Experimental assessment of the Tenti S6 model for combustion-relevant gases and filtered Rayleigh scattering applications

Abstract

Accurate descriptions of Rayleigh–Brillouin scattering (RBS) spectra for gas-phase species are important in light scattering measurement techniques including filtered Rayleigh scattering (FRS). The current manuscript targets evaluation of the well-known Tenti S6 model for calculating the RBS spectra for combustion-relevant species over a broad range of temperatures with relevance towards FRS applications in reacting flows. In this work, testing of the Tenti S6 model was performed by comparing measured FRS signals to synthetic FRS signals generated through the combination of the Tenti S6 model and an experimentally verified I₂ absorption model. First, temperature-dependent FRS signals were measured for a number of individual gases including Ar, N₂, O₂, CH₄, H₂, CO, and CO₂ from 300 to 1400 K. Comparisons between the measurements and synthetic FRS signals show excellent agreement (< 4% average difference) over the full temperature range. For pure CO₂, rotational Raman scattering effects must be taken into account when comparing measured and synthetic FRS signals. FRS measurements in binary mixtures were performed to assess the commonly used (but not verified) assumption that the total FRS signal from a mixture can be treated as the mole fraction-weighted average of the FRS signals from each component. Measured FRS signals in mixtures with large variations in both molecular weight and Rayleigh scattering cross section show a linear relationship with constituent mole fraction, indicating that this assumption is valid within the kinetic regime. Finally, FRS measurements were performed in near-adiabatic H₂/air and CH₄/air flames. Comparisons between measured and synthetic FRS signals show excellent agreement over a broad range of equivalence ratios ($\phi$), which includes a temperature range of 1100 < T(K) < 2400 and large relative changes in species mole fractions. Overall, the results indicate that the predicted RBS lineshapes calculated using the Tenti S6 model are sufficiently accurate in the context of FRS measurements for the species and temperatures evaluated.

Appendices

Appendix 1: calculation of rotational Raman scattering spectra and FRS signal contributions

If molecules are free to rotate, “Rayleigh scattering” consists of rotational Raman scattering as well as the Placzek trace scattering. Rotational Raman scattering comprised of an un-shifted Q-branch in addition to spectrally shifted Stokes and anti-Stokes branches. The Q-branch corresponds to no change of the rotational state ($\Delta$J = 0), while the Stokes and anti-Stokes branches follow selection rules of $\Delta$J = ± 2. The J → J + 2 transitions are the Stokes lines, while the J → J − 2 transitions are the anti-Stokes lines. The Raman shifts for the Stokes and anti-Stokes lines can be approximated as

$$\Delta {\nu _{J \to J^{\prime}}}= - \left( {4{B_o} - 6{D_o}} \right)(J+3/2)+8{D_o}{\left( {J+3/2} \right)^3},$$

(13)

and

$$\Delta {\nu _{J \to J^{\prime}}}=\left( {4{B_o} - 6{D_o}} \right)(J - 1/2) - 8{D_o}{\left( {J - 1/2} \right)^3},$$

(14)

respectively, where B_o is the rotational constant for the lowest vibrational level and D_o is the centrifugal distortion coefficient.

The intensity of a single Stokes or anti-Stokes-shifted rotational Raman scattering line is given by

$${I_{J \to {J^\prime }}}=C{I_o}{n_i}{F_J}\sigma _{i}^{{J \to {J^\prime }}}.$$

(15)

If the gas is in thermal equilibrium at temperature T, the fraction of molecules in state J can be expressed as

$${F_J}={g_J}\left( {2J+1} \right)\exp \left( { - \frac{{{E_J}}}{{{k_B}T}}} \right)/Q,$$

(16)

where g_J is a nuclear degeneracy factor (dependent on nuclear spin), E_J is the rotational energy, and Q is the rotational partition function, determined by the normalization satisfying $\mathop \sum \limits_{{J=0}}^{\infty } {F_J}=1$. The rotational energy can be accurately approximated by

$${E_J}=hc[{B_o}J\left( {J+1} \right) - {D_o}{J^2}\left( {J+1{)^2}} \right],$$

(17)

where h is Planck’s constant. Finally, the differential rotational Raman scattering cross section, $\sigma _{i}^{{J \to J\prime }}$is expressed as

$$\sigma _{i}^{{J \to {J^\prime }}}=\frac{{64{\pi ^4}}}{{45}}{b_{J \to J^{\prime}}}{\left( {{\nu _o} - \Delta {\nu _{J \to {J^\prime }}}} \right)^4}{\gamma ^2},$$

(18)

where ${b_{J \to J\prime }}$is a Placzek–Teller coefficient and $\gamma$ is the anisotropy of the molecular polarizability tensor. The Placzek–Teller coefficients for the Stokes and anti-Stokes lines are

$${b_{J \to J+2}}=\frac{{3\left( {J+1} \right)\left( {J+2} \right)}}{{2\left( {2J+1} \right)\left( {2J - 3} \right)}},$$

(19)

and

$${b_{J \to J - 2}}=\frac{{3J\left( {J - 1} \right)}}{{2\left( {2J+1} \right)\left( {2J - 1} \right)}},$$

(20)

respectively.

For each rotational line, Doppler and collisional broadening mechanisms lead to an intensity distribution described by a Voigt profile, which is the convolution of Gaussian (G) and Lorentzian (L) profiles. For the current work, the Voigt profile is approximated using a pseudo-Voigt profile [56]

$$\begin{aligned} & V_{{\text{p}}}^{{J \to {J^\prime }}}\left( {{\nu _{\text{r}}},\Delta {\nu _{J \to {J^\prime }}},\Delta {\nu _{\text{T}}}} \right) \\ & \quad =\eta L({\nu _{\text{r}}},\Delta {\nu _{J \to {J^\prime }}},\Delta {\nu _{\text{T}}})+(1 - \eta )G({\nu _{\text{r}}},\Delta \nu J \to {J^\prime }),\Delta {\nu _{\text{T}}}), \\ \end{aligned}$$

(21)

where the Gaussian and Lorentzian profiles are calculated as

$$\begin{aligned} & G\left( {{\nu _{\text{r}}},\Delta {\nu _{J \to {J^\prime }}},\Delta {\nu _{\text{T}}}} \right) \\ & \quad =\frac{{2\sqrt {\ln (2)} }}{{\Delta {\nu _{\text{T}}}\sqrt \pi }}\exp \left[ { - 4\sqrt {\ln (2)} \frac{{{{({\nu _r} - \Delta {\nu _{J \to {J^\prime }}})}^2}}}{{\Delta {\nu _{\text{T}}}^{2}}}} \right] \\ \end{aligned},$$

(22)

and

$$L\left( {{\nu _{\text{r}}},\Delta {\nu _{J \to {J^\prime }}},\Delta {\nu _{\text{T}}}} \right)=\frac{{\Delta {\nu _{\text{T}}}}}{{2\pi }}\frac{1}{{{{\left( {{\nu _{\text{r}}} - \Delta {\nu _{J \to {J^\prime }}}} \right)}^2}+{{\left( {\Delta {\nu _{\text{T}}}/2} \right)}^2}}},$$

(23)

and η is a function of the total full width at half maximum (FWHM) parameter, $\Delta {\nu _{\text{T}}}$. An accurate formula for η is [57]

$$\eta =1.36603\left( {\Delta {\nu _{\text{L}}}/\Delta {\nu _{\text{T}}}} \right) - 0.47719{\left( {\Delta {\nu _{\text{L}}}/\Delta {\nu _{\text{T}}}} \right)^2}+0.11116{\left( {\Delta {\nu _{\text{L}}}/\Delta {\nu _{\text{T}}}} \right)^3},$$

(24)

where $\Delta v_{L}$ is the Lorentzian FWHM parameter (due to collisional broadening) and the total FWHM parameter is given by

$$\Delta {\nu _{\text{T}}}=\left[ {\Delta \nu _{{\text{D}}}^{5}+2.69269\Delta \nu _{{\text{D}}}^{4}\Delta {\nu _{\text{L}}}+2.42843\Delta \nu _{{\text{D}}}^{3}\Delta \nu _{{\text{L}}}^{2}+4.47163\Delta \nu _{{\text{D}}}^{2}\Delta \nu _{{\text{L}}}^{3}+0.07842\Delta {\nu _{\text{D}}}\Delta \nu _{{\text{L}}}^{4}+\Delta \nu _{{\text{L}}}^{5}} \right],$$

(25)

where $\Delta {\nu _{\text{D}}}$ is the Gaussian FWHM parameter (due to thermal broadening). The appropriate expressions for the FWHM of the Doppler (Gaussian) and collisional (Lorentzian) broadening profiles are

$$\Delta {\nu _{\text{D}}}=7.16 \times {10^{ - 6}}\sqrt {\frac{T}{{{M_{\text{w}}}}}\frac{{\left( {2\nu _{o}^{2}+2{\nu _o}\Delta {\nu _{J \to {J^\prime }}}} \right)}}{2}+{{\left( {\Delta \nu _{{J \to {J^\prime }}}^{2}} \right)}^2}},$$

(26)

and

$$\Delta {\nu _{\text{L}}}=P\sqrt {\frac{{{T_{{\text{ref}}}}}}{T}} \left( {AJ+B} \right),$$

(27)

where P is the pressure in atmospheres, T_ref is a reference temperature which is 300 K in the current work, and A and B are the empirical constants describing the rotational level-dependent collisional broadening. Both the Gaussian and Lorentzian profiles (Eqs. 22 and 23) are normalized such that

$$\mathop \int \limits_{\nu }^{{}} G\left( {\nu ,\Delta {\nu _{\text{T}}}} \right)=\mathop \int \limits_{\nu }^{{}} L\left( {\nu ,\Delta {\nu _{\text{T}}}} \right)=1.$$

(28)

Using the above expressions, the spectral lineshape for each individual Stokes and anti-Stokes rotational line is defined as

$$\mathcal{R}_{i}^{{J \to {J^\prime }}}\left( {{\nu _{\text{r}}},\Delta {\nu _{J \to {J^\prime }}}} \right)= V_{{\text{p}}}^{{J \to {J^\prime }}},$$

(29)

and a complete Stokes/anti-Stokes spectrum for species i is generated by summing each individual rotational line as

$${S_{{\text{S}}/{\text{AS}}}}\left( \nu \right)=\mathop \sum \limits_{{J=0}}^{\infty } {F_J}\sigma _{i}^{{J \to {J^\prime }}}\mathcal{R}_{i}^{{J \to {J^\prime }}}\left( {{\nu _r},\Delta {\nu _{J \to {J^\prime }}}} \right).$$

(30)

Using the results of Kattawar et al. [58] and Miles et al. [16], the relative magnitude of the intensity (differential scattering cross section) of the Q-branch to the S/AS components of the rotational Raman scattering is 1/3 or

$${I_{\text{Q}}}={\text{C}}{{\text{I}}_{\text{o}}}{n_i}\sigma _{i}^{Q}=\mathop \sum \limits_{{J=0}}^{\infty } {I_{J \to {J^\prime }}}/3=\frac{C}{3}{I_{\text{o}}}{n_i}\mathop \sum \limits_{{J=0}}^{\infty } {F_J}\sigma _{i}^{{J \to {J^\prime }}}.$$

(31)

The lineshape of the Q-branch is described by a Voigt profile just as the Stokes and anti-Stokes lines and the same procedure outlined above using Eqs. (21)–(29) is followed with the exception that $\Delta {\nu _{J \to {J^\prime }}}=0$ for the Q-branch:

$$V_{{\text{p}}}^{{\text{Q}}}\left( {{\nu _{\text{r}}},\Delta {\nu _{\text{T}}}} \right)=\eta L\left( {{\nu _{\text{r}}},{\nu _{\text{T}}}} \right)+\left( {1 - \eta } \right)G\left( {{\nu _{\text{r}}},\Delta {\nu _{\text{T}}}} \right),$$

(32)

$$G\left( {{\nu _{\text{r}}},\Delta {\nu _{\text{T}}}} \right)=\frac{{2\sqrt {\ln \left( 2 \right)} }}{{\Delta {\nu _{\text{T}}}\sqrt \pi }}\exp \left[ { - 4\sqrt {\ln \left( 2 \right)} \frac{{{{\left( {{\nu _{\text{r}}}} \right)}^2}}}{{\Delta {\nu _{\text{T}}}^{2}}}} \right],$$

(33)

$$L\left( {{\nu _{\text{r}}},\Delta {\nu _{\text{T}}}} \right)=\frac{{\Delta {\nu _{\text{T}}}}}{{2\pi }}\frac{1}{{{{\left( {{\nu _{\text{r}}}} \right)}^2}+{{\left( {\Delta {\nu _{\text{T}}}/2} \right)}^2}}},$$

(34)

$$\Delta {\nu _{\text{D}}}=7.16 \times {10^{ - 6}}{\nu _{\text{o}}}\sqrt {\frac{T}{{{M_{\text{w}}}}}},$$

(35)

$$\Delta {\nu _{\text{L}}}=p\sqrt {\frac{{{T_{{\text{ref}}}}}}{T}} B,$$

(36)

$$\mathcal{R}_{i}^{{\text{Q}}}\left( {{\nu _r}} \right)=V_{{\text{p}}}^{{\text{Q}}}.$$

(37)

The values used for B_o, D_o, g_J, γ², A, and B are taken from the literature [58,59,60,61,62,63] and extrapolated to 532 nm where necessary.

Figure 11a shows an example of a calculated light scattering spectrum for N₂ at T = 296 K and P = 1 atm that compares the relative intensity of each of the components: (1) Placzek track, (2) Q-branch rotational Raman, and (3) stokes and anti-Stokes rotational Raman scattering. Integrating the various components over frequency, rotational Raman scattering comprises approximately 1.4% (0.35% Q-branch; 1.05% S/AS) of the total signal. For the majority of the species considered, the contribution of the rotational Raman scattering signal to the total scattering signal is very small as in the case of N₂ and can be neglected. However, CO₂ is an exception with the rotational Raman scattering comprising more than 6% of the total scattering signal. Figure 11b shows a comparison of the Stokes and anti-Stokes spectra for CO₂ and N₂. The total differential Raman scattering cross section of CO₂ is approximately ten times larger than that of N₂, while the total differential Rayleigh scattering cross section only is 2.39 times larger.

Fig. 11

a Calculated light scattering spectrum for N₂ at T = 296 K and P = 1 atm. b Comparison of Stokes and anti-Stokes rotational Raman scattering for N₂ ad CO₂ at T = 296 K and P = 1 atm. c “Zoomed in” region of the spectra from b showing the interaction between the rotational Raman lines and the I₂ filter spectrum. For S/AS rotational Raman scattering, the differential scattering cross section is plotted as ${F_J}\sigma _{i}^{{J \to {J^\prime }}}$

For the current FRS experiment, there are two more considerations to take into account: (1) the overlap of the rotational Raman scattering spectra with the 532-nm bandpass filter (BPF) and (2) the overlap of the rotational Raman scattering spectra with the I₂ absorption spectra. Both of these factors will alter the fraction of the rotational Raman scattering component that is collected within the total FRS signal. Figure 11b shows the measured spectral distribution of the BPF transmission around the center laser frequency ($\Delta$ν = 0). It is clear that the transmission of the BPF changes significantly over the span of the rotational Raman spectra, where species such as CO₂ that have small rotational constants (and smaller spacing between adjacent rotational lines) have a higher fraction of their total signal transmitted through the BPF.

For FRS experiments, the interaction between the Rayleigh–Brillouin and I₂ spectra is well established, but because of the large spectral bandwidth of I₂, there will be overlap between the rotational Raman lines and the I₂ spectra. Figure 11c shows a “zoomed in” portion the rotational Raman spectra shown in Fig. 11b along with an overlay of a portion of the $B\left( {{}_{0}^{3}{\Pi _{{0^+}u}}} \right) \leftarrow X\left( {{}_{0}^{1}{\Sigma _g}^{+}} \right)$ electronic transition of iodine calculated with the code of Forkey et al. [34]. Figure 11c shows that the I₂ spectrum is quite dense and that many of the I₂ lines overlap with individual rotational Raman transitions. Considering the effects of the bandpass and I₂ filters, the total rotational Raman scattering signal transmitted to the detector can be written as

$${S_{{\text{RR}},i}}={{\text{CI}}_0}{n_i}\left[ {\sigma _{i}^{{\text{Q}}}\mathop \int \limits_{{{\varvec{\upnu}}}} \mathcal{R}_{i}^{{\text{Q}}}\left( {{\nu _r}} \right){\tau _{I2}}\left( \nu \right){\tau _{{\text{BPF}}}}\left( \nu \right){\text{d}}\nu +\mathop \sum \limits_{{J=0}}^{\infty } \sigma _{i}^{{J \to {J^\prime }}}{F_J}\mathop \int \limits_{\nu } \mathcal{R}_{i}^{{J \to {J^\prime }}}\left( {{\nu _r},\Delta {\nu _{J \to {J^\prime }}}} \right){\tau _{I2}}\left( \nu \right){\tau _{{\text{BPF}}}}\left( \nu \right){\text{d}}\nu } \right],$$

(38)

where ${\tau _{I2}}\left( \nu \right)\;{\text{and}}\;{\tau _{{\text{BPF}}}}\left( \nu \right)~$is the transmission of the I₂ cell and bandpass filter, respectively. Since the location and width of the individual rotational Raman lines are species-specific, the exact level of the rotational Raman scattering signal contribution to the total FRS signal collected by the detector depends on the individual species. For example, calculations for the current experimental conditions show that for N₂ at T = 296 K and P = 1 atm, approximately 36% and 30% of the Q-branch and Stokes/anti-Stokes rotational Raman scattering signal, respectively, transmit through the I₂ and bandpass filters, while only 22% of the Cabannes line transmits through the filters. This implies that the fractional percent of the rotational Raman scattering component (F_RR) increases for FRS as compared to traditional Rayleigh scattering (2% vs 1.4%). For CO₂ at T = 296 K and P = 1 atm, approximately 44% and 49% of the Q-branch and Stokes/anti-Stokes rotational Raman scattering signal, respectively, transmit through the I₂ and bandpass filters, while only 16% of the Cabannes line signal transmits through the filters and is collected. Thus, for CO₂, there is a significant increase in F_RR, increasing from 6% for traditional Rayleigh scattering to 16% for FRS.

The value of F_RR can vary as a function of temperature since the Cabannes linewidth, ${\text{~}}{\mathcal{R}_i}\left( {{\nu _r}} \right)$ and the Raman linewidths, $\mathcal{R}_{i}^{{\text{Q}}}\left( {{\nu _{\text{r}}}} \right)$, and $\mathcal{R}_{i}^{{J \to {J^\prime }}}\left( {{\nu _{\text{r}}},\Delta {\nu _{J \to {J^\prime }}}} \right)$ can change width (and shape) as a function of gas temperature, while the I₂ lines remain constant for a given I₂ filter cell setting. Figure 12 shows the variation of F_RR as a function of temperature for five gases. For all species considered, F_RR decreases with increasing temperature. This is due to the fact that the Cabannes linewidth, ${\text{~}}{\mathcal{R}_i}\left( {{\nu _r}} \right)$ increases with increasing temperature due to Doppler broadening, while the rotational Raman linewidths remain largely constant due to competing effects of Doppler broadening and collisional broadening. Thus, with increasing temperature, an increasingly larger fraction of the RBS signal (as compared to the rotational Raman scattering signal) transmits through the I₂ filter.

Fig. 12

Variations in calculated fraction percent (FRR) of total FRS signal due to rotational Raman scattering (Q-branch and Stokes/anti-Stokes) as a function of temperature

Appendix 2: temperature-dependent values of ${c_{\text{int} }}~$and ${\mu _{\text{B}}}$ for combustion-relevant species

Kinetic models of scattered light spectra require collision properties of gases that are used within the collision integral of the Boltzmann equation. For the Tenti S6 model [5], the collision integral is estimated using gas transport properties including the internal specific heat capacity (${c_{\operatorname{int} }}$) and the bulk viscosity (${\mu _{\text{B}}}$). Both ${c_{\operatorname{int} }}$ and ${\mu _{\text{B}}}$ depend on the relaxation rates of the internal degrees of freedom of the molecule.

2.1 Internal specific heat capacity (${c_{\text{tint} }}$)

In general, the internal specific heat capacity can be calculated as

$${c_{\operatorname{int} }}=\frac{{\left( {5 - 3\gamma } \right)}}{{2\left( {\gamma - 1} \right)}},$$

(39)

where γ is the ratio of specific heats, which can be expressed in terms of the total number of internal degrees of freedom $\left( f \right),$ as

$$\gamma =\frac{2}{f}+1,$$

(40)

and thus

$${c_{\operatorname{int} }}=\frac{{\left( {f - 3} \right)}}{2}.$$

(41)

The total number of degrees of freedom is written as f = f_t + f_r + f_v, where f_t = 3 is the number of translational degrees of freedom, f_r is the number of rotational degrees of freedom given by

$${f_{\text{r}}}=\left\{ {\begin{array}{*{20}{l}} 2&{\quad {\text{linear}}\;{\text{molecule}}} \\ 3&{\quad {\text{non-linear}}\;{\text{molecule}}} \end{array}} \right.,$$

(42)

and ${f_v}$ is the number of vibrational degrees of freedom, given by

$${f_{\text{v}}}=\left\{ {\begin{array}{*{20}{l}} {2\left( {3N - 5} \right)}&{\quad {\text{linear}}\;{\text{molecule}}} \\ {2\left( {3N - 6} \right)}&{\quad {\text{non-linear}}\;{\text{molecule}}} \end{array}} \right.,$$

(43)

where N is the total number of atoms in the molecule. Note, the factor of 2 preceding the parenthetical arguments in Eq. (42) corresponds to the fact that each active vibrational mode has 2° of freedom. Since the number of accessible vibrational modes increases with increasing temperature, $\gamma =\gamma \left( T \right)$ and thus the temperature dependence of ${c_{\operatorname{int} }}$ is embedded within $\gamma$.

If the relaxation time scale of any internal degree of freedom is much longer than the characteristic time scales of sound propagating through the media, then the internal motion associated with that particular degree of freedom remains frozen on the timescale of the density fluctuations. For light scattering experiments, the characteristic sound frequencies are on the order of 1 GHz and thus internal degrees of freedom with relaxation times scales significantly longer than 1 ns are not active in the light scattering. For N₂, O₂, H₂, CH₄, CO, and CO₂, the vibrational relaxation time scales are longer than 10^− 7 seconds, even at the highest temperatures, and therefore only rotational modes contribute to the scattering. Under these conditions, an effective value of γ (due to rotational motion only) is written as

$${\gamma _{\text{r}}}=\frac{{{f_{\text{r}}}+5}}{{{f_{\text{r}}}+3}},$$

(44)

and the internal specific heat capacity is determined as

$${c_{\operatorname{int} }}=\left\{ {\begin{array}{*{20}{l}} 1&{\quad {\text{linear}}\;{\text{molecule}}} \\ {3/2}&{\quad {\text{non-linear}}\;{\text{molecule}}} \end{array}} \right..$$

(45)

For H₂O the vibrational relaxation times range from ~ 10 ns at 300 K to < 1 ns seconds at flame temperatures and cannot be neglected based on timescale arguments. In this manner, ${c_{\operatorname{int} }}$ is calculated in the current work using Eq. (39), where $\gamma ={C_{\text{p}}}/{C_{\text{v}}}$ and ${C_{\text{p}}}$ and ${C_{\text{v}}}$ are the heat capacity at constant pressure and volume, respectively. The values of ${C_{\text{p}}}$ are calculated using temperature-dependent polynomial curve fits from Kee et al. [64], and ${C_{\text{v}}}$ is calculated as ${C_{\text{v}}}={C_{\text{p}}}-R$, where $R$ is the universal gas constant.

When all vibrational modes are active, H₂O has 12 total internal degrees of freedom (3 translational, 3 rotational, and 6 vibrational), which leads to 3/2 ≤ ${c_{\operatorname{int} }}$ ≤ 9/2 over a full range of temperatures. The calculation of ${c_{\operatorname{int} }}$ for H₂O using Eq. (39) at any given temperature is an estimation, thus the sensitivity of the synthetic FRS signals to the specified value of ${c_{\operatorname{int} }}$ is assessed. Figure 13 shows calculated synthetic FRS signals from pure H₂O vapor at P = 1 atm over the full range of possible values of ${c_{\operatorname{int} }}$ (solid lines) at four different temperatures. The synthetic FRS signals for each value of ${c_{\operatorname{int} }}$ are normalized by the average synthetic FRS signal over the full range of ${c_{\operatorname{int} }}$ values. The symbols show the value of ${c_{\operatorname{int} }}$ calculated using Eq. (39) at each temperature. The results show minimal sensitivity of the calculated synthetic FRS signals to the specified value of ${c_{\operatorname{int} }}$ at 1 atm. For example for T > 700 K, varying ${c_{\operatorname{int} }}$ over the entire possible range of values results in variations of ${S_{{\text{FRS}},{{\text{H}}_2}{\text{O}}}}$ of less than 0.5%. At higher pressures (or increased y parameter values), the sensitivity of ${S_{{\text{FRS}},{{\text{H}}_2}{\text{O}}}}$ to variations in ${c_{\operatorname{int} }}~$is expected to increase.

Fig. 13

Variations in calculated water vapor FRS signals at $P=1~$ atm as a function of specified ${c_{{\text{int}}}}$ values

Table 2 Temperature-dependent species properties

2.2 Bulk viscosity (µ _B)

Temperature-dependent values of the bulk viscosity can be estimated from relaxation time measurements found within the literature. Assuming that rotational and vibrational modes relax independently, each with a single time scale, the bulk viscosity can be written as

$${\mu _{\text{B}}}={\mu _{{\text{B}},{\text{r}}}}+{\mu _{{\text{B}},{\text{v}}}},$$

(46)

where ${\mu _{{\text{B,r}}}}$ and ${\mu _{{\text{B,v}}}}$ are the rotational and vibrational contributions to the bulk viscosity and given by the following expressions:

$${\mu _{{\text{B,r}}}}={\left( {\gamma - 1} \right)^2}\frac{{{f_{\text{r}}}}}{2}p{\tau _{\text{r}}},$$

(47)

$${\mu _{{\text{B,v}}}}=~{\left( {\gamma - 1} \right)^2}\left( {\frac{{Cv}}{R} - \frac{{{f_{\text{r}}}+3}}{2}} \right)p{\tau _{\text{v}}},$$

(48)

where p is the pressure, τ_r is the rotational relaxation time, and ${\tau _v}$ is the vibrational relaxation time. As discussed above, when the vibrational relaxation times are long compared to the characteristic time scale of the density fluctuations, there is no vibrational–translational energy exchange and the vibrational modes are assumed to be “frozen”. For N₂, O₂, H₂, CH₄, CO, and CO₂, it is assumed that only rotation contributes to µ_B and the bulk viscosity is calculated using Eq. (47) with ${\gamma _{\text{r}}}$ replacing $\gamma$. For water vapor, both rotational and vibrational contributions are included and Eqs. (46)–(48) are used to calculate the bulk viscosity. The rotational and vibrational relaxation times for the considered species at various temperatures are taken from sources within the literature as reported in Table 2. The values of $p{\tau _{\text{r}}}$ and $p{\tau _{\text{v}}}$ are fit to either power law or logarithmic expressions that serve as convenient fits to the data. There is sufficient relaxation time data for all species at room temperature, but for many species the relaxation time data are sparse at higher temperatures. For these cases, existing bulk viscosity models were used to help guide the choice of fit (power law or logarithmic), but were not used as data to determine the fit. When compiling all the entirety of the relaxation rates, $p{\tau _{\text{r}}}$ and $p{\tau _{\text{v}}}$ (for water vapor) were fit to

$$p\tau =A{T^{\text{B}}}+C\ln T+D~.$$

(49)

Table 2 lists the constants for each species, the range of temperatures in which relaxation times were obtained, and the suggested range of temperatures over which the fit of Eq. (49) is valid. For H₂O, the first row of constants correspond to $p{\tau _{\text{r}}}$ and the second row of constants correspond to $p{\tau _{\text{v}}}$. Table 2 also lists the calculated values of ${\mu _{\text{B}}}/\mu$ at 300 K. For N₂, O₂, H₂, CH₄, CO, and CO₂, the values of µ_B/µ derived from Eqs. (47) and (49) compare very favorably with those reported within the literature, including the values inferred from direct measurements of the RBS spectra. For example, the current fit yields a value of ${\mu _{\text{B}}}/\mu$ = 0.74 for N₂, while Gu et al. [14] and Gu and Ubachs [13] determined ${\mu _{\text{B}}}/\mu$ = 0.79 at room temperature. For CO₂, the current fit yields a value of ${\mu _{\text{B}}}/\mu$ = 0.30, while Pan et al. [48], Lao et al. [52], Meijer et al. [53], and Gu et al. [12] determined ${\mu _{\text{B}}}/\mu$ = 0.25, 0.31, 0.39, and 0.38, respectively.

Because of the limited amount of bulk viscosity data, it is not possible to evaluate the accuracy of the temperature-dependent expressions for µ_B at elevated temperatures. Figure 14 shows example synthetic FRS signals as a function of temperature for four different species, N₂, CH₄, CO₂, and H₂O using (1) a constant ${\mu _{\text{B}}}/\mu$ ratio determined at room temperature (solid black line), implying that the temperature dependence of µ_B is exactly that of µ and (2) a temperature-dependent µ_B/µ ratio (solid red line), where µ_B is determined using Eqs. (46)–(49). In Fig. 14, the values of ${\mu _{\text{B}}}/\mu$ are shown with dashed lines. The results shown in Fig. 14 indicate little observable difference between the synthetic FRS signals calculated with the two different sets of ${\mu _{\text{B}}}/\mu$ values, even when there are significant differences in ${\mu _{\text{B}}}$ as is the case for CH₄.

Fig. 14

Normalized synthetic FRS signal curves as a function of temperature for constant µ_B/µ (solid black line) and a temperature-dependent value of µ_B/µ (solid red line) according to Eqs. (46)–(49). The ratio of µ_B/µ for each calculation is given as the dashed lines and shown on the secondary y-axis. The reference condition is N₂ at T = 296 K

To further investigate the sensitivity of the synthetic FRS signals to the chosen value of ${\mu _{\text{B}}}$, the synthetic FRS signals are calculated for all species as a function of temperature for varying values of ${\mu _{\text{B}}}$ The percent difference between the synthetic FRS signal calculated with an arbitrary value of the bulk viscosity, $\mu _{{\text{B}}}^{*}$, and the synthetic FRS signal calculated with the bulk viscosity determined using Eqs. (46)–(49) is defined by

$${\epsilon _{{\mu _{\text{B}}}}}=\frac{{\left| {{S_{{\text{FRS}}}}\left( {\mu _{B}^{*}} \right) - {S_{{\text{FRS}}}}\left( {{\mu _B}} \right)} \right|}}{{{S_{{\text{FRS}}}}\left( {{\mu _B}} \right)}}.$$

(50)

Figure 15 shows the results of varying the value of the bulk viscosity by a factor of 100 (0.1 ≤ $\mu _{{\text{B}}}^{*}/\mu$ ≤ 10) for temperatures of 300 K, 700 K, 1200 K, and 2000 K. For higher temperatures (T ≥ 700 K), the variation of the bulk viscosity leads to negligible variations in the calculated FRS signal over the full range of bulk viscosities tested. Only at room temperature (T = 300 K), do very small values of the bulk viscosity lead to notable changes in the calculated FRS signal. However, over the range of bulk viscosities reported within the literature, variations in µ_B lead to changes in the synthetic FRS signal by less than 1%. Overall, the results shown in Figs. 14 and 15 demonstrate that the calculated FRS signal is insensitive to the value of bulk viscosity, as long as µ_B does not approach zero. However, it should be noted that the FRS signal will likely become increasingly sensitive to the values of µ_B as pressure (y parameter) increases.

Fig. 15

Sensitivity of the synthetic FRS signals to changes in values of bulk viscosity for a T = 300 K, b T = 700 K, c T = 1200 K, and d T = 2000 K