Radiant transport equation

The radiation intensity Ix is described by the radiant transport equation (RTE), which in cylindrical geometry with the axis of symmetry Oz and neglecting scattering has the form... 

While equation (42) is valid for one-component systems without radiant transport, for binary and multicomponent mixtures there are other effects besides thermal conduction that contribute to the heat flux q. 

To represent the partially premixed turbulent combustion of a refinery gas in the heater, a combination of the flamelet formulations for premixed and nonpremixed combustion was used [16]. The standard k-e model was used for turbulent flow calculations. The effect of turbulence on the mixture fraction was accounted for by integrating a beta-PDF derived from the local mixture fraction and mixture fraction variance, which were in turn obtained by solving their respective transport equations. A relatively simple approach was used to compute radiant heat transfer—a diffusion model with a constant absorption coefficient (0.1 m i). 

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,... 

The radiant flux

optical system from the source to the detector may be described by the following equation ... 

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,... 

The condensed phase density p, specific heat C, thermal conductivity A c, and radiation absorption coefficient Ka are assumed to be constant. The species-A equation includes only advective transport and depletion of species-A (generation of species-B) by chemical reaction. The species-B balance equation is redundant in this binary system since the total mass equation, m = constant, has been included the mass fraction of B is 1-T. The energy equation includes advective transport, thermal diffusion, chemical reaction, and in-depth absorption of radiation. Species diffusion d Y/cbfl term) and mass/energy transport by turbulence or multi-phase advection (bubbling) which might potentially be important in a sufficiently thick liquid layer are neglected. The radiant flux term qr... 

The heat transport due to conduction and that due to radiation are not readily separable from the experimental data. Curve A of Fig. 4 shows the measured temperature distribution through a typical sample containing 29 shields per inch. Curve B shows the temperature distribution expected if each sheet of aluminum foil were a floating radiation shield. These results were obtained from Fig. 1. Curve C shows the temperature distribution througji an ideal sample, whose thermal conductivity would be independent of temperature. The observed result is probably a combination of radiation heat transfer and the change in thermal conductivity of the insulation with temperature. The thermal conductivity of most disordered dielectrics is approximately proportional to the first power of the temperature, but the temperature dependence of multiple contacts is not well understood. The fact that the temperature distribution for a sample of this type can be accounted for by a temperature-dependent thermal conductivity is sufficient justification for using Eq. (3), a particular solution of the Fourier equation, rather than Eq. (1), the heat flux equation for radiant heat transport, to represent our results. 

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