Radiative Properties of Soot Particles
Following from: The Mie solution for spherical particles, The scattering problem for cylindrical particles, Rayleigh scattering
Soot particles are produced in fuel-rich flames, or fuel-rich parts of flames, as a result of incomplete combustion of hydrocarbon fuels. Soot consists of nearly monodisperse spherical primary particles that collect into mass fractal aggregates having a broad size distribution. The primary soot particles are usually very small. By combustion of gaseous or liquid fuel, the diameters of primary soot particles are usually in the range between 5 and 80 nm. One can observe considerably more large particles in a mazut flame as a result of the thermal decomposition of fuel droplets (Blokh, 1988). The primary soot particles produced in gas flames are nearly spherical, but beyond a certain point in their evolution they grow by agglomeration, forming chain-like aggregates composed of essentially spherical primary particles. The formation of soot, the chemical composition, and the morphology of soot particles have been studied experimentally over the years and there is a large body of research on this subject (Haynes and Wagner, 1981; Wagner, 1981; Santoro et al., 1987; Kent and Honnery, 1990; Köylü and Faeth, 1992; Köylü et al., 1995; Köylü, 1997; Zhang and Megaridis, 1998; Smooke et al., 1999; Kronenburg et al., 2000; Appel et al., 2001; Manzello and Choi, 2002; Allouis et al., 2003; Kim et al., 2004, 2006; Pickett and Siebers, 2004; van der Wal and Tomasek, 2004; Tsurikov et al., 2005; Yang et al., 2005; Crookes, 2006; Lombaert et al., 2006; Jensen et al., 2007; Lapuerta et al., 2007; Lockett and Woolley, 2007; Tree and Svensson, 2007; Leylegian, 2008; Watanabe et al., 2008). One can also remember some attempts of theoretical modeling of soot formation (Brookes and Moss, 1999; Wen et al., 2003; Tao et al., 2004; Balthasar and Frenklach, 2005; Lignell et al., 2007; Paterson and Kraft, 2007).
Soot in combustion products of hydrocarbon fuels has a complex chemical composition. In soot, 97-99% of the mass is carbon, but soot also contains hydrocarbons and other substances. The chemical composition and structure of particles are connected in a complicated manner with type of fuel, as well as with the combustion conditions. As a result, the optical constants of soot may be different and may not coincide with the optical constants of pure amorphous carbon. The experiments done by Manzello and Choi (2002) concerning buoyancy-induced flows in normal-gravity flames and some microgravity droplet flames have demonstrated that the morphological properties of soot produced in microgravity droplet flames are significantly different from those measured in normal-gravity conditions. The formation of soot particles in diesel engines is also very sensitive to the engine operating conditions and physical parameters of the gas medium (Pickett and Siebers, 2004; Lapuerta et al., 2007).
Airborne soot and soot-containing aerosols are important components of the atmosphere (Harrison and van Grieken, 1998; Brown et al., 2003; Gelencser, 2005). These particles may affect the weather and climate on our planet mainly due to absorption and scattering of solar radiation and infrared thermal radiation of the earth’s surface and the atmosphere. For this reason, researchers working in the field of atmospheric optics and ecology are also involved in studies of morphology and radiative properties of soot particles (Pueschel, 1996; Gorbunov et al., 2001; Hasegawa and Ohta, 2002; Schnaiter et al., 2003; Sorokin and Arnold, 2004; Kis et al., 2006; Chakrabarty et al., 2007; Liu and Mishchenko, 2007).
Of course, the role of soot particles in combustion remains as one of the main motivations of both experimental and theoretical studies of soot radiative properties. Since soot particles are very small, they are generally at the same temperature as ambient gas and, therefore, strongly emit thermal radiation in a continuous spectrum over the infrared region. Experiments have shown that soot emission is often considerably stronger than the emission from combustion gases. Due to the practical importance of the heat transfer problems in furnaces and in the flame emission investigations, much attention has been given to soot optical constants in the visible and near-infrared spectral ranges (Stull and Plass, 1960; Millikan, 1961; Howarth et al., 1966; Boynton et al., 1968; Foster and Hovarth, 1968; Dalzell and Sarofim, 1969; Medalia and Richards, 1972; Graham, 1974; Janzen, 1979; Pluchino et al., 1980; Bard and Pagni, 1981; Lee and Tien, 1981; Ben Hamadi et al., 1987; Charalampopoulos and Felske, 1987; Chang and Charalampopoulos, 1990; Wu et al., 1997; Krishnan et al., 2000).
In one of the first studies, Stull and Plass (1960) assumed that the values of n and κ for amorphous carbon were used as that for soot. The following dispersion equations were obtained for a temperature of 2250 K:
(1a) |
(1b) |
where ω = 2πc / λ is the radiation frequency. Equations (1a) and (1b) were used by a number of authors for calculations of the soot radiative properties not only for the usual flames, but also for rocket engine exhaust jets (Boynton et al., 1968).
It should be noted that the soot optical constants may differ from the optical constants of pure carbon due to the presence of hydrocarbons in soot. This statement is confirmed by the experimental data present by Millikan (1961), Foster and Hovarth (1968), and Dalzell and Sarofim (1969). Dalzell and Sarofim (1969) obtained the optical constants of soot by using the measurements of polarized light reflection from a specimen surface. Soot was collected from a cooled copper plate placed in acetylene or propane flame. Small portions of soot were pressed in tablets used in the measurements. Analysis of the carbon-hydrogen composition was conducted for every soot type. The values of n and κ for acetylene soot in the infrared range appeared on the average higher than the corresponding values for propane soot. This is explained by a three times higher consistence of hydrogen in propane soot. The following dispersion model for propane soot with the coefficients obtained from the experimental results has been proposed:
(2a) |
(2b) |
Here, F is the number of effective electrons per unit volume, g is the electron damping constant, ω_{j} is the natural frequency of electron of j type, and γ = e^{2} / mε_{0} = 3.183 × 10^{3} m^{3}/s^{2} (where e and m are the electron charge and mass, respectively; and ε_{0} is the electric permittivity). The subscript “c” refers to conduction electrons and “j” refers to bound electrons. Limiting by j ≤ 2, Dalzell and Sarofim (1969) obtained the dispersion model constants presented in Table 1. These constants are used both at room temperature and at combustion temperatures. In general, the temperature dependence of the optical constants of soot is insignificant (about 1% per 100 K), and it is usually ignored (Howarth et al., 1966).
TableÂ 1.Â Constants in dispersion model (2)
Electron | F (m^{-3}) | g (s^{-1}) | ω (s^{-1}) |
C | 4.06 × 10^{27} | 6.00 × 10^{15} | - |
1 | 2.69 × 10^{27} | 6.00 × 10^{15} | 1.25 × 10^{15} |
2 | 2.86 × 10^{28} | 7.25 × 10^{15} | 7.25 × 10^{15} |
In the review of the soot optical properties presented by Bard and Pagni (1981), these authors noted that the data obtained by Dalzell and Sarofim (1969) may be in error due to cavities in soot tablets. According to Medalia and Richards (1972), the pressed soot in the experiments by Dalzell and Sarofim (1969) were comprised of one-third air. As estimated by Graham (1974), this could result in 20% smaller values of n and κ. Bard and Pagni (1981), considering a number of other publications, drew the conclusion that the data obtained by Lee and Tien (1981) can be recommended as representing more reliable average values of the soot optical constants for a large number of various hydrocarbon fuels. The effect of the soot formation conditions on the soot optical constants was indirectly confirmed by Charalampopoulos and Felske (1987), who studied the dependences of refraction and absorption indices on the distance from the burner.
Blokh (1988) obtained the following approximate formula for the complex index of refraction of soot in the wavelength region from 1 to 6 μm:
(3) |
where λ is expressed in microns. In a more recent study, Chang and Charalampopoulos (1990) provided the following polynomial expressions, valid for a wide wavelength range 0.4 ≤ λ ≤ 30 μm:
(4a) |
(4b) |
(4c) |
where λ is expressed in microns. The comparison of some different data for the optical constants of soot is presented in Fig. 1. One can see that approximation (3), suggested by Blokh (1988), is in good agreement with the data obtained by Stull and Plass (1960), and the data by Lee and Tien (1981) are just the same as that calculated by dispersion equations (2a) and (2b), derived by Dalzell and Sarofim (1969). Approximations (4a)-(4c) are also close to the data obtained by Dalzell and Sarofim (1969). Concluding a short review of the soot optical constants, one can say that for the more interesting wavelength range from 0.4 to 4 μm the refraction and absorption indices increase with the wavelength (irrespective of temperature) from values n = 1.5 - 2 and κ = 0.5 - 0.7 to n = 2.5 - 3 and κ = 1.5 - 2.5. The above review of soot optical constants is not complete. However, it is sufficient for the determination of optical constant variation ranges, which is necessary for the computational investigation of soot radiative properties.
FigureÂ 1.Â Indices of refraction and absorption for soot based on different studies: 1, Stull and Plass (1960), 2, Dalzell and Sarofim (1969); 3, Lee and Tien (1981); 4, Blokh (1988); 5, Chang and Charalampopoulos (1990); 6 - Krishnan et al. (2000).
A majority of primary soot particles satisfy Rayleigh conditions (1) from the article Rayleigh scattering. The efficiency factors of absorption and scattering for homogeneous spherical particles can be calculated by simple relations (8) from that article. The absorption cross-section of a particle in the Rayleigh region is proportional to the particle volume; i.e., the value Q_{a} / x = R(m) does not depend on the particle radius and the effect of spectral optical constants can be characterized by the coefficient
(5) |
It is also interesting to consider the specific absorption coefficient of soot defined as
(6) |
where α_{λ} is the absorption coefficient of a soot cloud, ρ is the mass concentration (density) of the cloud, and ρ_{s} is the density of the particle material. The calculated spectral dependences R(λ) and E_{a}(λ) are plotted in Fig. 2. The conventional value of ρ_{s} = 1100 kg/m^{3} was used in calculation of E_{a}. One can see that the difference between the results based on the very different data obtained by Stull and Plass (1960) and Dalzell and Sarofim (1969) is not large, especially in the most important wavelength range from 1.5 to 3 μm, and one can use simple approximation (3) for the estimates.
FigureÂ 2.Â Spectral coefficient R and specific absorption coefficients of soot E_{a} in the Rayleigh region: 1, Stull and Plass (1960); 2, Dalzell and Sarofim (1969); 3, Blokh (1988).
The properties of the single primary particles of soot are very simple because these particles are spherical and small, compared with the radiation wavelength. The problem for agglomerated particles containing numerous primary particles is much more complicated. There is a large body of research on this subject (Lee et al., 1982; Lee and Tien, 1983, Drolen and Tien, 1987; Mackowski et al., 1987; Charalampopoulos and Hahn, 1989; Ku and Shim, 1991; Köylü and Faeth, 1993; Draine and Flatau, 1994; Lou and Charalampopoulos, 1994; Mackowski, 1995, 2006; Xu, 1995; Farias et al., 1996, 1998; Mackowski and Mishchenko, 1996; Wriedt, 1998; Comberg and Wriedt, 1999; Shu and Charalampopoulos, 2000; Kimura, 2001; Krishnan et al., 2001; Sorensen, 2001; Eymet et al., 2002; Auger et al., 2003; Klusek et al., 2003; Ayranci et al., 2007; Sabouroux et al., 2007; Liu et al., 2008; Zhao and Ma, 2009), where the experimental technique and specific methods for calculating both the absorption and scattering characteristics of soot agglomerates have been elaborated and applied to practical problems. We give here only the references to these papers to help the reader gather the models and data for soot radiation. However, it is interesting to consider the general effect of the particle shape on the radiative properties of soot. The latter can be done on the basis of a relatively simple model. Following Lee and Tien (1983) and Mackowski et al. (1987), consider the properties of longitudinal soot particles by using the solution for long cylinders. In the Rayleigh region, absorption is predominant in comparison with scattering and the absorption efficiency factor is proportional to the diffraction parameter. At normal incidence of randomly polarized radiation, we have the following equation (see article Rayleigh scattering):
(7) |
Two terms in Eq. (7) correspond to the incident radiation components with different linear polarizations. The first term obtained from Q_{a}^{H} does not depend on the angle of incidence, whereas the value of Q_{a}^{E} responsible for the second term in Eq. (7) is a function of this angle. It is of interest to compare the radiative properties of soot with spherical and cylindrical particles. This comparison for particles with a small radius (when |m|x << 1) is shown in Fig. 3(a), and for larger particles it is shown in Fig. 3(b). In our calculations, we used dispersion equation (2) by Dalzell and Sarofim (1969) for the optical constants of soot. The specific coefficient of absorption E_{a} and specific transport coefficient of scattering E_{s}^{tr} for monodisperse cylindrical particles were determined as follows:
FigureÂ 3.Â Absorption and scattering characteristics of soot: (a) specific absorption coefficient for particles of small radius (1, spherical particles; 2, cylindrical particles at normal incidence; 3, cylindrical particles randomly oriented in space); (b) specific absorption coefficient and transport scattering coefficient of monodisperse soot in the case of large spherical particles (I) and randomly oriented cylindrical particles (II) of the same radius: solid lines, a = 50 nm; dashed lines, a = 100 nm.
(8) |
In the Rayleigh region, the value of E_{a} does not depend on the particle radius and is expressed as
(9) |
One can see in Fig. 3(a) and in the long-wave part of curves E_{a}(λ) in Fig. 3(b) that the radiation absorption by cylindrical particles is much greater than the absorption by spherical particles in the Rayleigh region. It is important to note that a decrease of E_{a} with the wavelength for cylinders is slower than that for spheres. This effect is explained by the quite different dependence of the absorption efficiency factor on optical constants. It is sufficient to compare expressions (5) and (7) in the case of n = κ >> 1, when R ~ 6 / n and R_{cyl} ~ πn^{2} / 2. The physical explanation of this difference is evident. It is the absorption of radiation polarized in a plane parallel to the axis of the cylindrical particle. It is interesting that the curve E_{a}(λ) for randomly oriented cylindrical particles is similar to the curve E_{a}(λ) for those particles at normal incidence. This result is explained by a weak spectral variation of the dependence of the absorption efficiency factor on the angle of incidence.
The absorption coefficient of larger particles is also sensitive to the particle shape [see Fig. 3(b)], but the value of E_{a} for randomly oriented cylinders may be less than that for spherical particles (at least in the visible range). The transport scattering coefficient of soot is very small compared with the absorption coefficient. The latter statement is true both for spherical and cylindrical particles. Of course, the cylindrical particles of soot cannot be considered as an adequate model of soot agglomerates. It is important that aggregated soot particles may scatter infrared radiation much more intensively than in the case of single primary particles of various shapes.
It is worth mentioning here an interesting study by Mackowski et al. (1989), who considered the case when soot generated during the gas-phase combustion of volatile matter forms a concentric “mantle” about the parent coal particle. The results of the computational study indicated that for coal particle diffraction parameter x > 20 the radiative transfer formulation of the problem considered by Choi and Kruger (1985) is in excellent agreement with the analysis based on the modified Mie solution for the core-mantled sphere with an effective complex index of refraction of the soot cloud obtained from Maxwell-Garnett theory, as discussed by van de Hulst (1957, 1981) and Bohren and Huffman (1983).
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