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Research Papers

A Revised Approach for One-Dimensional Time-Dependent Heat Conduction in a Slab

[+] Author and Article Information
D. Angeli

e-mail: diego.angeli@unimore.it

G. S. Barozzi

DIEF - Dipartimento di Ingegneria “Enzo Ferrari”,
Università di Modena e Reggio Emilia,
Via Vignolese, 905,
41125 Modena, Italy

S. Polidoro

Dipartimento di Scienze Fisiche,
Informatiche e Matematiche,
Università di Modena e Reggio Emilia,
Via G. Campi, 213/b,
41125 Modena, Italy

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received April 16, 2012; final manuscript received October 23, 2012; published online February 14, 2013. Assoc. Editor: Giulio Lorenzini.

J. Heat Transfer 135(3), 031301 (Feb 14, 2013) (8 pages) Paper No: HT-12-1167; doi: 10.1115/1.4007982 History: Received April 16, 2012; Revised October 23, 2012

Classical Green’s and Duhamel’s integral formulas are enforced for the solution of one dimensional heat conduction in a slab, under general boundary conditions of the first kind. Two alternative numerical approximations are proposed, both characterized by fast convergent behavior. We first consider caloric functions with arbitrary piecewise continuous boundary conditions, and show that standard solutions based on Fourier series do not converge uniformly on the domain. Here, uniform convergence is achieved by integrations by parts. An alternative approach based on the Laplace transform is also presented, and this is shown to have an excellent convergence rate also when discontinuities are present at the boundaries. In both cases, numerical experiments illustrate the improvement of the convergence rate with respect to standard methods.

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Figures

Grahic Jump Location
Fig. 1

Schematic diagram of the problem

Grahic Jump Location
Fig. 2

Solutions to Eq. (3) with boundary conditions (24), computed on the [0,1]×[0,1] domain, according to: (a) Eq. (22); (b) Eq. (29); (c) Eq. (36). All the finite sums are evaluated up to a number of terms N = 30. Left panels: profiles of ϑ(z,τ) for selected z-values. Middle panels: surface visualization of ϑ(z,τ). Right panels: profiles of ϑ(z,τ) for selected τ-values. Dashed lines in left and right panels denote a numerical solution obtained by second-order finite differencing.

Grahic Jump Location
Fig. 3

Numerical assessment of convergence rates: (a) infinity norm of the n-th series terms and (b) ratio between the norms of successive terms versus n in Eqs. (29) and (36)

Grahic Jump Location
Fig. 4

Solutions to Eq. (3) with boundary conditions (37), computed on the [0,1]×[0,1] domain according to: (a) Eq. (22); (b) Eq. (29); (c) Eq. (36). All the finite sums are evaluated up to a number of terms N = 30. Left panels: profiles of ϑ(z,τ) for selected z-values. Middle panels: surface visualization of ϑ(z,τ). Right panels: profiles of ϑ(z,τ) for selected τ-values. Dashed lines in left and right panels denote a numerical solution obtained by second-order finite differencing.

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