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Showing papers on "Radiative transfer published in 1992"


01 Jan 1992
TL;DR: In this article, the authors present a model of the photosphere of the black body and its radiation, and the measurement of stellar continua, spectral lines, and photospheric pressure.
Abstract: 1. Background 2. Fourier transforms 3. Spectroscopic tools 4. Light detectors 5. Radiation terms and definitions 6. The black body and its radiation 7. Radiative and convective energy transport 8. The continuous absorption coefficient 9. The model photosphere 10. The measurement of stellar continua 11. The line absorption coefficient 12. The measurement of spectral lines 13. The behavior of spectral lines 14. The measurement of stellar radii and temperatures 15. The measurement of photospheric pressure 16. Chemical analysis 17. Velocity fields in stellar photospheres 18. Stellar rotation.

1,109 citations


Journal ArticleDOI
TL;DR: In this paper, a correlated k(absorption coefficient)-distribution (CKD) method for radiative transfer in nonhomogeneous atmospheres, in terms of the physical and mathematical conditions under which this method is valid is discussed.
Abstract: The paper discusses the correlated k(absorption coefficient)-distribution (CKD) method for radiative transfer in nonhomogeneous atmospheres, in terms of the physical and mathematical conditions under which this method is valid. The CKD approach is theoretically formulated, and the mathematical and physical conditions under which this method is valid are identified. The parameterization for incorporating pressure and temperature effects in the k distribution is presented. The errors of CKD for absorption bands in the solar and thermal infrared spectra are examined, and the errors due to both the CKD method and the use of the optimum number of quadrature points are identified. Finally, an overall treatment for various absorption bands is developed, which will provide high efficiency and accuracy.

873 citations


Journal ArticleDOI
TL;DR: In this paper, a comprehensive scheme for the parameterization of radiative transfer in numerical weather prediction (NWP) models has been developed, which is based on the solution of the δ-two-stream version of the RTS equation incorporating the effects of scattering, absorption, and emission by cloud droplets, aerosols, and gases in each part of the spectrum.
Abstract: A comprehensive scheme for the parameterization of radiative transfer in numerical weather Prediction (NWP) models has been developed. The scheme is based on the solution of the δ-two-stream version of the radiative transfer equation incorporating the effects of scattering, absorption, and emission by cloud droplets, aerosols, and gases in each part of the spectrum. An extremely flexible treatment of clouds is obtained by allowing partial cloud cover in any model layer and relating the cloud optical properties to the cloud liquid water content. The latter quantity may either be a prognostic or diagnostic variable of the host model or specified a priori depending on cloud type, height, or similar criteria. The treatment of overlapping cloud layers is based on realistic assumptions, but any different approach requires only minor modifications of the code. The scheme has been tested extensively in the framework of the intercomparison of radiation codes in climate models (ICRCCM, WMO 1984, 1990). Rad...

789 citations


Book
01 Feb 1992
TL;DR: In this paper, Monte Carlo surface transfer is used for nonisothermal transfer in Nonscattering, homogeneous media and diffusion with anisotropic, multiple scattering, coupled with Conduction and Convection.
Abstract: The Nature of Thermal Radiation. Radiative Properties and Simple Transfer. Diffuse Surface Transfer. Electromagnetic Theory Results. Classical Dispersion Theory. Monte Carlo Surface Transfer. Radiative Transfer Equation. Thermal Radiation Properties of Gases. Radiative Properties of Particles. Radiative Transfer in Nonscattering, Homogeneous Media. Nonisothermal Transfer: Radiative Equilibrium and Diffusion with Isotropic Scattering. Radiative Transfer with Anisotropic, Multiple Scattering. Radiative Transfer Coupled with Conduction and Convection. Monte Carlo in Participating Media. Appendices. Index.

723 citations


Journal ArticleDOI
TL;DR: In this paper, a line-by-line radiance model FASCODE has been developed and applied to the calculation of cooling rates for atmospheric water vapor, which achieves computational accuracies for the longwave upwelling and downwelling fluxes of the order of 0.2%.
Abstract: A model for the accelerated calculation of clear sky fluxes based on the line-by-line radiance model FASCODE has been developed and applied to the calculation of cooling rates for atmospheric water vapor. The model achieves computational accuracies for the longwave upwelling and downwelling fluxes of the order of 0.2%, an accuracy well within current limitations imposed by uncertainties in the spectral parameters, the line shape, and the associated continua. For the same treatment of line shape, the Voigt profile with a 10 cm−1 cutoff and no continuum, the results from the present model are in acceptable agreement with those from two other line-by-line models reported as part of the intercomparison of radiation codes used in climate models (ICRCCM). For this line profile and the mid-latitude summer atmosphere, the largest difference between the results from our model and the Goddard Laboratory for Atmospheres (GLA) model occurs for the downwelling flux at the surface, with the present model providing a value greater than that from GLA. The differences are generally consistent with greater atmospheric opacity from the present model, attributable to the inclusion of a self-broadening component for the half width for water and to finer spectral sampling in the lower-pressure regime. Utilization of the line shape and associated continuum model included in FASCODE gives results that are significantly different from those provided by the other two models. This radiance model, including contributions from the foreign continuum as well as from a modified self-continuum, has received extensive validation against measured radiance spectra, an example of which is provided. For the mid-latitude summer atmosphere the principal contribution from the foreign continuum occurs in the upper troposphere in the 250–350 cm−1 spectral region, whereas the contribution from the self-continuum, dependent on the square of the water vapor density, is greatest in the lower troposphere. For the mid-latitude summer atmosphere the foreign continuum contributes 0.4 Kd−1 or 20% to the cooling in the upper troposphere and the self-continuum contributes 1.9 K d−1 to the cooling rate at the surface due to water vapor. The latter is 0.17 K d−1 less than the cooling rate from the GLA model which is principally due to a modification of the self-continuum. A significant result that has developed from the present work is the insight into atmospheric radiative processes provided by spectral profiles of the cooling rate. In the spectral domain there exists a mapping between the altitude and the molecular absorption strength as weighted by the Planck function. The extremely high correlation between the outgoing spectral radiance at the top of the atmosphere and the spectral cooling rate profile suggests that measurement of the outgoing spectral radiance can provide important information about atmospheric state that is not available from spectrally integrated quantities. Our results also indicate the critical importance of the spectral region from 100 to 600 cm−1 for the radiative transfer associated with atmospheric water vapor.

710 citations


Journal ArticleDOI
TL;DR: In this paper, an inversion code of Stokes line profiles is presented, which allows the recovery of the stratification of the temperature, the magnetic field vector, and the line of sight velocity through the atmosphere.
Abstract: An inversion code of Stokes line profiles is presented. It allows the recovery of the stratification of the temperature, the magnetic field vector, and the line of sight velocity through the atmosphere, and the micro- and macro-turbulence velocities, which are assumed to be constant with depth. It is based on the response functions, which enter a Marquardt nonlinear least-squares algorithm in a natural way. Response functions are calculated at the same time as the full radiative transfer equation for polarized light is integrated

699 citations


Journal ArticleDOI
TL;DR: A concise theoretical treatment is developed for the calculation of mean time, differential pathlength, phase shift, modulation depth and integrated intensity of measurements of light intensity as a function of time on the surface of tissue, resulting from either the input of picosecond light pulses, or radio frequency-modulated light.
Abstract: A concise theoretical treatment is developed for the calculation of mean time, differential pathlength, phase shift, modulation depth and integrated intensity of measurements of light intensity as a function of time on the surface of tissue, resulting from either the input of picosecond light pulses, or radio frequency-modulated light. The treatment uses the Green's function of the diffusion approximation to the radiative transfer equation, and develops this and its Fourier transform in a variety of geometries. Detailed comparisons are made of several of these parameters in several geometries, and their relation to experimentally measured clinical data. The limitations of the use of phase measurements is discussed.

657 citations


Journal ArticleDOI
TL;DR: In this article, a critical review of the existing data on ionization and dielectronic recombination and present new computations of radiative recombination rate coefficients of Fe(+14) through Fe(−25) using the photoionization cross sections of Clark et al.
Abstract: In the past few years important progress has been made on the knowledge of ionization and recombination rates of iron, an astrophysically abundant heavy element and a major impurity in laboratory fusion devices. We make a critical review of the existing data on ionization and dielectronic recombination and present new computations of radiative recombination rate coefficients of Fe(+14) through Fe(+25) using the photoionization cross sections of Clark et al. (1986). We provide analytical fits to the recommended data (direct ionization and excitation-autoionization cross sections; radiative and dielectronic recombination rate coefficients). Finally we determine the iron ionic fractions at ionization equilibrium and compare them with previous computations as well as with observational data.

633 citations


Journal ArticleDOI
TL;DR: An early version of a physical model was modified and expanded to derive surface solar irradiance from satellite observations as mentioned in this paper, which can produce both direct and diffuse spectral components in the 0.2-4.0-micron interval.
Abstract: An early version of a physical model was modified and expanded to derive surface solar irradiance from satellite observations. The model is based on radiative transfer theory, and can produce both direct and diffuse spectral components in the 0.2-4.0-micron interval. Attention is given to the absorption and scattering processes in the atmosphere and the interaction of radiation with the surface. The bidirectional nature of the exiting radiation at the top of the atmosphere is also accounted for. In this paper the emphasis will be on describing the current status of the model and its implementation on a global scale with the International Satellite Cloud Climatology Project C1 data.

629 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present new radiative Rosseland mean opacity tables calculated with the OPAL code developed independently at LLNL, which allow accurate interpolation in temperature, density, hydrogen mass fraction, as well as metal mass fraction.
Abstract: For more than two decades the astrophysics community has depended on opacity tables produced at Los Alamos. In the present work we offer new radiative Rosseland mean opacity tables calculated with the OPAL code developed independently at LLNL. We give extensive results for the recent Anders-Grevesse mixture which allow accurate interpolation in temperature, density, hydrogen mass fraction, as well as metal mass fraction

594 citations


Journal ArticleDOI
TL;DR: In this article, a 2D hydrodynamics code was proposed to model the dynamics of the wind and shock structure formed by the collision in early-type binary systems, which self-consistently accounts for radiative cooling and represents a significant improvement over previous attempts to model these systems.
Abstract: The dynamics of the wind and shock structure formed by the wind collision in early-type binary systems is examined by means of a 2D hydrodynamics code, which self-consistently accounts for radiative cooling, and represents a significant improvement over previous attempts to model these systems. The X-ray luminosity and spectra of the shock-heated region, accounting for wind attenuation and the influence of different abundances on the resultant level and spectra of X-ray emission are calculated. A variety of dynamical instabilities that are found to dominate the intershock region is examined. These instabilities are found to be particularly important when postshock material is able to cool. These instabilities disrupt the postshock flow and add a time variability of order 10 percent to the X-ray luminosity. The X-ray spectrum of these systems is found to vary with the nuclear abundances of winds. These theoretical models are used to study several massive binary systems, in particular V444 Cyg and HD 193793.

Journal ArticleDOI
TL;DR: In this article, an accurate and efficient convolution of the H2O vapor spectrum with water droplet clouds is presented that yields good agreement with available line-by-line (LBL) calculations for single-layer clouds.
Abstract: Motivated by the desire for a more flexible and general solar radiation calculation in the NCAR community climate model (CCM), the δ-Eddington approximation has been employed in the CCM version 2 (CCM2). Eighteen spectral intervals span the solar spectrum from 0.2 to 5.0 μm. Absorption parameterizations for H2O, O3, CO2, and O2 were developed by making use of the latest theoretical calculations. Water droplet scattering and absorption are parameterized as shown by Slingo (1989). An accurate and efficient convolution of the H2O vapor spectrum with water droplet clouds is presented that yields good agreement with available line-by-line (LBL) calculations for single-layer clouds. A simple and efficient method to simulate partial cloud cover and cloud overlap is included. The simulated albedo-solar zenith angle dependence agrees very well with adding/doubling scattering calculations. The CCM2 δ-Eddington method will make possible many interesting applications of CCM2 in the years to come.

Journal ArticleDOI
TL;DR: In this article, an agreement was achieved between helicopter and satellite measures of surface reflectance to within +/- 1 percent absolute for both the visible and the NIR bands, which translates into a +/- 10 percent error in the estimation in Fpar.
Abstract: Utilizing single-view-angle visible and NIR satellite observations, it was possible to make the necessary radiometric corrections to produce reliable estimates of the simple ratio and the normalized difference vegetation index as inputs to surface evaporation models. Agreement was achieved between helicopter and satellite measures of surface reflectance to within +/-1 percent absolute for both the visible and the NIR bands. A theoretical calculation utilizing radiative transfer models indicates that this translates into a +/-10 percent error in the estimation in Fpar.

Journal ArticleDOI
TL;DR: In this paper, the authors evaluated several procedures, including radiative transfer codes (RTCs) with simulated atmospheres, image-based procedures and dark-object subtraction (DOS), by comparing surface reflectance factors derived from Landsat Thematic Mapper (TM) digital data with low-altitude, aircraft-based measurements for seven dates over a 1-year period.

Book
01 Jan 1992
TL;DR: In this paper, the authors provide a systematic discussion of the transfer of solar and thermal infrared radiation in the atmosphere, and of aspects of cloud processes pertinent to radiative transfer, focusing largely on the physical principles and approximations that are required to develop important topics in atmospheric radiation, cloud physics, and thermal equilibrium.
Abstract: Description The field of atmospheric radiation and remote sensing has seen significant advances in recent years. This monograph addresses those advances and fills the need for an up-todate, cohesive integration of radiative transfer and cloud physics, allowing for the better forcast of weather on medium and long-range levels. The author here offers a systematic discussion of the transfer of solar and thermal infrared radiation in the atmosphere, and of aspects of cloud processes pertinent to radiative transfer. The book focuses largely on the physical principles and approximations that are required to develop important topics in atmospheric radiation, cloud physics, and thermal equilibrium. Aspects of the interactions and feedbacks of radiation-cloud and dynamic and climate processes are discussed using a hierarchy of atmospheric models. Radiometric data obtained from the ground, the air, and space are frequently employed to illustrate physical processes in the atmosphere as well as to cross check theoretical results. The author also cogently discusses the application of the radiative transfer principle to remote sensing of atmospheric and cloud parameters. This book for the first time bridges the gap between cloud-radiation and dynamic processes in the atmosphere. It will be welcomed by meteorlogists, earth scientists, and researchers interested in atmospheric modelling and radiation.

Journal ArticleDOI
TL;DR: Comparisons of results obtained using this microwave scattering model with measurements from deciduous and coniferous forests show good agreement at several frequencies for both like and cross polarizations.
Abstract: A microwave scattering model was developed for layered vegetation based on an iterative solution of the radiative transfer equation up to the second order to account for multiple scattering within the canopy and between the ground and the canopy. The model is designed to operate over a wide frequency range for both deciduous and coniferous forest and to account for the branch size distribution, leaf orientation distribution, and branch orientation distribution for each size. The canopy is modeled as a two-layered medium above a rough interface. The upper layer is the crown, containing leaves, stems, and branches. The lower layer is the trunk region, modeled as randomly positioned cylinders with a preferred orientation distribution above an irregular soil surface. Comparisons of results obtained using this model with measurements from deciduous and coniferous forests show good agreement at several frequencies for both like and cross polarizations. >

Journal ArticleDOI
TL;DR: In this article, a summary of the importance of ion-molecule reactions in the chemistry of diffuse and dense interstellar clouds and a nice overview of the different routes to the synthesis of molecules detected in interstellar clouds is given in this thematic issue of Chemical Reuiews.
Abstract: I . Introduction As was originally discussed by Herbst and Klemperer,' the low density and low temperature of interstellar regions restricts the gas-phase chemistry to exothermic binary reactions without activation energy. Since most of the neutral reactions are hindered by barriers, ionic gas-phase reactions play a dominant role in the synthesis of interstellar molecules. A summary of the importance of ion-molecule reactions in the chemistry of diffuse and dense interstellar clouds and a nice overview of the different routes to the synthesis of molecules detected in interstellar clouds is given in the review by D. Smith in this thematic issue of Chemical Reuiews.2 Since in some important steps in the synthetic pathways not only neupal reactions but also ionmolecule reactions founder because of their endothermicity or a barrier, several groups suggested in the 1970s the possibility of a radiative association process for overcoming such problems.l,*5 One prominent example is the first step in the chain of hydrocarbon chemistry: a collision of C+ with HP cannot lead to CH+ + H since this reaction requires 0.4 eV, but the strongly bound CH2+ excited collision complex can stabilize by the spontaneous emission of a photon. There are many other reaction systems where the intermediate collision

Journal ArticleDOI
TL;DR: In this paper, a solar radiation routine is developed for use in climate studies that includes absorption and scattering due to ozone, water vapor, oxygen, carbon dioxide, clouds, and aerosols.
Abstract: A solar radiation routine is developed for use in climate studies that includes absorption and scattering due to ozone, water vapor, oxygen, carbon dioxide, clouds, and aerosols. Rayleigh scattering is also included. Broadband parameterization is used to compute the absorption by water vapor in a clear atmosphere, and the k-distribution method is applied to compute fluxes in a scattering atmosphere. The reflectivity and transmissivity of a scattering layer are computed analytically using the delta-four-stream discrete-ordinate approximation. The two-stream adding method is then applied to compute fluxes for a composite of clear and scattering layers. Compared to the results of high spectral resolution and detailed multiple-scattering calculations, fluxes and heating rate are accurately computed to within a few percent. The high accuracy of the flux and heating-rate calculations is achieved with a reasonable amount of computing time. With the UV and visible region grouped into four bands, this solar radiation routine is useful not only for climate studies but also for studies on photolysis in the upper atmosphere and photosynthesis in the biosphere.

Journal ArticleDOI
TL;DR: In this article, a two-temperature radiation hydrodynamics code was used to model the UV burst from SN 1987A using a two temperature radiation hydrogrographic code.
Abstract: Detailed models for the UV burst from SN 1987A are calculated using a two-temperature radiation hydrodynamics code The sequence of events which occur during shock breakout is described, including the formation of a radiative precursor, the disappearance of the shock, and the growth of a thin, dense shell above the photosphere which gives rise to a new viscous, isothermal shock Continuum spectra are calculated under the assumption that the emergent radiation is a dilute blackbody with a color temperature equal to the radiation temperature at the thermalization depth This color temperature is 2-3 times higher than the effective temperature Light-travel-time corrections have been included in the light curves and the spectra Quantities most relevant to accurate modeling of the fluorescing ring around SN 1987A are highlighted

Journal ArticleDOI
TL;DR: In this article, it was shown that the broad permitted and semipermitted emission lines of the broad-line region (BLR) of radio-quiet active galactic nuclei (AGN) can be generated by strongly radiative (rapidly cooling) supernova remnants expected to occur in the central regions of early-type galaxies undergoing a starburst.
Abstract: It is shown that the broad permitted and semipermitted emission lines of the broad-line region (BLR) of radio-quiet active galactic nuclei (AGN) can be generated by strongly radiative (rapidly cooling) supernova remnants expected to occur in the central regions of early-type galaxies undergoing a starburst. Supernova remnants interacting with a circumstellar medium with a density of about n ∼ 10 7 cm -3 become strongly radiative while still expanding at several thousand km/s, and miss the adiabatic Sedov track. Radiative cooling becomes important well before the thermalization of the ejecta is completed, and the shocked matter undergoes a fast condensation behind both the outgoing forward shock and the reverse shock. Two concentric, high-density, and fast-moving thin shells are then formed

Journal ArticleDOI
TL;DR: In this paper, the thermal transport in polystyrene (PS) and polyurethane (PU) foam insulations is described, with special emphasis on the radiative transfer, and the total thermal conductivity of PU foams, including condensation effects of the blowing agent R 11, is modeled.

Journal ArticleDOI
TL;DR: In this article, an investigation of the radiative and non-radiative properties of the well known ZBLAN glass composition doped with individual rare earth elements (Pr3+, Nd3+, Dy3, Dy3+, Ho3+, Er3+ and Tm3+) at levels between 0.1 and 4 mol%.
Abstract: An investigation has been made of the radiative and non-radiative properties of the well known ZBLAN glass composition doped with individual rare earth elements (Pr3+, Nd3+, Dy3+, Ho3+, Er3+ and Tm3+) at levels between 0.1 and 4 mol%. The optical absorption and fluorescence spectra of the samples were measured and also the lifetimes for some transitions. Using the Judd-Ofelt model, the radiative decay rates between the different levels, the lifetimes and the branching ratios were calculated. These results and comparison of calculated and measured lifetimes are discussed with respect to radiative and non-radiative rates and quantum efficiencies. The multiphonon rates below 3200 cm−1 are found to be less in ZBLAN than in other host materials (ZBLA, SiO2, YAG).

Journal ArticleDOI
TL;DR: In this paper, a simple one-dimensional radiative-convective equilibrium model is used to investigate the relationship between the surface temperature and the outgoing infrared radiation at the top of the atmosphere.
Abstract: A simple one-dimensional radiative–convective equilibrium model is used to investigate the relationship between the surface temperature and the outgoing infrared radiation at the top of the atmosphere. The model atmosphere has a gray infrared absorption coefficient and is composed of a radiative equilibrium stratosphere and a moist adiabat troposphere. An upper limit of the outgoing infrared radiation is found to exist. The existence of the upper limit is characterized by the radiation limits that appear when the optical depth of the entire atmosphere becomes sufficiently deep and the temperature structure around the levels where the optical depth is about unity approaches a fixed profile. This appearance of an upper limit differs from that found by Komabayashi and Ingersoll, which is obtained from the constraint of the stratospheric radiation balance. As one of those radiation limits, the outgoing infrared radiation has an asymptotic limit as the surface temperature increases. This is caused by ...

Journal ArticleDOI
TL;DR: In this article, the finite volume method is used to solve three-dimensional radiation problems in cylindrical enclosures, where the medium is considered to be gray, and absorption, emission, and either isotropic or nonisotropic scattering are included.
Abstract: This article shows how the finite volume method can be implemented to solve three-dimensional radiation problems in cylindrical enclosures. The medium is considered to be gray, and absorption, emission, and either isotropic or nonisotropic scattering are included. For the special case of axisymmetric radiation, a mapping is described that yields a complete solution by solving the intensity in a single azimuthal direction. The method is shown to rapidly converge to the solution of the radiation transfer equation as the spatial and directional grid is refined. Results from the solution of axisymmetric bench mark problems show that the method is stable, accurate, and computationally efficient. 25 refs.

Journal ArticleDOI
TL;DR: In this paper, a model for predicting the structure and received thermal radiation around a turbulent reacting jet discharging into a cross-flow has been made using a finite-difference scheme for solving the fluid dynamic equations.

Book
24 Sep 1992
TL;DR: In this article, the role of radiation and cloud processes in atmospheric models is discussed, and the model is extended to include cloud processes and radiative transfer in clouds, as well as complex refractive indices of water and ice.
Abstract: Introduction Theory and parameterization of thermal infrared radiative transfer Theory and parameterization of solar radiation transfer Theory, observation and modeling of cloud processes in the atmosphere Radiative transfer in clouds Atmospheres in radiative and thermal equilibrium The role of radiation and cloud processes in atmospheric models Appendices: Physical constants Standard atmospheric profiles Complex refractive indices of water and ice.

Journal ArticleDOI
TL;DR: In this article, the authors reviewed solutions to inverse problems for the estimation of optical properties or the thickness of a medium, or the presence of a spatially distributed source or an obscured object.
Abstract: Recent solutions to inverse problems are reviewed for the estimation of optical properties or the thickness of a medium, or the presence of a spatially distributed source or an obscured object. Tim...

Journal ArticleDOI
TL;DR: In this article, a time-dependent, non-local mixing-length prescription is employed for convection, and the Eddington approximation to radiative transfer is applied to model solar envelopes.
Abstract: It is not currently known what excites solar five-minute oscillations. Of the two most plausible possibilities, thermal overstability and stochastic excitation by turbulent convection, the single most important discriminating factor is the intrinsic stability of the pulsation modes. In view of this fact, the problem of the linear stability of model solar envelopes is addressed. A time-dependent, non-local mixing-length prescription is employed for convection, and the Eddington approximation to radiative transfer. The calculations reveal that low-degree acoustic modes are damped. Moreover, the theoretical damping rates compare well with measurements of solar oscillation line widths. Turbulent pressure fluctuations play a critical role in stabilizing the pulsations

Journal ArticleDOI
Andrea Brignole1
TL;DR: In this article, the radiative corrections to the masses of the neutral Higgs bosons in the minimal supersymmetric standard model were computed by finding the propagator poles, instead of approximating them with the second derivatives of the affective potential.

Journal ArticleDOI
TL;DR: A simple model based on the simplified, monodimensional equation of Schuster for radiative transfer is discussed, which provides a simple way to determine a working illuminated volume in which growth occurs, therefore allowing indentification of kinetic parameters.
Abstract: The study of the interactions between physical limitation by light and biological limitations in photobioreactors leads to very complex partial differential equations. Modeling of light transfer and kinetics and the assessment of radiant energy absorded in photoreactors require an equation including two parameters for light absorption and scattering in the culture medium. In this article, a simple model based on the simplified, monodimensional equation of Schuster for radiative transfer is discussed. This approach provides a simple way to determine a working illuminated volume in which growth occurs, therefore allowing indentification of kinetic parameters. These parameters might then be extended to the analysis of more complex geometries such as cylindrical reactors. Moreover, this model allows the behavior of batch or continuous cultures of cyanobacteria under light and mineral limitations to be predicted. © 1992 John Wiley & Sons, Inc.