Mid-infrared laser absorption tomography for quantitative 2D thermochemistry measurements in premixed jet flames

Abstract

A tomographic laser absorption spectroscopy technique, utilizing mid-infrared light sources, is presented as a quantitative method to spatially resolve species and temperature profiles in small-diameter reacting flows relevant to combustion systems. Here, tunable quantum and interband cascade lasers are used to spectrally resolve select rovibrational transitions near 4.98 and 4.19 $\upmu$m to measure CO and ${\mathrm{CO}_{2}}$, respectively, as well as their vibrational temperatures, in piloted premixed jet flames. Signal processing methods are detailed for the reconstruction of axial and radial profiles of thermochemical structure in a canonical ethylene–air jet flame. The method is further demonstrated to quantitatively distinguish between different turbulent flow conditions.

Appendix: uncertainty analysis

In this paper, we report values of species concentration and temperature, but it is important to note the uncertainty in these values due to various factors. For Eqs. 1–4 as well as those in this section (unless otherwise noted), we follow the Taylor series method (TSM) of uncertainty propagation [35], in which the uncertainty of a variable r, $\Delta r$, is given by:

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

(10)

where $x_i$ are dependent variables and $\Delta {x_i}$ are their respective uncertainties. As indicated by Eq. 4, mole fraction of an absorbing species $X_{\mathrm{abs}}(r)$ depends on line strength $S_j(T(r))$ and reconstructed absorption coefficient $K_j(r)$. In turn, $S_j(T(r))$ depends on temperature T(r), which itself depends on R(r), which is also dependent on $K_{j}(r)$. We present a discussion which propagates uncertainty from initial intensity measurements $I_t$ and $I_0$ through these equations to obtain uncertainty in T(r) and $X_{\mathrm{abs}}(r)$.

The systematic error in $I_t$ and $I_0$ is assumed to be the same because the same system is used to measure both signals; thus, the only uncertainty considered for each of these signals is the random uncertainty among all the scans averaged within a spatial segment dr (in this case, the distance associated with 105 direct-absorption scans). For each spatial segment, the standard deviations of both the incident background $I_0$ and the absorbance signals $I_t$ are calculated, which are both used to determine the 95% confidence interval of the signals, represented by $\Delta I_0$ and $\Delta I_t$. To obtain the variation specifically in absorbance, $\Delta \alpha _\nu$, we subtract $\Delta I_0$ from $\Delta I_t$ and use the resulting value as a bound on absorbance signal, $I_t \pm (\Delta I_t - \Delta I_0)$. We then calculate the resulting variation in $\alpha _\nu$, $\Delta \alpha _\nu$, by propagating the uncertainty in Eq. 1. This is shown in Figs. 4 and 5 as gray-shaded regions. In turn, $\Delta A_{j,\mathrm{proj}}$ is calculated by propagating the uncertainty $\Delta \alpha _\nu$ in Eq. 2, generating an upper and lower bound on $A_{j,\mathrm{proj}}$. This process occurs for each spatial interval dr across the radius of the burner r. The resulting upper and lower bounds of $A_{j,\mathrm{proj}}(r)$ are smoothed with a Savitzky–Golay filter [33] and plotted alongside the averaged $A_{j,\mathrm{proj}}(r)$ as shown in Fig. 6.

The uncertainty in $K_j(r)$, $\Delta K_j(r)$, is determined numerically via tomographic reconstruction of the upper and lower bounds of $A_{j,\mathrm{proj}}(r)$. Applying Eq. 10 to Eq. 3, we can calculate the uncertainty in R(r), $\Delta R(r)$:

$$\begin{aligned} \left( \frac{\Delta R(r)}{R(r)}\right) ^2 = \left( \frac{\Delta K_A(r)}{K_A(r)}\right) ^2 + \left( \frac{\Delta K_B(r)}{K_B(r)}\right) ^2. \end{aligned}$$

(11)

The ratio R(r) is used to determine temperature T(r) via the following equation:

$$\begin{aligned} T(r) = \frac{\frac{hc}{k_B}\left( E_A'' - E_B''\right) }{\ln {(R(r))} + \ln {\left( \frac{S_B(T_0)}{S_A(T_0)}\right) } + \frac{hc}{k_B}\frac{(E_A'' - E_B'')}{T_0}}. \end{aligned}$$

(12)

Here, h [J$\cdot$s] is the Planck constant, c [cm/s] is the speed of light, $k_B$ [J/K] is the Boltzmann constant, and $E_j''$ [cm$^{-1}$] is the lower state energy for the two lines A and B. Since T(r) is a function of R(r), there is an associated uncertainty in temperature, $\Delta T(r)$. Using Eq. 10 in Eq. 12, $\Delta T(r)$ is given by:

$$\begin{aligned} \frac{\Delta T(r)^2}{T(r)^2} = \frac{\left( \Delta R(r)/ R(r)\right)^2 + \left( \Delta S_B(T_0)/S_B(T_0)\right)^2 + \left( \Delta S_A(T_0)/S_A(T_0)\right)^2}{\left( \ln {(R(r))} + \ln {\left( \frac{S_B(T_0)}{S_A(T_0)}\right) } + \frac{hc}{k_B}\frac{(E_A'' - E_B'')}{T_0}\right) ^2}. \end{aligned}$$

(13)

As mentioned, $S_j(T(r))$ is function of T(r) [19]:

$$\begin{aligned} S_j(T) = S_j(T_0)\frac{Q(T_0)}{Q(T)}\frac{T_0}{T} \exp {\left[ - \frac{h c E_j''}{k_B}\left( \frac{1}{T} - \frac{1}{T_0}\right) \right] } \times \left[ 1 - \exp \left( -\frac{hc\nu _{0,j} }{k_BT} \right) \right] \left[ 1 - \exp \left( -\frac{hc\nu _{0,j} }{k_BT_0} \right) \right] ^{-1}, \end{aligned}$$

(14)

where it is understood that T is T(r). Q is the partition function for the internal energy modes of the molecule. Therefore, $\Delta T(r)$ (from the uncertainty in $\Delta R(r)$) affects $S_j(T(r))$ which is used to calculate mole fraction. The following expression can be obtained for the uncertainty in line strength due to uncertainty in observed temperature, $\Delta T(r)$:

$$\begin{aligned} \Delta S^2_{j,T}(T) = S^2_j(T)\Delta T^2 \left( -\frac{\partial Q(T)/\partial T}{Q(T)} - \frac{1}{T} + \frac{hc E_j''}{k_BT^2} + \frac{hc \nu _{0,j}}{k_BT^2} \left( \frac{\exp \left( -hc\nu _{0,j}/k_BT\right) }{1 - \exp \left( -hc\nu _{0,j}/k_BT\right) } \right) \right) ^2. \end{aligned}$$

(15)

This expression is consistent with the analysis presented by Ouyang and Varghese [36]. For our experimental results, the line strength uncertainty is proportional to the line strength itself. Additionally, the HITRAN database reports inherent uncertainty in $S_j(T_0)$ (2% for all spectral lines in this work), which we refer to here as $\Delta S_j(T_0)$. Thus, the total uncertainty in line strength can be calculated:

$$\begin{aligned} \Delta S^2_j(T) = \Delta S^2_{j,T}(T) + \Delta S^2_j(T_0). \end{aligned}$$

(16)

Now, mole fraction is given by:

$$\begin{aligned} X_{\text {abs}}(r) = \frac{K_j(r)}{S_j(T(r))P}. \end{aligned}$$

(17)

For our ${\mathrm{CO}_{2}}$ measurements, ambient ${\mathrm{CO}_{2}}$ in the atmosphere (approximately 400 ppm at the time of the experiment) influences the background measurement of $I_0$ slightly, since there is ambient ${\mathrm{CO}_{2}}$ in the ${\mathrm{H}_{2}}$/air co-flow flame. Thus, the maximum expected uncertainty due to ambient concentrations of the species, $\Delta X_{\mathrm{abs, amb}}$, is 400 ppm for ${\mathrm{CO}_{2}}$, or $\Delta X_{\mathrm{CO}_{2}}$ = 0.0004, which is less significant than the other sources of uncertainty. Utilizing Eq. 10, the uncertainty in mole fraction, excluding uncertainty in total pressure P, is:

$$\begin{aligned} \left( \frac{\Delta X_{\mathrm{abs}}(r)}{X_{\mathrm{abs}}(r)}\right) ^2 = \left( \frac{\Delta K_j(r)}{K_j(r)}\right) ^2 + \left( \frac{\Delta S_j(T(r))}{S_j(T(r))}\right) ^2 + \left( \frac{\Delta X_{\mathrm{abs, amb}}(r)}{X_{\mathrm{abs}}(r)}\right) ^2. \end{aligned}$$

(18)

Thus, the uncertainties in $\Delta K_j(r)$ and $\Delta S_j(T(r))$ accounted for. Eqs. 13 and 18 are used to calculate the error bounds in Figs. 8, 9, and 10.