An Estimate of P_{1} Approximation Error for Optically Inhomogeneous Media
Following from: P_{1} approximation of spherical harmonics method, The simplest approximations of double spherical harmonics, Solutions for one-dimensional radiative transfer problems, Radiation of a isothermal plane-parallel layer
Leading to: Radiation of isothermal volumes of scattering medium: An error of the diffusion model
Analysis of the accuracy of the P_{1} approximation for the model one-dimensional problems of thermal radiation of an isothermal layer of a homogeneous medium and the radiative equilibrium in the medium layer showed that the usual P_{1} error is not greater than 5-10%. It is known that this error increases significantly in two- and three-dimensional radiative transfer problems (Ou and Liou, 1982; Dombrovsky and Barkova, 1986; Dombrovsky, 1996a,b,c; Modest, 2003). In addition, such an increase in the error in the P_{1} approximation can also be seen in one-dimensional problems in the case of significant optical inhomogeneity of the medium in the computational region. This effect is expected to be more pronounced in the absence of reflecting and radiating boundary surfaces. The simplest examples are provided by cylindrical or spherical volume radiation passing through a layer of a relatively cold nonscattering medium (Dombrovsky, 1997). These problems can be considered as qualitative one-dimensional models of radiation of a hot jet flowing out from a nozzle (Dombrovsky and Barkova, 1986) or a fireball from an explosion in the atmosphere (Surzhikov, 1997).
In the case of the cylindrical or spherical volume of an absorbing and scattering hot medium, the P_{1} boundary-value problem is formulated as follows:
(1a) |
(1b) |
where n = 1 for cylindrical or n = 2 for spherical volume, D_{λ} = 1/(3β_{λ}^{tr}) is the spectral radiation diffusion coefficient, γ = 1 corresponds to Marshak’s boundary condition, and γ = 2/(4 - √3) corresponds to Pomraning’s condition (the P_{1m} approximation). In the case of homogeneous isothermal medium, problems (1a) and (1b) can be written in the dimensionless form (hereafter, for brevity we omit the subscript λ):
(2a) |
(2b) |
where I_{0} = I_{0}(πB) and ζ = √α/D. The corresponding analytical solution and expressions for radiation flux, hemispherical emissivity, and the radiation flux divergence are
(3a) |
(3b) |
(3c) |
(3d) |
where
(4a) |
(4b) |
In Eq. (4a), I_{0}(x) is the modified Bessel function of the first kind (Abramowitz and Stegun, 1965) (do not confuse this function with the radiation energy density). The emissivity of a spherical volume of a nonscattering medium calculated in the P_{1} approximation by formula (3c) is compared in Table 1 with an exact solution that was obtained by direct integration of the radiative transfer equation (RTE) along a set of rays. The optical thickness in Table 1 is defined as τ_{0} = α r_{1}. One can see that the P_{1} error is not large and does not exceed 7.2% for both types of boundary conditions. The error is practically the same as that for a plane-parallel layer of the medium (see the article Radiation of an Isothermal Plane-Parallel Layer). This result is explained by the angular dependence of the radiation intensity (see Fig. 1).
Figure 1. Angular dependence of thermal radiation intensity of an isothermal sphere containing a homogeneous nonscattering medium: (a) on the surface of the sphere [(1) τ_{0} = 0.1, (2) τ_{0} = 0.5, (3) τ_{0} = 1.0, (4) τ_{0} = 2.0, (5) τ_{0} = 5.0] and (b) at various distances from the sphere [(1) r_{2} = 2r_{1}, (2) r_{2} = 3r_{1}] at optical thicknesses τ_{0} = 1 (red lines) and τ_{0} = 5 (black lines).
Table 1. Hemispherical emissivity of a sphere containing a homogeneous nonscattering medium
τ_{0} | ε_{0} | ||
P_{1} | P_{1m} | Exact solution | |
0.1 | 0.125 | 0.124 | 0.124 |
0.2 | 0.234 | 0.230 | 0.231 |
0.5 | 0.482 | 0.467 | 0.472 |
1 | 0.720 | 0.687 | 0.703 |
2 | 0.903 | 0.852 | 0.886 |
5 | 1.010 | 0.946 | 0.980 |
∞ | 1.072 | 1.000 | 1.000 |
A completely different situation takes place when radiation from a volume is observed from a considerable distance of r_{2} > r_{1}, when the angular size of the source of radiation is small. This case is illustrated by the exact calculation for the radiation intensity of a spherical volume, the results of which are presented in Fig. 1(b). It is expected that the P_{1} error, which is formally applicable in the region up to the observation point (it is sufficient to introduce an almost transparent medium in the region r_{1} < r < r_{2}), will increase significantly.
For a quantitative analysis of this effect, let us consider the problem of thermal radiation of a homogeneous isothermal region r < r_{1}, with α = α_{1} and D = D_{1}, surrounded by a layer of a homogeneous cold medium r_{1} < r < r_{2}, with α = α_{2} and D = D_{2}. The equations for the P_{1} approximation for the subregions take the following form:
(5a) |
(5b) |
where subscripts 1 and 2 indicate the central and peripheral regions, respectively. It is obvious that functions I_{0i} must satisfy the following conditions at the inner boundary:
(6) |
The boundary conditions at r = 0 and at the outer boundary r = r_{2} of the region are
(7) |
It is not difficult to obtain the following analytical solution to boundary-value problems (5)-(7) (Dombrovsky, 1997):
(8a) |
(8b) |
where
(9a) |
(9b) |
(9c) |
Here, K_{0}(x) is the modified Bessel function of the second kind (Abramowitz and Stegun, 1965). The emissivity of the central region and the radiation flux divergence are
(10a) |
(10b) |
It is interesting to compare the values of ε_{1} and Q_{1}(r) obtained by Eqs. (10a) and (10b) in the limiting case of a transparent nonscattering medium in the peripheral region (β_{tr2} = α_{2} → 0, ζ_{2},x_{2},x_{3} → 0, ζ_{2}D_{2} = 1/√3) with the values of ε_{0} and Q_{0}(r). After the obvious transformations, we obtain the final expression:
(11) |
It follows from Eq. (11) at γ_{2} = γ = 1 that maximum value ε_{1}/ε_{0} = 2 is obtained at ε_{0} = 1 and r_{2}/r_{1} →∞. In all of the remaining cases (i.e., at ε_{0} < 1 and any finite value of r_{2}/r_{1}), we obtain ε_{1}/ε_{0} < 2. That is, the P_{1} approximation does not overestimate the radiation flux and its divergence by more than a factor of 2. The Pomraning boundary condition used at r = r_{1} only worsens the result for r_{2}/r_{1} >> 1 in comparison with the Marshak boundary condition.
The dependences of the ratio ε_{1}/ε_{0} on r_{2}/r_{1} for different optical thicknesses of the central region and a wide variation of the relative absorption coefficient of the surrounding cold nonscattering medium are shown in Fig. 2. The calculations, by use of Eqs. (3c), (4), (9), and (10a), were performed at γ_{2} = γ = 1. One can see that the P_{1} error increases with the optical thickness of the hot region. This result contradicts the frequently made statement about the role of optical thickness. Of course, the P_{1} error also increases with the contrast between the medium properties in the hot and cold zones.
Figure 2. Relative error in the emissivity of a spherical (I) and cylindrical (II) volume of a nonscattering medium involving a peripheral region with a cold absorbing medium: (a) τ_{0} = 0.5 and (b) τ_{0} = 5; (1) α_{2}/α_{1} = 1, (2) α_{2}/α_{1} = 0.1, (3) α_{2}/α_{1} = 0.01, and (4) α_{2}/α_{1} = 0.001.
Assuming (as earlier) that γ = 1, let us turn our attention to the role of the coefficient γ_{2} in Eq. (11). It is not difficult to see that ε_{1} = ε_{0} at
(12) |
The physical meaning of Eq. (12) is obvious because this formula in fact takes into account the narrowing of the angle range in which the radiation intensity of a distant volume is other than zero [see Fig. 2(b)]. Expression (12) is related only to the simple case of a transparent nonscattering medium in the peripheral region. The role of parameter γ_{2} upon a change of the relative value of absorption coefficient α_{2}/α_{1} is shown in Fig. 3. One can see that the use of the modified boundary condition appears to be sufficiently effective in the practically important range r_{2}/r_{1} < 3.
Figure 3. Relative error in emissivity of a two-layer spherical volume: (a) ω = 0 and (b) ω = 0.8; (I) γ_{2} = 1 and (II) γ_{2} = (r_{2}/r_{1})^{2}; and (2) α_{2}/α_{1} = 0.1 and (3) α_{2}/α_{1} = 0.01.
Of course, the conditions of thermal radiation transfer in applied problems (even for a similar form of the computational region) are not the same as those in the model problem considered. As a result, it is difficult to select a radiating core and absorbing cold layer in a realistic problem. At the same time, the use of Eq. (12) with variation of the value of r_{2}/r_{1} in accordance with the possible schematization of the problem may be useful in estimating the expected error in the P_{1} calculations.
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