Anisotropic Scattering Phase Function Research Articles

Characterization of directional and anisotropic scattering dependency of emissivity for fibrous heat shields under non-isothermal conditions

The spectral and directional dependence of emissivity along with radiative emissions of scattering media has been characterized as functions of the optical thickness, scattering albedo, and anisotropic scattering phase function. Emissivity is found to be a strong function of the scattering albedo and the anisotropic scattering phase function of the medium, and a generalized relation between spectral emissivity and scattering behavior for thick optical media is developed through the use of Bézier curves. A comparison of the newly developed relation with the spectral emissivities of silica- and carbon-based fibrous media demonstrated excellent agreement, with a root mean square error of 0.19%. The characterization of directional emissivity through scattering media indicates that directional emissivity is higher than hemispherical emissivity in directions normal to the surface, but drops at polar angles greater than 50° because of increased reflection. A novel model is also developed to capture emissions from a medium with a non-uniform temperature distribution using the same emissivities derived under isothermal conditions, but with an effective blackbody temperature, which differs from the surface temperature of the medium. The new emission model exhibited excellent agreement compared to the true emissions under various conditions, with a root mean square error of 1.27%. This new approach offers a practical computationally tractable solution for the radiation energy balance at surfaces that can be used in current state-of-the-art computational fluid dynamics and thermal response simulations.

Parameterizing anisotropic reflectance of snow surfaces from airborne digital camera observations in Antarctica

Abstract. The surface reflection of solar radiation comprises an important boundary condition for solar radiative transfer simulations. In polar regions above snow surfaces, the surface reflection is particularly anisotropic due to low Sun elevations and the highly anisotropic scattering phase function of the snow crystals. The characterization of this surface reflection anisotropy is essential for satellite remote sensing over both the Arctic and Antarctica. To quantify the angular snow reflection properties, the hemispherical-directional reflectance factor (HDRF) of snow surfaces was derived from airborne measurements in Antarctica during austral summer in 2013/14. For this purpose, a digital 180∘ fish-eye camera (green channel, 490–585 nm wavelength band) was used. The HDRF was measured for different surface roughness conditions, optical-equivalent snow grain sizes, and solar zenith angles. The airborne observations covered an area of around 1000 km × 1000 km in the vicinity of Kohnen Station (75∘0′ S, 0∘4′ E) at the outer part of the East Antarctic Plateau. The observations include regions with higher (coastal areas) and lower (inner Antarctica) precipitation amounts and frequencies. The digital camera provided upward, angular-dependent radiance measurements from the lower hemisphere. The comparison of the measured HDRF derived for smooth and rough snow surfaces (sastrugi) showed significant differences, which are superimposed on the diurnal cycle. By inverting a semi-empirical kernel-driven bidirectional reflectance distribution function (BRDF) model, the measured HDRF of snow surfaces was parameterized as a function of solar zenith angle, surface roughness, and optical-equivalent snow grain size. This allows a direct comparison of the HDRF measurements with the BRDF derived from the Moderate Resolution Imaging Spectroradiometer (MODIS) satellite product MCD43. For the analyzed cases, MODIS observations (545–565 nm wavelength band) generally underestimated the anisotropy of the surface reflection. The largest deviations were found for the volumetric model weight fvol (average underestimation by a factor of 10). These deviations are likely linked to short-term changes in snow properties.

A new perspective to the conventional solar pond technology to increase the thermal efficiency

AbstractIn this study, a new perspective was introduced to conventional solar pond technology to spread its commercial application. In order to make it more useful with preserving energy efficiency, the brine layers were replaced with the normal freshwater by using a separated transparent partition. Structurally modified three different solar ponds were proposed and numerical investigations were conducted by using discrete ordinate method (DOM). For this, four types of solar ponds were modeled, the first one is the conventional salt‐gradient solar pond and three others are modified versions which are named as “Pure Water Solar Pond.” For all models, a comprehensive finite element method was developed to investigate the daily performance of solar ponds by using commercial software COMSOL. Salt‐gradient solar pond consists of seven layers where one layer is the heat storage zone, five layers are nonconvective zone, and one is the upper convective zone. However, in the modified models, solar ponds consist of two glass layers and one to three pure water zones depending on the model types. The numerical method includes a novel approach to calculate all the zone where absorption, emission, scattering, transmission, convection, and radiation are taken into account. Numerical results for all models were compared with an experiment to correlate the numerical accuracy of the DOM with anisotropic scattering phase function. Results indicate there is a good correlation between these four model approach and the experiment.

Discrete unified gas kinetic scheme for multiscale anisotropic radiative heat transfer

In this work, a discrete unified gas kinetic scheme (DUGKS) is developed for radiative transfer in anisotropic scattering media. The method is an extension of a previous one for isotropic radiation problems [1]. The present scheme is a finite-volume discretization of the anisotropic gray radiation equation, where the anisotropic scattering phase function is approximated by the Legendre polynomial expansion. With the coupling of free transport and scattering processes in the reconstruction of the flux at cell interfaces, the present DUGKS has the nice unified preserving properties such that the cell size is not limited by the photon mean free path even in the optical thick regime. Several one- and two-dimensional numerical tests are conducted to validate the performance of the present DUGKS, and the numerical results demonstrate that the scheme is a reliable method for anisotropic radiative heat transfer problems.

Influence of Scattering Phase Function on Estimated Thermal Properties of Al2O3 Ceramic Foams

Open ceramic foams are usually constituted of three-dimensional networks with randomly interconnected solid struts and fluid within pores. The heat transport within this material can be understood as the coupling of conduction, convection as well as radiation. The cooperative control of different heat transfer mechanisms is critical to successful design and optimization of components working at high temperatures. An answer to this problem usually requires good knowledge and understanding of the corresponding thermal properties in a wide range of temperatures. In the present paper, an inverse identification method was developed to determine coupled thermal properties from transient thermal measurements at temperatures up to 900 K for full description of conduction/radiation heat transports of foam media with absorbing, emitting, and anisotropic scattering effects. The influence of postulated phase function on the identified equivalent extinction coefficient, scattering albedo, anisotropic scattering factor, and two-phase thermal conductivity was discussed for better understanding of thermal behavior within ceramic foams. The estimated thermal properties under each postulated phase function of the sample at transient temperature profiles were used to calculate equivalent thermal conductivities, which were then compared with the measured results at more than 1000 K. The accordance between them indicated that linear anisotropic scattering phase function demonstrates superiority in description of radiation behavior within Al2O3 ceramic foam.

New treatment of strongly anisotropic scattering phase functions: The Delta-M+ method.

The treatment of strongly anisotropic scattering phase functions is still a challenge for accurate radiance computations. The new Delta-M+ method resolves this problem by introducing a reliable, fast, accurate, and easy-to-use Legendre expansion of the scattering phase function with modified moments. Delta-M+ is an upgrade of the widely-used Delta-M method that truncates the forward scattering peak with a Dirac delta function, where the '+' symbol indicates that it essentially matches moments beyond the first M terms. Compared with the original Delta-M method, Delta-M+ has the same computational efficiency, but for radiance computations the accuracy and stability have been increased dramatically.

A Discrete Transfer Method for Radiative Transfer through Anisotropically Scattering Media

A discrete transfer formulation for analyzing radiative heat transfer through participating medium with highly anisotropic scattering phase function is presented. This formulation used the finite volume method to evaluate integral of the radiative heat transfer problem. Hence, participating media anisotropic scattering phase function along a centre scattered sub-solid angle of the discrete transfer direction is defined by an average on the incident sub-solid angle. Numerical simulations on the efficiency of the proposed discrete transfer formulation for one-dimensional radiative heat transfer through participating, and anisotropic scattering medium under diffuse incidence is carried out. Results indicate that present discrete transfer radiative heat fluxes, transmittance and reflectance predictions have accuracy comparable to available analytical and numerical literature benchmark solutions. Excellent agreements are noted for different anisotropic scattering phase functions.

Sub-diffusive scattering parameter maps recovered using wide-field high-frequency structured light imaging.

This study investigates the hypothesis that structured light reflectance imaging with high spatial frequency patterns [Formula: see text] can be used to quantitatively map the anisotropic scattering phase function distribution [Formula: see text] in turbid media. Monte Carlo simulations were used in part to establish a semi-empirical model of demodulated reflectance ([Formula: see text]) in terms of dimensionless scattering [Formula: see text] and [Formula: see text], a metric of the first two moments of the [Formula: see text] distribution. Experiments completed in tissue-simulating phantoms showed that simultaneous analysis of [Formula: see text] spectra sampled at multiple [Formula: see text] in the frequency range [0.05-0.5] [Formula: see text] allowed accurate estimation of both [Formula: see text] in the relevant tissue range [0.4-1.8] [Formula: see text], and [Formula: see text] in the range [1.4-1.75]. Pilot measurements of a healthy volunteer exhibited [Formula: see text]-based contrast between scar tissue and surrounding normal skin, which was not as apparent in wide field diffuse imaging. These results represent the first wide-field maps to quantify sub-diffuse scattering parameters, which are sensitive to sub-microscopic tissue structures and composition, and therefore, offer potential for fast diagnostic imaging of ultrastructure on a size scale that is relevant to surgical applications.

Empirical model for target depth estimation used in the time-domain subsurface imaging

Monte Carlo simulations were performed in order to obtain reflectance measurements from phantoms typically used in biomedical optics when either unpolarized or circularly polarized incident light is used. Phantoms contain spherical targets of different diameters, placed at different depths, with higher absorption than the surrounding medium, which are detected using a coaxial setup of laser and detector. The considered turbid media have highly anisotropic scattering phase functions, so detected light for the considered times of flight is not diffuse, but rather in the multiple-scattering regime. Therefore, the target reconstruction methods typically used in diffuse optical imaging cannot be employed. However, spatially resolved reflectance measurements in the time domain allow use of a novel reconstruction method based on the approximation of average photon trajectories, which are functions of the separation distance from the point of incidence and of the time of flight. With the approximated average photon trajectories, one can estimate the depth of the target.

ANALYTIC MODELS FOR ALBEDOS, PHASE CURVES, AND POLARIZATION OF REFLECTED LIGHT FROM EXOPLANETS

New observational facilities are becoming increasingly capable of observing reflected light from transiting and directly imaged extrasolar planets. In this study, we provide an analytic framework to interpret observed phase curves, geometric albedos, and polarization of giant planet atmospheres. We compute the observables for non-conservative Rayleigh scattering in homogeneous semi-infinite atmospheres using both scalar and vector formalisms. In addition, we compute phase curves and albedos for Lambertian, isotropic, and anisotropic scattering phase functions. We provide analytic expressions for geometric albedos and spherical albedos as a function of the scattering albedo for Rayleigh scattering in semi-infinite atmospheres. Given an observed geometric albedo our prescriptions can be used to estimate the underlying scattering albedo of the atmosphere, which in turn is indicative of the scattering and absorptive properties of the atmosphere. We also study the dependence of polarization in Rayleigh scattering atmospheres on the orbital parameters of the planet-star system, particularly on the orbital inclination. We show how the orbital inclination of non-transiting exoplanets can be constrained from their observed polarization parameters. We consolidate the formalism, solution techniques, and results from analytic models available in the literature, often scattered in various sources, and present a systematic procedure to compute albedos, phase curves, and polarization of reflected light.

Arbitrary-Order Spherical Harmonics Method for Coupled Heat Transfer in Semitransparent Materials

An advanced numerical model and its application in thermal processing of nonlinearly anisotropic scattering semitransparent materials are presented in this paper. The decomposition of the radiative transfer equation is deduced using spherical harmonics (PN-approximation) method. Considering the transient problem, coupled radiative and conductive heat transfer model based on arbitrary-order PN-approximation method is established. With the coupled model, the calculation accuracy of high-order PN-approximation method is verified through comparison with theoretical solution and by the simulation results of other methods. Comparison shows that the arbitrary-order PN-approximation method reproduces results with a high accuracy for linear and non-linear anisotropic scattering, while it can easily treat the nonlinear anisotropic scattering phase functions with desired order. Keywords: semitransparent material, coupled heat transfer, numerical simulation, spherical harmonics method

Self-Adaptive Differential Evolution Based on the Concept of Population Diversity Applied to Simultaneous Estimation of Anisotropic Scattering Phase Function, Albedo and Optical Thickness

Self-Adaptive Differential Evolution Based on the Concept of Population Diversity Applied to Simultaneous Estimation of Anisotropic Scattering Phase Function, Albedo and Optical Thickness

Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions

The phase function for solar light scattering by large particles such as cloud droplets is strongly anisotropic due to very strong peaking in the forward direction. This creates numerical difficulties when attempting to calculate accurate reflected and transmitted radiances, which are important for remote sensing of atmospheric and surface properties. A popular approach uses the delta function to approximate the forward-scattering peak in a fraction of energy and a limited number of polynomial terms or a geometrically truncated function for the remaining fraction (so-called truncation approximations). This article compares and discusses several methods for fast and accurate calculations using truncation approximations. When using a single truncation approximation for all scattering orders, large biases appear in directions near the solar and anti-solar points. As shown here, high accuracy can be obtained using different truncation approximations depending on the order of scattering. Of particular importance is the use of phase functions close to the exact phase functions for the first few orders of scattering. Applying the method in combination with the Monte Carlo (MC) method, in which the truncation fraction for a scattering order depends on the scattering angle at the previous scattering event, obtains accurate radiance calculations under almost all geometrical and optical conditions, including in directions near the solar point. Because the method also reduces computational noise due to the MC sampling of radiance, it is useful for fast and accurate radiance calculations for cloudy atmospheres.

Comprehensive assessment of the two-flow model’s shape factors in aquatic environments

The applicability of the two-flow model to problems is dependent on assumptions that are used to reduce the model to two equations with two unknowns. Such a reduction requires proper parameterization of the shape factors. An extensive data set that reveals the relationships of shape factors to scattering-to-absorption ratios (0.1 < or = b/a < or = 30) and to the mean cosine of scattering (0.66 < or = micro(s) < or = 0.95) as obtained with a Monte Carlo model is presented. The sensitivity of the computed irradiance reflectance to the shape factors is also addressed. Results show increasing sensitivity of irradiance reflectance to the shape factors with increasing scattering-to-absorption ratio and with increasing anisotropy of the scattering phase function. For waters with highly anisotropic scattering phase functions, high scattering-to-absorption ratios, or both, the proper parameterization of the shape factors must be considered. A lookup table of the shape factors' depth-dependent values for various optical conditions is also presented.

Scaling anisotropic scattering in radiative transfer in three-dimensional nonhomogeneous media

Radiative heat transfer in three-dimensional nonhomogeneous participating medium was investigated by using REM 2 method. The anisotropic scattering phase function was dealt with the scaling technique based on delta function approximation. The three-dimensional scaled isotropic results were compared with the published anisotropic scattering computations. A good agreement between the scaled isotropic approaches and the anisotropic solutions was found. The effects of scattering albedo, forward fraction of phase function, and wall emissivity were discussed. It was found that, with the increase of the scattering albedo, the radiative heat flux increases for forward scattering media, but decreases for backward scattering media. The radiative heat flux is increased with the increases of forward fraction of phase function and wall emissivity. The emissive power at the center of a cubical nonhomogeneous medium in radiative equilibrium with gray diffuse walls equals to the averaged blackbody emissive power of the six walls.

RADIATIVE HEAT TRANSFER IN FINITE CYLINDRICAL HOMOGENEOUS AND NONHOMOGENEOUS SCATTERING MEDIA EXPOSED TO COLLIMATED RADIATION

Radiative heat transfer is studied in a finite axisymmetrical cylindrical enclosure exposed to collimated radiation. The integral equations for radiative transfer are solved by the YIX method and the quadrature method for comparison. Integrated intensity and radiative heat flux are presented in homogeneous and nonhomogeneous scattering media exposed to both uniform and Gaussian distributions of normal collimated incident radiation. The effects of aspect ratio, different incident radiation, and anisotropic scattering phase function as well as nonhomogeneous property distribution are discussed. Ray effects appear in the YIX solution for the case of a nonhomogeneous step change in the extinction coefficient. In order to eliminate the ray effect, an adaptive angular quadrature scheme is described and applied.

Chebyshev series solution for radiative transport in a medium with a linearly anisotropic scattering phase function

One‐dimensional radiative heat transfer is considered in a plane‐parallel geometry for an absorbing, emitting, and linearly anisotropic scattering medium subjected to azimuthally symmetric incident radiation at the boundaries. The integral form of the transport equation is used throughout the analysis. This formulation leads to a system of weakly‐singular Fredholm integral equations of the second kind. The resulting unknown functions are then formally expanded in Chebyshev series. These series representations are truncated at a specified number of terms, leaving residual functions as a result of the approximation. The collocation and the Ritz‐Galerkin methods are formulated, and are expressed in terms of general orthogonality conditions applied to the residual functions. The major contribution of the present work lies in developing quantitative error estimates. Error bounds are obtained for the approximating functions by developing equations relating the residuals to the errors and applying functional norms to the resulting set of equations. The collocation and Ritz‐Galerkin methods are each applied in turn to determine the expansion coefficients of the approximating functions. The effectiveness of each method is interpreted by analyzing the errors which result from the approximations.

RADIATIVE TRANSFER THROUGH AN IN HOMOGENEOUS FLY-ASH CLOUD: EFFECTS OF TEMPERATURE AND WAVELENGTH DEPENDENT OPTICAL PROPERTIES

The effect of spectral radiative properties of fly-ash particles on radiative heat transfer through an inhomogeneous planar layer near the walls of a combustion system is investigated. The spectral absorption and scattering coefficients, as well as the scattering phase function of the particles, are obtained from the Lorenz-Mie theory as functions of temperature and particle diameter. These data are employed in the solution of the radiative transfer equation, which is modeled using a double spherical harmonics approximation. In the calculations, a linearly anisotropic scattering phase function is used. It is shown that, if the medium temperature distribution is nonuniform, the divergence of the radiative flux term calculated using the spectral fly-ash properties differs significantly from that obtained using constant properties. This means that if fly-ash properties are assumed wavelength independent, a coupled radiation-convection heat transfer analysis would underestimate the gas temperature near the wa...

Computational attributes of the integral form of the equation of transfer

Difficulties can arise in radiative and neutron transport calculations when a highly anisotropic scattering phase function is present. In the presence of anisotropy, currently used numerical solutions are based on the integro-differential form of the linearized Boltzmann transport equation. In this paper, we depart from classical thought and present an alternative numerical approach based on application of the integral form of the transport equation. Use of the integral formalism facilitates the following steps: a reduction in dimensionality of the system prior to discretization, the use of symbolic manipulation to augment the computational procedure, and the direct determination of key physical quantities which are derivable through the various Legendre moments of the intensity. Our approach is developed in the context of radiative heat transfer in a plane-parallel geometry, and results are presented and compared with existing benchmark solutions. Encouraging results are presented to illustrate the potential of the integral formalism for computation. The integral formalism appears to possess several computational attributes which are well-suited to radiative and neutron transport calculations.

Radiation transfer for linearly anisotropic phase functions

Radiative transfer equation in a plane-parallel medium with isotropic boundary conditions for linearly anisotropic scattering phase function is considered. Two coupled integral equations for total density of radiation and total radiation flux are obtained. The Galerkin method is used to solve these equations. Numerical results for the radiative fluxes at the boundaries show that the Galerkin method yields accurate results compared well with other exact methods.