Gray Approximation Research Articles

Information-theoretical approach to radiative transfer

The maximum entropy formalism is used to obtain the radiation and matter distribution functions for radiative systems is steady nonequilibrium states, under the gray approximation. The radiation distribution function is expanded in a smallness parameter, which vanishes at equilibrium. In the first near-equilibrium approximation, we derive the results of near-equilibrium diffusion theory. This may be regarded as an analogue to the kinetic-theoretical result, according to which in the first Enskog approximation, the Fourier heat conduction equation is obtained. The theory is also developed up to the second order, leading to results which apply to situations further away from equilibrium than those corresponding to near-equilibrium diffusion theory. A simple application is analyzed.

Mass-Luminosity Relationship and Lithium Depletion for Very Low Mass Stars

We derive mass-luminosity relationships for very low mass stars (0.06 < m/M☉ < 0.6) for different metallicities. Calculations are conducted with the different nongray atmosphere models presently available to illustrate the uncertainties in the stellar models. The theoretical mass-magnitude relation reproduces accurately the observed relationship in the V band (Henry & McCarthy) but still underestimates the flux in the K band by ~0.5 mag, a consequence of the still inaccurate water opacities. Calculations based on the gray approximation are shown to yield incorrect results over the entire mass range of interest. We also show that the fate of objects near the hydrogen-burning limit depends significantly on the treatment of the atmosphere. Nongray effects yield cooler and less luminous objects for a given mass near the stellar/substellar transition, and thus a lower hydrogen-burning minimum mass, m ≈ 0.07 M☉, for solar metallicity.

Mathematical modelling of the coupled reacting boundary layer equations

AbstractA new numerical scheme for reacting axisymmetric jet flows formed between a fuel jet and co‐flowing air has been developed. The model is mathematically described by a set of non‐linear parabolic partial differential equations in two space dimensions, i.e. the boundary layer equations. The numerical scheme that the programme uses for solving the fully coupled conservation equations of mass, momentum, energy and species is a generaliztion of the discretization technique recently developed by Villasenor (J. Math. Comput. Simul., 36, 203–208 (1994)). Chemical production (and destruction) of the species is allowed to occur through N elementary reversible (or irreversible) reactions involving k species, although in the present model the reaction rates are evaluated with a simplified kinetic mechanism for a one‐step global reaction. Thermal radiation is considered assuming an optically thin limit and adopting the grey medium approximation. Allowances are made for natural convection effects and variable thermodynamic and molecular transport properties. The performance of the model in solving the coupled aerodynamic and finite rate chemistry effects is tested by comparing model predictions with experimental data of Mitchell et al. (Combust. Flame, 37, 227–244 (1980)) for a buoyant, laminar, diffusion axisymmetric methane‐air flame.

Nonisothermal Nongray Absorbing-Emitting-Scattering Suspension of Fe3O4 Particles Under Concentrated Solar Irradiation

Radiation transfer within a cloud of magnetite (Fe3O4) particles contained in an infinite slab is considered. The particulate cloud is modeled as a pseudo-continuous, nongray, nonisothermal, absorbing, emitting, and anisotropically scattering medium. The energy source is concentrated solar irradiation, which is assumed to be diffusely and uniformly distributed over a circular opening and has a 5780 K blackbody spectrum. Mie-scattering theory is applied to calculate the spectrally and directionally dependent optical properties of the particles. The Monte Carlo ray-tracing method is used to calculate the attenuation characteristics of the cloud and the temperature distribution under radiative equilibrium. The Monte Carlo simulation is optimized by incorporating the appropriate cumulative probability density functions via Bezier surfaces. The effect of spectral and directional dependency is investigated by comparing the results with those obtained for a gray and isotropic-scattering medium under diffuse or perpendicular incident radiation. It is found that a cloud of Mie-scattering Fe3O4 particles under perpendicular incident radiation requires approximately twice as much optical thickness to obtain the same attenuation (of radiation at all wavelengths and directions) as a cloud of isotropic-scattering particles under diffuse incident radiation. It is demonstrated that the gray approximation using Planck mean values can lead to considerable error in the temperature solution because the spectral absorption coefficient is higher in the region of longer wavelengths where the peak emission by Fe3O4 particles occurs.

Slip flow in a two-component plasma model with radiative heat transfer

The paper studies the flow of a low density thermally radiating two-component plasma in the presence of mass transfer and Hall current. Since the density is low, an optically thin grey gas approximation is adopted for the radiative heat transfer. Also, at high temperatures, the Arrhenius activation energy is important. For fully developed flow, the problem is modelled by coupled nonlinear differential equations from the incompressible Navier-Stokes equations, together with slip boundary conditions. Exact solutions are inconceivable, but various asymptotic solutions are derived, and these solutions are discussed quantitatively.

Statistical mechanical-atomistic determination of vacancy formation free energies in Cu-Ni alloys

Abstract Atomistic and statistical mechanical methods are combined to determine the vacancy formation free energy in binary solid solutions. The first-shell grey model is based on the assumption that all the atoms are effective or mean-field atoms, but with the concentration of the first shell different from the bulk. Comparison of the vacancy formation free energy and the average local concentration profile around the vacancy obtained from the first-shell grey model and the more accurate but much less computationally efficient first-shell black-white model (where the mean-field approximation is not used) suggest that only the composition of the first atomic shell adjacent to the vacancy must be determined. The temperature dependence of the vacancy formation free energy for a Cu-Ni system can be efficiently determined using the first-shell grey atom approximation. Our results show that the vacancy has a strong tendency to form in Cu-rich regions of the Cu-Ni alloy at low temperatures and low Cu bulk concentrations. This preference of vacancies to form in regions of composition fluctuations (of one sign) is the most important effect to include that goes beyond the standard mean-field theories of vacancy formation thermodynamics in solid-solution alloys.

EFFECTS OF MICROSCOPIC DIFFUSION AND ROTATIONAL MIXING ON STELLAR MODELS

ABSTRACT Evolutionary tracks and isochrones were calculated with alpha-enhanced compositions which cover the entire globular cluster metallicity range and include the effects of the diffusion of 4He. Including the effects of helium diffusion has a negligible effect (< 0.5 Gyr) on the derived ages of globular clusters. Regardless of the inclusion of helium diffusion, a significant age spread of ~5 Gyr exists among the globular clusters. The oldest globular cluster studied was M92 with an age of 17 ± 2 Gyr old. The stellar models may be tested by comparing the Li depletion and surface rotation rates to observations in young clusters stars. The observed Li abundances clearly indicate that standard or diffusive models do not deplete enough Li. Instabilities induced by rotation provide an additional mixing mechanism. For this reason the stellar evolution code was modified to include the combined effects of diffusion and rotational mixing on 1H, 4He and the trace elements 3He, 6Li, 7Li, and 9Be. The calibrated solar models have a convection zone depth of 0.709 - 0.714~Rodot, in excellent agreement with the observed depth of (0.713 ± 0.003)~Rodot. The rotational mixing inhibits the diffusion in the outer parts of the models, leading to a decrease in the envelope diffusion by 30 - 50%. The combined models are able to simultaneously match the Li abundances observed in the Pleiades, UMaG, Hyades, NGC 752 and M67. They also match the observed rotation periods in the Hyades. However, these models are unable to explain the presence of the rapidly rotating G and K stars in the Pleiades. Observations of Li abundances in extremely metal poor halo stars provide another test of the stellar models. All models which use Kurucz (1992) model atmospheres to determine the surface boundary conditions are unable to match the observed Li depletion in cool halo stars. Models which use the gray atmosphere approximation provide a much better fit to the data. Standard models do a good job of fitting the observed Li abundances and predict a primordial Li abundance of log NLi = 2.24 ± 0.03. In models which include microscopic diffusion, but not rotational mixing, too much Li depletion occurs in the hot stars, and the models do not match the observations. The [Fe/H] = -2.28 stellar models which include both diffusion and rotational mixing provide an excellent match to the observations and predict a primordial Li abundance of log NLi = 3.13 ± 0.1. However, the [Fe/H] = -3.28 Li destruction isochrones reveal problems with the combined models. These low metallicity isochrones predict a trend of decreasing log NLi with increasing effective temperature which is clearly not present in the observations.

Surface Temperature Measurement Using Infrared Radiometer. (2nd Report, Applicability of Pseudo Gray Body Approximation).

In the previous study, we clarified various fundamental characteristics, such as radiation temperature, radiosity coefficient, emissivity and reflectivity, of 3 kinds of infrared radiometers to measure temperature distribution on a non metal surface, on which gray-body approximation is applicable. However, gray-body approximation is not always applicable to all material surfaces. For instance, its approximation is not appropriate on metal surfaces owing to the effect of the directive and spectral properties. Therefore, in the present study, applicability of pseudo-gray-body approximation on a metal surface enclosed by black body walls at constant temperature was studied and analyzed to establish a simpler and more general measurement technique using an infrared radiometer.

Camera virtual equivalent model 0.01 pixel detectors

In computer vision, a standard approach to eliminate spatial distortion is to use a calibrated grid. We developed an original method using such a grid to establish an equivalence between any camera with inherent technical limitations (optic, solid-state sensor, digitizer) and a perfect virtual pinhole camera model. This bijection based on interpolation concepts takes all the deformations generated by the optical and electronical equipment into account (distortion, decentration, scale, and affinity ratios) without the need for their explicit determination. We propose a detection method of this grid image by the use of grey levels approximation surfaces particularly adapted to this problem. We obtained a good precision on real grid images with modelization errors standard deviation of about 0.007 pixels with an off-the-shelf 512 × 512 camera using 256 grey levels. Three interpolation methods are compared: bilinear, cross-ratio in projective coordinates and bicubic splines. We conclude from this study that, in distorted images, the use of the bicubic splines interpolation method is a good improvement.

Radiation Hydrodynamics in Pulsating Stars

AbstractWe present nonlinear radiation hydrodynamical calculations of Cepheid envelopes carried out using the method described in Dorfi &amp; Feuchtinger (1991, A&amp;A 249, 414). The radiative transport is treated within the grey Eddington approximation and all equations are discretised in conservative form on an adaptive mesh which is solved simultaneously with the physical equations. The resulting system of nonlinear algebraic equations is solved implicitly to avoid the restrictive Courant-Friedrichs-Lewy time step condition. In order to treat schock waves we use the artificial tensor viscosity developed by Tscharnuter &amp; Winkler (1979, Comp.Phys.Comm. 18, 171) which is suitable for spherically symmetric problems. For the opacity we employ the latest OPAL tables (Rogers and Iglesias, 1992, ApJ, submitted); the equation of state corresponds to the standard evolution calculations according to Baker &amp; Kippenhahn (1962, Zeitschr. f. Astrophys. 54,114).We start from an initial hydrostatic model whose parameters correspond to a typical Cepheid (5 M⊙, 3000 L⊙, 5600 K). The subsequent dynamical evolution is initiated by the excitation of oscillations at the ionization zones in the stellar atmosphere where the opacity changes by several orders of magnitude. After an initial relaxation caused by discretisation errors and numerical noise, a periodic pulsation develops. Each pulsation is accompanied by a shock wave which propagates through the stellar atmosphere. A typical pulsation cycle is covered by about 70 time steps and the obtained periods lie in the range of 1 to 40 days.We discuss in detail the excitation of the oscillations and the structure of the model during a pulsation cycle.

An analysis of the combined conductive-radiative heat transfer between a surface and a gas-fluidized bed at high temperature

A theoretical investigation of the combined conductive radiative wall-to-fluidized bed heat transfer is presented. The packet of emulsion is assumed to be an absorbing, emitting, and scattering nongray medium. Equations of energy and radiative transfer in the packet are solved simultaneously using an iterative numerical method. The bubble is modelled as a hemisphere of gas enclosed by the emulsion and the nongray heal transfer surface. Heat transfer coefficients were calculated for different bed systems in the temperature range 300–1000 °C. The predictions by the present analysis were in good agreement with available experimental data. The radiative contribution of as much as 35% varies directly with bed temperature, particle size and emissivity, surface temperature and emissivity. Both isotropic scattering and gray medium approximations were found to be acceptable for practical applications.

The Nitrogen Spectra of WN Stars: The WN6 “Standard” Star HD 192163 (WR136)

Hitherto our quantitative analyses of WR spectra [2][5] have been based on pure-helium models [1][6]. Now we further improved our non-LTE calculations by including a complex model atom of nitrogen (90 energy levels, 351 line transitions; with low-temperature dielectronic recombination) into our model atmospheres in order to synthesize adequately the spectra of WN subtypes (Wessolowski et al., in preparation). Together with the nitrogen (the most important “metal” in WN atmospheres), we introduced an improved temperature structure into our model calculations [3], now accounting for non-grey radiative equilibrium instead of the grey approximation applied so far. Moreover we took into account the line overlap of the considered elements (here: helium, nitrogen) and also their blanketing effects on the continuous radiation field.

Results from 2-D Numerical Simulations of Solar Granules

We have carried out detailed numerical simulations of solar granular convection cells of different horizontal dimension. The calculations account for the basic physics of compressible convection, including the ionization of H I, He I and He II, and H2 molecule formation as well as non-local, multi-dimensional radiative transfer (grey approximation in LTE). A more detailed description of the simulations has been given by Steffen and Muchmore (1988) and by Steffen et al. (1989).

The Premixed Flame in a Radiatively Active Porous Medium

Abstract The present study considers the thermal structure variation in a porous medium when the absorption coefficient of the porous medium that is the most important radiative property, the physical dimension of porous medium and the equivalence ratio of premixed gas mixture are changed. The unsteady radiation model was incorporated in a differential form using the two-flux gray approximation. The maximum value of the conductive heat transfer through both gas phase and porous medium was found to be less than 6% compared with that of the radiative heat transfer. The reaction zone shifted upstream and the flame thickness decreased as either the geometrical length of porous medium increased or the absorption coefficient decreased.

Some constraints on a greenhouse atmosphere for Triton

We explore the likelihood of a thick atmosphere around Neptune's satellite Triton. The infrared optical depth in the grey atmosphere approximation is computed for a range of possible surface compositions, albedos, and gravities. If molecular nitrogen and/or hydrogen are present, a self‐sustaining optically‐thick atmosphere is possible. As such an atmosphere develops, the reradiation of insolation tends toward a uniform global distribution, which provides a strong negative feedback. Bimodal behavior of Triton's atmosphere is therefore possible as seasonal effects and volatile distribution alter the distribution of thermal emission.

On the use of the Rosseland and Planck mean absorption coefficients in the non-equilibrium radiative transfer equation

The use of the Rosseland and Planck mean absorption coefficients in the non-equilibrium (time-dependent) radiative transfer equation is investigated in the gray approximation for a prescribed temperature distribution. The corresponding gray radiative flux at the surface of a halfspace is compared to an exact (multi-frequency) solution obtained from multiple collision theory. An improved mean absorption coefficient is then obtained using the exact angle integrated intensity at the surface, and the corresponding gray approximation is shown to be markedly accurate.

The evaluation of flame behavior by spectrum analysis of a flame. 2nd report Study of an application to a large oil flame.

The application of spectrum analysis in the visible wavelength region of a large flame was studied theoretically and experimentally. Theoretical study of a flame model, which has a temperature distribution with some variations of optical thickness, indicated that the apparent mean emissivity of continuum emission, calculated on a hypothesis of gray approximation, was applicable to large industrial flames to evaluate flame conditions. An experimental study using a large (12 MW) oil flame also supported the appropriateness of the theoretical study and the apparent mean emissivity was evaluated to correlation well with NOx emissions corresponding to the various kinds of flame conditions.

Effects of the interaction between radiation and a disintegrating vitreous material

This article considers the disintegration of a vitreous semitransparent material at the critical point in a stream of radiating gas. A study is made of the effect on the rate of disintegration of the indeterminacy of the effect of interaction between the radiation and the gaseous disintegration products and of the spectral distribution of the density of the radiation flux of the gas with respect to the semitransparancy of the material. An analysis of the error of the gray approximation with calculation of the disintegration of materials with a semitransparency window is also given.

The Use of the Milne-Eddington Absorption Coefficient for Radiative Heat Transfer in Combustion Systems

The applicability of the Milne-Eddington absorption coefficient approximation is discussed in relation to the calculation of radiative transport involving the two distinct types of species produced in combustion systems—gases and soot particles. The approximation is found to apply well to hydrocarbon soot particles and as a result analytical closed-form solutions are derived for the radiative heat transfer inside one-dimensional slab shaped soot clouds. (The applicability of the gray approximation to soot is also discussed.) For the calculation of total band radiation from gases, however, the Milne-Eddington approximation is found to be questionable. The meaning of its assumption is discussed in light of an established Curtis-Godson wide band scaling approximation. Its usefulness for real gases is then assessed through the calculation and comparison of slab radiation by both techniques.

Thermal Performance Analysis of the Stationary Reflector/Tracking Absorber (SRTA) Solar Concentrator

The performance of a novel solar energy concentrating system consisting of a fixed, concave spherical mirror and a sun-tracking, cylindrical absorber is analyzed in detail. This concentrating system takes advantage of the spherical symmetry of the mirror and its linear image which, when taken together, form a tracking, solar-concentrating system in which only the small cylindrical absorber need move. The effects of mirror reflectance, concentration ratio, heat transfer fluid flow rate, radiative surface properties, incidence angle, an evacuated absorber envelope, and insolation level upon thermal performance of the concentrator are studied by means of a mathematical model. The simulation includes first order radiation and convection processes between the absorber and its concentric glass envelope and between the envelope and the environment; radiation processes are described by a dual-band, gray approximation. The energy equations are solved in finite difference form in order that heat flux and temperature distributions along the absorber may be computed accurately. The results of the study show that high-temperature heat energy can be collected efficiently over a wide range of useful operating conditions. The analysis indicates that mirror surface reflectance is the single most important of the principal governing parameters in determining system performance. Efficiency always increases with concentration ratio although the rate of increase is quite small for concentration ratios above 50. High fluid flow rate (i.e., lower operating temperature), an evacuated envelope, or a highly selective surface can enhance performance under some conditions. The conclusion of the study is that high-temperature heat energy can be generated at high efficiency by the present concentrator with present technology in sunny regions of the world.