Temperature and Thermal Diffusivity Diagnostics in Laminar Methane Flames Using Infrared Four-Wave Mixing Techniques

Theory of LIGS and Simulation of Signal Waveform

The fundamental theory of LIGS can be found in Kiefer and Ewart ⁸ and Hemmerling et al., ³⁴ and only a brief description is included here. Two pump laser beams with the same wavelength of λ_pump crossing at an angle θ create an interference pattern, an intensity fringe pattern with characteristic grating spacing Λ = λ_pump/(2sin(θ/2)). When the pump beams are resonant with molecular transitions present in the crossing volume, the spatially periodic intensity pattern will lead to a spatial modulation of the excited molecular state population. Collisional quenching redistributes the laser energy absorbed by the excited molecules into heat, leading to a spatially modulated refractive index distribution with the same grating spacing Λ, i.e., a laser-induced thermal grating (LITG) is formed. The temporal behavior of this thermal grating reflects the superposition of a stationary temperature grating (decays exponentially due to thermal diffusion) and a standing wave oscillation (caused by the density modulation due to the rapid change of temperature and gas density in the LITG, which decays as the acoustic waves travel out of the measurement volume). The standing wave oscillates at the modulation frequency f_osc = c_s/Λ, where c_s is the speed of sound within the measured volume. In addition to the resonant grating, this density modulation, thus standing wave oscillation can also be generated by non-resonant pump laser beams due to the presence of a strong electric field gradient modulating the gas density through electrostriction. However, this process requires higher pump laser energies (typically 10–100 times stronger compared to resonant pump laser beams) and is usually negligible in cases where there is strong resonant absorption.^28,35 In this work, resonant pump laser beams are utilized and the enhanced LITG is generated.

The formation and evaluation of the LITG result from the processes of laser excitation, excitation and thermal energy deposition, density perturbation, and its associated modulation of the refractive index. Thermodynamic parameters such as the speed of sound, thermal diffusivity, and thermal energy transfer rate, will characterize the grating evaluation and thus affect the temporal signal waveform. The dynamic evaluation of the LITG can be probed by the laser-like signal beam, which is the scatter of the probe beam aligned to cross the induced LITG at the Bragg angle. Abundant information can be revealed from the signal intensity level, oscillation frequency, and signal temporal waveform. It is also possible to generate scattering signals by aligning the probe laser at higher-order Bragg angles. If the LITG forms a perfect sinusoidal grating, there would be no scattering at higher-order Bragg angles; however, if the absorption of the pump beams is enough to saturate the LITG, it can generate overtones in the shape of the LITG, making scattering at higher-order Bragg angles possible, although the efficiency of the scattering will be reduced compared to the first-order Bragg angle. However, it is a viable option in this paper, where the first-order Bragg angle is so small that the practical limitations of the setup make it very difficult to separate the probe and signal beams for interference-free signal detection.

To derive these thermodynamic parameters from the experimental LIGS signal quantitatively, the theoretical description of the LIGS signal waveform is needed. Analytical descriptions for the time evolution of a laser-induced grating can be found in, e.g., the works of Cummings, ³⁶ Cummings et al., ³⁷ Paul et al., ³⁸ and Hemmerling and Kozlov. ³⁹ In this work, Hemmerling's theory is utilized to fit the measured LITGS signals, with the modification of neglecting the signal contribution from electrostriction and the viscous damping. This is because the infrared laser pulse energy is relatively weak, and the viscous damping is slow enough to be negligible compared to the acoustic transit. Considering the above modification, the signal intensity S_LIGS is given by

S LIGS ( t ) ∝ [ S i ( t ) + S f ( t ) ] 2

(1)

S i ( t ) = M i [ cos ( 2 π f osc t ) exp ( − t 2 τ tr 2 ) − exp ( − t τ th ) ]

(2)

S f ( t ) = M f { sin ( 2 π f osc t ) exp ( − t 2 / τ tr 2 ) 2 π f osc t τ f − [ exp ( − t / τ th ) − exp ( − t / τ f ) ] 1 − ( τ f / τ th ) }

(3)

Thermometry Using Degenerate Four-Wave Mixing

A detailed theory of DFWM can be found in Abrams and Lind, ⁴⁰ and only the conclusions from their model used in this work are briefly introduced here. Abrams and Lind derived an analytical solution for DFWM signal intensity I_sig by assuming a phase-conjugate geometry alignment, as well as stationary two-level absorbers, an optically thin medium, and monochromatic laser beams, all the beams being polarized in the same direction and the probe beam having non-saturating intensity, which can be expressed as:

I sig = α 0 2 L 2 1 + δ 2 4 ( I pump / I sat ) 2 ( 1 + 4 I pump / I sat ) 3 I probe

(4)

Although Eq. 4 is based on many assumptions that may not always be satisfied under practical experimental conditions, it is fairly successful in analyzing the experimental DFWM signal intensity in many situations.^6,15 Therefore, in this work, a simplified empirical equation is used to calculate the DFWM signal intensity, based on Abrams and Lind's model and previous measurements of DFWM signal intensity:^30,41

$$\begin{gathered} I sig ( ν ) ∝ σ ( ν ) 2 N 2 ∝ [ ∑ i j S i j ( T ) g ( ν ) ] 2 N 2 \\ \end{gathered}$$

(5)

The temperature dependence of the line strength S_ij(T) can be expressed as:

S i j ( T ) = S i j ( T ref ) Q ( T ref ) Q ( T ) exp ( − c 2 E ″ / T ) exp ( − c 2 E ″ / T ref ) 1 − exp ( − c 2 v i j / T ) 1 − exp ( − c 2 v i j / T ref )

(6)

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Table I.

Hot water lines information near 3231 cm^–1 from HITEMP 2010. ⁴²

Line Group	Transition wavenumber, ν (cm–1)	Line intensity at 1800 K, Sij (cm·molecule–1)	Lower-state energy of transition, E'’ (cm–1)
I	3230.9830	6.79 × 10–22	1789.04
I	3230.9833	20.17 × 10–22	1789.04
II	3231.3206	19.43 × 10–22	5035.13
II	3231.3316	7.31 × 10–22	5713.25
II	3231.3316	21.95 × 10–22	5713.25

Equation 4 shows that the DFWM signal intensity depends on the degree of excitation saturation. In the saturation region where I_pump≥ I_sat, the signal intensity I_sig  ∝  μ²N²L²T₂/T₁ is less sensitive to laser intensity fluctuation and collisional quenching effects, and is generally preferred in practical measurements. However, it is critical to note that the saturation intensity I_sat is varying for different transitions when performing DFWM spectrum scanning because of the different transition dipole moments. The degrees of saturation for the two H₂O line groups I and II are often different and affected by the experimental arrangement and laser energy level, and the signal intensity ratio versus temperature thus becomes dependent on the exact degree of saturation of each line. Since the signal intensity ratio for the two H₂O lines at different degrees of saturation is difficult to simulate with simple models, a calibration of the function between signal intensity ratio and temperature is required for accurate DFWM thermometry. To achieve that, the accurate temperature measurements in lean flames using LITGS are used as reference temperature for the DFWM line ratios, and the detailed discussion of this process can be found in the following contents. The advantage of DFWM thermometry over LITGS is that prior information on composition species is not needed, thus it is more applicable in situations where the gas composition is difficult to predict, such as in fuel-rich flames.

Materials and Methods

The IR-DFWM/LITGS experimental setup used in this work is shown in Figure 1. The LITGS signal was first recorded with the laser wavelength set at the peak of line group I at 3230.98 cm^–1 and after this, the DFWM excitation scan was recorded in the same flame. Since stable laminar flames are studied in this work, the LITGS and DFWM were not performed simultaneously here for simplicity. In situations where the combustion is slowly varying over time, such as biomass combustion where the temperature varies over several minutes from ignition to when the fuel has been consumed, ⁶ it is possible to record the single-shot LITGS signals simultaneously with either the DFWM excitation scan or the on-line DFWM thermometry approach. ³² However, this simultaneous DFWM/LITGS temperature measurement can hardly be used for turbulent flames, since the temporal resolution is mainly limited by the wavelength scanning time required in DFWM.

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Figure 1.

Schematic of IR-DFWM/LITGS experimental setup. FDU: frequency doubling unit, BD: beam dump, BS: beam splitter, DFG: difference frequency generator, OPA: optical parameter amplifier, DB: dichromatic beam splitters, BCP: BoxCARS plates, DPSS: diode pump solid state, PMT: photomultiplier tube, OSC: oscilloscope, NBF: narrow bandpass filter.

Laser Diagnostics System

A tunable pulsed laser near 3 µm is used for IR-DFWM and as the pump laser for IR-LITGS, producing laser pulses with a repetition rate of 10 Hz and single pulse energy of 6–8 mJ, and full width half-maximum (FWHM) linewidth estimated to be 0.025 cm^–1. ²⁴ A more detailed description of this infrared laser system can be found in the previous literature. ²⁸ For IR-DFWM, four parallel laser beams with equal intensity were generated using two BCARS plates. ¹⁵ The two pump lasers and one probe laser were focused on one single point using a CaF₂ lens with a focal length of +500 mm, giving a probe volume of 0.2(width) × 0.2(height) × 10(length) mm³. The IR-DFWM signal propagates in a direction satisfying phase-matching and is collected by an InSb detector (Teledyne Judson Technologies, J10D-M204-R04M-60) with a PA-9 transimpedance pre-amplifier. In this work, no infrared bandpass filter was used since the interference from flame radiation is minor after the 4 m length path to the detector. For IR-LITGS, pump nos. 1 and 2 were used as the infrared pump beams for LITGS signal generation, and a continuous-wave laser (Laserglow Tech., LRS-0671) with a wavelength of 671 nm and an output power of 1000 mW was used as probe laser, which was aligned to cross the grating at the second-order Bragg angle. It is a choice by necessity since the optical layout of probe beam for first-order Bragg angle is limited by the non-adjustable short distance between the two pump beams, which is determined by the thickness of BoxCARS plates used to split the pump beams. Although the LITGS signal using second-order scatter is expected to be much weaker than using first-order scatter (approximately two orders of magnitude), in this work, sufficiently high S/N LITGS signals are obtained in all flames using the second-order Bragg angle. There is no requirement for the wavelength of the probe laser, however, the probe laser duration should be long enough to cover the formation and temporal evolution of the grating. Also, the signal intensity increases linearly with the probe intensity, thus higher probe laser power is needed to improve the S/N. Continuous-wave lasers are easier to obtain and have better power stability than long-pulse lasers, thus it is used in this work. The visible signal scattered from the LITG is collected by a photomultiplier tube (PMT, Hamamatsu H6780-04) with an optical filter (FL670-10, Thorlabs Inc., center at 670 ± 2 nm with FWHM of 10 ± 2 nm) to reduce noise from background light.

Flames

Measurements were performed at 6 mm height above the McKenna-type burner (HAB) in 15 different atmospheric CH₄/N₂/O₂ laminar flame product zones with equivalence ratios from 0.6 to 1.5. The operating conditions for different flames can be found in Table II. The burner has a plug diameter of 64 mm and the water-cooled burner head was utilized to provide a flat premixed flame. The combustion products at HAB = 6 mm of different flames are simulated using ANSYS Chemkin-Pro 2020 R2 (with Gri-Mech 3.0) ⁴³ for the species composition information within the measured volume required by IR-LITGS thermometry.

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Table II.

Operating conditions for flames 1–15. ϕ is the equivalence ratio, and Q is the flow rate (L·min^–1). T (K), M/γ (g·mol^–1), α (mm²·s^–1), and c_s (m·s^–1) are the simulated flame temperature, mass-to-specific-heat ratio, thermal diffusivity, and speed of sound, respectively, in the products zone simulated using Chemkin.

Flame number	Operating conditions				Flames simulations
	ϕ	Q CH4	Q O2	Q N2	T	M/γ	α	c s
1	0.6	0.79	2.64	7.00	2003.46	22.77	546.66	855.40
2	0.6	1.15	3.83	7.00	2283.05	23.10	686.42	906.45
3	0.7	1.30	3.71	9.00	2229.83	22.88	658.54	900.08
4	0.7	1.60	4.57	10.00	2297.21	22.94	695.30	912.52
5	0.8	1.82	4.55	10.00	2387.81	22.86	749.45	931.96
6	0.8	2.15	5.37	12.00	2375.40	22.84	742.43	929.86
7	0.9	2.25	5.00	12.00	2401.68	22.68	763.18	938.23
8	1.0	2.60	5.20	15.00	2319.98	22.45	722.55	926.84
9	1.1	2.40	4.36	15.00	2141.28	22.02	642.38	899.13
10	1.2	1.65	2.75	13.00	1855.38	21.39	518.19	849.18
11	1.3	1.30	2.00	9.00	1838.08	21.03	528.68	852.49
12	1.3	1.60	2.46	8.00	2032.55	21.30	625.06	890.68
13	1.4	1.50	2.15	7.00	1980.05	20.89	619.63	887.76
14	1.5	1.40	1.86	7.00	1858.89	20.38	577.28	870.91
15	1.5	1.10	1.47	5.00	1884.64	20.54	586.61	873.47

Laser-Induced Thermal Grating Spectroscopy Temperature Measurement in Flames

The oscillation frequency of the LITGS signal can be determined either by fitting the experimental LITGS signal waveform to the simulation model, or by a Fourier transform. The extracted oscillation frequency is used to determine the speed of sound in the crossing volume as c_s= f_oscΛ. The grating spacing Λ, which depends on the exact crossing angle between the two pump beams, can be accurately obtained by a calibration measurement in an environment with a known speed of sound. Figure 2 shows the LITGS signal of 2000 parts per million (ppm) C₂H₂ diluted in N₂ at 298 K. The oscillation frequency f_osc was determined to be 5.638 MHz in the calibration measurement. By using the speed of sound of pure N₂ c_s = 353.9 m·s^–1 at 298 K and 1 atm, ⁴⁴ the LITG spacing of this setup is determined as Λ = c_s/f_osc = 62.77 µm. Note that the geometry of the crossing pump laser beams would normally give a grating spacing of about 125 μm but aligning the probe laser at the second Bragg angle causes diffraction of an overtone of the LITG, which behaves as a LITG with half the grating spacing of the original LITG.

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Figure 2.

500 shot-averaged LITGS signal of 2000 ppm C₂H₂ diluted in N₂ at room temperature with the pump lasers aimed at 3230.98 cm^–1. Insert (a) zoom in the signal from 0 to 1 µs and insert (b) is the FFT of the signal. The Gaussian fit of FFT indicated a peak at f_osc = 5.638 MHz.

Since quite a high S/N LITGS signal in flames can be achieved using 100 shot-averaging and longer time averaging has little effect on the oscillation frequency extraction from the temporal waveform, 100 shot-averaged LITGS signals were recorded in all flames for temperature measurement and the example signals at flame no. 1 (ϕ = 0.6) and flame no. 8 (ϕ = 1) are shown in Figure 3. Compared to the signal waveform in diluted C₂H₂ at room temperature, the flame conditions signal decay much faster due to the increased sound speed and thermal diffusivity, which is why the theoretical model is used to determine the oscillation frequency in the flame signals with better precision than the Fourier transform method. ²⁹ Generally, longer-lasting signals give higher precision, which is why the LITGS thermometry precision increases at high pressures. ³² The contrast of the oscillations in the LITGS signals is caused by the rates of vibrational energy transfer of these two species: ³⁹ A fast initial energy transfer gives strong oscillations as seen in the H₂O signals in Figure 3, while a slower energy transfer gives small oscillations on top of a stationary background as in the calibration signals in Figure 2.

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Figure 3.

100 shot-averaged LITGS signal of water at flame temperature with the pump lasers aimed at 3230.98 cm^–1 and theoretical temporal profile fitting of LITGS signal. Insert is the FFT of the signal.

As shown in Figure 3, the LITGS signal at stoichiometric flame has a higher oscillation frequency due to the faster speed of sound at higher temperatures. With the knowledge of pre-calibrated LITG spacing Λ, the speed of sound c_s can be accurately determined from the LITGS oscillation frequency f_osc in various flames, as shown in Figure 4a. The precision of the c_s measurement, indicated by the error bars in Figure 4a, was estimated using the standard deviation of 10 repeated LITGS measurements, each recorded as an average of 100 single-shot signals. For these measurements, the precision is better than 0.5% (≤ 4.5 m·s^–1).

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Figure 4.

(a) Speed of sound and temperature measured with LITGS at 6 mm HAB for different flames, together with the simulated temperature. (b) Simulated M_gas/γ and H₂ concentration with Chemkin.

For ideal gases, the relation between temperature T and sound speed can be expressed as T = c_s²M_gas/γR_u, where R_u is the universal gas constant and M_gas/γ = (∑ _i χ_iM_i)/(∑ _i χ _i γ _i ) is the mass-to-specific-heat ratio for gas mixtures. Thus, the gas temperature can be derived from c_s if the gas constants in the measurement volume can be estimated. In this work, the mole fractions of major combustion products χ _i in different flames were taken from Chemkin simulation results based on Gri-Mech 3.0, ⁴³ together with heat capacity ratio for each species γ _i as a function of temperature and molecular weight for each species M_i, as shown in Figure 4b.

The uncertainty in the gas composition, and hence the calculated M_gas/γ in the measured volume, will affect the temperature measurement accuracy. For fuel-lean methane flames with equivalence ratios from 0.6 to 1.0 (flame no. 1 to flame no. 8), the mean value of M_gas/γ of the major combustion products mixture shows little variation (changing from 22.77 g·mol^–1 to 22.45 g·mol^–1 from flame no. 1 to flame no. 8 according to simulation) since the major products have similar molecular weight and heat capacity ratio. Based on this, we assume a 1.5% uncertainty in the gas constants in the lean flames. This can be even further improved if M_gas/γ in the measured volume can be more accurately determined. The flame temperature measurement precision in the product zone of the lean flames was estimated to be better than 0.8% (≤16 K), which was calculated by considering the standard deviation of 100 repeated temperature measurements. This high precision and accuracy demonstrate the suitability of IR-LITGS thermometry in lean flames. However, for fuel-rich methane flames with equivalence ratios from 1.1 to 1.5 (flame no. 9 to flame no. 15), the value of M_gas/γ in the product zone decreases dramatically with higher ϕ, mainly due to the increasing H₂ concentration. As a combustion product in fuel-rich flames, H₂ has a very small molecular weight and makes a dominating contribution to the change in M_gas/γ, as shown in Figure 4b. This means an accurate determination of the H₂ concentration is crucial for accurate LITGS temperature measurements in rich flames.

Acoustic Transit Time τ_tr and Thermal Diffusion Decay Time τ_th

In addition to the oscillation frequency, time constants such as acoustic transit time and thermal diffusion decay time can also be extracted from LITGS waveform with very high precision better than 2% (≤3 ns) thanks to the high S/N (∼1000) of the signal and the signal waveform's insensitivity to the pulse laser intensity fluctuation, as shown in Figure 5a.

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Figure 5.

(a) The acoustic transit time τ_tr and thermal diffusion decay time τ_th measured in different flames. (b) Measured and simulated thermal diffusivity.

The acoustic transit time τ_tr = w/( 2 c_s) will be inversely proportional to the speed of sound since the beam radius did not change during measurement. However, the determined beam radius w =  2 c_sτ_tr changes from 139 to 156 µm in these measurements. This may be caused by flame fluctuations affecting the crossing and beam diameter of the pump beams in the measurement volume but is more likely related to the uncertainty in the fitting model, which reduces the accuracy of determining the transit time due to uncertain influence from other fitting parameters. ⁴⁵

With the measured thermal diffusion decay time τ_th, LITGS provides an accurate direct method for measuring the thermal diffusivity α = (Λ/2π)²/τ_th in flame environments, as shown in Figure 5b. Compared to the simulated value of α in different flames shown in Table II, the results measured by LITGS show an underestimation of 20%. This is reasonable since the thermal diffusivity is positively correlated with temperature, and the simulation calculates α based on the adiabatic flame temperatures, which are higher than the actual flame temperatures in this case. The thermal diffusivity measured from the LITGS signal can be used for accurate, in situ pressure measurements. ⁴⁵ Since the thermal diffusivity α is also temperature dependent, as can be seen in Figure 5(b), this could potentially provide a secondary high-precision flame temperature measurement from the LITGS signal independent of the oscillation frequency. This could be useful in high-temperature, low-pressure situations where the fast decay of the signal decreases the number of visible oscillations, and thereby decreases the precision of the measured speed of sound from the oscillations.

Temperature Measurements in Rich Flames Using Degenerate Four-Wave Mixing

Figure 6a shows the recorded IR-DFWM spectrum at 6 mm height above the burner in flame nos. 1 and 8, which demonstrates the feasibility of the DFWM water line thermometry by showing the dramatic change in the relative signal intensity between line groups I and II for different flame temperatures. Quantitative measurement of temperature using a DFWM signal intensity ratio requires a calibration of the degree of excitation saturation, and the measured temperatures in the lean flames (flame no. 1 to flame no. 8) using LITGS are taken as reference temperatures. The calibration of the relation between signal intensity ratio and temperature is achieved by recording the DFWM signal ratios at the same point in the laminar lean flames, as shown in Figure 6b, which will be further used for the quantitative DFWM temperature measurement in rich flames.

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Figure 6.

(a) DFWM spectrum of H₂O at flame no. 1 and flame no. 8, scanned near 3231 cm^–1. The stems indicate the individual transitions in line groups I and II specified in Table I, and the height represents relative line strength at 1800 K taken from HITEMP 2010. ⁴² (b) The DFWM intensity ratio of line group I/II as a function of the flame temperature measured by LITGS. The error bars of temperature indicate the standard deviation of 100 repeat LITGS measurements, and the error bars of the area ratio are the standard deviation of 5 repeated DFWM scans over line group I/II.

The degree of saturation during DFWM measurement in flames is also studied under this setup. As shown in Figure 7, within the temperature range involved in this work, when the laser pulse energy ≥ 2 mJ (in this work, the laser pulse energy is 3.3 mJ) the line ratio approaches a constant value with increasing laser energy, indicating the saturation condition is satisfied and the temperatures at lean flames can be used as calibration for the line ratio also in the rich flames.

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Figure 7.

Saturation measurement in DFWM shows the signal ratio dependency on the total laser pulse energy at flame no. 1 and flame no. 8. The laser pulse energy used in the DFWM thermometry in this work is 3.3 mJ.

Flame Temperature Measurements in Lean and Rich Flames Using Combined LITGS/DFWM

The flame temperatures measured with LITGS (flame no. 1 to flame no. 8) and DFWM (flame no. 9 to flame no. 15) are shown in Figure 8. As a comparison, the flame temperature was also measured with a thermocouple. It can be seen that the measured temperature in different flames is in good agreement with the trend shown in the thermocouple measurements, especially for the LITGS data in lean flame. The thermocouple seems to underestimate the flame temperatures compared to the LITGS and DFWM temperatures. As the thermocouple used does not have high accuracy, and the LITGS temperatures have proved to be very accurate in lean flames, ²⁹ this is assumed to be due to a systematic error in the thermocouple measurement.

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Figure 8.

Simulated and measured flame temperatures by LITGS (flame nos. 1–8), DFWM (flame nos. 9–15), and thermocouple.

The DFWM temperature measurements show less precision (≤90 K) compared with LITGS temperature data. This is partly due to laser intensity fluctuations during the DFWM excitation scan during scanning (one scan typically takes several minutes), and partly due to the shot-to-shot variations in the mode structure of the infrared laser beam, which originates in the dye laser. These fluctuations during the scan result in fluctuations in the DFWM line intensity ratio in different scans which is why the DFWM temperature is calculated as the average of five consecutive scans. The LITGS method, which measures temperature based on temporal profile and frequency, is less sensitive to the fluctuation of laser energy and mode structure, which leads to higher precision in the LITGS temperature measurement.