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ABSTRACT
The analysis of thermal radiation in multisurface enclosures typically involves multiple integrals with four, five, or six variables. Major difficulties for the numerical evaluation of these integrals are nonconvex geometries and singular integral kernels. This paper presents hierarchical visibility concepts and coordinate transformations which eliminate these difficulties. In the transformed form, standard quadrature techniques are applied, and accurate results are achieved with moderate numerical effort. In benchmark example problems, the effectiveness and accuracy of the method are validated. A simulation model of an industrial furnace demonstrates the applicability of the method for large-scale problems.
Nomenclature
| Bα | = | bounding box |
| b | = | radiosity, W/m2 |
| C(α) | = | function for computing the visibility |
| D | = | integration domain |
| d | = | side length, distance, m |
| f | = | continuous function or part of the kernel k, m4 |
| g | = | part of the integral kernel k, 1/m2 |
| H | = | heat flux density absorbed by a volume zone, W/m2 |
| h | = | irradiance, W/m2 |
| I | = | total intensity, W/(m2 sr) |
| K | = | absorption coefficient, 1/m |
| k | = | integral kernel, 1/m4 |
| M | = | order of quadrature |
| NS | = | number of surface zones |
| NV | = | number of volume zones |
| = | net heat flow into volume zone, W | |
| = | net heat flow into surface zone, W | |
| r | = | distance coordinate along a ray, m |
| rj, rij | = | geometrical length of a ray, m |
| = | optical length of a ray | |
| 𝒮α | = | set of barycenters of surfaces zones |
| S(α) | = | functions for generating a hierarchy of bounding boxes |
| S | = | surface zone, surface area, m2 |
| s | = | evaluation point of Gaussian quadrature |
| = | surface–surface direct exchange area, m2 | |
| = | surface–volume direct exchange area, m2 | |
| T | = | gas temperature, K |
| t | = | surface temperature, K |
| V | = | volume zone, volume, m3 |
| Vref | = | reference volume zone |
| = | volume–surface direct exchange area, m2 | |
| = | volume–volume direct exchange area, m2 | |
| W | = | integration domain |
| w | = | weight of Gaussian quadrature |
| = | coordinates in volume zone, m | |
| α | = | tuple |
| = | normalized coordinates in reference volume zone | |
| = | relative coordinates | |
| ε | = | emissivity |
| θ | = | incident or emergent angle, rad |
| Λ | = | coordinate transformation |
| σ | = | Stefan–Boltzmann constant, W/(m2 K4) |
| Φ | = | function for a transformation of integration domains |
| Subscripts | = | |
| i, j | = | index of zone or index value |
| k, n | = | index value |
Nomenclature
| Bα | = | bounding box |
| b | = | radiosity, W/m2 |
| C(α) | = | function for computing the visibility |
| D | = | integration domain |
| d | = | side length, distance, m |
| f | = | continuous function or part of the kernel k, m4 |
| g | = | part of the integral kernel k, 1/m2 |
| H | = | heat flux density absorbed by a volume zone, W/m2 |
| h | = | irradiance, W/m2 |
| I | = | total intensity, W/(m2 sr) |
| K | = | absorption coefficient, 1/m |
| k | = | integral kernel, 1/m4 |
| M | = | order of quadrature |
| NS | = | number of surface zones |
| NV | = | number of volume zones |
| = | net heat flow into volume zone, W | |
| = | net heat flow into surface zone, W | |
| r | = | distance coordinate along a ray, m |
| rj, rij | = | geometrical length of a ray, m |
| = | optical length of a ray | |
| 𝒮α | = | set of barycenters of surfaces zones |
| S(α) | = | functions for generating a hierarchy of bounding boxes |
| S | = | surface zone, surface area, m2 |
| s | = | evaluation point of Gaussian quadrature |
| = | surface–surface direct exchange area, m2 | |
| = | surface–volume direct exchange area, m2 | |
| T | = | gas temperature, K |
| t | = | surface temperature, K |
| V | = | volume zone, volume, m3 |
| Vref | = | reference volume zone |
| = | volume–surface direct exchange area, m2 | |
| = | volume–volume direct exchange area, m2 | |
| W | = | integration domain |
| w | = | weight of Gaussian quadrature |
| = | coordinates in volume zone, m | |
| α | = | tuple |
| = | normalized coordinates in reference volume zone | |
| = | relative coordinates | |
| ε | = | emissivity |
| θ | = | incident or emergent angle, rad |
| Λ | = | coordinate transformation |
| σ | = | Stefan–Boltzmann constant, W/(m2 K4) |
| Φ | = | function for a transformation of integration domains |
| Subscripts | = | |
| i, j | = | index of zone or index value |
| k, n | = | index value |