NONUNIFORM ABSORPTION OF THERMAL RADIATION OF LARGE SEMI-TRANSPARENT PARTICLES AT ARBITRARY ILLUMINATION OF THE POLYDISPESE SYSTEM
Following from: Radiative transfer problems in nature and engineering
The disperse systems of separate spherical particles or droplets randomly positioned in vacuum or gas are considered in this section. It is assumed that the average distance between the particles is large in comparison with their size and the wavelength of the external thermal radiation. In this case, one can employ the radiation transfer theory to calculate the radiation field in the medium and absorption of the radiation by particles. In such calculations, the particles are usually assumed to be isothermal and the volume distribution of the absorbed radiation power inside the particles is not considered (Viskanta and Mengç, 1989; Dombrovsky, 1996; Viskanta, 2005).
An analysis of the nonuniform radiation field inside semi-transparent particles is important not only in the case of considerable thermal radiation from nonisothermal particles (Fiszdon, 1979; Dombrovsky and Ignatiev, 2003; Tseng and Viskanta, 2006a,b; Dombrovsky and Dinh, 2008) but also in the case of comparatively cold particles when absorbed radiation affects chemical or phase conversions in the particle. One should mention the problem of heating and evaporation of fuel droplets in diesel engines (Lage and Rangel, 1993b; Lage et al., 1995; Sirignano, 1999; Dombrovsky et al., 2001; Abramson and Sazhin, 2005, 2006; Tseng and Viskanta, 2005; Sazhin, 2006) and similar problems involving water droplets in fire suppression by water sprays and water spray curtains used in fire shielding (Harpole, 1980; Ravigururajan and Beltran, 1989; Coppalle et al., 1993; Grant et al., 2000; Yang et al., 2004; Buchlin, 2005; Sacadura, 2005; Tseng and Viskanta, 2006c; Collin et al., 2007). The thermal radiation from water or fuel droplets is negligible and the solution is divided into two stages: the ordinary spectral calculation of radiation transfer in a disperse system and the calculation of volume distribution of absorbed radiation power in semi-transparent droplets.
The monochromatic radiation field inside a spherical particle can be calculated by consideration of the incident radiation as a combination of plane electromagnetic waves of different amplitudes from different directions. The angular dependence of the radiation illuminating single particle is known from the solution of the radiation transfer problem for the disperse system as a whole.
The calculations based on the general Mie solution for the radiation field in a spherical particle are very time consuming. Fortunately, the integral characteristics of radiation absorption by large particles can be calculated in the geometrical optics approximation (Harpole, 1980; Liu et al., 2002). This approximation is inapplicable to local values near the caustics (van de Hulst, 1957, 1981; Bohren and Huffman, 1983; Lock and Hovenac, 1991; Choudhury et al., 1992). The latter limitation is important for laser illumination (Prishivalko, 1983a,b; Park and Armstrong, 1989; Sitarski, 1990; Tuntomo and Tien, 1992; Astafieva and Prishivalko, 1994,1998; Longtin et al., 1995; Foss and Davis, 1996) but not for diffuse thermal radiation considered in this section. Following the papers by Dombrovsky and Sazhin (2003, 2004) and Dombrovsky (2004) we consider an approximate analytical model of radiation absorption in large semi-transparent particles in the range of applicability of the geometrical optics approximation. The model should be applicable in a wide spectral range and for arbitrary illumination of a polydisperse system.
Approximate Description of Asymmetric Illumination of a Single Particle
In the general case, the angular dependences of radiation intensity are very complex and different for various spectral intervals. To simplify the problem for a single particle, one can describe the illumination of the particle as axisymmetric and uniform in two hemispheres:
(1) |
where functions I_{λ}^{-} and I_{λ}^{+} are determined by the following relation (see the article, Two-flux approximation):
(2) |
The approximation suggested simplifies radically the calculations for a single particle: one should solve only the problem for uniform illumination of the particle from a hemisphere. This solution is described by function p_{λ}^{h}(r,μ) (the absorbed power per unit spectral interval), where 0 ≤ r ≤ a is the radial coordinate measured from the center of the particle, μ = cosθ (where θ is measured from the spectral radiation flux direction). The corresponding function for illumination from two hemispheres is as follows:
(3) |
Equation (3) can be rewritten in the form:
(4) |
where A_{λ} = I_{λ}^{-}/ I_{λ}^{+} is the asymmetry parameter of illumination (0 ≤ A_{λ} ≤ 1) and p_{λ}^{sph}(r) = p_{λ}^{h}(r,μ) + p_{λ}^{h}(r,-μ) is the power absorption function for spherically symmetric illumination of the particle.
In the general case, the direction of spectral radiation flux is different for various spectral intervals and one can write subscript λ at angular parameter μ: P_{λ}(r,μ_{λ}). As a result, the integral power distribution will be three-dimensional:
(5) |
The absorption distribution in a particle may be axisymmetric only in the case of not too complex picture of the radiation transfer in a disperse system.
Solution Based on the Mie Theory
The Mie solution for the internal field in a spherical particle illuminated by a plane linearly polarized electromagnetic wave can be found in the article, Thermal radiation from nonisothermal spherical particles, where the case of a spherically symmetric problem was considered and the analytical expression for the radial profile of the absorbed power p _{λ}^{sph}(r) was given. In the case of uniform illumination of the particle from a hemisphere, the absorbed power per unit volume can be found from the relation derived by Lage and Rangel (1993a):
(6) |
where
(7) |
Since S_{h}(r,μ) does not depend on azimuth angle ϕ, one can rewrite Eq. (7) as
(8) |
Note that S_{h}(r,μ) + S_{h}(r,-μ) = S(r) and S_{h}(r,0) = S(r)/ 2.
The radiation power absorbed by a droplet as a whole is characterized by the efficiency factor of absorption Q_{a}, which can be calculated independently by use of the known equations of the Mie theory. For large semi-transparent particles, one can also use an analytical approximation. For this reason, it is sufficient to consider the normalized distribution of the absorbed power. In the case of spherically symmetric illumination of particles, this distribution is characterized by the following function:
(9) |
In the case when the particle is illuminated from a hemisphere, the normalized distribution of the absorbed power depends on two variables:
(10) |
It was shown by Dombrovsky (2000) that function w(r) for large semi-transparent particles can be calculated in the geometrical optics approximation; solution of the radiation transfer equation inside the particle at x ≥ 20 gives almost the same profiles of absorbed power as those calculated by using the Mie theory (see, also, the article, Thermal radiation from nonisothermal spherical particles). In the case of spherically symmetric illumination of a particle, the radial distribution of absorbed power coincides with the profile of radiation power generated by the isothermal particle.
Approximate Solution for Symmetric Illumination
An approximate but fairly accurate solution for the absorbed radiation power can be obtained by use of the MDP_{0} approximation (see the article, Thermal radiation from nonisothermal spherical particles). In the case of illumination of a particle by external radiation, the boundary-value problem in the MDP_{0} approximation for function g_{0}(τ) can be written in the following form:
(11a) |
(11b) |
where
(11c) |
where τ_{0} = 2κx is the spectral optical thickness of the particle and Θ is the Heaviside unit step function. The dimensionless absorbed radiation power per unit spectral interval is determined as
(12) |
The problem [Eqs. (11a)-(11c) and (12)] for a single particle can be easily solved numerically (Dombrovsky, 2000). At the same time, further simplification of the problem is important for possible implementation of the solution in multidimensional CFD codes and engineering calculations by taking into account combined heat and mass transfer processes. The simplest analytical approximation has been suggested by Dombrovsky and Sazhin (2003, 2004) separately for small and large optical thickness τ_{0}. This approximation is not good for intermediate values of the optical thickness, but it appears to be rather good for droplets of diesel fuel, which have wide regions of semi-transparency and sharp peaks of absorption. A more reliable approximate analytical solution for the whole range of the particle optical thickness was derived by Dombrovsky (2004). To obtain this solution one can use the following approximate expression for the radiation diffusion coefficient at the periphery of the particle (τ > τ_{*}):
(13) |
Strictly speaking, this relation is correct only near the particle surface for n^{2} >> 1. Similarly, one can replace also the coefficient (1 -μ_{*}) in the second term of Eq. (11a) with D_{λ}^{(1)}. As a result, Eq. (11a) at τ > τ_{*} is simplified radically as
(14) |
Keeping in mind the continuity of function g_{0} and the derivative g'_{0} at τ = τ_{*} one can find the following analytical solution:
(15a) |
(15b) |
where
(15c) |
(15d) |
For large optical thickness, Eq. (15) can be considerably simplified:
(16a) |
(16b) |
The last equations can be used in calculations at τ_{0} > 5. The normalized spectral profiles of absorption are calculated as
(17) |
The calculated radial profiles of the absorbed radiation power w(r) are presented in Fig. 1. Keeping in mind the possible applications of this research to droplets of water or typical fuels, the values of the refraction index, n = 1.3 and n = 1.5, are considered. The Mie theory calculations at x = 50 are also shown in Fig. 1. Remember that the Mie solution at x > 20 is close to the geometrical optics limit. Note that the kink in the curves at r = r _{*} can be easily explained in terms of the geometrical optics (Lai et al., 1991; Dombrovsky, 2000).
Figure 1. Normalized radial profiles of the absorbed radiation power inside droplets in the case of their symmetric illumination: (a), (b); I, Mie calculations at; II, numerical solution in approximation; III, approximate analytical solution; 1 - τ_{0} = 0.2; 2 - τ_{0} = 1; 3 - τ_{0} = 2; 4 - τ_{0} = 5.
One can see that the approximate analytical solution [Eq. (15)] gives the absorption profiles, which practically coincide with those obtained by the numerical solution to the boundary-value problem [Eq. (11)]. Equations (15) derived by Dombrovsky (2004) are applicable to the arbitrary optical thickness of the particle.
Approximation of Mie Calculations for Illumination from a Hemisphere
In the analysis of the calculations for illumination from a hemisphere, it is convenient to use the relation S_{h}(r,0) = S(r)/ 2 and consider only the function S_{h}(r,μ) = S_{h}(r,μ)/ S_{h}(r,0). The results of the Mie calculations for x = 50 at several fixed values of r are presented in Figs. 2 and 3. In the case of small optical thickness (Fig. 2), the radiation is absorbed mainly in the shadowed part of the particle. The strongest angular dependence of absorption is predicted near the particle surface (r > r_{*}) in the vicinity of the plane, μ = 0. The increase in the optical thickness leads to an increase in radiation absorption in the illuminated part of the particle and the dependence of the radiation absorption on μ becomes monotonic. In the case of large optical thickness (Fig. 3), the radiation is absorbed mainly near the illuminated surface of the particle and the dependence on μ becomes almost linear.
Figure 2. Angular distribution of the absorbed radiation power inside optically thin droplet in the case of its illumination from a hemisphere at (a,b) - τ_{0} = 0.2, (c,d) - τ_{0} = 0.5, (e,f) - τ_{0} = 1; 1 - r = 0.2, 2 - r = 0.4, 3 - r = 0.6, 4 - r = 1/n, 1 - r = 1.
Figure 3. Angular distribution of the absorbed radiation power inside an optically thick droplet in the case of its illumination from a hemisphere at (plots I) and (plots II): (a) - τ_{0} = 2, (b) - τ_{0} = 5; 1 - r = 0.25, 2 - r = 0.5, 3 - r = 0.75, 4 - r = 1.
Note that S_{h}(r,μ) - 1 can be approximated by an odd function of μ (see Figs. 2 and 3). This approximation simplifies considerably the integration of function S_{h}(r,μ):
(18) |
As a result, one can write:
(19) |
The analytical approximation of function S_{h}(r,μ) can be found in the form
(20) |
At the optical thickness of the particle τ_{0} ≥ 2, and the function f can be treated as independent of refraction index n and angular coordinate μ (see Fig. 3):
(21) |
Equation (21) gives a correct result in the limit of great optical thickness. At small optical thickness (τ_{0} < 2) the following approximation was suggested:
(22a) |
(22b) |
(22c) |
Approximate relations (20)-(22) derived by Dombrovsky (2004) can be used at the arbitrary optical thickness of the particle. The results of the approximate calculations at the same parameters as in Figs. 2 and 3 are presented in Figs. 4 and 5. One can see that Eqs. (20)-(22) give a reasonable approximation to the exact calculations.
Figure 4. Approximate angular distribution of the absorbed radiation power inside an optically thin droplet in the case of its illumination from a hemisphere: calculations using Eqs. (20)-(22); designations, see Fig. 2.
Figure 5. Approximate angular distribution of the absorbed thermal radiation power inside an optically thick droplet in the case of its illumination from a hemisphere: calculations using Eqs. (20)-(22); designations, see in Fig. 3.
Some Results for Water and Diesel Fuel Droplets
Consider a one-dimensional model problem for an optically thick homogeneous disperse system in the region with a flat boundary surface illuminated by diffuse thermal radiation. In this case, one can use the simplest DP_{0} approximation for the radiation transfer calculation. The corresponding boundary-value problem for function g_{0} = I_{λ}^{-}(y) + I_{λ}^{+}(y) (hereafter, y is the coordinate measured from the illuminated surface of the disperse system) is as follows (Dombrovsky, 1996):
(23) |
where q_{λ}^{e} is the spectral external radiation flux, and α and β_{tr} are the absorption and transport extinction coefficients of the medium. The spectral radiation flux is determined as follows:
(24) |
The analytical solution of the problem (23)-(24) is as follows:
(25) |
and one can obtain a very simple equation for the asymmetry parameter of illumination:
(26) |
Note that A_{λ} does not depend on the coordinate and coincides with the reflection coefficient of the disperse system.
For polydisperse systems of large droplets of water or diesel fuel, the value of ξ_{λ} can be calculated in a monodisperse approximation using the Sauter droplet radius a_{32} (see the article, Radiative properties of polydisperse systems of independent particles):
(27) |
where Q_{s}^{tr} and Q_{tr} = Q_{a} + Q_{s}^{tr} are the transport efficiency factors of scattering and extinction. The approximate relations (3) from the article, Radiative properties of semi-transparent spherical particles, can be used for both water and fuel droplets (Dombrovsky, 2002; Dombrovsky et al., 2003). One should know the spectral data for optical constants n(λ) and κ(λ) to calculate A_{λ}. The data for water reported by Hale and Querry (1973) and the approximate relations for a typical diesel fuel suggested by Dombrovsky et al. (2004) were used in the calculations (see Fig. 6). The calculated functions A_{λ}(τ_{0}) are presented in Fig. 7. The wavelength step Δλ = 0.02 μm was used in the range of 0.5 ≤ λ ≤ 6 μm. The values A_{λ} > 0.1 correspond to the short-wave range (λ < 2.3 μm for water and λ < 3 μm for fuel), where the droplets are almost transparent.
Figure 6. Spectral optical constants of water (1) and diesel fuel (2) used in the calculations.
Figure 7. Asymmetry parameter of droplet illumination in a monodisperse optically thick medium. Calculations for water (1, 2) and diesel fuel droplets (3, 4) using Eqs. (26) and (27), and approximation (3) from the article, Radiative properties of semi-transparent spherical particles: 1,3 - a = 10 μm; 2, 4 - a = 50 μm.
For various droplets, the function A_{λ}(τ_{0}) can be approximated as
(28) |
where b = 4 for small fuel droplets, b = 5 for large fuel droplets and small water droplets, and b = 6 for large water droplets. Note that the approximate relations by Dombrovsky et al. (2004) for optical constants of diesel fuel give the minimal evaluation for κ(λ) in the important spectral range of 1.1 < λ < 2 μm. Additional measurements reported by Sazhin et al. (2007) showed greater values of the absorption index in this spectral range. As a result, the values of parameter b for fuel droplets are almost the same as those for water droplets. Remember that one should use the value of τ_{0} for the equivalent radius of droplets a_{32} in Eq. (28), and the value of A_{λ} is the same for droplets of different radii.
In the case of a blackbody spectrum of external thermal radiation, the normalized radiation power absorbed in a particle is calculated as follows:
(29) |
The results of the calculations for water and fuel droplets at a_{32} = 50 μm and T_{e} = 1500 K are presented in Fig. 8. The plots for μ = 0 coincide with the absorption profiles for spherically symmetric illumination. One can see that thermal radiation is absorbed mainly in the central zone and in a thin layer close to the droplet surface. Increased absorption in the central zone is related to the contribution of radiation in the semi-transparency ranges, where the droplet thickness is small. Significant absorption near the droplet surface facing the external radiation is related to the contribution of radiation near the absorption peaks of the droplet substance. The main peak of absorption for diesel fuel is placed at the wavelength λ = 3.4 μm, which is in the more long-wave region in comparison with the absorption peak for water (see Fig. 6). It is far from the maximum of the external radiation (λ = 2 μm). As a result, the absorption of radiation near the droplet surface facing the external radiation is not as strong for fuel droplets as it is for droplets of water. At the same time, the absorption in the central zone of the diesel fuel droplets is not symmetric; it is greater at the “shadow” side facing the droplet system.
Figure 8. Normalized radial profiles of the absorbed thermal radiation power inside droplets of water (a), (b) and diesel fuel (c), (d) in the case of an optically thick disperse system: (a,c) - a = 10 μm; (b,d) - a = 50 μm; 1 - μ = 1 (the side facing the disperse system), 2 - μ = -0.5, 3 - μ = 0, 4 - μ = 0.5, 5 - μ = 1 (the side facing the external radiation).
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- Abramson, B. and Sazhin, S., Droplet vaporization model on the presence of thermal radiation, Int. J. Heat Mass Transfer, vol. 48, no. 9, pp. 1868-1873, 2005.
- Abramson, B. and Sazhin, S., Convective vaporization of a fuel droplet with thermal radiation absorption, Fuel, vol. 85, no. 1, pp. 32-46, 2006.
- Astafieva, L. G. and Prishivalko, A. P., Heating of metallized particles by high-intensity laser radiation, J. Eng. Phys. Thermophys., vol. 66, no. 3, pp. 304-308, 1994.
- Astafieva, L. G. and Prishivalko, A. P., Heating of solid aerosol particles exposed to intense optical radiation, Int. J. Heat Mass Transfer, vol. 41, no. 2, pp. 489-499, 1998.
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