 # Thermal conductivity

Conductivity

## Thermal conductivity in general

Thermal conductivity is the transfer of thermal energy without physically moving the material. Gasses and liquids transfer heat directly by collisions between molecules, whereas solids transfer through vibrations in the lattice structure. The energy's movement through the lattice is described using phonons i.e. quanta of lattice vibrations. Metals have the additional property that mobile electrons participate in the the transfer of heat, which makes them better thermal conductors than other materials.

Thermal conductivity, κ, for a specific material is

 ΔQ = −κ · ΔT Δt · A Δx

 Q = The amount of heat T = Temperature t = Time A = Cross section area x = Distance

The formula works for experimental determination of the specific thermal conductivity, under specific circumstances, but in practice, other formulas are used, depending on the material, that are more appropriate.

When working with thermal conductivity for insulation, e.g. for construction, the specific thermal conductivity, κ, is called the λ value instead. For this specific application, you also use what is called the U-value. The U-value is the resulting thermal conductivity per area unit (W/m²·K), through the combined materials, e.g. roofs: roof tile + stone wool or insulated concrete walls: concrete wall + PUR foam + concrete wall.

The differences in conductivity can easily become somewhat unclear, so, for the sake of clarity and a getting a sense of materials and relative values (at around 25 °C):

 Material κ (W/m·K) CO2 0.0146 Air 0.0262 Stone wool 0.045 Milk 0.53 Water 0.606 Concrete 0.8-1.28 Mercury 8.3 Aluminum 237 Copper 401

## Thermal conductivity in solids

Thermal conductivity in solids is a combination of vibrations in the lattice structure and electrons moving, if possible.

Heat makes the atoms vibrate, the the vibrations propagate through the material by moving from atom to atom. This can be enhanced using uniform vibrations, called fonons. Fonons is a quantum mechanical description of a particular type of vibrational movements, in which the lattice oscillates uniformly with the same frequency. This is equivalent to normal mode in classical mechanics.

In practice, you don't calculate thermal conductivity as a function of temperature for non-metallic solids, as the variability in the materials is too big. Instead you measure the thermal conductivity under specific conditions. In practice this is done by exposing the material to a constant effect per area unit, q, and at constant temperature on both sides of the material, the difference in temperature is measured (T2 - T1) along with the material thickness, d, and thermal conductivity (with the unit W/m·K) is calculated as:

 κ = d · q |T2 - T1|

For solid inorganic materials (not metals) the following correlation is used for thermal conductivity, κ, as a function of the temperature T (in Kelvin):

κ = A + BT + CT2

where A, B and C are regression coefficients for the material. The same formula is used for inorganic liquids. The correlation is a model based on experimental experience, so it should be used with care and only within a limited temperature range. The size of the temperature interval depends on the material.

For the metallic solids, on the other hand, the electric and thermal conductivities are proportional, but while the electrical conductivity decreases with increasing temperature, the thermal conductivity increases. This proportionality is described by Wiedemann-Franz' Law (sometimes called Wiedemann-Franz-Lorenz' Law):

 L · T = κ σ

 κ = Thermal conductivity σ = Electrical conductivity L = The Lorenz number T = Temperature

where the Lorenz number, L, is the proportionality factor. The relationship is based on the fact that both thermal and electrical conductivity involves free electrons in the metal. Thermal conductivity increases with increasing particle speed, as the particle speed increases the energy transport. For the electrical conductivity it is the opposite. Here the conductivity decreases, because the collisions prevent the electron's/charges forward motion. This means that the ratio between thermal and electrical conductivity depends on the average speed squared, which is proportionate with the kinetic temperature.

## Thermal conductivity in liquids

Thermal conductivity in liquids is one of the things that we still don't understand very well. The conductivity has some similarities with gasses, especially heavy gasses, in regards to the molecules being in motion, and moving faster at increasing temperatures. But, where the thermal conductivity increases with increasing temperatures for gasses, it is mostly decreasing for liquids. It is however mostly a general rule of thumb, so you have to test it for the individual liquids. The thermal conductivity for water, for instance, increases undtil around 140 °C, after which it decreases. Therefore: for liquids you have to determine the thermal conductivity as a function of temperature experimentally.

Obviously, models have been made for thermal conductivity as a function of temperature. For liquid inorganic compounds, the following correlation is sometimes used for the thermal conductivity, κ, as a function of the temperature T (in Kelvin):

κ = A + BT + CT2

where A, B and C are regression coefficients for the material. The same formula is used for solids.

The equivalent correlation for liquid organic compounds is:

 log10 κ = A + B· ⎛⎝ 1 + T ⎞⎠ 2/7 C

The correlations are models based on experimental experience, so they have to be used with care and within a limited temperature interval. The size of the interval depends on the material.

Another model being used is a modified version of Bridgman's equation, where the liquid molecules are regarded as positioned in a cubic lattice, where the energy is transferred from one lattice plane to another, at the speed of sound, through the liquid. The formula looks like this:

 κ = k · vs · 2.8 · ⎛⎝ NA ⎞⎠ 2/3 Vm

 k = Boltzmann's constant = 1.38065 · 10−23 J/K vs = The speed of sound in the material NA = Avogadro's number = 6.02214076 · 1023 mol−1 Vm = The molar volume of the material

## Thermal conductivity in gasses

Thermal conductivity in gasses is generally poor. The is a property used e.g. for insulation of houses, where the main part of the insulating property comes from making the molecules in the air move as little as possible.

For an ideal gas, the speed of the heat transfer is proportionate with the number of molecules per volume unit, the average particle speed and the mean free path of the gas.

 κ = n · ❬v❭ · λ · cv 3NA

 κ = Thermal conductivity n = Particles per volume unit ❬v❭ = Average particle speed λ = The mean free path, i.e. the average distance a moving particle travels between successive impacts cv = Molar heat capacity

In practice this means that e.g. lighter molecules are better at conducting heat than the heavy ones, simply they are easier to get in motion. For modern thermo glass windows, gasses with poor thermal conductivity are used between the sheets of glass, contrary to earlier, where the air was removed, thereby leaving fewer molecules to collide. For the Dewar flask in thermos (the inner glass container), pumping the air out of the double walled flask has also been the traditional solution.