eFunda: Theory of Pyrometers
engineering fundamentals Pyrometers: Theory
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Pyrometry literally means "fire" (pyros) "measuring" (metron). Pyrometers manipulate the fact that all objects above absolute zero temperature 0 K (-273.15 °C; -459.67 °F) radiate and absorb thermal energy. If the relationship between the radiation intensity and wavelength and the temperature can be established, the temperature can be found from the radiation.

Two principal theories are employed by pyrometry: Planck's law and the Stefan-Boltzmann law. Planck's law is used in narrow-band pyrometers, where only one or a few specific wavelengths are targeted. The Stefan-Boltzmann law is used in broad-band pyrometers, where a wide range of wavelengths are measured.

Planck's law

The radiation intensity and wavelength from a blackbody at a given temperature T is governed by Planck's law. For radiation from a flat suface onto a hemisphere, the law takes the form,

where:  Wb is the radiant power intensity from a blackbody (W/m3),
  l is the wavelength (m),
  T is the absolute temperature (K)
  h is Planck's constant, 6.625 × 10-34 J·s,
  c is the speed of light, 3 × 108 km,
  k is Boltzmann's constant, 1.380 × 10-23 J/K.

If the radiation intensity Wb at a specific wavelength l is measured, the temperature T is the only unknown in the equation.

If lT < (hc/k)/2.5 ~ 5.76×10-3 m·K, then e(hc/klT) » 1 and Planck's law can be simplified to Wein's radiation law,

As shown, the radiation intensity is a function of wavelength l and temperature T. For a given temperature, there is a particular wavelength associated with the maximum radiation intensity. An approximate equation that relates them is Wein's displacement law,

lpT = 2891 µm·K

where lp is the wavelength associated with the maximum radiation intensity for a given temperature T.

This equation tells us what the most effective wavelength should be for measure a certain temparature T.

Stefan-Boltzmann law

In broad-band pyrometers, the photodectors are collecting radiation intensity over a wide range of wavelengths. In other words, they are summing the radial power intensity Wb across the full wavelength range. Equivalently, they are integrating the Wb-l curve at a particular temperature T, resulting in the Stefan-Boltzmann law,

where s is the Stefan-Boltzmann constant (5.6697×10-8 W/m2·K). Note that the temperature T is the absolute temperature.

Emittance and Emissivity

When electromagnetic radiation impinges upon a surface of an object, the radiation is partially absorbed, partially reflected, and partially transmitted. If all of the electromagnetic radiation is absorbed by the object, the object is called a blackbody.

Planck's law and the Stefan-Boltzmann law are derived assuming blackbody properties. In reality, most common materials are not 100% black. As a result, the measured temperature Tm (assuming blackbody properties) is slighter smaller than the true temperature T.

The emittance e is introduced to describe the difference in radiation absorption between common objects and blackbodies, according to the equation,

If the emittance is independent of wavelength (i.e. el = e = constant) the object is called a graybody.

Another term called emissivity is similar to emittance in describing the difference in radiation intensity between real materials and blackbodies. However, emissivity is a material property usually defined only for highly polished surfaces or controlled conditions. In other words, emittance is the general concept of the radiation mismatch between an object and a blackbody, whereas the emissivity is the emittance of a particular material under a certain condition.

Error Correction

As a result of differences in emittance, different objects at the same temperature may radiate differently. The emissivity of a single object may change over time, due to oxidation for example. Thus, the same object at the same temperature may still radiate differently at two different times, and the emittance should be considered in the calibration of pyrometers.

Including the emittance in the above equations results in,

Planck's law:
Wein's radiation law:
Stefan-Boltzmann law:

Finally, the geometry of the probe tip and/or the detecting angle may also affect the radiation reading. In that case, a geometric correction factor fg should be introduced in the above equations as a multiplier.
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