Contents lists available at ScienceDirect International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt Macro-voxel algorithm for adaptive grid generation to accelerate grid traversal in the radiative heat transfer analysis via Monte Carlo method Hooman Naeimi a,⁎ , F. Kowsary b a Department of Mechanical Engineering, University of Bojnord, P.O. 9453155111, Bojnord, North Khorasan, Iran b School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran ARTICLE INFO Keywords: Heat transfer Radiation Monte Carlo Ray tracing Macro-voxel algorithm Radiation furnace ABSTRACT In the thermal radiation analysis via Monte Carlo method, the ray tracing algorithm often consumes a significant fraction of CPU time. As such, an efficient grid traversal algorithm can considerably affect the performance of the Monte Carlo method. This paper presents a new grid traversal acceleration algorithm by merging adjacent small empty voxels in a preprocessing step due to the fact that larger empty space, named “macro-voxel”, allows for traversing a ray over a large distance at a smaller cost. The proposed algorithm is validated theoretically, and the results are examined for a gray box with diffuse surfaces. Timing results of the new algorithm are compared with the USD method in a typical 3D radiation furnace with concave geometry and the speedup ratio of both the macro-voxel algorithm and the USD method with respect to direct method are calculated for an optimal grid of voxels. For the considered geometry, the macro-voxel algorithm is found to be clearly superior to the USD even if the size of the problem is large and the geometry is not convex. 1. Introduction Radiation is the dominant mode of energy transfer in high tem- perature environments including combustion chambers and furnaces and in the semiconductor industry for thermal processing of wafers [1–2]. The Monte Carlo method [3–4] is one of the most versatile and widely used numerical tools in calculation of the radiative distribution factors [4] among enclosure surfaces. Practical applications of en- closures such as radiation furnaces involve complex three-dimensional geometries and surfaces with complicated surface properties. Currently due to rapid growth in computer speed, memory and availability, the Monte Carlo method has evolved from an expensive and approximate estimation tool to a more feasible accurate and cost-effective approach. As each ray bundle can independently be considered in the Monte Carlo calculations, the method is quite suitable for parallel programming with today's more powerful computers. The disadvantage of this method is that, as a statistical method, it is subject to statistical error. Monte Carlo method is widely used in solar energy applications, as well. Zhou and Qiu [5] utilized the Monte-Carlo integral method to calculate the direct exchange area in the zone method for the modeling and simulation of the radiation transfer in an industrial furnace. The Monte Carlo method was used by Mazumder and Kersch [1] to model radiative transport in rapid thermal processing (RTP) and thermal chemical vapor deposition (RTCVD) reactors. The basic algorithm and a modified form of the binary spatial partitioning (BSP) algorithm was implemented to speed up ray tracing by at least a factor of 3. Wang [6] developed an accurate stochastic algorithm to estimate view factors between canyon facets in the presence of shade trees and considered the potential of shade trees in mitigating canyon surface temperatures as well as saving of building energy use. In the other work, Yi et al. [7] developed the Monte Carlo method for solving transient radiative transfer in one-dimensional scattering media with arbitrary distributions of refractive index exposed to a collimated short pulse-laser irradiation at one of its boundaries in which time shift and superposition principle was applied. Also, Kovta- nyuk et al. [8] applied Monte Carlo method in the coupled radiati- ve–conductive heat transfer mode in a chamber by two specularly and diffusely reflecting boundaries with anisotropic scattering medium. In this case a recursive Monte Carlo method was proposed and then the diffusion approximation of the radiative transfer equation was utilized to solve radiative heat transfer equation and an equation of the con- ductive heat exchange. Mirhosseini and Saboonchi [9] used the Monte Carlo method to determine view factor for the plate including strip elements to circular cylinder as a case in heating and cooling processes in material processing. The analysis displayed the differences between the numerical results obtained and analytical solutions and they in- dicated that smaller elements require more effort to obtain an accurate view factor. http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.06.020 ⁎ Corresponding author. E-mail addresses: hnaeimi@ut.ac.ir, h.naeimi@ub.ac.ir (H. Naeimi). International Communications in Heat and Mass Transfer 87 (2017) 22–29 0735-1933/ © 2017 Elsevier Ltd. All rights reserved. MARK