Radiation transport

A one-dimensional mesh through time (temporal mesh) is constructed as the calculation proceeds. The new time step is calculated from the solution at the end of the old time step. The size of the time step is governed by both accuracy and stability. Imprecisely speaking, the time step in an explicit code must be smaller than the minimum time it takes for a disturbance to travel across any element in the calculation by physical processes, such as shock propagation, material motion, or radiation transport [18], [19]. Additional limits based on accuracy may be added. For example, many codes limit the volume change of an element to prevent over-compressions or over-expansions. 

The models proposed to represent radiation transport process can be grouped into two classes. The first and simpler approach is to use some form of the Stefan-Boltzmann equation for radiant exchange between opaque gray bodies,... 

By far, the most widely used model in calculating hemodynamic response is based on the classic Beer-Lambert law. The Beer-Lambert law is derived from solution to radiation transport equation under several simplifying assumptions [91]. It describes a linear relationship between absorbance, A, of light through a medium and wavelength dependent extinction coefficient, e(A). This relationship is given by Equation (1)... 

Fast neutron detection sometimes uses a hydrogenous moderator to slow down the neutrons and then employs a low-energy neutron detector as described above. One common fast neutron detector is a Bonner sphere. In this detector, a scintillator is placed in the center of a polyethylene sphere. Radiation transport calculations are used to produce efficiency curves that depend on the energy of the incident neutron. Another common fast neutron detector is a long counter. This detector uses a slow neutron detector (originally a BF3 chamber) at the center of a cylindrical moderator designed so that the detector is only sensitive to neutrons incident from one side. 

Various computer codes exist which are used to simulate nuclear weapons effects on various targets. Variations of codes on radiation transport, shielding and cross sections also can be considered. A directory of currently used codes was compiled by Martin, Reitz and Root (Ref 23), which for the most part is a rather complete tabulation of computer programs applied to the numerical simulation of nuclear weapon expls phenomenology and effects... 

Since no astronomical standard candle is known - all proposed objects have been shown to be essentially non-uniform in one way or another - we nowadays have to calculate and plot the distance modulus for the objects. The scatter around the linear expansion line is less than 0.2 magnitudes or 20% Tonry et al. 2003. Independent of our ignorance of the exact explosion mechanism or the radiation transport in the explosions this proves that SNe la can reliably be used as a distance indicator in the local universe. This situation is very much comparable to the Cepheid stars, where the period-luminosity relation is based on empirical data of objects in the Magellanic Clouds. 

In the current state-of-the-art, one-dimensional models can best be used to look in detail at the coupling of a very large number of species interactions in a geometry that is an approximation to reality. Processes such as radiation transport, turbulence, or the effects of heterogeneity of materials can be included either as empirically or theoretically derived submodels. 

In enclosure fires, radiation may be the dominant mode of heat transfer. For flames burning in an open atmosphere, the radiative fraction of overall heat transfer ranges from less than 0.1 to 0.4, depending both on the fuel type and the fire diameter [45], Owing to the important role that radiation plays in fires, all fire CFD models have a radiation model that solves the radiation transport equation (RTE) [46,48] ... 

To evaluate the heating, a relativistic 1-D Fokker-Planck code was used. The configuration space is 1-D but the momentum space is 2-D, with axial symmetry. This code is coupled to a radiation-hydrodynamic simulation in order to include energy dissipation via ionization processes, hydrodynamic flow, the equation-of-state (EOS), and radiation transport. The loss of kinetic energy from hot electrons is treated through Coulomb and electromagnetic fields. 

The radiation-hydrodynamic simulation includes the Quotidien EOS [29] and Ion EOS based on the Cowan model [30], For the electron component, a set of fitting formulae derived from the numerical results from the Thomas-Fermi model and a semi-empirical bonding correction [31] are adopted. The effective Z-number of the partially ionized plasma is obtained from the average atom model. Radiation transport is treated by multigroup diffusion. 

The transport coefficients appearing in equations (5)-(7) are given in Appendix E. The external forces are specified (not derived). The radiant flux, is also viewed here as specified it is found fundamentally through the integro-differential equation of radiation transport (see Appendix E). The reaction rates in equation (4) are determined by the phenomonological expressions of chemical kinetics,... 

Calculation of q necessitates consideration of radiation transport [31]-[33]. The spectral intensity 7 (x,f2, t) is defined as the radiant energy per unit area per second, traveling in a direction defined by the unit vector ft, per unit solid angle about that direction, per unit frequency range about... 

In view of the complexity associated with equation (48), approximate methods are needed for applications. References [6] and [33]-[38] may be consulted for these approximations. While scattering may be important in combustion situations involving large numbers of small condensed-phase particles, often the effects of scattering may be approximated as additional contributions to emission and absorption, thereby eliminating the integral term. Two classical limits in radiation-transport theory are those of optically thick and optically thin media the former limit seldom is applicable in combustion, while the latter often is. In the optically thin limit, gas-phase... 

Roo is the reflectance of an infinitely thick sample (in the near-infrared, this means an approximate 5-mm thickness and more). The theory was recently revisited by Loyalka and Riggs, ° who reinvestigated the accuracy of the Kubelka-Munk equations. They found that the coefficient k must be replaced by k = 2a with the absorption coefficient a = In(lO) ec, as derivable from Beer s law for the latter equation In(lO) = 2.303, e the molar absorptivity, and c the molar concentration. Such a dependency for k was stated earlier by other researchers when comparing more refined radiation transport theories for biomedical applications, e.g., Ref.[ l... 

The discrete ordinates method in a S4-approximation is used to solve the radiation transport equation. Since the intensity of radiation depends on absorption, emission and scattering characteristics of the medium passed through, a detailed representation of the radiative properties of a gas mixture would be very complex and currently beyond the scope of a 3D-code for the simulation of industrial combustion systems. Thus, contributing to the numerical efficiency, some simplifications are introduced, even at the loss of some accuracy. The absorption coefficient of the gas phase is assumed to have a constant value of 0.2/m. The wall emissivity was set to 0.65 for the ceramic walls and to a value of 0.15 for the glass pane inserted in one side wall for optical access. 

Solar Radiation in the Atmosphere Solar radiation is modified considerably on its path from the top of the atmosphere down to the sea surface. The simple radiation transport model used here follows mainly Bodin (1979), modified by Meier et al. (1999). The radiation at the sea surface is described as... 

J. S. Truelove, Discrete-Ordinates Solution of the Radiation Transport Equation, ASME Journal of Heat Transfer, 109(4), pp. 1048-1051,1987. 

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