True contactless temperature measurement or pyrometry is closely connected to the knowledge of the material's emissivity which is generally a function of the material surface, wavelength, temperature and angle. The present paper will show how this indispensable parameter for contactless thermography can be deduced from real temperature data without any knowledge of and any demands on the material's chemical or physical structure. An algorithm was developed to deduce the unknown emissivity value by an iterative process involving random data taken with a virtual ratio or two-colour pyrometer within a temperature regime of 1100 K +/- 100 K. Possible influences on the simulation, namely level of true emissivity, emissivity ratio, measurement uncertainty and detection wavelength, were studied to find an increasing mismatch between the simulated and true emissivity for larger uncertainties and wavelengths which had been expected following Planck's Law of Thermal Radiation. An increased error has also been found for larger values of the true emissivity whereas no significant influence could be shown for the emissivity ratio of the two channels of a ratio pyrometer. The simulation was considered to be completed for results corresponding to a maximum deviation of 8 K from a given true temperature of 1100 K.
1.1
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Each body emits thermal radiation due to its own temperature and pyrometers
measure this radiation in different spectral regimes (spectral pyrometers) or across the
entire spectrum (broadband pyrometers). The detection regime, however, usually only
stretches from the visual to the mid-infrared regime due to typical temperatures
between 0 °C and 3000 °C. Planck’s Radiation Law [
=
connects the object’s temperature T to the detected thermal radiance M as a
funcand directional absorptivity αs,d equals εs,d at thermal equilibrium:
tion of the detection wavelength λ. h and kB denote the Planck and Boltzmann
constant, respectively. However, this only holds for black bodies (here denoted by bb),
i.e. objects that absorb and – according to Kirchhoff’s Law [
This is true for all objects, but hypothetical black bodies show a constant emissivity over wavelength and angle. Real bodies, however, are so called “coloured”, which means that their absorption and thus emission actually differs as a function of wavelength (cf. (2)). For those bodies (1) is incomplete, it needs to be expanded by the emissivity ε. This is a very crucial parameter as it determines a true temperature measurement which makes it the most desired material parameter for pyrometry. In addition to the variables named in (2) ε also depends on the sample surface. Due to diffuse reflection which leads to an increased probability that incoming radiation is absorbed and thus to a greater absorptivity emissivity is larger for rough than for smooth surfaces. Moreover, emissivity may be a function of temperature itself. Further, dust or smoke may change the effective emissivity for the pyrometer. This shows the importance of the emissivity for temperature determination beyond its existence as a material parameter. Without the knowledge of an effective emissivity for the measurement pyrometry is rather radiometry of thermal radiation. 1.2
If an object’s emissivity is smaller than one and constant independently of the spectral position, the object is said to be “grey”. However, this reduction in emittance may also be effectively caused by any undesired surrounding appearances such as smoke that is assumed to act like a neutral density filter, i.e. reducing the power density and thus the apparent emittance of the material by a single factor also independently of the considered wavelength. 0,5 ∗ 1 0,5 ∗ 2 constant damping to 50 %
If two detectors 1 and 2 are applied to measure at the same position on the object’s
surface, but at different spectral regimes, and the corresponding emissivities and
do not change during the measurement then their ratio remains unaffected by a
constant damping as it is visualized in Fig. 1. The damping factor can be cancelled out
from the emissivity ratio thus allowing for emissivity-independent temperature
measurement. [
Soon this idea had been applied to grey bodies which for a detector behave in the
exact same way, namely reducing the emittance at all spectral positions by a constant
factor which finally also leads to emissivity-corrected temperature measurement.
Theoretically, it works for coloured bodies similarily well if the two detecting
wavelengths are as close to each other as possible. In theoretically infinitely close vicinity
any change in emissivity as a function of wavelength approaches zero which means
that for an infinitely small spectral range any coloured body is also grey. However,
this places high demands on measurement technology which these days cannot be
met, yet, at justifiable costs. Current solutions include assumptions on the relationship
between both temperature and emissivity variations [
Finally, ratio pyrometers may be used if a hot sample is smaller than the detection spot of the pyrometer. Assuming a “cold” background, i.e. a surrounding temperature that is much smaller than the object’s temperature, then temperature measurement will only be marginally affected by the spot size mismatch as it corresponds effectively to a constant damping. Please note, that the temperature reading would significantly change if a single-wavelength pyrometer was used. 1.3
There have been a few approaches to determine the true sample temperature using
pyrometry without possessing any knowledge on the emissivity. A review may be
found here [
However, considering widely commercially available ratio pyrometers all of the approaches that do not utilize additional measurements (e.g. reflectometry) rely on further information to give a true temperature. Indeed, it is possible to determine any object’s temperature within the limits to be discussed if despite the ratio channel the two underlying single channels are simultaneously measured. An iterative algorithm that will be presented then gives the only possible combination of yields a true temperature reading. 2
If the detection spot is fully filled by the hot object whose temperature is to be measured and no damping needs to be considered then both the single-wavelength signals as well as the ratio measurement from a two-colour pyrometer may be used to determine the material emissivity and thus the object’s true temperature without any additional information. Further, this iterative algorithm assumes an exact overlap of the detection spot for all involved wavelength regimes. If these basic requirements are met, than all three (single-wavelength and ratio) channels of a two-colour pyrometer should “see” the same temperature, i.e. all three readings should match. This is mathematically described by the following equation system: , ( ) = ( ) = ( ) ( ) ( ) = ( ) ( ) (3) (4) (5) This threefold equation system – consisting of the emissivities correspond to the two single-wavelength-channels at and , of the respective real spectral radiances body equivalents in a true statement. ( ), of their ratio ,
( ) as well as of their black ( ) – can be solved by an iterative algorithm. It is straightforward to show that for the right combination of and equation (5) results Take and radiance
from measurement START
Choose =
/ Planck`s Law of Thermal
Radiation (cf. (1)) Take measurement of
= 2 / 2 , = / () false 2 1 1,2 (T) = 1 2 true
STOP all three attainable channels from a two-colour pyrometer. obtained. Finally, the ratio of
Even if no information is given on the material properties the true temperature can be gained by assuming a starting value for . An iterative approach (cf. Fig. 2) follows which may be shortened by an educated guess on . Consequently, ( ) may be calculated taking the measured data from ( ) and knowing the detecting wavelength of the first channel. As it must be safe to assume is equal for all channels ( ) can be easily gained and using the measured data of ( ) is readily and , ( ) are compared to ( ) / () . If this comparison does not satisfy equation (5) the iteration is repeated with a different starting emissivity until a true statement is achieved.
The above described iteration process was run for nine different hypothetically real emissivity ratios at a wavelength of (1.0 +/- 0.1) µm for temperatures of (1100 +/100) K and a measurement uncertainty of 25 K.
The results presented in Fig. 3 do not show a significant influence of the emissivity ratio on the simulation results. There seems to be an increased error in temperature due to a rise in the simulation uncertainty for larger nominal emissivities. In fact, this is a consequence of the exponential relationship of thermal radiation and temperature.
Table 1 shows the maximum temperature variation for the maximum deviation of the simulated and true emissivity at a temperature of 1100 K and at a detecting wavelength of 1µm. It can be seen that despite the small standard deviation at a true emissivity of 0.1 the resulting temperature error is indeed larger because the percentage deviation has equally increased which is the decisive contribution. Moreover, the presented errors attribute to less than 1.2 % for an assumed object temperature of 1100 K which is in the order of the set temperature uncertainty of 25 K. However, the stopping condition which accepts a maximum temperature error of 8 K corresponding to 0.7 % at a temperature of 1100 K, needs to be beared in mind although this still does not exceed the afore mentioned uncertainty of 25 K. Farther, these results at a wavelength regime of (1.0 +/- 0.1) µm were compared to two further spectral positions at (2.0 +/- 0.1) µm and at (4.0 +/- 0.1) µm (cf. Fig. 4).
An analysis of variances shows that the deviations are insignificant by a probability of more than 99 %. Larger standard deviations may be attributed, however, to the behaviour of Planck’s Law of Thermal Radiation for larger wavelengths. Towards larger wavelengths the maximum spectral emittance slowly declines in an asymptotic manner. The distance between different temperatures decreases likewise and even a small error in the assumed emissivity will result in comparatively large errors in temperature reading and vice versa. That means, the given temperature uncertainty of 25 K causes larger deviations in the emissivity simulation for larger wavelengths.
Finally, the measurement uncertainty was increased from 25 K to 100 K. As to be expected this led to a comparatively enormous increase both in the mean values and standard deviations (cf. Fig. 5). This is due to an increase in possible solutions to the equation system (3-5) which may be compensated by a stricter stopping condition rising the number of iterations at the same time. Again no significant influence of the emissivity ratio could be found.
Ratio pyrometers can be implemented in different ways. One may be the use of
two actually individual detectors. They may be either arranged at different positions
or stacked one behind the other (“sandwich detector”). The latter may lead to an
undesired crosstalk behaviour [
For practical use, it is important to point out that a real detector cannot operate at a single wavelength. On the other hand, this detector only gives integral values of the thermal radiance within its spectral range. This excludes an analytical determination of the temperature T and asks for a secondary simulation to find an M(T)’ which converges towards the measured value M(T). In the end, a balance needs to be found between simulation time and accuracy. 4
statement.
The present work has presented an algorithm to determine an object’s emissivity without any knowledge on the material properties. The sole prerequisite is that both “detectors” – may this be physically different devices or one detector with different filters – view the exact same position on the object’s surface such that one temperature can be assumed for all channels at a time. In this case the emissivities at both spectral positions
and the true temperature form a threefold equation system that – once the correct combination of has been found – returns a true
A dependence on the emissivity ratio has not been found and resulting temperature errors have not shown a significant influence of the nominal value of the true temperature although an increase in standard deviation for rising emissivities can be seen. The latter is, however, in good accordance with Planck’s Law of Thermal Radiation.
Though it could be seen that larger wavelengths result in a higher uncertainty whether the true emissivity had been found in the simulation as predicted by Planck’s Law the means have not shown to differ significantly. However, raising the pyrometer’s measurement uncertainty from 25 K to 100 K was found to increase the simulation error by almost an order of magnitude.
Considering real measurement conditions the present paper has proven the concept of evaluating an object’s true emissivity by a numerical solution of the equation system that results from the working principle of a ratio pyrometer.
List of Nomenclature
, αs,d (λ,γ)T εs,d (λ,γ)T thermal radiance of a black body at a detection wavelength λ Planck constant speed of light detection wavelength Boltzmann constant temperature absorptivity spectral and directional absorptivity as a function of detection emissivity wavelength λ and angle γ at a temperature spectral and directional absorptivity as a function of detection wavelength λ and angle γ at a temperature ratio of the real-body spectral radiances