Analyses of non-Fourier heat conduction in 1-D cylindrical and spherical geometry – An application of the lattice Boltzmann method Subhash C. Mishra ⇑ , Harsh Sahai Department of Mechanical Engineering, IIT Guwahati, Guwahati 781039, India article info Article history: Received 29 March 2012 Received in revised form 20 June 2012 Accepted 6 July 2012 Available online 27 July 2012 Keywords: Non-Fourier heat conduction Cylindrical and spherical geometry Lattice Boltzmann method Finite volume method abstract This article deals with the implementation of the lattice Boltzmann method (LBM) for the analyses of non-Fourier heat conduction in 1-D cylindrical and spherical geometries. Evolution of the wave like tem- perature distributions in the medium is obtained, and analysed for the effects of different sets of thermal perturbations at the inner and the outer boundaries of the geometry. The LBM results are validated against those available in the literature, and those obtained by solving the same problems using the finite volume method (FVM). Results of the LBM are in excellent agreement with those reported in the litera- ture, and with the results from the FVM. Computationally, the LBM has an advantage over the FVM. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Nothing is instantaneous. The maximum speed of propagation is limited by the speed of light, i.e., c = 3.0  10 8 ms 1 . If the effects of transport of thermal radiation (light is part of the thermal radi- ation) are investigated at time scale of the O distance speed of light  10 9 s   or lower, even the transport by the fastest mode of heat transfer, i.e., radiation, becomes a transient process. Thus, a time lag C between the cause and its associated effects exists. If the imposition of tem- perature difference DT / across a depth Dr / is the cause, the effect, the propagation of energy (heat flux) by conduction at any location r / in a medium with thermal conductivity k will be as below: q  ðr  ; t  þ CÞ¼krT  ðr  ; t  Þ ð1Þ It is to be noted that the time lag C represents the difference in time between the appearance of a temperature gradient and induction of the corresponding heat flux. This time lag allows the system to accommodate a finite speed of propagation of thermal signals, as suggested by the theory of relativity. In the aforementioned equation, if the thermal lag time C ¼ a C 2  ! 0, where a is the thermal diffusivity and C is the speed of propagation of the heat wave, Eq. (1) takes the familiar form of the governing law of heat conduction, known as Fourier’s law of heat conduction. q  ðr  ; t  Þ¼krT  ðr  ; t  Þ ð2Þ Thus, the Fourier’s law of heat conduction (Eq. (2)) is based on the assumption that there is no time lag between the cause and the ef- fect. In other words, as soon as the temperature gradient is imposed, the effect of conduction heat transfer will be felt instantaneously at all locations in the medium. Therefore, the non-Fourier heat con- duction equation (Eq. (1)) reduces to Fourier’s law of heat conduc- tion (Eq. (2)) if heat is assumed to propagate at an infinite speed. If the left hand side (LHS) of Eq. (1) is expanded into Taylor’s series, and second and higher order terms are neglected, we get the following: C @q  @t  þ q  ¼krT  ð3Þ Thus, the assumption that the effects of the perturbations is felt instantaneously at all locations does not hold true when a phenom- enon is investigated at lower system time scales defined as the ratio of the characteristic length dimension to the speed of propagation of the perturbation. In most commonly encountered materials, the time lag C is small enough to justify the omission of higher order Taylor terms. The situation is encountered fairly often (the ‘Telegrapher’s equation’) and has application in areas other than conduction. The hyperbolic heat conduction equation obtained by ignoring the higher order terms, has been known to be fairly accu- rate for materials with the large time lag too. Almost 136 years after Fourier proposed the law of heat con- duction (Eq. (2)), in 1958, Cattaneo [1] and Vernotee [2] suggested a revision to accommodate the theory of relativity. They proposed a form of heat conduction equation wherein there exists a constant thermal time lag between the cause and its effects, thus generaliz- ing the heat conduction equation (Eq. (3)). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.07.014 ⇑ Corresponding author. Tel.: +91 361 2582660; fax: +91 361 2690762. E-mail address: scm_iitg@yahoo.com (S.C. Mishra). International Journal of Heat and Mass Transfer 55 (2012) 7015–7023 Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt