Heat Conduction

Heat conduction is the flow of internal energy from a region of higher temperature to a region of lower temperature by the interaction of the adjacent particles (for example, atoms, molecules, ions, or electrons) in the intervening space. This flow can be viewed as the transfer of energy from the more energetic particles to the less energetic particles of a substance due to interactions between the particles.

Conduction takes place in all forms of matter: solids, liquids, gases, and plasmas. In solids, heat conduction is due to the combination of vibrations of the molecules in a lattice and diffusion of free electrons. In gases and liquids, heat conduction is due to collisions between molecules and the diffusion of molecules during their random motion. Conduction is greater in solids because of the relatively close fixed spatial relationships between atoms, whereas fluids and gases are less conductive due to the large distance between atoms.

The law of heat conduction, also known as Fourier’s law, provides a means of calculating the local heat flux. This law can be stated in two equivalent forms:

• The integral form, which considers the amount of energy flowing into or out of a body as a whole.
• The differential form, which considers the flow rates or fluxes of energy locally:

  $$\mathbf{\text{q}}″=-k\nabla T$$

  (195)

  In this expression, $\mathbf{\text{q}}″$ $[W/m^2]$ is the local heat flux vector, $k$ $[W/mK]$ is the thermal conductivity of the material, and $\nabla T$ $[K/m]$ is the temperature gradient. The thermal conductivity of the material can be a strong function of temperature, pressure (for gases), spatial location, and can exhibit anisotropic behavior (that is, it varies with orientation and is represented by a second-order tensor). The negative sign in the expression is because $k>0$ and heat always moves from a high temperature region to a low temperature (as defined in the Second Law of Thermodynamics).

Thermal conductivities vary from material to material, being the largest for metallic solids, smaller for nonmetallic solids, very small for liquids, and extremely small for gases.

The major objective of performing a simulation that includes conductive heat transfer is to determine the temperature field in a medium, taking into account the conditions that are imposed on the boundaries. The First Law of Thermodynamics (that is, Conservation of Energy) and Fourier’s law can be used to derive the Heat Diffusion Equation (a second-order partial differential equation in temperature, or enthalpy, or energy) from which the temperature field can be calculated. This temperature solution can be steady (a function of space only and not changing with time), or unsteady (the temperature field varies in both space and time as, for example, in metal quenching applications).