Radiation of an Isothermal Plane-Parallel Layer
Following from: P_{1} approximation of the spherical harmonics method, The simplest approximations of double spherical harmonics, Solutions for one-dimensional radiative transfer problems
Leading to: An estimate of P_{1} approximation error for optically inhomogeneous media, Radiation of a nonisothermal layer of scattering medium, Diffusion approximation in multi-dimensional problems
The radiative transfer equation (RTE) for a homogeneous plane-parallel layer of an emitting, absorbing, and scattering isothermal medium with an azimuthally symmetric radiation field can be written in the following dimensionless form:
(1a) |
where
(1b) |
(1c) |
Dimensionless quantity ω is called the albedo of a medium, or Schuster number. For brevity, subscript λ is omitted in designations of I, τ, ξ, and ω. The boundary conditions in symmetry plane τ = 0 and on layer surface τ = τ_{0} are:
(1d) |
The solution to problem (1) can be obtained by use of various methods, both exact and approximate. In the case of nonscattering medium, the RTE is radically simplified:
(2) |
and the analytical solution can be written as
(3) |
A dimensionless spectral radiation flux is expressed by exponential integrals E_{3} (Abramowitz and Stegun, 1965):
(4) |
It is convenient to introduce the spectral hemispherical emissivity of the medium layer:
(5) |
The exact analytical solution to the problem under consideration appears to be very complicated for the scattering medium. However, with the differential approximation being used, the scattering can be regarded without additional difficulties. In the diffusion approximation, we have the following boundary-value problem for dimensionless spectral radiation energy density :
(6a) |
(6b) |
(6c) |
Here, N_{appr} = 0 corresponds to the DP0 approximation; DP_{0} corresponds to the P_{1} approximation; the value N_{mod} = 0 corresponds to Marshak’s boundary condition; and N_{mod} = 1 corresponds to Pomraning’s condition. One can find the obvious analytical solution to problem (6):
(7a) |
(7b) |
(7c) |
The solution obtained can be easily generalized for the practically important case of the medium layer bounded by two identical walls having temperature T_{w} and spectral emissivity ε_{w}. The corresponding boundary condition is given by
(8) |
and the system emissivity, which is defined as
(9) |
can be determined by the equation
(10) |
where the medium layer emissivity is determined by formula (7c).
According to Eq. (1) from the article The Simplest Approximations of Double Spherical Harmonics, the dimensionless equations of the DP_{1} approximation for linear anisotropic scattering can be written as
(11a) |
(11b) |
where
(11c) |
(11d) |
The dimensionless radiation flux
(12) |
Boundary conditions (2) from the article The Simplest Approximations of Double Spherical Harmonics can be expressed as
(13a) |
(13b) |
An analytical solution to the coupled vector Eqs. (11) with boundary conditions (13) was obtained by Dombrovsky (1972). The emissivity of the medium layer is
(14a) |
where
(14b) |
(14c) |
(14d) |
The simple problem of the isothermal layer radiation can be used for estimating the accuracy of the differential approximations. A comparison of the emissivity values calculated by use of different approximate solutions with the exact solution for nonscattering medium is presented in Table 1 and Fig. 1. One can see that DP_{0} is more accurate than P_{1}, and the DP_{1} approximation gives ε_{m}, which differs from the exact values by no more than 3% at an arbitrary optical thickness of the layer. Using the Pomraning boundary condition (P_{1m}) improves the accuracy of P_{1} to a large extent: it becomes more precise than DP_{0} and gives the correct result ε_{m} = 1 in the limit of the large optical thickness of the layer.
FIG. 1: Hemispherical emissivity of an isothermal plane-parallel layer of nonscattering medium: (1) P_{1} approximation; (2) P_{1m}, (3) DP_{0}, (4) DP_{1}, and (5) exact solution.
Table 1. Hemispherical emissivity of a plane-parallel layer of a nonscattering medium
τ_{0} | ε_{m} | ||||
P_{1} | P_{1m} | DP_{0} | DP_{1} | Exact solution | |
0.1 | 0.3306 | 0.3234 | 0.3297 | 0.3060 | 0.2961 |
0.2 | 0.5557 | 0.5357 | 0.5507 | 0.4932 | 0.4854 |
0.5 | 0.8935 | 0.8431 | 0.8647 | 0.7762 | 0.7806 |
1 | 1.0406 | 0.9728 | 0.9817 | 0.9375 | 0.9397 |
2.5 | 1.0716 | 0.9998 | 1.000 | 0.9986 | 0.9982 |
In the case of an isotropically scattering medium, the exact solution to the problem considered can be obtained numerically by use of the method proposed by Love and Grosh (1965). Representing the spectral radiation intensity in the form
(15) |
and introducing functions
(16) |
one can rewrite RTE (1a) in the form of the following equations:
(17a) |
(17b) |
and boundary conditions (1d) become
(18) |
After the transition to one equation for function g, we have
(19a) |
(19b) |
The integral term on the right-hand side of Eq. (19a) can be presented as a sum by the use of the Gaussian quadrature (Korn and Korn, 1968, 2000), as is usually done in the discrete ordinates method:
(20) |
Using Eq. (20a) for every direction μ_{i} and introducing designation g_{i}(τ) = g(τ, μ_{i}), we obtain the following system of the ordinary differential equations:
(21a) |
with boundary conditions
(21b) |
The analytical solution to boundary-value problem (21) is
(22) |
where λ_{k} and B_{i}^{k} are the characteristic values and vectors of matrix
(23) |
where δ_{ij} is the Kronecker delta, and coefficients C_{k} are determined from the system of linear algebraic equations
(24) |
The spectral hemispherical emissivity of the medium layer is
(25) |
The convergence of the presented numerical method with increasing of quadrature order m is illustrated by the data given in Table 2. Note that the solutions obtained are equivalent to approximation DP_{m}-1 of the double spherical harmonics method (Case and Zweifel, 1967). Table 3 and Fig. 2(a) give some results of an accuracy comparison for isotropically scattering medium. At any combination of τ_{0} and ξ, the error of the DP_{1} approximation does not exceed 3%. All of the methods under consideration give highly accurate results for an optically thin layer at high albedo of the medium (ξ = 1-ω << 1). In the case of intermediate and large optical thicknesses at considerable absorption (ξ ≥ 0.5), approximations P_{1}, DP_{0}, and P_{1m} have subsequently increasing accuracy.
FIG. 2: Hemispherical emissivity of an isothermal plane-parallel layer of a medium with isotropic scattering ( μ = 0) and linear anisotropic scattering ( μ = 0.5): (I) τ = 0.1, (II) τ = 0.5, and (III) τ = 0.5; (1) P_{1}, (2) DP_{0}, and (3) DP_{1}.
Table 2. Hemispherical emissivity of a plane-parallel layer of an isotropically scattering medium
m | ε_{m} | |||||
ξ = 0.1 | ξ = 0.5 | ξ = 0.9 | ||||
τ_{0} = 0.5 | τ_{0} = 2.5 | τ_{0} = 0.5 | τ_{0} = 2.5 | τ_{0} = 0.5 | τ_{0} = 2.5 | |
2 | 0.17219 | 0.46959 | 0.55609 | 0.84711 | 0.74309 | 0.97626 |
3 | 0.17247 | 0.47019 | 0.55861 | 0.55861 | 0.74686 | 0.97619 |
4 | 0.17255 | 0.47023 | 0.55922 | 0.84815 | 0.74761 | 0.97625 |
5 | 0.17254 | 0.47024 | 0.55914 | 0.84816 | 0.74741 | 0.97625 |
10 | 0.17254 | 0.47024 | 0.55913 | 0.84817 | 0.74740 | 0.97626 |
Table 3. Hemispherical emissivity of a plane-parallel layer of an isotropically scattering medium
Approximation | ε_{m} | |||||
ξ = 0.1 | ξ = 0.5 | ξ = 0.9 | ||||
τ_{0} = 0.5 | τ_{0} = 2.5 | τ_{0} = 0.5 | τ_{0} = 2.5 | τ_{0} = 0.5 | τ_{0} = 2.5 | |
P_{1} | 0.1778 | 0.4858 | 0.6165 | 0.8968 | 0.8509 | 1.0453 |
P_{1m} | 0.1757 | 0.4704 | 0.5921 | 0.8460 | 0.8050 | 0.9769 |
DP_{0} | 0.1765 | 0.4503 | 0.6019 | 0.8276 | 0.8244 | 0.9736 |
DP_{1} | 0.1722 | 0.4696 | 0.5561 | 0.8471 | 0.7431 | 0.9762 |
Exact solution | 0.1725 | 0.4702 | 0.5591 | 0.8482 | 0.7474 | 0.9762 |
The simplest anisotropic scattering is the Rayleigh scattering occurring in the case of extremely small particles (van de Hulst, 1957, 1981; Bohren and Huffman, 1983) (also see the article Rayleigh Scattering for further details). The Rayleigh scattering function
The spectral hemispherical emissivity of the medium layer is
(26) |
when substituted into Eq. (1a) can be immediately integrated:
(27) |
and the following extension of Eq. (19a) for Rayleigh scattering is obtained:
(28) |
This equation, similarly to Eq. (19a), can be converted by use of the Gaussian quadrature to a system of ordinary differential equations having the form of Eq. (21a), where A_{j} is replaced by
(29) |
Since Eq. (17b) is also true for Rayleigh scattering, boundary conditions (21b) remain unchangeable. A comparison of the exact numerical solutions obtained by use of the high-order quadrature for isotropic and Rayleigh scattering is presented in Table 4. One can see that the value of ε_{m} for Rayleigh scattering is smaller, but its difference from the case of isotropic scattering is negligible. This result is interesting due to the fact that simple differential approximations “do not distinguish” the symmetrical scattering functions (μ = 0) from the isotropic scattering.
Table 4. Hemispherical emissivity of a plane-parallel layer: Comparison of exact solutions for isotropic (I) and Rayleigh (R) scattering
τ_{0} | ε_{m} at ξ = 0.1 | ε_{m} at ξ = 0.5 | ||
I | R | I | R | |
0.5 | 0.1725 | 0.1723 | 0.5591 | 0.5576 |
1 | 0.2973 | 0.2966 | 0.7478 | 0.7457 |
2 | 0.4362 | 0.4353 | 0.8392 | 0.8376 |
3 | 0.4910 | 0.4901 | 0.8515 | 0.8502 |
5 | 0.5181 | 0.5174 | 0.8534 | 0.8522 |
It is also worth considering anisotropic scattering with model linear scattering function f (μ_{0}) = 1+3μμ_{0}. The linear approximation of real scattering functions has a physical sense only at μ ≤ 1/3, although its formal application also can result in accurate enough radiation flux at μ > 1/3. The linear scattering function, like the Rayleigh one, can be integrated, and as a result we obtain the following equation:
This equation, similarly to Eq. (19a), can be converted by use of the Gaussian quadrature to a system of ordinary differential equations having the form of Eq. (21a), where A_{j} is replaced by
(30) |
which transforms into Eq. (19a) when μ = 0. Unfortunately, the relation between g and h appears to be more complex than Eq. (17b) for isotropic and Rayleigh scattering:
(31a) |
(31b) |
Therefore, the boundary condition at τ = τ_{0} is also complicated:
(32) |
The algorithm of the problem numerical solution remains the same. Computational results for the linear scattering function are presented in Table 5 as well as in Figs. 2(b) and 3. The anisotropy of scattering with the positive asymmetry factor (μ > 0) has a considerable influence on the medium emissivity at optical thickness τ > 1. This effect is well described by the DP_{1} approximation. Note that the error level of P_{1} and DP_{0} remains the same as for that of isotropic scattering.
FIG. 3: Effect of linear anisotropic scattering on hemispherical emissivity of the medium layer. Calculations in the DP _{1} approximation: (I) ω = 0.9 and (II) ω = 0.5; (1) μ = 0, (2) μ = 0.2, (3) μ = 0.4, (4) μ = 0.6, and (5) μ = 0.8.
Table 5. Hemispherical emissivity of a plane-parallel layer: Comparison of exact solutions for isotropic (I) and Rayleigh (R) scattering
τ_{0} | ξ = 0.1 | ξ = 0.5 | ||
μ = 0.4 | μ = 0.8 | μ = 0.4 | μ = 0.8 | |
ε_{m}, exact solution | ||||
0.5 | 0.1739 | 0.1752 | 0.5664 | 0.5740 |
1 | 0.3051 | 0.3134 | 0.7714 | 0.7973 |
2 | 0.4684 | 0.5076 | 0.8835 | 0.9369 |
3 | 0.5468 | 0.6251 | 0.9017 | 0.9652 |
5 | 0.5980 | 0.7353 | 0.9053 | 0.9724 |
ε_{m}, DP_{1} approximation | ||||
0.5 | 0.1735 | 0.1748 | 0.5632 | 0.5707 |
1 | 0.3041 | 0.3124 | 0.7674 | 0.7926 |
2 | 0.4677 | 0.5065 | 0.8821 | 0.9349 |
3 | 0.5460 | 0.6241 | 0.9003 | 0.9634 |
5 | 0.5972 | 0.7343 | 0.9036 | 0.9703 |
It is interesting to compare the DP_{1} calculations for linear and transport approximations of the scattering function. The corresponding results for an optically thick medium layer (τ » 1), when the effect of scattering on the emissivity is maximum, are given in Table 6. One can see that the computational results for different scattering functions with the same asymmetry factor are practically coincident. The transport approximation gives smaller values of ε_{m} and provides the correct result in the limiting case of strong forward scattering: ε_{m} = 1 when μ = 1. Under the same conditions, the error of the DP_{0} approximation can be estimated by using the data presented in Fig. 4.
Table 6. Effect of anisotropic scattering on hemispherical emissivity of an optically thick layer: Calculations in DP1 for linear (L) and transport (T) scattering functions
μ | ε_{m} | |||||
ξ = 0.1 | ξ = 0.5 | ξ = 0.9 | ||||
L | T | L | T | L | T | |
0.2 | 0.559 | 0.58 | 0.876 | 0.875 | 0.983 | 0.982 |
0.4 | 0.609 | 0.607 | 0.904 | 0.900 | 0.987 | 0.987 |
0.6 | 0.679 | 0.607 | 0.935 | 0.929 | 0.992 | 0.991 |
0.8 | 0.789 | 0.779 | 0.971 | 0.962 | 0.997 | 0.995 |
1.0 | 1.029 | 1.000 | 1.013 | 1.000 | 1.002 | 1.000 |
FIG. 4: Effect of scattering anisotropy on hemispherical emissivity of an optically thick medium layer. Calculation in transport approximation: (1) ω = 0.5, (2) ω = 0.8, and (3) ω = 0.9; black lines, DP_{1}; and red lines, DP0.
Let us consider the most general case of anisotropic scattering when the applicability of simple approximations for the scattering function is not evident. It is convenient to represent the scattering function as a truncated series in terms of the Legendre functions:
Therefore, the boundary condition at τ = τ_{0} is also complicated:
(33) |
The particular cases of k = 0 and k = 1 correspond to the isotropic and linear anisotropic scattering approximations. The coefficients in Eq. (33) can be determined by numerical integration of the scattering function (Abramowitz and Stegun, 1965):
(34) |
Taking into account the properties of the Legendre functions, one can derive the following equation:
(35) |
which enables us to reduce RTE (1a) to a system of differential equation
(36a) |
(36b) |
where
(37a) |
(37b) |
Omitting bulky transformations, which give a solution similar to the one obtained for simple scattering function approximations, we will consider some computational results for the complex scattering function. The scattering function of a single spherical particle having diffraction parameter x = 2πa/λ = 5 (where a is the particle radius and λ is the radiation wavelength) and complex index of refraction m = 1.5-0.01i is considered. The approximation of this scattering function by a truncated series according to Eq. (33) is illustrated in Fig. 5 (at small values of k) and in Table 7 (at large values of k); the asymmetry factor of scattering μ = 0.731. One can see that convergence of approximation (33) is very slow, where one needs about 10 terms of the series for a good presentation of the scattering function.
FIG. 5: Expansion of the model scattering function [(1) black curve] in the Legendre functions: (2) k = 1, (3) k = 2, and (4) k = 3 (where k is the number of terms taken into consideration).
Table 7. Approximation of scattering function by a truncated series [Eq. (33)]
μ | Φ | |||
k = 5 | k = 8 | k = 10 | k = ∞ | |
−1 | −1.30 | 1.08 | 0.54 | 0.43 |
−0.8 | 0.92 | 0.34 | 0.32 | 0.29 |
−0.6 | 0.10 | 0.18 | 0.06 | 0.09 |
−0.4 | −0.53 | −0.14 | 0.18 | 0.15 |
−0.2 | −0.12 | 0.34 | 0.12 | 0.14 |
0 | 0.73 | 0.33 | 0.15 | 0.16 |
0.2 | 0.99 | −0.12 | 0.29 | 0.26 |
0.4 | 0.20 | 0.51 | 0.34 | 0.37 |
0.6 | −0.56 | 1.26 | 1.01 | 0.98 |
0.8 | 2.32 | 0.46 | 0.93 | 0.97 |
1 | 16.00 | 23.73 | 25.77 | 25.94 |
It is important that not all of the details of the scattering function are necessary for reliable calculations of the medium thermal radiation. This statement is illustrated by the data given in Table 8, where the results obtained in the transport approximation are also presented. The value of ξ = 0.5, corresponding to the presence of an absorbing medium besides the particles, was assumed in the calculations. Data presented in Table 8 enable us to speak about the high accuracy of the transport approximation in determination of the thermal radiation flux from the isothermal layer regardless of its optical thickness.
Table 8.Hemispherical emissivity of a layer of anisotropically scattering medium
τ_{0} | ε_{m} | |
Complete scattering function | Transport approximation | |
0.1 | 0.1693 | 0.1684 |
0.2 | 0.3002 | 0.2984 |
0.5 | 0.5627 | 0.5595 |
1.0 | 0.7781 | 0.7743 |
2.0 | 0.9144 | 0.9113 |
3.0 | 0.9436 | 0.9412 |
5.0 | 0.9518 | 0.9500 |
As a result of the comparison of the approximate solutions with the exact solution for the problem considered, one can observe high accuracy of the radiation flux determination on the layer surface: in the diffusion approximation (P_{1}, DP_{0}) the usual error is not greater than 5–10%, and employing DP_{1} decreases the error to about 1–3% even for considerable anisotropy and the complex scattering function. The efficiency of the transport approximation should be particularly emphasized. Such a result is explained by the integral character of the radiation field characteristics under consideration. The angular distribution of the radiation intensity may be approximated too roughly, but the radiation energy density and radiation flux are obtained with insignificant errors.
A successful use of the simple approximate radiation transfer description does not certainly mean that such an approximation is also accurate to the same degree in more complicated cases, especially for multidimensional problems with nonhomogeneous angular and spatial radiation intensity distributions (Dombrovsky, 1996; see also the articles An Estimate of P_{1} Approximation Error for Optically Inhomogeneous Media and Diffusion Approximation in Multi-Dimensional Problems).
REFERENCES
Abramowitz, M. and Stegun, I. A., eds., Handbook of Mathematical Functions, New York: Dover, 1965.
Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, New York: Wiley, 1983.
Case, K. M. and Zweifel, P. F., Linear Transport Theory, Reading, MA: Addison-Wesley, 1967.
Dombrovsky, L. A., Calculation of radiation heat transfer in a plane-parallel layer of absorbing and scattering medium, Fluid Dyn., vol. 7, no. 4, pp. 691–695, 1972.
Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, New York: Begell House, 1996.
Korn, G. A. and Korn, T. M., Mathematical Handbook for Scientists and Engineers, 2nd ed., New York: McGraw-Hill, 1968.
Korn, G. A. and Korn, T. M., Mathematical Handbook for Scientists and Engineers, 2nd ed., New York: Dover, 2000.
Love, T. J. and Grosh, R. J., Radiative heat transfer in absorbing, emitting and scattering media, ASME J. Heat Transfer, vol. 87, no. 2, pp. 161–166, 1965.
van de Hulst, H. C., Light Scattering by Small Particles, New York: Wiley, 1957.
van de Hulst, H. C., Light Scattering by Small Particles, New York: Dover, 1981.
References
- Abramowitz, M. and Stegun, I. A., eds., Handbook of Mathematical Functions, New York: Dover, 1965.
- Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, New York: Wiley, 1983.
- Case, K. M. and Zweifel, P. F., Linear Transport Theory, Reading, MA: Addison-Wesley, 1967.
- Dombrovsky, L. A., Calculation of radiation heat transfer in a plane-parallel layer of absorbing and scattering medium, Fluid Dyn., vol. 7, no. 4, pp. 691â€“695, 1972.
- Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, New York: Begell House, 1996.
- Korn, G. A. and Korn, T. M., Mathematical Handbook for Scientists and Engineers, 2nd ed., New York: McGraw-Hill, 1968.
- Korn, G. A. and Korn, T. M., Mathematical Handbook for Scientists and Engineers, 2nd ed., New York: Dover, 2000.
- Love, T. J. and Grosh, R. J., Radiative heat transfer in absorbing, emitting and scattering media, ASME J. Heat Transfer, vol. 87, no. 2, pp. 161â€“166, 1965.
- van de Hulst, H. C., Light Scattering by Small Particles, New York: Wiley, 1957.
- van de Hulst, H. C., Light Scattering by Small Particles, New York: Dover, 1981.