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Measurement of gas temperature profile using spectral intensity from CO2 4.3 μm band
International Journal of Thermal Sciences 41 (2002) 883–890 www.elsevier.com/locate/ijts
Measurement of gas temperature profile using spectral intensity from CO2 4.3 µm band Su-Wan Woo, Tae-Ho Song ∗ Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Kusong-dong 373-1, Yusong-gu, Taejon, 305-701 South Korea Received 26 May 2000; accepted 23 January 2002
Abstract Spectral remote sensing (SRS) method for determining the temperature profile along a line-of-sight is investigated experimentally. Quartz tube, within which combustion gas flows, is used as the test section. The inversion procedure is carried out with a line-by-line (LBL) method and a CK-based WNB model. The optimal gas temperature profile that minimizes the error between the measured narrow band intensities around the CO2 4.3 µm band and the calculated ones is obtained as the result of the inversion process. The gas temperature is also measured with a shielded thermocouple and corrected for the error. The results show that the front, center and back temperatures are within errors of 1%, 4% and 12%, respectively. SRS technique shows poor performance in measuring cold gas temperatures behind a hot layer. The reconstructed temperature profile in the front region is, however, in good agreement with the thermocouple reading. The potential applicability of SRS is positively demonstrated and current technical limitations are also discussed. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Temperature profile measurement; Spectral remote sensing; Line-of-sight; Combustion gas; CO2 4.3 µm band
1. Introduction Measurement of temperature in combustion gases is difficult due to the complicated radiative characteristics of hot gases and the difficulty in installing a sensor. For this reason, various nonintrusive optical methods have been introduced. They are coherent anti-Stokes Raman spectroscopy (CARS) [1], computed tomography (CT) [2] and SRS [3] to name a few. Among them, the SRS is a very promising technique since it can measure the temperature profile along a line-of-sight in any geometry. This method begins with measuring spectral intensities from hot combustion gases. Then, the temperature profile is calculated by an inversion process. The narrow band intensities calculated from a temporary temperature profile are compared with the measured values, and the temporary profile is refined to reduce the discrepancy. Therefore, a good calculation model of spectral intensities is a prerequisite for application of the SRS. Yang [4] reports the absorption coefficients database of CO2 4.3 µm band. This band does not overlap with other absorption bands of CO2 or H2 O, and CO2 is a main * Correspondence and reprints.
E-mail address: [email protected] (T.-H. Song).
component of most combustion gases. Yang uses the spectral coefficients of Chedin [5] for the molecular coefficients of CO2 and the method of Scutaru [6] for hot band added to the cold band database HITRAN(92) [7]. Since it takes much time when using the line-by-line (LBL) method [8], many researchers have been looking for an efficient band modeling. There are the statistical narrow-band (SNB) developed by Mayer and Goody [9], the weighted-sum-ofgray-gases (WSGG) model of Hottel [10], the spectral group model (SGM) of Song and Viskanta [11] and the WSGGMbased narrow band (WNB) model of Kim and Song [12] and so on. In this research, the correlated-k (CK)-based WNB model of Yang and Song [13] for the 4.3 µm band of CO2 is selected, since it has been optimally modeled for the CO2 4.3 µm band. This model is shown [4] to impose an error of 10 K due to modeling inaccuracy at the gas temperature of 1600 K in 1 m layer of combustion gas, while SNB gives error of 30 K. Application of the SRS inversion in an engineering problem is first made by Krakow [14] in the late 1960s. And Cutting [15] obtains the inside temperature of furnace using intensities measured from both sides of furnace. In the late 1970s, Buchele [16] uses matrix calculation method, and Hommert, Viskanta and Mellor [3] determine temperatures
1290-0729/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 9 0 - 0 7 2 9 ( 0 2 ) 0 1 3 8 1 - 9
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Nomenclature A B E g I L R s T
first derivative of the residue R (Eq. (9)) second derivative of the residue R (Eq. (12)) modeling error cumulative k-distribution radiation intensity path length residue of intensity (Eq. (8)) coordinate along the radiation path temperature of medium
Greek symbols ε η
spectral emissivity wavenumber
with an optimization technique. Their results are, however, short of practical applicability possibly due to the poor database of CO2 radiative properties at that time. In this study, the spectral intensities from combustion gas flowing through a quartz tube are measured. The detailed experimental technique and the results of temperature inversion are shown. The limitations of SRS are also discussed.
2. Experimental study 2.1. Apparatus A schematic diagram of the experimental system is shown in Fig. 1. All components of the experimental system except the computer and the lock-in-amplifier are in a darkroom with a dimension of 2 m × 1 m × 1 m (W × L × H ). A quartz tube with open ends at both sides and a cooling annular channel, through which cold air flows, is
κ ξ
absorption coefficient coordinate transformation variable
Subscripts 0 c i, j L m n t η
at the exit of quartz tube at the highest temperature position indices of gray gas or narrow band at the end of quartz tube model or measurement power of absolute term in the assumed profile true value wavenumber
used as the test section. As shown in Fig. 2, the channel is 0.8 m long and its inner and outer diameters are 0.035 m and 0.050 m, respectively. The inlet slit of the monochromator is centered along the axis of quartz tube through two in-line iris diaphragms with very small aperture to block radiation from the inner wall of the quartz tube. For accurate alignment, the quartz tube, iris diaphragms and chopper are installed on an optical rail. An indium antimonide infrared detector is selected for the measurement of 4.3 µm band radiation for its sensing range is roughly between 2 to 5 µm with high sensitivity at 4.3 µm. The active size (diameter) of the IR detector is only 1 mm so the detector must be precisely mounted on the monochromator. A sensitive signal cannot be obtained when the detector is not exactly fitted to the exit slit of the monochromator. A series of alignments are performed with a diode laser. The IR detector is cooled with LN2 to increase the sensitivity. The dark-room is free of carbon dioxide from the combustion gas to minimize the radiation noise.
Fig. 1. Schematic diagram of the experimental system.
S.-W. Woo, T.-H. Song / International Journal of Thermal Sciences 41 (2002) 883–890
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Fig. 2. Quartz tube for the experiment. Table 1 Mole fraction of the propane-burned products
Fig. 3. Result of detector calibration.
2.2. Experimental procedure Preliminary calibrations and measurements are performed for the IR detector, the thermocouple and gas chromatography system. The blackbody radiation from a calibration furnace is measured by the monochromator and IR detector. As shown in Fig. 3, the result of calibration shows an almost linear relationship between the detector signal and the blackbody emissive power. Note that the blackbody furnace opening is located exactly at the same location as the quartz tube opening so that the attenuation ratio of the signal radiative intensity by the intervening atmospheric CO2 from the openings to the detector is maintained constant and included in the calibration procedure. The hot gas temperature is measured with a thermocouple shielded in a T-tube and it is corrected considering the radiative heat loss to the cold black sidewalls. Compensating radiation loss, reading of 1000 K is estimated to be 1090 ± 10 K. The last preliminary experiment is to apply the gas chromatography (GC) to measure the concentration of CO2 . GC is applied at the exit of the quartz tube. Note that the gas in the quartz tube comes in a burnt state so that there is no change of concentration along the tube. The fuel is propane. The result of measurement shown in Table 1 underestimates water vapor concentration because
Species
Concentration (%)
CO2 CO O2 H2 O
5.4 9.6 1.9 13.7
part of the water vapor in the hot combustion gas is condensed during quantification at room temperature. Fairly large amount of carbon monoxide is also detected, possibly due to the quenching effect of the quartz tube and the subsequent incomplete combustion. Including other minor measurement errors such as condensation of water vapor during sampling, we estimate ±0.5% absolute error in the final CO2 concentration. The final inversion is reliable, however, as far as the mole fraction of carbon dioxide is precisely measured. Although the SRS technique can be applied to simultaneous measurement of concentration and temperature, the current research is limited to temperature only. The current scheme may be used for boilers and furnaces where the flame occupies relatively small volume and the concentration is well-known. The intensity measurement begins when the experiment comes to a steady state. The temperature change at the exit of tube is monitored and it is assumed to have reached a steady state when it varies within ±0.3 ◦ C. Now, the intensity measurement is made from 2045 to 2451 cm−1 at an interval of 7 cm−1 . The reading takes about 10 min.
3. Inversion procedure 3.1. CK-based WNB model Various band models have been investigated for the application to SRS since the ‘exact’ intensity calculation using the line-by-line method requires long computing time even in the relatively narrow spectral interval of interest. Kim and Song [12] suggests the WSGGM-based narrow band (WNB), but it is not very accurate for the CO2 4.3 µm band. Instead, we use the CK-based WNB model of Yang
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and Song [13]. In the correlated k-distribution (CK) model of Marin and Buckius [17], the starting point of Yang and Song’s model, the re-ordered spectrum is used instead of the actual spectrum, in which the cumulative k-distribution function is defined as: 1 H (κ − κη ) dη (1) g(T , κ) = η η
The function H (κ − κη ) is the Heaviside unit function, which is unity when κη is smaller than κ, and 0 otherwise. Yang and Song use the g function of CK model for the expressions of the absorption coefficients because the WNB model of Kim and Song does not handle the dependence on temperature well. The cumulative k-distribution is used to obtain the tabulated absorption coefficients of gray gases using the following: κη (η, T , P , Ys ) = φ(T , P , Ys )ϕ(η)
(2)
Then the following relation is satisfied between the cumulative k-distribution functions of j th gray gas in two different temperatures, T and T ∗ . g(T , κj ) = g T ∗ , κj∗ (3) When the value of the function is given at a reference temperature T ∗ and absorption coefficient κj∗ , the corresponding absorption coefficient κj is obtained implicitly at another temperature T using the above equation. The absorption coefficient κj∗ at the reference temperature is determined by minimizing the error function given as, εt εm + −2 (4) E= εm εt L
In the CK-based WNB model [13], the narrow-band mean emissivity is defined as: 1 − e−κj L Wj (η) (5) εηm (T , L) = j
Note that Wj (η) here means the weight of j th gray gas in a narrow band and it does not change from one temperature to another since the variation of blackbody intensity in a narrow band is negligible. This simplification comes from Eq. (2), and it is taken to avoid the Leibniz terms that appear when the wavenumber boundaries of a gray gas change in inhomogeneous gas layers [18]. The detailed database for Wj (η) is available from Ref. [13]. When the modeling is finished, we can calculate the spectral intensity (averaged over ith narrow band) Ii,c at s = 0 as L s e− 0 κj (s ) ds Wj κj (s) ds (6) Ii,c (0) = Ibη (s) 0
j
3.2. Inversion procedure We take a few representative nodal temperatures Tj and take a smooth interpolation between them to compute the
intensity numerically. A very general and robust inversion scheme with many nodal temperatures is surely desirable in the long run. This method is a first step to demonstrate the inversion. In the inversion procedure, initial temperatures are updated by the following equation: [T ]new = [T ]old + [δT ]
(7)
The absolute value of the residue Ri defined by Eq. (8) converges to a minimum when the inverted temperatures are optimally approximated to the real ones. Ri =
Ii,c − Ii,m Ii,m
(8)
where Ii,m is the measured intensity of the ith narrow band at s = 0 and Ii,c is calculated from [T ]old . The temperature increase [δT ] at the next step is determined by Eq. (9) using the residue calculated by Eq. (8).
T δT = [A]T [R] A A + (p/q)Q (9) T where the i, j th entry of matrix A is ∂Ri/∂Tj [16]. The right side of Eq. (8) reaches a minimum when the calculated temperature profile approaches the real one. The matrix Q is diagonal and its diagonal elements are the same as those of [AT A]. And q, p and [B] are defined as: 1/2 q= qi2 (10) i
p=
i
1/2 pi2
bi,j = Tj2
(11)
∂ 2 Ri ∂Tj2
(12)
where qi and pi denote the diagonal elements of AT A and B T R, respectively. The right side hand of Eq. (12) is calculated by: +j
bi,j =
−j
0 +I Ii,c − 2Ii,c i,c
0.012Ii,m +j
(13)
−j
0 respectively mean the intensities where Ii,c , Ii,c and Ii,c calculated with 1% increase of Tj , 1% decrease and the unperturbed one. The following equation is the temperature profile between nodes used in this study. n ξ 0 ξ ξc T = Tc − (Tc − T0 ) − 1 , ξ c n (14) 1 + ξ − 2ξc T = Tc − (Tc − TL ) − 1 , ξc ξ 1 1 − ξc
where T0 , Tc and TL mean the temperatures at the exit of quartz tube, at the gas inlet and at the dead end of quartz tube, respectively. The assumed profile has a maximum at the gas inlet point. And ξ is the normalized coordinate (0 at the gas exit and 1 at the dead end). And n is a fixed parameter. The effect of profile parameter n will be investigated also.
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4. Results and discussion Fig. 4 shows a measured intensity distribution from 2045 to 2451 cm−1 . The general radiative characteristics of CO2 generally appear above 2200 cm−1 while some irregular intensities emerge below 2192 cm−1 due to overlap with the CO band and the error of measurement. Also the intensity above 2400 cm−1 reveals the background radiation (0.18 ± 0.02 W·m2 ·cm−1 ·sr) and it is compensated in all the narrow bands equally. This intensity distribution is used in the inversion. In this study, the nodal temperatures are calculated by changing two parameters, i.e., temperatures are obtained with variation of the assumed temperature profile and the employed narrow bands. In addition, temperatures calculated with the CK-based WNB model and those measured with thermocouples are compared with each other. 4.1. Results varying the assumed temperature profile The inversion calculation is performed first by varying the power n in Eq. (14). As shown in Fig. 5, the inversion
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result has good agreement with the real profile regardless of the power n. The fifty-nine narrow bands between 2045 and 2451 cm−1 are all used to determine the profile. The typical calculation time for the inversion is 18 s with a Pentium III (450 MHz) processor for 20 iterations for the temperature update. The inverted temperature with n = 1/2.5 is better than the other cases. And Tc slightly increases as the power becomes lower. Table 2 shows errors of T0 , Tc and TL when varying n. In all situations, T0 , Tc and TL are within errors of 1%, 8% and 14%, respectively. It is recognized that the cold back temperature behind the hot center temperature has large error compared with the low front temperature near the sensor. It is because hot center gas emits strong radiation and at the same time it actively absorbs the feeble emission from the cold gas behind. It is difficult to improve it as long as only one side measurement is performed. Nevertheless, the absolute error is maintained within 50 ◦ C. Fig. 6 shows an intensity distribution as calculated by the inverted temperature profile with n = 1/2.5. The result indicates that the intensities below 2192 cm−1 have large errors whereas those from 2200 to 2395 cm−1 show reasonable agreement with the measurement. The large discrepancy below 2192 cm−1 is considered to be due to overlap with the CO 4.7 µm band. Next, a parabolic profile shown below is applied to the inversion procedure. T = −4(Tc − T0 )ξ 2 + 4(Tc − T0 )ξ + T0 , 0 ξ ξc T = −4(Tc − TL )ξ 2 + 4(Tc − TL )ξ + TL , ξc ξ 1
(15)
Table 2 Errors of T0 , Tc and TL when varying n in Eq. (14)
Fig. 4. Measured intensity distribution from 2045 to 2451 cm−1 .
Fig. 5. Result of inversion calculation using Eq. (14) and the CK-based WNB model along various powers.
Measured temp.
T0 = 608.4
Tc = 1105.7
TL = 355.9
n = 1/1.5 n = 1/2.5 n = 1/3.5
604.7 (0.61%) 609.7 (0.21%) 613.8 (0.90%)
1050.7 (4.97%) 1145.4 (3.59%) 1186.7 (7.33%)
314.7 (11.6%) 310.5 (12.8%) 308.5 (13.3%)
Fig. 6. Intensity calculated from the temperature inverted with the power n = 1/2.5 using the CK-based WNB model.
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Fig. 7. Result of inversion calculation using a different profile, Eq. (15). Fig. 9. Measured intensity and those calculated from the inverted temperature with various employed wavenumber ranges.
Fig. 8. Kernel distribution from the measured temperature distribution between 2045 and 2451 cm−1 .
As shown in Fig. 7, the result is quite different from the real profile. This error is due to the inherent error in the assumed profile shape of the temperature. These results show that the inversion is not very much affected by the detailed profile parameters, however, the overall shape must be properly assumed. Hereafter, n is consistently taken as 1/2.5. 4.2. Results varying the employed narrow bands
− s κj (s ) ds In the inversion procedure, the kernel e 0 Wj κj (s) in Eq. (6) is a weighting factor of the local temperature. It is different from one narrow band to another. Some show greater emphasis on the closer side, while some on the farther side. One should properly select the narrow bands to give well-balanced accuracy in the overall inverted temperature profile. Fig. 8 shows the kernel distribution using the measured temperature profile. As can be seen from this information, narrow bands at 2290 and 2374 cm−1 participate relatively more in the calculation of temperature near the sensor and that at 2255 cm−1 is strongly affected by
Fig. 10. Measured and inverted temperature profiles with various employed wavenumber ranges.
temperatures at all position. And narrow bands from 2192 to 2206 cm−1 are engaged within the central high temperature region. Effects of the other narrow bands are very weak and thus they may not be used as the sensing narrow bands. It is worthy of notice that almost all kernel values are low near the dead end. To make things clear, however, we try three different wavenumber ranges in which all the included narrow bands are employed in the inversion process. Ranges 1, 2 and 3 mean intensities from 2045 to 2451 cm−1 , from 2199 to 2451 cm−1 and from 2199 to 2395 cm−1 , respectively. Intensities calculated from the finally inverted temperature profiles are shown in Fig. 9. Error of intensity is relatively large compared with that of temperature as shown in Fig. 10. This is common to any employed wavenumber range. We can see that the error of temperature is significantly reduced despite the appreciable error in the intensity measurement, as has been confirmed by Hommert et al. [3].
S.-W. Woo, T.-H. Song / International Journal of Thermal Sciences 41 (2002) 883–890
Table 3 Errors of T0 , Tc and TL with variation of employed wavenumber ranges Measured temp. (K)
T0 = 608.4
Tc = 1105.7
TL = 355.9
Range 1 Range 2 Range 3
609.7 (0.21%) 614.0 (0.93%) 613.3 (0.82%)
1145.4 (3.59%) 1076.1 (2.68%) 1089.0 (1.51%)
310.5 (12.8%) 309.9 (12.9%) 310.1 (12.9%)
Fig. 11. Measured and inverted temperature profiles with LBL method and CK-based WNB model.
As shown in the figure, regardless of the selection of range, all results generally have good agreement with the real profile. Table 3 indicates that T0 , Tc and TL are accurate within the error of 1%, 4% and 13%, respectively. For the case of range 1, errors of Tc are larger compared with the other employed ranges. Nevertheless, it is still small for any employed range on the whole because the kernel is large in any range except under 2122 cm−1 . For T0 , all results are excellent because the kernel is large near T0 in any range (see Fig. 8). The large error of TL can be understood similarly. The only difference between ranges 2 and 3 is the region from 2402 to 2451 cm−1 . The two results are similar to each other because the kernel values in the excluded wavenumbers in the range 3 are very small. The final verification is made regarding the validity of the current CK-based WNB model by comparing the results using the original LBL database, i.e., HITRAN(92) with Scutaru’s hot bands. As shown in Fig. 11, these two inversion results are almost identical to each other, while the computation time for inversion is 20 times longer for the LBL. It shows that the current spectral model is fast for real time applications without significantly sacrificing the accuracy.
5. Conclusion SRS method using CO2 4.3 µm band is applied to determine the line-of-sight temperature distribution in a quartz tube. The inversion procedure is performed with the CKbased WNB model. In the calculation, three nodal temper-
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atures are taken as the unknowns and realistic temperature distribution functions between them are set up. A good assumption of temperature profile improves the inversion accuracy, which calls for a more robust and general inversion scheme in the future. The results of the inversion agree reasonably with the measurement within 50 ◦ C error. For spectral modeling, the CK-based WNB model concurs with the LBL results with very small difference and it is proved suitable for SRS techniques. The error of rear temperature behind a hot spot is significantly large because the hot gas absorbs information from the cold gas behind. This problem can be corrected by measurements on both sides. This paper proves the applicability of SRS for actual industrial applications. Such examples are boilers and reactors where most of the gas volume is occupied by burned products and the walls are cold. For further study, research regarding the effects of wall emission and reflection, particle scattering and hotter gas may be made for more general furnaces. Even further application with proper concentration measurement technique may be made for intense combustors and chemical reactors where not only the temperature but also the concentration varies from one location to another.
Acknowledgements This work has been supported by the Critical Technology 21 Project of the Ministry of Science and Technology, Korea and also by a grant from Korea Science and Engineering Foundation (grant No. KOSEF 995-1000-013-2).
References [1] A.C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, Abacus Press, Turnbridge Wells, 1988. [2] P. Beckord, G. Hofelmann, H.O. Luck, D. Franken, Temperature and velocity flow fields measurements using ultrasonic computer tomography, Internat. J. Heat Mass Transfer 33 (1997) 395–404. [3] P.J. Hommart, R. Viskanta, A.M. Mellor, Flame temperature measurements by spectral remote sensing, Combust. Flame 30 (1977) 295– 308. [4] S.S. Yang, Radiation modeling of CO2 4.3 µm band for temperature inversion, Ph.D. Thesis, Korea Advanced Institute of Science and Technology, Korea, 1999. [5] A. Chedin, J. Teffo, The carbon dioxide molecule: a new derivation of the potential, spectroscopic, and molecular constants, J. Mol. Spectrosc. 107 (1984) 333–342. [6] D. Scutaru, L. Rosenmann, J. Taine, Approximate intensities of CO2 hot bands at 2.7, 4.3 and 12 µm for high temperature and medium resolution applications, J. Quant. Spectrosc. Radiat. Transfer 52 (1994) 765–781. [7] L.S. Rothman, R.L. Hawkins, R.B. Wattson, R.R. Gamache, Energy levels, intensities, and line widths of atmospheric carbon dioxide bands, J. Quant. Spectrosc. Radiat. Transfer 48 (1992) 537–566. [8] J.M. Hartman, R. Levi Di Leon, J. Taine, Line-by-line and narrowband statistical model calculations for H2 O, J. Quant. Spectrosc. Radiat. 32 (1982) 119–127.
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[9] R.M. Goody, Atmospheric Radiation, Clarendon Press, Oxford, 1964. [10] H.C. Hottel, A.F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. [11] T.H. Song, R. Viskanta, Development and application of a spectralgroup model to radiation heat transfer, ASME Paper (1986) 86WA/HT-36. [12] O.J. Kim, T.H. Song, Implementation of the weighted-sum-of graygases model to a narrow band: application and validity, Numer. Heat Transfer 30 (1997) 453–468. [13] S.S. Yang, T.H. Song, An improved WSGGM-based narrow band model for the CO2 4.3 µm band, Internat. J. Therm. Sci. 38 (1999) 228–238.
[14] B. Krakow, Spectroscopic temperature profile measurements in inhomogeneous hot gases, Appl. Opt. 5 (1966) 201–209. [15] R.D. Cutting, C. Mc Stewart, Furnace temperature profiles: measurements by spectroscopic methods, Appl. Opt. 14 (1975) 2707–2711. [16] D.R. Buchele, Computer program for calculation of a gas temperature profile by infrared emission-absorption spectroscopy, NASA, 1977, TM-73848. [17] O. Marin, R.O. Buckius, Wide band approach to thermal radiative transport on nonhomogeneous media, J. Heat Transfer T. ASME 119 (1997) 719–729. [18] M.K. Denison, B.W. Webb, A spectral line-based-sum-of-gray-gases model for arbitrary RTE solvers, J. Heat Transfer T. ASME 115 (1993) 1004–1012.
Measurement of gas temperature profile using spectral intensity from CO2 4.3 µm band Su-Wan Woo, Tae-Ho Song ∗ Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Kusong-dong 373-1, Yusong-gu, Taejon, 305-701 South Korea Received 26 May 2000; accepted 23 January 2002
Abstract Spectral remote sensing (SRS) method for determining the temperature profile along a line-of-sight is investigated experimentally. Quartz tube, within which combustion gas flows, is used as the test section. The inversion procedure is carried out with a line-by-line (LBL) method and a CK-based WNB model. The optimal gas temperature profile that minimizes the error between the measured narrow band intensities around the CO2 4.3 µm band and the calculated ones is obtained as the result of the inversion process. The gas temperature is also measured with a shielded thermocouple and corrected for the error. The results show that the front, center and back temperatures are within errors of 1%, 4% and 12%, respectively. SRS technique shows poor performance in measuring cold gas temperatures behind a hot layer. The reconstructed temperature profile in the front region is, however, in good agreement with the thermocouple reading. The potential applicability of SRS is positively demonstrated and current technical limitations are also discussed. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Temperature profile measurement; Spectral remote sensing; Line-of-sight; Combustion gas; CO2 4.3 µm band
1. Introduction Measurement of temperature in combustion gases is difficult due to the complicated radiative characteristics of hot gases and the difficulty in installing a sensor. For this reason, various nonintrusive optical methods have been introduced. They are coherent anti-Stokes Raman spectroscopy (CARS) [1], computed tomography (CT) [2] and SRS [3] to name a few. Among them, the SRS is a very promising technique since it can measure the temperature profile along a line-of-sight in any geometry. This method begins with measuring spectral intensities from hot combustion gases. Then, the temperature profile is calculated by an inversion process. The narrow band intensities calculated from a temporary temperature profile are compared with the measured values, and the temporary profile is refined to reduce the discrepancy. Therefore, a good calculation model of spectral intensities is a prerequisite for application of the SRS. Yang [4] reports the absorption coefficients database of CO2 4.3 µm band. This band does not overlap with other absorption bands of CO2 or H2 O, and CO2 is a main * Correspondence and reprints.
E-mail address: [email protected] (T.-H. Song).
component of most combustion gases. Yang uses the spectral coefficients of Chedin [5] for the molecular coefficients of CO2 and the method of Scutaru [6] for hot band added to the cold band database HITRAN(92) [7]. Since it takes much time when using the line-by-line (LBL) method [8], many researchers have been looking for an efficient band modeling. There are the statistical narrow-band (SNB) developed by Mayer and Goody [9], the weighted-sum-ofgray-gases (WSGG) model of Hottel [10], the spectral group model (SGM) of Song and Viskanta [11] and the WSGGMbased narrow band (WNB) model of Kim and Song [12] and so on. In this research, the correlated-k (CK)-based WNB model of Yang and Song [13] for the 4.3 µm band of CO2 is selected, since it has been optimally modeled for the CO2 4.3 µm band. This model is shown [4] to impose an error of 10 K due to modeling inaccuracy at the gas temperature of 1600 K in 1 m layer of combustion gas, while SNB gives error of 30 K. Application of the SRS inversion in an engineering problem is first made by Krakow [14] in the late 1960s. And Cutting [15] obtains the inside temperature of furnace using intensities measured from both sides of furnace. In the late 1970s, Buchele [16] uses matrix calculation method, and Hommert, Viskanta and Mellor [3] determine temperatures
1290-0729/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 9 0 - 0 7 2 9 ( 0 2 ) 0 1 3 8 1 - 9
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S.-W. Woo, T.-H. Song / International Journal of Thermal Sciences 41 (2002) 883–890
Nomenclature A B E g I L R s T
first derivative of the residue R (Eq. (9)) second derivative of the residue R (Eq. (12)) modeling error cumulative k-distribution radiation intensity path length residue of intensity (Eq. (8)) coordinate along the radiation path temperature of medium
Greek symbols ε η
spectral emissivity wavenumber
with an optimization technique. Their results are, however, short of practical applicability possibly due to the poor database of CO2 radiative properties at that time. In this study, the spectral intensities from combustion gas flowing through a quartz tube are measured. The detailed experimental technique and the results of temperature inversion are shown. The limitations of SRS are also discussed.
2. Experimental study 2.1. Apparatus A schematic diagram of the experimental system is shown in Fig. 1. All components of the experimental system except the computer and the lock-in-amplifier are in a darkroom with a dimension of 2 m × 1 m × 1 m (W × L × H ). A quartz tube with open ends at both sides and a cooling annular channel, through which cold air flows, is
κ ξ
absorption coefficient coordinate transformation variable
Subscripts 0 c i, j L m n t η
at the exit of quartz tube at the highest temperature position indices of gray gas or narrow band at the end of quartz tube model or measurement power of absolute term in the assumed profile true value wavenumber
used as the test section. As shown in Fig. 2, the channel is 0.8 m long and its inner and outer diameters are 0.035 m and 0.050 m, respectively. The inlet slit of the monochromator is centered along the axis of quartz tube through two in-line iris diaphragms with very small aperture to block radiation from the inner wall of the quartz tube. For accurate alignment, the quartz tube, iris diaphragms and chopper are installed on an optical rail. An indium antimonide infrared detector is selected for the measurement of 4.3 µm band radiation for its sensing range is roughly between 2 to 5 µm with high sensitivity at 4.3 µm. The active size (diameter) of the IR detector is only 1 mm so the detector must be precisely mounted on the monochromator. A sensitive signal cannot be obtained when the detector is not exactly fitted to the exit slit of the monochromator. A series of alignments are performed with a diode laser. The IR detector is cooled with LN2 to increase the sensitivity. The dark-room is free of carbon dioxide from the combustion gas to minimize the radiation noise.
Fig. 1. Schematic diagram of the experimental system.
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Fig. 2. Quartz tube for the experiment. Table 1 Mole fraction of the propane-burned products
Fig. 3. Result of detector calibration.
2.2. Experimental procedure Preliminary calibrations and measurements are performed for the IR detector, the thermocouple and gas chromatography system. The blackbody radiation from a calibration furnace is measured by the monochromator and IR detector. As shown in Fig. 3, the result of calibration shows an almost linear relationship between the detector signal and the blackbody emissive power. Note that the blackbody furnace opening is located exactly at the same location as the quartz tube opening so that the attenuation ratio of the signal radiative intensity by the intervening atmospheric CO2 from the openings to the detector is maintained constant and included in the calibration procedure. The hot gas temperature is measured with a thermocouple shielded in a T-tube and it is corrected considering the radiative heat loss to the cold black sidewalls. Compensating radiation loss, reading of 1000 K is estimated to be 1090 ± 10 K. The last preliminary experiment is to apply the gas chromatography (GC) to measure the concentration of CO2 . GC is applied at the exit of the quartz tube. Note that the gas in the quartz tube comes in a burnt state so that there is no change of concentration along the tube. The fuel is propane. The result of measurement shown in Table 1 underestimates water vapor concentration because
Species
Concentration (%)
CO2 CO O2 H2 O
5.4 9.6 1.9 13.7
part of the water vapor in the hot combustion gas is condensed during quantification at room temperature. Fairly large amount of carbon monoxide is also detected, possibly due to the quenching effect of the quartz tube and the subsequent incomplete combustion. Including other minor measurement errors such as condensation of water vapor during sampling, we estimate ±0.5% absolute error in the final CO2 concentration. The final inversion is reliable, however, as far as the mole fraction of carbon dioxide is precisely measured. Although the SRS technique can be applied to simultaneous measurement of concentration and temperature, the current research is limited to temperature only. The current scheme may be used for boilers and furnaces where the flame occupies relatively small volume and the concentration is well-known. The intensity measurement begins when the experiment comes to a steady state. The temperature change at the exit of tube is monitored and it is assumed to have reached a steady state when it varies within ±0.3 ◦ C. Now, the intensity measurement is made from 2045 to 2451 cm−1 at an interval of 7 cm−1 . The reading takes about 10 min.
3. Inversion procedure 3.1. CK-based WNB model Various band models have been investigated for the application to SRS since the ‘exact’ intensity calculation using the line-by-line method requires long computing time even in the relatively narrow spectral interval of interest. Kim and Song [12] suggests the WSGGM-based narrow band (WNB), but it is not very accurate for the CO2 4.3 µm band. Instead, we use the CK-based WNB model of Yang
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and Song [13]. In the correlated k-distribution (CK) model of Marin and Buckius [17], the starting point of Yang and Song’s model, the re-ordered spectrum is used instead of the actual spectrum, in which the cumulative k-distribution function is defined as: 1 H (κ − κη ) dη (1) g(T , κ) = η η
The function H (κ − κη ) is the Heaviside unit function, which is unity when κη is smaller than κ, and 0 otherwise. Yang and Song use the g function of CK model for the expressions of the absorption coefficients because the WNB model of Kim and Song does not handle the dependence on temperature well. The cumulative k-distribution is used to obtain the tabulated absorption coefficients of gray gases using the following: κη (η, T , P , Ys ) = φ(T , P , Ys )ϕ(η)
(2)
Then the following relation is satisfied between the cumulative k-distribution functions of j th gray gas in two different temperatures, T and T ∗ . g(T , κj ) = g T ∗ , κj∗ (3) When the value of the function is given at a reference temperature T ∗ and absorption coefficient κj∗ , the corresponding absorption coefficient κj is obtained implicitly at another temperature T using the above equation. The absorption coefficient κj∗ at the reference temperature is determined by minimizing the error function given as, εt εm + −2 (4) E= εm εt L
In the CK-based WNB model [13], the narrow-band mean emissivity is defined as: 1 − e−κj L Wj (η) (5) εηm (T , L) = j
Note that Wj (η) here means the weight of j th gray gas in a narrow band and it does not change from one temperature to another since the variation of blackbody intensity in a narrow band is negligible. This simplification comes from Eq. (2), and it is taken to avoid the Leibniz terms that appear when the wavenumber boundaries of a gray gas change in inhomogeneous gas layers [18]. The detailed database for Wj (η) is available from Ref. [13]. When the modeling is finished, we can calculate the spectral intensity (averaged over ith narrow band) Ii,c at s = 0 as L s e− 0 κj (s ) ds Wj κj (s) ds (6) Ii,c (0) = Ibη (s) 0
j
3.2. Inversion procedure We take a few representative nodal temperatures Tj and take a smooth interpolation between them to compute the
intensity numerically. A very general and robust inversion scheme with many nodal temperatures is surely desirable in the long run. This method is a first step to demonstrate the inversion. In the inversion procedure, initial temperatures are updated by the following equation: [T ]new = [T ]old + [δT ]
(7)
The absolute value of the residue Ri defined by Eq. (8) converges to a minimum when the inverted temperatures are optimally approximated to the real ones. Ri =
Ii,c − Ii,m Ii,m
(8)
where Ii,m is the measured intensity of the ith narrow band at s = 0 and Ii,c is calculated from [T ]old . The temperature increase [δT ] at the next step is determined by Eq. (9) using the residue calculated by Eq. (8).
T δT = [A]T [R] A A + (p/q)Q (9) T where the i, j th entry of matrix A is ∂Ri/∂Tj [16]. The right side of Eq. (8) reaches a minimum when the calculated temperature profile approaches the real one. The matrix Q is diagonal and its diagonal elements are the same as those of [AT A]. And q, p and [B] are defined as: 1/2 q= qi2 (10) i
p=
i
1/2 pi2
bi,j = Tj2
(11)
∂ 2 Ri ∂Tj2
(12)
where qi and pi denote the diagonal elements of AT A and B T R, respectively. The right side hand of Eq. (12) is calculated by: +j
bi,j =
−j
0 +I Ii,c − 2Ii,c i,c
0.012Ii,m +j
(13)
−j
0 respectively mean the intensities where Ii,c , Ii,c and Ii,c calculated with 1% increase of Tj , 1% decrease and the unperturbed one. The following equation is the temperature profile between nodes used in this study. n ξ 0 ξ ξc T = Tc − (Tc − T0 ) − 1 , ξ c n (14) 1 + ξ − 2ξc T = Tc − (Tc − TL ) − 1 , ξc ξ 1 1 − ξc
where T0 , Tc and TL mean the temperatures at the exit of quartz tube, at the gas inlet and at the dead end of quartz tube, respectively. The assumed profile has a maximum at the gas inlet point. And ξ is the normalized coordinate (0 at the gas exit and 1 at the dead end). And n is a fixed parameter. The effect of profile parameter n will be investigated also.
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4. Results and discussion Fig. 4 shows a measured intensity distribution from 2045 to 2451 cm−1 . The general radiative characteristics of CO2 generally appear above 2200 cm−1 while some irregular intensities emerge below 2192 cm−1 due to overlap with the CO band and the error of measurement. Also the intensity above 2400 cm−1 reveals the background radiation (0.18 ± 0.02 W·m2 ·cm−1 ·sr) and it is compensated in all the narrow bands equally. This intensity distribution is used in the inversion. In this study, the nodal temperatures are calculated by changing two parameters, i.e., temperatures are obtained with variation of the assumed temperature profile and the employed narrow bands. In addition, temperatures calculated with the CK-based WNB model and those measured with thermocouples are compared with each other. 4.1. Results varying the assumed temperature profile The inversion calculation is performed first by varying the power n in Eq. (14). As shown in Fig. 5, the inversion
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result has good agreement with the real profile regardless of the power n. The fifty-nine narrow bands between 2045 and 2451 cm−1 are all used to determine the profile. The typical calculation time for the inversion is 18 s with a Pentium III (450 MHz) processor for 20 iterations for the temperature update. The inverted temperature with n = 1/2.5 is better than the other cases. And Tc slightly increases as the power becomes lower. Table 2 shows errors of T0 , Tc and TL when varying n. In all situations, T0 , Tc and TL are within errors of 1%, 8% and 14%, respectively. It is recognized that the cold back temperature behind the hot center temperature has large error compared with the low front temperature near the sensor. It is because hot center gas emits strong radiation and at the same time it actively absorbs the feeble emission from the cold gas behind. It is difficult to improve it as long as only one side measurement is performed. Nevertheless, the absolute error is maintained within 50 ◦ C. Fig. 6 shows an intensity distribution as calculated by the inverted temperature profile with n = 1/2.5. The result indicates that the intensities below 2192 cm−1 have large errors whereas those from 2200 to 2395 cm−1 show reasonable agreement with the measurement. The large discrepancy below 2192 cm−1 is considered to be due to overlap with the CO 4.7 µm band. Next, a parabolic profile shown below is applied to the inversion procedure. T = −4(Tc − T0 )ξ 2 + 4(Tc − T0 )ξ + T0 , 0 ξ ξc T = −4(Tc − TL )ξ 2 + 4(Tc − TL )ξ + TL , ξc ξ 1
(15)
Table 2 Errors of T0 , Tc and TL when varying n in Eq. (14)
Fig. 4. Measured intensity distribution from 2045 to 2451 cm−1 .
Fig. 5. Result of inversion calculation using Eq. (14) and the CK-based WNB model along various powers.
Measured temp.
T0 = 608.4
Tc = 1105.7
TL = 355.9
n = 1/1.5 n = 1/2.5 n = 1/3.5
604.7 (0.61%) 609.7 (0.21%) 613.8 (0.90%)
1050.7 (4.97%) 1145.4 (3.59%) 1186.7 (7.33%)
314.7 (11.6%) 310.5 (12.8%) 308.5 (13.3%)
Fig. 6. Intensity calculated from the temperature inverted with the power n = 1/2.5 using the CK-based WNB model.
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Fig. 7. Result of inversion calculation using a different profile, Eq. (15). Fig. 9. Measured intensity and those calculated from the inverted temperature with various employed wavenumber ranges.
Fig. 8. Kernel distribution from the measured temperature distribution between 2045 and 2451 cm−1 .
As shown in Fig. 7, the result is quite different from the real profile. This error is due to the inherent error in the assumed profile shape of the temperature. These results show that the inversion is not very much affected by the detailed profile parameters, however, the overall shape must be properly assumed. Hereafter, n is consistently taken as 1/2.5. 4.2. Results varying the employed narrow bands
− s κj (s ) ds In the inversion procedure, the kernel e 0 Wj κj (s) in Eq. (6) is a weighting factor of the local temperature. It is different from one narrow band to another. Some show greater emphasis on the closer side, while some on the farther side. One should properly select the narrow bands to give well-balanced accuracy in the overall inverted temperature profile. Fig. 8 shows the kernel distribution using the measured temperature profile. As can be seen from this information, narrow bands at 2290 and 2374 cm−1 participate relatively more in the calculation of temperature near the sensor and that at 2255 cm−1 is strongly affected by
Fig. 10. Measured and inverted temperature profiles with various employed wavenumber ranges.
temperatures at all position. And narrow bands from 2192 to 2206 cm−1 are engaged within the central high temperature region. Effects of the other narrow bands are very weak and thus they may not be used as the sensing narrow bands. It is worthy of notice that almost all kernel values are low near the dead end. To make things clear, however, we try three different wavenumber ranges in which all the included narrow bands are employed in the inversion process. Ranges 1, 2 and 3 mean intensities from 2045 to 2451 cm−1 , from 2199 to 2451 cm−1 and from 2199 to 2395 cm−1 , respectively. Intensities calculated from the finally inverted temperature profiles are shown in Fig. 9. Error of intensity is relatively large compared with that of temperature as shown in Fig. 10. This is common to any employed wavenumber range. We can see that the error of temperature is significantly reduced despite the appreciable error in the intensity measurement, as has been confirmed by Hommert et al. [3].
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Table 3 Errors of T0 , Tc and TL with variation of employed wavenumber ranges Measured temp. (K)
T0 = 608.4
Tc = 1105.7
TL = 355.9
Range 1 Range 2 Range 3
609.7 (0.21%) 614.0 (0.93%) 613.3 (0.82%)
1145.4 (3.59%) 1076.1 (2.68%) 1089.0 (1.51%)
310.5 (12.8%) 309.9 (12.9%) 310.1 (12.9%)
Fig. 11. Measured and inverted temperature profiles with LBL method and CK-based WNB model.
As shown in the figure, regardless of the selection of range, all results generally have good agreement with the real profile. Table 3 indicates that T0 , Tc and TL are accurate within the error of 1%, 4% and 13%, respectively. For the case of range 1, errors of Tc are larger compared with the other employed ranges. Nevertheless, it is still small for any employed range on the whole because the kernel is large in any range except under 2122 cm−1 . For T0 , all results are excellent because the kernel is large near T0 in any range (see Fig. 8). The large error of TL can be understood similarly. The only difference between ranges 2 and 3 is the region from 2402 to 2451 cm−1 . The two results are similar to each other because the kernel values in the excluded wavenumbers in the range 3 are very small. The final verification is made regarding the validity of the current CK-based WNB model by comparing the results using the original LBL database, i.e., HITRAN(92) with Scutaru’s hot bands. As shown in Fig. 11, these two inversion results are almost identical to each other, while the computation time for inversion is 20 times longer for the LBL. It shows that the current spectral model is fast for real time applications without significantly sacrificing the accuracy.
5. Conclusion SRS method using CO2 4.3 µm band is applied to determine the line-of-sight temperature distribution in a quartz tube. The inversion procedure is performed with the CKbased WNB model. In the calculation, three nodal temper-
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atures are taken as the unknowns and realistic temperature distribution functions between them are set up. A good assumption of temperature profile improves the inversion accuracy, which calls for a more robust and general inversion scheme in the future. The results of the inversion agree reasonably with the measurement within 50 ◦ C error. For spectral modeling, the CK-based WNB model concurs with the LBL results with very small difference and it is proved suitable for SRS techniques. The error of rear temperature behind a hot spot is significantly large because the hot gas absorbs information from the cold gas behind. This problem can be corrected by measurements on both sides. This paper proves the applicability of SRS for actual industrial applications. Such examples are boilers and reactors where most of the gas volume is occupied by burned products and the walls are cold. For further study, research regarding the effects of wall emission and reflection, particle scattering and hotter gas may be made for more general furnaces. Even further application with proper concentration measurement technique may be made for intense combustors and chemical reactors where not only the temperature but also the concentration varies from one location to another.
Acknowledgements This work has been supported by the Critical Technology 21 Project of the Ministry of Science and Technology, Korea and also by a grant from Korea Science and Engineering Foundation (grant No. KOSEF 995-1000-013-2).
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