Thermal conductivity

The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by ${\displaystyle k}$, ${\displaystyle \lambda }$, or ${\displaystyle \kappa }$.

Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal conductivity. For instance, metals typically have high thermal conductivity and are very efficient at conducting heat, while the opposite is true for insulating materials like Styrofoam. Correspondingly, materials of high thermal conductivity are widely used in heat sink applications, and materials of low thermal conductivity are used as thermal insulation. The reciprocal of thermal conductivity is called thermal resistivity.

The defining equation for thermal conductivity is ${\displaystyle \mathbf {q} =-k\nabla T}$, where ${\displaystyle \mathbf {q} }$ is the heat flux, ${\displaystyle k}$ is the thermal conductivity, and ${\displaystyle \nabla T}$ is the temperature gradient. This is known as Fourier's Law for heat conduction. Although commonly expressed as a scalar, the most general form of thermal conductivity is a second-rank tensor. However, the tensorial description only becomes necessary in materials which are anisotropic.

Definition

Simple definition

Thermal conductivity can be defined in terms of the heat flow ${\displaystyle q}$ across a temperature difference.

Consider a solid material placed between two environments of different temperatures. Let ${\displaystyle T_{1}}$ be the temperature at ${\displaystyle x=0}$ and ${\displaystyle T_{2}}$ be the temperature at ${\displaystyle x=L}$, and suppose ${\displaystyle T_{2}>T_{1}}$. A possible realization of this scenario is a building on a cold winter day: the solid material in this case would be 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 in an attempt to equalize the temperature difference. This is quantified in terms of a heat flux ${\displaystyle q}$, which gives the rate, per unit area, at which heat flows in a given direction (in this case the x-direction). In many materials, ${\displaystyle q}$ is observed to be directly proportional to the temperature difference and inversely proportional to the separation:[1]

${\displaystyle q=-k\cdot {\frac {T_{2}-T_{1}}{L}}.}$

The constant of proportionality ${\displaystyle k}$ is the thermal conductivity; it is a physical property of the material. In the present scenario, since ${\displaystyle T_{2}>T_{1}}$ heat flows in the minus x-direction and ${\displaystyle q}$ is negative, which in turn means that ${\displaystyle k>0}$. In general, ${\displaystyle k}$ is always defined to be positive. The same definition of ${\displaystyle k}$ can also be extended to gases and liquids, provided other modes of energy transport, such as convection and radiation, are eliminated.

For simplicity, we have assumed here that the ${\displaystyle k}$ does not vary significantly as temperature is varied from ${\displaystyle T_{1}}$ to ${\displaystyle T_{2}}$. Cases in which the temperature variation of ${\displaystyle k}$ is non-negligible must be addressed using the more general definition of ${\displaystyle k}$ 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 ${\displaystyle \mathbf {q} (\mathbf {r} ,t)}$, which gives the heat flux at position ${\displaystyle \mathbf {r} }$ and time ${\displaystyle t}$. According to the second law of thermodynamics, heat flows from high to low temperature. Hence, it is reasonable to postulate that ${\displaystyle \mathbf {q} (\mathbf {r} ,t)}$ is proportional to the gradient of the temperature field ${\displaystyle T(\mathbf {r} ,t)}$, i.e.

${\displaystyle \mathbf {q} (\mathbf {r} ,t)=-k\nabla T(\mathbf {r} ,t),}$

where the constant of proportionality, ${\displaystyle k>0}$, is the thermal conductivity. This is called Fourier's law of heat conduction. In actuality, it is not a law but a definition of thermal conductivity in terms of the independent physical quantities ${\displaystyle \mathbf {q} (\mathbf {r} ,t)}$ and ${\displaystyle T(\mathbf {r} ,t)}$.[2][3] As such, its usefulness depends on the ability to determine ${\displaystyle k}$ for a given material under given conditions. The constant ${\displaystyle k}$ itself usually depends on ${\displaystyle T(\mathbf {r} ,t)}$ 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.[4]

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:

${\displaystyle \mathbf {q} (\mathbf {r} ,t)=-{\boldsymbol {\kappa }}\cdot \nabla T(\mathbf {r} ,t)}$

where ${\displaystyle {\boldsymbol {\kappa }}}$ is symmetric, second-rank tensor called the thermal conductivity tensor.[5]

An implicit assumption in the above description is the presence of local thermodynamic equilibrium, which allows one to define a temperature field ${\displaystyle T(\mathbf {r} ,t)}$.

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 ${\displaystyle k}$, area ${\displaystyle A}$ and thickness ${\displaystyle L}$, the conductance is ${\displaystyle kA/L}$, measured in W⋅K−1.[6] 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.[6] It is a convenient measure to use in multicomponent design since thermal resistances are additive when occurring in series.[7]

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.[8] In ASTM C168-15, this area-independent quantity is referred to as the "thermal conductance".[9] The reciprocal of the heat transfer coefficient is thermal insulance. In summary, for a plate of thermal conductivity ${\displaystyle k}$, area ${\displaystyle A}$ and thickness ${\displaystyle L}$, we have

• thermal conductance = ${\displaystyle kA/L}$, measured in W⋅K−1.
• thermal resistance = ${\displaystyle L/(kA)}$, measured in K⋅W−1.
• heat transfer coefficient = ${\displaystyle k/L}$, measured in W⋅K−1⋅m−2.
• thermal insulance = ${\displaystyle L/k}$, 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.[citation needed]

An additional term, thermal transmittance, quantifies the thermal conductance of a structure along with heat transfer due to convection and radiation.[citation needed] 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 ${\displaystyle \alpha }$ combines thermal conductivity with density and specific heat:[10]

${\displaystyle \alpha ={\frac {k}{\rho c_{p}}}}$.

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.[11]