### What is thermal effusivity?

Thermal effusivity is the measure of a material’s ability to exchange heat with its surroundings, and determines the temperature between two objects when they contact one another. Thermal effusivity is a transient quantity, and is important when two systems initially connect. For ideal systems, thermal effusivity can be used to determine contact temperature at any point after the initial correspondence. However, for real world systems, the measurement of thermal effusivity is used only for the initial moment of contact.

### Measurement of thermal effusivity

Thermal effusivity can be calculated, like diffusivity, by properly solving the relationship between volumetric heat capacity, and thermal conductivity, as such \( e = \sqrt{(k c \rho)} \). Despite \( e \propto \sqrt{K} \), different materials with the same conductivity can have widely different effusivities, such as potassium and cobalt. These elements have nearly the same conductivity, but the effusivity of potassium is more than twice of cobalt.

Measuring thermal effusivity can be done directly under transient conditions, with the transient plane source and modified transient plane source techniques. Indirectly, thermal effusivity can be calculated using the basic equation from above, \( e = (k c \rho)^{1/2} \).

### Mechanisms of thermal effusivity

Thermal effusivity is a quantitative measurement of temperature, but qualitatively, it is what causes objects to feel “cold” or “warm”. When resting your hand on an object with high effusivity, the temperature of the interface will tend toward that of the effusive object, rather than the temperature of your hand. If two materials have the same thermal effusivity, the heat transition behavior of the interface will equal a unique value, as if no junction exists between the two objects.

### Mathematical determination of thermal effusivity

Quantitatively, the inclination of the interface to tend toward the temperature of the more effusive object follows the equation

\[ T_{int}=\frac{T_{1}e_{1}+T_{2}e_{2}}{e_{1}e_{2}} \]

Where *T is *equal to the temperatures of the objects in contact, and *e *represents the effusivities of the materials. This equation is an idealization of real world applications, but it serves as a good approximation for initial contact. Notice there is no direct dependence on conductivity, only on temperatures and effusivities. An idealization closer to reality is an object placed into contact with a thermal reservoir, where the initial heat flow would be

\[ q=\frac{e(T_{res}-T_{obj})}{\sqrt{\pi t}} \]

Where *q *is equal to the initial heat flow, *e *is the effusivity of the object, *T *is equal to temperature, and *t *is the time passed since the initial contact. As time increases, the heat flow will adjust until equilibrium is met. Notice, once again, that there is no direct dependence on thermal conductivity.

It is useful to know thermal effusivity when there is a heat source of oscillating strength. Thermal effusivity not only determines the heat flow from this constantly changing source, but it also describes the way a “thermal wave” will act at another interface. In these situations, thermal effusivity replaces mechanical impedance, even though the “wave” does not match all the characteristics of physical waves.

### Internationally recognized standards

Methods used to measure thermal effusivity are manufactured in accordance with the ASTM standard D7984-16, which is the test method for measuring the effusivity of fabrics with the modified transient plane source instrument.

### Additional literature

The following piece of literature provides additional information on background, theory, and the importance of thermal effusivity.

- Thermal Physics Concepts: The Role of the Thermal Effusivity. Author: E. Marin (2006).
*The Physics Teacher.* - Thermal Effusivity Measurements of Insulating Liquids using Microsized Hot Strip Probes. Authors: M. Gustavsson, H. Nagai, T. Okutani (2003).
*Review of Scientific Instruments*74(10):4542. - TCi Principles of Operation. Author: Michael Emmanuel (2006).
*C-Therm Application Note,*1-7. - Thermal Effusivity Testing with the Hot Disk Thermal Constants Analyser. Authors: Dale Hume, Chris Vail, Andrey Sizov, Daniel Cederkrantz, Yi Ma, Besira Mihiretie, Mattias Gustavsson, Silas Gustafsson (2017). International Thermal Conductivity Conference.