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Abstract
This manuscript presents the design and initial application of a mid-infrared laser-absorption-imaging (LAI) technique for two-dimensional (2D) measurements and tomographic reconstruction of gas temperature and CO in laminar flames. In this technique, the output beam from a quantum-cascade laser (QCL) is expanded, passed through the test gas, and imaged in 2D using a high-speed mid-infrared camera. The wavelength of the QCL is scanned across the P(0,20) and P(1,14) transitions of CO near 4.8 μm at 50 Hz to provide 2D measurements of path-integrated gas temperature and CO column density across over 3,300 lines-of-sight simultaneously. This enabled the first sub-second (0.1 s), high-resolution (140 μm), 2D laser-absorption measurements and tomographic reconstruction of flame temperature and CO mole fraction using mid-infrared wavelengths. Prior to entering the test gas, the beam was reflected off two diffusers spinning at 90,000 RPM (≈9400 rad/s) to break the laser coherence and prevent diffraction-induced image artifacts. This technique was validated with measurements of CO in an isothermal jet and then demonstrated in laminar, partially premixed, oxygen-ethylene flames despite large background emission from soot and combustion products.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser-absorption spectroscopy (LAS) is widely used to provide quantitative, calibration-free measurements of chemical species and thermodynamic conditions (e.g., temperature, pressure, velocity) in a wide range of combustion applications [1]. Typically in LAS, wavelength-tunable laser light with intensity Io is directed through an absorbing gas and the transmitted light intensity (It) is measured by a photodetector. Beer’s Law is used to calculate the spectral absorbance (α = −ln(It/Io)) which is then compared with that predicted by spectroscopic models to provide path-integrated measurements of gas properties along a line-of-sight (LOS).
While useful, many combustion applications demand multi-dimensional measurements of chemical species and thermodynamic conditions, for example, to resolve the thermochemical structure of flames and/or, more generally, the spatiotemporal evolution of combustion. To meet this need, LAS diagnostics employing multiple LOS [2–8] or mechanical scanning of the LOS [9–12] have been used with and without tomographic reconstruction algorithms to quantify how gas properties vary in space and time. In the simplest form, this approach is performed to provide spatially resolved, path-integrated measurements of species and thermodynamic conditions. If absorption transitions with appropriate lower-state energy are used, path-averaged properties (e.g., absorbing-species column density, absorbing-species-weighted path-averaged temperature) can be obtained despite highly non-uniform conditions along the LOS [13]. For example, Goldenstein et al. [9] and Spearrin et al. [10] developed mid-infrared LAS sensors for absorbing-species-weighted path-averaged temperature and absorbing-species column density of H2O, CO, and CO2 in a model-scramjet combustor. The authors’ mechanically scanned the LOS in two dimensions (x–y) to map out path-averaged gas properties. While effective and convenient to execute in environments with somewhat limited optical access, this approach can be time consuming (often requiring minutes per 2D dataset), provides modest spatial resolution (order 1–10 mm) and is not spatially resolved along the LOS.
Alternatively, numerous researchers have developed laser-absorption tomography (LAT) techniques to provide tomographic reconstruction of 2D temperature and species fields within the measurement plane [5, 7, 11, 12, 14–19]. This approach provides the added benefit of resolving the gas conditions along the line-of-sight; however, it comes at the expense of more complicated data processing routines and steep demands for many lines-of-sight (typically 10s to 1000s of LOS depending on the level of spatial resolution desired [14]). For these reasons, LAT techniques developed to date have primarily used near-infrared wavelengths to exploit relatively inexpensive telecommunication-grade fiber optics (e.g., fibers, multiplexers, splitters) and detectors [5, 7, 11, 12, 14, 15]. Unfortunately, confinement to the near infrared significantly limits the number of chemical species that can be monitored at combustion-relevant concentrations, conditions, and spatial scales. As such, the majority of prior LAT work in combustion applications has focused on detecting H2O via its combination and overtone bands near 1.4 μm [5, 8, 11, 12]. Additionally, some work has been done to tomographically image hydrocarbons via their overtone C–H stretch absorption band near 1.7 μm [7] and NH3 via its overtone and combination bands near 1.5 μm [20]. To overcome this limitation several researchers have developed LAT techniques employing mid-infrared wavelengths for tomographic imaging of temperature, CO, and CO2 [16–19], however these works have relied on mechanical scanning of the line-of-sight, thereby significantly limiting the temporal and spatial resolutions to the order of seconds to minutes and mm, respectively.
That being said, there remains a critical need to develop mid-infrared LAS and LAT techniques with higher temporal resolution and spatial resolution (e.g., less than 1 second and 1 mm, respectively). Recent work by Wei et al. [21] demonstrated significant progress towards achieving this goal through the use of laser-absorption imaging (LAI) with a high-speed mid-IR camera. The authors demonstrated the ability to acquire LAS measurements of temperature, CO, CO2, and C2H6 along ≈400 lines-of-sight (in a single plane) simultaneously to provide 1D measurements and reconstructions of gas temperature and species concentrations. Diffraction-induced noise and image artifacts prevented use of the entire focal plane array and, thus, mechanical scanning of the measurement plane (in the axial direction) was required to achieve 2D measurements and reconstructions of flame structure. Using this technique the authors’ were able to acquire LAS measurements with a spatial resolution of near 50 and 125 μm in the horizontal and vertical directions respectively, and a 2D image of flame properties was acquired in < 10 seconds [21]. As a result, this approach provided ≈ 10× smaller spatial resolution and ≈ 100× better temporal resolution compared to more conventional LAT techniques that rely on translating the LOS in both the x- and y-direction.
Here we present the development and application of the first mid-infrared laser-absorption imaging technique capable of providing 2D-measurements and -tomographic reconstruction of flame temperature and CO concentration without mechanical scanning of the measurement plane. This was achieved by performing high-speed, 2D imaging of scanned-wavelength, mid-infrared laser light that was reflected off two ground-glass diffusers spinning at 90,000 RPM prior to passing through the flame. Spinning the diffusers was required to break the optical coherence and, thus, prevent diffraction-induced image artifacts. This approach enabled the following advancements in mid-IR LAI: 1) a 8× increase in the number of simultaneous measurement paths (≈3,300 LOS here compared to 400 LOS in [21]) and 2) a ≈20× improvement in 2D-measurement time (0.1 seconds here compared to ≈2 seconds in [21]) by avoiding the need to mechanically scan the measurement plane. We demonstrate that this approach enables 2D measurements of thermochemical flame structure on near-cm scales with a projected pixel size of ≈ 140 μm and a time resolution as small as 0.1 seconds (i.e., 10 Hz). The remainder of this manuscript is primarily devoted to presenting the design and evaluation of the optical setup and describing how this LAI technique was applied to provide 2D-measurements and -tomographic reconstruction of flame temperature and carbon monoxide concentration in flames.
2. Technical approach and equipment
Figure 1 illustrates the experimental setup used for LAI of temperature and CO in partially premixed, oxygen-ethylene flames. A continuous-wave (CW) quantum-cascade laser (QCL) providing 30 mW of optical power at wavelengths near 4.8 μm was used to provide 2D measurements of absorbance spectra. The wavelength of the laser was scanned across the P(0,20) and P(1,14) absorption transitions of CO near 2059.9 and 2060.3 cm−1, respectively, at 50 Hz via injection-current tuning with a 700 mV peak-to-peak triangle wave. The QCL’s current was scanned below its threshold current to enable the background and flame emission to be measured in each pixel immediately before each scan. The wavelength scanning of the QCL was characterized using a solid germanium etalon with a free-spectral range near 0.0163 cm−1. It is worth noting that the absorption transitions employed here have recently been used to measure gas temperature and CO in several other combustion environments (e.g., scramjet combustor [10], pulse-detonation engine [3], and propellant flames [22]). We refer the reader to [10] for additional details regarding why these absorption transitions are well suited for studying combustion gases.
Fig. 1 (a) Optical setup used to break the laser light’s coherence and provide diffraction-free LAI, (b) coflow burner, and (c) HDR image of partially premixed (ϕ = 6.43) oxygen-ethylene flame studied with LAI. The red dotted line indicates the approximate field of view of the LAI diagnostic.
A custom built optical assembly (see Fig. 1(a)) was used to expand the laser beam and break its coherence to prevent diffraction-induced image artifacts (see Section 3) that formed by passing coherent laser light through apertures in the beam-shaping optics and camera. The laser beam was immediately expanded by a 12.7 mm diameter, AR-coated, CaF2, concave lens with a focal length of −18 mm. The expanded beam was then directed to a gold-coated, ground-glass diffuser (ThorLabs DG10-120-M01) with 120 grits/inch (≈ 50 grit/cm). The gold coating provides > 97% reflectance at an angle-of-incidence (AOI) of 12° for wavelengths near 4.8 μm. The diffuse reflection was then collimated by a 25.4 mm diameter, AR-coated, silicon lens with a focal length of 25.4 mm and the beam was then focused onto a second diffuser using a 25.4 mm diameter, AR-coated, CaF2 lens with a focal length of 40 mm. The AOI for both of these diffusers was approximately 20°. When aligning the diffusers, the beam was directed to the outer edge of each diffuser to concentrate the beam onto regions of each diffuser with the highest velocity (while spinning), thereby providing more rapid scrambling of the laser light’s coherence. After the second diffuser, the laser light was partially collimated by a 50.8 mm diameter, concave mirror with a focal length of 100 mm. Close to the viewing plane, the beam was passed through a final 25.4 mm diameter lens (CaF2, focal length of 200 mm) to collimate a portion of the beam which was directed through the flame and into the high-speed IR camera.
Both of the diffusers were mounted in 25.4 mm diameter lens tubes which were mounted to a custom aluminum adapter (using two set screws) in order to attach each diffuser to a motor (Castle Creations 1406 Sensored 4-Pole Brushless Motors, 7700kV). The motors were powered by Mamba X sensored motor controllers. The motors for the first and second diffusers were supplied with 26 V DC and 5 and 7 A of current, respectively. With the diffusers mounted to the motors, both motors were capable of spinning at 90,000 RPM (≈9400 rad/s). Due to the high angular velocity, care was taken to balance the diffuser assembly during machining and shim stock was used to properly seat the diffuser in the diffuser holder. Failure to properly balance the diffusers can cause extreme mechanical vibration which can misalign the optical setup or, at best, introduce unwanted noise and oscillations in the light intensity imaged by a given pixel.
The laser light was imaged by a Telops Fast-IR 2K high-speed infrared camera which employs a cryogenically cooled InSb focal-plane array with 320 × 256 pixels. At full resolution, the camera is capable of operating at nearly 2000 frames-per-second (FPS). However, during all LAI experiments the field of view was reduced to 64 × 52 pixels to enable frame rates of 24 kFPS to be used. This frame rate was required to sufficiently resolve the absorbance spectrum measured in each pixel when employing the scan rates and amplitudes used here. A bandpass filter centered near 2060 cm−1 with a full-width at half-maximum of 40 cm−1 was used to prevent pixel saturation from background flame emission which was pronounced in the high-temperature and sooting flame studied here.
Flames were produced by a custom-made coflow burner (see Fig. 1(b)). The outer body of the burner is made from a stainless-steel pipe (330 mm long, 19 mm outer diameter) with weld neck flanges on each end. A smaller tube (3.2 mm (1/8″) outer diameter and 2.16 mm (0.085″) inner diameter) concentric with the outer body of the burner carried a mixture of oxygen and ethylene to the flame. Streams of ethylene (0.045 L/min) and oxygen (0.072 L/in) were mixed ≈150 diameters upstream of the burner exit via a mixing T-junction that was connected to 1/4″ diameter gas lines made of 316 stainless steel. A coflow of air entered the burner orthogonally through the main outer body 51 mm above the bottom flange. The air flowed through the annulus between the body wall and central tube and passed through a 6-cm deep bed of 3-mm diameter glass beads located 72 mm downstream of the coflow inlet. The coflow then passed through a 12.7 mm thick honeycomb flow straightener (0.88 mm cell size) at the exit plane to reduce the turbulence level and lateral velocity components. The exit of the fuel supply line is flush with the exit of the honeycomb flow straightener. To mitigate risk of flashback, the burner was first lit without coflow and without premixing to establish a pure-ethylene-air diffusion flame. Once the diffusion flame was established the flow rate of oxygen was gradually increased until the desired equivalence ratio was reached. The C2H4-O2 jet exit velocity and jet Reynolds number were 0.8 m/s and 61.0, respectively. The coflow velocity was adjusted until the flame appeared steady, resulting in an approximate exit velocity and Reynolds number of 0.5 m/s and 331.6, respectively.
Figure 1(c) shows the partially premixed (ϕ=6.43) oxygen-ethylene flame studied here using LAI. The image shown in Fig. 1(c) was acquired using high dynamic range (HDR) imaging to highlight flame structure and avoid camera saturation. This was required due to the extreme luminosity (too bright to view without eye protection) of this flame. The HDR images were acquired using a Nikon D3200 camera with an AF-S Nikkor 18–55 mm lens. For each HDR image, 4 photos were taken with varying shutter speeds and then combined with Luminance HDR software to create the final image. The shutter speed for each image was chosen to be optimal for a unique region of the flame.
3. Motivation for optical design
Experiments were conducted without diffusers to illustrate the challenges associated with performing mid-IR LAI with coherent laser light. Without diffusers, pronounced Airy-disk patterns were observed in images of the QCL’s laser beam (see Fig. 2(a)). These and other diffraction-induced patterns were altered by the presence of the flame and varied in time during scanned-wavelength experiments. For example, Fig. 2(b) illustrates the signal time history measured in a single pixel while scanning the wavelength and intensity of the QCL. Pronounced oscillations in signal intensity (analogous to optical fringes formed by conventional etalons) were found in pixels located in the vicinity of diffraction patterns. These oscillations severely complicate in situ determination of the non-absorbing light intensity (Io). This leads to spatially dependent errors in the best-fit spectroscopic parameters (e.g., integrated absorbance) obtained from the spectral-fitting routine, which leads to spatially correlated errors in the measured gas conditions. For example, Fig. 2(c) shows a single image of the path-integrated temperature field (calculated from the two-color ratio of integrated absorbances) that is severely compromised by a clear Airy-disk pattern in the middle of the flame.
Fig. 2 (a) Image of coherent laser light near 4.8 μm with Airy-disks. (b) Single-pixel time-history illustrating how diffraction-induced oscillations vary in time and wavelength space. (c) Image of oxygen-ethylene-flame temperature acquired using 2D LAI with coherent laser light which illustrates how diffraction patterns can compromise image quality.
One approach to avoiding these diffraction-induced image artifacts is to destroy the coherence in the laser light by reflecting the beam off spinning diffusers. This technique has been used in a variety of low-speed visible-imaging applications employing laser backlighting [23–25], most similarly, to perform low-speed (<30 frames per second) imaging of soot-volume-fraction in flames via 2D imaging of visible (632.8 nm) laser extinction [23]. However, implementing this strategy into a high-speed scanned-wavelength laser-absorption-spectroscopy technique has, to our knowledge, never been done previously and this presented several challenges. Most notably, high angular velocities (approaching 100,000 RPM) were needed to scramble the speckle pattern created by the diffusers (see Figs. 3(a) and 3(b)) and sufficiently homogenize the local light field on the timescale of the short camera integration times required here (order of 10 μs). For example, Fig. 3(c) compares single-pixel time histories measured during a scanned-wavelength experiment with stationary and spinning diffusers. When the diffusers are stationary the light field consists of a speckle pattern (see Fig. 3(a)) and large amplitude, temporally structured noise (see Fig. 3(c)) persists in each pixel (albeit with altered characteristics). The latter likely originates from wavelength-dependent optical interference. However, by spinning the diffusers the speckle pattern can be sufficiently homogenized on the timescale of the camera integration time (see Fig. 3(b)). This leads to the signal in each pixel consisting of relatively random (i.e., unstructured) and lower amplitude noise (see Fig. 3(c)) which is much less problematic for the spectroscopic fitting routine. It is worth noting that spinning both diffusers provides only a small gain in SNR compared to spinning a single diffuser. However, it was critical to reflect the light off two diffusers (e.g., 1 stationary, 1 spinning) to achieve an SNR sufficient for 2D LAI in the flames studied here. Figure 3(d) illustrates that using a longer camera integration time (tint) reduces the noise level. This is because it enables the time-varying speckle pattern to be averaged over a longer duration. The measurement of SNR vs integration time was least-squares fit to a power-law function of the form . The best-fit values of a and b were 4.82 and 0.51, respectively, and the coefficient of determination (i.e., R2) was 0.999 indicating that the model captures the effect of integration time on SNR well. From this analysis, it is clear that using the highest motor speed and longest integration time possible for a given frame rate is desirable.
Fig. 3 (a) Image of the speckle pattern created by reflecting the laser light off two stationary diffusers. (b) Image of laser beam profile with the diffusers rotating at 90,000 RPM which illustrates how spinning the diffusers homogenizes the speckle pattern. (c) 15-scan-averaged signal time-history measured by a single pixel for laser light reflected off two diffusers that are stationary or spinning at 90,000 RPM. (d) Scan-averaged SNR for a single scan as a function of camera integration time.
It is important to note that the results shown in Fig. 3 are dependent on the laser wavelength and diffuser grit size. Gold-coated ground-glass diffusers with 120, 220, and 1500 grits/inch were all tested. The diffusers with 120 grits/inch performed best (provided a diffuse beam and dense speckle pattern) when using laser light near 4.8 μm. Conversely, the diffusers with 1500 grits/inch performed best for laser light near 1.4 μm, but performed poorly (providing near-specular reflections with little speckling) for laser light near 4.8 μm.
4. 2D Laser-absorption measurements
4.1. Validation of LAI diagnostic
The accuracy of the LAI diagnostic was validated by studying a non-reacting, isothermal, laminar jet of 49% CO, 49% Ar and 2% H2 flowing into ambient air. The initial jet diameter was 2 mm and the exit velocity of the jet was 3.32 m/s, yielding a jet Reynolds number of 217.6. The spatial resolution of the LAI diagnostic was estimated by imaging a wire mesh backlit with the incoherent mid-infrared laser light (see Fig. 4). The known spacing between wires was used to determine that the projected pixel size was 140 μm. In all experiments the camera was focused onto the central axis of the jet (determined by imaging a wire mesh). The QCL was scanned across the P(0,20) transition at 50 Hz and the laser light was imaged at 24 kFPS with a resolution of 64 × 52 pixels. The QCL’s current was scanned below the threshold current to enable the background emission in each pixel to be measured. The measured background emission was subtracted from each pixel and the baseline incident light intensity was then determined for each pixel by least-squares fitting a 3rd-order polynomial to the non-absorbing regions of the intensity scan. The absorbance spectrum measured by each pixel (e.g., see Fig. 5(a)) was then calculated using Beer’s law. A Voigt profile was least-squares fit to the measured absorbance spectrum in each pixel with the transition linecenter, integrated absorbance (i.e., integrated area) and collisional width as free parameters and the Doppler width fixed at the value corresponding to the known temperature (296 K). With the temperature and pressure known, the integrated absorbance corresponding to the best-fit Voigt profile was then used to calculate the mole fraction or column density of CO using Eq. (1).
Here Aproj,j [cm−1] is the integrated absorbance of transition j, Sj(T) [cm−2/atm] is the linestrength of transition j at temperature T, P [atm] is the pressure of the gas, χCO is the mole fraction of CO, and L [cm] is the path length through the absorbing gas. Figure 5(b) shows that the measured CO mole fraction in the jet core (calculated assuming a constant optical path length equal to the initial jet diameter) agrees within 3% of the known value (i.e., χCO = 0.49) and the spatial standard deviation in the jet core is 3.1% of the known value. The image also reveals the expected structure of a laminar jet, with a near constant mole fraction in the core of the jet and increasing (in the axial direction) transport of CO into the boundary layer. Figure 5(c) illustrates how the measured column density of CO varies across the jet and how it compares to the value predicted assuming a constant χCO = 0.49 (i.e., no mixing) and the known initial jet geometry. At the jet exit, the measured profile of χCOL agrees exceptionally well with that predicted assuming constant χCO and similar agreement exists further downstream (y = 4 mm and 8.5 mm) in the middle of the jet (i.e., −0.75 mm < r < 0.75 mm), thereby supporting the accuracy of this diagnostic. The 95% confidence interval (obtained from the spectral-fitting routine) of χCOL varied between 1% and 3% across the jet. The radial profiles indicate increasing transport of CO into the boundary layer and this likely results from a combination of diffusion and laboratory air currents, the latter of which introduced an obvious asymmetry in the jet profile on other occasions.Fig. 5 (a) 15-scan average of measured absorbance spectrum with best-fit Voigt profile acquired via LAI of a 2 mm diameter jet of CO/Ar/H2 (0.49/0.49/0.02 by mole). (b) Corresponding 2D image of CO mole fraction (calculated assuming a constant absorbing path length of 2 mm). (c) Comparison between measured and predicted (for constant jet geometry and CO mole fraction) CO column density across the jet at various axial locations (error bars are too small to be seen).
4.2. Path-integrated measurements of 2D flame structure
2D LAI measurements of path-integrated temperature and CO column density were acquired in laminar, partially premixed, oxygen-ethylene flames. Figures 1(b) and 1(c) illustrate the burner and flame studied, respectively. The LAI measurements were acquired using a 64 × 52 pixel window with a frame rate of 24 kFPS and an integration time of 16.4 μs. Using this exposure time and a bandpass filter near 2060 cm−1, the flame produced ≈12,000 counts of background emission, which was measured with the laser off before each scan and then subtracted from each pixel’s time history prior to calculating the absorbance corresponding to each scan. Some slight instabilities in the flame were observed which introduced additional noise in the signal measured by each pixel. As a result, multiple scans were averaged together (after background subtraction) to further improve SNR. Figure 6(a) illustrates how averaging multiple scans together improves the SNR. Specifically, the scan-averaged SNR improves from 24.7 to 50.4 for 15- and 50-scan averages, respectively. Figure 6(b) shows the corresponding absorbance spectrum (15-scan average) measured by a single pixel imaging the middle of the flame. The absorbance spectrum exhibits a 1−σ absorbance noise level of 0.008. For each spectrum, the transition lineshapes were modeled by a Voigt profile and the best-fit spectrum was determined using a least-squares fitting routine with the transition linecenters, integrated absorbances, and collisional widths as free parameters. The Doppler width of each transition was fixed to the value given by the temperature determined from the two-color ratio of integrated absorbance, R (given by Eq. (2)), obtained from the previous iteration.
Here, Aproj,P(0,20) and Aproj,P(1,14) are the integrated absorbance of the P(0,20) and P(1,14) absorption transitions determined by the spectral-fitting routine. With R and spectroscopic constants known, the path-integrated temperature was determined using Eq. (3) [26]: where h [J· s] is Planck’s constant, c [cm/s] is the speed of light, kB [J/K] is the Boltzmann constant, E″ [cm−1] is the lower-state energy of a given transition (taken from the HITRAN2012 Database [27]), and To is the reference temperature (296 K). Here subscripts 1 and 2 refer to the P(1,14) and P(0,20) transitions, respectively.Fig. 6 (a) Transmission spectrum with varying amounts of averaging measured by a single pixel and (b) measured and best-fit absorbance spectra of CO’s P(0,20) and P(1,14) transitions. Measurements were acquired in a partially premixed, oxygen-ethylene flame with ϕ = 6.43. The measurement location corresponds to the center of the field of view shown in Fig. 1(c).
With the pressure and path-integrated temperature known, the integrated absorbance of the P(0,20) transition was used to calculate the column density of CO (χCOL) using Eq. (1) where the linestrength of the P(0,20) transition was evaluated at the measured path-integrated temperature using Eq. (4).
Here, νo is the transition linecenter frequency [cm−1] and Q is the partition function. It should be noted that this approach is strictly valid in the limit that the linestrength of each transition exhibits a linear dependence on temperature across the range of temperatures that exist along the line-of-sight [13].Figures 7(a) and 7(b) show images of path-integrated flame temperature for a 15- and 50-scan average, respectively, and Figures 7(c) and 7(d) show images of CO column density for a 15-and 50-scan average, respectively. The spectral-fitting routine was performed in all pixels where the peak absorbance on the P(0,20) transition was greater than 0.05, thereby ignoring pixels with an absorbance SNR lower than 6.25. The images illustrate that 2D LAI is capable of resolving the thermochemical structure of laboratory scale laminar flames. The images reveal that the temperature is lowest in the fuel-rich core near the burner exit and increases in both the radial and axial directions, and peaks near the boundary with the air coflow. These observations are consistent with the expected structure of a laminar diffusion flame. The image of CO column density is more difficult to interpret given that the flame diameter increases in the flow direction. Regardless, these images illustrate that, in general, the amount of CO present along a given line-of-sight increases in the flow direction. All of the images indicate that the flame is approximately axisymmetric, thereby enabling a tomographic reconstruction to be performed to provide greater detail regarding the 2D flame structure.
Fig. 7 Images of path-integrated flame temperature (a,b) and CO column density (c,d) obtained via either a 15-scan (a,c) or 50-scan (b,d) average.
5. Tomographic reconstruction of 2D flame structure
Figure 8 shows tomographic reconstructions of gas temperature and CO mole fraction in the flame’s radial plane for a 5-scan (Figs. 8(a) and 8(c)) and 50-scan (Figs. 8(b) and 8(d)) average. Radial profiles of temperature and CO mole fraction are shown in Fig. 8(e) at select axial locations to further illustrate the thermochemical flame structure. These results were obtained by applying the tomographic reconstruction algorithms developed by Daun et al. [28] to the path-integrated measurements of integrated absorbance obtained via LAI. These alogrithms have previously been applied to numerous flame studies [14,19,21]. The results shown in Fig. 8 more clearly reveal the flame structure (compared to path-integrated results) and illustrate temperature and species profiles that are consistent with that expected in a fuel rich, partially premixed flame. The temperature is lowest in the core of the flame and, in general, increases in both the axial and radial directions. In comparison to the images of path-integrated temperature, the tomographic reconstruction reveals that the temperature rise in the axial direction within the flame core is more modest, increasing from 740 K to 1320 K between 1 mm and 9 mm above the burner exit, respectively. The temperature is highest (reaching a maximum near 2200 K) near the boundary between the fuel-rich C2H4/O2 stream and air coflow where oxygen supplied by the coflow can reach the unburnt fuel and intermediate species. The reconstructions illustrate that very little CO is present near the burner exit and that the mole fraction of CO increases significantly in both the axial and radial directions before falling off in the boundary layer.
Fig. 8 Tomographic reconstructions of flame temperature (a,b) and CO mole fraction (c,d) within the flame’s radial plane using either a 5-scan/10 Hz average (a,c) or a 50-scan/1 Hz average (b,d). The images shown in (b) and (d) have been enhanced by 10× via cubic-spline interpolation to highlight image quality. (e) Radial profiles of temperature and CO mole fraction at axial locations of y = 1, 4.5 and 9 mm for the reconstructions obtained from a 50-scan average with (denoted by lines) and without (denoted by dots) a 10× enhancement in image resolution.
The remainder of this section is devoted to describing the pertinent details of the tomographic reconstructions. Tomographic reconstructions were performed assuming each pixel images a unique line-of-sight. Flame-induced beam steering could cause this assumption to breakdown and ultimately reduce the spatial resolution of the diagnostic. Since tomographic reconstruction is sensitive to noise in the input image, a 2×2-pixel average was applied to all images prior to averaging 5, or 50 measured spectra together (i.e., prior to performing time averaging). This significantly reduced the noise induced by beam steering and the rotating diffusers. Next, the absorbance spectra were processed using a two-line Voigt fit, as described previously, to determine the integrated absorbance (Aproj) of the P(0,20) and P(1,14) absorption transitions corresponding to each pixel. The tomographic reconstruction algorithm employed here relies on assuming that the flame is axisymmetric. As a result, Aproj only needs to be resolved along the flame radius. Since Aproj was measured here along both halves of the flame, the results for each half were simply averaged together, however it should be noted that this has a small effect on image quality (see Fig. 9). The image (i.e., matrix) of each transition’s integrated absorbance was then passed to the tomographic reconstruction algorithm as described next.
Fig. 9 Reconstruction of χCO in the flame’s radial plane for varying amounts of regularization: (a) α0 = 0, (b) α0 = 0.1, (c) α0 = 0.1 without 2×2 pixel-averaging, and (d) α0 = 1. Each image also compares how averaging both halves of the image together (prior to reconstruction) slightly improves final image quality, but does not significantly alter the observed structure.
The tomographic reconstruction algorithm requires the assumption that the flame can be divided into N concentric rings of thickness Δr with a homogeneously distributed field variable, k, which is given by Eq. (5) for a specific absorption transition.
k effectively represents the local integrated absorbance per unit path length and the integral of k along a line-of-sight represents the back-projected data (i.e., the measured integrated absorbance: Aproj). N is given by the number of pixels that resolve the flame radius and Δr is given by the projected pixel size.First, a process called onion-peeling deconvolution is used to generate a set of equations of the form Ax = b. Here, A is the coefficient matrix which contains the path length of light through each annular section (i.e., the absorbing path length of the line-of-sight imaged by a given pixel) and is given by Eq. (6), x contains the to-be-solved-for field variable (k), and b contains the back-projected data (i.e., the integrated absorbance, Aproj, measured by each pixel).
Solving this system with back substitution may return a solution that is sensitive to noise in the back-projected data which manifests as errors or noise in the reconstruction of k. This is because onion-peeling deconvolution yields an ill-conditioned coefficient matrix which makes it difficult to find a well defined global minimum [28]. To overcome this, Tikhonov regularization is employed to increase the condition of the matrix [28]. In doing so, the equations shown below are used to provide a more robust solution. where and Here, k̃* is the field variable that yields the global minimum residual after Tikhonov regularization (i.e., the solution that is ultimately used to calculate images of gas properties). For the zeroth order regularization used for the data presented in this manuscript, L0 is given by the N × N matrix shown in Eq. (10). The parameter α0 is called the regularization parameter and it controls the balance between solution accuracy and smoothness/stability [28]. This works by approximating solutions to the ill-conditioned set of equations with those of a similar well-conditioned set. A high value of α0 will return a solution which is smooth, but less faithful to the original problem, and lower values return high-fidelity but often oscillatory solutions. As α0 → 0, k̃* → k* where k* is the field variable that yields the true minimum residual. Although methods exist for determining an optimum value for α0 [29], the value is often chosen at the user’s discretion to obtain an acceptable balance of accuracy and smoothness [28]. Figure 9 illustrates how increasing α0 from 0 to 1 improves image quality but reduces spatial resolution. In addition, Fig. 9 also illustrates how averaging both halves of the absorbance data together (prior to reconstruction) and using a 2×2 pixel average both slightly improve image quality without significantly altering the final images of χCO. The images presented in Fig. 8 were obtained using α0 = 0.1 since this produced solutions with acceptable stability while preserving the vast majority of flame structure that was revealed using α0 = 0.The solution for each individual transverse plane (i.e., row of pixels) was stable, however differences between adjacent planes gave the final 2D images a somewhat noisy appearance. This noise was attenuated by applying a 6-point moving average filter along the axial direction of the 2D images of k̃* for each absorption transition. Figure 10 illustrates the row-to-row oscillations in k̃* for both absorption transitions near the flame’s central axis and how the 6-point moving average is effective at removing the oscillations while preserving flame structure.
Fig. 10 Values of and along the flame’s central axis and corresponding 6-point moving average used to smooth plane-to-plane oscillations in the reconstructions shown in Figs. 8(b) and 8(d).
After reconstructing the 2D image of k̃* for each absorption transition, the two-color ratio of k̃*, Rk̃*, was calculated using Eq. (11)
The final 2D image of temperature was obtained by replacing R with Rk̃* in Eq. (3). Next, the 2D image of χCO was obtained using Eq. (5) with k replaced by and using the 2D image of temperature to calculate the corresponding image of the P(0,20) transition’s linestrength.The SNR of the measured absorbance for the P(1,14) transition is low in the center of the flame near the exit of the burner since the gas is relatively cool and little CO is present there. This prevented a reliable reconstruction of the 2D temperature field in a 1.7 mm wide region located 0 to 3 mm above the burner exit. As a result, 224 pixels of the temperature data in this region were replaced with values extrapolated from a 2nd-order polynomial which was obtained via least-squares fitting to the remainder of the data set. A total of 224, or 6.7% of the temperature image presents a value that was obtained via extrapolation.
6. Conclusion
This work has demonstrated, for the first time, that mid-infrared LAI is capable of providing 2D measurements and tomographic reconstructions of flame temperature and carbon monoxide in axisymmetric flames without mechanical translation of the line-of-sight. This was achieved by reflecting scanned-wavelength laser light off two ground-glass diffusers spinning at 90,000 RPM prior to the light traversing the flame. This was done to break the coherence in the laser light and, therefore, prevent the formation of diffraction-induced image artifacts which have limited previously developed LAI techniques to 1D (simultaneously) imaging. Ultimately this approach enabled measurements of CO’s absorbance spectra to be acquired at 50 Hz across ≈3,300 lines-of-sight simultaneously. While additional time averaging of spectra was done to improve the final image quality, this work demonstrated that this technique can provide high-fidelity 2D measurements and tomographic reconstructions of temperature and CO mole fraction on near-cm scales with a projected pixel size of 140 μm and a time resolution of 0.1 to 1 seconds. In comparison to conventional mid-IR LAT techniques employing mechanical translation of the line-of-sight, this technique offers a reduction in data collection time of ≈1,000× and potential for a ≈10× gain in spatial resolution compared to LAT methods that are limited by the laser beam diameter. In short, the capabilities of this 2D mid-IR LAI technique represent a significant step towards achieving high-speed, high-resolution measurements of thermochemical flame structure via LAT.
Funding
Air Force Office of Scientific Research (FA9550-18-1-0210).
Acknowledgments
The authors would like to thank the Air Force Office of Scientific Research with Dr. Mitat Birkan as program manager for supporting this work.
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