Multi-line Boltzmann regression for near-electronvolt temperature and CO sensing via MHz-rate infrared laser absorption spectroscopy

Abstract

A mid-infrared laser absorption technique is developed for sensing of temperature and carbon monoxide (CO) number density from 2000 K to above 9000 K. To resolve multiple rovibrational lines, a distributed-feedback quantum cascade laser (DFB-QCL) is modulated across 80% of its current range using a trapezoidal waveform via a bias-tee circuit. The laser attains a spectral scan depth of 1 cm$^{-1}$, at a scan frequency of 1 MHz, which allows for simultaneous measurements of four isolated CO transitions near 2011 cm$^{-1}$ (4.97 $\mu$m) with lower-state energies spanning 3,000 to 42,000 cm$^{-1}$. The number density and temperature are calculated using a Boltzmann regression of the four population fractions. This method leverages the information contained in each transition and yields a lower uncertainty than using a single line pair. The sensor is validated in shock tube experiments over a wide range of temperatures and pressures (2300–8100 K, 0.3–3 atm). Measurements behind reflected shock waves are compared to a kinetic model of CO dissociation up to 9310 K and are shown to recover equilibrium conditions. The high temperature range of the sensor is able to resolve rapid species and temperature evolution at near electronvolt conditions making it suitable for investigations of high-speed flows, plasma applications, and high-pressure detonations.

Appendices

A Uncertainty analysis

In this section, we derive the uncertainty in the temperature and number density measurements with respect to the spectroscopic parameters of a given line pair and the experimental noise. The uncertainty analysis follows and expands the work presented in [20]. The uncertainty of f, function of $x_i$ variables, can be calculated using a Taylor expansion:

$$\begin{aligned} df(x_1, x_2, \cdots ) = \frac{\partial f}{\partial x_1}dx_1 + \frac{\partial f}{\partial x_2}dx_2 + \cdots \end{aligned}$$

(21)

Assuming the measured variables, $x_i$, are independent of one another, and that the errors in the measured variables, $\delta x_i$, are independent of one another [50], we get the uncertainty of f, $\delta f$, as a function of its partial derivatives:

$$\begin{aligned} \left( \delta f\right) ^2 = \left( \frac{\partial f}{\partial x_1}\delta x_1\right) ^2 + \left( \frac{\partial f}{\partial x_2}\delta x_2\right) ^2 + \cdots \end{aligned}$$

(22)

In the following sections, the relations of temperature with the linestrengths and areas of a line pair are derived and Eq. (22) is applied.

1.1 A.1 Temperature uncertainty

The temperature uncertainty for a line pair is derived from Eqs. 7 and 8, which are repeated here for clarity. The linestrength of a single line i is a function of temperature, T, the partition function Q(T), the ground state energy of the transition $E_{i}''$, the wavenumber of the transition $\nu _{i}$, the Boltzmann constant $k_B$, and $T_0$ = 296 K [31, 34].

$$\begin{aligned} S_{i}(T) = S_{i}(T_0) q(T, \nu _0) \exp \left[ -\frac{hcE_{i}''}{k_B} \left( \frac{1}{T} - \frac{1}{T_0}\right) \right] \qquad \quad (7) \end{aligned}$$

The term $q(T, \nu _0)$ accounts for partition function variation multiplied by a stimulated emission factor:

$$\begin{aligned} q(T, \nu _0) \approx q(T) = \frac{Q(T_0)}{Q(T)} \frac{ 1 - \exp \left( -\frac{hc}{k_BT}\nu _0\right) }{ 1 - \exp \left( -\frac{hc}{k_BT_0}\nu _0\right) } \qquad \qquad (8) \end{aligned}$$

The ratio of two linestrengths, R, is equal to the ratio of the integrated absorbance of two lines A and B, $A_\text {A}$ and $A_\text {B}$:

$$\begin{aligned} R(T) = \frac{S_\text {A}(T)}{S_\text {B}(T)} = \frac{A_\text {A}}{A_\text {B}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (3) \end{aligned}$$

The q(T) terms cancel out in Eq. (3) because $\nu _0$ variation has a weak impact on q(T) for two neighboring lines and the function R(T) can be expressed as:

$$\begin{aligned} R(T) = \frac{S_\text {A}^{0}}{S_\text {B}^{0}}\exp \left[ -\frac{hc}{k_\text {B}} (E_\text {A}'' - E_\text {B}'')\left( \frac{1}{T} - \frac{1}{T_0}\right) \right] \end{aligned}$$

(23)

Differentiating Eq. (3), using Eq. (23), and setting $\varDelta E = E_A'' - E_B''$, we get:

$$\begin{aligned} \left( \frac{dR}{R} = \right) \frac{dS_{A}^{0}}{S_{A}^{0}} - \frac{dS_{B}^{0}}{S_{B}^{0}} + \frac{hc}{k_B} \frac{\varDelta E}{T^2}dT = \frac{dA_A}{A_A} - \frac{dA_B}{A_B} \end{aligned}$$

(24)

which can be written:

$$\begin{aligned} \frac{dT}{T} = \frac{k_B}{hc} \frac{T}{\varDelta E} \left( - \frac{dA_A}{A_A} + \frac{dA_B}{A_B} + \frac{dS_{A}^{0}}{S_{A}^{0}} - \frac{dS_{B}^{0}}{S_{B}^{0}} \right) \end{aligned}$$

(25)

From here, the terms $\partial A_i / \partial T$, $\partial S_i^0 / \partial T$ can be identified and, substituting Eq. (24) into Eq. (22) with T as f, the temperature uncertainty is obtained in Eq. (26):

$$\begin{aligned} \frac{\delta T}{T} = \frac{k_B}{hc} \frac{T}{\varDelta E} \sqrt{ \sum _{i=1}^{2} \left[ \left( \frac{\delta S^0_i}{S^0_i}\right) ^2 + \left( \frac{\delta A_i}{A_i}\right) ^2 \right] } \end{aligned}$$

(26)

In this derivation, we neglected the contribution of the energy level uncertainty, $\delta E''$, which is close to the line position uncertainty and found to be $\delta E'' / E'' \le 10^{-6}$.

The area uncertainty, $\delta A_i$, is now related to the peak absorbance, $\alpha ^{pk}$. The area of the line can be approximated by the peak absorbance multiplied by the FWHM, i.e. $A_i \approx \alpha ^\text {pk}_i \varDelta \nu$. The line area uncertainty can be estimated by a rectangle of height $\delta \alpha$, the absorbance noise, and width $\varDelta \nu$, the FWHM of the line, i.e., $\delta A \approx \delta \alpha \varDelta \nu$. An illustration of the absorbance area and area uncertainty is provided in Fig. 13.

$$\begin{aligned} \frac{\delta A_i}{A_i} \approx \frac{\delta \alpha \varDelta \nu }{\alpha ^\text {pk}_i \varDelta \nu } = \frac{\delta \alpha }{\alpha ^\text {pk}_i } \end{aligned}$$

(27)

Fig. 13

Illustration of the uncertainty analysis performed assuming simplified areas, $A = \alpha ^p \varDelta \nu$, and area uncertainties $\delta A = \delta \alpha \varDelta \nu$. The inset shows the uncertainty optimum reached for $\alpha =1$, the minimum of $\exp (\alpha )/\alpha$

The absorbance is related to the measured amplitude of the non-absorbed laser intensity, $I_0$, and the transmitted intensity, $I_\text {t}$, see Eq. (1):

$$\begin{aligned} \alpha (\nu ) = - \ln \left(\frac{I_\text {t}}{I_0}\right)_\nu \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1) \end{aligned}$$

Differentiating Eq. (1) we obtain:

$$\begin{aligned} d\alpha = - \frac{dI_\text {t}}{I_\text {t}} + \frac{dI_0}{I_0} \end{aligned}$$

(28)

By using Eq. (22), with $\alpha$ as f, the absorbance noise $\delta \alpha$ can be related to the noise in the transmitted intensity, $dI_\text {t}$, and the noise in the background intensity, $dI_0$:

$$\begin{aligned} (\delta \alpha )^2 = \left( \frac{\delta {I_\text {t}}}{I_\text {t}}\right) ^2 + \left( \frac{\delta {I_0}}{I_0}\right) ^2 \end{aligned}$$

(29)

The transmitted intensity noise, $\delta I_t$, is the root mean square of the oscilloscope voltage and hence includes the effective sum of the laser fluctuations, the detector noise and the oscilloscope noise. Unlike the analysis of Ref. [51], in this work, the noise of the background intensity is greatly reduced via signal averaging, such that ${\delta }I_0\ll {\delta }I_\text {t}$, which means that Eq. (29) simplifies to:

$$\begin{aligned} \delta \alpha = \frac{{\delta }I_\text {t}}{I_\text {t}} \end{aligned}$$

(30)

This expression can be divided by $\alpha$ to find the relative noise in the absorbance measurement:

$$\begin{aligned} \frac{\delta \alpha }{\alpha }=\frac{1}{\alpha }\frac{{\delta }I_\text {t}}{I_\text {t}}=\frac{1}{\alpha }\frac{{\delta }I_\text {t}}{I_0\exp (-\alpha )} \end{aligned}$$

(31)

The signal-to-noise ratio of the raw laser intensity is defined as SNR = $I_0 / \delta I_t$ and hence:

$$\begin{aligned} \frac{\delta \alpha }{\alpha }= \frac{1}{\text {SNR}}\frac{\exp (\alpha )}{\alpha } \end{aligned}$$

(32)

Equation 32 indicates the level of noise to be expected in an absorption measurement. When $\alpha ^\text {p}$ is substituted for $\alpha$ in the denominator of both sides of the equation, the expected normalized residual r can be predicted for an absorbance measurement. This expression is approximately equal to the uncertainty in the area measurement, as indicated by Eq. (27). The ratio $\exp (\alpha )/\alpha$ indicates how much the relative noise in the raw intensity (1/SNR) is amplified to obtain the relative absorbance noise. As shown in the inset of Fig. 13, this function is minimized at $\alpha = 1$ at a value of e, indicating that at best, the absorbance noise is approximately 2.72 times the noise in the raw signal. When $\alpha$ is low, representing the optically-thin limit, the numerator is approximately 1, and the absorbance noise is inversely proportional to the absorbance. When $\alpha$ is high, the numerator dominates and the absorbance noise grows non-linearly with $\alpha$, representing the optically-thick limit where $I_\text {t}$ is close to 0 and below the noise level.

As indicated above, to approximate the uncertainty in an area measurement, the $\alpha$ in the denominator of Eq. (31) should be replaced by $\alpha ^\text {pk}_i$. The numerator is still a function of $\alpha$, which is maximized at the peak absorbance, where the transmitted laser intensity is the lowest. To provide a conservative estimate of the area uncertainty, this peak absorbance noise is also used for the numerator, such that:

$$\begin{aligned} \frac{\delta {A_i}}{A_i}\approx \frac{1}{\text {SNR}}\frac{\exp (\alpha ^\text {pk}_i)}{\alpha ^\text {pk}_i} \end{aligned}$$

(33)

The substitution of Eq. (33) in Eq. (26) gives the relative uncertainty in T for a given line pair given in Eq. (4).

$$\begin{aligned} \frac{\delta T}{T} = \frac{k_B}{hc} \frac{T}{\varDelta E} \sqrt{ \sum _{i=1}^{2} \left[ \left( \frac{\delta S^0_i}{S^0_i}\right) ^2 + \left( \frac{1}{\text {SNR}}\frac{\exp (\alpha ^\text {pk}_i)}{\alpha ^\text {pk}_i}\right) ^2 \right] } \quad (4) \end{aligned}$$

1.2 A.2 Number density uncertainty

Substituting Eq. (5) to Eq. (22), we get:

$$\begin{aligned} \frac{\delta n_\text {CO}}{n_\text {CO}} = \sqrt{\left( \frac{\delta A_i}{A_i}\right) ^2 + \left( \frac{\delta L}{L}\right) ^2 + \left( \frac{\delta S_i}{S_i}\right) ^2 } \end{aligned}$$

(34)

The uncertainty in the linestrength due to the temperature uncertainty must be separated from the reference linestrength uncertainty. The variation in the linestrength S is convoluted with the variation of the partition function, see Eq. (7), and its derivation is not straightforward to generalize to any molecule. In this work, we numerically evaluate the linestrength derivative with T and deduce:

$$\begin{aligned} \left( \frac{\delta S_i}{S_i}\right) ^2 = \left( \frac{1}{S_i(T)}\frac{\partial S_i}{\partial T} \delta T\right) ^2 + \left( \frac{\delta S^0_i}{S^0_i}\right) ^2 \end{aligned}$$

(35)

We obtain the number density uncertainty as:

$$\begin{aligned}&\left( \frac{\delta n_\text {CO}}{n_\text {CO}}\right) ^2 = \left( \frac{1}{\text {SNR}}\frac{\exp (\alpha ^\text {pk}_i)}{\alpha ^\text {pk}_i}\right) ^2 + \left( \frac{\delta L}{L}\right) ^2 + \\&\left( \frac{1}{S_i(T)}\frac{\partial S_i}{\partial T}\delta T\right) ^2 + \left( \frac{\delta S^0_i}{S^0_i}\right) ^2 \qquad \qquad \qquad \qquad \qquad \qquad (6) \end{aligned}$$

In this work, the pathlength is known within $\delta L / L = 1\%$. The uncertainty $\delta n / n$ is calculated in Fig. 3 using three line pairs available in the 2010.6 - 2011.6 cm$^{-1}$ region. To minimize the overall uncertainty, the area from P(1,25) transition is taken because its peak absorbance is typically the closest to 1. This line also offers the lowest uncertainty in the reference linestrength, $\delta S^0_{P(1,25)}$ = 1–2 %. Note that in the experimental results presented in this work, the uncertainty of the Boltzmann population fit are employed. The line pair uncertainties presented in Fig. 3 are only used to illustrate relative measurement accuracy of the different line pair combinations leveraged by the Boltzmann population fit method across the temperature range.

1.3 A.3 Non-isolated lines: perturbations due to neighboring features

Fig. 14

Full CO spectrum calculated at $T =$ 8000 K, $p =$ 1 atm, and $X_{\text {CO}}$ = 10% based on HITEMP database. The lineshapes are assumed to be triangles to calculate $A_\text {over}$, the overlap area of the P(3,14) and P(0,31) and ultimately estimate the error in fitting these lines

The line selection used in this work presents large $\varDelta E$ which improves the temperature uncertainty but also offers four spectrally separated lines. This is not the case for the line selection at 2008 cm$^{-1}$ used in previous work, where the P(3,14) line overlaps with the P(0,31) line. This effect is evident above 3500 K, see Fig. 15, when the P(3,14) line has a peak absorbance similar to P(0,31). The contribution of the P(3,14) line can be subtracted using iterative Voigt fitting but it is difficult to estimate how the uncertainty of this correction propagates to the temperature and number density measurements [20, 29]. In this subsection, we numerically calculate the impact of the uncertainty generated by the subtraction of a neighboring line, taking as an example the line selection at 2008 cm$^{-1}$. The lineshape is assumed to be triangular to simplify the calculations, with the FWHM of the triangle set equal to that of the actual Voigt lineshape. A representation of this approximation is shown in Fig. 14. The overlapping area of the two lines, $A_\text {over}$ in Fig. 14, can be calculated mathematically given the line peak absorbance, FWHM, and positions. When accounting for the P(3,14) line, the main source of uncertainty is estimated to arise from an erroneous allocation of the overlapping area to either P(0,31) and P(3,14). Assuming that 10% of $A_\text {over}$ is wrongly attributed to P(0,31), another contribution to $\delta A_i/A_i$ in Eq. (26) and Eq. (34) leads to an increase of $\delta T$ and $\delta n$. For a 2-% CO mole fraction adequate for measurements in combustion environments, Fig. 15 shows that, at 2000 K, a 10% error in the allocation of $A_\text {over}$ leads to a negligible increase in temperature and number density uncertainty, 0.01% and 0.05%, respectively. This confirms that the uncertainty arising due to the P(3,14) line was negligible in previous work [20, 29]. However, at temperatures higher than 3500 K and 10% CO mole fraction, the temperature and number density uncertainty increase by several percents, which further motivates the use of the new line selection at those temperatures.

Fig. 15

$\delta T$ and $\delta n_\text {CO}$ calculated assuming the P(3,14) line area overlap, $A_\text {over}$, is perfectly taken into account, full line, and if 10% of $A_\text {over}$ is erroneously attributed to the P(0,31) line, dashed line. The effect is calculated for 2% and 10% CO mixtures at 1 atm and $L =$ 10 cm

B Uncertainties in the reflected shock conditions

The method to calculate the uncertainty on $p_5$ and $T_5$ is given in this appendix. As described earlier, the reflected shock conditions are calculated using normal shock relations processed through a MATLAB code [41]. The normal shock relations require the knowledge of (1) $p_1$, the fill pressure, (2) $T_1$, the initial temperature, (3) $x_i$, the composition of the shock-heated gas, and (4) $v_\text {end}$, the speed of the shock on the endwall. The uncertainty of these parameters is propagated to the calculated $T_5$ and $p_5$ through a Taylor expansion method, see Eq. (22). The fill pressure uncertainty is assumed to be equal to the last digit of the most precise manometer that can be employed during the fill procedure, i.e. $\delta p_1 =$ 0.1, 0.01 or 0.001 Torr. The fill temperature uncertainty is assumed to be $\delta T_1 =$ 1 K. The mixture is prepared by subsequent filling of CO and Ar at increasing pressure. The species mole fraction uncertainty is calculated by accounting for the manometer precision in this step-by-step procedure and is negligible in this work, i.e. $\delta x_i / x_i \le 0.05 \%$. The shock position is determined through the reading of pressure transducers along the shock tube. The shock speed uncertainty is calculated by propagating the uncertainty in the shock position versus time through a York linear fit [37] accounting for the uncertainty in the pressure sensing positions ($\delta x = 1/16$ in.) and the time of arrival at each sensor (near 1 $\mu$s). In the speed range of this work, these two parameters contribute within the same order of magnitude to the speed uncertainty. As illustrated in Fig. 16, the resulting uncertainty on the speed is typically 10-30 m/s for shock speeds of 2000 m/s ($\le 1.5\%$).

Fig. 16

Shock speed measured using five pressure transducers. The end-wall speed is calculated using the linear York fit of the data [37]

We also included the uncertainty due to the shock tube leak rate in the calculation, by adding air to the mixture. The air leaked in the shock tube represents less than 0.5% of the mixture for the lowest $p_1$ of this study.

For the shock used as illustration in Fig. 16, the reflected shock heated conditions are $T_5$ = 8013 K (± 3.1%) and $p_5$ = 0.53 atm (± 4.5%). The contribution from each of the aforementioned sources to the total uncertainty in pressure and temperature is illustrated in Fig. 16. The uncertainties on $p_1$ and $T_1$ have a minor contribution to the total uncertainty and would be challenging to improve further. The shock speed measurement dominates the overall uncertainty. It can also be noted that the mixture uncertainty and the leak rate have negligible impact on $p_5$ and $T_5$, which indicates that our experimental procedure is adequate, see Fig. 17.

Fig. 17

Contributions to the $p_5$ and $T_5$ uncertainties

C Non-uniform path-integrated measurements

In this section, the robustness of this line selection to non-uniform temperature distribution is assessed. Let’s define $\overline{S(T)}$ and $\overline{T}$ as the path-averaged linestrength and temperature weighted by the number density of the absorbing species, $n_\text {CO}(l)$, that varies with the position along the beampath l [20, 52]:

$$\begin{aligned} \overline{S(T)}&= \frac{\int ^L_0 n_\text {CO}(l) S(T(l)) dl}{\int ^L_0 n_\text {CO}(l) dl} \end{aligned}$$

(36)

$$\begin{aligned} \overline{T}&= \frac{\int ^L_0 n(l) T(l) dl}{\int ^L_0 n(l) dl} \end{aligned}$$

(37)

Assuming a linear gradient of 1000 K across a constant pressure slab in Fig. 18, the linestrength deviation $\varDelta S = \overline{S(T)} - S(\overline{T})$ is below 2% of $\overline{S(T)}$ for the three primary lines in the spectrum. This result indicates that the linestrength variation in temperature is close to linear using these lines. Hence, the temperature measurements should be close to the CO number density-weighted path-averaged temperature. This is verified by calculating $\varDelta T$ defined in Eqs. 38, where $R^{-1}$ is the inverse function of R defined in Eq. (3) and relating the ratio of linestrengths to temperature:

$$\begin{aligned} \varDelta T = R^{-1}(\overline{S(T)}) - R^{-1}(S(\overline{T})) \end{aligned}$$

(38)

The results, shown in Fig. 18, indicate that $\varDelta T / \overline{T}$ is within 1% using the ratio of R(10,115), R(8,24), and P(4,7) with P(1,25) and assuming a uniform 1000-K gradient. Performing this calculation with a Boltzmann population fit, (in green in Fig. 18), the difference drops below 0.2%. An extreme-case estimate with a 3000-K gradient is calculated using a Boltzmann population fit and shows a maximum error of 4% compared to the average temperature. In conclusions, the Boltzmann fit of the four lines in the present wavenumber range is more robust to the bias induced by temperature gradients than the individual pair of lines taken separately. This robustness is essentially due to the R(10,115) transition, which positively bias non-uniform temperature measurements and compensates for the negatively bias temperature measurements induced by the other transitions. Therefore, the values reported in Fig. 12 during the 4-$\mu$s window displaying the propagation of the oblique shock are representative of the path-average temperature and CO number density and can be used for future studies of the oblique shock layer.

Fig. 18

(Top) Relative difference between $\overline{S(T)}$ and $S(\overline{T})$ assuming a linear gradient of 1000 K across an absorption slab. (Bottom) Relative error in temperature determination after fitting the integrated linestrength using the ratio of R(10,115), R(8,24), and P(4,7) with P(1,25), colored full lines. Two Boltzmann distribution fits have been performed at 1000 K (green dashed) and 3000 K (grey dashed)