Abstract

Surface temperatures are estimated with high precision based on a multitemperature method for Fourier-transform spectrometers. The method is based on Planck’s radiation law and a nonlinear least-squares fitting algorithm applied to two or more spectra at different sample temperatures and a single measurement at a known sample temperature, for example, at ambient temperature. The temperature of the sample surface can be measured rather easily at ambient temperature. The spectrum at ambient temperature is used to eliminate background effects from spectra as measured at other surface temperatures. The temperatures of the sample are found in a single calculation from the measured spectra independently of the response function of the instrument and the emissivity of the sample. The spectral emissivity of a sample can be measured if the instrument is calibrated against a blackbody source. Temperatures of blackbody sources are estimated with an uncertainty of 0.2–2 K. The method is demonstrated for measuring the spectral emissivity of a brass specimen and an oxidized nickel specimen.

© 1996 Optical Society of America

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References

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1995

1993

J. Lohrengel, R. Todtenhaupt, M. Ragab, “Determination of the directional spectral emissivity of solids in the spectral range from 2.5 μm to 45 μm at temperatures between 80 °C and 350 °C” (in German), Wärme- Stoffübertrag. 28, 321–327 (1993).

1992

1991

1988

1981

1963

D. W. Marquardt, “A method for the solution of certain nonlinear problems in least squares,” Q. Appl. Math. 2, 164 (1963).

Ballico, M.

Brown, S. D.

Buijs, H.

Chase, D. B.

Compton, S.

de Haseth, J. A.

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1986), Chap. 5, pp. 202–203.

DeBlase, F. J.

Dietl, H.

Eckersdorf, K.

L. Michalski, K. Eckersdorf, J. McGhee, Temperature Measurement (Wiley, Chichester, 1991), Chap. 10, pp. 317–350.

Griffiths, P. R.

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1986), Chap. 5, pp. 202–203.

Haschberge, P.

Haschberger, P.

E. Lindermeir, V. Tank, P. Haschberger, “Contactless measurement of the spectral emissivity and temperature of surfaces with a Fourier transform infrared spectrometer,” in Thermosense XIV: An International Conference on Thermal Sensing and Imaging Diagnostic Applications, J. K. Eklund, Proc. SPIE1682, 354–364 (1992).

Howell, H. B.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1992), Appendix D, pp. 1038–1039.

Jones, T. P.

LaPorte, D. D.

Lindermeir, E.

E. Lindermeir, P. Haschberge, V. Tank, H. Dietl, “Calibration of a Fourier transform spectrometer using three black-body sources,” Appl. Opt. 31, 4527–4533 (1992).
[CrossRef] [PubMed]

E. Lindermeir, V. Tank, P. Haschberger, “Contactless measurement of the spectral emissivity and temperature of surfaces with a Fourier transform infrared spectrometer,” in Thermosense XIV: An International Conference on Thermal Sensing and Imaging Diagnostic Applications, J. K. Eklund, Proc. SPIE1682, 354–364 (1992).

Lohrengel, J.

J. Lohrengel, R. Todtenhaupt, M. Ragab, “Determination of the directional spectral emissivity of solids in the spectral range from 2.5 μm to 45 μm at temperatures between 80 °C and 350 °C” (in German), Wärme- Stoffübertrag. 28, 321–327 (1993).

Marquardt, D. W.

D. W. Marquardt, “A method for the solution of certain nonlinear problems in least squares,” Q. Appl. Math. 2, 164 (1963).

McGhee, J.

L. Michalski, K. Eckersdorf, J. McGhee, Temperature Measurement (Wiley, Chichester, 1991), Chap. 10, pp. 317–350.

Michalski, L.

L. Michalski, K. Eckersdorf, J. McGhee, Temperature Measurement (Wiley, Chichester, 1991), Chap. 10, pp. 317–350.

Monfre, S. L.

Ragab, M.

J. Lohrengel, R. Todtenhaupt, M. Ragab, “Determination of the directional spectral emissivity of solids in the spectral range from 2.5 μm to 45 μm at temperatures between 80 °C and 350 °C” (in German), Wärme- Stoffübertrag. 28, 321–327 (1993).

Rathmann, O.

O. Rathmann, “A nonlinear least squares fitting program,” Department of Energy Technology, Risø National Laboratory, Denmark (personal communication, November1992).

Redgrove, J.

J. Redgrove, Thermophysical Properties Section, National Physical Laboratory, Teddington TW11 0LW, UK (personal communication, 1992).

Revercomb, H. E.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1992), Appendix D, pp. 1038–1039.

Smith, W. L.

Sromovsky, L. A.

Tank, V.

E. Lindermeir, P. Haschberge, V. Tank, H. Dietl, “Calibration of a Fourier transform spectrometer using three black-body sources,” Appl. Opt. 31, 4527–4533 (1992).
[CrossRef] [PubMed]

E. Lindermeir, V. Tank, P. Haschberger, “Contactless measurement of the spectral emissivity and temperature of surfaces with a Fourier transform infrared spectrometer,” in Thermosense XIV: An International Conference on Thermal Sensing and Imaging Diagnostic Applications, J. K. Eklund, Proc. SPIE1682, 354–364 (1992).

Todtenhaupt, R.

J. Lohrengel, R. Todtenhaupt, M. Ragab, “Determination of the directional spectral emissivity of solids in the spectral range from 2.5 μm to 45 μm at temperatures between 80 °C and 350 °C” (in German), Wärme- Stoffübertrag. 28, 321–327 (1993).

Appl. Opt.

Appl. Spectrosc.

Q. Appl. Math.

D. W. Marquardt, “A method for the solution of certain nonlinear problems in least squares,” Q. Appl. Math. 2, 164 (1963).

Wärme- Stoffübertrag

J. Lohrengel, R. Todtenhaupt, M. Ragab, “Determination of the directional spectral emissivity of solids in the spectral range from 2.5 μm to 45 μm at temperatures between 80 °C and 350 °C” (in German), Wärme- Stoffübertrag. 28, 321–327 (1993).

Other

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1992), Appendix D, pp. 1038–1039.

J. Redgrove, Thermophysical Properties Section, National Physical Laboratory, Teddington TW11 0LW, UK (personal communication, 1992).

L. Michalski, K. Eckersdorf, J. McGhee, Temperature Measurement (Wiley, Chichester, 1991), Chap. 10, pp. 317–350.

E. Lindermeir, V. Tank, P. Haschberger, “Contactless measurement of the spectral emissivity and temperature of surfaces with a Fourier transform infrared spectrometer,” in Thermosense XIV: An International Conference on Thermal Sensing and Imaging Diagnostic Applications, J. K. Eklund, Proc. SPIE1682, 354–364 (1992).

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1986), Chap. 5, pp. 202–203.

O. Rathmann, “A nonlinear least squares fitting program,” Department of Energy Technology, Risø National Laboratory, Denmark (personal communication, November1992).

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Figures (10)

Fig. 1
Fig. 1

Experimental setup for measuring the spectral emissivity of a sample: 1, emission port; 2, optional emission port; a, 25-mm aperture; b, focusing mirror (f = 152.4 mm); c, sample; d, insulated heated aluminum block; e, thermocouple for measuring the temperature of the aluminum block; f, aluminum disk.

Fig. 2
Fig. 2

Measured offset-corrected calibration spectra from the blackbody at four temperatures: a, 473 K; b, 673 K; c, 873 K; d, 1073 K. The ambient temperature was 296.7 K. The spectra are measured with a DTGS detector with a resolution of 8 cm−1. Bands from H2O and CO2 are visible in the spectra.

Fig. 3
Fig. 3

Ratio L′(T)/L′(1073 K) of the blackbody calibration spectra plotted with the theoretical curves based on Planck’s radiation law for T = 473 K (lower curve) in steps of 100 K to T = 973 K (upper curve). The ambient temperature of 296.7 K and the blackbody readings are used for plotting the theoretical curves. Deviations of 1–1.5% are observed for wave numbers greater than 3500 cm−1 with low radiance from the blackbody.

Fig. 4
Fig. 4

Overall stability of instrumentation of 10 h with a Mikron blackbody source at 1023 K. A reference spectrum was measured 1 h after the InSb detector was filled with liquid nitrogen (0 h on the abscissa). The mean, minimum, and maximum values are calculated as the ratio of the measured spectrum to the reference spectrum in the interval from 2550 to 3200 cm−1.

Fig. 5
Fig. 5

Influence of noise on estimating temperature with the multitemperature method on simulated spectra. The points connected with the solid lines are calculated with two spectra (M = 2), and the points connected with the dashed lines are calculated with three spectra (M = 3). The standard deviation is based on 100 calculations for each temperature set.

Fig. 6
Fig. 6

Temperature error on estimated temperatures by the multitemperature method in the experiment on the low-temperature blackbody. The accuracy of the temperature estimate increases with the number of spectra applied in the calculation. The ambient temperature is measured by a thermocouple to be 292.2 K.

Fig. 7
Fig. 7

Estimated temperatures from measurements on the Mikron blackbody. The spectra were offset-corrected with a spectrum as measured with a temperature reading of the blackbody at 1073 K.

Fig. 8
Fig. 8

Estimated temperatures from the spectra of the brass sample and temperatures measured with the thermocouple. The spectra were offset-corrected with a spectrum measured at ambient temperature (296.0 K).

Fig. 9
Fig. 9

Estimated normal spectral emissivity of the brass sample in the spectral range from 1750 to 3250 cm−1: (a) T1 = 323.5 K, T2 = 370.1 K, and T3 = 418.4 K (M = 3); (b) T1 = 324.6 K and T2 = 372.3 K (M = 2); (c) T1 = 325.8 K, T2 = 374.6 K, T3 = 424.5 K, and T4 = 478.1 K (M = 4). The ambient temperature is 296.0 K.

Fig. 10
Fig. 10

Calculated normal spectral emissivity of an oxidized nickel specimen: (a) T1 = 319.3 K and T2 = 351.7 K (1750–2500 cm−1); (b) T1 = 319.6 K and T2 = 352.3 K (1750–3500 cm−1); (c) T1 = 320.2 K, T2 = 353.4 K, and T3 = 389.0 K (1750–3250 cm−1). The ambient temperature is 296.2 K. The two dots indicate the values measured by the NPL at 773 K.

Tables (1)

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Table 1 Calibration of the Mikron M360 Blackbody (1-m Distance)

Equations (20)

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ɛ ( ν , T ) L s ( ν , T ) L ( ν , T ) ,
Δ ɛ ɛ ( Δ T ) = Δ L s L s + Δ L L 2 1 1 - exp ( - C 2 ν T ) C 2 ν T 2 Δ T 2 C 2 ν T 2 Δ T ,
f ( ν , T ) = R ( ν ) exp [ i ϕ ( ν ) ] { ɛ ( ν , T ) L ( ν , T ) E + I ( ν ) ρ ( ν , T ) E + G ( ν , T I ) E I exp [ i ψ ( ν ) ] } ,
f ( ν , T ) = ½ - F ( x ) exp ( - i 2 π ν x ) d x ,
I ( ν ) = L ( ν , T amb ) + O ( ν ) .
ρ ( ν , T ) = 1 - ɛ ( ν , T ) .
f ( ν , T ) - f ( ν , T amb ) = R ( ν ) exp [ i ϕ ( ν ) ] × { ɛ ( ν , T ) [ L ( ν , T ) - L ( ν , T amb ) ] + O ( ν ) [ ɛ ( ν , T ) - ɛ ( ν , T amb ) ] } .
f ( ν , T ) - f ( ν , T amb ) = R * ( ν ) exp [ i ϕ ( ν ) ] [ L ( ν , T ) - L ( ν , T amb ) ] ,
R * ( ν ) R ( ν ) ɛ ( ν , T ) .
f i k = f ( ν i , T k ) = R * ( ν i ) [ L ( ν i , T k ) - L ( ν i , T amb ) ]             i = 1 , , N             k = 1 , , M .
min χ 2 = i = 1 N k = 1 M ( S i k - f i k ) 2 W i k ,
χ 2 R i * = 0 i = 1 , , N , χ 2 T k = 0 k = 1 , , M .
Q R R i r i + k = 1 M Q T R i k t k = X R i             i = 1 , , N ,
i = 1 N Q T R i k r i + Q T T k t k = X T k             k = 1 , , M ,
Q R R i = k = 1 M W i k [ L i k - L i ( T amb ) ] 2 , Q T R i k = W i k R i * [ L i k - L i ( T amb ) ] L i k T k , Q T T k = i = 1 N W i k ( R i * L i k T k ) 2 ,
X R i = k = 1 M W i k ( S i k - f i k ) [ L i k - L i ( T amb ) ] , X T k = i = 1 N W i k ( S i k - f i k ) R i * L i k T k .
l = 1 M P k l t l = Y T k             k = 1 , , M ,
P k l = δ k l Q T T k - i = 1 N Q T R i k P T R i l             k , l = 1 , , M , Y T k = X T k - i = 1 N Q T R i k Y R i             k = 1 , , M ,
P T R i l = Q T R i l Q R R i ,
Y R i = X R i Q R R i = r i .

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