Thermal wave

The term thermal wave describes a spatially and temporally variable temperature field that is caused by a time-dependent heating in a medium. Mathematically, a thermal wave is a solution to the heat conduction equation , a diffusion equation . Without a rest position and restoring force , as is the case with systems that obey a wave equation , there is also no oscillation, which is why the (internationally common) term “thermal wave” is sometimes referred to as inappropriate. An oscillatory external or local excitation only works over a short distance; there is no far field as with real waves.

In nature, thermal waves can be experienced if you look at the temperature in the ground, which fluctuates over the course of the day or year. The temperature fluctuations during the day can be felt near the surface, while at greater depths only seasonal and climatic fluctuations. Applications include drawing conclusions from the precisely measured temperature field on the temperature profile over time, as well as from the development of the surface temperature over time on inhomogeneities in heat conduction, see heat flow thermography .

history

Fourier already carried out a mathematical analysis of the heat waves. Ångström used heat waves experimentally to determine thermal conductivity. Bell and Röntgen used the photoacoustic effect, in which sometimes heat waves also play a role. A broad application began only in 1972 with the work of A. Rosencwaig, after powerful detection technology became available.

Mathematical description

Using the example of the thermal wave, which is excited in a homogeneous, semi-infinite body that absorbs only on the surface with harmonious-periodic and large-area irradiation, the most important properties of thermal waves can be described and the physical quantities and parameters recognized by means of thermal waves of the measurement are accessible. The temperature variation caused by the power absorbed at the surface is ${\displaystyle P_{0}}$ ${\displaystyle \Delta T(x,t)}$

(1) . ${\displaystyle \Delta T(x,t)=\Delta T_{0}(x)\cdot \cos(2\pi ft+\Delta \Phi (x))}$

Where:

(2) ${\displaystyle \Delta T_{0}(x)={\frac {P_{0}}{e{\sqrt {2\pi f}}}}\exp(-x/\mu )}$

(3) ${\displaystyle \Delta \Phi (x)=-{\frac {x}{\mu }}-{\frac {\pi }{4}}}$

(4) . ${\displaystyle \mu ={\sqrt {\frac {\alpha }{\pi f}}}}$

The abbreviations (thermal effusivity, thermal penetration coefficient) and (thermal diffusivity, thermal conductivity ) depend on the thermal conductivity k , the mass density and the specific heat capacity c . ${\displaystyle e={\sqrt {k\rho c}}}$ ${\displaystyle \alpha ={\frac {k}{\rho c}}}$ ${\displaystyle \rho }$

The amplitude of the thermal wave (equation 1) decreases exponentially with depth , starting from the heated surface ( x = 0). The penetration depth that can be detected by the measurement is in the order of magnitude of the thermal diffusion length µ. Due to the frequency dependence of the thermal diffusion length μ (equation 4), the penetration depth can be adjusted by deliberately varying the modulation frequency f of the heater.

In the one-dimensional case there is an analogy to the electromagnetic skin depth for the electromagnetic wave.

Excitation and detection of heat waves

In principle, all effects that locally release heat can be used to stimulate heat waves. The classic excitation method uses optical radiation in the form of modulated or pulsed light from solar radiation, lamps or lasers. However, excitations from the entire electromagnetic spectrum including X-rays and particle beams as well as elastic vibrations, ohmic heat and hot air are known. Particularly deep heat waves are stimulated by climatic fluctuations over geological time periods.

A large spectrum of temperature-dependent effects was used to detect heat waves. A classic technique is the conversion of the heat wave into sound with the help of the photoacoustic effect. The detection of temperature fluctuations by means of infrared detectors of various kinds is of greater importance. Other techniques use the temperature-dependent change in optical reflection and the mirage effect . With infrared cameras, a high-resolution and non-contact detection of the heat wave is achieved. For climate research, measurements are made in boreholes with thermometers.

Applications

In research, thermal waves are used interdisciplinary in physics, biology and chemistry. Thermal waves are used on scales from meters (building physics) to sub-micrometers (nanomicroscopy). An international conference deals with thermal waves.

Industrial applications can be found in non-contact layer thickness measurement, in the characterization of semiconductors, in infrared spectroscopy, in gas sensors and in non-destructive material testing and characterization, see heat flow thermography .

literature

• DP Almond: Photothermal Science and Techniques . Chapman & Hall London, 1996, ISBN 0-412-57880-8 .
• A. Mandelis: Diffusion-Wave Fields . Springer-Verlag New-York, 2001, ISBN 0-387-95149-0 .
• European standard EN 15042-2 . Beuth-Verlag Berlin, 2006.

Web links

• ttz-bremerhaven.de: "On the track of wind turbine defects with thermal technology" ( Memento from October 11, 2007 in the Internet Archive ) Explanation of the use of thermal waves for testing wind turbines in the section "Active thermography" (accessed on 24. February 2009).
• "Infrared measuring techniques for solar cells and solar materials." Mathematical description of thermal waves in the section "Physical principles". (PDF; 1.9 MB) mpi-halle.mpg.de, accessed on February 24, 2009 . 

Individual evidence

1. ↑ Harris & Chapman, Geothermics and climate change… doi : 10.1029 / 97JB03296
2. ↑ Gerhard Busse: Raster image process with optically generated heat waves in non-destructive material testing. 1984 (habilitation thesis, University of Stuttgart)
3. ↑ International Conference on Photoacoustic and Photothermal Phenomena, ICPPP