Using a High-Order Spatial/Temporal Scheme and Grid Adaptation with a Finite-Volume Method for Radiative Heat Transfer

In this work an algorithm using the finite-volume method along with a second-order-accurate spatial/temporal scheme and the h-refinement technique is evaluated in the parallelized computation of radiative heat transfer in absorbing-emitting and scattering gray media. The second-order spatial scheme is based on the reconstruction of radiative intensity's values, jointed with a slope limiter to maintain monotonicity. Additionally, the h-refinement method is incorporated to enrich targeted areas of the mesh during the solution procedure. The numerical results reveal the mitigation of false scattering and consequently the improvement of accuracy, employing these techniques in coarse unstructured hybrid grids.     

RADIATION ELEMENT METHOD FOR TRANSIENT HYPERBOLIC RADIATIVE TRANSFER IN PLANE-PARALLEL INHOMOGENEOUS MEDIA

In this study the radiation element method is formulated to solve transient radiative transfer with light radiation propagation effect in scattering, absorbing, and emitting media with inhomogeneous property. The accuracy of the method is verified by good agreement between the present calculations and Monte Carlo simulations. The sensitivity of the method against element size, ray emission number, and time increment size is examined. The transient effect of radiation propagation is essential in short-pulse laser radiation transport when the input pulse width is not considerably larger than the system radiation propagation time. The transient characteristics of radiative transfer are investigated in the media subject to collimated laser irradiation and/or diffuse irradiation withtemporal Gaussian and/or square profiles. The inhomogeneous profile of extinction coefficient of the medium affects strongly the transient radiative flux divergence inside the medium.     

Meshless method for radiation heat transfer in graded index medium

Because ray goes along a curved path determined by the Fermat principle, curved ray tracing is very difficult and complex in graded index media. To avoid the difficult and complex computation of curved ray trajectories, a meshless local Petrov–Galerkin approach based on discrete-ordinate equations is developed to solve the radiative transfer problem in multi-dimensional absorbing–emitting–scattering semitransparent graded index media. A moving least square approximation is used to construct the shape function. Two particular test problems in radiative transfer are taken as examples to verify this meshless approach. The predicted temperature distributions and the dimensionless radiative heat fluxes are determined by the proposed method and compared with the other benchmark approximate solutions. The results show that the meshless local Petrov–Galerkin approach based on discrete-ordinate equations has a good accuracy in solving the radiative transfer problems in absorbing–emitting–scattering semitransparent graded index media.     

Scaling anisotropic scattering in radiative transfer in three-dimensional nonhomogeneous media

Radiative heat transfer in three-dimensional nonhomogeneous participating medium was investigated by using REM² method. The anisotropic scattering phase function was dealt with the scaling technique based on delta function approximation. The three-dimensional scaled isotropic results were compared with the published anisotropic scattering computations. A good agreement between the scaled isotropic approaches and the anisotropic solutions was found. The effects of scattering albedo, forward fraction of phase function, and wall emissivity were discussed. It was found that, with the increase of the scattering albedo, the radiative heat flux increases for forward scattering media, but decreases for backward scattering media. The radiative heat flux is increased with the increases of forward fraction of phase function and wall emissivity. The emissive power at the center of a cubical nonhomogeneous medium in radiative equilibrium with gray diffuse walls equals to the averaged blackbody emissive power of the six walls.     

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.     

Comparison of Quadrature Schemes in DOM for Anisotropic Scattering Radiative Transfer Analysis

The commonly implemented level-symmetric S _N quadrature set for the discrete-ordinates method suffers from a limitation in discrete direction number to avoid physically unrealistic weighting factors. This limitation can have an adverse impact for determining radiative transfer, as directional discretization results in angular false scattering errors due to distortion of the scattering phase function in addition to the ray effect. To combat this limitation, several higher-order quadrature schemes with no directional limitation have been developed. Here, four higher-order quadrature sets (Legendre-equal weight, Legendre-Chebyshev, triangle tessellation, and spherical ring approximation) are implemented for determination of radiative transfer in a 3-D cubic enclosure containing participating media. Heat fluxes obtained at low direction number are compared to the S _N quadrature and Monte Carlo predictions to gauge and compare quadrature accuracy. Investigation into the reduction/elimination of angular false scattering with increase in direction number, including heat flux accuracy with respect to Monte Carlo and computational efficiency, is presented. It is found that while the higher-order quadrature sets are able to effectively minimize angular false scattering, the number of directions required is extremely large, and thus it is more computationally efficient to implement proper phase-function normalization to obtain accurate results.     

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.     

Meshless local Petrov–Galerkin approach for coupled radiative and conductive heat transfer

A meshless local Petrov–Galerkin approach is employed for solving the coupled radiative and conductive heat transfer in absorbing, emitting and scattering media. The meshless local Petrov–Galerkin approach with upwind scheme for radiative transfer is based on the discrete ordinate equations. The moving least square approximation is used to construct the shape function. Three particular test cases for coupled radiative and conductive heat transfer are examined to verify this new approximate method. The dimensionless temperatures and the dimensionless heat fluxes are obtained. The results are compared with the other benchmark approximate solutions. By comparison, the results show that the meshless local Petrov–Galerkin approach has a good accuracy in solving the coupled radiative and conductive heat transfer in absorbing, emitting and scattering media.     

Combined conductive and radiative heat transfer in an anisotropic scattering participating medium with irregular geometries

This paper deals with the numerical solution for the steady state combined conductive–radiative heat transfer in an anisotropic participating medium within the irregular geometries using the blocked-off method in Cartesian coordinates. The walls of the enclosures were considered to be opaque, diffuse and gray having specified heat flux and temperature boundary conditions. The finite-volume method has been adopted to solve the energy equation and the discrete ordinates method has been employed to solve the radiative transfer equation. The radiative and radiative–conducive models were validated by comparison with the results of specific test cases taken from the literature. The results showed very satisfactory predictions compared with the benchmarked results. As the degree of enclosure complexity (with curved or skewed walls) increased, finer grids were required. Based on this method, the effects of various influencing parameters such as the conduction–radiation parameter, scattering albedo and extinction coefficient have been considered.     

Transient thermal analysis of semitransparent composite layer with an opaque boundary

Transient coupled radiative and conductive heat transfer in a two-layer, absorbing, emitting, and isotropically scattering non-gray slab is investigated by the ray tracing method in combination with Hottel's zonal method. One outer boundary is opaque, and another is semitransparent. The radiative energy transfer process in a semitransparent composite is divided into two sub-processes, one of which considers scattering, the other does not. The radiative transfer coefficients of the composite are deduced under specular and diffuse reflection and combined specular and diffuse reflection, respectively. The radiative heat source term is calculated by the radiative transfer coefficients. Temperature and heat flux are obtained by using the full implicit control-volume method in combination with the spectral band model. The method presented here needs only to disperse the space position, instead of the solid angle. A comparison with previous results shows that the results are more accurate.     

RADIATIVE HEAT TRANSFER IN FINITE CYLINDRICAL HOMOGENEOUS AND NONHOMOGENEOUS SCATTERING MEDIA EXPOSED TO COLLIMATED RADIATION

Radiative heat transfer is studied in a finite axisymmetrical cylindrical enclosure exposed to collimated radiation. The integral equations for radiative transfer are solved by the YIX method and the quadrature method for comparison. Integrated intensity and radiative heat flux are presented in homogeneous and nonhomogeneous scattering media exposed to both uniform and Gaussian distributions of normal collimated incident radiation. The effects of aspect ratio, different incident radiation, and anisotropic scattering phase function as well as nonhomogeneous property distribution are discussed. Ray effects appear in the YIX solution for the case of a nonhomogeneous step change in the extinction coefficient. In order to eliminate the ray effect, an adaptive angular quadrature scheme is described and applied.     

Transient coupled heat transfer in multilayer composite with one specular boundary coated

By the ray tracing?node method, the transient coupled radiative and conductive heat transfer in absorbing, scattering multilayer composite is investigated with one surface of the composite being opaque and specular, and the others being semitransparent and specular. The effect of Fresnel’s reflective law and Snell’s refractive law on coupled heat transfer are analyzed. By using ray tracing method in combination with Hottel and Sarofim’s zonal method and spectral band model, the radiative intensity transfer model have been put forward. The difficulty for integration to solve radiative transfer coefficients (RTCs) is overcame by arranging critical angles according to their magnitudes. The RTCs are used to calculated radiative heat source term, and the transient energy equation is discretized by control volume method. The study shows that, for intensive scattering medium, if the refractive indexes are arranged decreasingly from the inner part of the composite to both side directions respectively, then, the total reflection phenomenon in the composite is advantageous for the scattered energy to be absorbed by the layer with the biggest refractive index, so at transient beginning a maximum temperature peak may appear in the layer with the biggest refractive index.     

Neural-finite difference method (NFDM) in development of improved differential approximation (IDA) and its application for coupled conduction and radiation heat transfer in a square enclosure: An experimental validation

The present numerical and experimental analysis addresses coupled conduction and radiation heat transfer (CCR) in differentially heated vertical isothermal walls and horizontal insulated walls of a square enclosure with absorbing, emitting and isotropic scattering participating gray medium. The P₁ approximation solution is utilized as the input signal to the neuron model. The computational domain is treated by the neural-finite difference method (NFDM) with ray tracing technique of ray emission model (REM) for the development of improved differential approximation (IDA). The output results are validated with the results of DOM. The practical implementation of IDA for wide range of radiative parameters are illustrated and examined. Experiments have been performed in a square enclosure with solid isothermal walls made of aluminum and insulated walls with bakelite, thus forming air filled cavity. Finally, the consistence of isotherm pattern of the numerical work with the interferogram captured by Mach–Zehnder interferometer corroborates the IDA theory and its realistic approach.     

Evaluation of the FTn Finite Volume Method for Transient Radiative Transfer in Anisotropically Scattering Medium

In this paper, we formulated, applied, and tested the FTn Finite Volume Method (FTn FVM) for transient radiative transfer in three-dimensional absorbing, emitting, and anisotropically scattering medium. Both the STEP and the Curved-Line Advection Method (CLAM) are introduced for spatial discretization of the transient radiative transfer equation. The results show that FTn FVM reduces largely the ray effects and it is more accurate than the standard FVM. Also, using both STEP and CLAM schemes, FTn FVM has smaller convergence time than the standard FVM for all cases. On the contrary, the STEP scheme is faster than the CLAM scheme but it has less accuracy. Then, the effects of optical thickness, scattering albedo, and anisotropy factor on incident radiation and radiative flux are presented and discussed.     

Radiative heat transfer in a participating medium with specular–diffuse surfaces

The results obtained by ray-tracing method can be regarded as benchmarks for its good accuracy. However, up to now, this method can be only used to solve radiative transfer within medium confined between two specular surfaces or two diffuse surfaces. This article proposes a hybrid ray-tracing method to solve the radiative transfer inside a plane-parallel absorbing–emitting–scattering medium with one specular surface and another diffuse surface (S–D surfaces). By the hybrid ray-tracing method, radiative transfer coefficients (RTCs) for S–D surfaces are deduced. Both surfaces of the medium under consideration are considered to be semitransparent or opaque. This paper examines the effects of scattering albedo, opaque surface emissivity and anisotropically scattering on steady-state heat flux and transient temperature fields. From the results it is found that the effects of anisotropic scattering is more for a bigger optical thickness medium; and keeping other optical parameters unchanged, anisotropic scattering affects transient temperature distributions so much in a small refractive index medium.     

Chebyshev collocation spectral method for one-dimensional radiative heat transfer in graded index media

Chebyshev spectral collocation method based on discrete ordinates equation is developed to solve radiative transfer problems in a one-dimensional absorbing, emitting and scattering semitransparent slab with spatially variable refractive index. For radiative transfer equation, the angular domain is discretized by discrete ordinates method, and the spatial domain is discretized by Chebyshev collocation spectral method. Due to the exponential convergence of spectral methods, a very high accuracy can be obtained even using few nodes for present problems. Numerical results by the Chebyshev collocation spectral-discrete ordinates method (SP-DOM) are compared with those available data in references. Effects of refractive index gradient on radiative intensity are studied for space dependent scattering media. The results show that SP-DOM has a good accuracy and efficiency for solving radiative heat transfer problems in even spatially varying absorbing, emitting, scattering, and graded index media.     

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.     

Inverse radiation analysis of simultaneous estimation of temperature field and radiative properties in a two-dimensional participating medium

An inverse radiation analysis is presented for simultaneous estimation of temperature field and radiative properties including absorption and scattering coefficients in a two-dimensional rectangular, absorbing, emitting and scattering gray medium from the knowledge of the exit radiative energy received by charge-coupled device (CCD) cameras at boundary surfaces. The backward Monte Carlo method was introduced to describe the radiative heat transfer for its efficiency. The inverse problem is formulated as an optimization problem and solved by the least-square QR decomposition (LSQR) method. The effects of measurement errors, optical thickness and search step length on the accuracy of the estimation were investigated and the results show that the temperature field and radiative properties can be reconstructed accurately for the exact and noisy data.     

An optimized and accurate Monte Carlo method to simulate 3D complex radiative enclosures

The modeling of radiative heat transfer in complex radiant enclosures is a particularly challenging subject. This simulation is often best treated by calculating distribution factors through the Monte Carlo method. In order to enhance performance of the Monte Carlo method, efficient algorithms to find location of emission and direction of emission in the original Monte Carlo method are implemented. Next, the best ray tracing algorithm is introduced by comparing timing results of the USD, the BSP, the Simplex and the VVA acceleration ray tracing algorithms to make it numerically efficient as possible. Also, the constrained maximum likelihood estimation is used to enhance accuracy of the Monte Carlo by smoothing inherent random errors in the estimated distribution factors to simultaneously satisfy both of the reciprocity and summation rules. Accuracy of the Monte Carlo method is tested for a classical problem, namely a 3D box, with diffuse gray walls. For efficiency study, the optimized Monte Carlo method is then tested for two real radiative enclosures with convex and concave geometries. All ray tracing algorithms are found to result in computational gains, with respect to direct calculations that do not employ any acceleration technique. In the considered test cases, the VVA and the USD algorithms are found to be clearly superior to the BSP and the Simplex algorithms, particularly for concave geometries that have some obstructions within the computational domain.     

Radiative and conductive transfer for a real gas in a cylindrical enclosure with gray walls

The purpose of this study is to examine the interaction of radiative and conductive transfer for a radiatively participating real gas stagnant in a cylindrical enclosure with gray diffuse walls. Consideration of reflecting boundaries represents an extension of previous black wall studies. Examination of radiative transfer was made by the zone method with gas radiative properties furnished by the weighted sum of gray gases model. Directed flux areas are expressed as the weighted sum of gray gas total exchange areas which are evaluated using the matrix formulation method from direct exchange areas. Axial and radial gas temperatures are examined along with wall heat flux or temperature for respective cases of either specified wall temperatures or heat fluxes. Emphasis is placed on examining results to show the effects of wall emittance and duct diameter. Results for heat generation within the gas are also presented.