Technical Note A closed-form solution of differential approximation for radiative transfer in a planar refractive medium Ming-Feng Hou, Chih-Yang Wu ⇑ , Yi-Bin Hong Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC article info Article history: Received 3 June 2014 Received in revised form 17 November 2014 Accepted 1 December 2014 Keywords: Radiative transfer Variable refractive index Analytical solution Differential approximation abstract The closed-form solution of differential approximation (DA) for radiative transfer in a planar, scattering, refractive medium can be derived when the medium is in radiative equilibrium or purely scattering. In this work, linearly and exponentially spatial variations of refractive index are considered. The results obtained by the DA agree well with those obtained by the discrete ordinates method (DOM) and the Monte Carlo method (MCM) for the optically thick cases and the discrepancy between the DA results and the accurate numerical solutions obtained by the DOM and the MCM increases as the optical thick- ness decreases. The discrepancy between the results obtained by the DA and by the DOM also increases as the gradient of the linear refractive index increases. Besides, the influence of the scattering albedo and the scattering phase function coefficient on the accuracy of the analytical DA solution is less noticeable. Since the internal reflection due to the exponential decay of refractive index increases with the variation range of refractive index, the dimensionless radiative flux decreases with the increase of variation range of refractive index in a purely scattering medium. The present results show that the effect of the internal reflection is stronger in a thinner medium with an exponentially decaying refractive index. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction The radiative transfer problems incorporating the variable refractive index have been of increasing interest in recent years. In 1999, Ferwerda [1] derived the transient radiative transfer equa- tion (RTE) for a scattering medium with a spatially varying refrac- tive index. Due to the continuous variation of the refractive index in the medium, the radiation streams in curved paths rather than straight lines. Thus, the solution of radiative transfer in a medium with a varying refractive index is more difficult than that of radia- tive transfer in a medium with a constant refractive index. In the following year, Ben Abdallah and Le Dez [2] developed a curved ray-tracing method to solve the quasi-steady radiative transfer in an absorbing–emitting slab with a spatially varying refractive index. The method was further extended to solve the quasi-steady radiative transfer in rectangular and spherical media with a vary- ing refractive index by Ben Abdallah and co-workers [3,4]. Further discussion on the RTE has been presented by Tualle and Tinet [5] and Premaratne et al. [6]. More methods, including a combined curved ray-tracing and pseudo-source adding method [7], the dis- crete ordinates method (DOM) [8], a discrete curved ray-tracing method [9], the finite element method [10], the diffusion approxi- mation and the Monte Carlo method (MCM) [11], the discrete transfer method [12], the modified finite volume method [13] and the numerical solutions of integral equations [14], have been developed for solving quasi-steady radiative transfer in media with various spatial variations of refractive index. As for the transient radiative transfer problem, Tualle and Tinet [5] reported the results of the diffusion approximation and the Monte Carlo simulation, Khan and Thomas [15] developed the spherical harmonics approx- imation and Wu [16] presented a discrete ordinates solution. Then, Shendeleva and Molloy [17] compared the results of the diffusion approximation and the Monte Carlo simulation for a spherical medium and Wu [18] presented the Monte Carlo simulation for the cases with a laser pulse irradiation. Recently, Wang et al. [19] applied the DRESOR method to transient radiative transfer in graded medium. In spite of the good success, most of them resort to numerical computations which are usually CPU-time consum- ing. Among the large number of published methods there are still only a few methods [11,17] that provide analytical solutions. In this work, we aim to derive the analytical solution of a differential approximation (DA) for radiative transfer in a planar, scattering, refractive medium which is in radiative equilibrium or purely scattering. To exemplify the present solution, linearly and exponentially spatial variations of refractive index are considered. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.12.004 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +886 6 2757575 62151; fax: +886 6 2352973. E-mail address: cywu@mail.ncku.edu.tw (C.-Y. Wu). International Journal of Heat and Mass Transfer 83 (2015) 229–234 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt