An accurate evaluation of geometric view factors for modelling radiative heat transfer in randomly packed beds of equally sized spheres

Abstract This paper proposes an effective general numerical procedure for the accurate evaluation of the geometric view factors between equally sized spheres in a randomly packed assembly. The procedure is a novel combination of three key techniques: the Tanaka integral expression for the view factor between two equal spheres, the product form of the Fibonacci integration scheme for integrals on spheres, and the non-uniform variable scaling. The Tanaka integral not only provides a simple integral to numerically compute the exact value of the view factor between two unit spheres without a blockage, but also plays a fundamental role in improving the computed shaded view factor between two spheres subject to any blockage. The product Fibonacci integration scheme, essentially a ray-tracing method, appears to be a low cost but sufficiently accurate cubature for evaluating the dual-integral defining the view factor. The use of a particular non-uniform scaling function very favourably modulates the properties of the view factor integrals, and thus substantially improves the solution accuracy of the Fibonacci integration scheme. Several practical issues associated with the accuracy and efficiency of the procedure are discussed, and test cases are provided to illustrate the applicability of the procedure.

[1]  Eckart Laurien,et al.  Thermal hydraulic analysis of a pebble-bed modular high temperature gas-cooled reactor with ATTICA3D and THERMIX codes , 2012 .

[2]  C. L. Tien,et al.  Radiative Transfer in Packed Spheres , 1974 .

[3]  E. F. M. Van Der Held,et al.  The contribution of radiation to the conduction of heat , 1952 .

[4]  Avram Sidi,et al.  A New Variable Transformation for Numerical Integration , 1993 .

[5]  Pablo R. Rubiolo,et al.  Modeling of the radiative contribution to heat transfer in porous media composed of spheres or cylinders , 2002 .

[6]  A. Steinfeld,et al.  Numerical and experimental study of gas–particle radiative heat exchange in a fluidized-bed reactor for steam-gasification of coal , 2007 .

[7]  Sadao Tanaka.  Exact view-factor analysis for radiation from a sphere to another sphere linked with a coaxial cylinder(MARITIME STUDIES, AND SCIENCE AND ENGINEERING) , 2008 .

[8]  Y. T. Feng,et al.  Discrete thermal element modelling of heat conduction in particle systems: Basic formulations , 2008, J. Comput. Phys..

[9]  David R. Owen,et al.  Discrete thermal element modelling of heat conduction in particle systems: Pipe-network model and transient analysis , 2009 .

[10]  Gilles Flamant,et al.  3D modelling of radiative heat transfer in circulating fluidized bed combustors: influence of the particulate composition , 2005 .

[11]  Massoud Kaviany,et al.  Modelling radiative heat transfer in packed beds , 1992 .

[12]  V. I. Lebedev,et al.  Quadratures on a sphere , 1976 .

[13]  J. R. Howell,et al.  RADIATIVE HEAT TRANSFER THROUGH A RANDOMLY PACKED BED OF SPHERES BY THE MONTE CARLO METHOD. , 1982 .

[14]  I. Sloan.  Lattice Methods for Multiple Integration , 1994 .

[15]  Stuart W. Churchill,et al.  Radiant heat transfer in packed beds , 1963 .

[17]  Correlated radiative transfer through a packed bed of opaque spheres , 2005 .

[18]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[19]  Didier Bouvard,et al.  A ray tracing method for evaluating the radiative heat transfer in porous media , 1996 .

[20]  David R. Owen,et al.  An augmented spatial digital tree algorithm for contact detection in computational mechanics , 2002 .

[21]  A. Yu,et al.  A boundary element method for evaluation of the effective thermal conductivity of packed beds , 2007 .

[22]  M. Pinar Mengüç,et al.  Thermal Radiation Heat Transfer , 2020 .

[23]  J. Hannay,et al.  Fibonacci numerical integration on a sphere , 2004 .