# Macroscopic Heat Conduction Formulation

• Leandro A. Sphaier
• Jian Su
Reference work entry

## 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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© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Leandro A. Sphaier
• 1
Email author
• Jian Su
• 2
• 3
1. 1.Laboratory of Thermal Sciences – LATERMO, Department of Mechanical Engineering – TEM/PGMECUniversidade Federal FluminenseNiteroiBrazil
2. 2.Nuclear Engineering Department – PEN and Nanoengineering Department – PENT, COPPEFederal University of Rio de JaneiroRio de JaneiroBrazil
3. 3.Universidade Federal do Rio de Janeiro – UFRJRio de JaneiroBrazil

## Section editors and affiliations

• Renato M. Cotta
• 1
1. 1.Universidade Federal do Rio de Janeiro, Department of Mechanical EngineeringUFRJ Politécnica/COPPERio de JaneiroBrazil