Nongray radiation from gas and soot mixtures in planar plates based on statistical narrow-band spectral model

Abstract

The nongray behavior of combustion products plays an important role in various areas of engineering. Based on the statistical narrow-band (SNB) spectral model with an exponential-tailed inverse intensity distribution and the ray-tracing method, a comprehensive investigation of the influence of soot on nongray radiation from mixtures containing H₂O/N₂+soot, CO₂/N₂+soot, or H₂O/CO₂/N₂+soot was conducted in this paper. In combustion applications, radiation transfer is significantly enhanced by soot due to its spectrally continuous emission. The effect of soot volume fraction up to 1×10^-6 on the source term, the narrow-band radiation intensities along a line-of-sight, and the net wall heat fluxes were investigated for a wide range of temperature. The effect of soot was significant and became increasingly drastic with the increase of soot loading.

Keywords

soot / combustion / SNB model / nongray radiation

Introduction

Radiative heat transfer is the dominant heat transfer mode in combustion systems involving boilers, industrial furnaces, and combustors. It is also very important in relatively small-scale laminar flames. The gray gas model, which assumes the absorption coefficient to be independent of wavelength, generally gives rise to large errors in the calculation of radiation from combustion products containing H₂O, CO₂, CO, and soot [1, 2]. In reality, the absorption coefficients of CO, CO₂, and H₂O vary greatly with wavelength. Therefore, the nongray radiation properties of real gases must be considered to attain good accuracy in the calculations of radiation heat transfer in flames and combustion systems.

Rapid variation in the radiative properties of real gases with wavelength makes it difficult to predict gas radiation heat transfer accurately and efficiently. Nevertheless, a great deal of effort has been made in the last decade. In addition, several spectral models have been developed. These models can be classified into two categories [3]. The first category includes various band models (narrow or wide) that provide gas transmissivity but are not compatible with the differential radiative transfer equation (RTE). Methods belonging to the second category, such as the weighted-sum-of-gray-gases (WSGG) model [4] and the line-by-line (LBL) model, provide the gas absorption coefficient and are compatible with an arbitrary RTE solver. The LBL approach can give the most accurate results for radiative heat transfer in real gases, but it requires enormous computer resources and relies on a molecular absorption database, such as HITRAN [5] (high-resolution transmission) and its extension HITEMP [6] (high temperature). Even with today’s supercomputers, this type of calculation is a formidable task for large-scale three-dimensional problems. The LBL model is only used to generate benchmark solutions for the validation of approximate models.

Statistical narrow-band (SNB) models have been proven to be highly accurate and reliable. A full account of these can be found in Modest [1]. These models were developed with the assumption that the spectral lines are not equally spaced or of equal strength; instead, they are of random strength and are randomly distributed across the narrow band. The Goody SNB model [7] (with an exponential-tailed line intensity distribution) and the Malkmus SNB model [8] (with an exponential-tailed inverse intensity distribution) are the two best known and successful SNB models. Both models are based on the hypothesis that the positioning of independent lines within subdivisions of the infrared spectrum is random with a Lorentz profile. Malkmus SNB model is found to be somewhat superior to the Goody model [1,3,9]. Recently, the two SNB models were compared, with the Malkmus SNB model again found to be superior to the Goody SNB model [10]. Therefore, the Malkmus SNB model was employed in this work.

Combustion products often contain not only radiating gases but also soot, which primarily contains elemental carbon with very small amounts of hydrogen and oxygen and appears as fractal aggregates formed by nearly spherical primary particles whose diameters range from 10 to 60 nm. Primary particles fall in the Rayleigh regime for thermal radiation at a temperature relevant to combustion. Unlike gases, soot is continuously absorbed and emitted across the entire spectrum. It has been well established that the existence of soot significantly enhances thermal radiation. In recent years, various studies have been conducted to investigate the effects of soot on radiation heat transfer, flame properties, and soot diagnostics [1,11–19]. Liu et al. [14] employed an SNB correlated-K method for the radiative properties of CO, H₂O, CO₂, and soot in order to investigate their respective effects on the computed soot field and flame structure. The luminous emission of continuum radiation from incandescent soot particles is one of the more distinctive features of nonpremixed hydrocarbon flames. Brookes and Moss [17] focused on the intimate coupling between soot production rate and flame radiative heat loss for well-documented jet diffusion flames, burning methane at atmospheric and elevated pressures. Based on the RADCAL program, Yan and Holmstedt [18] developed the FASTNB model to study radiation from H₂O, CO₂, and soot. However, detailed analyses of the role of soot are still lacking, especially its influence on spectrally resolved narrow-band radiation intensities. Nevertheless, this study makes an attempt to gain further understanding on the effect of soot on nongray radiation, including narrow-band radiation intensities.

In this paper, attention is given to the influence of soot on nongray total and spectrally resolved (narrow-band) radiation intensities at different levels of soot loading using the Malkmus SNB model and the ray-tracing method. The methodology of the ray-tracing/SNB model has been previously used by Marakis [3] and Liu et al. [20]. Based on the SNB spectral model with an exponential-tailed inverse intensity distribution and the ray-tracing method, a comprehensive investigation of the influence of soot on nongray radiation from gas mixtures (H₂O/N₂, CO₂/N₂, or H₂O/CO₂/N₂) was conducted. In combustion applications, radiation transfer is significantly enhanced by soot due to its spectrally continuous emission. The effect of soot volume fraction up to 1×10^-6 on the source term, the narrow-band radiation intensities along a line-of-sight, and the net wall heat fluxes were investigated for a wide range of temperature. The effect of soot was significant and became increasingly drastic with the increase of soot loading.

Formulation

Narrow band formulation

For emitting-absorbing media, the RTE and its corresponding boundary condition for the spectral radiation intensity at a diffuse wall can be written as follows [21]:

(1){\displaystyle \frac{\partial I_{\nu }\left(s,\mathrm{\Omega }\right)}{\partial s}}=-{\kappa }_{\mathrm{a}\nu }\left(s\right)I_{\nu }\left(s,\mathrm{\Omega }\right)+{\kappa }_{\mathrm{a}\nu }\left(s\right)I_{\mathrm{b}\nu }\left(s\right),

I_{\nu }\left(s_{\mathrm{w}},\mathrm{\Omega }\right)={ϵ}_{\mathrm{w}\nu }{I}_{b\mathrm{w}\nu }+{\displaystyle \frac{\left(\mathrm{1}-{ϵ}_{\mathrm{w}\nu }\right)}{\mathrm{\pi }}}

·{\displaystyle {\int }_{\hat{n}\cdot {\mathrm{\Omega }}^{\prime }}\left|\widehat{n}\cdot {\mathrm{\Omega }}^{\prime }\right|}{I}_{\nu }\left(s_{\mathrm{w}},{\mathrm{\Omega }}^{\prime }\right)\mathrm{d}{\mathrm{\Omega }}^{\prime },

(2){\displaystyle }\mathrm{f}\mathrm{o}\mathrm{r}\left|\widehat{n}\cdot {\mathrm{\Omega }}^{\prime }\right|>\mathrm{0},

where {\kappa }_{\mathrm{a}\nu }\left(s\right) is the absorption coefficient, I_{\mathrm{b}\nu }\left(s\right) is the blackbody radiative intensity, and {ϵ}_{\mathrm{w}\nu } is wall emissivity.

When the Malkmus SNB model is used to provide gas radiative properties, the RTE must be transformed into an integral form for the band averaged radiation intensity to make use of the gas transmissivity. The narrow-band averaged RTE has been presented by Kim et al. [22]:

(3{{\rm\tf="fp1"\char"0061}}){\displaystyle \frac{\partial {\overline{I}}_{\nu }\left(s,\mathrm{\Omega }\right)}{\partial s}}=-\overline{{\kappa }_{\mathrm{a}\nu }\left(s\right)I_{\nu }\left(s,\mathrm{\Omega }\right)}+{\overline{\kappa }}_{\mathrm{a}\nu }\left(s\right){\overline{I}}_{\mathrm{b}\nu }\left(s\right),

where

\overline{{\kappa }_{\mathrm{a}\nu }\left(s\right)I_{\nu }\left(s,\mathrm{\Omega }\right)}=\overline{I_{\mathrm{w}\nu }\left(s_{\mathrm{w}},\mathrm{\Omega }\right){\kappa }_{\mathrm{a}\nu }\left(s\right){\tau }_{\nu }\left(s_{\mathrm{w}}\to s\right)}

+{\displaystyle {\int }_{s_{\mathrm{w}}}^s\overline{{\kappa }_{\mathrm{a}\nu }\left(s\right){\kappa }_{\mathrm{a}\nu }\left(s^{\prime }\right){\tau }_{\nu }\left(s^{\prime }\to s\right)}}{\overline{I}}_{\mathrm{b}\nu }\left(s^{\prime }\right)\mathrm{d}{s}^{\prime },

(3{{\rm\tf="fp1"\char"0062}})

(3{{\rm\tf="fp1"\char"0063}}){\tau }_{\nu }\left(s^{\prime }\to s\right)=\exp \left(-{\displaystyle \underset{s^{\prime }}{\overset{s}{\int }}{\kappa }_{\mathrm{a}\nu }\left(s^{″}\right)}\mathrm{d}{s}^{″}\right),

where the overbar symbol indicates quantities averaged over a narrow-band, s_w is the starting point along the line-of-sight at the wall, s is either the end point along the line-of-sight in the absorbing-emitting medium or an indefinitely small path length along the line-of-sight, {\tau }_{\nu } is the spectral transmittance, and I_{\mathrm{w}\nu } is the spectral radiation intensity at the boundary wall.

Following the development of Kim et al. [22], the narrow-band averaged RTE, in terms of transmissivity, can be written as

(4){\displaystyle \frac{\partial {\overline{I}}_{\nu }\left(s,\mathrm{\Omega }\right)}{\partial s}}={\left({\displaystyle \frac{\partial \overline{{\tau }_{\nu }\left(s^{\prime }\to s\right)}}{\partial s^{\prime }}}\right)}_{s^{\prime }=s}{\overline{I}}_{\mathrm{b}\nu }\left(s\right)+{\overline{I}}_{\mathrm{w}\nu }\left(s_{\mathrm{w}},\mathrm{\Omega }\right){\displaystyle \frac{\partial }{\partial s}}\left[{\overline{\tau }}_{\nu }\left(s_{\mathrm{w}}\to s\right)\right]+{\displaystyle {\int }_{s_{\mathrm{w}}}^s{\displaystyle \frac{\partial }{\partial s}}}\left({\displaystyle \frac{\partial \overline{{\tau }_{\nu }\left(s^{\prime }\to s\right)}}{\partial s^{\prime }}}\right){\overline{I}}_{\mathrm{b}\nu }\left(s^{\prime }\right)\mathrm{d}{s}^{\prime },

where Ω is the solid angle along the line-of-sight, and a simplification of

\overline{I_{\mathrm{w}\nu }\left(s_{\mathrm{w}},\mathrm{\Omega }\right){\kappa }_{\mathrm{a}\nu }\left(s\right){\tau }_{\nu }\left(s_{\mathrm{w}}\to s\right)}\approx {\overline{I}}_{\mathrm{w}\nu }\left(s_{\mathrm{w}},\mathrm{\Omega }\right)\overline{{\kappa }_{\mathrm{a}\nu }\left(s\right){\tau }_{\nu }\left(s_w\to s\right)}

is made for high-emissivity walls \left({ϵ}_w\to \mathrm{1.0}\right).

Along a line-of-sight for high-emissivity surrounding walls, the discretized form of Eq. (4) is given as follows (see Kim et al. [22]):

(5){\overline{I}}_{\nu ,n,i+\mathrm{1}}={\overline{I}}_{\nu ,n,i}+\left(\mathrm{1}-{\overline{\tau }}_{\nu ,n,i\to i+\mathrm{1}}\right){\overline{I}}_{\mathrm{b}\nu ,i+\mathrm{1}/\mathrm{2}}+{\overline{C}}_{\nu ,n,i+\mathrm{1}/\mathrm{2}},

where

{\overline{C}}_{\nu ,n,i+\mathrm{1}/\mathrm{2}}={\overline{I}}_{\mathrm{w}\nu ,n,\mathrm{1}}\left({\overline{\tau }}_{\nu ,n,\mathrm{1}\to \mathrm{}i+\mathrm{1}}-{\overline{\tau }}_{\nu ,n,\mathrm{1}\to \mathrm{}i}\right)

+{\displaystyle \sum _{k=\mathrm{1}}^{i-\mathrm{1}}[\left({\overline{\tau }}_{\nu ,n,k+\mathrm{1}\to \mathrm{}i+\mathrm{1}}-{\overline{\tau }}_{\nu ,n,k+\mathrm{1}\to \mathrm{}i}\right)}

(6){\displaystyle -\left({\overline{\tau }}_{\nu ,n,k\to \mathrm{}i+\mathrm{1}}-{\overline{\tau }}_{\nu ,n,k\to \mathrm{}i}\right)\backslash bigr]}{\overline{I}}_{\mathrm{b}\nu ,k+\mathrm{1}/\mathrm{2}}.

Equation (5) has the same boundary condition expression as that given in Eq. (2).

Once the spectral radiation intensity is calculated, the total (spectrally integrated) net radiative flux can be given by

(7)q\left(x_i\right)={\displaystyle \sum _{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{\Delta }v}\left({\displaystyle \sum _{n=\mathrm{1}}^N\mu }{\overline{I}}_{\nu ,n,i}{w}_n\right)}\mathrm{\Delta }v.

Furthermore, the radiative source term, or the divergence of the heat flux, of this one-dimensional problem can be obtained by

(8)-{\displaystyle \frac{\mathrm{d}q}{\mathrm{d}x}}=-{\displaystyle \frac{q_{i+\mathrm{1}}-q_i}{x_{i+\mathrm{1}}-x_i}}.

Statistical narrow-band model

For an isothermal and homogeneous path length L at a total pressure p and gas molar fraction f_{\mathrm{g}}, the narrow-band averaged transmittance is given as (see Modest [1] and Ludwig et al. [23])

(9){\tau }_{\mathrm{g},\nu }\left(L\right)=\exp \left(-W\right),

where W is the equivalent bandwidth, which is expressed for the Malkmus SNB model as

(10)W={\displaystyle \frac{\mathrm{\pi }B}{\mathrm{2}}}\left(\sqrt{\mathrm{1}+{\displaystyle \frac{\mathrm{4}SL}{\mathrm{\pi }B}}}-\mathrm{1}\right),

where B={\mathrm{2}{\overline{\beta }}_{\nu }/\mathrm{\pi }}^{\mathrm{2}}, S={\overline{k}}_{\nu }{f}_{\mathrm{g}}p, and {\overline{\beta }}_{\nu }=\mathrm{2}\mathrm{\pi }{\overline{\gamma }}_{\nu }/{\overline{\delta }}_{\nu }. The mean narrow-band parameters {\overline{\gamma }}_{\nu }, {\overline{\delta }}_{\nu }, and {\overline{k}}_{\nu } for H₂O and CO₂ have been given in [23,24], but these databases are out of date and may yield inaccurate results. More recently, an updated dataset of these parameters has been made available by Soufiani and Taine [25] for CO₂, H₂O, and CO for a much wider temperature range of 300–2900 K. The bandwidth is 25 cm^-1 for wavenumbers between 150 and 9300 cm^-1. Further details of this dataset can be found in Soufiani and Taine [25].

For a nonisothermal and/or an inhomogeneous path, the Curtis-Godson technique [26] is employed to obtain the equivalent band parameters. The overlapping band is treated in a manner as adopted by Kim et al. [22].

Soot particles are sufficiently small to be considered in the Rayleigh regime, where the absorption by soot is proportional to its volume fraction, and its scattering can be neglected. Thus, the absorption coefficient for the soot can be calculated using a model proposed by Hottel and Sarofim [4]:

(11)k_{\mathrm{s},\mathrm{\nu }}=C{f}_{\mathrm{s}}v,

where C is a constant.

Buckius and Tien [27] suggested that C is equal to 5.5 and is not fuel-dependent, and f_{\mathrm{s}} is the soot volume fraction. The value of C is not as critical as f_{\mathrm{s}}. Hottel and Sarofim [4] suggested that the soot volume fraction varies in the range of 10^-8 to 10^-6 for typical hydrocarbon flames. In the present work, the effects of soot for soot volume fractions are investigated for the 0–10^-6 range. Furthermore, soot transmissivity can be expressed as

(12){\tau }_{\mathrm{s},\nu }=\exp \left(-\mathrm{5.5}{f}_{\mathrm{s}}vL\right).

The spectral transmissivity for a mixture of gases and soot can then be written as follows:

(13){\tau }_{\nu }={\tau }_{\mathrm{g},\nu }{\tau }_{\mathrm{s},\nu }.

Results and discussion

In this paper, thermal radiation in a parallel-plane space filled with absorbing-emitting gas mixtures (H₂O/N₂, CO₂/N₂, or H₂O/CO₂/N₂) and soot was studied. The wall surfaces were assumed to be black, and the total gas pressure was kept at one atmosphere for all test cases. The same spatial and angular discretizations were adopted for all test cases. The planar geometry was divided into 20 sublayers, and the polar angle was divided into 20 intervals.

Effect of soot on source term and narrow-band radiation intensities

Isothermal homogeneous medium (Case 1)

An isothermal homogeneous medium filled with water vapor between planar plates bounded by cold black walls was first investigated. In this case, the two walls were held at 0 K. The medium between the two planar plates was filled with a mixture containing 20% water vapor and 80% nitrogen (mole basis) at a uniform temperature of 1500 K. Four levels of soot volume fractions (i.e., 10^-9, 10^-8, 10^-7, and 10^-6), in addition to vanishing soot volume fraction, were considered. The wall separation distance was 2 m.

The calculated radiative source term distribution is illustrated in Fig. 1. For the radiative source term with vanishing soot volume fraction, the results are in good agreement with the LBL and the SLW models, validating the SNB nongray gas model. The results for the LBL and SLW models were obtained by Denison [28] and Solovjov [29], but the predictions of both models are only for a gas mixture with vanishing soot volume fraction. With the increase of soot volume fraction, the gas/soot mixture becomes more absorbing and emitting than the pure gas case. As expected, the radiative source term increasingly deviates from that of the gas-only case with an increase in the soot volume fraction.

Figure 2 illustrates the variation of narrow-band intensities along a line-of-sight (at X = L and along the positive x direction) with different soot volume fractions between 0 and 10^-6. The narrow-band intensities also increase with the increase in soot volume fraction. In the range of 150–1800 cm^-1, as presented in Fig. 2(b), the increase in narrow-band intensity is small for the soot volume fraction range considered. For other wavenumber ranges, as demonstrated in Figs. 2(c) and 2(d), the variation of narrow-band radiation intensity with soot volume fraction is much stronger because the absorption coefficient of soot is proportional to the wavenumber, as shown in Eq. (11).

Nonisothermal homogeneous medium (Case 2)

In this case, a nonisothermal but homogeneous medium was considered, which has a cosine temperature profile given by T=T_{\mathrm{a}\mathrm{v}\mathrm{e}}+\left[\mathrm{\Delta }T\cos \left(\mathrm{\pi }x/L\right)\right]/\mathrm{2} with T_ave = 1500€K and ∆T = 1000 K. The medium contained 10% water vapor balanced by N₂; the two walls were black at 2000 K (left wall) and 1000 K (right wall). The separation distance was 2 m.

The distribution of the radiative source term and narrow-band intensities is shown in Figs. 3 and 4, respectively, for different levels of soot volume fraction. For this case, the contribution of soot is also very important, especially at higher soot volume fractions. In this nonisothermal but homogeneous case, the effect of soot on the source term and narrow-band becomes significant even at moderate soot volume fractions.

Isothermal homogeneous medium (Case 3)

In the third case, the CO₂ mole fraction was uniform at 10% with a uniform temperature of 1500 K and bounded by cold black walls. Soot volume fractions of 10^-9, 10^-8, and 10^-7 (in addition to vanishing soot volume fraction) were studied. The wall separation distance was 0.5 m.

In Figs. 5 and 6, similar conclusions as those for case 1 for water vapor can be reached. There are three peaks for narrow-band intensities of CO₂ in Fig. 6. CO₂ has 96 radiating narrow bands in four spectral regions: 450–1200€cm^-1 (31 bands), 1950–2450 cm^-1 (21 bands), 3300–3800 cm^-1 (21 bands), and 4700–5250 cm^-1 (23 bands). Fig. 6 shows that the three peaks are in three of the four spectral regions of CO₂-radiating bands and that soot does not alter the wavenumbers where the narrow-band intensities peak but strengthens their intensities instead.

Nonisothermal homogeneous medium (Case 4)

In this case, the temperature distribution was similar to that in Case 2, but with T_ave = 1250 K and ΔT = 1000 K. The left wall was hot at 1750 K, whereas the right wall was at 750 K. The medium between the two planar plates was filled with 30% CO₂ balanced by N₂. The layer depth was 0.2 m.

Predictions for this case are shown in Figs. 7 and 8. The same profiles for radiative source and narrow-band intensities as those in Case 2 for water vapor can be observed.

Isothermal inhomogeneous medium (Case 5)

For the last case, the same temperature and medium as in the previous case (Case 4) were considered. The soot volume fractions 10^-9, 10^-8, 10^-7, and 10^-6 (in addition to vanishing soot volume fraction) were studied for a gas mixture of 10% H₂O and 10% CO₂. The wall separation distance was 1 m.

The computational results are presented in Figs. 9 and 10. In Solovjov and Webb [13], the effect of soot was considered (Fig. 9), and their results were used in this paper €as €benchmarks €to €validate €the €calculation. €Figures 9(a), 9(b), 9(c), and 9(d) show that the results are in good agreement. When the soot volume fraction varies, the strength of the radiative source and the narrow-band intensities also vary considerably.

Effect of soot on net wall fluxes

The net wall fluxes at the boundary for Case 5 were tested. The impact of soot on radiative transfer was quantified using the following relation (see Solovjov and Webb [13]):

(14){\mathrm{\Phi }}_{\mathrm{s}\mathrm{o}\mathrm{o}\mathrm{t}}={\displaystyle \frac{q_{\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h},\mathrm{s}\mathrm{o}\mathrm{o}\mathrm{t}}-q_{\mathrm{w}/\mathrm{0},\mathrm{s}\mathrm{o}\mathrm{o}\mathrm{t}}}{q_{\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h},\mathrm{s}\mathrm{o}\mathrm{o}\mathrm{t}}}}\times \mathrm{100}\mathrm{\%}.

The three-dimensional surface contours of {\mathrm{\Phi }}_{\mathrm{s}\mathrm{o}\mathrm{o}\mathrm{t}} are shown in Fig. 11. Note that the net wall fluxes are sensitive to soot volume fraction (in the range of 0–10^-6), and this conclusion was also obtained in Solovjov and Webb [13].

Conclusions

Radiation transfer in a parallel-plane space filled with absorbing and emitting mixture of H₂O, CO₂, and soot for wide ranges of temperature and soot volume fraction (0–10^-6) was studied in this paper. The radiative source term, the wall fluxes, and the narrow-band intensities along a line-of-sight were investigated. Results obtained in this study indicate that soot has a very important impact on thermal radiation heat transfer in high-temperature systems. For all cases, the influence of soot on radiative transfer from a gas mixture is significant. The contribution of soot increases significantly along with the increase in its volume fraction.

In this paper, the assumption of black walls was made, although the boundaries were not always black. This means that the present method needs further improvement. Therefore, the effect of nonblack boundaries should be considered in the future.

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