Modeling the 3-D radiation of anisotropically scattering media by two different numerical methods

An original model and code for 3-D radiation of anisotropically scattering gray media is developed where radiative transfer equation (RTE) is solved by finite volume method (FVM) and scattering phase function (SPF) is defined by Mie Equations (ME). To the authors’ best knowledge this methodology was not developed before. Missing the benchmark, another new 3-D model and code, which solve the same problems, based on a combination of zone method (ZM) and Monte Carlo method (MC), as a solution of RTE, is developed. Here SPF is also calculated by Mie Equations. The conception ZM + MC is numerically expensive and is used and recommended only as a benchmark. The 3-D rectangular enclosure and the spherical geometry of particles are considered. The both models are applied: (i) to an isotropic and to four anisotropic scattering cases previously used in literature for 2-D cases and (ii) to solid particles of several various coals and of a fly ash. The agreement between the predictions obtained by these two different numerical methods for coals and ash is very good. The effects of scattering albedo and of wall reflectivity on the radiative heat flux are presented. It was found that the developed 3-D model, where FVM was coupled with ME, is reliable and accurate. The methodology is also suitable for extension towards: (i) mixture of non-gray gases with particles and (ii) incorporation in computational fluid dynamics.     

3D modelling of radiative heat transfer in circulating fluidized bed combustors: influence of the particulate composition

A three-dimensional model is developed to predict the bed-to-wall radiative heat transfer coefficient in the upper dilute zone of circulating fluidized bed (CFB) combustors. The radiative transfer equation is solved by the discrete ordinates method and Mie scattering theory is applied to calculate the absorption and scattering efficiency factors of particles existing in CFB combustors. Empirical correlations calculate both spacial variation of solid volume fraction and temperature distribution at the wall. The model considers the influences of the particle properties (including particle size distribution, particle optical constants and solid composition) on the radiative heat transfer coefficient. Simulation results show that the particle properties have significant influences on the bed-to-wall radiative heat transfer coefficient in CFB combustors. A very good agreement of predicted results is shown with experimental data.     

燃烧产物辐射特性的误差对炉内辐射传热计算精度的影响

本文用离散坐标法对含吸收散射性介质矩形空腔内的３维辐射传递过程进行了模拟，并编写了相应的数值计算程序。利用该程序分析了介质的吸收系数、散射系数、相函数、光谱特性及壁面灰渣沉积层黑度的不确定性对矩形燃烧室内烟气温度及热流计算精度的影响。结果表明计算精度很大程度上取决于燃烧产物辐射特性的取值精度，特别是壁面灰渣沉积层黑度的取值精度。在煤粉燃烧室中，介质的散射不宜忽略。     

Assessment of gas radiative property models in the presence of nongray particles

In this study, a radiation code based on the method of lines solution of the discrete ordinates method for the prediction of radiative heat transfer in nongray gaseous media is developed by incorporation of two different spectral gas radiative property models, banded spectral line-based weighted sum of gray gases (banded SLW) and gray wide band (GWB) approximation in the presence of nongray absorbing–emitting–scattering particles. The aim is to introduce an accurate and CPU efficient spectral gas radiation model, which is compatible with spectral fuel/ash particle property models. Input data required for the radiation code and its validation are provided from two combustion tests previously performed in a 300 kWt atmospheric bubbling fluidized bed combustor test rig burning low calorific value Turkish lignite with high volatile matter/fixed carbon (VM/FC) ratio in its own ash. The agreement between wall heat fluxes and source term predictions obtained by global and banded SLW models reveal that global SLW model can be converted to an accurate wide band gas model (banded SLW) which can directly be coupled with spectral particle radiation. Furthermore, assessment of GWB approximation by benchmarking its predictions against banded SLW model shows that GWB gives reasonable agreement with a higher CPU efficiency when the particle absorption coefficient is at least in the same order of magnitude with the gas absorption coefficient.     

Modeling of the radiative contribution to heat transfer in porous media composed of spheres or cylinders

A theoretical model for evaluating the radiative conductivity tensor of a porous media is developed in this paper. The porous media is composed of a transparent fluid and opaque particles with characteristic lengths longer than the radiation wavelength. The main features of the proposed approach are (i) take into account the interaction between conduction and radiation heat transfers, (ii) allow the modeling of the radiative transfer in anisotropy system such as an assembly of cylinders, and (iii) have an easy numerical implementation into the energy equations of the porous media. In order to study the accuracy of the approach, the paper evaluates the model for porous media composed of spheres or cylinders. The predictions of the model agree well with experimental data and with results obtained from finite element simulations. The numerical results also show that the radiative conductivity can be strongly influence by the effect of temperature distribution across the particle surface and by the effect of the multiple scattering of radiation in the porous media.     

TRANSIENT RADIATION HEAT TRANSFER WITHIN A NONGRAY NONISOTHERMAL ABSORBING-EMITTING-SCATTERING SUSPENSION OF REACTING PARTICLES UNDERGOING SHRINKAGE

ABSTRACT

A nonisothermal, nongray, absorbing, emitting, and anisotropically scattering suspension of reacting particles exposed to concentrated thermal radiation is considered. The steam gasification of coal is selected as the model thermochemical reaction. The unsteady energy equation that couples the radiative heat flux with the chemical kinetics is solved by means of a numerical model that incorporates Monte Carlo ray tracing, the finite-volume method, and an explicit Euler time integration scheme. Two modeling approaches are applied: (1) a quasi-continuous model that assumes a homogeneous medium and utilizes its macroscopic radiative properties (absorption and scattering efficiencies and scattering phase function), and (2) a particle-discrete model that assumes an ensemble of randomly positioned particles and traces the interaction of radiation with each particle by geometric optics. Temperature profiles and reaction extent are computed using both approaches. The quasi-continuous approach is superior in accuracy at the expense of lower spatial resolution, while the particle-discrete approach gives detailed information for every single particle in the suspension at the expense of larger stochastic errors.     

An application of Spectral line-based weighted sum of grey gases (SLW) model with geometric optics approximation for radiative heat transfer in 3-D participating media

A three-dimensional radiation code based on method of lines (MOL) solution of discrete ordinates method (DOM) coupled with spectral line-based weighted sum of grey gases (SLW) model and geometric optics approximation for particles is developed and its predictive ability is tested by applying it to the freeboard of a 0.3 MW_t Atmospheric Bubbling Fluidized Bed Combustor (ABFBC) containing a non-grey, absorbing, emitting and isotropically scattering particle laden flue gas and comparing its predictions with measurements and former predictions obtained by the grey gas model with Mie theory for particles. The MOL of DOM with SLW and geometric optics assumption are found to provide more accurate solutions for incident radiative heat flux than grey gas model with Mie theory particularly for high particle loading. Parametric studies are also carried out to investigate the effect of size parameter and presence of particles on fluxes. MOL–SLW predictions are found to be sensitive to both the size parameter and particle load.     

Conduction and radiation in a rectangular isotropic scattering medium with black surfaces by the RTNAM

The ray tracing-node analyzing method (RTNAM) has been successfully developed to solve 1-D coupled heat transfer in isotropic and anisotropic scattering media in the past, and in this paper it is further extended to solving the 2-D coupled heat transfer in a rectangular isotropic scattering medium. Using the control-volume method, the partial transient energy equation is discretized in implicit scheme. The effect of radiation on heat transfer is considered as a radiative source term (RST) in the discretized energy equation, and in combination with spectral band model, the RST is calculated using the radiative transfer coefficients (RTCs), which are deduced by the ray tracing method. The Patankar’s linearization method is used to linearize the RST and the opaque boundary condition, and the linearized equations are solved by the ADI method. Before solving the RTCs for isotropic scattering media, the RTCs without considering scattering must be solved at first. And then, the RTCs without considering scattering are normalized according to their integrality relationships. In addition, the correctness of the results obtained by the RTNAM is validated, and effects of scattering albedo and refractive index on transient temperature distribution are investigated.     

Numerical simulation of laser irradiation to a randomly packed bimodal powder bed

Bimodal packing arrangements can be used to overcome the balling phenomena in selective laser sintering (SLS). However, little attention has been paid to the effects of bimodal packing structures on radiative transfer in the SLS powder beds. In this study, a sequential addition packing algorithm is firstly employed to generate 3-D random packing of opaque, diffusively or specularly reflecting spherical particles with the same or different sizes. Then, a Monte Carlo based ray tracing algorithm is formulated, for the first time, to simulate the radiative transfer in the bimodal random packing structures composed of particles with different sizes and emissivities. The credibility of the computer code is verified with published experimental data. A comparison is also made between the calculating results and those obtained by two-flux model. By using the present algorithm, the radiative heat fluxes at both levels of particle and entire bed as well as the transmitted laser energy are statistically evaluated. The influences of bimodal size distribution and particle surface emissivity on the radiative transfer process are examined. Such information is expected to be helpful for optimizing the SLS process.     

Tomography based pore-level optimization of radiative transfer in porous media

A non-energy-partitioning Monte Carlo Ray Tracing (MCRT) model is employed to optimize radiative transfer in porous media. The pore level geometry is incrementally modified using 3D equivalents of image manipulation algorithms such as erosion, dilation, opening, and closing. Subsequently, direct, pore-level analysis of radiative transfer is carried out for each modification step to optimize the pore-level geometry for maximum absorptance. Results have been obtained for an opaque, diffusely or specularly reflecting solid phase within a non-participating void phase. Model media studied are: (i) reticulate porous ceramics (RPCs) and (ii) packed beds of CaCO₃ particles. The extinction coefficient and the forward scattering fraction have been determined for the media via a two-flux model of radiative transfer. Optimum porosities for maximizing absorptance at given medium thicknesses are then obtained from the analytical model. For the RPC, the forward scattering fraction varies between 0.38 and 0.57, and the extinction correlation coefficient varies between 9.56 and 7.03. For the packed CaCO₃ particle bed, the forward scattering fraction varies between 0.6 and 0.72, and the extinction coefficient varies between and 2.84 and 2.14.     

Physico-chemical processes occurring inside a degrading two-dimensional anisotropic porous medium

A lumped-parameter kinetic model is applied to simulate the pyrolysis of lignocellulosic particles, exposed to a high temperature environment. Physical processes account for radiative, conductive and convective heat transport, diffusion and convection of volatile species and pressure and velocity variations across a two-dimensional (2-D) , anisotropic, variable property medium. The dynamics of particle degradation are found to be strongly affected by the grain structure of the solid. A comparison is made between the total heat transferred to the virgin solid (conduction minus convection) along and across the grain. Notwithstanding the lower thermal conductivities, because of the concomitant slower convective transport (lower gas permeabilities) , the largest contribution is that across the solid grain. The role played by convective heat transport is successively less important as the particle size is increased. Finally, the 2-D and the widely applied one-dimensional (1-D) predictions are compared.     

Investigation on the effects of fly ash particles on the thermal radiation in biomass fired boilers

Ash is produced in combustion of biomass. Some part of this matter is called fly ash and is carried by the flow and causes not only air pollution and erosion, but also can affect the thermal radiation. The effects of fly ash particles on the thermal radiation are considered in this investigation. By analyzing sampled data in an electrostatic filter, a realistic particle size distribution is found. Although the optical data on biomass fly ash are not available, however, similarity between coal and biomass ash compositions showed that the optical constants of the low Fe coal fly ash can be applied for the biomass fly ash. The Mie theory is used to predict scattering and absorption coefficients and phase function. The mean Planck scattering and absorption coefficients and phase function are predicted by averaging over the particle size distribution and Planck function, respectively. The effects of fly ash particles on thermal radiation are evaluated by a three-dimensional test case. It is assumed that the medium is a mixture of non-grey gases and different level of particle loading. Predicted results from the test case showed that the fly ash can be influential on the thermal radiation. In addition, in selected fly ash volume fractions, the effect of scattering by particles is not so important on the radiative heat source and radiative heat flux to the wall whereas their absorption effect is important and can increase the radiative heat source and wall heat fluxes.     

Application of a Particle Swarm Algorithm for Parameter Retrieval in a Transient Conduction-Radiation Problem

This article addresses the application the particle swarm optimization (PSO) algorithm as an optimization tool for retrieval of parameters in a combined mode 1-D transient conduction-radiation heat transfer problem. In the chosen problem, the participating medium is absorbing, emitting, and scattering. The boundaries are taken to be diffuse gray. In both direct and inverse methods, the energy equation is solved using the lattice Boltzmann method (LBM) and the finite volume method (FVM) is used to compute the radiative information. In the inverse method, the objective function is minimized using the PSO algorithm. The objective function considered in the inverse formulation is an error function evaluated with the exact and inverse temperature fields for the simultaneous retrieval of the extinction coefficient and the scattering albedo. The inverse analysis constituted the effect of measurement errors on solution efficacies. In addition, the effect of important PSO parameters such as swarm size, inertia factor and constriction factor on the parameter retrieval is considered. For the chosen problem, it is found that the PSO with 20 discrete particles and 50 iterations is adequate for accurate parameter retrieval. The PSO has been found to provide a better accuracy than the genetic algorithm.     

On the discrete ordinates method for radiative heat transfer in anisotropically scattering media

In the discrete ordinates method (DOM), the normalized condition for the numerical quadrature of some complex scattering phase functions may not be satisfied. In this paper, a revised discrete ordinates method (RDOM) is developed to overcome this problem, in which a renormalizing factor is added into the numerical quadrature of in-scattering term. The RDOM is used to solve the radiative transfer problem in one-dimensional anisotropically scattering media with complex phase function. The radiative heat fluxes obtained by the RDOM are compared with those obtained by the conventional discrete ordinates method (CDOM) and Monte Carlo method. The results show the RDOM can overcome the false scattering resulted from the numerical quadrature of in-scattering term and improve largely the accuracy of solution of the radiative transfer equation by comparison with the CDOM.     

A formal averaging procedure for radiation heat transfer in particulate media

Radiative heat transfer in participating particulate media is modeled using a formal volume averaging procedure. The multiphase medium is composed of emitting-absorbing-scattering phases, i.e., a gas phase and several particle phases. Each particle phase contains large, opaque, gray, diffuse, and spherical particles having locally the same geometrical, thermophysical, and radiative properties. The resulting multiphase radiative transfer equation (MRTE) is solved using the discrete ordinates method. The present computed results are found to be in good agreement with those obtained using the Monte-Carlo theory and with the available experimental results. The coupling effect of the MRTE with the averaged energy equations in a three-dimensional cavity which is differentially heated or which contains a volumetric heat source is studied. A parametric study is performed for particle-phase and gas properties, and wall emissivity.     

Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction-radiation problem

A lattice Boltzmann method (LBM) is used to solve the energy equation in a test problem involving thermal radiation and to thus investigate the suitability of scalar diffusion LBM for a new class of problems. The problem chosen is transient conductive and radiative heat transfer in a 2-D rectangular enclosure filled with an optically absorbing, emitting and scattering medium. The energy equation of the problem is solved alternatively with a previously used finite volume method (FVM) and with the LBM, while the radiative transfer equation is solved in both cases using the collapsed dimension method. In a parametric study on the effects of the conduction-radiation parameter, extinction coefficient, scattering albedo, and enclosure aspect ratio, FVM and LBM are compared in each case. It is found that, for given level of accuracy, LBM converges in fewer iterations to the steady-state solution, independent of the influence of radiation. On the other hand, the computational cost per iteration is higher for LBM than for the FVM for a simple grid. For coupled radiation-diffusion, the LBM is faster than the FVM because the radiative transfer computation is more time-consuming than that of diffusion.     

Experimental investigation of the two-phase theory in a fluidized-bed combustor

Though the two-phase theory of fluidization is well-accepted, no direct experimental measurements of the different gas concentrations predicted to occur in bubble and particulate phases could be found in the literature. For the first time, theoretical predictions of these different gas concentrations have been validated experimentally, using a combined oxygen/bubble probe. Based on the two-phase theory, a mathematical model was developed for the combustion of a batch of char particles in a fluidized-bed combustor. The experimental oxygen concentration in the particulate phase as a function of time was well predicted by the model. Slight discrepancies for the bubble phase values were eliminated when low-oxygen-concentration bubbles were excluded from the data, attributed to some char combustion occurring in bubbles being contrary to the model assumption. The temperature difference between char and bed particles (ΔT) was the only adjustable parameter in the model. A value of 20°C fitted the burnoff times measured by visual observation of the top of the bed, for both 5 and 10 g char batch masses. Model predictions of the oxygen concentrations were not sensitive to ΔT during the first half of burnoff, when mass transfer controlled the combustion rate, so the mass transfer processes were predicted correctly by the model effectively with no adjustable parameter. The ΔT value of 20°C was significantly lower than experimental measurements of maximum burning char particle temperatures, reported to be 70°C for the small-diameter bed particles used in this work. The discrepancy was attributed to two factors: (i) the decrease in char particle temperature towards the end of the burnoff, when kinetics significantly affected the combustion rate; and (ii) a lower char particle temperature in the particulate phase than in the bubble phase, with experimental char particle temperature measurements biased towards the higher bubble phase values. It was inferred: (i) that the maximum values of ΔT measured experimentally are too high for calculation of the char particle combustion rate during the kinetic-controlled latter stage of burnoff and (ii) that reported values of the heat transfer coefficient from burning char particles to the particulate phase deduced from these particle temperature measurements may have been underestimated.     

Investigation of radiative contribution in a high temperature fluidized-bed using the alternate-slab model

The modified alternate-slab model of Gabor is examined for the prediction of radiative contribution to the total heat transfer from a high temperature fluidized-bed system of air-sand to an immersed surface. The results are compared with the predictions of other models and experimental data on average heat transfer coefficient, and percentage radiative contribution as a function of various influencing parameters. The heat transfer coefficients are overestimated by the model within reasonable limits and approach the experimental data for high values of heat transfer surface temperature. The percentage radiative contribution is substantial for large values of particle diameter, surface and bed temperatures. The model is found reliable and simple to handle over a wide temperature range. Results are also presented for air-ash and air-dolomite systems in view of their practical significance.     

Analysis of Radiative Transport in a 2-D Cylindrical Participating Medium Subjected to Collimated Radiation

This article deals with the numerical analysis of radiative transport in a 2-D axisymmetric cylindrical enclosure containing absorbing, emitting, and scattering medium. The participating medium receives collimated radiation from the top boundary of the enclosure. Attenuation of the collimated radiation in the medium gives rise to the diffuse radiation. Thus, the governing radiative transfer equation accounts for both collimated and diffuse radiation. The radiative transfer equation is solved using the modified discrete ordinate method. Effects of extinction coefficient, scattering albedo, and aspect ratio on radial and axial distributions of heat flux and incident radiation are studied. In all cases, results are validated against those available in the literature. Modified discrete ordinate method has been found to provide accurate results.     

Benchmarking grey particle approximations against nongrey particle radiation in circulating fluidized bed combustors

Investigation of the effect of grey/nongrey particle property models on radiative heat fluxes and source terms is performed in the dilute zone of the lignite-fired 150?kW Middle East Technical University circulating fluidized bed combustor test rig. Predictive accuracy and computational economy of several grey particle models, geometric optics approximation (GOA) with average particle reflectivity (GOA2), GOA with Fresnel solution for particle reflectivity (GOA3), and Planck mean particle properties from spectral Mie solution are tested by benchmarking their predictions against spectrally banded solution of radiative transfer equation (RTE). Comparisons reveal that all grey models lead to accurate and CPU efficient radiative heat flux predictions. On the other hand, only GOA3 and Planck mean properties are in favorable agreement with the benchmark solution for both incident fluxes and source terms. These findings indicate that grey particle approximation with GOA3 is a more practical choice in solution of RTE as it eliminates the need for spectral calculations.