General procedure for calculation of diffuse view factors between arbitrary planar polygons Sandip Mazumder ⇑ , Mahesh Ravishankar Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA article info Article history: Received 27 May 2012 Received in revised form 19 July 2012 Accepted 24 July 2012 Available online xxxx Keywords: View factor Form factor Contour integration Arbitrary polygon Radiation transport Light transport abstract Diffuse view factors (or form factors) between two planar polygonal surfaces placed at arbitrary orienta- tion relative to each other serve as inputs for surface-to-surface radiation transport or other similar cal- culations. In this article, a general formulation and accompanying numerical procedures are presented to perform such calculations. The formulation uses a parametric vector representation of the two surfaces along with the well-known contour integral method to develop an integral formula that finally requires a numerical quadrature scheme to evaluate. The use of the parametric vector representation of the sur- faces overcomes the difficulty of defining limits to the integrals in the contour integral method. Six test cases are considered, and the results are compared either to exact analytical solutions (when available) or to Monte Carlo results, which are also generated as part of this study. It is found that irrespective of the case studied, the present method matches analytical results perfectly (up to six decimal places accuracy) with a 10-point or higher Gaussian quadrature scheme. The errors in the Monte Carlo results increase when the two surfaces are either far from each other and/or at grazing angles, although the actual errors are quite small (less than 0.5%). For the cases when analytical solutions are not available, the Monte Carlo results and those generated by the present method were found to be in close agreement. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Diffuse view factors, also known as form factors or shape fac- tors, are required as inputs for the treatment of surface-to-surface exchange formulations [1,2] for radiation transport. In the radia- tion literature, they are sometimes also referred to as radiation configuration factors [1,2]. Most textbooks on radiation present a table of formulae to cal- culate diffuse view factors for a large number of scenarios for which closed-form analytical expressions can be derived. A com- prehensive catalog of radiation exchange factors for which closed-form analytical solution is derivable has been assembled by Howell [3], and is also available online. While the number of scenarios covered by such catalogs is large, practical engineering calculations require much more. In fact, the shapes encountered in practical engineering applications such as combustors, rapid thermal processing chambers, or nuclear reactors are far too com- plex, and do not necessarily fit any of the scenarios available through catalogs. For such cases, one has to either make approxi- mations to the actual geometry, or revert back to the basic definition of the view factor and compute the double area integrals numerically. Numerical computation of view factors can be performed either using deterministic methods or Monte Carlo methods. The Monte Carlo method is the most general and powerful method—especially since it can also handle obstructions, and has been employed extensively for this purpose [1,2,4–10]. Unfortunately, they are computationally expensive. Furthermore, solutions generated by Monte Carlo calculations inherently contain statistical noise, espe- cially when the solid angle subtended by one surface on the other is small, which results in small view factor values. This usually oc- curs when the two surfaces are placed far from each other or at grazing angles of incidence, as will also be demonstrated in this article. Some methods have been suggested to smooth fluctuations due to statistical errors [11]. Nonetheless, because of their compu- tational cost, the Monte Carlo method is prohibitive for large scale three-dimensional calculations that are demanded by practical applications. One of the most general deterministic methods to compute view factors numerically from first principles in three-dimensional geometries is the so-called contour integral method [1,2], first pro- posed by Sparrow [12]. In this method, the double surface area integrals in the definition of the view factor are converted to dou- ble line integrals through use of the Stokes’ theorem. This method has found prolific use in the computation of view factors between 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.07.066 ⇑ Corresponding author. Address: Department of Mechanical and Aerospace Engineering, The Ohio State University, Suite E410, Scott Laboratory, 201 West 19th Avenue, Columbus, OH 43210, USA. Tel.: +1 (614) 247 8099; fax: +1 (614) 292 3163. E-mail address: mazumder.2@osu.edu (S. Mazumder). International Journal of Heat and Mass Transfer xxx (2012) xxx–xxx Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt Please cite this article in press as: S. Mazumder, M. Ravishankar, General procedure for calculation of diffuse view factors between arbitrary planar poly- gons, Int. J. Heat Mass Transfer (2012), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.07.066