1. Introduction

Scalar measurements in flames are needed for understanding combustion phenomenon, validating chemical kinetic models, and verifying numerical simulations. Intrusive measurements, such as temperature sensing via thermocouples or gas sampling, are straightforward to implement at discrete locations. Measurements at multiple locations or simultaneous determination of temperature and species (at a location) are more challenging and further disturb the flow field. Laser-based methods provide highly effective, non-intrusive means to interrogate both laminar and turbulent flow fields, and are the cornerstone of combustion diagnostics [1]. These techniques often require sophisticated experimental arrangements with multiple optical access points. Simultaneous measurement of multiple scalar quantities (i.e., temperature and mole fractions of various flame species) typically requires multiple laser sources, and mapping these with high spatial resolution can be arduous.

Flame emission measurements are another class of nonintrusive diagnostics that complement laser-based techniques. Multiple line-of-sight Fourier-transform spectrometer (FTS) measurements, when paired with appropriate tomographic deconvolution algorithms, can be used to simultaneously determine temperature and mole fractions of major flame species [2, 3]. High-speed infrared cameras with various band-pass filters have been used to map spatial and temporal variations in radiant intensity and relate these to the spatial distribution of scalar values [4] and to various measures of turbulence (e.g. integral length and time scales) [5, 6].

A new flame emission technique using an imaging Fourier-transform spectrometer (IFTS) can provide spatially resolved, detailed (e.g. δν̃ = 0.25cm⁻¹) wide-band (e.g. the mid-IR, 1.5 μm to 5.5 μm) spectra in a single measurement. Such highly resolved spectral measurements across the mid-IR can be used to monitor rotation-vibration emissions from fuels, intermediates, and major combustion products. High-resolution spectra are valuable for tomography algorithms (e.g. [7]) since variations in different scalar values (i.e., temperature and species concentrations) produce distinct and nearly unambiguous changes in the observed spectral emissions. Previously, IFTS has been used to identify pollutants and quantify species concentrations in the non-reacting turbulent exhaust plume from a coal-fired power plant smokestack [8]. More recently, IFTS has been employed in the measurement and qualitative assessment of a turbulent jet flame [9] and to study plumes arising from laser-material interactions [10]. Tomography techniques have not yet been adapted to IFTS flame measurements. This is significant because the high-resolution spectra, collected simultaneously at multiple lines-of-sight, have the potential to allow significantly improved accuracy in determining scalar values compared to other infrared emission based techniques.

Considering the value of nonintrusive emission-based measurement techniques, and the potential for using an IFTS to determine multiple scalar values in three-dimensions, this paper lays the groundwork for such efforts through the study and analysis of a laminar flame. More specifically, the objectives of this work are as follows: (1) collect spectral information from a laminar flame produced by a canonical burner; (2) develop a novel calibration technique needed for mid-IR flame measurements; (3) pair the latest high-temperature spectroscopic databases with a line-by-line radiative transfer model to estimate two-dimensional scalar values from the spatially-resolved spectra; (4) apply Monte Carlo techniques to understand systematic calibration errors on spectrally-retrieved scalars; (5) compare scalar estimates to those reported in the literature to assess capabilities and limitations of this new technique. This work is a key step in the development of IFTS for three-dimensional scalar estimation and should serve as a benchmark for future applications of IFTS to the study of reacting flows.

2. Experimental

In this work, a well-characterized, partially-premixed ethylene (C₂H₄) flame was studied to allow comparisons with scalar values reported in the literature. The flame was produced by a Hencken burner, which has a 25.4mm × 25.4mm square burner with an array of fuel tubes arranged within a honeycomb mesh through which the oxidizer flows. Immediately surrounding this is a similar honeycomb arrangement 6.4 mm wide consisting of tubes which carry an inert co-flow gas (N₂) to improve flame stability. The fuel and oxidizer mix shortly above the burner, resulting in a partially premixed flame. Additional details regarding the burner can be found in the literature [11, 12]. The Hencken burner was placed in a three-sided enclosure to minimize room disturbances. To minimize reflections, the enclosure was coated in a textured diffuse black paint. A schematic of the experimental set-up and detailed view of the Hencken burner is provided in Fig. 1. The flame was produced by flowing 12.21 standard liters per minute (SLPM) of air and 0.69 SLPM to 1.11 SLPM of ethylene (C₂H₄) through the burner. This produced equivalence ratios between 0.8 and 1.3. Temperature and OH mole fractions from a similarly configured Hencken burner were previously measured via laser absorption measurements [12]. 12.0 SLPM of N₂ co-flow was used to minimize entrainment of lab air and matched the conditions reported in the literature. Flow rates were controlled using MKS 1480A ALTA mass flow controllers which have an accuracy of ±1% of the set point flow rate. Laboratory conditions were steady with a temperature of 297 K, pressure of 976 hPa, and an 18.5% relative humidity.

 

Fig. 1 Schematic of the experimental arrangement. Important relative distances are provided as the image is not to scale. The Telops camera was placed on a rotation platform for fast and accurate transitions between the flame and calibration sources. An expanded view of the burner surface is provided to show the fuel tube and honeycomb mesh arrangement.

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A Telops Hyper-Cam MW-E imaging Fourier-transform spectrometer (IFTS) was used to capture mid-infrared (MWIR, 1.5 μm to 5.5 μm) hyperspectral radiation emitted by the Hencken burner flame. The IFTS features a traditional Michelson interferometer coupled to a high-speed 320 × 256 Indium Antimonide staring focal-plane array (FPA) via f/2.5 imaging optics. The pixel pitch is 30 μm and the optics produce a mean RMS spot size of 14 μm across the array. Instrument details can be found in the literature [9, 13, 14]. An external 0.25× telescope expanded the field-of-view and provided an effective focal length of 21.5 mm. The IFTS was located a distance d = 373mm from the center of the flame with its optical center located approximately u₀ = 60mm above the burner. The spatial resolution of each pixel was 0.52mm × 0.52mm at the flame (the image plane). Under the paraxial, thin-lens approximation, the depth of focus is 43 mm. This exceeds the detectable flame width, and due to its symmetry, indicates that each pixel’s signal corresponds to photons emitted along the full line-of-sight through the flame. For this experiment, the FPA captured the two-dimensional data with a 40 μs integration time on a 256 × 128 pixel sub-window. The spectral resolution was set to δν̃ = 0.75cm⁻¹ where δν̃ is the full-width at half-maximum of the instrument line shape (ILS). Symmetric interferograms were obtained by continuously scanning the Michelson interferometer at a uniform speed (0.21 cm s⁻¹) to a maximum optical path difference (OPD) of x_m = 0.8cm, with images captured every 632.8 nm. This corresponds to a camera frame rate of 3.4 kHz. Each interferometric “cube” consisting of 25,278 images was acquired in 7.5 s, and corresponds to an interferometric cube rate of 0.13 Hz. Each cube requires more than 1.5 GB of storage. To balance the needs of capturing a statistically sufficient sampling of the flame and efficiently storing and processing a large data set, 50 interferometric cubes of each flame were acquired and averaged.

Data for radiometric calibration was collected using 2 in and 6 in square CI Systems black-body sources placed at distances of 5 cm and 22 cm, respectively, from the camera. This ensured each over-filled the instrument’s field-of-view (FOV). A standard [15] two-point calibration—modified for noise suppression (see Sec. 3.2)—was performed to remove both system response and instrument self-emission from the raw signal by measuring the 2 in and 6 in blackbodies at 200 °C and 580 °C, respectively. Correction of the spectral axis due to dispersion in the imaging system was performed as previously described [13]. Temperatures for the 2 in and 6 in are accurate to within ±0.3% and ±1.0%, respectively, of the set-point temperatures. Spectral emissivities for the 2 in and 6 in blackbodies are reported to be 0.980 ± 0.004 and 0.96 ± 0.02, respectively. The manufacturer-reported blackbody uncertainties were taken to represent 95% confidence intervals (CIs), and were used to perform a Monte Carlo error analysis of both the calibrated radiances and spectrally-retrieved scalar values. This analysis is presented in a later section. Calibrated radiance spectra L_i(ν̃) are reported at each pixel (i) with units μW/(cm² sr cm⁻¹). The spectral variable ν̃ = λ⁻¹ is expressed in wavenumbers (reciprocal wavelength, cm⁻¹).

3. Radiation model

3.1. Interferogram formation

Here, great care is taken in describing the modeling, approach, and calibration of data since this work develops a new technique that will serve as a benchmark for future studies. A Michelson imaging interferometer splits the incident radiation into two beams and then recombines them so that an interference pattern is produced at the focal plane array (FPA). The measured signal at each pixel (i) of the FPA, I_i(x), varies with the optical path difference, x = vt, as the interferometer scans in time, t, at a constant speed, v. For a static scene with a radiance spectrum L_i(ν̃), the measured signal after passing through an ideal interferometer is given by

In the present work, the flame becomes unsteady about 20 mm above the burner due to buoyancy effects. This caused substantial, systematic variation in scene radiance so that L_i = L_i(ν̃, t) with a fluctuation timescale much shorter than the interferometric measurement time scale. The DC term is no longer a constant, but captures the broadband radiance fluctuations caused by buoyancy effects in the flame. Furthermore, the AC term is no longer the cosine transform of a static spectrum, and its Fourier-transformation is difficult to interpret quantitatively. If the dynamic source is ergodic and a statistically sufficient number of data cubes are acquired, then the mean interferogram corresponds to the mean source radiance. The Fourier-transform is a linear transform mapping the interferogram to the spectrum and thereby preserves this association. However, the radiance spectrum is a nonlinear function of the flame scalar values, so interpretation of the flame radiance must be performed carefully.

There is significant information content about the unsteady fluctuations encoded in a single interferogram. To appreciate this, note that the Michelson encodes spectral information by modulating the intensity at frequencies above f_b.g. = vν̃_b.g.. In these measurements, the mirror velocity was v = 0.21cm s⁻¹ yielding f_b.g. ≃ 390Hz. Thus, at frequencies below f_b.g., the intensity fluctuations can be attributed to radiance fluctuations. Application of an appropriate low-pass filter removes the intensity modulations caused by the action of the Michelson, thereby yielding a high-speed infrared intensity images (i.e., a movie) of the unsteady flow. This will facilitate interpretation of the spectra presented in this work, and more generally enables flow field analysis similar to what is currently performed by infrared cameras [4, 5]. It is also possible to take advantage of the DC fluctuations and produce “quantile interferograms” which map to spectra associated with the flame at various integrated intensity quantiles [16]. We plan to adapt this approach to the study of this and other unsteady flames in a follow-on effort.

3.2. Instrument calibration

Radiometric calibration of FTS is well established for low temperature participating media [8, 13, 15]. However, high flame temperatures required the development of a new calibration approach. We first review the standard FTS calibration methodology, which starts by assuming the instrument responds linearly to incident radiation. Observation of a blackbody source at two distinct temperatures affords the pixel-wise (i) determination of the system response (G_i(ν̃), gain) and self-emission ( $L_i^I(\tilde{\nu })$, offset) which define the linear transform mapping scene radiance L_i(ν̃) to instrument response Y_i(ν̃):

This calibration approach has practical limitations for this work. The flame of interest has a nominal temperature of ∼2400 K and exhibits strong, selective emission between 2100 cm⁻¹ to 2400 cm⁻¹ and between 3100 cm⁻¹ to 4200 cm⁻¹. Elsewhere, the flame emission is negligible, and the integrated radiance across the IFTS’s bandpass produces a signal comparable to that produced by a 550 °C blackbody. Thus, to ensure the flame integrated radiance is bracketed by the calibration integrated radiances, the higher-temperature blackbody should exceed this temperature. However, the integrated radiance corresponding to this blackbody temperature is constrained by the upper limit of the camera’s dynamic range. In the present work, a wide-area blackbody set to 580 °C was used, and filled ∼90% of the FPA dynamic range at the zero path difference (ZPD, x = 0). For comparison, the brightest part of the flame used ∼85% of the dynamic range at ZPD. Between 3250cm⁻¹ and 4250cm⁻¹, the signal-to-noise ratio (SNR) of a typical flame measurement can exceeded the SNR of the high-temperature blackbody by a factor of three. The blackbody curve at this temperature monotonically decreases with ν̃ across the IFTS bandpass. Moreover, the number of photons per unit energy emitted by the blackbody decreases as 1/ν̃ across the detector bandpass. These effects reasonably limit the achievable SNR of the gain term at higher wavenumbers. Since the calibrated scene radiance is obtained via division by G_i(ν̃), a low-SNR gain can introduce errors in L_i(ν̃) that are non-normally distributed, complicating quantitative interpretation of flame spectra.

The InSb quantum efficiency, optics transmittances, and other factors affecting system response are such that the gain magnitude, |G_i(ν̃)|, exhibits a smooth, slow variation with ν̃. A possible exception to this is the effect that the atmospheric transmittance profile would have on |G_i(ν̃)|, if it is not accounted for when modeling the at-sensor radiance from a distant black-body source. Our calibration methodology accounts for atmospheric absorption, and leverages the expectation of smoothness of the resulting gain term. First, Mertz phase correction [17, 18] is performed when converting calibration (and flame) interferograms to raw spectra. Then a least-squares smoothing spline is applied to each pixel’s gain magnitude and the gain phase is set to zero. (Phase correction ensured the gain phase was zero-mean noise.) It was determined via graphical inspection that fifteen spline points adequately captured the variation in |G_i(ν̃)| without fitting to the noise. The imaginary component of the calibrated flame radiance is not discarded as it can be used to assess calibration quality and estimate SNR.

Single-pixel gain curves obtained via the standard calibration method and our modified approach are compared in Fig. 2. The gain determined from the standard calibration without atmospheric compensation is shown in grey. Overlaid with this curve is the modified gain obtained using atmospheric compensation and a spline fit. The residual difference between the standard gain and the smooth gain, multiplied by the atmospheric transmittance profile, is also shown. The largely unstructured residuals are dominated by noise, indicating the effectiveness of the atmospheric compensation and spline smoothing.

 

Fig. 2 Representative single-pixel gain curve magnitude computed from standard calibration method (grey) compared with the gain curve magnitude obtained after atmospheric correction and spline smoothing (black). Residual difference between standard gain curve and the product of the smooth gain curve with atmospheric transmittance function is provided, offset by $-1\raisebox{1ex}{$\text{a.u.}$}\!\left/ \!\raisebox{-1ex}{$\text{r.u.}$}\right.$. Atmospheric absorption features are annotated. Here, a.u. represents arbitrary units and r.u. represents radiometric units.

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3.3. Spectral radiation model for determining scalar values

A simple model describing the apparent line-of-sight flame radiance was used to estimate temperature and relative species concentrations from the measured spectra. The spectral radiance L_i(ν̃) from a non-scattering source in local thermodynamic equilibrium (LTE) can be approximated by

The phenomenological absorption cross-section σ_i for the i^th species represents a sum over discrete spectral emission lines, each with its own line intensity, S_ij, and line shape, ϕ_ij(ν̃), via:

Spectral estimates of the flame temperature and fractional column densities were determined at each pixel by fitting Eq. (4) to the corresponding measured spectrum. This was accomplished by minimizing the sum of squared differences between the data and model parameters, T and q⃗ = (q_H2O, q_CO2, q_CO), using a Nelder-Mead direct search followed by a Levenberg-Marquardt gradient-based error minimization. Atmospheric H₂O and CO₂ concentrations were also model parameters to ensure the best possible estimate of τ. In the Φ = 0.8 and Φ = 0.9 flames, a small baseline oscillation was observed in spectral regions absent of line emission from the combustion gases. As discussed later, this only occurred in certain flame locations and is likely the result of insufficient temporal averaging over the flame unsteadiness. To mitigate the impact of the baseline oscillation on the spectral fit results, a 4^th-order polynomial was added to the model for the Φ = 0.8 and Φ = 0.9 measurements. The spectrum at each pixel contains 3000 unique radiance values, ensuring the six-parameter or eleven-parameter nonlinear model is highly over-determined.

This model makes a few notable simplifications to the actual radiative transfer problem, the most significant being the assumption of flame homogeneity along each pixel’s line-of-sight. This is a reasonable approximation near the base of the Hencken burner where—as supported by flame images presented in the next section—the mixing layer is a small fraction of the total line-of-sight. This assumption is also made in the analysis of OH laser absorption measurements of the similarly configured Hencken flame against which the present results will be compared [12]. The width of the mixing layer steadily increases with height above the flame, and the homogeneity approximation above ∼20 mm breaks down. As the mixing layer grows, temperature and density gradients become non-negligible and will systematically bias the fit parameters due to the variation in T and ξ_i along the line-of-sight. Given these limitations, this model can only be used to estimate core flame temperature and concentrations near the base of the burner and along lines-of-sight that are not dominated by the mixing layer, i.e., u < 20mm and |v| < 10mm. Here u and v represent the flame coordinate system, and are defined in Fig. 3. A multi-layer deconvolution approach can be developed for flames or regions where mixing is significant, as discussed previously. Within this region, absolute concentrations are determined via ξ_i = q_i/l where l = 25.4mm/cos(θ (u)) and $\theta (u)={\text{tan}}^{-1}\left(\raisebox{1ex}{$(u_0-u)$}\!\left/ \!\raisebox{-1ex}{$d$}\right.\right)$ is a small angle which accounts for the line-of-sight through the flame for rays a distance y₀ −y from the optical center of the camera. Elsewhere in the flame, the biased estimates, due to mixing, of T and q⃗ are to be interpreted as path-averaged quantities. Note that they are not true path-averaged quantities due to the nonlinear dependence T and ξ_i have on measured radiance. A multilayer deconvolution method is being developed to estimate 3-D distributions of temperature and concentrations.

 

Fig. 3 Split imagery of the symmetric flame for each of the four Φ values tested. Mean camera intensity values are on the left and coefficient of variation (CoV) values are on the right. The top and bottom of the color bar correspond to the mean intensity in 1,000’s of counts and CoV values, respectively.

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Other model simplifications include neglecting the transport of background radiation through the plume and atmospheric path radiance generated between the flame and sensor. However, these quantities are negligible compared to the flame radiance. The impact of ignoring collisional self-broadening in the Voigt profile has not been assessed, however its impact is expected to be small as typical line widths are narrower than the width of ILS(ν̃).

4. Results

4.1. Data overview

Figure 3 compares the broadband imagery captured by the IFTS in a single interferometric measurement at each equivalence ratio. The imagery was obtained by applying a low-pass filter along the OPD dimension with a cut-off frequency below f_b.g.. The left half of each image corresponds to the time-averaged intensity in raw camera counts. Near the base of each burner, the intensity rapidly increases with height, peaking at about 10 mm, and then decaying more gradually with increasing distance from the burner. The most fuel lean trial with Φ = 0.8 exhibits the swiftest decay in intensity with height and the fuel rich trial with Φ = 1.1 shows the slowest intensity decay with height. Traversing each flame axially, the transition from flame emissions to background occurs near v = ±13mm near the base of the burner. This mixing layer remains small up to about u = 10mm. Due to the symmetry of the burner, it follows that the mixing layer represents a small contribution to the measured line-of-sight radiance near the burner. The mixing layer increases in size with additional height above the burner due to entrainment of the N₂ co-flow.

The right half of each image shows the intensity coefficient of variation (CoV), which is the standard deviation of intensity normalized by the mean intensity. Near the base of the burner, each flame is steady, yielding intensity values on the order of 25,000 to 30,000 counts with minimal variations (CoV < 2%). Around 20 mm above the burner, buoyancy effects produce unsteady flow, causing intensity variations which increase with height and are more than 15% of the mean intensity by 40 mm. Unsteady intensity fluctuations were largest for the Φ = 0.9 trial with CoV values exceeding 22% in the mixing layer near 60 mm above the burner. Interestingly, CoV values between 40mm ≤ u ≤ 90mm and v = 0 were substantially larger for the fuel lean-flames compared with the fuel-rich flames.

Center-flame spectra at 10 mm, 60 mm and 100 mm above the burner are presented in Fig. 4. The largest emission feature near 2250cm⁻¹ is from the asymmetric stretching mode of CO₂, as well as combination bands at nearly resonant frequencies. Careful inspection of the region between 2000 cm⁻¹ to 2150 cm⁻¹ reveals the P-branch of CO (see inset plot). The R-branch overlaps with the strong CO₂ emission and becomes difficult to discern. The line emission between 3000 cm⁻¹ to 4200 cm⁻¹ is primarily due to H₂O rotational fine structure associated with transitions between several vibrational states. Also within this region is weaker broadband emission from CO₂, and very weak spectral emission from OH. The spectral radiance decreases with height due to cooling brought on by mixing with the N₂ co-flow and surrounding air. The entrainment of air enables oxidation of CO to occur with increasing distance from the burner, and by 100 mm CO emission lines are substantially diminished.

 

Fig. 4 Three center-flame spectra corresponding to heights 10 mm, 60 mm and 100 mm above the base of the Φ = 1.1 flame. The inset plots presents a detailed view of the P-branch corresponding to the fundamental 1 → 0 emission from CO. Odd numbered rotational levels are marked.

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4.2. 2-D spectral estimates of scalar values

4.2.1. Single pixel results

Fitting Eq. (4) enables simultaneous retrieval of T and mole fractions ξ_H2O, ξ_CO2 and ξ_CO. In the top panel of Fig. 5, the emission spectrum between 1975 cm⁻¹ to 4225 cm⁻¹ is shown for a single pixel at location (u, v) = (10mm, 0mm) for the Φ = 1.1 trial. Also provided is the model prediction corresponding to the best-fit parameters of T = (2318 ± 19)K, ξ_H2O = (12.5 ± 1.7)%, ξ_CO2 = (10.1 ± 1.0)% and ξ_CO = (3.8 ± 0.3)%. The measured temperature compares favorably with recent laser diagnostics results [12], (2348 ± 115) K, as well as with NASA CEA [23] equilibrium flame temperature of 2389 K. H₂O, CO₂ and CO mole fractions are in good agreement with the CEA equilibrium values of 12.8%, 9.9% and 4.1% respectively. We are not aware of other experimental determinations of these same species mole fractions for a similarly configured Hencken burner to compare against.

 

Fig. 5 Top: Ethylene Φ = 1.1 center-flame spectrum 10 mm above burner (· black) is compared with a model fit ( gray). Fit residuals, offset by −150μW/(cm² sr cm⁻¹), and instrument noise level, offset by −350μW/(cm² sr cm⁻¹), are provided. Bottom: Ratio of the flame path length, l, to the calculated mean free path of a photon, l_MFP, under the conditions estimated by the model fit.

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Reported fit parameter errors are the 95% CIs due to measurement noise and the propagation of calibration uncertainties. They do not account for systematic errors in the model. Details on the computation of parameter uncertainties is postponed until Sec. 4.3. The difference between the data and model are also presented in Fig. 5, denoted Residuals, and compared with the imaginary part of the spectrum, denoted Noise. The fit residuals are mostly unstructured with a root-mean-square (RMS) value of 10.1 μW/(cm² sr cm⁻¹). The RMS value of the fit residuals is only 1.5 times larger than the RMS value of instrument noise, indicating that the model describes the flame spectrum well overall. However, some systematic errors are present. This is most evident from the observed structure in the residuals between 2150 cm⁻¹ to 2400 cm⁻¹. Within this range, the fit residual RMS value is 5.9 times larger than the instrument noise RMS value. However, the noise in this region is small, and the RMS fit residual in this band is only 3.5% of the RMS signal. The quality of this single pixel spectral fit is representative of the fit quality across the hyperspectral image.

Mixing layer effects may contribute to the systematic fitting errors observed between 2150 cm⁻¹ to 2400 cm⁻¹. The bottom panel of Fig. 5 shows the ratio of the flame path length, l, and the calculated mean free path of a photon, l_MFP (Eq. (5)), under the conditions estimated by the model fit. This ratio is less than 3% across most of the spectrum, but approaches 40% within the strong CO₂ asymmetric-stretch emission band. This suggests that while the plume is optically thin across most of the spectrum, some optical trapping occurs within this CO₂ band. It follows that emission from the mixing layer will have a larger effect within this band in comparison with the optically thin spectral regions. This interpretation is consistent with the observation that fit residual magnitudes relative to the signal increase with height (i.e. mixing layer width). While the homogeneous assumption is adequate across most of the spectrum and yields spectral retrievals in good agreement with experimental and theoretical predictions, the high fidelity measurements may contain information about small mixing layer effects.

A second contributing factor for systematic fit errors may be associated with limitations of the spectroscopic databases used in this work. Recent measurements [24, 25] of high-temperature (2000 K to 5000 K) CO₂ produced via microwave discharge suggests that systematic errors in the spectral radiance between 2100 cm⁻¹ to 2450 cm⁻¹ can be up to 10–30%. Similarly, in a recent study involving laser irradiation of graphite targets [26], an empirical emissivity correction factor was needed to adequately reproduce the observed spectral emissions from the hot, CO₂-rich plume above the target within this same spectral region. However, in both of these recent experiments, CO₂ temperatures were much larger than the flame temperatures observed in this work and, in some cases, exceeded the useful temperature range of the CDSD-4000 database. Moreover, the radiative transfer models required to accurately describe the data were considerably more complicated than what is needed to model the Hencken burner flame. Thus, while limitations of the spectroscopic databases cannot be ruled out, their effect on the scalar estimates is likely small given both the good fit quality and the excellent agreement between spectrally-retrieved scalar values with Meyer’s results and CEA thermodynamic calculations.

Finally, the point-spread function of the instrument causes subtle mixing of spectral emissions from nearby pixels, and our instrument model does not yet account for these effects. As our future efforts will focus on three-dimensional scalar field reconstruction, these small effects will need to be properly modeled. Given the possible systematic errors due to instrumentation, model simplifications, and limitations of high-temperature spectroscopy databases, actual scalar value uncertainties may exceed those presented here by an additional factor of two. Despite this important caveat, the small fit uncertainties exemplify the benefit of a highly-resolved emission spectrum across a wide band pass for temperature estimation. 3000 unique data points between 1975cm⁻¹ ≤ ν̃ ≤ 4225cm⁻¹ sample myriad H₂O, CO₂ and CO emission lines, each representing a transition between pairs of internal energy levels. When local thermodynamic equilibrium prevails, the population of each internal energy level is governed by the same temperature via the Boltzmann distribution.

4.2.2. Radial and axial fit results

Spectrally-estimated scalars were uniform across the flame near the base of the burner. Figure 6 shows the results for the Φ = 1.1 trial at u = 10mm. The mean temperature obtained for all pixels within 5mm of flame center was T = 2317K with a standard deviation of 2 K.

 

Fig. 6 Spectrally-retrieved scalar values of ethylene flame 10 mm above the burner for the Φ = 1.1 condition. Error bars indicate the 95% confidence interval and only every other bar is shown for clarity. For comparison, the temperature value obtained by OH-laser absorption measurements ( ) and NASA CEA values ( , , ) are provided.

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Note that the pixel-to-pixel temperature variance is approximately ten times smaller than the fit errors associated with the individual spectral estimates (nominally 20 K). For comparison, the corresponding laser-based temperature measurement is also provided. The transition to lines-of-sight dominated by radiance from the mixing layer occurs around ±10mm from the center of the burner; beyond this, the retrieved temperature rapidly decays with distance. While the H₂O and CO₂ mole fraction profiles are qualitatively similar in shape to that of temperature, the CO profile is less flat across the center of the flame. Mean values within 5mm of flame center for H₂O, CO₂, and CO were 12.5% and 10.1%, and 3.8%, respectively. Pixel-to-pixel standard deviations (0.1%, 0.08%, and 0.08%, respectively) were much smaller than measurement uncertainties. The reduction in both temperature and mole fraction in the mixing layer result in SNRs too low to support spectral retrievals beyond v = ±15mm.

Temperatures estimated at u = 10mm for each equivalence ratio are compared with the laser absorption measurements [12] and equilibrium calculations in Fig. 7. The results are in very good agreement. IFTS temperature estimates are well within the error bars of the laser-based measurements, and don’t differ by more than 30K. (The error bars provided by Meyer et al. are not clearly defined and may not represent 95% CIs.) However, with the exception of the Φ = 1.3 case, spectrally-estimated temperatures are lower than those reported by Meyer.

 

Fig. 7 Variation of spectrally-retrieved average temperature with equivalence ratio in ethylene flame at a height of 10 mm above the burner. Comparison values of temperature measured with OH-laser absorption and chemical equilibrium are from Meyer et al. [12].

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The left panel of Fig. 8 presents the spectrally-estimated temperature profiles for all pixels with a sufficient signal-to-noise ratio at heights of 5 mm, 10 mm, 20 mm and 40 mm above the burner for the Φ = 0.9 flame. Error bars are not displayed to improve visualization, but errors are nominally ±1%. The temperature profiles at heights of 5 mm and 10 mm are flat and rapidly decay within the thin mixing layer near v = ±12.5mm. This nearly top-hat profile is consistent with the approximation of the flame as a single, homogeneous layer. However, at u = 20mm, the mixing layer has widened slightly, and by u = 40mm, the flame core only spans |v| ≤ 4mm. The right panel of Fig. 8 compares spectrally-retrieved temperature of the Φ = 0.9 flame traversing vertically through the centerline (v = 0) with Meyer’s OH absorption results [12]. Temperature increases rapidly above the base of the flame and peaks near 10 mm. The spectrally-estimated results are in good agreement with the laser measurements at each height investigated in Meyer’s work.

 

Fig. 8 Left panel: Spectrally-retrieved temperature of ethylene flame at heights of 5 mm, 10 mm, 20 mm and 40 mm above the burner for the Φ = 0.9 condition. Error bars are not shown for clarity but have nominal half-widths of approximately 20 K. Right panel: Comparison of the spectrally-estimated temperatures (IFTS) with laser absorption measurements (OH-LA) at various heights along the centerline of the Φ = 0.9 flame. Error bars are omitted at every other point for clarity.

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The variation in scalar values with distance from the burner is presented for all equivalence ratios in Fig. 9. The results should be interpreted as path-averaged scalar values to recognize the significance of temperature and concentration gradients along the line-of-sight which become more prominent with increasing height. The cooler, outer edge of the flame contributes less to the path-integrated signal, so path-averaged temperatures will be lower than the center flame temperature. Additionally, the increasing size of the mixing layer makes it difficult to estimate absolute concentrations, so column densities are reported instead. The current two-dimensional path-averaged results are valuable for qualitative understanding of the flame. However, quantitative interpretation at heights u > 10mm is not recommended since the impact of the mixing layer—which widens with height—is not accounted for. Retrieving three-dimensional scalar fields using a deconvolution technique is the ultimate goal of this effort.

 

Fig. 9 Spectrally-retrieved scalar values of the C₂H₂/air flame, along the centerline (v = 0), for Φ = 0.8 ( red), Φ = 0.9 ( blue), Φ = 1.1 (• black), and Φ = 1.3 ( green). Error bars are omitted for clarity. Fit results obtained without the polynomial baseline correction for Φ = 0.8 ( red) and Φ = 0.9 ( blue) are annotated.

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Of particular interest is the variation in scalar values with equivalence ratio. All scalar values immediately increase with u. At each equivalence ratio, temperature, q_H2O (u), and q_CO2 (u) rapidly increase and peak between 9mm ≤ u ≤ 12mm. The initial behavior of q_CO(u) is similar, but it reaches a maximum between 2mm ≤ u ≤ 4mm. After reaching their peak values, the scalar quantities exhibit a gradual decay with u in a manner that is highly dependent on Φ. In the fuel lean flames (Φ = 0.8, 0.9), the oxygen rich environment enables quick conversion of CO to CO₂, and the CO₂ profiles follow an axial profile similar to H₂O. In the fuel rich flames (Φ = 1.1, 1.3), the oxygen lean environment results in much larger CO concentrations near the burner and a more gradual conversion of CO to CO₂. While the N₂ co-flow limits diffusion of surrounding air into the flame, above u = 30mm, the buoyancy forces lead to flame unsteadiness, and the entrainment of atmospheric O₂ is likely. (Recall the CoV map in Fig. 3.) It is reasonable that this increases the rate of CO → CO₂ conversion, as evidenced by a slight increase in slope of the q_CO2 (u) curve for Φ = 1.1 and a substantial increase in slope for Φ = 1.3. The enhanced oxidation of CO at this height may be responsible for the observable change in the corresponding T (u) curve. Between 10mm ≤ u ≤ 30mm, the T (u) slope is −3.9 K/mm and −5.5 K/mm for Φ = 1.1 and Φ = 1.3, respectively. Between 40mm ≤ u ≤ 70mm, the slopes increase to −3.4 K/mm and −3.3 K/mm for Φ = 1.1 and Φ = 1.3, respectively.

As mentioned briefly before, spectra between 40mm ≤ u ≤ 90mm for Φ = 0.8, and to a lesser extent for Φ = 0.9, exhibit a subtle, low-frequency baseline oscillation. This oscillation produced a small bias in the T and q_CO fit parameters, and prompted the pragmatic addition of the 4^th-order polynomial to the model for the fuel-lean flames. This was the minimum order needed to visually account for the baseline oscillation; however, polynomials up to 7^th-order did not significantly affect the fit results. The baseline oscillation is a small fraction of the total signal and the polynomial never represented more than 1.5% of the peak spectral radiance. The baseline oscillation is likely a result of insufficient averaging over the intensity fluctuations arising from unsteady behavior in this flame region. It was not perceptible in the fuel-rich cases, and this is consistent with the smaller CoV values at v = 0 observed in Fig. 3. For comparison, the fit results obtained without using the 4^th-order polynomial to model the baseline are provided in Fig. 9 for context. The difference in q_CO2 (u) is imperceptible. While the baseline polynomial removes anomalous behavior in q_CO(u) and T (u) within the unsteady region, it also systematically lowers the q_H2O (u) column density where the flame is highly stable (u < 40mm).

4.2.3. Scalar maps

Figure 10 compares two-dimensional (2-D) maps of temperature and column densities obtained from the Φ = 0.8 and Φ = 1.1 flames. These maps represent the most complete, spatially resolved, line-of-sight scalar value measurements of Hencken burner ethylene flames to date, and demonstrate the utility of IFTS for combustion diagnostics. The 2-D scalar profiles reveal the differences in combustion under fuel-lean and fuel-rich conditions. Most notable is the rapid conversion of CO to CO₂ in the Φ = 0.8 flame, because of the abundance of available oxygen. This quick conversion of CO yields near identical H₂O and CO₂ maps in the Φ = 0.8 flame. The Φ = 1.1 flame has a large concentration of CO near the base. Diffusion and entrainment of surrounding air enables the gradual conversion of CO to CO₂, and the corresponding exothermicity of this reaction reduces the rate at which the path-averaged temperature decays with height, u.

 

Fig. 10 Split imagery comparison of path-averaged scalar values for Φ = 0.8 (left) and Φ = 1.1 (right). Left panel: Temperature. Center-left panel: H₂O column density. Center-right panel: CO₂ column density. Right: Scaled CO column density. CO values are tripled to use the full range of the common color axis. The top and bottom color bar scales represent temperature and column density, respectively.

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4.3. Uncertainty estimation

An important source of systematic error in the spectrum, and consequently the estimation in scalar values, arises from imperfections in the calibration blackbody sources. High-temperature, wide-area blackbodies are necessary for calibrating flame measurements, but these are not as accurate as smaller blackbodies designed for calibrating lower temperature scenes. To assess the impact of absolute temperature and spectral emissivity uncertainty on the spectrally-retrieved scalar values, a Monte Carlo error analysis was performed. The manufacturer-specified 95% CIs for set-point temperatures and emissivities defined normal distributions from which random values were drawn. 2,000 iterations were performed, each time calibrating using black-body temperatures and emissivities drawn from their respective uncertainty distributions. With each iteration, the spectral fits were performed leading to a distribution of scalar values which capture the uncertainty due to calibration. In general, these systematic calibration errors (δ_cal) were larger than the statistical uncertainty determined from the non-linear regression (δ_nlr). In this work, scalar value uncertainties represent the root quadrature sum of these two sources, i.e. ${\delta }_{\text{total}}=\sqrt{{\delta }_{\text{nlr}}^2+{\delta }_{\text{cal}}^2}$.

The uncertainty distributions of T, ξ_H2O, ξ_CO2 and ξ_CO obtained from the Monte Carlo error analysis of the Φ = 1.1 flame spectrum at (u, v) = (10mm, 0mm) are presented in Fig. 11. Continuous probability distribution functions were estimated via kernel density estimation using the Epanechnikov kernel. The 95% CI for temperature spans 2307 K to 2326 K, which is more than 5 times larger than the regression uncertainty of 3.5 K. Adding in quadrature both the statistical fit error and the calibration systematic error yields a spectrally-estimated temperature of T = (2318 ± 19)K. Calibration errors had a much larger impact on the uncertainty of concentrations. For example, the H₂O mole fraction 95% CI (11.85 % to 13.64 %) is 15 times larger than the statistical fit uncertainty (0.12 %). Quadrature addition yields ξ_H2O = (12.6 ± 0.8)%. The impact on the other flame species was similar and is reported in Fig. 11. Uncertainties of a few percent in blackbody temperature and emissivity affect the absolute radiometric accuracy much more than the relative spectral shape. Since the spectrally-retrieved temperature is strongly influenced by the relative line heights, whereas the column densities are dominated by the absolute line heights, the relative errors in temperature are much smaller than relative errors in column density.

 

Fig. 11 Uncertainty distributions estimated from a 2,000 iteration Monte Carlo analysis to propagate calibration source uncertainties to the spectrally-retrieved scalar values. The shaded areas are centered at the mean value and correspond to the 95% confidence interval. Previous experimental and equilibrium results are provided for context.

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5. Conclusions

This investigation of an ethylene flame (Φ = 0.8 − 1.3) sought to establish IFTS as a useful combustion diagnostic. Spectrally-determined temperatures at all Φ values agree with previous laser absorption flame measurements of a similarly-configured Hencken burner. The large number of ro-vibrational emission lines and band structures arising from multiple species lead to statistical temperature uncertainties less than 50K in the homogeneous flame region. Additionally, the retrieved H₂O, CO₂ and CO mole fractions are in excellent agreement with equilibrium predictions. The 2-D scalar fields obtained enable both the visualization and quantitative comparison of the Φ-dependent chemistry throughout the flame.

IFTS offers several unique advantages for combustion diagnostics. First, it is portable and can be set up and collecting calibrated spectral imagery in about an hour. Second, it enables the measurement of a moderate resolution spectrum (up to 0.25cm⁻¹) across a wide band pass (1.5 μm to 5.5 μm). This represents highly constraining data which can be used to benchmark computational fluid dynamics simulations. As demonstrated, the spectra are readily interpreted in terms of 2-D path-averaged temperature and column density maps. Moreover, the adaptation of existing deconvolution algorithms to the high-fidelity hyperspectral flame images may enable the retrieval of the 3-D scalar fields. The high-speed imagery existing within the interferometric measurement enables visualization of flame dynamics, and this enhances interpretation of 2-D scalar fields derived from the time-averaged spectra. Additionally, existing flow field analyses currently performed by infrared cameras can be readily adapted to the broadband imagery captured within the IFTS measurement. However, as a passive measurement, IFTS does not have the same sensitivity to trace species that lasers enjoy, and homonuclear diatomic molecules (e.g. H₂) cannot be observed. Additionally, the slow speed (0.1 Hz to 10 Hz) at which typical IFTS measurements are captured requires statistically significant numbers of observations (100–1000) to properly average over temporal fluctuations in the scalar fields.