Modified collapsed dimension method for radiative heat transfer problems

Abstract

Introduction C OLLAPSED dimensionmethod (CDM) is one of the ray tracing methods used for the solution of radiative transfer problems. Development of this method is partly based on the works of Shih and Chen1 and Shih and Ren2 on the discretized intensity method. Some light on the method was thrown by Blank. Subsequent developmentof the method was made by Blank andMishra.4 Detailed description of this method is available in Ref. 5. In CDM, three-dimensionalradiative information is mapped into a two-dimensionalsolutionplane in terms of effective intensity (EI) andoptical thicknesscoefŽ cient (OTC). Thus, unlikeothermethods, analysis and computations in this method are performed in a twodimensional solution plane instead of three-dimensionalspace. CDMhas beenused for the solutionof radiativetransferproblems with highaccuracy.4;5 Thismethodhasbeenfoundtowork for awide rangeof optical thickness (very low to high optical thickness). Furthermore, CDM has also been found to work well for the conjugate mode heat transfer problems. In CDM, for determinationof heat  ux and temperature information, at each point of interest EIs have to be integrated over planar angle in the solutionplane.Theseangularintegrationsareperformed by dividing the planar angle into intervals of equal sizes. In each subinterval, EI is assumed isotropic. Hence, for higher accuracy, CDM requiresmore EIs and, thus, higher computational time. In the presentwork,with the objectiveofmakingCDMmore economical, the method is modiŽ ed. In the modiŽ ed CDM (MCDM), angular integrations of the EIs are performed differently. The discretizationand integrationsperformed in thismethod originatefrom the concept of the discrete ordinate method (DOM). Here, the planar angle is divided into a Ž nite number of subintervals according to the number of Gaussian quadraturepoints considered.Hence, in MCDM, angular subintervals are unequal and instead of considering average values of interval, the weighted mean corresponding to the Gauss points is considered. With this, the method becomes more realistic, and it derives computational efŽ ciency with a more realistic representationof the radiative transfer process. In the present work, improvements in MCDM over CDM are tested by solving radiative transfer problems in oneand twodimensional Cartesian enclosures with gray and homogeneous, absorbing, emitting, and anisotropically scattering medium. Both radiative and nonradiative equilibrium situations are considered. MCDM and CDM results are compared with results from the exact method and DOM.