Conduction

Heat conduction is the transfer of internal energy through interactions between the microscopic particles that compose a conducting substance. Energy flows in the direction of high temperature to low temperature as those particles in high temperature regions are more energetic than their low temperature neighbors.

Conduction takes place in all forms of matter: solids, liquids, gases, and plasmas. In solids, heat conduction is occurs due to the vibrations of molecules in a lattice in conjunction with diffusion of free electrons. In gases and liquids, heat conduction is due to collisions between molecules and the diffusion. Conduction is greater in solids because of the relatively close spatial relationships between atoms, whereas fluids and gases are characterized by larger distances 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:

  $$\dot{\mathbf{\text{q}}}″=-k\nabla T$$

  (1661)

  In this expression, $\dot{\mathbf{\text{q}}}″$ $[W/m^2]$ 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 required because k is greater than zero, yet heat always moves from a high temperature region to a low temperature region.

Thermal conductivities vary from material to material. They are 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 that satisfies the boundary conditions. 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).