The Thermal Properties of Matter

To be able to use Fourier’s law, the thermal conductivity of the material must be known. This property, referred to as a transport property, gives an indication of the rate at which energy is transferred by the diffusion process and it depends on the physical structure of matter, atomic and molecular, which is related to the state of the matter.

Thermal Conductivity

From Fourier’s law (equation 6 from this blog entry), we can define the thermal conductivity associated with conduction in the x-direction as:

Analogous definitions are associated with thermal conductivities in the y- and z-directions (k_y, k_z), but for an isotropic material, the thermal conductivity is independent of the direction of transfer, k_x = k_y = k_z ≡ k.

Thus, for a given temperature gradient, the conduction heat flux increases with increasing thermal conductivity.

In general, the thermal conductivity of a solid is larger than that of a liquid, which is larger than that of a gas i.e., k_solid > k_liquid > k_gas. The thermal conductivity of a solid may be more than four orders of magnitude larger than that of a gas. This is largely due to differences in intermolecular spacing for the two states.

The Solid State: A solid may be comprised of free electrons and atoms bound in a periodic arrangement called a lattice. Therefore, transport of thermal energy may be due to two effects: the migration of free electrons and lattice vibrational waves (phonons). In pure metals, the electron contribution to conduction heat transfer dominates, whereas in nonconductors and semiconductors, the phonon contribution is dominant.

The Fluid State: This includes both liquids and gases.  The thermal energy transport is less effective in fluids due to the much larger intermolecular spacing and more random motion of molecules as compared to the solid state. Thus, the thermal conductivity of gases and liquids is generally smaller than that of solids.

The kinetic theory of gases can be used to explain the effect of temperature, pressure and chemical species on the thermal conductivity of a gas. From this theory, we know that thermal conductivity is directly proportional to the density of the gas,

From this theory it is known that the thermal conductivity is directly proportional to the density of the gas, the mean molecular speed c, and the mean free path λ_mfp, which is the average distance traveled by a molecule before experiencing a collision:

For an ideal gas, the mean free path may be expressed as:

Where: k_B is the Boltzmann’s constant, k_B = 1.381 x 10^-23 J/K and d is the diameter of the gas molecule.

As is expected, the mean free path is small for high pressure or low temperature due to the densely packed molecules. The mean free path also depends on the diameter of the molecule where larger molecules are more likely to experience collisions than small molecules; in the rare case of an infinitesimally small molecule, the molecules cannot collide, resulting in an infinite mean free path.

Other Relevant Properties

In the analysis of heat transfer problems, it is necessary to use several properties of matter. These properties are often referred to as thermophysical properties. These properties include two distinct categories:

• Transport Properties: includes the diffusion rate coefficients such as the thermal conductivity, k (for heat transfer), and the kinematic viscosity, ν (for momentum transfer).
• Thermodynamics Properties: These relate to the equilibrium state of a system. Examples include density (ρ) and specific heat (c_p). The volumetric heat capacity, ρc_p (J/m³K), measures the ability of a material to store thermal energy. Since substances, like solids and liquids, with large densities are characterized by small specific heats they are very good energy storage media while gases which have small densities are poor for thermal energy storage.

In heat transfer analysis, the ratio of the thermal conductivity to the heat capacity is an important property termed the thermal diffusivity α, with the units of m²/s:

Thermal diffusivity measures the ability of a material to conduct thermal energy relative to its ability to store thermal energy.

Materials of large α will respond quickly to changes in their thermal environment, while materials with small α will respond more slowly, taking longer to reach a new equilibrium condition.