Chih Yang Wu

Integral equation formulation for transient radiative transfer in an anisotropically scattering medium

Abstract The integral equation formulation for transient radiative transfer in a 3D absorbing and anisotropically scattering medium is developed. The method developed is applied to transient radiative transfer in 1D planar and 2D cylindrical linearly anisotropically scattering media exposed to pulse radiation. The integral equations for the two examples are solved by the quadrature method. The results by the present method agree quite well with those obtained by the Monte Carlo methods. The effects of various parameters are investigated.

Propagation of scattered radiation in a participating planar medium with pulse irradiation

Abstract The integral equation in terms of angle-distance integration is developed for transient radiative transfer in an absorbing and isotropically scattering planar medium with pulse irradiation. The exact integral equation is solved by an adaptation of the quadrature method. The results obtained by the present method are in excellent agreement with those obtained by the Monte Carlo simulation. Accurate results for the time-resolved hemispherical reflectivity and transmissivity of the slab are presented. The dependence of the propagation of scattered radiation on the scattering albedo and the optical thickness is investigated.

Transient two-dimensional radiative and conductive heat transfer in a scattering medium

Abstract This work considers transient radiative and conductive heat transfer in a rectangular, absorbing-emitting and isotropically scattering medium. An integral-equation method and a modified differential approximation are applied to the radiation part of the present problem. Then, the radiation part and the energy equation are solved by numerical methods simultaneously. Comparisons to results published elsewhere for the special cases of steady-state two-dimensional and transient one-dimensional heat transfer are made. The comparisons show that, along with a finite-difference scheme, the integral-equation method generates very accurate results. The influence of aspect ratios, scattering albedos and conduction-to-radiation parameters is investigated.

Differential approximations for transient radiative transfer through a participating medium exposed to collimated irradiation

First, we apply the modified differential approximation (MDA) suggested by Chandrasekhar to transient radiative transfer in a scattering planar medium exposed to collimated pulse irradiation. Next, a hybrid method of the P1/3 approximation suggested by Olson and the MDA is developed. The hybrid method may be referred to as the modified P1/3 approximation (MP1/3A) and is also applied to the same example. Comparisons of the results obtained by solving the MDA, the MP1/3A and the exact integral equation are made. The comparisons show that the temporal distribution of the transmissivity obtained by the MDA contains a small protuberance or an abrupt slope change, which decreases with the decrease of the scattering albedo. The results obtained by the MP1/3A are more accurate than those obtained by the MDA for most of the cases considered, because the MP1/3A corrects the propagation speed of the transmitted radiation.

Integral equation solution for hyperbolic heat conduction with surface radiation

Abstract Temperature distributions for the hyperbolic heat conduction in a semi-infinite medium with surface radiation are found from the solutions of a nonlinear Volterra equation for the surface temperature. The integral equation is obtained by the Laplace transform. This method has the advantage that the temperature distributions do not involve numerical oscillations around the thermal wave front.

Modelling hyperbolic heat conduction in a finite medium with periodic thermal disturbance and surface radiation

Abstract Modelling and subsequent solutions of hyperbolic heat conduction in a finite slab with surface radiation and periodic on–off heat flux are the subject of this work. The non-linear integral equation for the radiating surface temperature is derived by the Laplace transform and solved numerically. The results show that the present method solves the problem accurately. The surface radiation not only lowers the temperature, but also causes the non-symmetrical oscillation of the surface temperature for the case with periodic supplied heat flux around that for the case with stepwise supplied heat flux. When the period of the propagation–reflection cycle is not a multiple of the period of the supplied heat flux, the superposition of the reflected temperature wave in a finite slab results in the multiple jumps of the temperature wave. The jumps, the slope of the average temperature and the degree of the temperature oscillation increase with the increase of the amplitude.

Integral equation solutions for transient radiative transfer in nonhomogeneous anisotropically scattering media

The integral equation formulation for transient radiative transfer in two-dimensional cylindrical nonhomogeneous absorbing and linearly anisotropically scattering media with collimated pulse irradiation is presented. The integral equations are solved by the quadrature method. The effects of spatially variable properties on transient radiative transfer are investigated for various optical sizes and extinction coefficient distributions

Radiative transfer in a two-layer slab with Fresnel interfaces

Abstract The exact formulation for radiative transfer in an absorbing and isotropically scattering two-layer slab with Fresnel interfaces is developed. The resulting integral equations are solved by the Nystrom method. The present model allows refractive indices larger than one; directional interface reflectivities and multiple specular reflections are taken into account. The hemispherical reflectivity and transmissivity of the slab are determined. The present results are compared with the transparent-interface results and the single-layer results reported in the literature. Excellent agreement is found, and so the present solution procedure is valid. The effects of scattering albedos, optical thicknesses and refractive indices of the two layers on the hemispherical reflectivity and transmissivity are investigated.

Composite discrete-ordinate solutions for radiative transfer in a two-layer medium with fresnel interfaces

Abstract The discrete-ordiaate method using composite quadrature is extended to analyze radiative transfer in a two-layer scattering medium with Fresnet interfaces. Since the discrete ordinates used in the present method depend on the critical angles at the interfaces, the discrete ordinates for each of the two layers may be different. Thus a technique adopting the formal solution of the radiation intensity leaving the interface from one layer to determine the intensity on the discrete ordinates of the other is presented. We apply the technique to obtain the hemispherical reflectivity and transmissivity of a two-layer medium. The effects of albedos, refractive indices, and various reflecting models at the lower surface of the medium are investigated.

Time-resolved spatial distribution of scattered radiative energy in a two-dimensional cylindrical medium with a large mean free path for scattering

Abstract Transient radiative transfer in a 2-D finite cylindrical medium with collimated pulse irradiation and a large mean free path for scattering is considered thoroughly. Highly accurate solutions of integral equation for the transient radiative transfer reveal that the radiative energy of the medium core is less than the radiative energy of the medium boundary, after the attenuated pulse irradiation has passed through the medium. The distinction between the extraordinary results of the above case and the results of other cases is examined. It is found that influence of the decrease rate of radiative energy with the passage of time is larger than that of the extinction decay of the radiative intensity along a propagation path for transient radiative transfer in a 2-D medium with a large mean free path for scattering. Moreover, scattering coefficient and geometric size are the major factors determining the spatial distribution type of scattered radiation energy at large time and the temporal evolution of the spatial distribution type of radiation energy.