Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 703823, 11 pages

http://dx.doi.org/10.1155/2015/703823

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

^{1}School of Materials & Metallurgy, Northeastern University, Shenyang 110819, China^{2}Institute of Thermal Engineering, School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China^{3}Beijing 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.

#### Abstract

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.

#### 1. Introduction

In recent decades, the zonal method [1] has been being an outstanding tool for thermal radiation calculations, whether for the developments of the numerical methods themselves [2–10] or the applications in thermal engineering [11–15]. It is well known that the most heavy computation cost of zonal method comes from the direct exchange areas (DEAs), which include the radiative properties of the participating medium and boundaries, together with the geometrical relations. After the DEAs, the total exchange areas (TEAs) can be solved, then the solution of zonal method. Usually, the calculations of DEAs involve six-, five-, and fourfold integrals for three-dimensional systems, as can be seen in the following contents. Further, the analytical or exact solutions of DEAs are difficult or impossible for most cases. Therefore, considerable efforts have been made to reduce or simplify the computation cost of DEAs till now.

Larsen and Howell [3] defined the exchange factor method to reduce the computer cost using the reciprocity and symmetry. Sika [4] defined the DEAs of all kinds, say, surface to surface, surface to volume, volume to volume in axisymmetric cylindrical geometries with gray gas, and reduced multifold (four, five, and six) integrals to single- or double-fold integrals, and simplified formulae with high efficiency. Ma [5] illustrated that the GZM (generalized zoning method) can treat the anisotropic scattering using scattering-exchange areas; the GZM was used in one-dimensional case and compared with existing solutions. Goyhénèche and Sacadura [6] presented the explicit matrix relation for total exchange areas; the integration order can be reduced; then the TEAs in anisotropically scattering medium and anisotropically reflecting walls were obtained, and the weighted sum of gray gases model (WSGG) could be adopted in the zonal method. Tian and Chiu [7] deduced the formulas to reduce the multifold to less fold integrations by variable transformation for nonuniform zones in cylindrical systems. Two years later, Tian and Chiu [8] proposed a new approach to compute the DEAs by hybridization of the finite volume method (FVM) with the midpoint integration scheme; the FVM is for adjacent and overlapping zones; the midpoint integration scheme is for nonadjacent zones. El Hitti et al. [9] used zone method plus the thin layer approximation to simulate the glass sheets under high temperature condition; they evaluate the DEAs by flux planes approximation. At the same time, El Hitti et al. [10] evaluated the TEAs from DEAs of nongray system by replating algorithm.

The applications of zonal method can be found in many thermal processes, whether for offline analysis of parameter design or for online control of operating parameter optimization. Chapman et al. [11] performed a parametric study for a batch reheating furnace; the DEAs were evaluated by Monte Carlo method and the WSGG (weighted sum of gray gas) was adopted to consider different species of gases. Jeong and Ha [12] simulated the continuously annealing furnace using zonal method for radiation with the measured gas temperature and imaginary surface and discussed the correction of the soot effects on radiative properties. Barr [13] reviewed the thermal model of steady-state pusher-type reheating furnace; more details like the treatment of combination of radiation and forced convection were mentioned also. Chen and Jaluria [14] used the parallelization for DEA computation in their research. Steinboeck et al. [15] presented the thermal model of a slab reheating furnace, in which the heat conduction in the slab was solved by the weighted residual approach and the radiation was solved by the net radiation method based on zone method.

Recently, we have made the mathematical modeling of thermal processes both in conventional reheating furnace and regenerative reheating furnace [16] using zonal method to consider the radiative heat transfer. After cautious analyzing and comparing of above-mentioned approaches, the reduced integration scheme (RIS), which was described in detail in [7], is extended to simplify the computation of DEAs in rectangular systems, for uniform or nonuniform zone arrangement. Although Tian and Chiu [7] have already pointed out that this technique can be easily extended to three-dimensional rectangular enclosures, we still would like to give the detail deduction and results. More interestingly, the believable evidences of accuracy and efficiency will be given, especially for the treatment of singularities. This paper is organized as follows. In Section 2, the definition of DEAs for different zone pairs and detailed formulae are given. In Section 3, the detailed description of the RIS for DEAs is described. In Section 4, verifications and comparisons for specified cases are made. Finally, the last section gives the conclusions.

#### 2. Definitions of DEAs in Rectangular Enclosures

In fact, the definitions of DEAs of all kinds of zone pair, say, surface-surface, gas-gas (or volume-volume), and gas-surface (or volume-surface), can be found in [17, 18]. Here we present them briefly for the subsequent deductions. Figure 1 shows the radiative exchange between surface zones and , whose areas are and , respectively. Then the formula of DEA between them can be obtained with the guarantee of energy conservation for semitransparent medium of extinction coefficient . While, as indicated in the title, the media considered here are nonscattering, hereafter, only the absorption coefficient is used. Consider