Thermal Diffusivity: what is it and how to measure it?

The knowledge of the thermal conductivity and/or the thermal diffusivity is more and more required in many industrial fields. As a matter of fact, these are the most important parameters when heat transfer processes are involved. In the power generation industries, materials are selected primarily considering their thermal properties. In the automotive and aeronautical industries, CMC (ceramic matrix composite) materials for high performance brakes and heat shields are under development and manufacturers require, as essential information the thermal diffusivity [1]. Polymers and particulate composites are being increasingly used in heat insulation and dissipation applications. Accurate measurement of their thermal diffusivity is critical for reliable thermal designs

Figure 1: Heat transfer mechanism

In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure[2] It measures the rate of transfer of heat of a material from the hot end to the cold end. It has the SI derived unit of m²/s. Thermal diffusivity is usually denoted α but a,h,κ, K, and D are also used. Both properties are related by the following formula

• The specific heat capacity indicates how much of this heat is stored in the fabric.
• The density takes into account how much substance per cubic meter is present.
• Thermal conductivity describes the ability to conduct heat without the transport of particles or electromagnetic waves. It needs the presence of a material and is the predominant mechanism in solids. Heat is transmitted by vibrations of molecules, atoms or ions and their random collisions (Brownian motion ; lattice vibrations = phononic contribution) and by freely moving electrons (electronic contribution)

Flash Method [3]

The Flash Method is a method for the determination of the thermal diffusivity of different materials. For that, the sample is subjected to a high intensity short duration radiant energy pulse. The power source can be a laser or a flash lamp. The energy will then be absorbed by the specimen and emitted again on the top of the sample. This radiation results in a temperature rise on the surface of the sample. This temperature rise is recorded from an infrared (IR) detector.
The schematic measurement setup and procedure is shown in the following figure.

Figure 2: Laser Flash Method scheme

The detector signal shows the duration of the measurement and the normalized temperature rise on the surface of the sample, where the light pulse occurred at. For the calculation of the thermal diffusivity, it is necessary to determine the baseline and the maximum temperature rise. This is done by a suited fitting model. Additionally, the model determines the time at which half of the maximum temperature raise was reached.

The thermal diffusivity is calculated with the following formula where describes the sample thickness and the half time rise.

Figure 3: Thermal diffusivity calculation

TWI Method [4]

Thermal Wave Interferometry (TWI) is a well-established technique for the measure of the thickness and the thermal diffusivity of coatings and is specifically well suited for thermal barrier coatings When a thermal wave is generated within solid multilayered samples by intensity modulated optical excitation (typically a continuous wave laser), the propagation of this wave is affected by the presence of the thermal mismatch at the coating-substrate interface. In particular, as thermal waves behave like common waves, they are partially reflected and transmitted at the separation surface of the two different materials. The analysis of the temperature distribution on the sample surface gives information on both the coating thickness and its thermal diffusivity.

For a two layer sample heated uniformly by a temporally modulated optical source at an angular frequency ὠ, the ac-component of the surface temperature depends on both the coating thickness L and the thermal diffusivity, as follows:

Furthermore, from Equation it is possible to define both amplitude A and phase ὠ,as follows;

Figure 4: Phase (a) and amplitude (b) of the ac component of temperature versus the normalised thickness h for different values of the reflection coefficient R ranging from -1 to 1 (in the sense of the arrow).

Phase measurement is usually preferred to the amplitude one because it is less sensitive to the optical features of the sample surface as well as to laser power variation during the measurement
Experimentally measured phase of the ac component of surface temperature versus different modulation frequency, and the results are fitted to equation of phase

Figure 5: TWI experimental set-up

The Thermographic method I (spatially resolved method) [5]

The temperature on the front (z=0) and the rear (z=L) surfaces of a semi-infinite slab after an instantaneous spatially Gaussian shaped heating are respectively;

The next figure shows the spatial distribution of the temperature as a function of the distance r from the spot center for the rear slab surface at different times

Figure 6: Eq.T(r,L,t) as a function of the distance r. Curves refer to time values ranging from 1 to 10 seconds increasing in the direction of the arrow. Computations have been performed by using the energy Q=1 J for a 5.5 mm diameter Gaussian spot. After obtaining the results, the data are adjusted. The adjustment results in the desired values

The Thermographic method II (lateral thermal waves) [5]

The thermal model adopted to describe the proposed experimental set-up is the original one due to Ångström. It describes the temperature along a semi-infinite bar heated by a periodic source on one end, exchanging heat with the environment and being thermally thin. Therefore temperature varies only down to the bar (1D diffusion) according to the following equation:

where T(x,t) is temperature function depending on x coordinate along the bar and time t, Tenv the environment temperature, An the amplitude of the nth harmonic component, _the angular frequency, _ is the thermal diffusivity, heat exchange coefficient, p and S perimeter and cross-section area of the bar respectively, cp and _ specific heat and volumic mass, and ” n the initial phase of the nth harmonic component.
The goal of the data reduction procedure consists in determining the spatial phase velocity k2n and the attenuation coefficient of the harmonic component k1n. Hence, the diffusivity _ and heat exchange related parameter are given by:

Figure 7: Experimental set up for diffusivity measurements by thermoelectric generated thermal waves.

The Thermographic method III (one side flash method) [5]

It is very interesting to analyze the possibility to measure diffusivity in reflection mode by flash heating. The solution for the heated side of the slab is that given in the next equation.

but for the term (-1)n. For the rear side it gives the alternate signs of the series terms making the solution increasing monotonically. For the front side the lack of such a term makes the solution decreasing monotonically from the maximum temperature value after the flash to the T_ value. Unfortunately, the temperature reached during or immediately after the flash is not a reliable value to measure and therefore we cannot extract a noticeable time from the amplitude evolution. On the other hand it is possible to show that multiplying the solution of the front side by the cube root of the time, such a function presents a minimum (that can be measured more easily) at Fo=0.2656 (Fo=_t/L2) giving the following relation for the diffusivity measurement:

Diffusivity equation  /  Data and fitting function to obtain the time of minimum for diffusivity measurement.

Figure 8: Experimental set-up for one side diffusivity measurement by flash heating

References
[1]F. Cernuschi, P.G. Bison, S. Marinetti, A. Figari, L. Lorenzoni, E. Grinzato, Comparison of thermal diffusivity measurement techniques
[2] https://en.wikipedia.org/wiki/Thermal_diffusivity
[3] F. Cernuschi, P.G. Bison , S. Marinetti , A. Figari, L. Lorenzoni, E. Grinzato, Comparison of thermal diffusivity measurement techniques,
[4] F. Cernuschi, A. Figari, L. Fabbri, Thermal wave interferometry for measuring the thermal diffusivity of thin slabs, JOURNAL OF MATERIALS SCIENCE 35 (2000) 5891 – 5897
[5] F. Cernuschi, P.G. Bison°, A. Figari, S. Marinetti° and E. Grinzato, Thermal diffusivity measurements by photothermal
and thermographic techniques, International Journal of Thermophysics ,2004. DOI: 10.1023/B:IJOT.0000028480.27206.cb
Figures
Figure 1: https://www.linseis.com/en/wiki-en/what-does-thermal-conductivity-mean/
Figure 2: F. Cernuschi, P.G. Bison , S. Marinetti , A. Figari, L. Lorenzoni, E. Grinzato, Comparison of thermal diffusivity measurement techniques,
Figure 3: http://www.tainstruments.com/wp-content/uploads/BROCH-ThermalConductivityDiffusivity-2014-EN-1.pdf
Figures 4-5: F. Cernuschi, A. Figari, L. Fabbri, Thermal wave interferometry for measuring the thermal diffusivity of thin slabs, JOURNAL OF MATERIALS SCIENCE 35 (2000) 5891 – 5897
Figures 6-8: F. Cernuschi, P.G. Bison°, A. Figari, S. Marinetti° and E. Grinzato, Thermal diffusivity measurements by photothermal and thermographic techniques, International Journal of Thermophysics ,2004. DOI: 10.1023/B:IJOT.0000028480.27206.cb