Transient Radiation Element Method for Three-dimensional Scattering, Absorbing, and Emitting Media

In this study transient radiative heat transfer is investigated in scattering, absorbing, and emitting media. The radiation element method is formulated for the first time to solve the transient radiative transfer equation in 3-D geometries. The sensitivity and accuracy of the method are examined. A good agreement of temporal transmittance predicted by the present method and Monte Carlo method is found. The characteristics of transient analysis are investigated via various problems of radiative transfer in inhomogeneous cubes. It is found that the transmitted signals are strongly affected by the inhomogeneous properties of the media through which the radiation has passed. In the position where the radiation travels a larger optical thickness, the broadening of the transmitted pulse width is more obvious and the magnitude of the transmittance is smaller. INTRODUCTION In the analysis of thermal radiation, the time-derivative term in the radiative transfer equation is usually negligible even if the boundary conditions and/or the sources that are responsible for the radiative intensity vary with time (Siegel, 1998). Such a neglecting does not result in any practical errors for most traditional engineering applications because the imposed temporal variations at the boundaries are relatively slow when compared with the time scale associated with the propagation of radiation in the system at the speed of light. However, transient radiative heat transfer that accounts for radiation propagation speed is very important in small time scale systems such as in ultrashort pulsed laser transport (Kumar and Mitra, 1999) and in problems involving very large physical domain (Balsara, 1999). The short pulse laser systems are rapidly being deployed in a variety of nascent applications, such as laser material processing of microstructures (Tien et al., 1997), optical tomography (Yamada, 1995), laser tissue welding (Poppas et al., 2000) and ablation (Pettit and Sauerbrey, 1993), measurement of optical properties (Prahl et al., 1993), and remote sensing (Walker and McLean, 1999). In the simulation of short pulse laser radiation transport in such systems, the neglect of radiation propagation effect could lead to considerable errors. Several numerical models have been developed to simulate transient radiation transport through absorbing-scattering turbid media such as biological tissues. Previously, researchers in the biomedical field broadly used the transient diffusion equation of angle-independent photon flux (Flock et al., 1989; Yamada, 1995). Experiments have shown that such diffusion based approximation is only accurate for very thick samples, and it fails to match experimental data for thin and intermediate samples (Yoo et al., 1990). Further, it is inaccurate in the regions near laser incident spot and near surface even for thick samples (Richards-Kortum and Sevick-Muraca, 1996). In the 1 Copyright © 200 lby ASME solution of the complete transient radiative transfer equation, Kumar et al. (1996) employed the P1 model for 1-D parallel systems. The Laplace transform (Skocypec and Buckius, 1982) and adding-doubling methods (Prahl et al., 1993; Rackmil and Buckius, 1983) have also been introduced in the solution of 1D transient radiative transfer. Mitra and Kumar (1999) compared the various transient models for 1-D problems. Guo and Kumar (2001a) developed transient radiation element method for plane-parallel media of scattering, absorbing, and emitting and with collimated and/or diffuse irradiations. Mitra et al. (1997) considered 2-D transient radiative transfer using the Pi model. More recently, Wu and Wu (2000) developed an integral equation formulation for transient radiative transfer for 1and 2-D problems. Guo and Kumar (2001b) used the widely applied discrete ordinates method for the solution of timedependent radiative transfer equation in 2-D rectangular geometries. The Monte Carlo (MC) method is also widely used in the simulation of short pulse laser radiation transport (Flock et al., 1989; Hasegawa et al., 1991; Guo et al., 2000). The feasibility of applying various models to complex 3-D configurations is of major concern in practical applications. The MC method is feasible for arbitrary shapes. However, when optically thick media, such as living tissue, are considered, the MC method suffers from severe statistical errors due to finite samplings. Moreover, its drawback of longer CPU time limits it from the clinical use in imaging reconstruction. The radiation element method (REM) with ray tracing model was initially developed to deal with surface radiation problems with arbitrary shapes (Maruyama, 1993; Guo et al., 1997). It was further extended to analyze radiative heat transfer in participating media including nongray and inhomogeneous properties (Maruyama and Aihara, 1997; Guo and Maruyama, 2000). In this method, the configuration is handled by various radiation elements consisting of numerous polygons and polyhedrons so that arbitrary shapes can be easily handled. New concepts of absorption view factor and diffusely scattering view factor are introduced, and their values are obtained via ray tracing method. The analysis of radiative transfer is based on the net radiation method commonly used in surface radiation transfer. The method has been successful applied to radiative transfer in arbitrary 3-D configurations, such as in boilers (Guo and Maruyama, 2000) and Czochralski crystal growth furnaces (Maruyama, 1993; Guo et al., 1997). In this study, the REM has been extended to solve transient radiative transfer in scattering, absorbing, and emitting media in 3-D geometries with inhomogeneous properties. To assess the accuracy of the method, comparisons have been performed between the MC simulation and the transient REM (TREM) calculation. The influences of radiation element size, number of ray emission bundles, and time step size are examined. The characteristics of transient analysis are investigated via various problems of radiative transfer in inhomogeneous cubes. NOMENCLATURE A R = effective radiation area c = speed of light Eb = blackbody emissive power F.. A _ j _ = absorption view factor from element i to j F,J diffusely scattering view factor G = incident radiation I = radiation intensity /o = diffuse radiant intensity N = number of radiation elements Qr = heat transfer rate of emissive power Qj = heat transfer rate of diffuse radiosity Qx = net heat transfer rate of radiative heat generation S, 7 = path length between element from element i to j t = time t* = dimensionless time T = temperature x,y,z = coordinates At = time step Ax = element thickness 6 = emissivity, -1-.(2 = scattering phase function tr = Stefan-Boltzmann constant O'a = absorption coefficient <re = extinction coefficient o's = scattering coefficient /2 = albedo of a volume element or diffuse reflectivity of a surface element co = scattering albedo or solid angle