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.     

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.     

Analysis of Gas Radiative Transfer Using Box Model and Its Comparison with Gray Band Approximation

On the basis of a wide range survey of various models or treatment methods for the calculation of radiative properties of gases, box model, which is similar to the gray band approximation of spectral band model, was applied to evaluate the gas properties in this paper. In order to compare the accuracy of box model with that of gray band approximation of spectral band models, a typical one-dimensional gas radiation problem was analyzed using discrete ordinate method. Comparing with the widely used gray band approximation of narrow band model or exponential wide band model, box model can well evaluate the radiation source term of the radiative problem. It also has the advantages of simplicity and easy to code, so it is practicable and useful for some complex engineering problems.     

箱带模型结合DOM在非灰气体辐射换热计算中的应用

在总结现有计算非灰气体辐射特性的各种模型与方法的基础上，利用箱带模型与离散坐标法(DOM)相结合的方法计算了一维无限大平行平板间气体介质在均匀型、边界层型及抛物型三种温度分布下的辐射换热。研究表明，箱带模型与常用的统计窄带模型或指数宽带模型的灰带近似计算结果相比具有较好的准确性，且由于箱带模型具有简单、易于编程、计算速度快等优点，对一些较为复杂的工程计算问题是一种较好的可供选择的模型。     

A MONTE CARLO (MC) METHOD APPLIED TO THE MEDIUM WITH NONGRAY ABSORBING-EMITTING-ANISOTROPIC SCATTERING PARTICLES AND GRAY APPROXIMATION

A Monte Carlo (MC) method is applied to calculate radiative transfer in a nongray medium using spectral radiative exchange factor RD ij u . The creditability of the present MC model has been validated by comparing it with the results using other research methods. Meanwhile, the radiative transfer in an isothermal and nonisothermal medium with nongray absorbing-emitting-anisotropic ash particles is calculated by a nongray model and several gray approximation methods. A simplification from a nongray problem to a gray one by Rosseland's mean extinction coefficient, mean albedo y bar 2 , and Planck mean phase function is suggested.     

Nongray inverse boundary design problems in enclosures at radiative equilibrium

In this paper, an inverse analysis is used to find an appropriate heat flux distribution over the heater surface of radiant enclosures, filled with nongray media at radiative equilibrium from the knowledge of desired (prespecified) temperature and heat flux distributions over the given design surface. Regular and irregular 2D enclosures filled with nongray combustive gas products are considered. Radiation is considered the dominant mode of heat transfer and the medium temperature is obtained from the energy equation. To evaluate the nongray behavior of the participating gases properly, the spectral‐line weighted‐sum‐of‐gray‐gases (SLW) model with updated correlations is used. The dependence of absorption coefficients and the weights of the SLW model on the temperature of the medium makes the inverse problem nonlinear and difficult to handle. Here, the inverse problem is formulated as an optimization problem and the Levenberg‐Marquardt method has been used to solve it. The finite volume method is exploited for the discretization of the energy equation and the spatial discretization of the radiative transfer equation (RTE). The discrete ordinates method (TN quadrature) is used for the angular discretization of RTE. Five test cases, including homogeneous and inhomogeneous media, are investigated to prove the ability of the present methodology for achieving the desired conditions.     

Evaluation of emissivity correlations for H2OCO2N2/air mixtures and coupling with solution methods of the radiative transfer equation

This study is directed towards the limitations of applying total emissivity correlations in computational fluid dynamics (CFD) computer codes for flame modeling. The predictions of nine widely applied total emissivity models for H₂O---CO₂ homogeneous mixtures are compared with the exponential wide band model (EWBM) calculations. The comparison covers a range of total pressures, temperatures and path lengths which are suitable for the use of fine numerical grids in CFD simulations of atmospheric and high pressure combustors.

Attention is paid to coupling of the property models with the radiative transfer equation (RTE) and their performance in non-homogeneous applications. In this respect both the total transmittance non-homogeneous (TTNH) model and the spectral group model (SGM) are used. The latter model is combined with five weighted sum of gray gases models (WSGGM), the single line based sum of gray gases model (SLW) and the k-distribution model. The non-homogeneous validation tests used in situ total radiance measurements in two non-luminous natural gas flames representing two industrial situations, a water cooled furnace and a refractory lined furnace. The main conclusions are as follows. The spectral group model provides an elegant and accurate method of coupling WSGGM, k-distribution and SLW property models to the equation of radiative transfer. Both homogeneous and non-homogeneous tests indicate the advantage of using the Smith, Shen and Friedman weighted sum of gray gases model over polynomial correlations and the SLW model. It has been shown that in the near burner region of a natural gas diffusion flame, the water vapor to carbon dioxide partial pressure ratio departs significantly from the value expected for the complete combustion of methane in air. This finding emphasizes the limitation of existing WSGGM to H₂O to CO₂ partial pressure ratios of one and two only.     

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。     

Inverse Boundary Design Problems in Enclosures With Non-Gray Media

In this work, the inverse analysis is applied to radiative heat transfer boundary design problems with non-gray media. The objective of the inverse problem is to find the power of the heaters on the heater surface that produces the desired output, that is, temperature and heat flux distribution over the design surface. The inverse problem is formulated as an optimization problem for minimization of an objective function, which is defined by the sum of the squared difference between estimated and desired heat flux distributions over the design surface. The non-gray optimization problem is solved using the conjugate gradient method, which is a gradient-based optimization method. The spectral line weighted-sum-of-gray-gases model (SLW) is used to account for non-gray gas radiation properties. The radiative transfer equation is solved by the discrete ordinates method combined with two models for simulation of non-gray media. Enclosures with diffuse and gray walls are considered. Radiation is assumed the dominant mode of heat transfer. Example problems including homogeneous/nonhomogeneous, isothermal/nonisothermal media are considered. The results obtained using the SLW model and the gray model are compared.     

SLW model for computational fluid dynamics modeling of combustion systems: Implementation and validation

ABSTRACT

Spectral Line-Based Weighted Sum of Gray Gases (SLW) model was implemented to Computational Fluid Dynamics (CFD) Solver, ANSYS FLUENT. Discrete Ordinate Method (DOM) available in ANSYS FLUENT was used as Radiative Transfer Equation (RTE) Solver. ANSYS FLUENT with SLW was applied to the prediction of incident heat fluxes for three test problems; two containing isothermal homogenous/nonhomogenous water vapor and one isothermal water vapor/carbon dioxide mixture. Predictive accuracy of SLW in ANSYS FLUENT was assessed by benchmarking its predictions against those of ray tracing (RT) with Statistical Narrow-Band (SNB) and Method of Lines (MOL) solutions of DOM with SLW. Comparisons reveal that the results of CFD code are in good agreement with the benchmark solutions. This finding proves that the use of DOM with SLW in CFD codes would provide more accurate solutions in studies involving gas combustion, where accuracy in spectral radiative properties plays dominant role in heat flux predictions.     

Non-gray radiative convective conductive modeling of a double glass window with a cavity filled with a mixture of absorbing gases

Coupled radiation and natural convection heat transfer occurs in vertical enclosures with walls at different temperatures filled with gas media. In glass window thermal insulation applications in hot climates, infrared absorbing gases appear as an alternative to improve their thermal performance. The thermal modeling of glass windows filled with non-gray absorbing gases is somewhat difficult due to the spectral variation of the absorption coefficients of the gases and the phenomena of natural convection. In this work, the cumulative wavenumber (CW) model is used to treat the spectral properties of mixtures of absorbing gases and the radiative transport equation is solved using CW model and the discrete ordinates method. Due to the range of temperature variation, the mixture of gases is considered as homogeneous. The absorption coefficients were obtained from the database HITRAN. First, the natural convection in a cavity with high aspect ratio is modeled using a CFD code and the local and global Nusselt numbers are computed and compared with available empirical correlations. Also, the flow pattern for different Rayleigh numbers is analyzed. Then, the heat transfer in the gas domain is approximated by a radiative conductive model with specified heat flux at boundaries which is equivalent to convective transport at the walls surroundings. The energy equation in its two-dimensional form is solved by the finite volume technique. Three types of gas mixtures, highly absorbing, medium and transparent are investigated, to determinate their effectiveness in reducing heat gain by the gas ambient. Reflective glasses are also considered. The numerical method to solve radiative heat transport equation in gray and non-gray participant media was validated previously. The temperatures distributions in the gas and the glass domain are computed and the thermal performance of the gas mixtures is evaluated and discussed. Also, comparison with pure radiative conductive model is shown.     

Statistical modeling of thermal radiation transfer in buoyant turbulent diffusion flames

A statistical (Monte Carlo) method for radiative heat transfer has been incorporated in CFD modeling of buoyant turbulent diffusion flames in stagnant air and in a cross-wind. The model and the computational tool have been developed and applied to simulate both burner flames with controlled fuel supply rate and in self-sustained pool fires with burning rates coupled with flame radiation. The gas–soot mixture was treated either as gray (using the effective absorption coefficient derived from total emissivity data or the Planck mean absorption coefficient) or as non-gray (using the weighed sum of gray gases model). The comparison of predicted radiative heat fluxes indicates applicability of the gray media assumption in modeling of thermal radiation in case of high soot content. The effect of turbulence-radiation interaction is approximately taken into account in calculation of radiation emission, which is corrected to allow for temperature self-correlation and absorption-temperature correlation. In modeling buoyant propane flames in still air above 0.3 m diameter burner, extensive comparison is presented of the predictions with the measurements of gas species concentrations, temperature, velocity and their turbulent fluctuations, and radiative heat fluxes obtained in flames with different heat release rates. Similar to previously published experimental data, the predicted burning rate of flames above the acetone pools exposed to flame radiation increases with the pool diameter and approaches a constant level for large pool sizes. The magnitude of predicted burning rates is shown to be in agreement with the reported measurements. Augmentation of burning rate of the pool fire in a cross-wind because of increased net radiative heat flux received by the fuel surface and non-monotonic dependence of burning rate on cross-wind velocity, subject to the pool diameter, is predicted. The statistical treatment of thermal radiation transfer has been found to be robust and computationally efficient.     

On the application of the exponential wide band model to the calculation of radiative heat transfer in one- and two-dimensional enclosures

Various implementations of the exponential wide band model (EWBM) are used to model radiative heat transfer in one- and two-dimensional enclosures containing CO₂ and H₂O. These are, first, the original banded approach using the four-region approximation for the total band absorption, second, a numerical integration of the spectral transmittance, and third, the wide band correlated k-distribution method (CKM). A correlated and a non-correlated formulation are used to solve the radiative transfer equation. In two-dimensional enclosures, these formulations are implemented using a ray tracing method (RTM) and the discrete ordinates method (DOM), respectively. The wide band CKM is found to be the best choice concerning accuracy and computational effort.     

Modeling the radiation of anisotropically scattering media by coupling Mie theory with finite volume method

A new mathematical model and code for radiative heat transfer of particulate media with anisotropic scattering for 2-D rectangular enclosure is developed. The model is based on the coupling of (i) finite volume method for the solution of radiative transfer equation with (ii) Mie equations for the evaluation of scattering phase function. It has not been done before to the authors’ best knowledge. The predictions were compared against the only found results, published 15 years ago. For those results the S-N discrete ordinates method for the solution of radiative transfer equation and the Legendre polynomials expansions for the evaluation of scattering phase function were used. The agreement between the results is very good. The advantages of new model and code are in their straight forward application to any given particles parameters without the need for previously designed analytical expression for scattering phase function. In addition, that analytical expression, with generated expansion coefficients, is restricted and can be used only for that particular case of particle parameters. The new model was applied to the solid particles of several various coals and of an ash and the series of 2-D predictions are performed. The effects of particle size parameter and of scattering albedo on radiative heat flux and on incident radiation were analyzed. It was found that the model developed is reliable and very accurate and thus suitable for extension towards: (i) 3-D geometries, (ii) mixtures of non-gray gases with particles as well as for (iii) incorporation in computational fluid dynamics codes.     

EFFECTS OF THERMAL RADIATION ON THE SOUND WAVE PROPAGATION IN A GAS-PARTICLE TWO-PHASE MEDIUM

A study is made of the sound wave propagation through a radiating gas medium that contains solid particles in suspension. The relaxation models are introduced to describe the temporal momentum and thermal nonequilibrium interactions between gas and particles. The gray gas differential approximation is used for radiation. It is found that the radiation induces the attenuation mode, the position of which is varied with the absorption coefficient in addition to the immovable mode by suspended particles. The attenuation because of radiation is greatly influenced by the absorption coefficient of the radiative medium while the dispersion remains almost unchanged. As the absorption coefficient increases, the attenuation mode because of radiation shifts to the higher frequency zone. This radiation effect is significantly reduced as the particle mass loading increases, since the convection becomes much more dominant.     

Models for gaseous radiative heat transfer applied to oxy-fuel conditions in boilers

Models of gas radiation properties have been evaluated for conditions relevant to oxy-fired boilers, characterized by larger pressure path-lengths and possibly different ratios of H₂O/CO₂ compared to air-fired boilers. Statistical narrow band (SNB) models serve as reference. The other radiation models tested are the weighted-sum-of-grey-gases model, the spectral line-based weighted-sum-of-grey-gases model and two grey-gas approximations. The range of validity of the existing coefficients of the weighted-sum-of-grey-gases model is limited, and new coefficients have therefore been determined to cover the conditions of interest. Several assumed test cases, involving both uniform and non-uniform paths, have been studied to evaluate the accuracy of the models. Comparisons with experimental data are also included. The results show that a grey approximation can give accurate wall fluxes, but at the expense of errors in the radiative source term. The weighted-sum-of-grey-gases model with the new coefficients yields predictions within 20% of those of the reference model in most cases, while the spectral line-based weighted-sum-of-grey-gases model usually gives results within 10%. There are, however, discrepancies between the SNB models at high temperatures. The weighted-sum-of-grey-gases model with its low computational cost is recommended for computationally demanding applications where predictions of both wall fluxes and the radiative source term are important.     

Heat transfer model of semi-transparent ceramics undergoing laser-assisted machining

A three-dimensional, unsteady heat transfer model has been developed for predicting the temperature field in partially-stabilized zirconia (PSZ) undergoing laser-assisted machining. The semi-transparent PSZ is treated as optically thick within a spectral band from approximately 0.5 to 8 μm. After comparing the diffusion approximation and the discrete ordinates method for predicting internal radiative transfer, suitability of the diffusion approximation is established from a comparison of model predictions with surface temperature measurements. The temperature predictions are in good agreement with measured values during machining. Parametric calculations reveal that laser power and feedrate have the greatest effect on machining temperatures.     

Infrared radiative heat transfer in nongray nonisothermal gases

In the present paper, both nongray and nonisothermal behaviors of an infrared emittingabsorbing gas have been taken into account in radiative transfer analyses through the use of the nonisothermal band absorptance. Consideration is given specifically to a simple system consisting of a radiating medium bounded by two infinite parallel black surfaces of different temperatures. Solutions are presented for the cases of radiative equilibrium and combined conduction and radiation. Results based on different methods of evaluating the nonisothermal band absorptance are also compared among themselves. Differences in several fundamental features are exhibited in the nongray nonisothermal solutions as compared to those with nongray but isothermal properties.     

The Re-Plating Algorithm for Radiation Total Exchange Area Calculation

Great efforts have been made to date toward modeling nongray radiative heat transfer accurately. In this article, a new version of the plating algorithm, designated the re-plating algorithm, for total exchange areas (TEAs) calculation from direct exchange areas (DEAs) for nongray radiative problems is presented. The re-plating algorithm calculates TEAs for a given band number b from those of band number b ? 1 by performing successive re-plating procedures. The effectiveness of the new algorithm is demonstrated for thermal modeling of an aluminum brazing furnace and a glass treatment furnace. CPU requirements for TEA calculation were reduced significantly.     

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.