Thermal effusivity

In thermodynamics, a material's thermal effusivity, thermal inertia or thermal responsivity is a measure of its ability to exchange thermal energy with its surroundings. It is defined as the square root of the product of the material's thermal conductivity (${\displaystyle \lambda }$) and its volumetric heat capacity (${\displaystyle \rho c_{p}}$).^[1][2][3]

Thermal effusivity sensor typically used in the direct measurement of materials.

${\displaystyle e={\sqrt {\left(\lambda \rho c_{p}\right)}}}$

The SI units for thermal effusivity are ${\displaystyle {\rm {W}}{\sqrt {\rm {s}}}/({\rm {m^{2}K}})}$, or, equivalently, ${\displaystyle {\rm {J}}/({\rm {m^{2}K}}{\sqrt {\rm {s}}})}$.

Thermal effusivity is a parameter that emerges upon applying solutions of the heat equation to heat flow through a thin surface-like region.^[3] It becomes particularly useful when the region is selected adjacent to a material's actual surface. Knowing the effusivity and equilibrium temperature of each of two material bodies then enables an estimate of their interface temperature ${\displaystyle T_{m}}$ when placed into thermal contact.^[4]

${\displaystyle T_{m}={\frac {e_{1}T_{1}+e_{2}T_{2}}{e_{1}+e_{2}}}}$

Specialty sensors have also been developed based on this relationship to measure effusivity.

Thermal effusivity and thermal diffusivity are related quantities; respectively a product versus a ratio of a material's fundamental transport and storage properties. The diffusivity appears explicitly in the heat equation, which is an energy conservation equation, and measures the speed at which thermal equilibrium can be reached by a body.^[2] By contrast a body's effusivity (also sometimes called inertia, accumulation, responsiveness etc.) is its ability to resist a temperature change when subjected to a time-periodic, or similarly perturbative, forcing function.^[5][6]

Applications

Temperature at a contact surface

If two semi-infinite^[i] 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.

Heat sensed by human skin

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]

Planetary science

For planetary surface materials, thermal inertia is the key property controlling the diurnal and seasonal surface temperature variations and is typically dependent on the physical properties of 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).^[7] A rough approximation to thermal inertia is sometimes obtained from the amplitude of the diurnal temperature curve (i.e. maximum minus minimum surface temperature).^[5] The temperature of a material with low thermal inertia changes significantly during the day, while the temperature of a material with high thermal inertia 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.^[8]

On Earth, thermal inertia of the oceans is a major factor influencing climate commitment, the degree of global warming predicted to eventually result from a step change in climate forcing, such as a fixed increase in the atmospheric concentration of a greenhouse gas. Ocean thermal inertia is much greater than land inertia because of convective heat transfer throughout the upper mixed layer.^[9]

Measurement interpretation

When a material is measured from the surface with short test times by any transient method or instrument, the heat transfer mechanisms generally include conductivity, convection and radiation. Conductivity may dominate in solid bodies near and below room temperature. Also, a contact resistance (due to surface roughness, oxidation, impurities, etc.) between the sensor and sample may exist. All of these factors, along with the body's finite dimensions, must be considered during execution of measurements and interpretation of results.

Thermal effusivity of selected materials and substances

Thermal effusivity of selected materials and substances

at room temperature, unless otherwise indicated

Material	Thermal effusivity ( ${\displaystyle {\rm {kJ}}/({\rm {m^{2}K}}{\sqrt {\rm {s}}})}$  )	References
Air	0.006	^[10]
Wool	0.07	^[10]
Wood	0.26	^[11]
Plasterboard	0.38	^[11]
PVC (polyvinyl chloride)	0.6	^[10]
Brick	0.99-1.6	^[11]
Skin	1.0	^[10]
Quartz	1.5	^[10]
Water	1.6	^[10]
Concrete (dense)	2.0	^[11]
Ice	2.8	^[10]
Silicon	14.4	^[10]
Iron	15.9	^[10]
Aluminum	23.7	^[10]
Copper	36.9	^[10]

See also

• Heat capacity
• Heat equation
• Specific heat capacity
• Thermal diffusivity

References

1. i.e. their thermal capacity is sufficiently large that their temperatures will not change measurably owing to this heat transfer

2. Dante, Roberto C. (2016). Handbook of Friction Materials and Their Applications. Elsevier. pp. 123–134. doi:10.1016/B978-0-08-100619-1.00009-2.
3. Carslaw, H.S.; Jaeger, J.C. (1959). Conduction of Heat in Solids. Clarendon Press, Oxford.
4. Baehr, H.D.; Stephan, K. (2004). Wärme- und Stoffübertragung 4. Auflage. Springer. p. 172. doi:10.1007/978-3-662-10833-8. ISBN 978-3-662-10834-5.
5. Veto, M.S.; Christensen, P.R. (2015). "Mathematical Theory of Thermal Inertia Revisited" (PDF). 46th Lunar and Planetary Science Conference.
6. Williams, F. A. (2009). "Simplified theory for ignition times of hypergolic gelled propellants". J. Propulsion and Power. 25 (6): 1354–1357. doi:10.2514/1.46531.
7. Christensen, P.R.; et al. (25 October 2001). "Mars Global Surveyor Thermal Emission Spectrometer experiment: Investigation description and surface science results". Journal of Geophysical Research: Planets. 106 (E10). doi:10.1029/2000JE001370.
8. Price, John C. (20 June 1977). "Thermal Mapping: A New View of the Earth". Journal of Geophysical Research: Oceans and Atmospheres. 82 (18): 2582–2590. doi:10.1029/JC082i018p02582.
9. Michon Scott (24 April 2006). "Earth's Big Heat Bucket". NASA Earth Observatory.
10. Clemens J.M. Lasance (November 2007). "Materials Data". electronics-cooling.com.
11. Tim Dwyer (January 2013). "Simple thermal analysis for buildings". cibsejournal.com.

External links

• "Thermal heat transfer". HyperPhysics.