Thermal conductivity and resistivity

The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by,, or and, in SI units, is measured in W·m⁻¹·K⁻¹. It quantifies the proportionality between the heat flux and the temperature gradient in the direction of heat transport. The reciprocal of thermal conductivity is called thermal resistivity.
Materials with high thermal conductivity transfer heat more efficiently than those with low thermal conductivity. Heat transport can arise from different microscopic mechanisms: In metals, thermal conductivity is typically dominated by free electrons, whereas in dielectric materials such as diamond it is largely due to lattice vibrations. Materials with high thermal conductivity are used in heat sink applications, while materials with low thermal conductivity, such as mineral wool or Styrofoam, are used for thermal insulation.
The defining equation for thermal conductivity is, where is the heat flux, is the thermal conductivity, and is the temperature gradient. This is known as Fourier's law for heat conduction. Although is commonly treated as a scalar, it is a second-rank tensor in the general case. The tensorial description is necessary for anisotropic materials.

Definition

Simple definition

Consider a solid material placed between two environments of different temperatures. Let be the temperature at and be the temperature at, and suppose. An example of this scenario is a building on a cold winter day; the solid material in this case is the building wall, separating the cold outdoor environment from the warm indoor environment.
According to the second law of thermodynamics, heat will flow from the hot environment to the cold one as the temperature difference is equalized by diffusion. This is quantified in terms of a heat flux, which gives the rate, per unit area, at which heat flows in a given direction. In many materials, is observed to be directly proportional to the temperature difference and inversely proportional to the separation distance :
The constant of proportionality is the thermal conductivity; it is a physical property of the material. In the present scenario, since heat flows in the minus x-direction and is negative, which in turn means that. In general, is always defined to be positive. The same definition of can also be extended to gases and liquids, provided other modes of energy transport, such as convection and radiation, are eliminated or accounted for.
The preceding derivation assumes that the does not change significantly as temperature is varied from to. Cases in which the temperature variation of is non-negligible must be addressed using the more general definition of discussed below.

General definition

Thermal conduction is defined as the transport of energy due to random molecular motion across a temperature gradient. It is distinguished from energy transport by convection and molecular work in that it does not involve macroscopic flows or work-performing internal stresses.
Energy flow due to thermal conduction is classified as heat and is quantified by the vector, which gives the heat flux at position and time. According to the second law of thermodynamics, heat flows from high to low temperature. Hence, it is reasonable to postulate that is proportional to the gradient of the temperature field, i.e.
where the constant of proportionality,, is the thermal conductivity. This is called Fourier's law of heat conduction. Despite its name, it is not a law but a definition of thermal conductivity in terms of the independent physical quantities and. As such, its usefulness depends on the ability to determine for a given material under given conditions. The constant itself usually depends on and thereby implicitly on space and time. An explicit space and time dependence could also occur if the material is inhomogeneous or changing with time.
In some solids, thermal conduction is anisotropic, i.e. the heat flux is not always parallel to the temperature gradient. To account for such behavior, a tensorial form of Fourier's law must be used:
where is symmetric, second-rank tensor called the thermal conductivity tensor.
An implicit assumption in the above description is the presence of local thermodynamic equilibrium, which allows one to define a temperature field. This assumption could be violated in systems that are unable to attain local equilibrium, as might happen in the presence of strong nonequilibrium driving or long-ranged interactions.

Other quantities

In engineering practice, it is common to work in terms of quantities which are derivative to thermal conductivity and implicitly take into account design-specific features such as component dimensions.
For instance, thermal conductance is defined as 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 kelvin. For a plate of thermal conductivity, area and thickness, the conductance is, measured in W⋅K⁻¹. The relationship between thermal conductivity and conductance is analogous to the relationship between electrical conductivity and electrical conductance.
Thermal resistance is the inverse of thermal conductance. It is a convenient measure to use in multicomponent design since thermal resistances are additive when occurring in series.
There is also a measure known as the heat transfer coefficient: the quantity of heat that passes per unit time through a unit area of a plate of particular thickness when its opposite faces differ in temperature by one kelvin. In ASTM C168-15, this area-independent quantity is referred to as the "thermal conductance". The reciprocal of the heat transfer coefficient is thermal insulance. In summary, for a plate of thermal conductivity, area and thickness,

thermal conductance =, measured in W⋅K⁻¹.
*thermal resistance =, measured in K⋅W⁻¹.
heat transfer coefficient =, measured in W⋅K⁻¹⋅m⁻².
*thermal insulance =, measured in K⋅m²⋅W⁻¹.

The heat transfer coefficient is also known as thermal admittance in the sense that the material may be seen as admitting heat to flow.
An additional term, thermal transmittance, quantifies 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 also used.
Finally, thermal diffusivity combines thermal conductivity with density and specific heat:
As such, it quantifies the thermal inertia of a material, i.e. the relative difficulty in heating a material to a given temperature using heat sources applied at the boundary.

Units

In the International System of Units, thermal conductivity is measured in watts per meter-kelvin . Some papers report in watts per centimeter-kelvin .
In cgs units, thermal conductivity is measured in esu/. In imperial units, thermal conductivity is measured in BTU/.
The dimension of thermal conductivity is M¹L¹T⁻³Θ⁻¹, expressed in terms of the dimensions mass, length, time, and temperature.
Other units which are closely related to the thermal conductivity are in common use in the construction and textile industries. The construction industry makes use of measures such as the R-value and the U-value. Although related to the thermal conductivity of a material used in an insulation product or assembly, R- and U-values are measured per unit area, and depend on the specified thickness of the product or assembly. The textile industry has several units including the tog and the clo which express thermal resistance of a material in a way analogous to the R-values used in the construction industry.

Measurement

There are several ways to measure thermal conductivity; each is suitable for a limited range of materials. Broadly speaking, there are two categories of measurement techniques: steady-state and transient. Steady-state techniques infer the thermal conductivity from measurements on the state of a material once a steady-state temperature profile has been reached, whereas transient techniques operate on the instantaneous state of a system during the approach to steady state. Lacking an explicit time component, steady-state techniques do not require complicated signal analysis. The disadvantage is that a well-engineered experimental setup is usually needed, and the time required to reach steady state precludes rapid measurement.
In comparison with solid materials, the thermal properties of fluids are more difficult to study experimentally. This is because in addition to thermal conduction, convective and radiative energy transport are usually present unless measures are taken to limit these processes. The formation of an insulating boundary layer can also result in an apparent reduction in the thermal conductivity.

Experimental values

The thermal conductivities of common substances span at least four orders of magnitude. Gases generally have low thermal conductivity, and pure metals have high thermal conductivity. For example, under standard conditions the thermal conductivity of copper is over times that of air.
Of all materials, allotropes of carbon, such as graphite and diamond, are usually credited with having the highest thermal conductivities at room temperature. The thermal conductivity of natural diamond at room temperature is several times higher than that of a highly conductive metal such as copper.
Thermal conductivities of selected substances are tabulated below; an expanded list can be found in the list of thermal conductivities. These values are illustrative estimates only, as they do not account for measurement uncertainties or variability in material definitions.
For thermal conductivity determination of composites composed ofFe78Si9B13 microparticles and graphene nanoplatelets embedded in a transparent epoxy matrix Had-Hoc methods have been used based on the flash method, eliminating the porosity that these samples may present.

Substance	Thermal conductivity	Temperature
Air	0.026	25
Styrofoam	0.033	25
Water	0.6089	26.85
Concrete	0.92	–
Steel	45	18.05
Aluminium	237	18.05
Copper	384	18.05
Diamond	895–1350	26.85