ME608_Project_Rands_Westover - FINITE VOLUME METHOD APPLIED...

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FINITE VOLUME METHOD APPLIED TO RADIATIVE TRANSPORT IN PARTICIPATING MEDIA Tyler Westover School of Mechanical Engineering Purdue University West Lafayette, IN 47907 USA Email: [email protected] R. Curtis Rands School of Mechanical Engineering Purdue University West Lafayette, IN 47907 USA Email: [email protected] ABSTRACT A finite volume method is developed to solve the radiative transfer equation in rectangular domains for a quiescent, participating medium with conduction. Scattering in the medium is neglected. Discretization of the domain is accomplished using a structured orthogonal grid, and results are presented for various conditions. The effects of both Dirichlet and Neumann type boundary conditions are explored in different design configurations and where applicable, the results are in good agreement with previously published solutions. NOMENCLATURE A = cell face area vector g = geometric factor I = radiation intensity S h = source term, RTE source function S C = constant part of source term S P = coefficient part of source term T = temperature a = coefficient for discretized equation b = constant term for discretized equation b = vector of all b terms in linear system k = thermal conductivity s = direction vector = solid angle = prefix to indicate a difference or an extent δ = prefix to indicate a difference or an extent θ = polar angle κ = absorption coefficient ω = solid angle φ = azimuthal angle σ = Stefan-Boltzmann constant v = volume of cell Subscripts b = blackbody e = east face E = east cell neighbor f = face i = direction n = north face N = north cell neighbor nb = number of cell neighbors P = principal cell s = south face S = south cell neighbor w = west face W = west cell neighbor Superscripts * = dimensionless quantity 0 = current iterate value INTRODUCTION In the last decade, the discrete ordinates method has fostered increasing interest as a reliable method for solving the radiative transfer equation. However, serious drawbacks in the method, such as its non-conservative nature, have provided the motivation for a shift in focus to other methods [1]. One such method is the control-volume approach, also known as the finite volume method (FVM), which a conservative method widely used in the field of computational heat, mass, and momentum transfer. Incorporating the radiative transfer equation (RTE) into multimode heat transfer problems requires one of two approaches: employ the same spatial discretization for the energy equation and the RTE; or interpolate temperature and heat flux divergence between two different computational domains during an iterative process [3]. Although other methods exist which allow grid-sharing between the two equations, application of FVM to solve the RTE provides added convenience of sharing the same general formulation and solver without the need for special considerations. Of course, when properly formulated, finite volume methods provide the flexibility of using the same or different grids for the coupled solution of the RTE and the energy equation. Such an approach
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This note was uploaded on 12/29/2011 for the course ME 608 taught by Professor Na during the Fall '10 term at Purdue.

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ME608_Project_Rands_Westover - FINITE VOLUME METHOD APPLIED...

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