Integral Transfer Equation Research Articles

Solution of the Radiative Integral Transfer Equations in Rectangular Absorbing, Emitting, and Anisotropically Scattering Homogeneous Medium

Radiative integral transfer equations for a rectangular absorbing, non-emitting, and linearly anisotropically scattering homogeneous medium are solved numerically for the incident energy and the net heat flux components using the method of “singularity subtraction.” A benchmark problem is chosen as a rectangular homogeneous cold participating medium which is subject to externally uniform diffuse radiation on the bottom wall. The solutions for the incident energy and net heat flux components for selected points for a square domain are provided in tabular form for benchmarking purposes.

Solution of the radiative integral transfer equations in rectangular participating and isotropically scattering inhomogeneous medium

Radiative integral transfer equations for a rectangular participating and isotropically scattering inhomogeneous medium are solved numerically for the incident energy and the net partial heat fluxes using the method of “subtraction of singularity”. All the relevant single (surface integrals) and double integrals (volume integrals) are carried out analytically to reduce the computation time and numerical integration errors. The resulting system of linear equations are solved iteratively. A benchmark problem is chosen as a rectangular inhomogeneous cold participating medium which is subject to externally uniform diffuse radiation on the bottom surface. Solutions for linearly and quadratically varying scattering albedos are provided in tabular form.

Thermodynamic instabilities in one-dimensional particle lattices: a finite-size scaling approach.

One-dimensional thermodynamic instabilities are phase transitions, not prohibited by Landau's argument because the energy of the domain wall which separates the two phases is infinite. Whether they actually occur in a given system of particles must be demonstrated on a case-by-case basis by examining the properties of the corresponding singular transfer integral (TI) equation. The present work deals with the generic Peyrard-Bishop model of DNA denaturation. In the absence of exact statements about the spectrum of the singular TI equation, I use Gauss-Hermite quadratures to achieve a single-parameter-controlled approach to rounding effects; this allows me to employ finite-size scaling concepts in order to demonstrate that a phase transition occurs and to derive the critical exponents.

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 S 8 and the integral transfer equation solutions are provided.

Radiative transfer in absorbing, emitting and linearly anisotropic-scattering inhomogeneous cylindrical medium

Radiative integral transfer equations for one-dimensional solid cylinder with absorbing, emitting and linearly anisotropic-scattering inhomogeneous medium were derived by Abulwafa et al. (JQSRT 62 (1999) 755). The anisotropic terms in the integral equations and their results for anisotropic benchmark problems (JQSRT 66 (2000) 487) are inaccurate. In this study, the integral equations for absorbing, emitting and linearly anisotropic-scattering medium are rederived, and the integral equations for one-dimensional solid cylindrical medium are solved. The results are compared with selected cases using the discrete ordinates S 16 and the exact solutions available in the literature.

Line Formation in a Purely Scattering, Optically Thick Atmosphere

A model problem in the theory of line formation in an optically thick, purely scattering, stellar atmosphere is considered. The integral equation of radiation transfer at line frequencies is solved numerically for a two-level atom in the approximation of complete frequency redistribution in scattering. The numerical results are compared with those calculated from equations of the asymptotic theory. On the basis of the asymptotic theory, the positions of intensity maxima in a line are found for different absorption profiles.

The SKN approximation for solving radiation transport problems in absorbing, emitting, and scattering rectangular geometries

A high order approximation, the SK N method—a mnemonic for synthetic kernel—is introduced for solving radiation transfer problems in one- and two-dimensional geometries. The method relies on approximating the integral transport kernel by a sum of exponential kernels. The integral equation is then reducible to a set of coupled second-order differential equations— SK N equations. In this study, two types of boundary conditions have been proposed and explored. Naive boundary condition assumes one-dimensional slab boundary conditions on each boundary. Corrected boundary conditions; tackles the error terms that result from the approximation and is based on the minimization of the error term. The solutions of a test problem—incident radiation, outgoing intensities and heat fluxes at the boundaries—are compared with those of obtained by direct numerical solution of the integral transfer equation. Solutions are obtained for N=2,3,4, and 5 and are in agreement with the direct numerical solutions even for N=2 and 3. The corrected boundary condition gives better solutions than naive boundary condition in optically thin configurations; the relative errors for the intensities are below 1% while 2–4% errors are encountered in the radiation function solutions.

Statistical mechanics of nonlinear Klein–Gordon chains: the φ8-chain and the Gaussian double-well model

We study the statistical mechanics of two models of nonlinear Klein–Gordon chain: the ‘ φ 8-chain’ with the single-site potential v( y)= a( y 2−1) 4, and the Gaussian double-well model with v(y)= 1 2 ky 2+b exp(− 1 2 cy 2) where a, b, c and k are positive constants. The thermodynamics of the classical chains is investigated by the transfer-integral equation technique and the pseudo-Schrödinger equation approximation. The results for the heat capacity, the displacement correlation function, and the wave-vector-dependent susceptibility are compared with those of the familiar φ 4-chain. The partition functions of the quantum chains are calculated by the low-coupling effective potential method. The effects of quantum fluctuations on the low-temperature heat capacity are examined.

Theory of DNA melting based on the Peyrard-Bishop model

The DNA melting based on the Peyrard-Bishop (PB) model is systematically investigated. Our study on the eigenvalues and eigenvectors of the transfer integral equation for the original PB model points out that the eigenvectors are composed of two kinds: discrete bound states that constitute the internal states of a DNA molecule and the continuous unbound states that represent its dissociated states. Another process controlling the melting of DNA-the dissociation equilibrium between duplex and single-stranded DNAs-is introduced, which leads to an extended model applicable for a realistic DNA chain with a finite number of base pairs. Based on the expansion of kernels, the calculations of the thermodynamic quantities of the system are reduced to multiplication of matrix series. Calculations on model block DNAs show the method is much more efficient than molecular dynamic simulation and has enough high precision to handle the melting of a natural DNA with arbitrary sequence. The discreteness effect and nonlinear effect of the model are discussed based on the Gaussian model. Rigorous melting curves for periodic DNA with two and three base pairs in a unit cell and boundary effects are presented by the transfer integral approach.

Microstrip Slow-Wave Structures on the Bianisotropic Substrate

ABSTRACT An efficient spectral domain full-wave algorithm based on 4 × 4 transfer matrix and integral equation method is developed for the infinite periodic microstrip line on the multilayered bianisotropic substrate. A periodic microstrip line on the chiral substrate is considered. The slow-wave factor and wave admittance of the dominant mode in this structure versus the phase shift per period are presented. Dispersion characteristics for both one-stage and two-stage microstrip periodic structures on a chiral substrate are computed. The dependence of these characteristics upon the chirality parameter is studied.

Enhancement of heat transfer using porous convection-to-radiation converter for laminar flow in a circular duct

The convection-to-radiation converter (CRC) made of porous ceramics has been proven to be an effective means in the enhancement of heat transfer in burners, combustors, industrial furnaces and high temperature plug flow systems. The paper proposes a novel application using this CRC technique in a partial bypass flow system for potential heat transfer enhancement. Hydrodynamically fully-developed and thermally developing laminar flow is assumed in a circular duct which is fitted with a porous core to enhance both convective and radiative transfer from a high-temperature gaseous medium to a heat-absorbing wall. The Navier-Stokes and modified Darcy equations are applied, respectively, to the flow field in a gas layer and porous core region. The radiation field is modeled by a two-dimensional S-type integral equation of transfer for its accuracy and efficiency in the numerical calculations. The effects of heat transfer augmentation using a porous CRC are investigated numerically for various flow and thermal parameters. Numerical solutions which include velocity profiles and flow characteristics of the CRC system as well as isotherm patterns, parameterization of the heat fluxes, mean bulk temperature and Nusselt number distributions along the flow direction are presented. Copyright © 1996 Elsevier Science Ltd.

Wave propagation in a periodic microstrip line on a multilayered anisotropic substrate

An efficient spectral domain full-wave algorithm based on 4/spl times/4 transfer matrix and integral equation method is developed for the periodic microstrip line on the multilayered anisotropic substrate. The substrate may consists of arbitrary number of layers whose permittivity and permeability are tensors of general form. It is shown, that in some cases, the quasistatic approach for calculations of characteristics of the dominant mode may be invalid even at very small frequencies. The method may be used for simulation of integrated circuits, which include anisotropic layers.

Thermal radiation in layered participating cylindrical media using an integral equation method

Radiative transfer in a layered cylindrical medium is analyzed using a newly developed S-type integral equation of transfer in terms of incident radiation and net radiative heat flux. The physical systems in hollow and solid cylindrical geometry are modeled as nonhomogeneous participating media with stepwise variable properties. Thus, the necessity to establish the equation of transfer for each layer and combine these layers through the intensity continuity on interface is avoided. Numerical predictions from a collocation method are presented to illustrate the effects of radiation properties on the radiative transfer for the systems considered. Comparison of the results with other available data in literature shows that highly accurate results are obtainable by current method with simplicity.

The albedo-shifting method

We propose a new interpretation of the albedo-shifting method in the theory of radiative transfer, based on constructing a three-factor factorization of the initial operator of the transfer integral equation in a homogeneous half-space. Such an approach simplifies the albedo-shifting method while simultaneously widening its potential and limits of applicability.

Multidimensional radiative transfer in absorbing, emitting, and linearly anisotropic scattering cylindrical medium with space-dependent properties

The integral form of three-dimensional radiative transfer equation for an absorbing, emitting, and linear-anisotropic scattering medium with space-dependent properties is formulated. A product-integration method is subsequently applied to develop a numerical scheme for solving the corresponding integral transfer equations in a two-dimensional, axisymmetric and nonhomogeneous medium subjected to externally incident radiation or bounded by emitting and diffusely-reflecting walls. The numerical solutions for cases of constant, continuous, and stepwise variations of scattering albedo are presented to illustrate its accuracy and flexibility, and validated by comparing with results available in the literature.

On nonstationary radiative transfer in a spectral line in stellar atmospheres

We consider nonstationary radiative transfer in a line in stellar atmospheres simulated as a stationary semi-infinite plane-parallel medium. We assume complete frequency redistribution in the elementary act of scattering. We assume that the time a photon spends in the medium is determined only by the mean time spent in the absorbed state. We obtain an explicit expression for the resolvent of the nonstationary integral equation of transfer, which is a bilinear expansion with respect to the eigenfunctions found in [12] for the corresponding stationary transfer equation.

Inversion of Monte Carlo model for estimating vegetation canopy parameters

The Monte Carlo technique for estimating model parameters is presented. The bidirectional reflectance factor is calculated by the Monte Carlo method of solving the integral equation of radiation transfer in plant canopies. A general inversion technique for the estimation of canopy parameters is considered. The method of calculating derivatives of the bidirectional reflectance factor with respect to unknown parameters using the same photon trajectories as in the solution of the direct problem is developed. The leaf dimensions are taken into account using a new extinction coefficient. Finally, numerical results for estimating three optical canopy parameters (coefficients of leaf reflectance, transmittance, and soil albedo) and four geometrical canopy parameters (leaf area index, two parameters of leaf angle distribution function, and leaf size parameter) are briefly discussed.

Monte Carlo method and transport equation in plant canopies

Plant canopy reflectance is calculated using the governing equation for photon transport. The integral equation of transfer is solved by the Monte Carlo method. The main emphasis is on statistical estimation and simulation of the Marcov chain. The leaf dimensions are taken into account in obtaining the hot-spot effect of the canopy. Finally, numerical results for transport equation obtained by the Monte Carlo method are compared with those obtained by using the method of discrete ordinates and the geometrical Monte Carlo model.

Atmospheric Profiling of Water Vapor Density with a 20.5–23.5 GHz Autocorrelation Radiometer

A tropospheric water vapor density profiling system is presented. The hardware consists of an autocorrelation radiometer (CORRAD) operating over a frequency range from 20.5 to 23.5 GHz. The CORRAD directly measures the autocorrelation of downwelling thermal emission from the atmosphere. The 3 GHz predetection bandwidth of each measurement provides for extremely rapid decorrelation of the noise inherent in all radiometer measurements. This, in turn, allows for high temporal resolution of the water vapor dynamics. Fourier transformation of the raw data produces a brightness temperature spectrum with 100 MHz resolution across the frequency range. Inversion of the radiative transfer integral equation to solve for the water vapor distribution is constrained by the 31 equivalent frequency channels. Previous microwave profilers of the troposphere, with 2 to 5 frequency channels, were much less constrained and the inversion process was accordingly more sensitive to measurement noise. Water vapor profiles estimated by the inversion are in good agreement with coincident radiosonde measurements made by the National Weather Service.

Applications of the transfer integral techniques to two-dimensional lattices

The transfer integral formalism is used to study the statistical mechanical properties of a two-component field defined on a two-dimensional lattice. This field, taken to have anisotropic elasticity, is subject to both a nonlinear local potential and an external field. The free energy and magnetization are calculated using an approximate solution of the transfer integral problem. This solution employs a strong-coupling approximation to the transfer integral equation and a variational principle with correlated Gaussian trial functions. As a special case, the ø 4 model for a structural phase transition, in the absence of an applied field, is analyzed; a phase diagram consistent with previous calculations is obtained. The phase diagram for a two-component field, with anisotropic elasticity, a ø 4 local potential, and an external field, is also considered.