Thermal effusivity

If two semi-infinite[lower-roman 1] bodies initially at temperatures ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ are brought in perfect thermal contact, the temperature at the contact surface ${\displaystyle T_{m}}$ will be a weighted mean based on their relative effusivities.[4] This relationship can be demonstrated with a very simple "control volume" back-of-the-envelope calculation:

Consider the following 1D heat conduction problem. Region 1 is material 1, initially at uniform temperature ${\displaystyle T_{1}}$, and region 2 is material 2, initially at uniform temperature ${\displaystyle T_{2}}$. Given some period of time ${\displaystyle \Delta t}$ after being brought into contact, heat will have diffused across the boundary between the two materials. The thermal diffusivity of a material is ${\displaystyle \alpha =\lambda /(\rho c_{p})}$. From the heat equation (or diffusion equation), a characteristic diffusion length ${\displaystyle \Delta x_{1}}$ into material 1 is

${\displaystyle \Delta x_{1}\simeq {\sqrt {\alpha _{1}\cdot \Delta t}}}$, where ${\displaystyle \alpha _{1}=\lambda _{1}/(\rho c_{p})_{1}}$.

Similarly, a characteristic diffusion length ${\displaystyle \Delta x_{2}}$ into material 2 is

${\displaystyle \Delta x_{2}\simeq {\sqrt {\alpha _{2}\cdot \Delta t}}}$, where ${\displaystyle \alpha _{2}=\lambda _{2}/(\rho c_{p})_{2}}$.

Assume that the temperature within the characteristic diffusion length on either side of the boundary between the two materials is uniformly at the contact temperature ${\displaystyle T_{m}}$ (this is the essence of a control-volume approach). Conservation of energy dictates that

${\displaystyle \Delta x_{1}(\rho c_{p})_{1}(T_{1}-T_{m})=\Delta x_{2}(\rho c_{p})_{2}(T_{m}-T_{2})}$.

Substitution of the expressions above for ${\displaystyle \Delta x_{1}}$ and ${\displaystyle \Delta x_{2}}$ and elimination of ${\displaystyle \Delta t}$ yields an expression for the contact temperature.

${\displaystyle T_{m}=T_{1}+\left(T_{2}-T_{1}\right){\frac {e_{2}}{e_{2}+e_{1}}}={\frac {e_{1}T_{1}+e_{2}T_{2}}{e_{1}+e_{2}}}}$

This expression is valid for all times for semi-infinite bodies in perfect thermal contact. It is also a good first guess for the initial contact temperature for finite bodies.

Even though the underlying heat equation is parabolic and not hyperbolic (i.e. it does not support waves), if we in some rough sense allow ourselves to think of a temperature jump as two materials are brought into contact as a "signal", then the transmission of the temperature signal from 1 to 2 is ${\displaystyle e_{1}/(e_{1}+e_{2})}$. Clearly, this analogy must be used with caution; among other caveats, it only applies in a transient sense, to media which are large enough (or time scales short enough) to be considered effectively infinite in extent.

An application of thermal effusivity is the quasi-qualitative measurement of coolness or warmth "feel" of materials, also known as thermoception. It is a particularly important metric for textiles, fabrics, and building materials. Rather than temperature, skin thermoreceptors are highly responsive to the inward or outward flow of heat. Thus, despite having similar temperatures near room temperature, a high effusivity metal object is detected as cool while a low effusivity fabric is sensed as being warmer.[2]

For planetary surfaces, thermal inertia is a key phenomenon controlling the diurnal and seasonal surface temperature variations. The thermal inertia of a terrestrial planet such as Mars can be approximated from the thermal effusivity of its near-surface geologic materials. In remote sensing applications, thermal inertia represents a complex combination of particle size, rock abundance, bedrock outcropping and the degree of induration (i.e. thickness and hardness).[8]

A rough approximation to thermal inertia is sometimes obtained from the amplitude of the diurnal temperature curve (i.e. maximum minus minimum surface temperature).[6] The temperature of a material with low thermal effusivity changes significantly during the day, while the temperature of a material with high thermal effusivity does not change as drastically. Deriving and understanding the thermal inertia of the surface can help to recognize small-scale features of that surface. In conjunction with other data, thermal inertia can help to characterize surface materials and the geologic processes responsible for forming these materials.[9]

On Earth, thermal inertia of the global ocean is a major factor influencing climate inertia. Ocean thermal inertia is much greater than land inertia because of convective heat transfer, especially through the upper mixed layer.[10] The thermal effusivities of stagnant and frozen water underestimate the vast thermal inertia of the dynamic and multi-layered ocean.[11]

Thermographic inspection encompasses a variety of nondestructive testing methods that utilize the wave-like characteristics of heat propagation through a transfer medium. These methods include Pulse-echo thermography and thermal wave imaging. Thermal effusivity and diffusivity of the materials being inspected can serve to simplify the mathematical modelling of, and thus interpretation of results from these techniques.[12]