This file is raw output from pdftotext and may not be ideal for distribution. If you are a maintainer for Hackipedia, please sit down when you have time and clean this text version up. Source PDF: /mnt/fw-js/docs/Hardware/Electronics/Sensors/Diode-laser absorption sensor for line-of-sight gas temperature distributions.pdf Like all conversions the text below should be fully readable as UTF-8 unicode text. --------------------------------------------------------------- Diode-laser absorption sensor for line-of-sight gas temperature distributions Scott T. Sanders, Jian Wang, Jay B. Jeffries, and Ronald K. Hanson Line-of-sight diode-laser absorption techniques have been extended to enable temperature measure- ments in nonuniform-property flows. The sensing strategy for such flows exploits the broad wavelength- scanning abilities 1.7 nm 30 cm 1 of a vertical cavity surface-emitting laser VCSEL to interrogate multiple absorption transitions along a single line of sight. To demonstrate the strategy, a VCSEL- based sensor for oxygen gas temperature distributions was developed. A VCSEL beam was directed through paths containing atmospheric-pressure air with known and relatively simple temperature distributions in the 200 –700 K range. The VCSEL was scanned over ten transitions in the R branch of the oxygen A band near 760 nm and optionally over six transitions in the P branch. Temperature distribution information can be inferred from these scans because the line strength of each probed transition has a unique temperature dependence; the measurement accuracy and resolution depend on the details of this temperature dependence and on the total number of lines scanned. The performance of the sensing strategy can be optimized and predicted theoretically. Because the sensor exhibits a fast time response 30 ms and can be adapted to probe a variety of species over a range of temperatures and pressures, it shows promise for industrial application. © 2001 Optical Society of America OCIS codes: 300.1030, 300.6260, 280.3420, 140.2020, 120.1740, 120.6780. 1. Introduction flows. The sensing strategy is enabled by recently Nontomographic line-of-sight absorption sensors for developed vertical cavity surface-emitting laser gas temperature and species concentration tradition- VCSEL sources that provide rapid single-mode ally have been limited to flows with near-uniform scans over a broad wavelength range 1.7 nm 30 properties. However, because many flows of practi- cm 1 . Scanning approximately ten times farther cal interest contain strong nonuniformities, several than a traditional Fabry–Perot or distributed feed- researchers have advanced strategies for nonuniform back DFB diode laser, a single VCSEL can probe flows, particularly in the form of corrections for many absorption transitions, yielding expanded tem- boundary layer effects.1,2 More recently, a path- perature information. Although the temperature integrated sensor has been demonstrated for active distributions inferred from this expanded informa- control of a highly nonuniform flow,3,4 a strategy for tion are not fully resolved, they are sufficient for removing sensitivity to flow nonuniformities has many flow monitoring or control applications. Fur- been developed,5 and an indicator for temperature thermore, VCSEL-based temperature distribution nonuniformity along a line of sight has been sensors offer the simplicity, affordability, and time suggested.6 response necessary for such applications. Building on such previous research, in this paper Many practical systems contain significant nonuni- we describe a novel strategy for extending line-of- formities in temperature and species concentration sight temperature measurements to nonuniform along candidate lines-of-sight for optical flow diag- nostics. These nonuniformities are typically caused by heat transfer to surfaces, flow mixing, inhomoge- neous combustion zones, buoyancy, and phase The authors are with the High Temperature Gasdynamics Lab- change. Temperature nonuniformities usually af- oratory, Department of Mechanical Engineering, Stanford Univer- fect system performance, as processes such as chem- sity, Stanford, California 94305-3032. The e-mail address for S. T. Sanders is ssanders@stanford.edu. ical reaction, liquid evaporation, and material failure Received 2 January 2001; revised manuscript received 7 May are all strongly temperature dependent. To opti- 2001. mize a flow for a given purpose, one often has to 0003-6935 01 244404-12$15.00 0 minimize, maximize, or in some way control the tem- © 2001 Optical Society of America perature gradients. 4404 APPLIED OPTICS Vol. 40, No. 24 20 August 2001 The sensor described below interrogates multiple detailed previously.21 In brief, the integrated absor- transitions in the 0 – 0 atmospheric A band of the bance areas a1 and a2 cm 1 of the individual tran- molecular oxygen system O2 b1 g 4 O2 X3 g to sitions, given by infer temperature distributions in the 200 –700 K range; this approach is attractive because of the com- I mercial availability of single-mode VCSELs near 760 ai ln d , (1) nm and the ubiquity of oxygen. The portion of the I0 O2 A band useful for temperature measurements in this range has been well characterized are measured with a tunable, spectrally narrow spectroscopically.7–9 Other transitions in this sys- source. I and I0 represent the transmitted and in- tem could be chosen to tailor future sensors to differ- cident light intensities, respectively; and the integra- ent temperature ranges. tion is over frequency cm 1 , implicitly taken to Although several researchers have developed opti- exclude all but the ith transition. Each area can be cal sensors for temperature based on O2 A-band expressed as transitions,10 –13 and others have used VCSELs to in- terrogate these transitions,14 –16 to the best of our a1 S 1 T PxL, a2 S 2 T PxL, (2) knowledge no previous VCSEL absorption-based temperature sensors have been reported. Oxygen is prevalent in the 200 –700 K range in where T K and x are the absorber temperature and many flows where temperature uniformity is of great mole fraction, respectively. The pressure of the interest. For example, near the fuel injectors in a probed volume P atm and the path length L cm are gas turbine engine, gas temperature and oxygen con- both usually known. The line strengths Si cm 2 centration govern fuel droplet evaporation and fuel– atm 1 are given by air mixture stoichiometry, which in turn affect engine performance. When gas property uniformities are Q T0 T0 hcE i 1 1 Si T Si T0 exp maximized in this region, lower pollutant emissions Q T T k T T0 and higher combustion efficiency can be realized.17,18 1 Hence the sensor described here has potential for hc 0,i hc 0,i 1 exp 1 exp , direct use in active control systems for propulsion kT kT 0 and stationary gas turbines. Future sensors can (3) employ identical techniques by use of different oxy- gen transitions i.e., lines with different lower-state where T0 K is a reference temperature; Q is the energies for different temperature ranges or transi- absorber’s partition function; and 0,i cm 1 and Ei tions of different species to obtain the information cm 1 are the frequency and lower-state energy of required by a specific application. the ith transition, respectively. We begin by providing a theoretical basis for prop- The ratio of the areas a1 a2 thus can be reduced to erty distribution measurements in nonuniform flows. a function only of the absorber temperature T: Two techniques for reducing absorption data to prop- erty distributions, the discretization technique and the distribution fitting technique, are described. a1 S1 T0 R f T After providing this theoretical background, we a2 S2 T0 demonstrate the sensor’s performance when applied to two atmospheric-pressure test cases with con- hc 1 1 exp E1 E2 . (4) trolled nonuniformities: an air path composed of k T T0 two segments, each with a different temperature, and an air path containing a linear temperature distribu- Using the measured areas a1 and a2, researchers tion. Although these test cases are common enough typically solve Eqs. 2 for the two unknowns, T and to represent potential sensor applications, the sens- x, by finding T with Eq. 4 and X with one of Eqs. 2 . ing strategy is not limited to these simple cases. A discussion of the sensor’s capabilities when applied to B. Discretization Technique flows with more complex nonuniformities follows. When the path of interest contains nonuniformities, Finally, discussions of line-shape fitting and absorp- additional lines, each with a unique temperature de- tion line selection, both applicable to nonuniform- pendence, can be probed to determine an approxi- path measurements, are presented. mate temperature distribution along the path. As shown above, two area measurements yielded two 2. Theory of Temperature Distribution Measurements flow parameters, T and x; three area measurements could be used to obtain three flow parameters, for A. Review of Uniform-Path Measurements example, T1, T2, x, and so on. For the general prob- Along a uniform-property path, an absorber’s temper- lem, we can model the nonuniform-property path as ature and mole fraction have traditionally been mea- being composed of n nearly uniform property path sured by means of probing a pair of its absorption segments, each with a temperature Ti , pressure Pi , transitions.19,20 This two-line technique has been mole fraction xi , and path length Li . Denoting the 20 August 2001 Vol. 40, No. 24 APPLIED OPTICS 4405 product of the last three quantities as PxL i , we can generalize Eq. 2 as S1 T1 S1 T2 · · · S1 Tn PxL 1 a1 S2 T1 S2 T2 · · · S2 Tn PxL 2 a2 · · · · · · , · · · · · · · · · · · · Sm T1 Sm T2 · · · Sm Tn PxL n am (5) or Sx a, where m is the number of absorption lines probed in the measurement of the vector of integrated absor- bance areas a. If the possible temperatures T1, T2, . . . , Tn in the path are postulated and the spec- troscopic parameters in Eq. 3 are known, then the matrix S is known and the solution x can be found. Because each element of x must be positive, it is usually best to use a nonnegative least-squares NNLS algorithm to solve Eq. 5 . Typically, the pressure P is uniform and known and therefore can be divided out of the vector x, leaving the solution Fig. 1. Demonstration experiments: sensor applied to a xL 1, xL 2, . . . , xL n T. The product xL is termed uniform-temperature path measurement, b two-temperature the column density; in general, the discretization path measurement employing two lasers, and c path containing a technique determines the column density present in linear temperature distribution. Note that in case c the cell is each specified temperature bin defined by the postu- oriented vertically to stratify the air, providing a stable tempera- lated temperatures T1, T2, . . . , Tn . No informa- ture distribution. tion regarding the way in which the constituent column densities are arranged along the path is ob- tained. mum temperatures, T1 and Tn, are required . Its Depending on simplifications allowed by a specific major drawback is that the discretization may be too application, the column density solution vector can be coarse unless many bins are used, and too many bins reduced to more meaningful information. For exam- can result in an ill-conditioned problem. Therefore ple, if the absorber mole fraction is known to be uni- it is often advantageous for one to assume a general form, the mole fraction can be determined and form for the temperature and mole fraction distribu- divided out, leaving the vector L1, L2, . . . , Ln T. tions in the flow and obtain the best-fit distributions This vector can be divided by the total path length, for a given measurement of the area vector a using a L1 L2 ... Ln, to yield the fraction of gas nonlinear function minimization algorithm. Use of assigned to each temperature bin. Such simplifica- this technique requires that a suitable form for the tions are detailed below in Section 3. distributions can be assumed. Because the assump- The error in the solution vector x is governed by the tion of such a functional form effectively constrains condition number of S, which in turn depends on the the problem, the distribution fitting technique is typ- number of absorption transitions probed m, the spec- ically more stable than the discretization technique. troscopic properties of the absorption transitions The same area data a can be analyzed with either particularly the lower-state energies Ei , and the technique, as is demonstrated in Section 3. The res- number of temperature bins n. For example, if olution and accuracy of either technique increase many temperature bins are used, cond S is large, with the number of lines scanned, especially if lines and therefore the measurement error is relatively with appropriate lower-state energies are chosen see high. The condition number, and its utility for Section 4 . choosing absorption lines for temperature distribu- tion measurements, is detailed below in Subsection 3. Demonstration Experiments 4.C. A. Sensor Description C. Distribution Fitting Technique A temperature distribution sensor was demonstrated The discretization technique has two main advan- for three controlled test cases: a uniform- tages: It is fast because a linear system is solved, temperature path Fig. 1 a , a path containing two and it requires little a priori knowledge of the tem- temperatures Fig. 1 b , and a path with a linear perature distribution in the flow i.e., if a tempera- temperature distribution Fig. 1 c ; in this case the ture bin vector T1, T2, . . . , Tn is chosen with equally cell is oriented vertically to stratify the gas . Type S spaced bins, only the expected minimum and maxi- thermocouples are used in all cases to determine the 4406 APPLIED OPTICS Vol. 40, No. 24 20 August 2001 tized, and the two digital signals are then added to reconstruct the essentially bit-noise-free trans- mitted intensity signal. A sample transmitted sig- nal corresponding to the 292 K data of Fig. 3 b is shown in Fig. 2 a . This signal, corresponding to the transmitted intensity I of Eq. 1 , contains ten absorption lines. A fixed number of data points about each line center is deleted from the transmit- ted signal in Fig. 2 a , leaving the segmented trace labeled baseline fit regions. A third-order polyno- mial is fit to this trace and is called the baseline fit. The baseline fit approximates I0 in Eq. 1 . An initial determination of absorbance, termed quasi- absorbance because the value of I0 is imperfect, is then calculated as ln I I0 ; see Fig. 2 b . A sec- ond polynomial is then fit to the baseline fit regions Fig. 2. Raw data trace highlighting baseline fitting routines. Lines are numbered and can be referenced in Table 1. in Fig. 2 b and subtracted from the quasi- absorbance trace; the result is the true absor- bance, shown in all subsequent spectra in this actual air temperatures. The VCSEL CSEM760, paper containing measured spectra. Each mea- Centre Suisse d’Electronique et de Microtechnique, sured spectrum has a minimum detectable absor- Switzerland is held at a constant case temperature bance of 1 10 4. The known line positions, while triangle current modulation 175 Hz is ap- listed in Table 1, are used to convert the time axis plied. An off-axis parabolic mirror f 1 cm, not to absolute optical frequency. shown collimates the VCSEL light, forming a 3-mm- The data shown in Fig. 2 a are an average of 16 diameter beam. The VCSEL beam is directed consecutive traces, each digitally filtered to 150 kHz through the case-specific atmospheric-pressure air as are the traces in all subsequent figures ; thus the path, and the transmitted intensity is monitored by a time required to record the data for each temperature silicon detector. The VCSEL, air path, and detector distribution measurement presented is 16 1.9 ms are enclosed in a nitrogen-purged environment to re- 30 ms. The minimum detectable absorbance of 1 move room O2 interference. 10 4 is essentially the lower limit for these conditions To reduce bit noise, the amplified function gen- based on the relative intensity noise of the lasers.14 erator signal is subtracted from the detector signal If the technique is applied for stronger absorbers or before digitization on a 12-bit, 5-megasamples s over a longer path length eliminating the need for scope. The function generator signal is also digi- averaging , the temperature distribution measure- Table 1. Spectroscopic Data Used in the Computations a Line Number Positions S 296 105b S 296 105a Ea Assignmenta this paper cm 1 cm atm 1 2 cm 2atm 1 cm 1 NN JJ R Scan 1 13138.19389 20.07 19.95 42.224 R 5 Q6 2 13140.55687 18.33 18.27 81.581 R 7 R7 3 13142.57276 21.99 21.89 79.565 R 7 Q8 4 13144.53073 18.62 18.51 130.44 R 9 R9 5 13146.57050 21.53 21.41 128.40 R 9 Q 10 6 13148.12585 16.96 16.86 190.77 R 11 R 11 7 13150.18729 19.11 19.06 188.71 R 11 Q 12 8 13151.34015 14.13 14.05 262.58 R 13 R 13 9 13153.42196 15.62 15.62 260.50 R 13 Q 14 10 13154.17134 10.80 10.81 345.85 R 15 R 15 P Scan 11 13033.19673 1.895 1.910 791.40 P 23 Q 22 12 13031.39115 1.960 2.004 793.21 P 23 P 23 13 13023.07776 — 1.053 931.74 P 25 Q 24 14 13021.29000 — 1.099 933.53 P 25 P 25 15 13012.58414 — 0.545 1083.4 P 27 Q 26 16 13010.81402 — 0.567 1085.2 P 27 P 27 a From the HITRAN database.9 b From Brown and Plymate.7 20 August 2001 Vol. 40, No. 24 APPLIED OPTICS 4407 Fig. 4. Comparison of absorbance data recorded in a uniform- temperature path 496 K, L 20 cm with data recorded in a Fig. 3. a HITRAN prediction of O2 A-band absorption, showing two-temperature 296 K, L 20 cm; in series with 712 K, L 20 the R-scan and P-scan regions probed. b Data recorded in two cm path, each divided by the total path length. Although 496 K uniform-temperature path experiments P 1 atm, L 20 cm at is nearly the mean of 296 and 712 K, the traces are easily distin- temperatures of 292 and 710 K. guishable, graphically demonstrating the plausibility of the tem- perature distribution sensing strategy. ments can be made very rapidly because both the ness. As shown in Fig. 1 a , the VCSEL beam is VCSEL scan range and the scan rate are greatly directed through a quartz cell 20-cm path length enhanced over DFB lasers.14 We observed scan containing air at a known uniform to within 3% rates of up to 6 cm 1 s with the VCSELs used in temperature. The laser is tuned over the R-scan these demonstrations; this scan rate enables a tem- lines of the O2 A band, and the data recorded in two perature distribution measurement time of 5 s if a different uniform-temperature experiments T 292 30-cm 1 scan range is required. K, 710 K are shown in Fig. 3 b . Figure 4 shows Figure 3 a shows a theoretical O2 A-band spec- data for T 496 K open circles taken in the same trum calculated from HITRAN.9 The two VCSEL configuration, but plotted as absorbance per centime- scan ranges used in these demonstration experi- ter to facilitate a future comparison. We determined ments, labeled R scan and P scan, are highlighted in the areas of each line scanned for the three uniform- the figure. The spectroscopic data for the lines temperature cases T 292 K, 496 K, 710 K using probed by the scans are listed in Table 1, numbered 1–16 for clarity two S 296 data columns are pre- Voigt fits, which yielded the following three a vectors: sented in Table 1; we used the recent data of Brown and Plymate7 for our computations except for mea- 8.70 10 4 3.18 10 4 surements involving the P-scan lines; Ref. 7 data are 4 4 unavailable for P-scan lines, therefore we used the 8.08 10 3.18 10 4 4 HITRAN9 data. The R-scan lines were chosen for 9.67 10 3.85 10 4 4 their spectral proximity, allowing ten lines to be en- 7.99 10 3.59 10 4 4 compassed by a single VCSEL scan. These lines are 9.45 10 3.88 10 often adequate for O2 temperature distribution mea- a292 4 a496 4 7.27 10 3.60 10 surements in the 200 –700 K range. Although sen- 8.35 10 4 4.17 10 4 sitivity would be improved if we expanded the R scan 4 4 6.08 10 3.54 10 to the right on Fig. 3 a , encompassing lines with 4 4 higher E values, the complexities of the bandhead 6.78 10 3.61 10 4 4 discourage this, at least for the present demonstra- 4.48 10 3.02 10 tion experiments. Instead, to improve temperature 1.64 10 4 sensitivity, especially above 500 K, the P-scan lines 4 1.71 10 are added. The P-scan lines have higher E values 4 than the R-scan lines and thus improve the condition 1.89 10 4 number of the line-strength matrix S and the mea- 2.13 10 4 surement uncertainty, as explained in Section 4. 2.30 10 1 a710 4 cm . 2.11 10 4 B. Uniform Temperature Path: R Scan Only 2.20 10 4 2.28 10 Although the temperature distribution sensor is not 4 required for uniform flows, we discuss its application 2.15 10 4 to a uniform-temperature test case here for complete- 2.13 10 4408 APPLIED OPTICS Vol. 40, No. 24 20 August 2001 Fig. 6. Results of uniform-temperature path measurements with the discretization technique. Because none of the temperatures Fig. 5. Boltzmann plots for a uniform-temperature path mea- probed are exactly the bin center temperatures, a compromise surements Tactual 292, 496, and 710 K with R scan only and b between two bins is obtained in all cases. Bars corresponding to two-temperature path measurement Tactual 296, 712 K with the actual conditions would be centered at the Tactual and have a both R and P scans. Temperature labels correspond to best-fit height of 0.209. lines through the measurements. The first component of each vector is the area of line Figure 6 demonstrates that, with only the R scan 1, and the last is the area of line 10. active, the sensor performs best at relatively low tem- Before applying the discretization and distribution peratures. This trend is partly because the condi- fitting techniques to these data, we present a stan- tion number of S increases with temperature see dard technique for determining temperature: the Subsection 4.C and partly because the error in a Boltzmann plot of Fig. 5 a . The best-fit lines increases with temperature as shown in Fig. 5 a . through the data provide the measured temperatures The 292 K result is nearly perfect: The ideal result of 288.1, 478.3, and 699.8 K, which are within 1.3%, would be a bar of height 0.209 at T 292 K, but the 3.6%, and 1.4%, respectively, of the actual gas tem- discretization forces a compromise between the bins peratures. Figure 5 a shows the measurement er- at 200 and 300 K. The results for the higher- ror scatter about the best-fit lines increasing with temperature tests could be improved if we probed temperature, which is due to the reduced absorbance additional lines such as the P-scan lines. as shown in Fig. 3 b . Of course, because the path temperature is known The standard Boltzmann plot is not an appropriate to be uniform in this case, the binwise results of the means for determining temperature in nonuniform discretization technique shown in Fig. 6 are not as flows. Figure 5 b shows a Boltzmann plot for mea- meaningful as a single temperature and mole frac- surements presented below in Fig. 7 from the two- tion. As we illustrate below, the discretization tech- temperature path configuration Fig. 1 b . The nique is most useful for providing a snapshot of a temperature nonuniformity causes significant curva- path’s temperature content, particularly when little ture in the data; the actual cell temperatures are 296 is known about the path a priori. and 712 K, but the best-fit line determines an inter- The distribution fitting technique can be applied to mediate temperature of 403.5 K. Both the discreti- the same three a vectors. For this simple case, the zation and the distribution fitting techniques, when distribution fitting technique amounts to our finding applied to nonuniform-temperature flows, yield re- the single temperature and O2 mole fraction that best sults that exhibit the necessary curvature on the replicate each measured a vector. This problem is Boltzmann plot. much more constrained than the discretization prob- To apply the discretization technique to the above lem, and the results are good for all three tempera- area vectors a, we first generate a 10 lines 7 tures: The measured temperatures are 290, 479, equally spaced T bins matrix S with T1 200 K and and 695 K and are within 0.7%, 3.4%, and 2.4%, T7 800 K using the data of Table 1 Brown and respectively, of the actual values and in good agree- Plymate line-strength data . Using the three mea- ment with the values obtained from Fig. 5 a . The sured a vectors, we then solve Eq. 5 three times for measured mole fractions are 0.212, 0.196, and 0.208 and are within 1.4%, 6.2%, and 0.5%, respectively, of x using a NNLS algorithm. The x vectors are di- the actual value 0.209. vided by the known pressure of 1 atm to obtain the column density in each specified temperature bin; these results are shown in histogram form in Fig. 6. C. Two-Temperature Path: R Scan Only In this simple case, we can also divide by the known In the second configuration Fig. 1 b , the VCSEL path length of 20 cm to obtain the mole fraction of O2 beam is directed through two quartz cells in series in each temperature bin, also shown in Fig. 6. each with a 20-cm path length containing room air 20 August 2001 Vol. 40, No. 24 APPLIED OPTICS 4409 Fig. 8. Results of two-temperature path measurements with the discretization technique. Dashed bars correspond to actual con- Fig. 7. Data recorded in a two-temperature 296, 712 K path ditions and cannot be matched exactly by the sensor because the measurements with both the R-scan and the P-scan VCSELs. bin center temperatures are fixed. at temperatures of 712 and 296 K, respectively. The R-scan lines are probed, and the resulting absorbance additionally probe the P-scan lines the additional data are divided by the total path length of 40 cm to VCSEL beam could be time-division or phase-division obtain the absorbance per centimeter data shown as multiplexed if both beams were required to probe the a solid curve in Fig. 4. The open circles in Fig. 4 are same path . With the cells still maintained at 712 the data recorded in the 496 K uniform-temperature and 296 K, the data shown in Fig. 7 are acquired, path. Because the columns of S are linearly inde- from which the a vector is determined. Because we pendent, the two traces in Fig. 4 are easily distin- changed m by adding the P-scan lines, a new matrix guished, even though 496 K is approximately the S must now be formed: A 16 lines 9 equally mean of 296 and 712 K. Graphically, this is why spaced T bins matrix S with T1 100 K and T7 temperature distribution measurements are possi- 900 K is generated by use of the data of Table 1 ble. The lower the condition number of S, the less HITRAN line-strength data . This time the dis- data such as those compared in Fig. 4 will look alike cretization method yields an acceptable solution vec- and the lower the temperature distribution measure- tor x, shown in Fig. 8. The vertical scale on the right ment errors will be see Subsection 4.C . shows x divided by the known pressure of 1 atm to To measure the temperature distribution in the obtain column density. On the left, x is divided by two-temperature path, we first determine the area the cell path length of 20 cm for comparison with Fig. vector a from the data of Fig. 4 using the hybrid 6. Note that the actual path conditions, plotted for line-shape fitting method described in Section 4 . comparison in Fig. 8, could be obtained by the sensor The discretization and the distribution fitting tech- only if the bin center temperatures were allowed to niques can now be applied. However, with only the vary. R scan active, the results of the discretization tech- Figure 8 highlights the sensor’s potential for use in nique are extremely sensitive to the temperature bin engineering applications: The sensor can provide vector, and therefore they are not reported here. As rapid temperature distribution information every 30 demonstrated in Subsection 3.D, the addition of the ms in this case of the type shown. Whereas previ- P-scan lines improves the condition of S and enables ous sensors that use Fabry–Perot or DFB lasers have stable discretization technique results for this two- been limited to providing a single temperature and temperature path. Because it is able to impose more mole fraction along a line of sight, the VCSELs used constraints, the distribution fitting technique outper- by this sensor effectively allow a range of tempera- forms the discretization technique when only the R tures and mole fractions to be measured. To illus- scan is active. Specifying that the 40-cm path is trate the utility of such expanded information, let us really composed of two uniform-property paths, each consider applying the sensor to monitor flow proper- 20 cm long, and x is uniform throughout, we can use ties near the fuel injectors in a gas turbine engine. the distribution fitting technique to determine which Typically, maximum uniformity i.e., all the O2 in a two temperatures and single mole fraction three pa- single temperature bin rather than distributed as in rameters best reproduce the measured data. The Fig. 8 is desired in this region.17,18 If the major results are T1 293 K, T2 768 K, and xO2 0.209, temperature components in the flow are measured, which are within 1.4%, 7.8%, and 0.0%, respectively, the sensor could reveal unwanted hot spots, cold of the actual values. spots, and exceptionally broad temperature distribu- tions. In general, the sensor’s ability to rapidly re- D. Two-Temperature Path: R Scan Plus P Scan port data of the type shown in Fig. 8 should be useful To enhance the results of both techniques, we add a for controlling any system that is affected by signifi- second VCSEL shown in Fig. 1 b to the setup and cant changes in its temperature distributions. 4410 APPLIED OPTICS Vol. 40, No. 24 20 August 2001 Fig. 9. Data recorded in a linear temperature distribution mea- surement. Fig. 10. a Discretization technique and b distribution fitting technique results from a linear temperature distribution measure- ment. The measured linear temperature distribution b is every- where within 5% of the actual linear temperature distribution. With both the R-scan and P-scan VCSELs active, we can apply the distribution fitting technique again using known system information. This time we specify only that the measured system be composed of bottom end of the copper tube is cooled by dry ice, and two uniform-property paths, each 20 cm long, which the top end is heated by a furnace. relaxes a constraint from the three-parameter fit per- A sample absorbance trace, obtained by probing formed above because it allows nonuniform mole frac- the R-scan absorption lines, is shown in Fig. 9. Be- tion x. The two temperatures and two mole cause of the slightly longer path length and the cooler fractions four parameters that best fit the data are temperature portion, the peak absorbances are determined to be T1 284 K, T2 687 K, xO2,1 greater than in previous figures, resulting in a higher 0.188, and xO2,2 0.225. These best-fit tempera- signal-to-noise ratio. The a vector obtained from tures and mole fractions are within 4% and 10%, Fig. 9 is therefore very close within 1.4% to the respectively, of the actual values. It is interesting to theoretical a vector for this case. Therefore, even note that, if we use the traditional two-line technique though only the R-scan lines are probed, the discreti- employing lines 2 and 12 as a representative test zation technique yields good results if relatively few case to measure the temperature and mole fraction temperature bins are used. After generating a 10 along this path, we find T 380 K and xO2 0.189. lines 4 equally spaced T bins matrix S with T1 We successfully conditioned the line-strength ma- 100 K and T4 700 K using the data of Table 1 trix S to higher temperatures by adding the P-scan Brown and Plymate line-strength data , we obtained lines, improving the results of both data reduction the solution vector x using a NNLS algorithm. We techniques. Attempts to add even higher-energy found the column density, shown in Fig. 10 a right P-branch lines better conditioning S for combustion vertical axis , by dividing x by the known pressure of temperatures, for example will meet two challenges. 1 atm. To determine the total O2 mole fraction, the First, the line strengths will be lower than those used vector x is divided by the known pressure of 1 atm in the present experiments. Lower line strengths and the known total path length of 50 cm, and the could demand use of more sensitive wavelength mod- result is termed x . The sum of the elements of x is ulation spectroscopy techniques,14 depending on the the measured value of xO2, 0.226 in this case. The available path length. Second, room-temperature vector x , formed by dividing x by xO2, represents the line strengths Si 296 will be less well known; the fraction of absorber in each temperature bin; x is precise data of Brown and Plymate terminate at the plotted in Fig. 10 a on the left vertical axis. The P23–P23 line as shown in Table 1, and the HITRAN measured data shown in Fig. 10 a are compared with data become increasingly suspect above this point. the actual binwise discretization of the known linear Further fundamental studies of the room- temperature distribution. Because of the reduced temperature line strengths Si 296 of the higher- number of bins employed, the discretization is rela- energy lines may therefore be required. tively coarse, but the agreement between the mea- sured and the actual discretizations demonstrates E. Linear Temperature Distribution: R Scan Only the sensor’s fidelity. Even when coarsely dis- In the third configuration Fig. 1 c , the VCSEL cretized, the sensor is potentially useful for applica- beam is directed through a 50-cm-long cell containing tions requiring a broad sense of the temperature atmospheric-pressure dry air Praxair extra-dry nonuniformity. grade with a linear temperature distribution rang- When the distribution fitting technique is used to ing from 230 to 620 K. The cell is housed in a ver- interpret the same a vector, the best-fit uniform- tical copper tube that ensures this distribution: The mole-fraction linear temperature distribution is 20 August 2001 Vol. 40, No. 24 APPLIED OPTICS 4411 found to have Tlo 235 K, Thi 593 K, and xO2 0.227. This result is plotted in Fig. 10 b . The measured temperature distribution is everywhere within 5% of the actual thermocouple-verified dis- tribution. By imposing the known form for the tem- perature distribution, we made the same data that generated the coarse discretization of Fig. 10 a to generate the more meaningful results of Fig. 10 b . The above experiments demonstrate that, in flows containing simple nonuniformities, the distribution fitting technique is more robust than the discretiza- tion technique because it incorporates additional user-supplied information. The distribution fit- ting technique is especially useful whenever a simple and accurate functional form for the property distri- butions in the flow is known or can be assumed. Fig. 11. Hybrid line-shape fitting method used to determine the When such a form is not available i.e., in the case of area of line shapes recorded in nonuniform-temperature paths. unknown or complex flow properties , the discretiza- tion technique may be favorable; this is especially true if approximate information is sufficient i.e., if one is looking only for the appearance of a cold zone to combine the techniques of this paper with existing in a hot system or if computational cost must be tomographic techniques.22 minimized i.e., for rapid control applications . 4. Discussion B. Hybrid Line-Shape Fitting for Nonuniform-Temperature Paths A. Complex Nonuniformities in Temperature For the uniform-temperature path measurements de- and Mole Fraction scribed in Subsection 3.B, the determination of the a The demonstration experiments described above had vector from the data i.e., from Fig. 2 was straight- uniform mole fraction x. However, in Subsection forward. With isolated lines and a single tempera- 3.D, the distribution fitting problem was easily cast to ture, standard baseline and Voigt fitting routines allow mole fraction nonuniformity. The distribution provided accurate integrated absorbance areas ai . fitting technique is especially useful whenever coin- It is important to note that no line-shape broadening cident mole fraction and temperature nonuniformi- information was input to the fitting routine used to ties are expected along a path of interest. The determine the areas ai ; this is a great advantage discretization technique can still be applied, and, as because the broadening parameter 2 T, P, xj is usu- always, it provides the column density present in ally unknown. each temperature bin. However, the column density Complications arise when a must be determined in information becomes less meaningful as the mole flows with nonuniform temperatures. To our knowl- fraction nonuniformity becomes more severe i.e., a edge, few if any previous publications have discussed hot spot containing a low mole fraction of the ab- these complications or attempted to determine accu- sorber may go undetected because its contribution to rate line-shape areas in highly nonuniform flows. column density is small . The distribution fitting Here we discuss the complications, including the ad- technique can provide more meaningful results, es- ditional challenges caused by high pressures, and pecially if the local mole fraction can be expressed as offer a hybrid line-shape fitting method as a solution a single-valued function of local flow temperature. to determining a in nonuniform flows. For example, in a reacting flow with heat transfer to Consider first the two-temperature case of Fig. walls, one might specify a parabolic distribution for 1 b . If the two temperatures are considerably dif- temperature and also prescribe xabsorber c f T ferent, each line shape recorded will be poorly mod- everywhere along the probe path, where c is a pa- eled by a single Voigt profile. The proper profile rameter determined in the fit and f might return the would be a sum of two Voigt functions with different equilibrium mole fraction of the probed species at collisional widths, different Doppler widths, and dif- temperature T. Similar prescriptions could be made ferent line-center positions. Such a fit could be per- in cases in which the probed path interrogates the formed, but if we move to a continuously varying mixing of two flows. If the local temperature and temperature distribution such as the linear distribu- mole fraction are statistically dependent, and if in- tion of Fig. 1 c , this approach becomes unreliable telligent application-specific distribution function- unless 2 T, P, xj is known. Even if 2 T, P, xj is alities can be assumed, the distribution fitting known, the approach is computationally costly. problem can provide meaningful results even for Rather than attempting such complicated fitting pro- flows with somewhat complex nonuniformities. In cedures, we could numerically integrate the data as flows with exceptionally complex nonuniformities, or suggested by Eq. 1 . This approach, however, gen- when spatial information is required, it may be useful erally provides systematically low estimates of ai be- 4412 APPLIED OPTICS Vol. 40, No. 24 20 August 2001 cause the line-shape area contained in the far wings, where the signal-to-noise ratio is 1, is sacrificed. To solve this problem, we adopted the hybrid line- shape fitting method depicted in Fig. 11. The fea- ture shown in Fig. 11 is a closeup of the R13–R13 transition from the linear temperature distribution data shown in Fig. 9. The hybrid fitting method is performed as follows. The central 0.3 cm 1 of the data is temporarily ignored, and a Voigt fit is applied to the remaining wings, fixing the Doppler width at the appropriate value for the application-specific lowest expected temperature 200 K in this case . This resulting fit area is called aVoigt. The fit curve is then subtracted from the original data, yielding the residual. The residual is integrated numerically over the central 0.3 cm 1, and the result aresidual is added to aVoigt to obtain the desired line-shape area atotal. This hybrid method works because the fit to the Voigt wings maintains the baseline accuracy sac- Fig. 12. Condition number of the line-strength matrix S gener- rificed by numerical integration whereas the numer- ated from the R-scan transitions solid curve and from the R-scan ical integration handles the complicated behavior and P-scan transitions dotted curve , indicating that the addition near the line center where the signal-to-noise ratio is of the P-scan lines reduces the measurement error at high tem- peratures. The componentwise condition number estimate high. The method is used to find all areas ai in the dashed curve demonstrates that the measurement error can be nonuniform-temperature experiments described in bin specific. Such information is useful for the selection of the this paper. The method was found to perform well optimum absorption lines for a given measurement. and remains independent of a priori knowledge of 2 T, P, xj . Depending on the line-shape broaden- ing in a specific application, a Lorentzian or Doppler Typically, Eq. 5 should be cast as a linear least- fit can in many cases be substituted for the Voigt fit squares problem for x when we choose m n. The to enhance the speed of the hybrid fitting method. solution to Eq. 5 is then x S a, where S is the The hybrid fitting method can be used in nonuni- Moore–Penrose pseudoinverse of S. The sensitivity form flows to arbitrarily high pressures, as long as of the solution x to the uncertainties in S and a is the baseline fitting technique shown in Fig. 2 is not characterized by the condition number of S: compromised. For O2 A-band transitions in the P branch, this condition limits the technique to approx- max S imately 10 atm at room temperature and 30 – 40 atm cond S S 2 S 2 max S max S , min S at combustion temperatures. However, when spec- trally dense features common in polyatomic species (6) or near the bandhead in the A-band R branch of O2 where max S and min S are the maximum and must be probed, blended transitions often disable the minimum singular values of S. The relative error in baseline fitting near 1 atm. Such blended spectra the solution vector x, x 2 x 2 x xmeasured can be managed by a recently developed quasi- xactual , is bounded according to absorbance fitting procedure14 if the temperature is roughly uniform, but this procedure fails when the x 2 a 2 temperature is highly nonuniform. There is cur- cond S , (7) x 2 a 2 rently no simple scheme available to determine the ai in blended spectra when nonuniform temperatures so that the maximum relative error in x is simply the are present. However, if the broadening parameters relative error in the measured a vector, amplified by 2 T, P, xj are known, the problem is tractable, and the condition number of S. Hence absorption lines not only the ai but also the line-shape information should be selected to minimize cond S for a given measured can in principle be used to determine flow sensor application. properties. Figure 12 shows condition number calculations rel- evant to the measurements described in this paper. C. Condition Number and Absorption Line Selection Using a temperature bin vector with n 3 equally The framework of the discretization technique pro- spaced bins, we show cond S versus T1, with T3 set vides a basis for choosing the optimum lines for a equal to 4 T1, for the R-scan matrix S solid curve given temperature distribution measurement; line and the R-scan and P-scan matrix S dotted curve . selection can be based on the condition number of the In addition to this standard condition number line-strength matrix S. Although the line selection cond S , results from a statistical condition number strategy is derived from the discretization technique, estimate23 are shown for the R-scan matrix S dashed the strategy will generally optimize the performance curve . The statistical condition number provides of the distribution fitting technique as well. the relative sensitivity of individual components of x 20 August 2001 Vol. 40, No. 24 APPLIED OPTICS 4413 to the problem; the curve shown in Fig. 12 is for the mended to optimize and simulate sensor first component of x, x 1 and thus represents the performance. First, optimize the absorption line se- sensitivity for the first temperature bin only. The lection, guided first by Eq. 8 and then by the condi- componentwise and standard condition numbers dif- tion number of S as described above. With the fer, indicating that the elements of x have different optimized matrix S and an expected mole fraction sensitivities and that a single condition number is vector xexact, calculate a by use of Eq. 5 ; this can be somewhat inappropriate. called aexact. Generate realistic vectors ameasured by It is clear from Fig. 12 that the R-scan matrix S is the addition of random noise to aexact 1–10% peak- best suited to low temperatures for two reasons. to-peak noise is reasonable, depending on the appli- First, all curves have minima at very low tempera- cation , and solve for simulated vectors xmeasured by tures T1 30 K; liquefaction is ignored in Fig. 12 . use of a NNLS or nonlinear function minimization Second, the componentwise condition number for x 1 algorithm, depending on the technique chosen. Fi- is much smaller than the standard condition number nally, compare xmeasured to xexact to estimate the ex- cond S . This implies that the measurement error pected accuracy of the sensor. For all the in the low-temperature bins will be less than the measurements presented above, the sensor perfor- error in the higher-temperature bins, which is exactly mance predicted by such simulations agreed well what we observed, for example, in the measurement with the final results. results shown in Fig. 6. That the R-scan matrix S is conditioned for low 5. Conclusions temperatures is consistent with a result from the A simple absorption sensor for line-of-sight temper- two-line technique21: The peak temperature sensi- ature distribution measurements, enabled by re- tivity for a specific line pair occurs at cently developed VCSEL sources, has been demonstrated in flows containing controlled nonuni- hc formities in the 200 –700 K range. The current re- T E1 E2 0.72 E . (8) alization of the sensor utilizes O2 A-band transitions, 2k but future sensors of this type may be developed for For these ten lines see Table 1 , the maximum E is other species and temperature ranges. 304 cm 1, which means that each possible pair of Widely tunable VCSELs are responsible for the lines made from these ten, taken independently, will relative simplicity of the sensor. Each VCSEL used have a peak sensitivity at a temperature 0.72 in the sensor effectively replaces approximately ten 304 218 K. When the P-scan lines are included, multiplexed Fabry–Perot or DFB lasers; thus a tra- Eq. 8 predicts better sensor performance at higher ditional diode-laser sensor, retrofitted to use temperatures: The maximum E becomes 1043 VCSELs, can realize a tenfold increase in the mea- cm 1 and therefore each pair of lines made from the sured flow information. The increased information 16 total lines will have a peak sensitivity at a tem- has been used, in this case, to provide enhanced tem- perature 751 K instead of 218 K. Indeed, as perature measurements along a single line of sight. shown in Fig. 12, S becomes relatively well condi- Two techniques for reducing the increased infor- tioned at higher temperatures when the P-scan lines mation to temperature distributions have been devel- are included. This is why the P-scan lines were oped: the discretization technique and the needed to achieve the results of Fig. 8. distribution fitting technique. The discretization The results of Fig. 12 correspond to n 3 bins; technique is faster because it is based on a linear similar calculations show that, with each bin added, system. Also, it is applicable to flows with arbitrary the curves shown in Fig. 12 are shifted upward ap- nonuniformities: Only the expected minimum and proximately 1 order of magnitude. This dramatic maximum temperatures are required to find the dis- increase confirms the expected result, namely, that tribution of absorber column density versus temper- one should choose the smallest number of bins allow- ature. For simple flows, the results of column able for a specific sensor application. Choosing a density versus temperature can be reduced to more small number of bins, for example, enabled us to meaningful information. The distribution fitting obtain the results of Fig. 10 a even though we used technique is generally more accurate and robust. only the R-scan lines. However, it is applicable only to flows that have tem- The absolute value of the Fig. 12 condition num- perature and absorber mole fraction distributions bers overestimates the absolute error realized when a that can be accurately modeled by a general func- NNLS solver is used to solve Sx a; because the tional form. Although the discretization technique NNLS solver constrains the solution, excluding neg- is well suited to arbitrary flows and the distribution ative values in the solution vector x, the errors are fitting technique is most appropriate for well- reduced. The trends, however, remain as predicted, understood flows, the best technique depends and the condition number remains an appropriate strongly on the details of the specific sensor applica- figure of merit to optimize sensor performance. tion. Line selection based on the condition number has D. Simulating Sensor Performance been shown to be useful to optimize sensor perfor- When either the discretization or the distribution fit- mance for a given species and temperature range. ting technique is used, the following steps are recom- Both data reduction techniques can be optimized 4414 APPLIED OPTICS Vol. 40, No. 24 20 August 2001 with the same line selection strategy. The accuracy Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, of either technique always increases with the number J.-Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. of absorption lines probed, but increases most dra- Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular matically with the addition of lines having properly spectroscopic database and HAWKS HITRAN ATMO- chosen lower-state energies. SPHERIC WORKSTATION : 1996 edition,” J. Quant. Spec- The sensing strategy has been demonstrated for trosc. Radiat. Transfer 60, 665–710 1998 . two atmospheric-pressure test cases with controlled 10. L. C. Philippe and R. K. Hanson, “Laser-diode wavelength- nonuniformities: a two-temperature path and a modulation spectroscopy for simultaneous measurement of path containing a linear temperature distribution. temperature, pressure, and velocity in shock-heated oxygen The successful results obtained in the sensor demon- flows,” Appl. 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