Radiative transfer equation solutions

Up to now, the sdB/sdOB stars, the classical sdOs and the extremely helium-rich luminous sdOs have been analyzed for the most important (and accessible) metal abundances. The analyses usually require extensive non-LTE line formation calculations to solve the statistical equilibrium in detailed model atoms simultaneously with the radiative transfer equations for all relevant frequencies. With the advent of computer codes based on modern powerful solution algorithms (Auer and Heasley, 1976 Werner and Husfeld, 1985) it has now become possible to test (and eventually remove) approximations necessary in older computations. This and the availability of improved atomic data make the non-LTE predictions more reliable, and obstacles in obtaining accurate abundance determinations come now mainly from the observational side where high-quality spectra are needed to identify and to measure weak... 

Chapter 6 describes solar-powered photocatalytic reactors for the conversion of organic water pollutants. Nonconcentrating reactors are identified as some of the most energetically efficient units. It is reported that the absorption of radiation is a critical parameter in the efficiency reactor evaluation. The radiative transfer equation (RTE) solution under the simplified conditions given by the PI approximation is proposed for these assessments. 

Solution of the Radiative Transfer Equation at Wavelengths Longer than 3.5 pm Absorption and Emission of Infrared Radiation... 

The RTE is a simplified form of the complete Maxwell equations describing the propagation of an electromagnetic wave in an attenuating medium. The simplified RTE does not include the effects of polarization of the radiation or the influence of nearby particles on the radiation scattered or absorbed by other particles (dependent scattering or absorption). For example, if polarization effects are present (as they are when reflections occur at off-normal incidence from polished surfaces or in reflections from embedded interfaces), then the analyst should revert to complete solution of the Maxwell equations, which is a formidable task in complex geometries Delineating the bounds of applicability of the radiative transfer equation is an area of active research. 

Boundary Conditions for the RTE. The solution of the radiative transfer equation in a given geometry is subject to boundary conditions, which give the radiation intensity distribution on the boundaries. The boundary intensity is comprised of two components (1) contribution due to emission at the boundary surfaces and (2) contribution due to diffuse and specular reflection of radiation intensity incident on the boundaries. The radiation incident on the boundary is due to intensity emitted from all volume and surface elements in the medium. In mathematical terms, the general boundary condition on any surface element is written as [1,6] ... 

The radiative transfer equation (RTE) is an integro-differential equation it is difficult to develop a closed-form solution to it in general multidimensional and nonhomogeneous media. After introducing a number of approximations, however, reasonably accurate models of the RTE can be obtained. In all models, the objective is to solve the RTE, or a modified form of it, in terms of radiation intensity or its moments (such as flux) and then calculate the distribution of the divergence of radiative flux V q everywhere in the medium. In this section, we will discuss the approximate models of the RTE which can be extended to multidimensional geometries. 

As discussed previously, the radiative transfer equation is written in terms of radiation intensity, which is a function of seven independent parameters. The RTE is developed phenomenologically and is a mathematical expression of a physical model (i.e., the conservation of the radiative energy). It is a complicated integro-differential equation. There is no available analytical solution to the RTE in its general form. In order to solve it, physical and mathematical approximations are to be introduced individually or in tandem. 

It is important to realize that the solution of the radiative transfer equation is required only to obtain the divergence of the radiative flux vector that is a total quantity (i.e., inte-... 

In this chapter, we presented a general overview of radiative heat transfer. A number of practical models were included for the solution of the radiative transfer equation and to calculate the required radiative properties of particles, combustion gases, and surfaces. Even though the material presented can allow the reader to tackle a radiative transfer problem, it is not possible to claim that our coverage of the subject was comprehensive. We tried to list most significant references, and the reader is encouraged to consult the literature for more detailed and the most up-to-date analyses and data. 

W. A. Fiveland, Discrete-Ordinate Solutions of the Radiative Transfer Equation for Rectangular Enclosures, ASME Journal of Heat Transfer, 106, p. 699,1984. 

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 ... 

Method, Including Rigorous Solution of the Radiative Transfer Equation for Complex Geometric Structure 62... 

Thus, the construction of predictive models of photobioreactors requires careful formulation of radiative transfer within the reaction volume, in order to obtain the radiation field (cf step 1 in the earlier procedure). Such analysis is developed this chapter, starting in Section 2 with determination of the light scattering and absorption properties of photosynthetic-microorganism suspensions. Next, these properties are used in Section 3 for analysis of radiative transfer and in Section 4 for rigorous solution of the radiative transfer equation by the Monte Carlo method. Finally, the thermokinetic coupling between radiative transfer and photosynthesis is addressed in Section 5. It should be noted that Sections 2 and 4 mainly summarize works that have been already published elsewhere, whereas Sections 3 and 5 include extensive original work and results. 

The main steps in our model and their organization within this chapter are summarized in Fig. 1. Our practice of the Monte Carlo method extends beyond the solution of the radiative transfer equation in Section 4, we also argue that the Monte Carlo method is well suited for numerical implementation of the entire model, especially in research on photobioreactors with complex geometric structure. 

These results are further vaUdated in Dauchet et al. (2015), where the transmittance spectra that were recorded for microorganism suspensions were compared with those predicted by solution of the radiative transfer equation for the radiative properties presented in Fig. 5. In every configuration that has been tested so far, the description of the microorganism s shape increases the accuracy of the results. 

As we have seen, the intensity L expressed in energy units and the intensity L expressed in kinetic units obey the same radiative transfer equation. The solutions obtained for these two physical quantities thus have the same formulation and, in case of numerical calculation, either of these variables can be determined (with the same formula) depending on the unit chosen for expressing the incident flux or its spectral distribution The latter is an input parameter. If we express q in W m, the result determines L, but if we express in pmol s m, then the result determines L. For this... 

As we saw in Eq. (18), description of each frequency of the spectrum is completely independent of other frequencies. Thus, all the derivation of equations in the rest of this section, including the approximate solutions developed in Sections 3.3—3.5, are valid regardless of the wavelength of incident radiation. For this reason, we decided to omit the spectral dependencies in our notations. For numerical calculations, however, one should use the value of the radiative properties and the incident flux n,t/ corresponding to the wavelength in question. In this approach, the radiative transfer equation is solved for each frequency, and the spectral solution thus obtained is integrated over PAR in order to calculate the local absorption rate A according to Eq. (31). This approach can be implemented with approximate solutions from Sections 3.3-3.5. 

The radiative transfer equation is not invariant with this transformation, but we find this invariance in various situations for example, the diffusion equation obtained with the PI approximation is invariant with this transformation (see Section 3.4). In addition, we fbrmd that solution of this equivalent problem usually provides results that are very close to those obtained by solution of the original problem in the case of a photobioreactor. The approximate solutions that are derived and validated in Sections 3.3 and 3.4 are obtained by addressing this equivalent problem. Note that this transformation is also useful for comparison of very different situations, regardless of the form of the phase function in the field of transport theory research, when mentioning optical thickness, we are generally referring to c rather than Cj. 

Kxi defined in Section 3.2. Hereafter, we will express the diffusion coefficient in m rather than in m /s this approach is convenient for analysis of steady-state systems. Indeed, in this case, the solution of the radiative transfer equation is independent of the speed of light r, accordingly, it is customary to divide Eq. (70) by c ... 

It is also important to note that due to the linearity of the radiative transfer equation, the solutions for configurations illuminated on both sides (or for mixtures of collimated and difiuse illumination) are obtained simply by adding up the solutions obtained in this section. For example, for incident solar radiation with direct and diffuse components, the radiation field can be obtained by adding up solutions 2 and 3. For a photobioreactor illuminated... 

NUMERICAL IMPLEMENTATION OF PHOTOBIOREACTOR MODELS BY THE MONTE CARLO METHOD, INCLUDING RIGOROUS SOLUTION OF THE RADIATIVE TRANSFER EQUATION FOR COMPLEX GEOMETRIC STRUCTURE... 

In the earlier example, the local production rate was assumed to be known for the purposes of illustration. Nevertheless, as stated in Section 1 (and detailed in Section 5), is a function of the specific rate of photon absorption A. This is why photobioreactor studies require solution of the radiative transfer equation prior to estimation of the production... 

The intensity of maser radiation the molecular aind level population analysed by the concept of radiation brightness temperature Tg and solution of the radiative transfer equation (see for general discussion e.g. Winnewisser et al., 1979 maser specific reference Elltzur, 1982). The intensity of... 

The DO-FV model [ 17] solves the radiative transfer equation (RTF) as a field equation for a finite number of discrete solid angles each associated with a vector direction s fixed in the global Cartesian system. The procedure involves the solution of as many transport equations as there are solid angles. 

We need to a dataset for training of ALM. The training dataset was generated by numerical solution of radiative transfer equation. Radiative transfer equation simulates the radiative transfer in atmosphere-ocean system. 

The discussions of the equation of transfer and the solution of this equation in Chapter 2 rest entirely on concepts of classical physics. Such treatment was possible because we considered a large number of photons interacting with a volume element that, although it was assumed to be small, was still of sufficient size to contain a large number of individual molecules. But with the assumption of many photons acting on many molecules we have only postponed the need to introduce quantum theory. Single photons do interact with individual atoms and molecules. The optical depth, r (v), depends on the absorption coefficients of the matter present, which must fully reflect quantum mechanical concepts. The role of quantum physics in the derivation of the Planck function has already been discussed in Section 1.7. Both the optical depth and the Planck function appear in the radiative transfer equation (2.1.47). 

An alternate approach is to apply a relaxation technique similar to the method of temperature inversion discussed in Subsection 8.2.c. In this case the number of parameters describing the gas profile is chosen equal to the number of wavenumbers for which we have measurements. The radiance at each wavenumber v, is associated with a gas mole fraction qi at the atmospheric level to which the radiance is most sensitive, i.e., near the peak of the contribution function. A first guess, q (i = 1, m), is introduced and used in the radiative transfer equation to calculate a set of radiances, /°(v,). In order to carry out the radiance calculation it is necessary to adopt some form of interpolation between the levels for which q is initially specified. An improved solution g/ is then obtained using the relaxation relation... 

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