Thermal conductivity and resistivity

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 the most materials, there is no correlation between thermal conductivity and electrical conductivity, as shown in the table below, where highly electrically conductive silver is shown to be less thermally conductive than diamond, which is an electrical semiconductor. The reason for this difference is that phonons, not electrons, are the primary carriers of heat in diamond. In metals, thermal conductivity approximately tracks electrical conductivity, as the freely moving valence electrons transfer not only electric current but also heat energy.

Thermal conductivity is not a simple property, and depends intimately on structure and temperature. For instance, pure crystalline substances also 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 perpendicular to the c-axis of 2.6 W·m⁻¹·K⁻¹ at 373 K, and 6000 W·m⁻¹·K⁻¹ at 35 K for an angle of 36 degrees to the c-axis.

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)	1,000^ad	273^a	~10⁻¹⁶	Type I (98.1% of Gem Diamonds)
Silver (Ag), pure	406^d - 429^ag (418^f)	300^ag	61.35^g 62.89 - 63.01 × 10⁶	Highest electrical conductivity of any metal
Copper (Cu), pure	385^d - 401^g (386^f - 390^e)	273^g-373^g(293^e)	59.17^g 58.82 - 59.6 × 10⁶
Gold (Au), pure	314^d - 318^fg	273^g - 373^g	40.98 - 45.45^g × 10⁶
Aluminium (Al), pure	205^d - 237^eg (220^f)	293^eg	37.45^g 35.46 - 37.8 × 10⁶
Brass (Cu+(35-15)%Zn)	109^dg - 159^g (151^f)	296^g	12.82^g - 21.74^g × 10⁶
Iron (Fe), pure	71.8^f - 80.4^g (79.5^d - 80.2^a)	273^g-373^g(300^a)	9.901^g - 10.00 × 10⁶
Cast iron (Fe+(2-3.5)%C+(1-3)%Si)	55^f
Bronze (Cu+11%Sn)	42^g - 50^g ((25%Sn)26^f)	296^g	5.882^g - 7.143^g × 10⁶
Carbon Steel (Fe+(1.5-0.5)%C)	36^f - 54^f (50.2^d)
Lead (Pb), pure	34.7^d - 35.3^g (35^f)	273^g - 373^g	4.545 - 4.854^g × 10⁶
Stainless Steel (Fe+18%Cr+8%Ni)	14^a - 16.3^fg	273^a - 296^g	1.389^g - 1.429^g × 10⁶
Granite (Si+14%Al+4%K+3%Na)	1.73^b - 3.98^b			70.18%SiO₂
Marble	2.07^b - 2.94^b			Mostly CaCO₃
Sandstone	1.83^b - 2.90^b			Mostly SiO₂
Ice	1.6^d - 2.2^a (2.1^e)	273^a (293^e)
Limestone	1.26^b - 1.33^b			Mostly CaCO₃
Concrete	0.8^d - 1.28^e	293^e		Mostly CaO
Glass	0.8^d−0.93^e((96%SiO₂)1.2-1.4)^g	293^eg	(10⁻¹²)^g 10⁻¹⁴ - 10⁻¹⁰
Soil	0.17^c - 1.13^c
Water	0.6^de	293^de	5×(Pure)10⁻⁶-(Sweet)10⁻³⁽¹⁾-(Sea)1
Fibre-reinforced plastics	0.23^g - 0.7^g (1.06^e)	296^g (293^e)	(10⁻¹⁵ - 10⁰)^g
High-Density Polymers	0.33^g - 0.52^g	296^g	(10⁻¹⁶ - 10²)^g
Glycerol	0.29^e	293^e
Oak OR Wood	0.16^a - 0.4^e	298^a - 293^e
Low-Density Polymers	0.04^g - 0.33^g (0.16 - 0.25)^e	296^g (293)^e	(10⁻¹⁷ - 10⁰)^g
Rubber (92%)	0.16^a	303^a	~10⁻¹³
Alcohols OR Oils	0.1^e - 0.21^e	293^e
Snow (dry)	0.11^d
Cork (material)	0.04^d - 0.07^e	293^e
Fiberglass OR Foam OR Wool	0.04^d (0.03 - 0.045)^e	(293)^e
EPS/XPS (PS+Air+CO₂+C_nH_2n+x)	0.033^ad (0.1 - 0.13)^g	98^a-298^a (296)^g	(<10⁻¹⁴ - 10⁰)^g
Air (78%N+21%O+1%Ar) (1 atm)	0.024^d - 0.0262^a (0.025^e)	273^d-300^a(293^e)
Oxygen (O₂) (1 atm)	0.0238^d	293^d
Nitrogen (N₂) (1 atm)	0.0234^d - 0.026^a	293^d - 300^a
Silica Aerogel	0.003^a	98^a - 298^a		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

References and additional sources of k values

^aCRC handbook of chemistry and physics ^bMarble Institute ^cSoil Sci Journals ^dGeorgia State University - Hyperphysics
^eHukseflux Thermal Sensors ^fEngineers Edge ^gGoodFellow

Heat Conduction Calculator

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, incoporates 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.

External links

http://physics.nist.gov/Pubs/SP811/appenB9.html
http://www.npl.co.uk/thermal/faq_index.html#heat%20transfer%20property thermophysics FAQ5
http://www.ornl.gov/roofs+walls/research/detailed_papers/rastra/dynamic.htm
http://www.tak2000.com/data2.htm
http://thermophys.savba.sk
http://www.processassociates.com/process/convert/cf_tcn.htm (unit conversion and conversion factors)

References

Callister, William (2003). "Appendix B". Materials Science and Engineering - An Introduction. John Wiley & Sons, INC. p. 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)

Srivastava G. P (1990), "The Physics of Phonons." Adam Hilger, IOP Publishing Ltd, Bristol.