# EXTENSION TO COMPLEX GEOMETRIES OF THE HYBRID FINITE ELEMENT / FINITE VOLUME METHOD FOR THE SOLUTION OF THE RADIATIVE TRANSFER EQUATION

DOI: 10.1615/ICHMT.2008.CHT.1010
16 pages

Pedro J. Coelho
Instituto Superior Technico Department of Mechanical Engineering Av. Rovisco Pais, 1049-001 Lisbon Portugal

## Abstract

A hybrid finite element / finite volume method was recently developed to solve the radiative transfer equation. In this method, the radiation intensity is approximated as a linear combination of basis functions, dependent only on the angular direction. The coefficients of the approximation are unknown functions of the spatial coordinates. The spatial discretization is carried out using the finite volume method, like in the discrete ordinates and finite volume methods, transforming the differential equations into algebraic equations. The angular discretization is accomplished using a methodology similar to that employed in the finite element method. The Galerkin-like approximation of the radiation intensity is introduced into the radiative transfer equation. Then, this is multiplied by the nth basis function and integrated over all directions, yielding a set of differential equations. The number of equations is equal to the number of terms in the summation. The basis functions are taken as the bilinear basis functions used in the finite element method, and a classical polar/azimuthal discretization is carried out, like in the finite volume and discrete transfer methods. However, while in these methods the radiation intensity is constant over a control angle or a solid angle, respectively, in the present method the radiation intensity is a continuously varying function, because the basis functions vary continuously within the control angle elements. Previous development and application of the method was limited to Cartesian coordinates. In the present work, the method is extended to complex geometries using a structured body-fitted mesh. Radiative transfer is calculated for several two-dimensional enclosures containing emitting-absorbing, scattering, grey media, and the predicted results are compared with benchmark solutions published in the literature. It was found that the results are in good agreement with reference solutions, demonstrating the ability of the present method to handle complex geometries.

## ICHMT Digital Library

Bow shocks on a jet-like solid body shape. Thermal Sciences 2004, 2004. Pulsed, supersonic fuel jets - their characteristics and potential for improved diesel engine injection. PULSED, SUPERSONIC FUEL JETS - THEIR CHARACTERISTICS AND POTENTIAL FOR IMPROVED DIESEL ENGINE INJECTION
View of engine compartment components (left). Plots of temperature distributions in centreplane, forward of engine (right). CHT-04 - Advances in Computational Heat Transfer III, 2004. Devel... DEVELOPMENT AND CURRENT STATUS OF INDUSTRIAL THERMOFLUIDS CFD ANALYSIS
Pratt & Whitney's F-135 Joint Strike Fighter Engine under test in Florida is a 3600F class jet engine. TURBINE-09, 2009. Turbine airfoil leading edge stagnation aerodynamics and heat transfe... TURBINE AIRFOIL LEADING EDGE STAGNATION AERODYNAMICS AND HEAT TRANSFER - A REVIEW
Refractive index reconstructed field. (a) Second iteration. (b) Fourth iteration. Radiative Transfer - VI, 2010. Theoretical development for refractive index reconstruction from a radiative ... THEORETICAL DEVELOPMENT FOR REFRACTIVE INDEX RECONSTRUCTION FROM A RADIATIVE TRANSFER EQUATION-BASED ALGORITHM
Two inclusion test, four collimated sources. Radiative Transfer - VI, 2010. New developments in frequency domain optical tomography. Part II. Application with a L-BFGS associated to an inexa... NEW DEVELOPMENTS IN FREQUENCY DOMAIN OPTICAL TOMOGRAPHY. PART II. APPLICATION WITH A L-BFGS ASSOCIATED TO AN INEXACT LINE SEARCH