Equation of radiative transfer

The standard approach to modeling PDRs is to use a one-dimensional approach in which the radiation strikes perpendicularly. The region is divided into slabs, so that the equations of radiative transfer and chemical kinetics can be solved conveniently. The slabs can be homogeneous, or can have different gas densities. The radiation is scattered and absorbed by dust particles, but, in addition, both H2 and... 

Optical designers and specialists in heat transfer calculations in the chemical engineering and mechanical engineering sciences are familiar with the mathematical construct known as The Equation of Radiative Transfer, although most chemists and spectroscopists are not. The Equation of Radiative Transfer states that, disregarding absorbance and scattering, in a lossless optical system... 

Integrate the equation of radiative transfer to produce a synthetic spectrum as a function of assumed abundances of each element of interest. 

The equation of radiative transfer is an energy balance except for this concept, its physical content is slight. The phy.sical problems of interest enter through the extinction coefficient and the source function. Many papers and monographs have been written on its solution... 

Expressions for the intensity can be obtained by integrating the equation of radiative transfer (5.46) over the horizontal distance from the test object to the point of observ atioo. If b and J are not functions of s, the integration gives... 

Chandrasekhar, S. (1960) Radiative Transfer, Dover, New York. This is ihe dassic reference on the equation of radiative transfer and its solutions. 

Oxford, This monograph includes application of the theory of small particle scattering and the equation of radiative transfer to Earth s atmosphere. 

The radiation field in the atmosphere is determined from the equation of radiative transfer (Chandrasekhar, 1950 Kourganoff, 1952 Sobolev, 1963 Lenoble, 1977), which is an expression of the energy balance in each unit volume of the atmosphere, including absorption, scattering, and emission. In the case of a horizontally stratified medium, the following expression can be used to describe the radiative transfer in... 

Solution of the Equation of Radiative transfer for Wavelengths Less than 3.5 /im Multiple Scattering... 

Several analytic methods have been proposed to solve the equation of radiative transfer in an absorbing and scattering atmosphere, but they can only be applied for the most simple cases. To obtain quantitative solutions, numerical methods are generally used, such as the Monte-Carlo method, DART method, iterative Gauss, discrete ordinate method, etc. A complete summary of these techniques is provided by Lenoble (1977), and a detailed discussion of multiple scattering processes in plane parallel atmospheres is given in the book by Liou (2002). 

Y. Bayazitoglu and J. Higenyi, Higher Order Differential Equations of Radiative Transfer Py Approximation, AIAA Journal, 14, p. 424,1979. 

Local thermodynamic equilibrium in space and time is inherently assumed in the kinetic theory formulation. The length scale that is characteristic of this volume is i whereas the timescale is xr. When either L i, ir or t x, xr or both, the kinetic theory breaks down because local thermodynamic equilibrium cannot be defined within the system. A more fundamental theory is required. The Boltzmann transport equation is a result of such a theory. Its generality is impressive since macroscopic transport behavior such as the Fourier law, Ohm s law, Fick s law, and the hyperbolic heat equation can be derived from this in the macroscale limit. In addition, transport equations such as equation of radiative transfer as well as the set of conservation equations of mass, momentum, and energy can all be derived from the Boltzmann transport equation (BTE). Some of the derivations are shown here. 

It is clear that in the most general form as described in Eq. 8.45, the scattering terms pose difficulty for solving. Therefore, the relaxation-time approximation is usually made for convenience, in which case the equation of radiative transfer reduces to... 

The equation of radiative transfer will not be solved here since solutions to some approximations of the equation are well known. In photon radiation, it has served as the framework for photon radiative transfer. It is well known that in the optically thin or ballistic photon limit, one gets the heat flux as q = g T[ - T ) from this equation for radiation between two black surfaces [13]. For the case of phonons, this is known as the Casimir limit. In the optically thick or diffusive limit, the equation reduces to q = -kpVT where kp is the photon thermal conductivity. The same results can be derived for phonon radiative transfer [14,15]. 

The equation of radiative transfer for radiation in a direction 0 becomes... 

Analytic solutions to the radiative transfer equation (RTE) exist for simple cases however, for more realistic media with complex multiple scattering effects, numerical methods are required. The equation of radiative transfer simply states that as a beam of radiation travels, it loses energy to absorption, gains energy by emission and redistributes energy by scattering. The differential form of the equation for radiative transfer is ... 

Y. R. Sivathanu, J. P. Gore A Discrete Probability Function Method for the Equation of Radiative Transfer, J. Quant. Spectrosc. Radiat. Transfer 49(3), 269-280 (1993). 

It is usually assumed that the contributions of line and continuum attenuation are separable. The depth dependence of the photodestruction rates due to grain attenuation can be computed by solving the equations of radiative transfer for specified grain properties and geometry of the cloud. If the cloud is assumed to have a plane-parallel geometry with radiation Incident on both sides of the cloud, the depth-dependent rates depend strongly on the total extent of the cloud. The results for clouds with a total visual extinction are usually represented by single exponential decays to the centers... 

If diffuse reflection spectrometry is to be used for quality control purposes, it is essential to know the actual investigated sample volume, which is equivalent to the radiation penetration depth or the effective sample size meff- In a publication by Berntsson et al. [73], the effective sample size of pharmaceutical powders was investigated by the three-flux approximation presented above that they called the equation of radiative transfer (ERT) method, and an empirical method they called the variable layer thickness (VLT) method. In this publication, the effective sample size meg is defined as the mass per area of the sample at which its diffuse reflectance has reached 98% of the diffuse reflectance of a corresponding optically thick sample. 

P-n Radiation Model The P-n model (method of spherical harmonics) transforms the equations of radiative transfer into a set of partial differential equations that are simultaneously solved. Usually, low-order approximations, i.e. P-1, P-3 are used in which the angular dependence of the real radiation field is reduced to spherical functions. The low-order approximations are only accurate in optically thick media. 

The apparent brightness temperature of a deep atmosphere is related to the physical parameters of the atmosphere, such as pressure, temperature, and composition, through the equation of radiative transfer. To a good approximation at radio wavelengths, the equation of radiative transfer for a ray making an angle cos with the vertical in a lossy medium is... 

Given the thermal absorption coefficient and the boundary conditions on the heating, it is possible to determine the constants of temperature (To and T ) and phase ( ) in Eq. (21). The inverse problem is faced by the radio astronomer, namely to determine the thermal absorption coefficient from measurements of the thermal emission. This is done in the following manner. The temperature distribution given by Eq. (21) is used in the equation of radiative transfer... 

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