An analytic approach to the unsteady heat conduction processes in one-dimensional composite media F. de Monte * Dipartimento di Energetica, University of L’Aquila, Localit  a Monteluco, 67040 Roio Poggio, L’Aquila, Italy Received 3 August 2000; received in revised form 5 January 2001 Abstract The transient heat conduction problems in one-dimensional multi-layer solids are usually solved applying con- ventional techniques based on Vodicka’s approach. However, if the thermal diffusivity of each layer is retained on the side of the heat conduction equation modified from the application of the separation-of-variables method where the time-dependent function is collected, then the modified heat conduction equation by itself represents a transparent statement of the physical phenomena involved. This ‘natural’ choice so simplifies unsteady heat conduction analysis of composite media that thermal response computation reduces to a matter of relatively simple mathematics when compared with traditional techniques heretofore employed. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Composites; Conduction; Unsteady 1. Introduction An exact closed-form solution for the transient tem- perature response of multi-layer series composite solids was originally obtained by Vodicka [1]. He determined the solution applying the method of separation of vari- ables to the heat conduction partial differential equation. In separating the variables, Vodicka retained the ther- mal diffusivity on the side of the modified heat con- duction equation where the space-variable function is collected [1,2]. This choice makes the time-variable function which appears in the series solution of the problem explicitly independent of the thermal diffusiv- ity. For this reason the solution does not represent the physical reality of the problem, although it yields correct quantitative results, and the computation of the eigen- values and corresponding eigenfunctions results is a quite lengthy and difficult matter. After Vodicka, the analysis of unsteady-state heat conduction in composite solids has been under devel- opment for some 50 years and includes individual con- tributions of inspired quality. Noteworthy attempts are those of Mikhailov et al. [3], Carslaw and Jaeger [4], Huang and Chang [5] and Feng and Michaelides [6], Haji-Sheikh and Beck [7], and Yener and € Ozis ßik [8], to which correspond, respectively, the orthogonal expan- sion technique, the Laplace transform method, the Green’s function approach, the Galerkin procedure and the finite integral transform technique. The reader may refer to € Ozis ßik [9] for a quite complete review of the specialized literature. However, with the exception of Carslaw and Jaeger [4], who only considered regions of infinite and semi-infinite thickness, all attempts drew on Vodicka’ approach. Herein lies their mathematical dif- ficulty, which is mainly related to the determination of the eigenvalues and corresponding eigenfunctions. In fact, in transient heat conduction the thermal diffusivity acts straight on only the time-dependent function, and the method of Vodicka is ill equipped to do it. Con- cerning this, Nietzsche would say that a difficulty per- sists only for as long as it is subject to inappropriate (although outstanding) methods of attack [10]. As a matter of fact, another exact closed-form solu- tion for the transient thermal response of multi-layer composite media was independently derived by Tittle [11]. He applied the separation-of-variables method to the problem at issue and, with an appropriate choice concerning the position of the thermal diffusivity in the International Journal of Heat and Mass Transfer 45 (2002) 1333–1343 www.elsevier.com/locate/ijhmt * Tel.: +39-0862-434326; fax: +39-0862-434303. E-mail address: demonte@ing.univaq.it (F. de Monte). 0017-9310/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII:S0017-9310(01)00226-5