Thermal Radiation from Nonisothermal Spherical Particles
Following from: The Mie solution for spherical particles; Radiative properties of semitransparent spherical particles
Leading to: Nonuniform absorption of thermal radiation in large semitransparent particles at arbitrary illumination of the polydisperse system; Thermal radiation from nonisothermal particles in combined heat transfer problems
Thermal radiation of a nonisothermal particle is an especially interesting problem for particles of a semitransparent material (in particular, for metal oxide particles). The point is that materials that are semitransparent in the infrared spectral range are usually characterized by low thermal conductivity and, as a result, by a comparatively large temperature difference between the center and the surface of the particle. On the other hand, in the case of a small index of absorption, the solution to the problem is expected to be more complex because of possible considerable contribution of radiation emitted from the central core of the particle.
A rigorous statement of the problem must take into account effects of interference as is done in the Mie theory. The thermal radiation emitted by a homogeneous isothermal spherical particle of arbitrary material can be calculated by using the ordinary Mie solution for the radiation field at a large distance from the particle. The spectral emissivity of the particle is simply equal to the absorption efficiency factor Q_{a}. The radial distribution of the radiation power, or heat generation rate, in the particle can also be calculated by using the Mie solution for internal radiation field (van de Hulst, 1957; Kattawar and Eisner, 1970; Dusel et al., 1979; Bohren and Huffman, 1983; Prishivalko, 1983a; Prishivalko et al., 1984; Dobson and Lewis, 1989; Mackowski et al., 1990; Tuntomo et al., 1991; Dombrovsky, 1999, 2000). Both the particle emissivity ε_{λ} and the normalized profile of the spectral radiation power P_{λ}(r) depend on two parameters, namely, the diffraction parameter x = 2πa/λ and complex index of refraction m = n - iκ. Let us describe the evolution of ε_{λ} and P_{λ}(r) with the diffraction parameter for particles of a weakly absorbing material (κ << 1). In the range of Rayleigh-Gans scattering, when the phase shift is very small, 2x(n - 1) << 1, the heat generation in the particle is uniform and the particle emissivity is directly proportional to the diffraction parameter,
(1) |
For larger particles, in the so-called Mie scattering region, one can observe very complex resonance behavior of the heat generation field and numerous oscillations of function ε_{λ}(x) (Chýlek et al., 1985; Greene et al., 1985; Benincasa et al., 1987; Dombrovsky, 1996, 2000). It is the interference range where both index of refraction and diffraction parameter are very important parameters, whereas the effect of a small absorption index on radiation power distribution in the particle is insignificant. This statement was illustrated by numerical data reported by Dombrovsky (1999, 2000). The further increase in the diffraction parameter leads to degeneration of the general solution. This is a region of geometrical optics [x >> 1, 2x(n - 1) >> 1], where the solution does not depend on the diffraction parameter and the remaining parameters of the problem are index of refraction n and optical thickness of the particle τ_{λ} = 2κx = α_{λ}a. In this section, we consider large semitransparent particles, for which the temperature difference is more important. In the case of large particles, one can try to use the geometrical optics approximation and the radiation transfer theory. As was shown by Lai et al. (1991), Choudhury et al. (1992), and Velesco et al. (1997), the geometrical optics is a reasonable approximation for the calculating the energy density distribution inside large nonabsorbing spherical particles. It is important that the geometrical optics results show the main features of the Mie solution and provide a physical explanation of the electromagnetic interactions inside particles, which are large in comparison with the wavelength of radiation. The geometrical optics approximation is inapplicable for local values near caustics (van de Hulst, 1957; Lock and Hovenac, 1991; Choudhury et al., 1992), but this limitation is not important for thermal radiation considered in this section.
Note that one can also use the general Mie theory for calculating the thermal radiation flux and heat generation profiles in the case of nonisothermal particles with known radial variation of temperature. This problem was considered by Mackowski et al. (1990) on the basis of analytical solution for a multilayered spherical particle (Toon and Ackerman, 1981; Bhandari, 1985; Sitarski, 1987a). The material of this article is based on papers by Dombrovsky (1999, 2000, 2002, 2007), where one can find some additional details.
Mie Solution for Radiation Field In an Isothermal Particle
The thermal radiation emitted by a homogeneous isothermal spherical particle of arbitrary material can be calculated using the Mie solution. It is sufficient to determine the value of the absorption efficiency factor Q_{a}, which is usually calculated as a difference between the extinction and scattering efficiency factors. According to Eq. (2) from the article “The Mie solution for spherical particles,” one can write
(2) |
Equation (2) was derived from the expansions for electromagnetic field far from the particle (in the so-called radiation zone). As was shown by van de Hulst (1957), an alternative expression for the absorption efficiency factor can be obtained by integration of the internal electromagnetic field (see also the book by Prishivalko, 1983a) is
(3) |
It can be shown that Eqs. (2) and (3) are mathematically equivalent (Kattawar and Eisner, 1970).
The spectral emissivity of a particle can be determined from the solution of the fluctuation electrodynamics problem (Levin and Rytov, 1967). In this case, the final expression for ε_{λ} is identical to Eq. (3). This result confirms Kirchhoff’s law, i.e., ε_{λ} ≡ Q_{a}.
It is well known that absorption of the radiation by a particle is, generally speaking, nonuniform over the volume of the particle. In the case of interaction of a plane electromagnetic wave with a homogeneous spherical particle, the dimensionless amplitudes of the electric field components inside the particle (as referred to the electric field in the incident wave) are given by the following equations:
(4a) |
(4b) |
(4c) |
where
(5a) |
(5b) |
The value of the heat generation rate is equal to the power p_{λ} absorbed by a unit volume of the particle. The corresponding dimensionless value can be expressed as follows:
(6) |
After integration over the angles, one can derive the following relation for the angle-averaged heat generation rate of a spherical layer of unit thickness:
(7a) |
(7b) |
The spectral emissivity of the particle can be expressed as follows:
(8) |
Calculation by means of Eqs. (7) and (8) must give the same values of ε_{λ} as those calculated by Eqs. (1) or (2), which are simpler; however, (7) also gives radial profiles of the heat source inside the particle. Equation (8) can be used as a control of the accuracy of the heat source calculation.
It should be noted that the Mie solution for internal radiation field in a spherical particle can be employed not only in calculations for the spherically symmetric problem of the particle thermal radiation; the problem of nonuniform internal absorption of the radiation in the case of arbitrary illumination of the particle can also be solved on the same theoretical basis. A reader can find some solutions on this subject, including those for applied heat transfer problems, in other papers (Rosasco and Bennett, 1978; Prishivalko and Leiko, 1980; Prishivalko, 1980, 1983b, 1984; Prishivalko and Veremchuk, 1981; Astaf’eva and Prishivalko, 1987, 1994, 1998; Bott and Zdunkowski, 1987; Sitarski, 1987b, 1990; Prishivalko et al., 1988; Hunter et al., 1988; Park and Armstrong, 1989; Allen et al., 1991; Tuntomo and Tien, 1992; Lage and Rangel, 1992, 1993a,b; Lage et al., 1995; Qiu et al., 1995; Longtin et al., 1995; Foss and Davis, 1996; Astafieva, 1997; Ruppin, 1998; Widmann and Davis, 1998; Zemlyanov and Geints, 2004).
Geometrical Optics Approximation
The radiative transfer equation (RTE) for a spherical volume of an absorbing medium with the absorption coefficient α_{λ}(r), temperature profile T(r), and constant index of refraction n is as follows (Öziik, 1973; Siegel and Howell, 2002; Modest, 2003):
(9) |
where I_{λ}(r,μ) is the radiation intensity at point r in the direction -1 ≤ μ = cosθ ≤ 1 integrated over the azimuth, and B_{λ}(T) is the Planck function. The boundary conditions (symmetry at r = 0 and Snell’s law at the particle surface r = a) are
(10) |
where R(n,μ) is the Fresnel’s reflection coefficient for unpolarized radiation. In the case of a weakly absorbing medium (κ << n), one can use the following expressions for this coefficient:
(11a) |
(11b) |
(11c) |
The heat generation rate and the radiation flux from the particle are determined as follows:
(12) |
(13) |
We will also use the normalized dimensionless values
(14) |
where the average temperature of the particle is defined as
(15) |
In the particular case of a homogeneous isothermal medium (α_{λ}, T = const.), it is convenient to use dimensionless variables
(16) |
and introduce the following quantities:
(17) |
Note that the absorption coefficient and the optical thickness are related to the index of absorption and the diffraction parameter by the following simple expressions:
(18) |
In the new variables, the radiative transfer equation (9), the boundary conditions (10), and Eqs. (12) and (13) can be written as follows:
(19a) |
(19b) |
(19c) |
Presenting the radiation intensity in the form
(20) |
and introducing the functions
(21) |
we obtain the following, instead of Eq. (19):
(22a) |
(22b) |
(22c) |
(22d) |
where γ = (1 - R)/(1 + R). Replacing the variables (τ,μ) by (τ,y), where y = τ√1 - μ^{2}, leads to much simpler coupled equations,
(23) |
in the triangular computational region 0 ≤ τ ≤ τ_{0}, 0 ≤ y ≤ τ_{0}. After transition to one second-order equation for the function g(τ,y), we have
(24a) |
(24b) |
where γ = γ[n, √1 - (y/τ)^{2}]. The parabolic problem (24) does not contain derivative ∂g/∂y and may be considered to be a set of separate boundary value problems at different fixed values of y. After solving Eq. (24), one can find W_{λ}(τ) and ε_{λ} by integration,
(25) |
To calculate the value of ε_{λ} we can also use the following simple relation derived from the energy balance on the surface τ = τ_{0},
(26) |
Note that one can use a similar algorithm in the case of a nonisothermal volume. But the emissivity has no sense in this case, and the radiation flux should be calculated immediately.
The solution of the radiative transfer problem (9)-(14) or equivalent alternative formulations of the same problem can also be obtained by use of one of the other mathematical methods. Some variants of these solutions can be found in papers by Harpole (1980), Miliauskas (2001, 2003), Liu et al. (2002a,b,c), Cassel and Williams (2006), and Tseng and Viskanta (2005, 2006a,b,c). We do not discuss the quality of different methods. Of course, the geometrical optics calculations for large particles are much simpler than the Mie theory calculations. But the spectral geometrical optics calculations for a nonisothermal particle with temperature-dependent absorption coefficient of the particle substance are also time consuming. For this reason, following Dombrovsky (1999, 2000, 2002, 2007), we will consider a further simplification of the problem.
In this article, we do not consider the case of a nonsymmetric temperature field in the particle (this limitation may be important for some applied problems). A novel method for solving this problem has been suggested recently by Liu (2004). The method is based on a Monte Carlo ray-tracing technique.
Following early papers by Dombrovsky (1999, 2000), consider first the results for spectral emissivity of an isothermal particle. A comparison of the Mie solution and the radiative transfer calculations inside the particle for the most interesting range of optical thicknesses is presented in Fig. 1. One can see that the numerical solution of the RTE underestimates slightly the emissivity of the particle when κ < 0.01 and τ_{0} > 0.2; it does not essentially differ from the asymptotic solution in the limit when κ→0.
Figure 1. Spectral emissivity of spherical particle at (a) n = 1.5 and (b) n = 2: 1, RTE solution inside the particle; 2-5, Mie theory calculations (2, κ = 0.002; 3, κ = 0.005; 4, κ = 0.01; 5, κ = 0.02).
Exact calculations of particle emissivity using the geometrical optics approximation are very simple in the case of n = 1, when the analytical solution is known. Note that it is more convenient to consider absorption of the incident plane wave by a particle instead of thermal radiation from a particle. In this case, one can use the following formula for the efficiency factor of absorption derived by van de Hulst (1957) for optically soft particles (x >> 1, |m - 1| << 1):
(27) |
One can compare the numerical solution for the RTE with the values of ε_{λ} = Q_{a} given by Eq. (27). Because the angular dependencies of the radiation intensity are smooth, we can also employ the diffusion approximation and the following analytical solution (Dombrovsky, 2000):
(28) |
where D = 1/(4 - N_{a}) is the dimensionless radiation diffusion coefficient. N_{a} is equal to 0 for the DP_{0} approximation and to 1 for the P_{1} approximation; N_{m} = 0 for the Marshak boundary condition, and N_{m} = 1 for the Pomraning one (the P_{1m} approximation). A comparison of different calculations is presented in Table 1. Note that there is practically no difference between calculations using the Mie theory and those using the geometrical optics approximation for small values of the absorption index. We can also see that the diffusion approximation error is very small at an arbitrary optical thickness of the particle. It is interesting that all the theoretical models considered give the same result of ε_{λ} = 4τ_{0}/3 in the limit when κ, τ_{0} << 1 for uniform heat generation over the volume of the particle.
Table 1. Spectral emissivity of nonrefracting particles (n = 1)
Mie theory | Radiation transfer theory | Diffusion approximation, Eq. (28) | |||||
τ_{0} | κ = 0.01 | κ = 0.001 | Numerical solution | Exact analytical solution [Eq. (27)] | DP_{0} | P_{1} | P_{1m} |
0.2 | 0.2317 | 0.2306 | 0.226 | 0.2306 | 0.2331 | 0.2336 | 0.2300 |
0.5 | 0.4729 | 0.4716 | 0.464 | 0.4715 | 0.4768 | 0.4823 | 0.4672 |
1 | 0.7047 | 0.7030 | 0.695 | 0.7030 | 0.6990 | 0.7201 | 0.6870 |
2 | 0.8884 | 0.8865 | 0.881 | 0.8864 | 0.8576 | 0.9033 | 0.8518 |
5 | 0.9821 | 0.9801 | 0.978 | 0.9800 | 0.9473 | 1.0106 | 0.9465 |
To estimate the error of the RTE numerical solution, a comparison can be made with the tabulated data by Pinchuk and Romanov (1977) obtained by accurate integration in the following expression:
(29) |
where l(μ) = 2√1 - (1- μ^{2})/n^{2} (see Table 2). It is important that the computational error is much less than the difference between the geometrical optics approximation and the Mie theory solution (Fig. 1).
Table 2. Spectral emissivity of particles with index of refraction n > 1; calculations in geometrical optics approximation
Exact solution [Eq. (29)] | RTE numerical solution | MDP_{0} approximation | ||||
τ_{0} | n = 1.5 | n = 2 | n = 1.5 | n = 2 | n = 1.5 | n = 2 |
0.1 | 0.1582 | 0.1639 | 0.156 | 0.162 | 0.160 | 0.169 |
0.2 | 0.2889 | 0.2917 | 0.283 | 0.289 | 0.293 | 0.304 |
0.4 | 0.4790 | 0.4744 | 0.474 | 0.472 | 0.494 | 0.498 |
0.6 | 0.6100 | 0.5934 | 0.604 | 0.591 | 0.632 | 0.625 |
1 | 0.7626 | 0.7256 | 0.758 | 0.723 | 0.793 | 0.766 |
2 | 0.8831 | 0.8221 | 0.881 | 0.821 | 0.924 | 0.868 |
4 | 0.9074 | 0.8930 | 0.907 | 0.838 | 0.956 | 0.887 |
6 | 0.9082 | 0.8394 | 0.907 | 0.838 | 0.958 | 0.888 |
Let us consider heat generation profiles in an isothermal particle. In the case of n > 1, internal heat generation is not uniform even for optically thin particles (τ_{0} << 1) (except for the Rayleigh region when nx << 1); it depends essentially on the diffraction parameter. This contention is illustrated in Fig. 2, where calculations using Eqs. (7) and (8) are presented in the form of dimensionless profiles,
Figure 2. Heat generation profiles in particles of moderate size; Mie theory calculations at (a) x = 1 and (b) x = 10: I, κ = 0.01; II, κ = 0.02; 1, n = 1.5; 2, n = 2; 3, n = 2.5; 4, n = 3.
(30) |
The effect of the absorption index, when κ ≤ 0.02, is insignificant, whereas an increase of the refractive index, especially at n > 1.5, results in considerable deformation of the curves W_{λ}(r). This deformation strongly depends on the value of the diffraction parameter. As usually is the case, the most complex interference effects are observed in the resonance range 2x(n - 1) < 10. For larger particles, we can expect a satisfactory description of the dependencies W_{λ}(r) by radiation transfer theory. A comparison of the numerical solution to the problem (24) and the Mie theory calculations presented in Fig. 3 shows that for sufficiently large particles (x ≥ 20), the heat generation profile can be calculated without taking into account any interference effects for an arbitrary optical thickness of the particle. One can see that radiation transfer theory describes with sufficient accuracy the following special features of the internal radiation field: a displacement of the maximum of local heat generation from the center to the surface as the optical thickness of the particle increases; a change in the radial dependence of heat generation at the point r = 1/n; and a relative increase in the thermal radiation emitted from the central region r < 1/n with increasing the index of refraction. The latter features of the heat generation distribution inside weakly absorbing particles were found by Mackowski et al. (1990). The discontinuity at r = 1/n was also reported by Lai et al. (1991) for dielectric spheres at n = 1.33 and x ≥ 300. The kink in the curves W_{λ}(r) is explained by the internal reflection of radiation emitted by elementary volumes placed at r > 1/n. One can find the position of the kink from a simple geometrical consideration.
Figure 3. Heat generation profiles in large particles at (a) τ_{0} = 0.2, (b) τ_{0} = 2, and (c) τ_{0} = 5: I, n = 2; II, n = 3; 1, RTE solution; 2-4, Mie theory calculations (2, x = 20; 3, x = 50; 4, x = 100).
In spite of the comparatively simple physical model of radiation transfer, the calculated radiation field in a large refracting particle is rather complex, mainly due to the effect of the total internal reflection at μ ≤ μ_{c}. Some typical angular dependencies of the dimensionless radiation intensity are presented in Fig. 4, which illustrates the evolution of φ(r,θ) as r varies from 1/n to 1 and due to refraction on the particle surface. The complex shape of the angular curves when r > 1/n indicates that the ordinary diffusion approximation is inapplicable for describing radiation transfer in the problem under consideration.
(a) | (b) |
Figure 4. Angular dependencies of dimensionless radiation intensity at n = 2 for particles with τ_{0} = 0.2 (a) and τ_{0} = 2 (b): 1, r = 0.5; 2, r = 0.75; 3, r = 1 - δ; 4, r = 1 + δ (0 < δ << 1).
Modified Differential Approximation
An analysis of angular dependencies of the radiation intensity in the range τ_{0}/n < τ ≤ τ_{0} (see Fig. 4) shows that the following approximation of the function φ(τ,μ) may be rather good:
(31) |
Integrating the RTE over μ separately for intervals -1 ≤ μ < μ_{*} and μ_{*} < μ ≤ 1, and introducing the functions
(32) |
we obtain, after transformations, the following coupled equations and the boundary condition:
(33a) |
(33b) |
For simplicity, the coefficient R_{1} may be taken to be R(1). In the central region τ ≤ τ_{*}, according to the usual DP_{0} approximation, we have
(34) |
and the symmetry condition h_{0}(0) = 0. Finally, the boundary value problem in the modified DP_{0} approximation (MDP_{0}) for an unknown function g_{0}(τ) can be written as follows:
(35a) |
(35b) |
(35c) |
(35d) |
The MDP_{0} approximation is much simpler than the RTE. It is sufficient to note that the computational time on the same finite difference mesh decreases in two orders of magnitude when MDP_{0} is used. It goes without saying that the calculations using MDP_{0} are much faster than Mie calculations. At the same time, the error in MDP_{0} is not large (see Table 2 and Fig. 5). It is important that the heat generation profiles calculated in MDP_{0} may be considered to be a sufficiently accurate approximation for large particles. Note that the MDP_{0} approximation gives a correct value for ε_{λ} and a correct profile W_{λ}(τ), even in the case of an optically thin particle. The dependences ε_{λ}(τ_{0}) at n = 2 are presented in Fig. 6. In the range of moderate optical thickness, the error of MDP_{0} is considerably less than that of the DP_{0} approximation.
Figure 5. Heat generation profiles calculated (a) by numerical solving the RTE and (b) by use of MDP_{0} approximation at n = 2: 1, τ_{0} = 0.2; 2, τ_{0} = 2; 3, τ_{0} = 5.
Figure 6. Spectral emissivity of spherical particle at n = 2: 1, Mie theory calculations (1a, x = 50; 1b, x = 300); 2, RTE solution inside the particle; 3, MDP_{0} approximation; 4, DP_{0} approximation.
In the case of a nonisothermal particle, the MDP_{0} boundary value problem in dimensional variables can be formulated as follows (Dombrovsky, 2002):
(36a) |
(36b) |
(36c) |
(36d) |
(36e) |
To estimate an error of MDP_{0}, consider the case of parabolic temperature profile in the particle,
(37) |
with average volume temperature
(38) |
Assuming α_{λ} = const., we will use the value of optical thickness τ_{0} as a parameter of the problem. Some numerical results for particles with n = 2, T = 3000 K, and ΔT = ±500 K are presented in Fig. 7, where the following dimensionless quantities are used:
(a) | (b) |
Figure 7. Heat generation profiles and relative radiation flux for nonisothermal particles: 1, RTE solution; 2, MDP_{0} approximation; a, ΔT = -500 K; b, ΔT = 500 K.
(39) |
It is obvious that the accuracy of MDP_{0} remains rather high in the case of nonisothermal particles in a wide range of the optical thickness. Having in mind the known results on the accuracy of diffusionlike approximations, one should not expect a considerable increasing in the error of MDP_{0} for other temperature profiles. The applications of MDP_{0} approximation to the engineering problems are considered in the articles Nonuniform absorption of thermal radiation in large semitransparent particles at arbitrary illumination of the polydisperse system and Thermal radiation from nonisothermal particles in combined heat transfer problems.
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