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.     

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.     

PREDICTION OF SURFACE RADIATIVE HEAT TRANSFER USING THE MODIFIED DISCRETE TRANSFER METHOD

An ideal surface radiation model applied in many manufacturing and materials processing systems must be able to take into account specular, spectral, and shadowing effects with complex geometries, and must be computationally efficient to permit its inclusion in the fluid flow and heat transfer models. In this study, a novel surface radiative heat transfer method is developed to meet all these practical needs. The present model is based on the discrete transfer method (DTM). A direct application of the DTM to modeling surface radiative heat transfer may result in a large error due to strong ray effects. In order to eliminate these ray effects, the DTM is modified by considering radiation contribution from all surface cells intercepted by a control angle. Calculation of these surface cell areas represents one of the most important tasks in the modified DTM (MDTM), and it is described in detail in this study. To investigate the accuracy and efficiency of the MDTM, three benchmark problems covering different geometric and boundary conditions are considered, and the present solution is compared with the solutions from the exact approach, the DTM, and discrete ordinates method (DOM). For each problem, the accuracy of the MDTM, DTM, and DOM appears to be affected by the angular discretization. For a reasonable fine angular discretization, the solutions from the MDTM and DOM match the exact solution very well, while the solution from the DTM usually shows strong ray effects. The CPU times spent on the MDTM and DTM are very similar, but they are usually orders of magnitude less than that for the DOM. The present study indicates that the MDTM is not only accurate but also very efficient for modeling complicated surface radiation problems. Such a model will greatly benefit the simulation of many manufacturing and materials processing systems.     

Modeling of transient natural convection heat transfer in electric ovens

Prediction of transient natural convection heat transfer in vented enclosures has multiple applications such as understanding of cooking environment in ovens and heat sink performance in electronic packaging industry. The thermal field within an oven has significant impact on quality of cooked food and reliable predictions are important for robust design and performance evaluation of an oven. The CFD modeling of electric oven involves three-dimensional, unsteady, natural convective flow-thermal field coupled with radiative heat transfer. However, numerical solution of natural convection in enclosures with openings at top and bottom (ovens) can often lead to non-physical solutions such as reverse flow at the top vent, partly a function of initialization and sometimes dependent on boundary conditions. In this paper, development of a physics based robust CFD methodology is discussed. This model has been developed with rigorous experimental support and transient validation of this model with experiments show less than 3% discrepancy for a bake cycle. There is greater challenge in simulating a broil cycle, where the fluid inside the cavity is stably stratified and is also highlighted. A comparative analyses of bake and broil cycle thermal fields inside the oven are also presented.     

Performance evaluation of four radiative transfer methods in solving multi-dimensional radiation and/or conduction heat transfer problems

This article reports results of the four popular and widely used numerical methods, viz., the Monte Carlo method (MCM), the discrete transfer method (DTM), the discrete ordinates method (DOM) and the finite volume method (FVM) used to calculate radiative information in any thermal problem. Different classes of problems dealing with radiation and/or conduction heat transfer problems in a 2-D rectangular absorbing, emitting and scattering participating medium have been considered. In radiative equilibrium and non-radiative equilibrium cases, the MCM results have been used as the benchmark data for comparing the performances of the DTM, the DOM and the FVM. In the combined radiation and conduction mode problem, the energy equation has been formulated using the lattice Boltzmann method (LBM). To compare the performance of the DTM, the DOM and the FVM, the required radiative field data computed using these methods have been provided to the LBM formulation. Temperature distributions obtained using the four methods and those obtained from the LBM in conjunction with the DTM, the DOM and the FVM have been compared for different parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the wall emissivity, the aspect ratio and heat generation rate. In all the cases, results of these methods have been found in good agreements. Computationally, the DTM was found the most time consuming, and the DOM was computationally the most efficient.     

TWO-DIMENSIONAL COMBINED CONDUCTION AND RADIATION HEAT TRANSFER: COMPARISON OF THE DISCRETE ORDINATES METHOD AND THE DIFFUSION APPROXIMATION METHODS

Heat transfer by combined conduction and radiation in a two-dimensional semitransparent medium has been investigated. The discrete ordinates method (DOM) and the Rossland diffusion approximation are used to analyze radiative transfer. Glass is considered as an example of a radiation absorbing and emitting medium, and the spectral dependence of the absorption coefficient on wavelength is accounted for. The results predicted by the DOM are in good agreement with those based on the one-dimensional integral formulation. However, when the opacity of the medium is large, the DOM suffers from the numerical smearing, which distorts the radiative flux distribution. The diffusion approximation greatly underpredicts the temperature and flux distributions in the medium, particularly when the thickness or the opacity of the medium is small. The predictions of the diffusion approximation are only reasonable for the thick layer. Hence, the approximation should be used with extreme caution to obtain quantitatively accurate results.     

SOLUTION OF STEADY PERIODIC HEAT CONDUCTION BY THE FINITE-ELEMENT METHOD

This paper describes the use of the finite-element technique to solve problems of steady periodic heal conduction. The concept of a complex variable can be used to reduce the governing unsteady heat conduction equation to two noncoupled Poisson equations. The validity of the present method is confirmed with a one-dimensional problem for which an analytic solution exists. Numerical solutions for a three-dimensional problem are presented to illustrate the capability of the method.     

Second-order-accurate discrete ordinates solutions of transient radiative transfer in a scattering slab with variable refractive index

The discrete ordinates method (DOM) with a second-order upwind interpolation scheme is applied to solve transient radiative transfer in a graded index slab suddenly exposed to a diffuse strong irradiation at one of its boundaries. The planar medium is absorbing and anisotropically scattering. From the comparison of the results obtained by the first-order DOM, the second-order DOM, the modified DOM and the Monte Carlo method, it can be seen that the numerical diffusion in the transient solutions obtained by the second-order DOM is less than that in the solutions obtained by the first-order DOM, but the numerical diffusion is still noticeable, especially for optically thin and moderate cases. By contrast, for optically thick cases the numerical diffusion due to the finite difference of the advection term of the transient radiative transfer equation is minor. In general, it is still necessary to adopt a DOM with a higher order scheme to capture the wave front of transient radiative transfer accurately. Besides, the influence of numerical diffusion is a little less noticeable for the case with a larger gradient of refractive index, and the distribution of direction-integrated intensity around the irradiation boundary decreases and that around the other boundary increases with the increase of the anisotropically scattering coefficient.     

A FAST METHOD OF RADIATIVE HEAT TRANSFER ANALYSIS BETWEEN ARBITRARY THREE-DIMENSIONAL BODIES COMPOSED OF SPECULAR AND DIFFUSE SURFACES

The radiation element method by ray emission model (REM²) has been improved by using the law of reciprocity for the specular view factor and the incomplete Cholesky conjugate gradient (ICCG) method to reduce computational time. This improved method was applied to analyze the radiative heat transfer between arbitrary three-dimensional bodies composed of specular and diffuse surfaces. The accuracy of the improved method was evaluated by comparing analytical solutions. And the method was used to calculate radiative heat transfer between machine elements and heater panels. CPU time to calculate the radiative exchange for a model composed of 3193 elements was reduced to 1/430 of that by the previous numerical method using a decomposition method.     

NUMERICAL SIMULATION OF THE GAS FLOW AND MASS TRANSFER BETWEEN TWO COAXIALLY ROTATING DISKS

A problem of combined conductive and two-phase radiative heat transfer in a two-dimensional rectangular enclosure with two-phase (gas-particles) media is analyzed. A two-phase radiative transfer equation (RTE) considering radiation by both gas and particles is studied. Its nonlinear integrodifferential RTE is solved using the discrete ordinates method (DOM, or so-called S N method). To validate the program, we compare the solution in a two-dimensional rectangular black enclosure with others. The DOM is then applied to the unsteady thermal development in two-phase media contained in a rectangular enclosure. A parametric study is performed by changing the gas and particle absorption coefficients, particle number density, particle emissivity, wall emissivity, and aspect ratio of the enclosure. The results confirm a significant effect of the two-phase radiation on the thermal development in the geometry. However, it is found that the conduction is predominant near the hot wall.     

Theoretical development of a thermal model for the reheater of a power plant boiler

A three-dimensional numerical model for simulating flow and heat transfer in the reheater of a boiler is presented. The aim is to describe, as well as possible, the geometry of the reheater and to be able to perform different mass flows of steam along each of the tube serpentines. The model thus makes it possible to calculate the temperature of the tube surfaces along the reheater. The porosity concept is employed, along with empirical correlations for the convective heat transfer coefficient and the radiation heat transfer coefficients. The radiation equations consider most of the radiative effects of the gas: ash content, triatomic gases, type of fuel and temperatures, tube layout and distances and temperatures of other radiative surfaces. The model is proposed with a view to using the measured values of velocities, temperatures and gas composition in the reheater as boundary conditions. The equations are solved using a general purpose computational fluid dynamics (CFD) code in conjunction with specific calculations for the source terms.     

Assessment of the axisymmetric radiative heat transfer in a cylindrical enclosure with the finite volume method

The radiative heat transfer in an axisymmetric enclosure containing an absorbing, emitting, and scattering gray medium is investigated by using the finite volume method (FVM). Especially, formulations with the cylindrical base vectors are introduced and its characteristics is discussed by comparing with other solution methods in the finite volume category. By considering the three-dimensional procedure, the angular redistribution term, which appears in such curvilinear coordinates as axisymmetric and spherically symmetric ones, can be treated efficiently without any artifice usually introduced in the conventional discrete ordinates method (DOM). After a mathematical formulation and corresponding discretization equation for the radiative transfer equation (RTE) are derived, final discretization equation is introduced by using the directional weight, which is the key parameter in the FVM since it represents the inflow or outflow of radiant energy across the control volume faces depending on its sign. The present approach is then validated by comparing the present results with those of previous works. All the results presented in this work show that the present method is accurate and valuable for the analysis of cylindrically axisymmetric radiative heat transfer problems.     

MODELING THE EFFECT OF EGR AND MULTIPLE INJECTION SCHEMES ON I. C. ENGINE COMPONENT TEMPERATURES

A three-dimensional, finite-element-based, transient heat conduction in components (HCC) code was successfully developed and used to calculate the temperature distribution of the piston, head, and cylinder liner of a heavy-duty diesel engine. The HCC code was used in an iterative sequence with the KIVA3V engine CFD code. The methodology was used to study the effect of EGR and multiple injection schemes on metal component temperatures for a Caterpillar 3401 engine. Single, double, and triple injections were studied. Increasing the EGR level was shown to increase the average and peak component temperatures with the peak temperature occurring at the lip where the piston bowl and face meet, in the vicinity of the axis of the fuel spray. Retarding the injection timing had the effect of decreasing both the average and peak component temperatures.     

The DOM approach to the collapsed dimension method for solving radiative transport problems with participating media

A new formulation of the collapsed dimension method (CDM), called the modified collapsed dimension method (MCDM) whose approach is similar to the discrete ordinate method (DOM), has been proposed. In the MCDM, the time consuming procedures of ray tracing and source term evaluation are not required, as a result of which the method becomes computationally efficient. To validate the formulation, test problems dealing with radiative heat transfer with absorbing, emitting and scattering medium have been considered. To compare the performance of the MCDM, the same problems have also been solved using the CDM and the DOM. Results have been compared against the benchmark results. For the same level of accuracy, MCDM has been found faster than the CDM and the DOM. However, the number of iterations required for the converged solution in the MCDM and the DOM has been found to be almost the same.     

ITERATIVE METHOD FOR THE NUMERICAL PREDICTION OF HEAT TRANSFER IN PROBLEMS INVOLVING LARGE DIFFERENCES IN THERMAL CONDUCTIVITIES

Heal exchange that occurs between materials with largely differing thermal conductivities is commonly encountered in engineering practice. Conventional iterative solution methods perform poorly for the numerical solution of such problems. A block correction procedure, designed for enhancing the convergence of iterative solution methods, is used in conjunction with the line-by-line iterative solution method. The overall solution algorithm is a multigrid strategy with two grid levels. Results of computations for test problems indicate that the proposed solution procedure enables efficient solution of heat transfer problems with large conductivity differences for which the conventional line-by-line method proves ineffective.     

SPATIAL DISCRETIZATION ERRORS IN THE HEAT FLUX INTEGRAL OF THE DISCRETE TRANSFER METHOD

The discrete transfer method (DTM) is a widely used algorithm for the computation of radiative heat transfer in enclosures. The truncation error in the heat flux integral associated with the method is the difference between the actual heat flux and its DTM approximation. Estimates were presented by Versteeg et al. [1, 2] of the error associated with the discretization of the hemisphere around irradiated points in enclosures filled with transparent and nonscattering participating media. In this article we quantify the errors due to the spatial discretisation of the enclosure surface and medium conditions. We have studied radiation problems in enclosures with nonhomogeneous absorbing/emitting media with nonuniform surface intensity based on Hsu and Farmer's benchmark case E1 [3]. Our error estimates are found to be in excellent agreement with the actual DTM errors and have been used successfully to predict the convergence rates of the DTM as the control-volume mesh is refined.     

Numerical predictions of two-dimensional conduction,convection, and radiation heat transfer. II. Validation

This paper presents several test problems that were used to validate the formulation and implementation of a CVFEM for combined-mode heat transfer in participating media. The objective here is to demonstrate that the proposed CVFEM can be used to solve combined modes of heat transfer in media that emit, absorb, and scatter radiant energy in regularly- and irregularly-shaped geometries. The paper first assesses briefly the CVFEM for the solution of convection–diffusion problems, then attention is focused on radiative heat transfer problems. Finally, several combined mode problems are investigated. Results show that the proposed method adequately solves the governing equations (energy and RTE): as the solutions compare favorably with those obtained with other acknowledged methods or with analytical solutions, when available. The proposed CVFEM could, however, be improved to broaden its scope of application and enhance its numerical efficiency. In a near future, the method will be combined with a code already established for fluid flow calculations.     

Performance Evaluation of Two Heat Transfer Models of a Walking Beam Type Reheat Furnace

Two different heat transfer models for predicting the transient heat transfer characteristics of the slabs in a walking beam type reheat furnace are compared in this work. The prediction of heat flux on the slab surface and the temperature distribution inside the slab have been determined by considering thermal radiation in the furnace chamber and transient heat conduction in the slab. Both models have been compared for their accuracy and computational time. The furnace is modeled as an enclosure with a radiatively participating medium. In the first model, the three-dimensional (3D) transient heat conduction equation with a radiative heat flux boundary condition is solved using an in-house code. The radiative heat flux incident on the slab surface required in the boundary condition of the conduction code is calculated using the commercial software FLUENT. The second model uses entirely FLUENT along with a user-defined function, which has been developed to account for the movement of slabs. The results obtained from both models have a maximum temperature difference of 2.25%, whereas the computational time for the first model is 3 h and that for the second model is approximately 100 h.     

USE OF GCG METHODS FOR THE EFFICIENT SOLUTION OF MATRIX PROBLEMS ARISING FROM THE FVM FORMULATION OF RADIATIVE TRANSFER

Preconditioned generalized conjugate gradient (GCG) iterative methods are applied to the solution of large, sparse, and unsymmetric linear algebraic equations resulting from the application of the finite-volume method to the problem of radiative heat transfer in an absorbing, emitting, and scattering gray medium, with the boundary surfaces reflecting radiation in both diffuse and specular regimes. The governing radiative transfer equation, which is a complicated integro-differential equation, has been discretized using the S N finite-volume method (FVM). Different variants of GCG methods have been tested on a problem of 2-D radiation in a cylinder, and efficiencies of the methods have been compared. Numerical results indicate that preconditioning suggested in the article dramatically improves the performance of the GCG methods. Results on test problems based on S 8 FVM agree well with exact results reported in the literature.     

Parallel Computation for Radiative Heat Transfer Using the DOM in Combustion Applications: Direction,Frequency, Subdomain Decompositions,and Hybrid Methods

In the framework of coupled large-eddy/discrete ordinates method (LES/DOM) computations of turbulent combustion problems, various decompositions for parallel calculations of the radiative heat transfer based on the DOM are investigated. The methods analyzed are: (A) a task decomposition on the discrete directions and frequencies with two numeric strategies: Message Passing Interface (MPI) with distributed memory and OpenMP with shared memory for the direction decomposition; (B) a new algorithm for a DOM subdomain decomposition, which is proposed and tested using MPI; and (C) hybrid methods combining an OpenMP strategy for direction and MPI for tasks and subdomain decomposition. It is shown for the case of coupled simulations that the convergence and the parallel efficiency of the domain decomposition (B) are optimal. This method is limited in this work to 25 sub-domains, at which point the efficiency stagnates. Combining the directions with frequency and/or domain decompositions in a hybrid method (C) results in very good efficiency up to 1,200 processors. This hybrid strategy is also very efficient in terms of memory usage. This work shows that the best way to perform massively parallel computation for radiative heat transfer with the DOM is to combine different decomposition levels. The analysis performed in this work shows the best parallel strategy to be used in coupled simulations between radiation and LES on massively parallel architectures.