# Thermal conductivity

In physics, thermal conductivity, k, is the intensive property of a material that indicates its ability to conduct heat.

It is defined as the quantity of heat, Q, transmitted in time t through a thickness L, in a direction normal to a surface of area A, due to a temperature difference ΔT, under steady state conditions and when the heat transfer is dependent only on the temperature gradient.

thermal conductivity = heat flow rate × distance / (area × temperature difference)
$k=\frac{Q}{t}\times\frac{L}{A\times\Delta T}$

## Examples

In metals, thermal conductivity approximately tracks electrical conductivity, as freely moving valence electrons transfer not only electric current but also heat energy. However, the general correlation between electrical and thermal conductance is broken in other materials, due to the relative importance of phonon carriers for heat in non-metals. As shown in the table below, highly electrically conductive silver is less thermally conductive than diamond, which is an electrical semiconductor.

Thermal conductivity depends on many properties of a material, notably its structure and temperature. For instance, pure crystalline substances exhibit highly variable thermal conductivities along different crystal axes, due to differences in phonon coupling along a given crystal dimension. Sapphire is a notable example of variable thermal conductivity based on orientation and temperature, for which the CRC Handbook reports a thermal conductivity of 2.6 W/m·K perpendicular to the c-axis at 373 K, but 6000 W/m·K at 36 degrees from the c-axis and 35 K.

Air and other gases are generally good insulators, in the absence of convection. Therefore, many insulating materials function simply by having a large number of gas-filled pockets which prevent large-scale convection. Examples of these include expanded and extruded polystyrene (EPS and XPS, both popularly referred to as "styrofoam") and silica aerogel. Natural, biological insulators such as fur and feathers achieve similar effects by dramatically inhibiting convection of air or water near an animal's skin.

Thermal conductivity is important in building insulation and related fields. However, materials used in such trades are rarely subjected to chemical purity standards. Several construction materials' k values are listed below. These should be considered approximate due to the uncertainties related to material definitions.

The following table is meant as a small sample of data to illustrate the thermal conductivity of various types of substances. For more complete listings of measured k-values, see the references.

## Some typical thermal conductivities (k values)

Material Thermal conductivity
(W·m−1·K−1)
Temperature
(K)
Electrical conductivity
(S·m−1=Ω−1·m−1
=kg−1·m−3·s3·A2
=W·m−1·V−2=A·m−1·V−1)
Notes
Purified Synthetic Diamond 2,000-2,500 (Lateral)10−16 - (Ballistic)108+
Diamond, impure (C+0.1%N) 1,000ad 273a ~10−16 Type I (98.1% of
Gem Diamonds)
Silver (Ag), pure 406d - 429ag (418f) 300ag 61.35g 62.89 - 63.01 × 106 Highest electrical
conductivity
of any metal
Copper (Cu), pure 385d - 401g (386f - 390e) 273g-373g(293e) 59.17g 58.82 - 59.6 × 106
Gold (Au), pure 314d - 318fg 273g - 373g 40.98 - 45.45g × 106
Aluminium (Al), pure 205d - 237eg (220f) 293eg 37.45g 35.46 - 37.8 × 106
Brass (Cu+(35-15)%Zn) 109dg - 159g (151f) 296g 12.82g - 21.74g × 106
Iron (Fe), pure 71.8f - 80.4g (79.5d - 80.2a) 273g-373g(300a) 9.901g - 10.00 × 106
Cast iron(Fe+(2-3.5)%C+(1-3)%Si) 55f
Bronze (Cu+11%Sn) 42g - 50g ((25%Sn)26f) 296g 5.882g - 7.143g × 106
Carbon Steel (Fe+(1.5-0.5)%C) 36f - 54f (50.2d)
Lead (Pb), pure 34.7d - 35.3g (35f) 273g - 373g 4.545 - 4.854g × 106
Stainless Steel(Fe+18%Cr+8%Ni) 14a - 16.3fg 273a - 296g 1.389g - 1.429g × 106
Granite (Si+14%Al+4%K+3%Na) 1.73b - 3.98b 70.18%SiO2
Marble 2.07b - 2.94b Mostly CaCO3
Sandstone 1.83b - 2.90b ~95%-71%SiO2
Ice 1.6d - 2.2a (2.1e) 273a (293e)
Limestone 1.26b - 1.33b Mostly CaCO3
Concrete 0.8d - 1.28e 293e ~61%-67%CaO
Glass 0.8d−0.93e((96%SiO2)1.2-1.4)g 293eg (10−12)g 10−14 - 10−10 <1% Iron oxides
Soil 0.17c - 1.13c
Water 0.6de 293de (Pure)10−6-(Sweet)10−3±1-(Sea)1 <3%(Na+Mg+Ca)
Fibre-reinforced plastics 0.23g - 0.7g (1.06e) 296g (293e) (10−15 - 100)g 10-40%GF or CF
High-Density Polymers 0.33g - 0.52g 296g (10−16 - 102)g
Glycerol 0.29e 293e
Low-Density Polymers 0.04g - 0.33g (0.16 - 0.25)e 296g (293)e (10−17 - 100)g
Rubber (92%) 0.16a 303a ~10−13
Alcohols OR Oils 0.1e - 0.21e 293e
Wood (+12% water) 0.09091h - 0.21h (0.16a - 0.4e) (298a - 293e) Species Variable
Wood (oven-dry) 0.07692h - 0.17h (0.16a - 0.4e) (298a - 293e) Cedar - Hickory
Snow (dry) 0.11d
Cork (material) 0.04d - 0.07e 293e
Fiberglass OR Foam OR Wool 0.04d (0.03 - 0.045)e (293)e
EPS/XPS (PS+Air+CO2+CnH2n+x) 0.033ad (0.1 - 0.13)g 98a-298a (296)g (<10−14 - 100)g
Air (78%N+21%O+1%Ar) (1 atm) 0.024d - 0.0262a (0.025e) 273d-300a(293e)
Oxygen (O2) (1 atm) 0.0238d 293d
Nitrogen (N2) (1 atm) 0.0234d - 0.026a 293d - 300a
Silica Aerogel 0.003a 98a - 298a Foamed Glass
Material Thermal conductivity
(W·m−1·K−1)
Temperature
(K)
Electrical conductivity
(S·m−1=Ω−1·m−1
=kg−1·m−3·s3·A2
=W·m−1·V−2=A·m−1·V−1)
Notes

## Measurement

For good conductors of heat, Searle's bar method can be used [1]. For poor conductors of heat, Lees' disc method can be used [2]. VELA is an old data logging machine. An alternative traditional method using real thermometers is described at [3]. A brief review of relatively new class of dynamic methods that are measuring thermal conductivity, thermal diffusivity and specific heat within a single measurement is available at [4]

A thermal conductance tester, one of the instruments of gemology, determines if gems are genuine diamonds using diamond's uniquely high thermal conductivity.

## Related terms

The reciprocal of thermal conductivity is thermal resistivity, measured in kelvin-metres per watt (K·m·W−1).

When dealing with a known amount of material, its thermal conductance and the reciprocal property, thermal resistance, can be described. Unfortunately there are differing definitions for these terms.

### First definition (general)

For general scientific use, thermal conductance is the quantity of heat that passes in unit time through a plate of particular area and thickness when its opposite faces differ in temperature by one degree. For a plate of thermal conductivity k, area A and thickness L this is kA/L, measured in W·K−1. This matches the relationship between electrical conductivity (A·m−1·V−1) and electrical conductance (A·V−1).

There is also a measure known as heat transfer coefficient: the quantity of heat that passes in unit time through unit area of a plate of particular thickness when its opposite faces differ in temperature by one degree. The reciprocal is thermal insulance. In summary:

• thermal conductance = kA/L, measured in W·K−1
• thermal resistance = L/kA, measured in K·W−1
• heat transfer coefficient = k/L, measured in W·K−1·m−2
• thermal insulance = L/k, measured in K·m2·W−1.

The heat transfer coefficient is also known as thermal admittance, but this term has other meanings.

### Second definition (buildings)

When dealing with buildings, thermal resistance or R-value means what is described above as thermal insulance, and thermal conductance means the reciprocal. For materials in series, these thermal resistances (unlike conductances) can simply be added to give a thermal resistance for the whole.

A third term, thermal transmittance, incorporates the thermal conductance of a structure along with heat transfer due to convection and radiation. It is measured in the same units as thermal conductance and is sometimes known as the composite thermal conductance. The term U-value is another synonym.

The term K-value is a synonym for thermal conductivity.

In summary, for a plate of thermal conductivity k, area A and thickness L:

• thermal conductance = k/L, measured in W·K−1·m−2;
• thermal resistance (R value, thermal resistivity in scientific terms) = L/k, measured in K·m2·W−1;
• thermal transmittance (U value) = 1/(Σ(L/k)) + convection + radiation, measured in W·K−1·m−2.

##  Textile industry

In textiles, a tog value may be quoted instead of thermal resistance.

## Origins

The thermal conductivity of a system is determined by how atoms comprising the system interact. There are no simple, correct expressions for thermal conductivity. There are two different approaches for calculating the thermal conductivity of a system. The first approach employs the Green-Kubo relations. Although this expression is exact*, in order to calculate the thermal conductivity of a dense fluid or solid using this relation requires the use of molecular dynamics computer simulation.

• The term exact is applied to mean that the equations are solvable.

The second approach is based upon the relaxation time approach. Due to the anharmonicity within the crystal potential, the phonons in the system are known to scatter. There are three main mechanisms for scattering:

• Boundary scattering, a phonon hitting the boundary of a system;
• Mass defect scattering, a phonon hitting an impurity within the system and scattering;
• Phonon-phonon scattering, a phonon breaking into two lower energy phonons or a phonon colliding with another phonon and merging into one higher energy phonon.

Further information can be found in the publication "The Physics of Phonons" by G P Srivastava.

## References

Callister, William (2003). “Appendix B”, Materials Science and Engineering - An Introduction. John Wiley & Sons, INC, 757. ISBN 0-471-22471-5.

Halliday, David; Resnick, Robert; & Walker, Jearl(1997). Fundamentals of Physics (5th ed.). John Wiley and Sons, INC., NY ISBN 0-471-10558-9.

TM 5-852-6 AFR 88-19, Volume 6 (Army Corp of Engineers publication)