![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
A local radial basis function meshless (LRBFM) method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional participating media, in which compact support radial basis functions (RBFs) augmented on a polynomial basis are employed to construct the trial function, and the radiative transfer equation (RTE) and energy conservation equation are discretized directly at nodes by the collocation method. LRBFM belongs to a class of truly meshless methods which require no mesh or grid, and can be readily implemented in a set of uniform or irregular node distributions with no node connectivity. Performances of the LRBFM is compared to numerical results reported in the literature via a variety of coupled radiative and conductive heat transfer problems in 1D and 2D geometries. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular node distribution, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. In addition, it is extremely simple to implement.
Nomenclature
| ai, bj | = | coefficients for RBF approximation |
| a, b | = | vector of coefficient |
| d | = | radius of the support domain, m |
| I | = | radiation intensity, W/m2 sr |
| i, j, k | = | general spatial indices |
| M | = | number of discrete directions |
| Nsol | = | total number of solution nodes |
| n | = | unit normal vector |
| G | = | interpolation matrix |
| D,B,M,N | = | tool matrix |
| p | = | vector of Legendre polynomial |
| pj | = | Legendre polynomial of jth order |
| q | = | radiative heat flux, W/m2 |
| r | = | distance between points x and xi, m |
| S | = | source term of the RTE |
| s | = | unit vector in a given direction |
| G | = | incident radiation |
| T | = | temperature, K |
| R | = | solution domain |
| w | = | weight function |
| x | = | vector of location |
| β | = | extinction coefficient, m−1 |
| μm, ηm, ξm | = | direction cosine in direction m |
| κa | = | absorption coefficient, m−1 |
| κs | = | scattering coefficient, m−1 |
| ε | = | wall emissivity |
| αsup | = | dimensionless parameter |
| σ | = | Stefan–Boltzmann constant, W/m2 K4 |
| τ | = | optical thickness |
| Φ | = | scattering phase function |
| Ω | = | vector of radiation transfer direction |
| Ω | = | solid angle, sr |
| ω | = | single scatting albedo |
| Subscripts | = | |
| b | = | black body |
| i, j | = | node index |
| m | = | direction index |
| j, k | = | Legendre polynomial order index |
| w | = | wall |
| Superscript | = | |
| m, m′ | = | direction index |
| T | = | transposition |
Nomenclature
| ai, bj | = | coefficients for RBF approximation |
| a, b | = | vector of coefficient |
| d | = | radius of the support domain, m |
| I | = | radiation intensity, W/m2 sr |
| i, j, k | = | general spatial indices |
| M | = | number of discrete directions |
| Nsol | = | total number of solution nodes |
| n | = | unit normal vector |
| G | = | interpolation matrix |
| D,B,M,N | = | tool matrix |
| p | = | vector of Legendre polynomial |
| pj | = | Legendre polynomial of jth order |
| q | = | radiative heat flux, W/m2 |
| r | = | distance between points x and xi, m |
| S | = | source term of the RTE |
| s | = | unit vector in a given direction |
| G | = | incident radiation |
| T | = | temperature, K |
| R | = | solution domain |
| w | = | weight function |
| x | = | vector of location |
| β | = | extinction coefficient, m−1 |
| μm, ηm, ξm | = | direction cosine in direction m |
| κa | = | absorption coefficient, m−1 |
| κs | = | scattering coefficient, m−1 |
| ε | = | wall emissivity |
| αsup | = | dimensionless parameter |
| σ | = | Stefan–Boltzmann constant, W/m2 K4 |
| τ | = | optical thickness |
| Φ | = | scattering phase function |
| Ω | = | vector of radiation transfer direction |
| Ω | = | solid angle, sr |
| ω | = | single scatting albedo |
| Subscripts | = | |
| b | = | black body |
| i, j | = | node index |
| m | = | direction index |
| j, k | = | Legendre polynomial order index |
| w | = | wall |
| Superscript | = | |
| m, m′ | = | direction index |
| T | = | transposition |