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Macroscopic Heat Conduction Formulation

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Handbook of Thermal Science and Engineering

Abstract

In this chapter, mathematical formulations of macroscopic heat conduction are derived from the First Law of Thermodynamics. Specific forms of the heat conduction equation in isotropic media are given in Cartesian, Cylindrical, and Spherical coordinates systems, as well as in a general orthogonal coordinate system. Heat conduction equations in anisotropic media and in heterogeneous media are then derived. Mathematical formulations of one-dimensional transient heat conduction with phase change and in multilayered composite media are presented. Finally, classical and improved lumped parameter formulations for transient heat conduction problems are analyzed more closely. The so-called Coupled Integral Equations Approach (CIEA) is reviewed as a problem reformulation and simplification tool in heat and mass diffusion. The averaged temperature and heat flux, in one or more space coordinates, are approximated by Hermite formulae for integrals, yielding analytic relations between boundary and average temperatures, to be used in place of the usual plain equality assumed in the classical lumped system analysis. The accuracy gains achieved through the improved lumped-differential formulations are then illustrated through a few typical examples.

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Notes

  1. 1.

    English translation reprint in (Fourier 1878).

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Authors and Affiliations

  1. Laboratory of Thermal Sciences – LATERMO, Department of Mechanical Engineering – TEM/PGMEC, Universidade Federal Fluminense, Niteroi, RJ, Brazil

    Leandro A. Sphaier

  2. Nuclear Engineering Department – PEN and Nanoengineering Department – PENT, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Jian Su

  3. Universidade Federal do Rio de Janeiro – UFRJ, Rio de Janeiro, RJ, Brazil

    Renato Machado Cotta

Authors
  1. Leandro A. Sphaier
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  2. Jian Su
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  3. Renato Machado Cotta
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Corresponding author

Correspondence to Leandro A. Sphaier .

Section Editor information

  1. Armour College of Engineering, Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, Chicago, Illinois, USA

    Sumanta Acharya

  2. Gas Technology Institute, Des Plaines, Illinois, USA

    Yaroslav Chudnovsky

  3. Universidade Federal do Rio de Janeiro - UFRJ, Rio de Janeiro, Rio de Janeiro, Brazil

    Renato Machado Cotta

  4. Department of Mechanical Engineering, Louisiana State University, Baton Rouge, Louisiana, USA

    Ram Devireddy

  5. Mechanical and Aerospace Engineering, University of California Los Angeles, Los Angeles, California, USA

    Vijay K. Dhir

  6. Cekmeköy Campus, Özyegin University, Çekmeköy - Istanbul, Turkey

    M. Pinar Mengüç

  7. Department of Mechanical and Industrial Engineering, Faculty of Applied Science + Engineering, University of Toronto, Centre for Advanced Coating Technologies, Toronto, Ontario, Canada

    Javad Mostaghimi

  8. Department of Mechanical Engineering, University of California, Riverside, California, USA

    Kambiz Vafai

  9. Universidade Federal do Rio de Janeiro, Department of Mechanical Engineering, UFRJ Politécnica/COPPE, Rio de Janeiro, Brazil

    Renato M. Cotta

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Sphaier, L.A., Su, J., Cotta, R.M. (2018). Macroscopic Heat Conduction Formulation. In: Handbook of Thermal Science and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-26695-4_3

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