Modified Discrete Ordinates Method Research Articles

The modified discrete ordinates method for radiative heat transfer in two-dimensional cylindrical medium

Ray effect is one of the major shortcomings of discrete ordinate method (DOM). It is a consequence of angular discretization, resulting in unphysical oscillations in the solutions, and often occurs in the problems where the temperature of walls is discontinuous. In this paper, the modified DOM (MDOM) is developed for the radiative heat transfer in an axisymmetric cylindrical enclosure, which contains emitting, absorbing and scattering medium, to overcome this shortcoming. The radiative intensity is decomposed into two components, the wall-related and the medium-related ones. The former is determined analytically, whereas the latter is still solved by DOM. In MDOM, the most important step is the evaluation of wall-related incident radiation and radiative heat flux. Two methods to evaluate them are comparatively given: one is the semi-analytical method based on coordinate transformation which is more accurate, and the other is the numerical method with segments integration to treat the discontinuity which is mathematically simpler. Results show that both are effective to mitigate the ray effect, and cost computational time comparable to DOM. Exact semi-analytic solutions for the pure absorbing cases are also given in tabular form for convenience. Besides, alternative boundary conditions for the MDOM proposed by Sakami and Charette (JQSRT, 64(2000) 275–298) are investigated. This type of boundary conditions is proved to have simplified solution procedure, but results in slight unphysical oscillations in the MDOM solutions.

Transient radiative transfer in a complex refracting medium by a modified Monte Carlo simulation

Abstract A modified Monte Carlo (MMC) method applied with time shift and superposition principle is developed for solving transient radiative transfer in one-dimensional scattering media with arbitrary distributions of refractive index exposed to a collimated short pulse-laser irradiation at one of its boundaries. We consider a multilayer medium and a graded index medium. The accuracy and computational efficiency of the MMC algorithm are examined firstly. Our results are compared with those obtained by the reverse Monte Carlo method (RMC), the modified discrete ordinates method (MDOM) and the lattice Boltzmann method (LBM), and the excellent agreements are achieved. With the time shift and superposition principle, the MMC method can greatly improve the computational efficiency. We have a parametric investigation on the transient radiative transfer in the three-layer systems and graded-index media, illustrating the effects of refractive index distribution, optical thickness, interface/surface reflection mode, scattering phase function and scattering albedo on the time-resolved signals of reflectance and transmittance. Results show that the reflection mode has an important influence on the transient radiative transfer; the specular reflection at interfaces/boundaries can enhance the effects of forward scattering, while the diffuse reflection can enhance the effects of backward scattering.

Application of modified discrete ordinates method to combined conduction‐radiation heat transfer problems in irregular geometries

PurposeThe solution of radiative heat transfer problems in participating media is often obtained using the standard discrete ordinates method (SDOM). This method produces anomalies caused by ray effects if radiative boundary conditions have discontinuities or abrupt changes. Ray effects may be mitigated using the modified discrete ordinates method (MDOM), which is based on superposition of the solutions obtained by considering separately radiation from the walls and radiation from the medium. The purpose of this paper is to study the role of ray effects in combined conduction‐radiation problems and investigate the superiority of the MDOM over SDOM.Design/methodology/approachThe MDOM has been used to calculate radiative heat transfer in irregular geometries using body‐fitted coordinates. Here, the blocked‐off region concept, originally developed in computational fluid dynamics, is used along with the finite volume method and SDOM or MDOM to solve combined conduction‐radiation heat transport problems in irregular geometries. Enclosures with an absorbing, emitting and isotropically or anisotropically scattering medium are analyzed.FindingsThe results confirm the capability of the MDOM to minimize the anomalies due to ray effects in combined heat transfer problems, and demonstrate that MDOM is more computationally efficient than SDOM.Originality/valueThe paper demonstrates the application of MDOM to combined conduction‐radiation heat transfer problems in irregular geometries using blocked‐off method.

Analysis of Conduction and Radiation Heat Transfer in a 2-D Cylindrical Medium Using the Modified Discrete Ordinate Method and the Lattice Boltzmann Method

This article deals with the analysis of radiative transport with and without conduction in a finite concentric cylindrical enclosure containing absorbing, emitting, and scattering medium. Isothermal medium as the radiation source confined between the cold cylinders and a nonisothermal medium with the inner cylinder as the radiation source are the two nonradiative and radiative equilibrium problems. They involve only calculation of radiative information. In the third problem, a conducting-radiating medium is thermally perturbed by raising the temperature of the inner cylinder. In all problems, radiative information is computed using the modified discrete ordinate method (MDOM), and in the third problem, the lattice Boltzmann method (LBM) is used to formulate and solve the energy equation. Depending on the problems, effects of various parameters such as the extinction coefficient, the scattering albedo, the boundary emissivity, the conduction-radiation parameter, and the radius ratio are studied on temperature and heat flux distributions. The MDOM and the LBM-MDOM results are compared with those available in the literature. To further establish the accuracy of the MDOM and the LBM-MDOM results, in all problems, comparisons are made with the results obtained from the finite volume method (FVM) and the finite difference method-FVM approach, in which FVM provides the radiative information. The selection of the FDM-FVM for the third problem is also with the objective that for this problem, not much work is reported in which the FVM is used to calculate the radiative information. MDOM and LBM-MDOM results are found to compare well with those available in the literature, and in all cases they are in excellent agreement with FVM and FDM-FVM approaches.

Application of the modified discrete ordinates method with the concept of blocked-off region to irregular geometries

The standard discrete ordinates method (SDOM) produces anomalies caused by ray effects if the temperature of the walls has discontinuities or abrupt changes. Ray effects may be mitigated using the modified discrete ordinates method (MDOM), which is based on superposition of the solutions obtained by considering separately radiation from the walls and radiation from the medium. The MDOM has been used to calculate radiative heat transfer in irregular geometries using body-fitted coordinates. Here, the blocked-off region concept is used along with the MDOM to solve radiative transfer problems in irregular geometries. Two-dimensional irregular enclosures with curved boundaries, obstacles or radiation shields and containing a participating medium are considered. The accuracy of the results calculated using the SDOM and the MDOM is assessed by comparing the predictions with those obtained by other researchers. The results confirm the capability of the MDOM to minimize the anomalies due to ray effects in various situations. It is concluded that the blocked-off region concept can be recommended as a good option to solve radiative transport problems in irregular geometries.

Lattice Boltzmann Method and Modified Discrete Ordinate Method Applied to Radiative Transport in a Spherical Medium with and without Conduction

This article deals with the application of the modified discrete ordinate method (MDOM) to calculate volumetric radiative information with and without conduction in a concentric spherical enclosure containing a participating medium. With radiative information known from the MDOM, the energy equation of the combined mode transient conduction and radiation heat transfer is formulated and solved using the lattice Boltzmann method (LBM). Without conduction, for pure radiation case, two benchmark problems, representing nonradiative and radiative equilibrium situations are taken up. In the case of non-radiative equilibrium, an isothermal medium is bounded by cold walls and medium is the source of radiation, while in the case of radiative equilibrium, nonisothermal medium is confined between a hot and a cold wall, and the hot (inner sphere) wall is the radiation source. Depending upon the problem, heat flux, energy flow rate, emissive power, and temperature distributions in the medium are calculated for different values of parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the boundary emissivity, and the radius ratio. To validate the MDOM and the LBM-MDOM formulations, problems are also solved using the finite volume method (FVM) and the finite-difference method (FDM)–FVM approach, in which the FVM is used to calculate the volumetric radiation and the energy equation is also solved using the FDM. Results of the MDOM, LBM–MDOM, FVM and FDM–FVM are also benchmarked against those available in the literature. MDOM and LBM–MDOM have been found to provide accurate results.

Analysis of radiative transport in a cylindrical enclosure—An application of the modified discrete ordinate method

Application of the modified discrete ordinate method (MDOM) proposed by Mishra et al. [ Mishra SC, Roy HK, Misra N. Discrete ordinate method with a new and simple quadrature scheme. J Quant Spectrosc Radiat Transfer 2006;101:249–262.] has been extended for calculation of volumetric radiative information in a cylindrical enclosure. Radiatively, the medium inside a diffuse gray 1-D concentric cylinder is absorbing, emitting and scattering. Three types of problems, viz., an isothermal medium representing non-radiative equilibrium case, a non-isothermal medium representing radiative equilibrium situation and the case of a combined mode conduction and radiation heat transfer have been used to test the robustness of the MDOM. Temperature/emissive power and heat flux/energy flow rate distributions in the medium have been found for the effects of various parameters like the extinction coefficient, the scattering albedo, the boundary emissivity and the conduction–radiation parameter. To check the accuracy of the results of the MDOM, results have been compared with those available in the literature and also by obtaining the radiative information using the finite volume method. MDOM has been found to provide accurate results.

Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate

In this work, we applied the discrete ordinates method (DOM) with a first-order spatial scheme and adapted a modified DOM (MDOM) to solve transient radiative transfer in a refractive, absorbing and scattering slab suddenly exposed to a diffuse strong irradiation at one of its boundaries. The other boundary is diffusely reflecting. From the comparison of the results obtained by the first-order DOM, the MDOM and the Monte Carlo method, it can be seen that the results obtained by the three methods are in excellent agreement as the time is long enough. Besides, the solutions of optically thin and moderate cases obtained by the first-order DOM include some early transmitted radiation due to numerical diffusion and those obtained by the MDOM do not show numerical diffusion in the beginning of a transient process. The reason is that the MDOM solves exactly the reduced incident intensity which dominates radiative transfer in the beginning of a transient process. The time-resolved hemispherical reflectance and transmittance of the slab are obtained for various linearly varying refractive indices, optical thicknesses, scattering albedos and substrate reflectivities. Effects of those parameters are investigated.

Radiative heat transfer between two concentric spheres separated by a two-phase mixture of non-gray gas and particles using the modified discrete-ordinates method

The radiative heat transfer between two concentric spheres separated by a two-phase mixture of non-gray gas and a cloud of particles is investigated by using the combined finite-volume and discrete-ordinates method, named modified discrete-ordinates method (MDOM), which integrates the radiative transfer equation (RTE) over a control volume and a control angle simultaneously like in the finite-volume method (FVM) and treats the angular derivative terms due to spherical geometry as the conventional discrete-ordinates method (DOM). The radiative properties involving non-gray gas and particle behavior are modeled by using the extended weighted sum of gray gases model (WSGGM) with particles. Mathematical formulation and final discretization equations for the RTE are introduced by considering the behavior of a two-phase mixture of non-gray gas and particles in a spherically symmetric concentric enclosure. The present approach is validated by comparing with the results of previous works including gray and non-gray radiative heat transfer. Finally, a detailed investigation of the radiative heat transfer with non-gray gases and/or a two-phase mixture is conducted to examine the dependence of the radiative heat transfer upon temperature ratio between inner and outer spherical enclosure, particle concentration, and particle temperature.

A modified version of the discrete ordinates method for radiative heat transfer modelling

The standard discrete ordinates method (SDOM) suffers from two major sources of inaccuracy, the ray effects and false scattering. False scattering may be significantly reduced using high order discretization schemes, while ray effects originated from abrupt changes of wall temperatures are mitigated by the modified discrete ordinates method (MDOM). However, the MDOM is insensitive to ray effects caused by sharp gradients of the emissive power of the medium. Therefore, a new version (NMDOM) is proposed to overcome this shortcoming. The accuracy of the SDOM, MDOM and NMDOM are investigated, along with the interaction between ray effects and false scattering. It is concluded that the NMDOM yields accurate solutions even for coarse spatial and angular discretizations.

Analysis of Radiative Heat Transfer in Complex Two-Dimensional Enclosures With Obstacles Using the Modified Discrete Ordinates Method

In this work, an extension of a modified discrete ordinates method recently proposed by other researchers is presented. It is intended to counter the ray effect inherent in this method. The media analyzed are absorbing, emitting and isotropically or anisotropically scattering and the enclosure geometry is arbitrary. Cases with obstructions are emphasized. The radiative intensity is broken into two parts: the wall-related intensity and the medium-related intensity. The former is treated separately by rigorous integration over the entire solid boundary. A new differencing scheme based on quadrangular grids is also proposed and used for the treatment of the medium-related intensity. Results confirm that the proposed method is a good general remedy for anomalies caused by the ray effect due to the geometry.

Photon transport in vegetation canopies with anisotropic scattering Part II. Discrete-ordinates/exact-kernel technique for one-angle photon transport in slab geometry

In this paper, we discuss a computational technique for photon transport in vegetation canopies involving anisotropic scattering. The exact kernel technique eliminates the need for finite Legendre polynomial expansions and thus the inherent expansion errors. The scattering phase function (see Part I) now takes the form of a transfer matrix, whose elements are the actual phase function evaluated at discrete quadrature angles. To use the non-expanded scattering cross-sections in transport calculations, a modified discrete-ordinates method is formulated. The proposed method is tested for energy conservation and its results are compared with tabulated (reflection and transmission) functions of some standard solutions in atmospheric photon transport studies. Finally, the method was used to study the one-angle problem of photon transport in vegetation canopies.