=Paper=
{{Paper
|id=Vol-2815/CERC2020_paper05
|storemode=property
|title=Development and Validation of an Algorithm for Emissivity-Corrected Pyrometry Independent of Material Properties
|pdfUrl=https://ceur-ws.org/Vol-2815/CERC2020_paper05.pdf
|volume=Vol-2815
|authors=Denise Reichel
|dblpUrl=https://dblp.org/rec/conf/cerc/Reichel20
}}
==Development and Validation of an Algorithm for Emissivity-Corrected Pyrometry Independent of Material Properties==
https://ceur-ws.org/Vol-2815/CERC2020_paper05.pdf
Data Computing and Artificial Intelligence
Development and Validation of an Algorithm for
Emissivity-Corrected Pyrometry Independent of Material
Properties
Denise Reichel1,2,
1
SRH Hochschule Heidelberg, Bonhoeffer Straße 11, 69123 Heidelberg, Germany
denise.reichel@srh.de
2
Steinbeis Transferzentrum Sensorik und Informationssysteme, Moltkestraße 30, 76133 Karls-
ruhe, Germany
denise.reichel@sensin.eu
Abstract. 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 pre-
sent paper will show how this indispensable parameter for contactless thermog-
raphy 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 simula-
tion, 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 in-
creased error has also been found for larger values of the true emissivity where-
as 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.
Keywords: Pyrometry, Thermography, Temperature Measurement, Emissivity,
Algorithm, Ratio Pyrometry, Two-colour Pyrometry
1 Introduction
1.1 Importance of Emissivity for Pyrometry
Sensors for contactless temperature measurement are indispensable for applications
that do not allow for direct contact due to surrounding circumstances, e.g. rotation
bodies, or due to purity requirements of the material, e.g. semiconductors for microe-
lectronic use. For this reason measuring devices (here: pyrometers) are needed which
are able to detect thermal radiation from a distance.
Copyright © 2020 for this paper by its authors.
Use permitted under Creative Commons License 89 CERC 2020
Attribution 4.0 International (CC BY 4.0).
Data Computing and Artificial Intelligence
2
Each body emits thermal radiation due to its own temperature and pyrometers meas-
ure 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 be-
tween 0 °C and 3000 °C. Planck’s Radiation Law [1]
𝑀 = (1)
connects the object’s temperature T to the detected thermal radiance M as a func-
tion of the detection wavelength λ. h and kB denote the Planck and Boltzmann con-
stant, respectively. However, this only holds for black bodies (here denoted by bb),
i.e. objects that absorb and – according to Kirchhoff’s Law [2] – emit 100 % of the
incident radiation (𝜀 = 1). Generally speaking, Kirchoff’s Law states that the spectral
and directional absorptivity αs,d equals εs,d at thermal equilibrium:
αs,d (λ,γ)T = εs,d (λ,γ)T (2)
This is true for all objects, but hypothetical black bodies show a constant emissivi-
ty 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 wave-
length (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. Fur-
ther, 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 meas-
urement pyrometry is rather radiometry of thermal radiation.
1.2 Ratio Pyrometry or Two-Colour Pyrometry
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.
CERC 2020 90
Data Computing and Artificial Intelligence
3
𝜀1 0,5 ∗ 𝜀1
ε1 ε2 =
𝜀2 0,5 ∗ 𝜀2
constant damping to 50 %
Fig. 1. Working principle of any ratio pyrometer for background-corrected temperature meas-
urement of a hot object (shown in red) if the background meant constant damping illustrated as
a dashed line. The two channels of a ratio pyrometer are represented by the green and blue
bodies respectively. The damping affects both channels equally and is thus cancelled out of the
quotient.
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 𝜀 and 𝜀
do not change during the measurement then their ratio remains unaffected by a con-
stant 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 meas-
urement. [3, 4]
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 wave-
lengths 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 [5, 6] or wavelength. The latter
includes – primarily for multi-wavelength-pyrometry – a linear and log-linear emis-
sivity model, which refer to the respective relationship between emissivity and wave-
length [7, 8].
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 effective-
ly to a constant damping. Please note, that the temperature reading would significant-
ly change if a single-wavelength pyrometer was used.
1.3 Status Quo
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 [9]. However, at present additional information is required. This may be
accomplished through reflectometry which yields a measure for the material’s emis-
91 CERC 2020
Data Computing and Artificial Intelligence
4
sivity using Kirchhoff’s Law and conservation of radiation if transmission can be
excluded [10, 11, 12]. Other approaches use a combination of ratio pyrometry for
grey bodies as described above and some knowledge on the material properties [13,
14]. In particular, ratio pyrometry is helpful if one can use literature data on the prop-
erties of the material assuming that the emissivity ratio does not change with modified
process conditions. Moreover, it may be successfully applied to surfaces that are or
become grey during thermal treatment, namely oxidizing steel surfaces [15]. Alt-
hough the time slot during oxide growth affects the measurement significantly while
the true surface temperature may stay the same, but the changing readings can be vice
versa interpreted in terms of increasing oxide thickness as a function of time [16].
Multi-wavelength pyrometry uses more than two detecting wavelengths to replicate
the course of the thermal radiation spectrum as the full spectrum unambiguously de-
termines the object’s temperature regardless of its emissivity [17, 18].
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 𝜀 and 𝜀 and
yields a true temperature reading.
2 Algorithm
If the detection spot is fully filled by the hot object whose temperature is to be meas-
ured 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 deter-
mine the material emissivity and thus the object’s true temperature without any addi-
tional 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 math-
ematically described by the following equation system:
𝜀 𝑀 (𝑇) = 𝑀 (𝑇) (3)
𝜀 𝑀 (𝑇) = 𝑀 (𝑇) (4)
𝜀 , 𝑀 (𝑇)
𝜀 𝐹 (𝑇) = (5)
𝑀 (𝑇)
This threefold equation system – consisting of the emissivities 𝜀 and 𝜀 which
correspond to the two single-wavelength-channels at 𝜆 and 𝜆 , of the respective real
CERC 2020 92
Data Computing and Artificial Intelligence
5
spectral radiances 𝑀 (𝑇) and 𝑀 (𝑇), of their ratio 𝐹 , (𝑇) as well as of their black
body equivalents 𝑀 (𝑇) and 𝑀 (𝑇) – can be solved by an iterative algorithm. It is
straightforward to show that for the right combination of 𝜀 and 𝜀 equation (5) results
in a true statement.
START
Choose 𝜀
Take 𝜆 and radiance 𝑀 𝑇 from measurement
𝑀 𝑇 = 𝑀 𝑇 /𝜀
𝜆
Planck`s Law of Thermal 𝑀 𝑇
Radiation (cf. (1))
Take measurement of 𝑀 𝑇
𝜆
𝜀 = 𝑀2 𝑇 / 𝑀𝑏𝑏2 𝑇
𝐹 , 𝑇 = 𝑀 𝑇 / 𝑀 (𝑇)
false 𝜆 true
𝜀2 1,2 𝑀𝑏𝑏1 𝑇
𝜀1 𝐹𝑅𝑎𝑡𝑖𝑜 (T) = 𝜆 STOP
𝑀𝑏𝑏2 𝑇
Fig. 2. Visualization of an algorithm determining the true emissivity of an object by evaluating
all three attainable channels from a two-colour pyrometer.
Even if no information is given on the material properties the true temperature can
be gained by assuming a starting value for 𝜀 . An iterative approach (cf. Fig. 2) fol-
lows which may be shortened by an educated guess on 𝜀 . Consequently, 𝑀 (𝑇)
may be calculated taking the measured data from 𝑀 (𝑇) and knowing the detecting
wavelength of the first channel. As it must be safe to assume 𝑇 is equal for all chan-
nels 𝑀 (𝑇) can be easily gained and using the measured data of 𝑀 (𝑇) 𝜀 is readily
𝜀 ,
obtained. Finally, the ratio of 𝜀 and 𝐹 (𝑇) are compared to 𝑀 (𝑇) / 𝑀 (𝑇). If
this comparison does not satisfy equation (5) the iteration is repeated with a different
starting emissivity 𝜀 until a true statement is achieved.
93 CERC 2020
Data Computing and Artificial Intelligence
6
3 Results and Discussion
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.
Fig. 3. Simulated emissivity as a function of a true emissivity ratio. The real object’s
emissivities that were used to obtain measurement data are circled in red. Please note
that only the measured results from the two respective single-wavelength channels
and from the ratio channel enter the simulation. The stopping condition was set to
differences in the calculated and measured ratio factor 𝐹 , (𝑇) of less than 1E-3.
This means that at a temperature of 1100 K a maximum error of 8 K needs to be ac-
cepted.
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 wave-
length of 1µm. It can be seen that despite the small standard deviation at a true emis-
sivity 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
CERC 2020 94
Data Computing and Artificial Intelligence
7
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.
Table 1. Maximum temperature error for the considered true emissivites according to Fig. 3 if
the maximally deviated simulation result is used for temperature determination
True Emissivity 0.1 0.2 0.4 0.6 0.8
12.71 K 8.94 K 8.79 K 9.44 K 9.53 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).
Fig. 4. Simulated emissivity as a function of a true emissivity ratio and the central wavelength.
The real object’s emissivities that were used to obtain measurement data are circled in red.
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 tem-
perature 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
95 CERC 2020
Data Computing and Artificial Intelligence
8
rising the number of iterations at the same time. Again no significant influence of the
emissivity ratio could be found.
Fig. 5. Simulated emissivity as a function of a true emissivity ratio and an assumed measure-
ment uncertainty of the pyrometer of 25 K and 100 K. The real object’s emissivities that were
used to obtain measurement data are circled in red.
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 un-
desired crosstalk behaviour [19]. Further, there may be only one detector that is fil-
tered at different wavelengths which will automatically lead to a temporal mismatch
between the channels. However, due to a finite sampling time a similar mismatch may
likewise occur even if different detectors are used. During the simulation such a mis-
match has been accounted for by the measurement uncertainty which acts randomly
on the temperature measurement of both channels. Moving objects, which are a com-
mon application of pyrometry are accounted for by a random change in temperature
of a total of 200 K. If larger changes are to be expected such as it would be the case
for active heating and cooling the simulation time may exceed a critical change in
temperature thus increasing the simulation error.
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 con-
verges towards the measured value M(T). In the end, a balance needs to be found
between simulation time and accuracy.
CERC 2020 96
Data Computing and Artificial Intelligence
9
4 Summary
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 tempera-
ture can be assumed for all channels at a time. In this case the emissivities at both
spectral positions 𝜀 and 𝜀 and the true temperature 𝑇 form a threefold equation sys-
tem that – once the correct combination of 𝜀 and 𝜀 has been found – returns a true
statement.
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 temper-
ature 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 pyrome-
ter’s measurement uncertainty from 25 K to 100 K was found to increase the simula-
tion 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 sys-
tem that results from the working principle of a ratio pyrometer.
List of Nomenclature
𝑀 thermal radiance of a black body at a detection wavelength λ
ℎ Planck constant
𝑐 speed of light
λ detection wavelength
𝑘 Boltzmann constant
𝑇 temperature
𝛼 absorptivity
αs,d (λ,γ)T spectral and directional absorptivity as a function of detection
wavelength λ and angle γ at a temperature 𝑇
ε emissivity
εs,d (λ,γ)T spectral and directional absorptivity as a function of detection
wavelength λ and angle γ at a temperature 𝑇
𝐹 , ratio of the real-body spectral radiances 𝑀 (𝑇) and 𝑀 (𝑇)
97 CERC 2020
Data Computing and Artificial Intelligence
10
References
1. Planck, M.: Über eine Verbesserung der Wien’schen Strahlungsgleichung. Verhandlungen
der Deutschen Physikalischen Gesellschaft (1900)
2. Kirchoff, G.: Über den Zusammenhang zwischen Emission und Absorption von Licht und
Wärme. Monatsberichte der Königlichen Preußischen Akademie der Wissenschaften zu
Berlin, 783 (1859)
3. Gibson, A. F.: A two-colour infra-red radiation pyrometer. Journal of Scientific Instru-
ments 28(5), 153 (1951)
4. Araújo, A.: Multi-spectral pyrometry – a review. Journal of Measurement Science and
Technology 28, 082002 (2017) DOI: https://doi.org/10.1088/1361-6501/aa7b4b
5. Sun, X. G.: Processing Method of Multi-Wavelength PyrometerData for Continuous Tem-
perature Measurements. International Journal of Thermophysics 26(4), 1255 (2005)
DOI: 0.1007/s10765-005-6724-6
6. Xing, J. et al.: A data processing algorithm for multi-wavelength pyrometry-which does
not need to assume the emissivity model in advance. Journal of the International Meas-
urement Confederation 67, 92 (2015)
DOI: https://doi.org/10.1016/j.measurement.2014.11.019
7. Wen, C.-D., Chai, T.-Y.: Examination of multispectral radiation thermometry using linear
and log-linear emissivity models for aluminum alloys. Journal of Heat and Mass Transfer
47, 847 (2011)
DOI: https://doi.org/10.1007/s00231-011-0769-9
8. Araújo, A., Silva, R.: Surface temperature estimation in determined multi-wavelength py-
rometry systems. Review of Scientific Instruments 91, 054901 (2020)
DOI: https://doi.org/10.1063/5.0005676
9. Reichel, D.: Temperature Measurement in Rapid Thermal Processing with focus on the
application to Flash Lamp Annealing. Critical Reviews in Solid State and Material Scienc-
es 36(2), 102-128 (2011)
DOI: https://doi.org/10.1080/10408436.2011.572757
10. Nulman, J. et al.: Pyrometric Emissivity Measurements and Compensation in an RTP
Chamber. In: MRS Proceedings, pp. 146 – 461. Cambridge University Press, Cambridge
(2011)
DOI: 10.1557/PROC-146-461
11. Sentenac, T.: Trichromatic thermoreflectometry for an improved accuracy of
true temperature field measurement on a multi-material part. International Journal of
Thermal Sciences 145, 105980 (2019)
DOI: https://doi.org/10.1016/j.ijthermalsci.2019.105980
12. Fu, T.: Measurements of the directional spectral emissivity based on a radiation heating
source with alternating spectral distributions. International Journal of Heat and Mass
Transfer 90, 1207 (2015)
DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2015.07.064
13. Madura, H., Piatkowski, T.: Emissivity compensation algorithms in double-band pyrome-
try. Journal of Infrared Physics and Technology 46(1-2), 185-189 (2004)
DOI: https://doi.org/10.1016/j.infrared.2004.03.024
14. Müller, B., Renz, U.: Development of a fast fiber-optic two-color pyrometer for the tem-
perature measurement of surfaces with varying emissivities. Review of Scientific Instru-
ments 72, 3366 (2001)
DOI: https://doi.org/10.1063/1.1384448
CERC 2020 98
Data Computing and Artificial Intelligence
11
15. Muller, M. Fabbro, R.: Temperature measurement of laser heated metals in highly oxidiz-
ing environment using 2D single-band and spectral pyrometry. Journal of Laser Applica-
tions 24(2), 022006 (2012)
DOI: https://doi.org/10.2351/1.3701400
16. Roebuck, B., Edwards, G., Gee, M. G.: Characterisation of oxidising metal surfaces with a
two colour pyrometer. Journal of Materials Science and Technology 21(7), 831 (2013)
DOI: https://doi.org/10.1179/174328405X46169
17. Coates, P.B.: Multi-Wavelength Pyrometry. Metrologia 17(3), 103 (1981)
DOI: https://doi.org/10.1088/0026-1394/17/3/006
18. Madura, H., Kastek, M., Piatkowski, T.: Automatic compensation of emissivity in three-
wavelength pyrometers. Journal of Infrared Physics and Technology 51(1), 1-8 (2007)
DOI: https://doi.org/10.1016/j.infrared.2006.11.001
19. Luryi, S., Xu, J., Zaslavsky, A.: Future Trends in Microelectronics: Up the Nano Creek. 1st
edn. John Wiley & Sons, Inc., Hoboken New Jersey (2007)
99 CERC 2020