Application of Synthetic Kernel (SKN) method to participating linearly anisotropically scattering solid spherical medium

The Synthetic Kernel (SK_N) method is applied to a solid spherical absorbing, emitting and linearly anisotropically scattering homogeneous and inhomogeneous medium. The SK_N method relies on approximating the integral transfer kernels by Synthetic Kernels. The radiative integral transfer equation is then reducible to a set of coupled second-order differential equations. The SK_N method, which uses Gauss quadratures, is tested against integral equation and the discrete-ordinates S₈ solutions for various optical radius and scattering albedo variations.     

Radiative transfer in one-dimensional hollow sphere with anisotropic scattering and variable medium properties

The effects of variable medium properties on radiation transfer in participating and anisotropically scattering one-dimensional spherical medium were investigated by Tsai et al. (JQSRT 42(3) (1989) 187). The discrete ordinates method solutions they provided for hollow spherical medium cases are incorrect. The correct DOM S₈ and the integral transfer equation solutions are provided.     

Effects of specular reflection on radiative heat transfer in an absorbing,emitting, and anisotropically-scattering medium

The effect of specular reflection in a one-dimensional, absorbing, emitting, and anisotropically-scattering medium is analyzed. The mathematical formulation is shown to involve the function F_n(x)= ∝¹_o(e^-x/μ/1 - ?₁?^-2e_2L/μ)υ^n-2 dμ, which can be readily evaluated as a fast-coverging, infinite series of exponential integral functions. Numerical solutions to the resulting integral equations are generated by the method of point allocation.Physically, the radiative energy within a participating medium bounded by specularly reflective surfaces is observed to experience more multiple reflection than the corresponding diffuse reflection case. This leads to some differences between the two cases in the heat transfer and temperature profile results. These differences, however, are generally quite minor for the considered one-dimensional planar system.     

A general integration method for radiative transfer in 3-D non-homogeneous cylindrical media with anisotropic scattering

A general set of integral equations is presented to solve 3-D radiative heat transfer problems in emitting, absorbing and linear anisotropic scattering finite hollow or solid cylinders with non-homogeneous media. By tracing a ray to compute the intensity,it is much easier to handle the spatial change properties including extinction coefficient. Both the continuous change property and step-change property are dealt with without difficulties. The solid angle integration in getting the incident radiation and heat fluxes is represented by the bounding surface integration. In order to avoid the singularity problem near the bounding surface, the surface integrations are transformed to new modified integral equations by mathematical methods. By doing so, we get more flexible general integral equations applicable to all cases (3-D solid cylinders, 3-D hollow cylinders, finite cylinders or infinite cylinders). This scheme has been verified by comparing the results with published data in the literature. It is believed that this method will be useful in combined radiation and convection heat transfer problems.     

Benchmark solutions of radiative transfer equation for three-dimensional rectangular homogeneous media

Three-dimensional steady-state radiative integral transfer equations (RITEs) for a cubic absorbing and isotropically scattering homogeneous medium are solved using the method of “subtraction of singularity”. Surface integrals and volume integrals are carried out analytically to eliminate singularities, to assure highly accurate solutions, and to reduce the computational time. The resulting system of linear equations for the incident energy is solved iteratively. Six benchmark problems for cold participating media subjected to various combinations of externally uniform/non-uniform diffuse radiation loads are considered. The solutions for the incident energy and the net heat flux components are given in tabular form for scattering albedos of ω=0, 0.5 and 1.     

Benchmark solutions of radiative integral transfer equations for three-dimensional emitting–absorbing and scattering media bounded by gray walls

Three-dimensional dimensionless radiative integral transfer equations (RITEs) for a cubic emitting–absorbing and isotropically scattering homogeneous medium of constant properties bounded by gray walls are solved using the method of “product integration”. The resultant system of linear equations for the incident energy is solved iteratively. Evaluation of the accuracy of the numerical solution is achieved by using the computer codes to make predictions for an idealized enclosure in which the exact analytical solution is possible. Comparison of the analytical and numerical values of medium temperatures and heat fluxes has shown that our method is logically correct and has good accuracy. Four benchmark problems for participating medium subjected to various combinations of internally uniform/non-uniform source terms are solved. To make the benchmark problems more complex and more general, different scattering albedo (ω=0.1, 0.5, 0.9), different wall emissivity (ε=0.1, 0.5, 0.9) and different optical thickness (τ=0.1, 1.0, 5.0) are considered. The solutions for the temperature of the medium and the heat flux components are given in tabular form and they may serve as standard values to assess the accuracy of other methods more conveniently.     

Heat transfer characteristics of a porous radiant burner under the influence of a 2-D radiation field

This paper deals with the heat transfer analysis of a 2-D rectangular porous radiant burner. Combustion in the porous medium is modelled as a spatially dependent heat generation zone. The gas and the solid phases are considered in non-local thermal equilibrium, and separate energy equations are used for the two phases. The solid phase is assumed to be absorbing, emitting and scattering, while the gas phase is considered transparent to radiation. The radiative part of the energy equation is solved using the collapsed dimension method. The alternating direction implicit scheme is used to solve the transient 2-D energy equations. Effects of various parameters on the performance of the burner are studied.     

Non-Fourier heat conduction with radiation in an absorbing, emitting, and isotropically scattering medium

This article numerically analyses the combined conductive and radiative heat transfer in an absorbing, emitting, and isotropically scattering medium. The non-Fourier heat conduction equation, which includes the time lag between heat flux and the temperature gradient, is used to model the conductive heat transfer in the medium. It predicts that a temperature disturbance will propagate as a wave at finite speed. The radiative heat transfer is solved using the P₃ approximation method. In addition, the MacCormack's explicit predictor-corrector scheme is used to solve the non-Fourier problem. The effects of radiation including single scattering albedo, conduction-to-radiation parameter, and optical thickness of the medium on the transient and steady state temperature distributions are investigated in detail. Analysis results indicate that the internal radiation in the medium significantly influences the wave nature. The thermal wave nature in the combined non-Fourier heat conduction with radiation is more obvious for large values of conduction-to-radiation parameter, small values of optical thickness and higher scattering medium. The results from non-Fourier-effect equation are also compared to those obtained from the Fourier equation. Non-Fourier effect becomes insignificant as either time increases or the effect of radiation increases.     

Radiation transfer in an anisotropically scattering plane-parallel medium with space-dependent albedo ω(x)

A method of analysis is presented for solving radiation-transfer problems involving space-dependent albedo ω(x) for an absorbing, emitting and anisotropically scattering plane-parallel medium with reflecting boundaries. The albedo is represented in terms of Legendre polynomials in the form ω(x) = Σ^R_r=0D_rP_r(x/L), where x is the optical variable, L is the half optical-thickness of the slab, P_r(x/L) are the Legendre polynomials and D_r are known expansion coefficients. The effects of spatial variation of albedo on the reflectivity and transmissivity of a medium having a slab geometry are examined for the cases of both forward and backward anisotropic scattering over a wide range of system variables. The effects of ω(x) on the angular distribution of radiation are also shown for some representative cases.     

A fundamental-source-function formulation of radiative transfer and the resulting fundamental reciprocity relations

An important problem in radiative transfer is finding the radiative fields produced by various illuminations (both external and internal) of a plane-parallel, inhomogeneous, absorbing, emitting, and anisotropically-scattering finite medium. One approach to a solution is to find the source function, which represents the rate of production of scattered radiation per unit volume and solid angle, at each point in the medium. The present study develops the existence of a Green's function, called the fundamental source function, which separates the optical properties of the medium from the driving illumination. Radiative linearity then allows the representation of all possible source functions as convolutions of the illumination with the fundamental source function. Parametric differentiation (invariant imbedding) is used to replace the governing linear integral equation for the fundamental source function with a set of differential equations appropriate for numerical integration. This approach for finding the fundamental source function leads naturally to the introduction of fundamental scattering and transmission functions. Our inclusion of anisotropic internal illumination (sources) allows us to develop four new reciprocity relations involving these functions. The reciprocity relations state general equivalences between an internally and an externally driven medium and thus greatly reduce the complexity of radiative transfer.     

磁偶极和电四极在磁各向异性介质中的辐射功率

在经典电动力学的框架下,研究了磁各向异性介质中的电磁辐射问题,得到了磁偶极和电四极在磁各向异性介质中的辐射功率表达式.进一步地,通过把各向同性介质中的μrii代入所得辐射功率表达式,得到了与文献相符合的结果,验证了所得结果的正确性.研究结果表明磁偶极和电四极在磁各向异性介质中的辐射功率大小与磁各向异性介质的μrii大小有关,对判断磁偶极和电四极在磁各向异性介质中的辐射效果有较大的帮助.     

Analysis of collimated radiation in participating media using the discrete transfer method

A general formulation of the discrete transfer method is provided to analyze radiative heat transfer problems in a participating medium subjected to collimated radiation. The formulation is validated by considering 1-D planar absorbing, emitting and anisotropically scattering gray medium in radiative equilibrium. Anisotropy of the medium is approximated by linear anisotropic phase function. For the purpose of comparison, the problem is also solved analytically. Results are obtained for different angles of incidence of the collimated radiation. At a given angle of incidence, results are obtained for forward, isotropic and backward scattering situations. Heat flux results are compared over a wide range of values of the extinction coefficient. Emissive power distributions in the medium are also obtained for some cases. The discrete transfer method results are found to compare very well with the analytic results.     

Pulse shaping via modulation of resonant absorption

Propagation of monochromatic linearly polarized plane electromagnetic wave through resonantly absorbing anisotropic medium with frequency-modulated response is studied analytically and numerically. Frequency modulation is assumed to be provided by means of either mechanical vibration of a solid sample or modulation of quantum transition frequency by a driving low-frequency electromagnetic field. Possibility of generation of train of pulses with polarization of the incident field as well as with orthogonal polarization is shown. For each polarization, optimal combinations of values of four parameters that provide maximal ratio of peak pulse intensity to the average output intensity is found numerically. Possible realization of this resonant method of pulse shaping in laser crystal Dy²⁺:CaF₂ is discussed.     

Exact formulation of multiple scattering in a three-dimensional cylindrical geometry

Exact expressions for the source function, flux, and scattered intensity normal to the surface are developed in cylindrical coordinates for a three-dimensional, absorbing, emitting, isotropically scattering medium exposed to both diffuse and collimated radiation. Simplifications of these expressions for certain important geometries and uniform loading are presented. Also, superposition of these equations and radiative equilibrium are discussed. The generalized three-dimensional equations are shown to reduce to the familiar one-dimensional results. Also, the equations for a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on an otherwise isotropic phase function are expressed in terms of the isotropic expressions.     

Radiation transport and internal reflection in a sphere

We construct an integral equation for the flux intensity in a scattering and absorbing medium using the integro-differential form of the radiative transfer equation in a sphere. The sphere is uniformly irradiated by an external source of arbitrary angular distribution. The Fresnel boundary conditions, which incorporate reflection and refraction, are used. For the special cases of a non-scattering medium, and in the limit of an optically transparent medium, we obtain exact solutions for specular and diffuse refection. Some numerical examples are given which give qualitative agreement with some recent work of Tian and Chiu (JQSRT, 2005).     

Radiative transfer in an isotropically scattering two-region slab with reflecting boundaries

The problem of radiative heat transfer in an absorbing, emitting, isotropically scattering two-layer slab with diffusely and specularly reflecting boundaries is solved by the F_N method and results are presented for the transmissivity and reflectivity of the slab.     

Calculation of phase diagrams for solid solutions of ferroelectrics

The integral equations for calculating ferroelectric and antiferroelectric phase transition temperatures, order parameters, and critical concentrations of solid solution components are derived. The electric dipoles randomly distributed in the system are treated as sources of random fields. The random field distribution function is constructed taking into account the contribution of nonlinear effects and the differences in the dipole orientations for different solid solution components. The dependence of the phase transition temperature on the composition of a binary solid solution in the ferroelectric-antiferroelectric and ferroelectric-paraelectric systems is calculated. Numerical calculations are carried out for the PbTi_xZr_1?x O₃ and BaZr_xTi_1?x O₃ solid solutions. The results obtained are in good agreement with the experimental phase diagrams of these systems. Analysis of the results indicates that any solid solution containing ferroelectric (antiferroelectric) and paraelectric components transforms into a relaxor state at sufficiently high concentrations of the paraelectric component.     

Generalized Landau-Lifshitz models on the interval

We study the classical generalized gln Landau-Lifshitz (L-L) model with special boundary conditions that preserve integrability. We explicitly derive the first non-trivial local integral of motion, which corresponds to the boundary Hamiltonian for the sl₂ L-L model. Novel expressions of the modified Lax pairs associated to the integrals of motion are also extracted. The relevant equations of motion with the corresponding boundary conditions are determined. Dynamical integrable boundary conditions are also examined within this spirit. Then the generalized isotropic and anisotropic gln Landau-Lifshitz models are considered, and novel expressions of the boundary Hamiltonians and the relevant equations of motion and boundary conditions are derived.     

Source function expansion method for radiative transfer in a two-layer slab

Radiative heat transfer in an isotropically scattering, absorbing and emitting two-layer slab with specularly reflecting boundaries is solved by expanding the source function by Legendre polynomials in the space variable in the integral form of the equation of radiative transfer. The reflectivity and the transmissivity of the slab for an externally incident isotropic radiation are determined. The S-1 solution yields results which are sufficiently accurate for most engineering applications.