population Biology, 30:143--160, 1986. Y. Sun and I. S. Wichman. On transient heat conduction in a one-dimensional composite slab. Int. J Heat Mass Transfer, 47:1555--1559, 2004. doi:10.1016/j.ijheatmasstransfer.2003.09.011. C. W. Tittle. Boundary value problems in composite media: Quasi-orthogonal functions. J. Appl. Phys., 36(4):1486--1488, 1965. doi:10.1063/1.1714335