![Sample our Engineering & Technology journals, sign in here to start your access, latest two full volumes FREE to you for 14 days]({"type":"image" , "src" : "/sda/5933/TOC-Subject-Sample-2021-Engineering-Tech.jpg"})
ABSTRACT
The discrete ordinates-collocation spectral method (DO-CSM) is successfully applied to solve the three-dimensional radiative transfer equation (RTE). The RTE is angular discretized by the DOM first and then solved by the CSM. Two examples are adopted to identify the performances of the DO-CSM for 3D RTE. The results that are obtained from the DO-CSM and the discrete ordinates method with a diamond scheme (DOM-DS), respectively, are compared with those of the exact solutions or benchmarks. These comparisons indicate that the DO-CSM can give more accurate intensity distribution in both space and direction than the DOM-DS. The false scattering of the DO-CSM is much less than that of the DOM-DS. The errors of the DO-CSM are primarily due to the “ray-effect” and the errors caused by the false scattering can even be ignored when the resolutions are fine enough. The errors due to the ray-effect are significant especially for an optical thin medium and the ray-effects are largely suppressed by the finer angular discretization. The results of the DO-CSM improved more quickly than those of the DOM-DS with the increase in the number of directions because the errors caused by the false scattering of the DO-CSM are much less than those of the DOM-DS.
Nomenclature
| Am | = | matrix defined in Eq. (10a) |
| Bm | = | matrix defined in Eq. (10b) |
| Cm | = | matrix defined in Eq. (10c) |
| D | = | first-order derivative matrix |
| Fm | = | matrix defined in Eq. (10d) |
| I(r, s) | = | dimensionless intensity of radiation at point r |
| I(r, s) | = | dimensionless intensity of radiation at point rw |
| Nx, Ny, Nz | = | resolutions (number of grid points) in three coordinates, respectively. |
| Nd | = | total number of directions |
| nw | = | unit vectors normal to the surface |
| s | = | unit vector of direction |
| T | = | dimensionless temperature |
| u | = | Chebyshev collocation points in x direction |
| v | = | Chebyshev collocation points in y direction |
| w | = | weight of discrete ordinates approximation |
| x, y, z | = | Cartesian coordinates |
| α | = | uniformed Chebyshev collocation points in x direction |
| β | = | uniformed Chebyshev collocation points in y direction |
| γ | = | uniformed Chebyshev collocation points in z direction |
| ϵ | = | wall emissivity |
| λx, λy, λz | = | uniform factors in x, y, and z directions, respectively |
| μ, η, ξ | = | direction cosines in x, y, and z directions, respectively |
| υ | = | Chebyshev collocation points in the z direction |
| τ | = | optical thickness |
| Φ(s′ → s) | = | scattering phase function of energy transfer from an incoming direction s′ to an outgoing direction s |
| ω | = | scattering albedo |
| Ω | = | solid angle |
| □(1), □(2), □(3) | = | matrix multiplication in x, y, z directions defined in Ref [Citation10]. |
| Subscripts | = | |
| 1, 2, 3, 4, 5, 6 | = | wall number |
| b | = | black body |
| g | = | values of medium |
| i, j, k | = | solution node index for x, y, and z directions, respectively |
| w | = | values of the wall |
| Superscripts | = | |
| m | = | angular direction of radiation |
Nomenclature
| Am | = | matrix defined in Eq. (10a) |
| Bm | = | matrix defined in Eq. (10b) |
| Cm | = | matrix defined in Eq. (10c) |
| D | = | first-order derivative matrix |
| Fm | = | matrix defined in Eq. (10d) |
| I(r, s) | = | dimensionless intensity of radiation at point r |
| I(r, s) | = | dimensionless intensity of radiation at point rw |
| Nx, Ny, Nz | = | resolutions (number of grid points) in three coordinates, respectively. |
| Nd | = | total number of directions |
| nw | = | unit vectors normal to the surface |
| s | = | unit vector of direction |
| T | = | dimensionless temperature |
| u | = | Chebyshev collocation points in x direction |
| v | = | Chebyshev collocation points in y direction |
| w | = | weight of discrete ordinates approximation |
| x, y, z | = | Cartesian coordinates |
| α | = | uniformed Chebyshev collocation points in x direction |
| β | = | uniformed Chebyshev collocation points in y direction |
| γ | = | uniformed Chebyshev collocation points in z direction |
| ϵ | = | wall emissivity |
| λx, λy, λz | = | uniform factors in x, y, and z directions, respectively |
| μ, η, ξ | = | direction cosines in x, y, and z directions, respectively |
| υ | = | Chebyshev collocation points in the z direction |
| τ | = | optical thickness |
| Φ(s′ → s) | = | scattering phase function of energy transfer from an incoming direction s′ to an outgoing direction s |
| ω | = | scattering albedo |
| Ω | = | solid angle |
| □(1), □(2), □(3) | = | matrix multiplication in x, y, z directions defined in Ref [Citation10]. |
| Subscripts | = | |
| 1, 2, 3, 4, 5, 6 | = | wall number |
| b | = | black body |
| g | = | values of medium |
| i, j, k | = | solution node index for x, y, and z directions, respectively |
| w | = | values of the wall |
| Superscripts | = | |
| m | = | angular direction of radiation |