Radiative Equilibrium in a 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: Radiative boundary layer
We will consider a situation in which the thermal radiation is the only mode of heat transfer in an absorbing and scattering medium between two flat parallel walls with different temperatures. Since there is no heat transfer by convection and conduction, one can speak about the radiative equilibrium in the medium, and the energy equation has the following form:
(1) |
where q is the integral thermal radiation flux and d is the distance between the walls.
First, we give a formulation of the radiative equilibrium problem in the case of a gray medium and gray isotropically radiating walls. The radiative transfer equation (RTE) for the integral radiation intensity and appropriate boundary conditions are
(2a) |
(2b) |
(2c) |
where τ = β z is the optical coordinate; τ_{0} = β d is the optical thickness of the medium layer; T_{w1}, T_{w2} are the temperatures of the walls; ε_{w1}, ε_{w2} are the hemispherical emissivities of the walls; for specular reflection; and for the diffuse (isotropic) reflection of the radiation by the walls. The radiation flux is defined as follows:
(3) |
In the above statement, the radiative equilibrium problem has no analytical solution. For isotropic scattering, the integration of Eq. (2a) with regard to Eqs. (1) and (3) leads to the following relations:
(4a) |
(4b) |
The first of these equations is the radiation energy conservation for an arbitrary gray medium; the second one is true only for isotropic scattering. One can see that the solution does not depend on the medium albedo if the scattering is isotropic. The relations, which are similar to Eq. (4), can also be obtained for the transport approximation of the arbitrary scattering function. It is sufficient to redefine optical coordinate τ through transport extinction coefficient β_{tr} instead of the ordinary extinction coefficient β. Therefore, the solution to the radiative equilibrium problem is technically identical for the nonscattering medium and the isotropically scattering medium and in the transport approximation for arbitrary anisotropic scattering.
Heaslet and Warming (1965) have found the exact solution for the problem considered at an arbitrary optical thickness. This solution was expressed in terms of tabulated functions. A more complicated problem of radiative equilibrium in a medium with linear anisotropic scattering was considered by Dombrovsky (1974) and by Modest and Azad (1980). The following analytical solution in the DP_{1} approximation was derived by Dombrovsky (1974):
(5a) |
(5b) |
(5c) |
Here, a_{1}(τ_{0},ε_{w1},ε_{w2}) and a_{2}(τ_{0},ε_{w1},ε_{w2}) are the functions equal to zero at ε_{w1} = ε_{w2} = 1. In the diffusion approximation, dimensionless functions Q and θ can be written in the form:
(6a) |
where τ_{tr} = β_{tr}z, τ_{tr}^{0} = β_{tr}d, and
(6b) |
Here, N_{appr} = 0 corresponds to the DP_{0} approximation; N_{appr} = 1 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 see that θ(0) ≠ 1 and θ(τ_{tr}^{0}) ≠ 0. This means that the medium temperature near the walls does not coincide with the wall temperature except in the limiting case of τ_{tr}^{0} → ∞. This effect, called the radiation slip, has been discussed in some detail by Deissler (1964) and by Sparrow and Cess (1978).
In the case of isotropic scattering, the results obtained by use of the approximate differential approximations can be compared with the exact solution. One can see in Fig. 1 and Table 1 that the P_{1} approximation is better for the optically thick layers, but DP_{0} is better for the optically thin layers of the medium. The modification P_{1m} gives the same error as P_{1} at large optical thickness τ_{tr}^{0}, but at τ_{tr}^{0} < 0.6 the error of P_{1m} is much higher (not to mention that the obvious result of Q = 1 at τ_{tr}^{0} = 0 is not obtained). The accuracy of the P_{1} approximation in radiative equilibrium calculations for anisotropically scattering medium was estimated by Modest and Azad (1980). The comparison of P_{1} with the exact solution for linear anisotropic scattering showed the same error level as that in the isotropic case.
Figure1. Dimensionless radiative flux in radiative equilibrium for medium layer with isotropic scattering: (1) P_{1}, (2) DP_{0}, (3) DP_{1}, and (4) exact solution.
Table1. Dimensionless radiation flux in the radiative equilibrium: Solution for plane-parallel layer of isotropically scattering medium
τ_{0} | P_{1} | P_{1m} | DP_{0} | DP_{1} | Exact solution(by Heaslet and Warming, 1965) |
0.1 | 0.9302 | 0.8271 | 0.9091 | 0.9127 | 0.9157 |
0.4 | 0.7692 | 0.6974 | 0.7143 | 0.7401 | 0.7458 |
1 | 0.5714 | 0.5308 | 0.5000 | 0.5510 | 0.5532 |
2 | 0.4000 | 0.3796 | 0.3333 | 0.3896 | 0.3900 |
3 | 0.3077 | 0.2955 | 0.2500 | 0.3015 | 0.3016 |
10 | 0.1176 | 0.1158 | 0.0909 | 0.1167 | 0.1167 |
As one could expect, the DP_{1} approximation is much more accurate than the P_{1} or DP_{0} approximation. According to Table 1, the deviation of DP_{1} from the exact solution for isotropic scattering does not exceed 1%. At large optical thickness, the solution for linear anisotropic scattering (5b) gives the following expression for dimensionless radiation flux:
(7) |
The exact solution obtained by Heaslet and Warming (1965) at μ = 0 gives a similar result:
(8) |
One can see from Eq. (14) that transport approximation is a good approach even for linear anisotropic scattering in the case of an optically thick medium layer.
The computational results presented above allow formulating this general conclusion: in radiative equilibrium, as in thermal radiation of an isothermal layer, the diffusion approximation gives satisfactory results for radiation flux, and the transport approximation of the scattering function is a fairly good model for the scattering anisotropy description.
REFERENCES
Deissler, R. G., Diffusion approximation for thermal radiation in gases with jump boundary conditions, ASME J. Heat Transfer, vol. 86, no. 2, pp. 240-246, 1964.
Dombrovsky, L. A., Radiative equilibrium in a plane-parallel layer of absorbing and scattering medium, Fluid Dyn., vol. 9, no. 4, pp. 663-666, 1974.
Heaslet, M. A. and Warming, R. F., Radiative transport and wall temperature slip in an absorbing planar medium, Int. J. Heat Mass Transfer, vol. 8, no. 7, pp. 979-994, 1965.
Modest, M. F. and Azad, F. H., The differential approximation for radiative transfer in an emitting, absorbing, and anisotropically scattering medium, J. Quant. Spectrosc. Radiat. Transf., vol. 23, no. 1, pp. 117-120, 1980.
Sparrow, E. M. and Cess, R. D., Radiation Heat Transfer, New York: McGraw-Hill, 1978.
References
- Deissler, R. G., Diffusion approximation for thermal radiation in gases with jump boundary conditions, ASME J. Heat Transfer, vol. 86, no. 2, pp. 240-246, 1964.
- Dombrovsky, L. A., Radiative equilibrium in a plane-parallel layer of absorbing and scattering medium, Fluid Dyn., vol. 9, no. 4, pp. 663-666, 1974.
- Heaslet, M. A. and Warming, R. F., Radiative transport and wall temperature slip in an absorbing planar medium, Int. J. Heat Mass Transfer, vol. 8, no. 7, pp. 979-994, 1965.
- Modest, M. F. and Azad, F. H., The differential approximation for radiative transfer in an emitting, absorbing, and anisotropically scattering medium, J. Quant. Spectrosc. Radiat. Transf., vol. 23, no. 1, pp. 117-120, 1980.
- Sparrow, E. M. and Cess, R. D., Radiation Heat Transfer, New York: McGraw-Hill, 1978.