Nongray Medium Research Articles

Improved Gold-SA algorithm for simultaneous estimation of temperature-dependent thermal conductivity and spectral radiative properties of semitransparent medium

Simultaneous estimation of temperature-dependent thermal conductivity and spectral radiative properties in semitransparent medium is investigated in this work. The finite volume method (FVM) is used to solve the transient coupled radiative-conductive problem in one-dimensional emitting, absorbing and scattering nongray medium in the forward model. The time-varying temperature and exit radiative intensity on the right boundary are used as the measurement information for the inverse analysis. A novel Golden Sine Algorithm (Gold-SA) is introduced and the improved Gold-SA (IGold-SA) is developed to solve the inverse problems of coupled radiation-conduction heat transfer for the first time. Findings show that the proposed IGold-SA has better performance than the original Gold-SA in terms of global searching ability and searching accuracy. A multi-thickness-combination model based on laser irradiation is proposed to obtain more measurement information. All the retrieval results show that the temperature-dependent thermal conductivity, spectral extinction coefficient and spectral scattering albedo based on five-band model can be accurately reconstructed by the IGold-SA combined with a three-thickness-combination model even with measurement error.

Gray and non-gray simulations of the combined conduction and radiation heat transfer in a complex enclosure utilizing FSK method considering the scattering influences

This research simulates the conduction-radiation problem in a complex enclosure for the gray/ non-gray assumptions with the presence of scattering influences to improve energy efficiency of thermal systems. The enclosure bottom wall includes an inclined recess with two diagonal walls. To convert these diagonal surfaces to the regular rectangular surfaces in Cartesian coordinates, the Blocked-off method is used. The air containing 10% CO2 and 20% H2O is regarded as participating medium in the enclosure. The absorption coefficient of this mixture is obtained across the spectrum using the information of HITRAN2008 database. Simulating the non-gray medium is done by employing the full-spectrum k-distribution (FSK) approach, whilst the gray medium simulation is performed utilizing the Planck mean absorption coefficient. The finite volume and discrete ordinates methods are respectively applied to calculate the energy and radiation transport equations. Impacts of surface emissivity on conductive, radiative and total heat fluxes are examined in details.

Point mean beam length, a new concept to enhance the computational efficiency of multidimensional, non-gray radiative heat transfer

A new concept of point mean beam length (PMBL) is introduced. For enclosures with simple geometry, this concept provides a fundamental self-consistent interpretation on the various different definition of the conventional mean beam length. The concept is further demonstrated to be effective in enhancing the computational efficiency for multidimensional radiative heat transfer in non-gray media. In the evaluation of radiative exchange between two perpendicular areas with a common edge, the use of PMBL leads to a factor of 100 to 400 reduction in computational effort compared to the direct integration approach. For practical applications, PMBL is combined with RADNNET (a neural network correlation for a one-dimensional CO2/H2O/soot combustion mixture) to generate two highly efficient and accurate solvers for the evaluation of exchange factors between two parallel or perpendicular rectangular areas of arbitrary dimensions with an intervening combustion mixture.

Benchmark Solutions of Three-Dimensional Radiative Transfer in Nongray Media Using Line-by-Line Integration

Abstract Nongray gas radiation calculations are conducted for four three-dimensional benchmarks using line-by-line (LBL) integration with the up-to-date high-resolution spectroscopic database HITEMP 2010. The radiative transfer equation (RTE) is solved using the finite volume method (FVM) over each wavenumber interval of the spectrum. A detailed mesh quality analysis assured the mesh independence of the solution. Accurate results for distributions of volumetric radiative heat source term and wall radiative heat flux are provided for four cases: (i) an isothermal pure water vapor medium at 1000 K; (ii) an isothermal and nonhomogeneous H2O–N2 mixture at 1000 K; (iii) a nonisothermal and homogeneous CO2–H2O–N2 mixture; and (iv) a nonisothermal and nonhomogeneous CO2–H2O–N2 mixture. These data can be useful to assess the accuracy of gas radiative property models.

Analysis of spectral radiative entropy generation in a non-gray planar participating medium at radiative equilibrium with two different boundary conditions

In the present study, spectral radiative entropy generation analysis is presented for radiative heat transfer between two parallel plates containing non-gray absorbing and emitting medium at radiative equilibrium. In order to evaluate the effect of boundary condition (BC) on the entropy generation, two configurations of BC are considered. In the first configuration, two bounding walls are assumed to be at prescribed temperatures. In the second configuration, one wall is assumed to be at a prescribed temperature and the other wall is assumed to have prescribed heat flux BC. The discrete ordinates method and the finite volume method are respectively used for the angular and spatial discretization of the radiative transfer equation and radiative entropy generation equations. While the correlated-k (CK) non-gray model is used to evaluate spectral radiative properties. Five cases are considered, including two homogeneous cases and two non-homogeneous cases with a single participating gas (i.e. H2O or CO2) and one homogeneous case with a mixture of H2O, CO2 and soot particles. Spectral, volumetric and total radiative entropy generations are calculated for each BC configuration and case. Also, the effects of the concentration of participating gas, wall emissivity and total gas pressure on radiative entropy generation are attentively analyzed. The results show that irreversible radiative transfer at the wall with lower temperature is the main source of radiative entropy generation in the system. Soot particles increase radiative entropy generation due to absorption and emission process but decrease radiative entropy generation due to walls radiation. The radiative entropy generation is approximately constant when the concentration of participating gas varies for both BC's. Total radiative entropy generation for the system with the prescribed temperature at walls significantly increases with the increase of wall emissivity; conversely, for the system with mixed BC, total radiative entropy generation linearly decreases. Furthermore, for the system with the prescribed temperature at the walls, radiative entropy generation slightly decreases when medium total pressure increases; while for the problems with mixed BC, increasing medium total pressure does not significantly effect on radiative entropy generation. Finally, results demonstrate that the wall emissivity has a considerable effect on the radiative entropy generation for radiative heat transfer in the non-gray medium in radiative equilibrium.

Nongray inverse boundary design problems in enclosures at radiative equilibrium

AbstractIn 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.

Line by line benchmark solutions for radiative heat transfer in 2D irregular enclosures with non-gray media

Thermal radiation heat transfer in two-dimensional complex (irregular) enclosures containing non-gray gases, namely CO2/H2O/N2 mixtures, is calculated. The meshless Natural Element Method (NEM) is used for spatial discretization and the Discrete Ordinates Method (DOM) is employed for angular discretization of the radiative transfer equation. The Line-by-Line (LBL) and the correlated-k model (CK) models are used to model the spectral radiative properties of the participating gases. To obtain the most accurate results, the updated HITEMP spectroscopic database i.e. HITEMP2010 is employed in LBL calculations and the updated parameters (based on the HITEMP2010) is used in the CK calculation. The objectives of this study are threefold: (1) to combine a meshless natural elements method with LBL and the CK spectral models for the solution of radiative transfer problems in irregular (complex) enclosures, (2) to evaluate the accuracy of the CK model by comparing its results against the LBL results, and (3) to provide benchmark solutions for radiative heat transfer in irregular enclosures with the HITEMP2010 as the most accurate spectral database at high temperatures. Calculations are carried out for isothermal/non-isothermal and homogeneous/inhomogeneous mixtures of CO2/H2O/N2 gases. The present LBL results can serve as benchmark results for validating other approximate models in gas radiation heat transfer calculations in 2D complex enclosures. Results show that the CK results are in good agreement with those of the LBL model.

Development and application of a simplified radiative transport equation in water curtain systems

In this study, a simplified radiative transfer equation (RTE) for water curtain is introduced. The scattering phase function, asymmetric factor, single scattering albedo, and efficiency factors are investigated with non-gray and gray medium, respectively. The prediction values by a simplified equation are compared with previous results by the two-flux, P3 and F9 methods in non-gray medium. The deviations of reduced intensities are less than 10%. As the droplet size increases more than 100 μm, the asymmetry factor, scattering phase function, and scattering albedo are converged as 0.975, 0.992 and 0.501, respectively. Thus, a more simplified equation for the reduced intensity of radiation can be determined with the exponential function of optical depth.To correlate the relations between the flow characteristics of water curtain and the optical depth, the droplet mean diameter and the number density are analyzed with a modified gamma distribution function. In addition, the FTIR spectrometer is calibrated to obtain the spectral intensity in the thermal infrared and large droplet region.Finally, the important engineering parameters such as the mass flow rate, the droplet mean diameter, the number density, the optical depth, and the reduced intensity are systematically investigated. It can be predicted that optical depth (τ > 0.89), number density (N > O(4 × 106 m−3)), droplet mean radius (rm < 180 μm) and flow rate (Qflow > 1.2 kg/s) are determined to reduce the radiation intensity less than 50% for the selected water curtain nozzle in this study.

A line-by-line hybrid unstructured finite volume/Monte Carlo method for radiation transfer in 3D non-gray medium

A hybrid method combing the unstructured finite volume method and the Monte Carlo method and incorporating the line-by-line model has been developed to simulate the radiative transfer in highly spectral and inhomogeneous medium. In this method, the unstructured finite volume method is adopted to solve the spectral radiative transfer equation at wave numbers or spectral locations determined by the Monte Carlo method. The Monte Carlo method takes effects by firstly defining the monotonic random number relations corresponding to the spectral emitted power density of every discretized element of the concerning medium, and then by reversing the spectral location through comparison of these relations with predefined random numbers. Through this Monte Carlo method, the actual number of spectral locations on which the spectral radiative transfer equations are solved may be reduced: only the spectral locations that have higher spectral emissive powers would be more possibly selected. To increase the performance of the presented method, the total variation diminishing scheme on unstructured grids is adopted in treating the spectral radiative intensity at interface between control volumes. And, the discretized radiative transfer equation is implicitly and iteratively solved by an algebraic multi-grid solution approach to accelerate the convergence of the equation. The presented method was applied to 3D homogeneous and inhomogeneous cases for the validation and performance studies. Results show that for both cases, the presented method agree well with pure Monte Carlo benchmark solutions with acceptable number of spectral locations and computing time.

Numerical analysis of radiative heat transfer in an inhomogeneous and non-isothermal combustion system considering H2O/CO2/CO and soot

PurposeH2O, CO2 and CO are three main species in combustion systems which have high volume fractions. In addition, soot has strong absorption in the infrared band. Thus, H2O, CO2, CO and soot may take important roles in radiative heat transfer. To provide calculations with high accuracy, all of the participating media should be considered non-gray media. Thus, the purpose of this paper is to study the effect of non-gray participating gases and soot on radiative heat transfer in an inhomogeneous and non-isothermal system.Design/methodology/approachTo solve the radiative heat transfer, the fluid flow as well as the pressure, temperature and species distributions were first computed by FLUENT. The radiative properties of the participating media are calculated by the Statistical Narrow Band correlated K-distribution (SNBCK), which is based on the database of EM2C. The calculation of soot properties is based on the Mie scattering theory and Rayleigh theory. The radiative heat transfer is calculated by the discrete ordinate method (DOM).FindingsUsing SNBCK to calculate the radiative properties and DOM to calculate the radiative heat transfer, the influence of H2O, CO2, CO and soot on radiation heat flux to the wall in combustion system was studied. The results show that the global contribution of CO to the radiation heat flux on the wall in the kerosene furnace was about 2 per cent, but that it can reach up to 15 per cent in a solid fuel gasifier. The global contribution of soot to the radiation heat flux on the wall was 32 per cent. However, the scattering of soot has a tiny influence on radiation heat flux to the wall.Originality/valueThis is the first time H2O, CO2, CO and the scattering of soot were all considered simultaneously to study the radiation heat flux in combustion systems.

Inverse Estimation of Main Parameters of Spectral Line-Based Weighted Sum of Gray Gases Model With Few Gray Gases to Simulate the Radiation in Nongray Media

The aim of this study is to present a reduced spectral line-based weighted sum of gray gases (SLW) model to simulate the radiation heat transfer in nongray media at high temperatures. Inverse approach is used to divide the absorption cross section band into a clear gas with one gray gas and two gray gases, which are called the S-1 and S-2 approaches, respectively. The unknown absorption cross sections are determined from the knowledge of measured total incident intensities received by wall surfaces. In order to simulate the exact solution of radiation heat transfer in nongray gaseous media, the discrete transfer method (DTM) in combination with S-20 model is used, where the nongray medium is replaced with a set of a clear gas and 20 gray gases. The inverse problem is formulated as an optimization problem to minimize a least square objective function, which is solved by the conjugate gradient method (CGM). The accuracy of the present method is verified by comparing with previous researches and the S-20 approach with a large number of gray gases. The effects of noisy data on the inverse solution are investigated by considering an extreme case with large measurement error. The results show that the unknown absorption cross sections are retrieved well, even for noisy data.

Inverse Design for Radiative Heat Source in Nongray Media

An optimization technique is performed to obtain the unknown heat source distribution to produce the known temperature and heat flux distribution over the design surface. The medium is considered as nongray, absorber, emitter, and nonscatter in radiative equilibrium. The discrete ordinates method in combination with the spectral line-based weighted sum of gray gases approach is used for solving the radiative transfer equation. The inverse problem involves the minimization of an appropriate objective function, which is minimized by the conjugate gradient method. The objective function is defined as the square residuals between the desired and estimated heat flux distribution over the design surface. Some examples are presented to prove the ability of the approach for achieving the desired thermal conditions.

Immersed Boundary Method for Radiative Heat Transfer Problems in Nongray Media With Complex Internal and External Boundaries

The immersed boundary method (IBM) is gaining attention in the computational fluid dynamics but its applications in the field of a conjugated radiative–conductive or radiative–convective heat transfer seem limited. Therefore, the paper presents extension of this method to heat transfer problems dominated by thermal radiation in a nongray medium. The present model enables simulation of heat and fluid flows in a domain with complex stationary or moving internal and external boundaries on a fixed Cartesian grid (FCG) by applying the finite volume method. The special attention is paid to modeling thermal radiation and optical phenomena at highly curved, opaque, or transparent boundaries which confine the computational domain or separate zones of different thermophysical and optical properties, e.g., different values of a refractive index. The model is limited to a 2D planar or axisymmetric spaces. Detailed verification procedure proves accuracy and correctness of the developed model and shows its potential application field. The model may be used for simulations of a conjugated radiative–conductive or radiative–convective heat transfer in a nongray medium in a complex domain with opaque or transparent curved internal or external boundaries without unstructured or body fitted mesh generation.

Inverse Boundary Design Problems in Enclosures With Non-Gray Media

ABSTRACTIn 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.

Numerical Solution of Radiative Transfer in a Real-Participating Media

In this paper, the control volume finite element method (CVFEM) is coupled with the weighted sum of gray gases model (WSGGM) to study the radiative heat transfer in a nongray medium. To the best of our knowledge, the CVFEM–WSGGM is applied for the first time to simulate real-gas. The accuracy of the proposed method is tested through one- and two-dimensional radiative heat transfer within an enclosure filled with a single composition (water vapor or carbon dioxide) or a mixture of H2O, CO2, and N2. Compared to the discrete ordinates method (DOM)–statistical narrow band model (SNBM), the proposed method, using the WSGG model parameters due to Smith or Farag, yields much accurate results than the zonal method (ZM)–WSGGM and DOM–WSGGM. In addition, the present method needs very less control volumes and angles, and consequently computational time, compared to the DOM and ZM coupled with WSGGM.

Reconstruction of Temperature Distribution in the Combustion Region of a Non-Gray Medium

Two-dimensional temperature distribution in the combustion region of a radiant furnace with non-gray gaseous products is reconstructed in an inverse manner. The discrete ordinate method is used to solve the radiative transfer equation, and the non-gray behavior of gaseous medium is modeled by the spectral line weighted sum of gray gas model. Conjugate gradient method is employed for the inverse analysis. The domain of interest is divided into two regions: the unknown- and known-temperature zones. The temperature distribution in the unknown-temperature zone, near the flame, is recovered in an inverse manner by the measurement of heat fluxes over the opposite walls in the direction of flame diffusion. Effects of the size of unknown-temperature zone, variable concentrations, and measurement errors in temperature reconstruction are discussed.

Natural element method for NON-GRAY radiation heat transfer problems

A radiation code based on the meshless Natural Element Method (NEM) for spatial discretization and on the Discrete Ordinates Method (DOM) for angular discretization of radiative transfer equation was developed for realistic calculation of the radiative heat transfer in two-dimensional complex geometries containing real gases with non-gray behavior (namely mixtures of H2O, CO2 and N2). The spectral line-based weighted-sum-of-gray-gases (SLW) model was used to calculate the radiative properties of the medium. The boundaries are assumed to be opaque, diffuse and gray. The shape functions in the NEM are constructed by interpolations from the natural neighbors, and the essential boundary conditions can be imposed directly. The predictive accuracy of the proposed method is assessed by applying it to five test problems and comparing its predictions with other numerical solutions reported in the literature. The predictions are also compared with those of the gray gas model. A good agreement was observed between the results of the NEM/DOM/SLW method and other numerical methods, and it is shown that the natural element method has good accuracy, and can be used for the solution of radiative heat transfer problems in irregular enclosures containing non-gray media.

Application of full-spectrum k-distribution method to combined non-gray radiation and forced convection flow in a duct with an expansion

This paper presents a numerical analysis of interaction between non-gray radiation and laminar forced convection flow in a duct with an expansion. Distributions of absorption coefficients across the spectrum (50 cm −1 < η < 20000 cm −1) are obtained from the HITRAN2008 database. The full spectrum k-distribution method (FSK) is used to deal with the non-gray part of the problem, while the gray radiation calculations are carried out using the Planck mean absorption coefficient. In addition, the results of non-gray medium are compared with the gray results in order to judge if the differences between these two approaches are significant enough to justify the usage of non-gray models. Results show that for air mixtures with different mole fractions of CO2 and H2O, use of gray model for the radiative properties may leads to considerable errors and should be eschewed.

Analysis of the conduction–radiation problem in absorbing, emitting, non-gray planar media using an exact method

Systems in which both conduction and radiation are the dominant modes of heat transfer are important in many engineering applications and various numerical methods exist to analyze such systems. An exact solution to the conduction–radiation problem in a one-dimensional, planar, absorbing, emitting, non-gray medium is presented. The method uses an integrating factor to solve the radiative transfer equation and variation of parameters is used to solve the energy equation. The model is verified by comparing the temperature profiles calculated from this work to those found using numerical methods for both gray and non-gray cases.

Inverse Boundary Design Radiation Problem Within Combustion Enclosures with Absorbing-Emitting Non-Gray Media

An inverse analysis is employed to estimate the unknown heat flux distribution over the heater surface of two-dimensional enclosures with irregular geometeries and non-gray participating media from the knowledge of desired temperature and heat flux distributions over the given design surface. The temperature distribution throughout the medium is specified. The radiative transfer equation is solved by modified discrete ordinates method combined with three models for simulation of non-gray media. The inverse problem is solved by the conjugate gradient method through an iterative procedure. The results obtained by all the models are verified and compared.