Calculations of narrow-band transimissities and the Planck mean absorption coefficients of real gases using line-by-line and statistical narrow-band models

Abstract

Narrow-band transmissivities in the spectral range of 150 to 9300 cm^-1 and at a uniform resolution of 25 cm^-1 were calculated using the statistical narrow-band (SNB) model with the band parameters of Soufiani and Taine, the more recent parameters of André and Vaillon, and the line-by-line (LBL) method along with the HITEMP-2010 spectroscopic database. Calculations of narrow-band transmissivity were conducted for gas columns of different lengths and containing different isothermal and non-isothermal CO₂-H₂O-N₂ mixtures at 1 atm. Narrow-band transmissivities calculated by the SNB model are in large relative error at many bands. The more recent SNB model parameters of André and Vaillon are more accurate than the earlier parameters of Soufiani and Taine. The Planck mean absorption coefficients of CO₂, H₂O, CO, and CH₄ in the temperature range of 300 to 2500 K were calculated using the LBL method and different versions of the high resolution transmission (HITRAN) and high-temperature spectroscopic absorption parameters (HITEMP) spectroscopic databases. The SNB model was also used to calculate the Planck mean absorption coefficients of these four radiating gases. The LBL results of the Planck mean absorption coefficient were compared with the classical results of Tien and those from the SNB model.

Keywords

transimissity / HITEMP / HITRAN / Planck mean absorption coefficients

Introduction

The extremely rapid variation of the spectral radiative properties of real gases, primarily CO₂ and H₂O in combustion, makes it difficult to conduct accurate and efficient calculations of radiation heat transfer and spectrally-resolved intensities over a narrow band with a bandwidth of the order of 25 cm^-1. Significant progress has been made in the last two decades to develop accurate and efficient non-gray gas radiation models for gas radiation heat transfer [1]. These models include the spectral-line based weighted-sum-of-gray-gases (SLW) model [2], the statistical narrow-band based correlated-k (SNBCK) models [3], the full-spectrum correlated-k (FSCK) model [4], and the spectral-line moment-based (SLMB) model [5]. For applications where low-resolution spectrally-resolved intensities are required, such as for concentration and temperature measurements by detecting flame emissions at a narrow band [6], the statistical narrow-band (SNB) [7], SNBCK, narrow-band correlated-k (NBK) [8], and the SLMB model [5] can be readily employed. Similar to the NBK model, the SLW model could also be developed for narrow-bands for problems where low-resolution spectrally-resolved intensities are required, though such development has not been made. Calculations of the most accurate line-by-line (LBL) model are conducted at a spectral resolution that nearly resolves individual spectral lines. LBL is the most accurate way to calculate spectrally-resolved and total radiation transfer, though it is in general only used to obtain benchmark solutions due to its very long computing time. Although theoretically LBL is the most accurate model, it is important to realize that its accuracy is dependent on the accuracy of the high-resolution spectroscopic database used in the calculations [9]. Chu et al. [9] have recently conducted a comparative study to investigate the effect of spectroscopic database on the LBL results in radiation heat transfer calculations in 1D parallel-plate enclosures using different versions of HITRAN (1996, 2004, 2008) and HITEMP (1995, 2010) for CO₂ and H₂O [10,11] and the CDSD-1000 database for CO₂ [12]. The SNB results were also included in the comparison using the band parameters of Soufiani and Taine published in 1997 [13], which have been a preferred choice to conduct SNB and SNBCK calculation in recent years. The study of Chu et al. [9] has shown that the LBL results are fairly sensitive to the spectroscopic database and the SNB results are in very good agreement with the LBL results obtained using the HITEMP-2010 database in all the cases considered, which suggests that the SNB data set of Soufiani and Taine [13] is accurate as far as radiation heat transfer is concerned. However, spectrally-resolved low-resolution quantities, such as narrow-band transmissivities or radiation intensities, were not investigated in Ref. [9]. In some applications, such as measurements of CO₂ concentration [6], spectrally-resolved narrow-band transmissivities are required. It is therefore important to validate the accuracy of the SNB model in the prediction of spectrally-resolved narrow-band transmissivity with the LBL results. Hence the first objective of the present study is to evaluate the accuracy of the SNB model in terms of the narrow-band gas tranmissivities at a uniform spectral resolution of 25 cm^-1 in the range of 150 to 9300 cm^-1. In SNB calculations, both the band parameters of Soufiani and Taine [13] and the more recently developed band parameters of André and Vaillon [14] were employed.

The Planck mean absorption coefficients of various radiating gases are an important quantity in radiation heat transfer, especially in situations where the optically thin assumption is valid. The Planck mean absorption coefficients can be calculated from both the LBL model and the SNB model. Zhang and Modest [15] reported Planck mean absorption coefficients of CO₂, H₂O, CO, CH₄, and few other trace species relevant to combustion based on LBL calculations using the HITRAN-1996 and HITEMP-1995 spectroscopic databases. Since in the study of Zhang and Modest [15], both the HITRAN and HITEMP databases have been updated, it is desirable to obtain updated Planck mean absorption coefficients based on these recently available spectroscopic databases, namely HITRAN-2004, HITRAN-2008, and HITEMP-2010. The second objective of the present study is to report Planck mean absorption coefficients of the most important radiating gases in combustion, namely CO₂, H₂O, CO, and CH₄, calculated from the LBL model using the three recent versions of the HITRAN and HITEMP databases as well as the carbon dioxide spectroscopic databank (CDSD) database (for CO₂ only) and to validate the Planck mean absorption coefficients calculated by the SNB model using the band parameters of Soufiani and Taine [13] and André and Vaillon [14].

Formulation

LBL spectral absorption coefficient

In the LBL approach, the spectral absorption coefficient at a wavenumber\eta,k_{\eta }, contributed by all individual lines is given as [16]

(1)k_{\eta }=N(T,p){\displaystyle \sum _i{S}_i(T)F_i(\eta )},

where N is the molecule number density of the radiating species under consideration, S_i is the line intensity of the ith line, andF_i(\eta ) is the Lorentzian line shape profile. Further details of the LBL approach can be found in Refs. [8,9,16]. The line parameters for a radiating species, such as H₂O and CO₂, can be obtained from a high resolution spectroscopic database. For a gas mixture containing more than one radiating species, the spectral absorption coefficient of the mixture is simply the summation of contribution from all the radiating species.

Following the recent study [9], a uniform spectral resolution of 0.02 cm^-1 was chosen to conduct LBL calculations. Furthermore, a spectral distance of 20 half-width at half-maximum (HWHM) between the spectral line center and the wavenumber under consideration was used as the cut-off distance, i.e., if the spectral distance between the center of a spectral line and the spectral location of interest is greater than 20 HWHM the contribution of this spectral line to the absorption coefficient at the wavenumber under consideration is neglected. To gain a quantitative understanding of the error caused by using a cut-off of 20 HWHM, additional calculations were conducted using a cut-off of 50 HWHM for the total tranmissivity of pure water at 1000 K and an optical path of 0.1 m∙atm. It was found that the difference between the two total transmissivity is only 2%, justifying the use of 20 HWHM as the cut-off spectral distance in the LBL calculations.

Narrow-band transmissivity of gases

LBL approach

In the LBL approach the spectral transmissivity over a non-isothermal and/or inhomogeneous path is given as [16]

(2){\tau }_{LBL,\eta }=\exp \left(-{\displaystyle \sum _i^M{k}_{\eta ,i}d{x}_i}\right),

where the summation is over the discretized intervals along the radiation path and the subscript i indicates the center of each discretized interval. The total number of intervals along the path is fixed at M = 20 in the present work.

The low-resolution narrow-band transmissivity can be obtained as

(3){\overline{\tau }}_{LBL,{\eta }_{\mathrm{0}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{\Delta }\eta }}{\displaystyle {\int }_{{\eta }_{\mathrm{0}}-(\mathrm{\Delta }\eta /\mathrm{2})}^{{\eta }_{\mathrm{0}}+(\mathrm{\Delta }\eta /\mathrm{2})}{\tau }_{LBL,\eta }}d\eta ,

where the integration was performed with a uniform wavenumber interval of 0.02 cm^-1, which is the spectral resolution used in the LBL calculations of the spectral absorption coefficient. Furthermore, \mathrm{\Delta }\eta =\mathrm{25}cm^-1 was chosen as the narrow band width to provide a direct comparison with the results from the SNB model, whose model parameters were compiled for a uniform bandwidth of 25 cm^-1. η₀ stands for the center of a narrow-band.

For a mixture of gases, the LBL transmissivity can be calculated as

(4){\overline{\tau }}_{LBL,{\eta }_{\mathrm{0}}}={\displaystyle \prod _{j=\mathrm{1}}^n{\overline{\tau }}_{LBL,{\eta }_{\mathrm{0}},j}}(j=\mathrm{1},\mathrm{2},\cdots ,n),

where the product is over all the radiating species present along the path and the subscript j stands for the jth radiating species.

SNB model

For an isothermal and homogeneous path-length L containing a radiating gas at a mole fraction X and total pressure p, the SNB model provides the narrow band averaged transmissivity given as [17]

(5){\overline{\tau }}_{SNB,\eta }(L)=\exp \left[-{\displaystyle \frac{\pi B}{\mathrm{2}}}\left(\sqrt{\mathrm{1}+{\displaystyle \frac{\mathrm{4}{\overline{k}}_{\eta }XpL}{\pi B}}}-\mathrm{1}\right)\right],

where L is the path length, B=\mathrm{2}{\overline{\beta }}_{\eta }/{\pi }^2, and {\overline{\beta }}_{\eta }=\mathrm{2}\pi {\overline{\gamma }}_{\eta }/{\overline{\delta }}_{\eta }. The mean narrow band parameters {\overline{\gamma }}_{\eta }, {\overline{\delta }}_{\eta } and {\overline{k}}_{\eta } for CO, CO₂, and H₂O have been reported by Soufiani and Taine [13] as an updated SNB model data set based on a high resolution spectral database developed at EM2C and more recently by André and Vaillon [14] based on LBL calculations using HITEMP-2010 [11] and CDSD-1000 [12].

For a non-isothermal and/or inhomogeneous path, the Curtis-Godson approximation [18] was employed. The overlapping band of gas mixture is treated in a manner as adopted by Kim et al. [19], i.e., using the multiplication property of transmissivity.

Planck mean absorption coefficient

LBL approach

The Planck mean absorption coefficient is defined as [1]

(6){\kappa }_P={\displaystyle \frac{{\displaystyle {\int }_{\mathrm{0}}^{\infty }{I}_{b\eta }{k}_{\eta }d\eta }}{{\displaystyle {\int }_{\mathrm{0}}^{\infty }{I}_{b\eta }d\eta }}}={\displaystyle \frac{\pi }{\sigma T^{\mathrm{4}}}}{\displaystyle {\int }_{\mathrm{0}}^{\infty }{I}_{b\eta }{k}_{\eta }d\eta },

where k_{\eta } and I_{b\eta } are the spectral absorption coefficient and the spectral blackbody intensity at a wave number\eta. Based on the definitions of the Planck mean absorption coefficient and the LBL spectral absorption coefficient, Zhang and Modest [15] showed that the Planck mean absorption coefficient can be calculated simply as

(7){\kappa }_{LBL,P}={\displaystyle \frac{\pi }{\sigma T^{\mathrm{4}}}}{\displaystyle {\int }_{\mathrm{0}}^{\infty }{I}_{b\eta }{k}_{\eta }d\eta }={\displaystyle \frac{N\pi }{\sigma T^{\mathrm{4}}}}{\displaystyle \sum _i{I}_{b\eta ,i}}{S}_i(T),

where the summation is over all the absorbing lines. The line intensity S_i was defined as

S_i={\displaystyle {\int }_{\mathrm{0}}^{\infty }{\kappa }_{\eta ,i}d\eta },

which is provided directly from a spectroscopic database. In the derivation of the second part of Eq. (7), it is implicitly assumed that the spectral blackbody intensity varies much slower than individual absorption line profile, which is evidently valid. The advantage of Eq. (7) over Eq. (6) is that there is no need to evaluate the spectral absorption coefficient to obtain the Planck mean absorption coefficient.

SNB model

The Planck mean absorption coefficient can also be obtained by the SNB model. In Ref. [20], Ju et al. have outlined an approach to calculate the Planck mean absorption coefficient using the SNB model. However, their approach requires calculations of radiation heat transfer between two parallel plates of a very small separation distance. In the present study, a convenient expression was used to calculate the Planck mean absorption coefficient based on the definition of the mean line-intensity to spacing ratio {\overline{k}}_{\eta }. The Planck mean absorption coefficient calculated from the SNB model can be written as

(8){\kappa }_{SNB,P}={\displaystyle \frac{\pi }{\sigma T^{\mathrm{4}}}}{\displaystyle {\int }_{\mathrm{0}}^{\infty }{I}_{b\eta }{k}_{\eta }d\eta }={\displaystyle \frac{\pi }{\sigma T^{\mathrm{4}}}}{\displaystyle \sum _{i=\mathrm{1}}^J{I}_{b\eta ,i}}{\overline{k}}_i\mathrm{\Delta }\eta ,

where \mathrm{\Delta }\eta =\mathrm{25}cm^-1, J is the total number of narrow bands, and {\overline{k}}_i is the mean line-intensity to spacing ratio of the ith narrow-band and is available directly from the SNB parameters developed by Soufiani and Taine at EM2C [13] or by André and Vaillon [14]. Equation (8) is derived from the definition of {\overline{k}}_i, which is related to the summation of intensity of spectral lines covered within the band width of \mathrm{\Delta }\eta =\mathrm{25}cm^-1 [21]. Both the SNB data set of Soufiani and Taine [13] and André and Vaillon [14] cover a wide temperature range from 300 to 2900 K for a uniform bandwidth of 25 cm^-1 between 150 and 9300 cm^-1. H₂O absorbs and emits radiation at all of the 367 narrow-bands while CO₂ has 96 radiating bands in the following four spectral regions: 450 to 1200 cm^-1 (31 bands), 1950 to 2450 cm^-1 (21 bands), 3300 to 3800 cm^-1 (21 bands), and 4700 to 5250 cm^-1 (23 bands). For CO, the centers of the contributing bands range from 1750 to 2325 cm^-1 (24 bands) and from 3775 to 4350 cm^-1 (24 bands). The total number of CO bands is 48. As an extension of the EM2C SNB parameters for CO, CO₂, and H₂O described in Ref. [13], the SNB parameters for CH₄ was later developed by Perrin and Soufiani [22] covering a temperature range of 300 to 2000 K for a uniform bandwidth of 25 cm^-1 between 25 and 6200 cm^-1. CH₄ absorbs and emits radiation at all of the 248 narrow-bands.

Results and discussion

Narrow-band transmissivity

The narrow-band gas transmissivities between two parallel-plates containing a mixture of CO₂, H₂O, and N₂ along the normal direction at a uniform spectral resolution of 25 cm^-1 in the range of 150 to 9300 cm^-1 were calculated using the LBL approach along with the most up-to-date HITEMP-2010 database and the SNB model with the band parameters of Soufiani and Taine [13] and André and Vaillon [14]. The purpose of conducting these calculations is to validate these two sets of SNB parameters in the prediction of spectrally-resolved quantities through a direct comparison between the SNB and LBL results.

Calculations were conducted in three representative cases to assess the accuracy of the SNB model in the prediction of narrow-band gas transmissivity. The total pressure in all three cases is 1 atm. The first two cases contain an isothermal and homogeneous medium: pure water at 1500 K in Case 1 and a mixture of CO₂ and N₂ with X_{{CO}_2}=\mathrm{0.}1, X_{N_2}=\mathrm{0.}9, and T = 1500 K in Case 2. In both cases a fairly long path-length of L = 2 m was used to better illustrate the deviations of the SNB results from the LBL ones. The third case deals with a non-isothermal and inhomogeneous mixture of CO₂, H₂O, and N₂ with details given in Ref. [9] for the oxy-fuel combustion scenario.

Figure 1 displays the narrow-band transmissivities calculated for Case 1 for a separation distance of L = 0.1, 1.0 and 2.0 m. It can be found in Fig. 1 that the differences become lager with the increasing of the separation distance. For L = 2.0 m, also plotted in Fig. 1 are the difference between the SNB and LBL results, i.e.,\left({\overline{\tau }}_{SNB,\eta }-{\overline{\tau }}_{LBL,\eta }\right), and the relative error of the SNB results. Since the relative errors are very large at some bands where the transmissivities are very small, the relative errors are displayed only in the range of - 50% to 50%. It is evident that there are large discrepancies between the SNB and LBL results, with the SNB results calculated using the more recent parameters of André and Vaillon [14] in general in much better agreement with those of the LBL results, except at few bands. The SNB results based on the parameters of Soufiani and Taine [13] are in large error at many bands. It is worth pointing out that the errors in the SNB transmissivity become much smaller for shorter path-lengths, especially when the parameters of André and Vaillon [14] are used. Nevertheless, the results shown in Fig. 1 indicate that narrow-bands must be carefully selected to ensure the accuracy of the SNB model, besides other considerations, for H₂O concentration measurement based on narrow-band transmissivity. The results shown in Fig. 1 also suggest that both sets of SNB parameters for H₂O are not accurate and are subject to improvement, especially those of Soufiani and Taine [13].

The narrow-band transmissivities for Case 2 are compared in Fig. 2 for a separation distance of L = 2 m. The results are plotted up to 5500 cm^-1 since the transmissivities are essentially unity at higher wavenumbers. The deviations of the SNB results obtained from those of the LBL model are again fairly large when the parameters of Soufiani and Taine [13] are used, especially at the 15 μm absorbing band (between approximately 400 and 1100 cm^-1). Errors of the SNB results caused by using the parameters of André and Vaillon [14] are, in general, smaller than those caused by using the parameters of Soufiani and Taine [13], especially in the 15 μm band, (Fig. 2(b)). However, at one narrow-band at η = 2400 cm^-1 the André and Vaillon parameters produce very large error in the SNB transmissivity. The CO₂ parameters of André and Vaillon at η = 2400 cm^-1 seem in serious error. The results shown in Figs. 1 and 2 indicate that the more recent SNB parameters of André and Vaillon [14] are, in general, more accurate than those of Soufiani and Taine [13].

Figure 3 compares the narrow-band transmissivities calculated using the SNB and LBL models, their difference, and the relative errors of the two SNB results for Case 3 with a separation distance of L = 0.5 m. The distributions of species concentrations and temperature which were taken from Ref. [9] are also plotted in Fig. 3(a) for convenience of the readers. Mainly because of the shorter path-length in Case 3, the SNB results are in much better agreement with those of the LBL model compared to the first two cases. In this case, the errors of the two SNB results are very similar in magnitude and are fairly large at small numbers of narrow-bands (See Figs. 3(c) and 3(d)). It should be pointed out that in this non-isothermal and inhomogeneous case the errors of the two SNB results originate from both the model parameters and the Curtis-Godson approximation.

Planck mean absorption coefficients

Figure 4 compares the Planck mean absorption coefficients of CO₂ and H₂O calculated using the LBL model along with several recent versions of the HITRAN and HITEMP spectroscopic databases, the CDSD-1000 database (for CO₂ only), the SNB model with the parameters of Soufiani and Taine [13] and André and Vaillon [14] over the temperature range of 300 to 2500 K. Also plotted in Fig. 4 are the results obtained by Tien [23]. For H₂O there is good agreement among the LBL results of different databases and between the SNB and the LBL results, except at T = 300 K where the two SNB results are lower. Taking the LBL results obtained using the HITEMP-2010 database as the benchmark solution, the SNB results are very accurate. The results of Tien are somewhat higher at temperatures below 1000 K and slightly lower at high temperatures above 2200 K. The LBL results obtained by using HITRAN-1996 are slightly lower at temperatures above 1500 K. Much larger discrepancies can be observed in the LBL Planck mean absorption coefficient of CO₂ calculated using different spectroscopic databases, especially at temperatures above 1000 K. Use of HITEMP-1995 results in much higher Planck mean absorption coefficients of CO₂ at high temperatures, in agreement with the results of Zhang and Modest [15]. Use of the three versions of the HITRAN database, i.e., 1996, 2004, and 2008, leads to smaller Planck mean absorption coefficient of CO₂ at high temperatures, due to the absence of hot-lines in these databases. It is noticed from Fig. 4 that there is very good agreement among the LBL results obtained by using HITEMP-2010 and CDSD-1000 and the two SNB results for CO₂. The close agreement between the LBL results of HITEMP-2010 and CDSD-1000 is expected based on the fact that the HITEMP-2010 CO₂ parameters are extracted from CDSD-1000 by rescaling intensities from 1000 K to 296 K [11]. The results obtained by Tien for CO₂ are, in general, higher than the benchmark solution (LBL results based on the HITEMP-2010 spectroscopic database), especially at temperatures below approximately 1600 K. The results shown in Fig. 4 suggest that the LBL results calculated by using HITRAN or HITEMP-1995 databases are in large error for the Planck mean absorption coefficients of CO₂. Both the SNB parameters of Soufiani and Taine [13] and André and Vaillon [14] for CO₂ and H₂O are accurate as far as their Planck mean coefficients are concerned.

The Planck mean absorption coefficients of CO and CH₄ are shown in Fig. 5. The Planck mean absorption coefficient of CO was calculated using the LBL model and the five spectroscopic databases considered, namely, HITRAN-1996, HITRAN-2004, HITRAN-2008, HITEMP-1995 and HITEMP-2010, the SNB model using the parameters of Soufiani and Taine [13] and André and Vaillon [14]. For methane, only the three versions of HITRAN were employed, since methane is not included in the two versions of HITEMP. The Planck mean absorption coefficient of CH₄ was also calculated using the SNB model and the parameters of Perrin and Soufiani [22]. The LBL results of the Planck mean absorption coefficient of CO obtained by using different versions of HITRAN or HITEMP were in excellent agreement. The results of Tien [23] are somewhat higher than the LBL results, especially at low and high temperatures. The two SNB results and those of the LBL model are also in good agreement, especially at temperatures above 1200 K. At lower temperature (below 1200 K), the SNB results calculated using the parameters of André and Vaillon [14] are in better agreement with those of the benchmark LBL results. It should be pointed out that the Planck mean absorption coefficients of CO reported by Ju et al. [20] are erroneous. The LBL results of the CH₄ Planck mean absorption coefficient calculated using HITRAN-2004 and HITRAN-2008 are in excellent agreement. The LBL results obtained by using HITRAN-1996 are somewhat lower than those based on HITRAN-2004 and HITRAN-2008, especially in the temperature range of 600 to 1600 K. The results of Tien [23] for CH₄ are significantly higher than the LBL results obtained by using HITRAN-2004 and HITRAN-2008, but in good agreement with those of the SNB results calculated by using the parameters of Perrin and Soufiani [22] at temperatures (above 1500 K). It is interesting to observe that the SNB results obtained by using the parameters of Perrin and Soufiani are in good agreement with those of the LBL model at low temperature (below 600 K) but in good agreement with those of Tien at high temperatures (above 1500 K). Nevertheless, the Planck mean absorption coefficients of CH₄ at temperatures above 600 K are subject to uncertainties and further studies are required.

Conclusions

Narrow-band transmissivities and the Planck mean absorption coefficients were calculated using the LBL and SNB models. The following conclusions can be reached from the present results:

1) For relatively long paths and at temperatures relevant to combustion the narrow-band transmissivities over an isothermal column containing H₂O or CO₂ calculated by the SNB model using the parameters of Soufiani and Taine are in large error at many narrow bands for H₂O and at mainly the 15 μm absorbing band for CO₂.

2) The recent SNB model parameters of André and Vaillon are more accurate than those of Soufiani and Taine for both CO₂, especially at the 15 μm absorbing band, and H₂O. However, both sets of SNB parameters are not sufficiently accurate to predict reliable narrow-band transmissivities over a long path containing H₂O of high concentrations and at high temperatures. For CO₂ the parameters of André and Vaillon at 2400 cm^-1 are in serious error. Both the SNB model parameters of Soufiani and Taine and André and Vaillon are subject to improvement.

3) The Planck mean absorption coefficients of CO₂ and H₂O calculated using the SNB model with the parameters of Soufiani and Taine and André and Vaillon are in good agreement with each other and with the benchmark solution generated using the LBL model with the HITEMP-2010 spectroscopic database.

4) The LBL Planck mean absorption coefficients of H₂O and CO are quite insensitive to the spectroscopic databases. However, the LBL Planck mean absorption coefficients of CO₂ and CH₄ exhibit much stronger dependence on the spectroscopic databases. The Planck mean absorption coefficients of CH₄ at temperatures above 600 K have not been well established.

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