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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 703823, 11 pages
Research Article

Extension of the Reduced Integration Scheme to Calculate the Direct Exchange Areas in 3D Rectangular Enclosures with Nonscattering Media

1School of Materials & Metallurgy, Northeastern University, Shenyang 110819, China
2Institute of Thermal Engineering, School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China
3Beijing Key Laboratory of Multiphase Flow and Heat Transfer for Low Grade Energy, North China Electric Power University, Beijing 102206, China

Received 20 August 2014; Accepted 5 November 2014

Academic Editor: Denis Lemonnier

Copyright © 2015 Guo-Jun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The evaluation of direct exchange areas (DEA) in zonal method is the most important task due to the heavy computer cost of multi-integrals together with the existing of singularities. A technique of variable transformation to reduce the fold of integrals, which was developed originally by Erkku (1959) to calculate the DEAs of uniformly zonal dividing cylindrical system, was extended by Tian and Chiu (2003) for nonuniformly zonal dividing cylindrical system with large thermal gradients. In this paper, we further extend the reduced integration scheme (RIS) to calculate the DEAs in three-dimensional rectangular system. The detail deductions of six-, five-, and fourfold integrals to threefold ones are presented; the DEAs in a rectangular system with assumption of gray medium are computed by the Gaussian quadrature integration (GQI) and the RIS comparatively. The comparisons reveal that the RIS can provide remarkable higher accuracy and efficiency than GQI. More interestingly and practicably, the singularities of DEAs can be decomposed and weakened obviously by RIS.