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Approximation Error RACKETT EQUATION RADAR RADIAL COMPRESSOR RADIAL ENERGY FLOWS RADIAL FANS RADIAL GAP SIZE RADIATION RADIATION ABSORPTION METHOD RADIATION BETWEEN PARALLEL PLATES RADIATION DIFFUSION APPROXIMATION, FOR COMBINED RADIATION AND CONDUCTION RADIATION DOSIMETRY RADIATION DRYING Radiation from semi-transparent oxide particles in thermal spraying Radiation heat transfer in a supersonic nozzle of a solid-propellant rocket engine Radiation heat transfer in solid-propellant rocket engines RADIATION IN ENCLOSURES Radiation in nanomanufacturing Radiation in production of carbon fibers Radiation of an isothermal plane-parallel layer Radiation of isothermal volumes of scattering medium: An error of the diffusion model Radiation of nonisothermal layer of scattering medium RADIATION SHIELDS RADIATION TO FURNACE TUBES Radiation transfer between the surfaces through a non-participating medium Radiation transfer in emitting, absorbing and scattering media Radiation transfer in combustion chambers Radiation transfer problems in nature and engineering Radiation transfer theory and the computational methods Radiation-turbulence interaction Radiative boundary layer Radiative cooling and solidification of core melt droplets Radiative cooling of particle flow in vacuum RADIATIVE DIFFUSION Radiative effects in semi-transparent liquid containing gas bubbles Radiative equilibrium in plane-parallel layer RADIATIVE EXCHANGE Radiative exchange between an isothermal gas and surrounding walls RADIATIVE HEAT FLUX Radiative heat transfer Radiative heat transfer in moving media RADIATIVE HEAT TRANSFER, IN POROUS MEDIA Radiative properties of gas bubbles in semi-transparent medium Radiative properties of metal particles in infrared and microwave spectral ranges Radiative properties of micro- and nanostructures Radiative properties of particles and fibers (theoretical analysis) Radiative properties of polydisperse systems of independent particles Radiative properties of semi-transparent fibers at arbitrary illumination Radiative properties of semi-transparent particles Radiative properties of single particles and fibers: the hypothesis of independent scattering and the Mie theory Radiative properties of soot particles Radiative properties of water droplets in near infrared RADIATIVE SPECTRAL INTENSITY Radiative transfer equation Radiative transfer equation: a general formulation Radiative transfer for coupled atmosphere and ocean systems Radiative transfer in combustion phenomena affected by radiation Radiative transfer in combustion systems Radiative Transfer in Coupled Atmosphere and Ocean Systems: Impact of Surface Roughness on Remotely Sensed Radiances Radiative transfer in coupled atmosphere and ocean systems: the adding and doubling method Radiative Transfer in Coupled Atmosphere and Ocean Systems: the Discrete Ordinate Method RADIATIVE TRANSFER IN COUPLED ATMOSPHERE AND OCEAN SYSTEMS: THE SUCCESSIVE ORDER OF SCATTERING METHOD Radiative transfer in glass production Radiative transfer in laminar flames Radiative transfer in laser processing Radiative transfer in medical laser treatment RADIATIVE TRANSFER IN MULTIDIMENSIONAL PROBLEMS: A COMBINED COMPUTATIONAL MODEL Radiative transfer in space applications Radiative transfer in the atmosphere Radiative transfer in turbulent flames Radiative transfer in two-phase combustion Radiative-conductive heat transfer in dispersed materials Radiative-conductive heat transfer in foam insulations RADIO FREQUENCY HEATING RADIO FREQUENCY, RF, DRYING RADIO WAVES RADIOACTIVE DECAY RADIUM RADIUS, HYDRAULIC RADON RAE RAFFINATE PHASE RAINBOW VOLUMIC VELOCIMETRY RAINFALL RAMAN SPECTROSCOPY RAMJET ENGINES RANDOM PROCESSES RANKINE CYCLE RANKINE DEGREE RANKINE VORTEX RANKINE, WJM RAOULT'S AND DALTON'S LAW RAOULT'S LAW RAREFACTION RAREFACTION WAVE RAREFIED GAS DYNAMICS RATE-CONTROLLED CONSTRAINED EQUILIBRIUM Ray effects and false scattering Ray optics and wave effects in radiation propagation Rayleigh equation, for bubble growth Rayleigh equation, for droplet formation Rayleigh formula Rayleigh law of scattering Rayleigh number Rayleigh scattering Rayleigh, Lord (1842-1919) Rayleigh-Gans scattering Rayleigh-Taylor instability REACTING GAS FLOW REACTION TURBINES REACTIVE CONTAMINANT TRANSPORT REACTOR PHYSICS Real gaseous spectra REATTACHMENT REATTACHMENT, OF BOUNDARY LAYER Reaumur Degree REBOILERS RECIPROCATING COMPRESSOR RECIRCULATION RECIRCULATION BOILERS RECONSTRUCTED WAVEFRONTS RECOVERY COEFFICIENT RECOVERY TEMPERATURE RECTANGULAR CHANNEL RECTANGULAR CYLINDERS RECTANGULAR DUCTS RECTANGULAR STENOTIC MODELS RECUPERATIVE HEAT EXCHANGERS REDLICH-KWONG EQUATION REDOX REACTIONS REDUCED GRAVITY CONDITIONS REDUCED INSTRUCTION SET COMPUTER, RISC REDUCED PROPERTIES REFINING REFLECTANCE REFLECTION COEFFICIENT (REFLECTANCE) REFLECTION COEFFICIENTS FOR EARTH'S SURFACE REFLECTIVITY REFLOOD REFLUX CONDENSATION REFLUX CONDENSER REFLUX RATIOS REFORMING REFRACTION REFRACTIVE INDEX REFRACTIVE INDICES FOR GASES AND LIQUIDS REFRACTORY MATERIALS, FOR ELECTRIC FURNACES REFRIGERANTS REFRIGERATION REGENERATIVE BURNER REGENERATIVE FEED HEATING REGENERATIVE GAS TURBINE REGENERATIVE HEAT EXCHANGERS REGULAR REGIME OF DRYING REHEATING REICHARDT'S FORMULA, FOR VELOCITY DISTRIBUTION IN TUBES REIMANN'S INTEGRAL REINER-RIVLIN FLUID RELATIVE HUMIDITY RELATIVE MOLAR MASS RELATIVE PERMEABILITY RELATIVE POWER DEMAND, RPD RELATIVE ROUGHNESS RELAXATION TIME RENEWABLE ENERGY RENEWABLE ENERGY SOURCES RESIDUAL ENTHALPY RESIDUAL GIBBS ENERGY RESINS RESISTANCE HEATING RESISTANCE THERMOMETERS RESISTANCE THERMOMETRY RESISTANCE, ELECTRICAL RESISTIVITY, ELECTRICAL RESONANCE FLUORESCENCE RETENTATE RETENTION INDEX RETROGRADE CONDENSATION RETURN TO NUCLEATE BOILING REVERSE OSMOSIS REVERSED HEAT ENGINE CYCLES REVERSIBILITY PRINCIPLE REVERSIBLE PROCESSES REWETTING REWETTING OF HOT SURFACES REYNOLDS ANALOGY Reynolds Number REYNOLDS NUMBER, CRITICAL, IN TUBES REYNOLDS STRESS REYNOLDS STRESS TRANSPORT MODELS REYNOLDS' AVERAGING REYNOLDS' EQUATIONS REYNOLDS, OSBORNE (1842-1912) RHEOLOGY RHEOMETERS RHEOPEPTIC FLUIDS RHODAMINE RICCATTY-BESSEL FUNCTIONS RICHARDSON NUMBER RIDEAL-ELEY MODEL, FOR HETEROGENEOUS CATALYSIS RIEDEL-PLANK-MILLER EQUATION RIEMAN WAVES RIGHT-ANGLE TRIANGULAR ENCLOSURE RIGID-WALLED CHANNEL RISC. REDUCED INSTRUCTION SET COMPUTER RISK ANALYSIS TECHNIQUES RISK ASSESSMENT ROASTING ROCKET PROPELLANTS ROCKETS ROD BAFFLES ROD BUNDLE TESTS ROD BUNDLES, FLOW IN ROD BUNDLES, HEAT TRANSFER IN ROD BUNDLES, PARALLEL FLOW IN ROD CLIMBING ROD-STABILIZED LAMINAR PREMIXED FLAME RODRIGUES FORMULA ROHRSCHNEIDER CONSTANT ROLL MOMENT ROLL WAVES ROOTS TYPE COMPRESSOR ROSIN-RAMMLER ROSIN-RAMMLER SIZE DISTRIBUTION ROSSBY NUMBER ROSSELAND COEFFICIENT ROTAMETERS ROTARY ATOMIZERS ROTARY DRYERS ROTARY KILNS ROTARY REGENERATORS ROTATED TUBE BANKS ROTATING CHANNEL WITH RIBS ROTATING CYLINDERS, CRITICAL SPEED ROTATING CYLINDERS, FLOW BETWEEN ROTATING CYLINDERS, FLOW OVER ROTATING DISC CONTACTOR ROTATING DISC SYSTEMS, APPLICATIONS ROTATING DISC SYSTEMS, BASIC PHENOMENA ROTATING DUCT SYSTEMS, ORTHOGONAL, HEAT TRANSFER IN ROTATING DUCT SYSTEMS, PARALLEL, HEAT TRANSFER IN ROTATING FLOW IN A POROUS LAYER ROTATING FLOW PASSAGE ROTATING PIPE FLOW ROTATING SURFACES ROTATIONAL DISCONTINUITIES Rotational Rayleigh number ROTATIONAL REYNOLDS NUMBERS ROUGH CHANNELS, FRICTION FACTOR IN ROUGH SURFACE FRICTION FACTORS ROUGH SURFACES ROUGH TUBES ROUGH TUBES, FLOW IN ROUGH TUBES, HEAT TRANSFER IN ROUGHNESS FACTORS ROYAL ACADEMY OF ENGINEERING, RAE ROYAL SOCIETY OF CHEMISTRY ROYAL SOCIETY, RS RS RSC RUBBER RUMFORD, COUNT, BENJAMIN THOMPSON RUSHTON TURBINE
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Radiative transfer equation

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This introductory material begins a set of articles on computational models employed in radiative transfer calculations for disperse systems. We will not consider here the nature and basic laws of thermal radiation because this general knowledge is given in the well-known textbooks by Sparrow and Cess (1978), Siegel and Howell (2002), and Modest (2003). Nevertheless, before proceeding to mathematical formulation of radiation transfer problems for scattering media, it is reasonable to recall some of the definitions of the main physical quantities.

The radiation energy in wavelength interval (λ, λ + dλ), passing per time dt in solid angle d near direction through area dσ located at point and oriented perpendicular to , is equal to Iλ(,) dλ dt dσ d, where function Iλ(,) is the spectral intensity of radiation. This function is the most general characteristic of the radiation field in the case of unpolarized (randomly polarized) radiation. The polarization of electromagnetic waves is usually not important in the problems concerning thermal radiation, and it is sufficient to use the scalar function Iλ(,) instead of the Stokes parameters (Born and Wolf, 1999). The details concerning description of polarized radiation can be found in the classic book by Chandrasekhar (1950, 1960) and in the monographs by van de Hulst (1957, 1981) and Bohren and Huffman (1983), which are very close to the problems under consideration and should be included by a reader in a short list of the most important handbooks. The spectral intensity of equilibrium (the so-called “black body”) thermal radiation of an isothermal medium is given by the Planck’s function: Iλ = Bλ(T). The black-body radiation is homogeneous and isotropic; i.e., independent of both coordinate and direction . For radiating medium, a deviation of the function Iλ(,) from the intensity of equilibrium radiation at local temperature T() is described by the radiative transfer equation.

Absorption and scattering of radiation in a medium are described by spectral coefficients αλ and σλ, respectively, by the extinction coefficient βλ = αλ + σλ and by the scattering function Φλ(', ), which is also called the scattering phase function or indicatrix of scattering. The latter function presents the angular intensity distribution for the radiation scattered by a small (elementary) volume of the medium by one act of scattering. The scattering function satisfies the normalizing condition:

(1)

Note that coefficients αλ, σλ, and βλ are also referred to as the medium elementary volume. It is assumed that the absorption and scattering characteristics of a small element of the medium can be determined on the basis of the so-called far-field single-scattering approximation. This assumption, which is also known as the independent scattering approximation, is correct in many applied problems concerning rarefied disperse systems when positions of single particles are random and uncorrelated and the average distances between the neighboring particles are greater than the particle size and radiation wavelength. The physical sense of this assumption has been considered in some detail by Mishchenko et al. (2004).

Note that the above definitions of the absorption and scattering characteristics of a medium correspond to the continuous model of the radiation propagation in the medium. It is a widely used approach, which appears to be fairly good in many practical problems concerning thermal radiation in disperse systems. However, there are some specific cases, such as particulate debris beds or large-scale cellular structures, when the classical continuum theory may not be appropriate and the discrete transfer models are physically more adequate to the real processes (Vortmeyer, 1978; Viskanta and Mengüç, 1989).

REFERENCES

Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, New York: Wiley, 1983.

Born, M. and Wolf, E., Principles of Optics, 7th ed. (expanded), New York: Cambridge University Press, 1999.

Chandrasekhar, S., Radiative Transfer, Oxford, UK: Oxford University Press, 1950.

Chandrasekhar, S., Radiative Transfer, New York: Dover, 1960.

Mishchenko, M. I., Hovenier, J. W., and Mackowski, D. W., Single scattering by a small volume element, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 71-87, 2004.

Modest, M. F., Radiative Heat Transfer, 2nd ed., New York: Academic, 2003.

Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 4th ed., New York: Taylor & Francis, 2002.

Sparrow, E. M. and Cess, R. D., Radiation Heat Transfer, New York: McGraw-Hill, 1978.

van de Hulst, H. C., Light Scattering by Small Particles, New York: Wiley, 1957.

van de Hulst, H. C., Light Scattering by Small Particles, New York: Dover, 1981.

Viskanta, R. and Mengüç, M. P., Radiative transfer in dispersed media, Appl. Mech. Rev., vol. 42, no. 9, pp. 241-259, 1989.

Vortmeyer, D., Radiation in packed solids, Proc. of 6th International Heat Transfer Conference, Toronto, vol. 6, pp. 525-539, 1978.

References

  1. Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, New York: Wiley, 1983.
  2. Born, M. and Wolf, E., Principles of Optics, 7th ed. (expanded), New York: Cambridge University Press, 1999.
  3. Chandrasekhar, S., Radiative Transfer, Oxford, UK: Oxford University Press, 1950.
  4. Chandrasekhar, S., Radiative Transfer, New York: Dover, 1960.
  5. Mishchenko, M. I., Hovenier, J. W., and Mackowski, D. W., Single scattering by a small volume element, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 71-87, 2004.
  6. Modest, M. F., Radiative Heat Transfer, 2nd ed., New York: Academic, 2003.
  7. Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 4th ed., New York: Taylor & Francis, 2002.
  8. Sparrow, E. M. and Cess, R. D., Radiation Heat Transfer, New York: McGraw-Hill, 1978.
  9. van de Hulst, H. C., Light Scattering by Small Particles, New York: Wiley, 1957.
  10. van de Hulst, H. C., Light Scattering by Small Particles, New York: Dover, 1981.
  11. Viskanta, R. and Mengüç, M. P., Radiative transfer in dispersed media, Appl. Mech. Rev., vol. 42, no. 9, pp. 241-259, 1989.
  12. Vortmeyer, D., Radiation in packed solids, Proc. of 6th International Heat Transfer Conference, Toronto, vol. 6, pp. 525-539, 1978.

Following from:

Computational methods for radiative transfer in disperse systems

Leading to:

Transport approximation
Monte Carlo simulation of radiative transfer
Discrete ordinates method
Differential approximations

This article belongs to the following areas:

Computational methods for radiative transfer in disperse systems in Fundamentals
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