Integral equation solutions based on exact ray paths for radiative transfer in a participating medium with formulated refractive index Chih-Yang Wu ⇑ , Ming-Feng Hou Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC article info Article history: Received 6 March 2012 Received in revised form 14 June 2012 Accepted 21 June 2012 Available online 24 July 2012 Keywords: Radiative equilibrium Directional emittance Analytical path length Variable refractive index Integral equations abstract The exact analytical path length of radiation traveling in a slab with formulated variable refractive index is derived. Based on the analytical path lengths, the integral equations in terms of intensity moments for radiative transfer in a participating slab with one of the family of spatially varying refractive indices are developed. We solve the integral equations for radiative transfer in a slab at radiative equilibrium or for radiative transfer in an isothermal slab. The boundaries are assumed to be black for the slab at radiative equilibrium and the index jumps at both boundaries for the isothermal slab are considered. For compar- ison purpose, we also solve the radiative equilibrium problems by the discrete ordinates method (DOM). The nondimensional emissive power and nondimensional radiative heat flux obtained by solving integral equations show an excellent agreement with those obtained by the DOM. For the slab at radiative equi- librium and with positive gradient of refractive index, the jump of the emissive power at bottom bound- ary decreases with the increase of optical thickness for the cases with slightly varying refractive index, but the trend may not hold for the cases with significantly varying refractive index. For the non-scattering slab with positive gradient of refractive index and fixed refractive indices at the boundaries, the direc- tional emittances at both boundaries for the case with linear refractive index are smaller than those for the case with a refractive index of slope-increasing profile. Effects of the scattering albedo and the scattering phase function coefficient are investigated too. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction When the refractive index continuously varies in a medium, the radiation streams travel in curved trajectories instead of straight lines. Knowing the curved trajectory of light as it propagates in a refractive medium is frequently of importance in many applica- tions, including gradient lenses and optical fibers [1,2], photother- mal deflection spectroscopy [3] and atmospheres [4]. There are some references in the literature that addresses ray tracing in refractive media; they can be classified into two types, analytical methods [1,5] and numerical methods [2,6–10]. The numerical methods are versatile, but suffer from numerical error due to finite step size. In contrast, the exact trajectory of light can be found ana- lytically for a limited number of formulated refractive indices [1]. Besides, there has been growing interest in radiative heat transfer in a refractive medium with scattering, because of the applications, such as thermal barrier coatings [11], the thermal blooming due to high energy beams [12] and the bioluminescence tomography [13]. In contrast to the methods solving the integrodifferential radiative transfer equation (RTE) [11,13–19], highly accurate solutions have been obtained by solving integral RTE [20,21] and by the Monte Carlo method (MCM) [22–24] based on exact analytical ray tracing. Both integral equation solutions (IES) and efficient Monte Carlo simulation take advantage of the exact analytical ray tracing developed for refractive media [20–24]. Here, we present the exact analytical trajectories and path lengths of radiation traveling in media with some formulated refractive indices. To the best of our knowledge, the exact analytical trajectories and path lengths are derived for the first time. Then, radiative transfer in a plane- parallel, gray, absorbing-emitting-scattering slab with the spatially varying refractive indices mentioned above is solved by employing the integral formulation of intensity moments [21]. Two cases serving as examples are radiative equilibrium problem and ther- mal emission problem. The radiative equilibrium problem has been studied by investigators for the cases with scattering [15–17,19] and without scattering [14,20], while the apparent emittance of a non-scattering slab have been obtained by integrating the thermal emission along a ray path in analytical form [25]. To verify the present method, comparison of the present results with those ob- tained by the discrete ordinates method (DOM) [14] is made. The emissive power and the radiative heat flux are calculated for the radiative equilibrium problem, while the apparent emittance emerging from an isothermal, graded slab is obtained for the cases 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.06.068 ⇑ Corresponding author. Tel.: +886 6 2757575 62151; fax: +886 6 2352973. E-mail addresses: cywu@mail.ncku.edu.tw (C.-Y. Wu), n1892116@ccmail .ncku.edu.tw (M.-F. Hou). International Journal of Heat and Mass Transfer 55 (2012) 6600–6608 Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt