Collapsed Dimension Method Research Articles

A new hybrid algorithm for solving transient combined conduction radiation heat transfer problems

A new algorithm based on the lattice Boltzmann method (LBM) and the Control Volume Finite Element Method (CVFEM) is proposed as an hybrid solver for two dimensional transient conduction and radiation heat transfer problems in an optically emitting, absorbing and scattering medium. The LBM was used to solve the energy equation and the CVFEM was used to compute the radiative information. The advantages of the proposed methodology is to avoid problems that confronted when previous techniques are used to predict radiative heat transfer, essentially, in complex geometries and when there is scattering and/or non-black boundaries surfaces. This method combination, which is applied for the first time to solve this unsteady combined mode of heat transfer, has been found to accurately predict the effects of various thermo-physical parameters such as the scattering albedo, the conduction-radiation parameter and the extinction coefficient on temperature distribution. The results of the LBM-CVFEM combination were found to be in excellent agreement with the LBM-CDM (Collapsed Dimension Method)this proposed numerical approach include, among others, simple implementation on a computer, accurate CPU time, and capability of stable simulation.

The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media

With the finite volume formulation (FVM) approach applied to the collapsed dimension method (CDM), this article deals with the application of the CDM to analyze radiative heat transfer problems in a participating medium subjected to a continuous diffuse or a continuous/short-pulse collimated boundary radiative loading. The planar medium contained between diffuse gray boundaries is absorbing, emitting and anisotropically scattering. With three categories of thermal boundary radiative loadings, for the four types of problems considered, the CDM results are compared for a wide range of radiative parameters with that of the FVM.

View factor calculation in the 2-D geometries using the collapsed dimension method

Usage of the collapsed dimension method (CDM) applied so far in solving radiative transport problems with participating media has been extended for the determination of view factors of the 2-D geometries. Unlike its application in the participating media, in view factor calculations, the CDM does not require tracing of rays, and thus its expressions are exact. To test the applicability of the CDM for the calculation of view factors, two geometries were considered. The CDM results were found to match precisely with the exact results. If in the CDM, the view factor expressions are not considered exact, the ray tracing becomes mandatory and, in this method too like the discrete ordinate method, the discrete transfer method and the finite volume method, ray effect manifests. For this situation, for a given number of control surfaces, effects of number rays were studied and trends were found similar to those found in other methods.

Heat transfer analysis of a two-dimensional rectangular porous radiant burner

In the present work, a 2-D rectangular porous burner is considered for investigation. Methane–air combustion with detailed chemical kinetics is used to model the combustion part. 164 chemical reactions with 20 species are considered. Separate energy equations for gas and solid phases are solved. The radiative part of the energy equation is modeled using the collapsed dimension method. The effects of the power density, equivalence ratio, extinction coefficient and volumetric heat transfer coefficient on temperature and concentration profiles are studied.

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.

Radiative heat transfer for irregular geometries with the collapsed dimension method

A blocked-off region procedure is implemented with the collapsed dimension method (CDM) to deal with radiative transport problems in irregular geometries. Different test problems are validated for radiative and non-radiative equilibrium situations in participating or non-participating media. Results are found to be satisfactory for all straight edged, inclined and curved boundaries. The blocked-off region procedure based on Cartesian coordinate is found to be very convenient for a ray-tracing method like the CDM. The same ray tracing algorithm for a rectangular enclosure could be effectively used for any kind of 2-D geometries. This significantly reduces the effort of developing different ray-tracing algorithm for different geometries. In addition, it is an alternative than to write an algorithm in curvilinear coordinate for irregular geometries which found to be complicated for a ray-tracing method like the CDM.

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.

Two-dimensional transient conduction and radiation heat transfer with temperature dependent thermal conductivity

This paper deals with the effect of the temperature dependent thermal conductivity on transient conduction and radiation heat transfer in a 2-D rectangular enclosure containing an absorbing, emitting and scattering medium. The thermal conductivity of the medium is assumed to vary linearly with temperature. The radiative part of the energy equation was solved using the collapsed dimension method. To facilitate solution of the energy equation, which is a highly nonlinear one, time linearization was done first and then the equation was solved using the alternating direction implicit scheme. Results for the effects of the variable thermal conductivity were found for temperature and heat flux distributions.

EFFECT OF ANGULAR QUADRATURE SCHEMES ON THE COMPUTATIONAL EFFICIENCY OF THE DISCRETE TRANSFER METHOD FOR SOLVING RADIATIVE TRANSPORT PROBLEMS WITH PARTICIPATING MEDIUM

This article deals with the effects of angular quadrature schemes on the computational efficiency of the discrete transfer method (DTM) used for solving radiative transport problems with a participating medium. Four different quadrature schemes are considered. The first scheme uses the ray directions and the associated weights as per the original formulation of the DTM. In the second and third schemes, the ray arrangements are different but the quadrature schemes are the same as that used in the collapsed dimension method. The fourth scheme uses the ray directions and the corresponding weights according to the imporved quadrature scheme employed in the discrete ordinates method (DOM). The effects of the four quadrature schemes on the computational efficiency are tested for four benchmark radiative transport problems. Results are compared with the exact methods and with those available in the literature. It has been found that if the ray directions and the weights are chosen as per the DOM, the computational efficiency of the DTM is the best.

Heat transfer characteristics of a porous radiant burner under the influence of a 2-D radiation field

This paper deals with the heat transfer analysis of a 2-D rectangular porous radiant burner. Combustion in the porous medium is modelled as a spatially dependent heat generation zone. The gas and the solid phases are considered in non-local thermal equilibrium, and separate energy equations are used for the two phases. The solid phase is assumed to be absorbing, emitting and scattering, while the gas phase is considered transparent to radiation. The radiative part of the energy equation is solved using the collapsed dimension method. The alternating direction implicit scheme is used to solve the transient 2-D energy equations. Effects of various parameters on the performance of the burner are studied.

Computational efficiency improvements of the radiative transfer problems with or without conduction––a comparison of the collapsed dimension method and the discrete transfer method

This paper deals with the performance evaluation of the collapsed dimension method (CDM) and the discrete transfer method (DTM) in terms of computational time and their abilities to provide accurate results in solving radiation and/or conduction mode problems in a 2-D rectangular enclosure containing an absorbing, emitting and scattering medium. For some pure radiation cases, studies were made for two representative benchmark problems dealing with radiative equilibrium and non-radiative equilibrium. For the combined mode, the transient conduction and radiation problem was solved. The alternating direction implicit scheme was used for the solution of the finite difference part of the energy equation. For the three types of problems considered, tests were performed for a wide range of aspect ratio, extinction coefficient, scattering albedo, conduction–radiation parameter and boundary emissivity. For pure radiation problems, results from the two methods were validated against the results from the Monte Carlo method. For the combined mode, some steady-state results were compared with results available in the literature. For the transient situations, results from the two methods were validated against each other. While both the methods were found to give the same results, the CDM was found to be much more economical than the DTM.

Collapsed Dimension Method Applied to Radiative Transfer Problems in Complex Enclosures with Participating Medium

Application of the collapsed dimension method is extended to solve radiative transfer problems in complex enclosures with participating medium. Formulations are provided for absorbing, emitting, and anisotropically scattering medium. Formulations are validated by solving some benchmark problems in rectangular as well as cylindrical geometries. In rectangular geometry, 2-D L-shaped, quadrilateral, and 3-D cubical enclosures are considered. In cylindrical geometry, problems in 1-D and 2-D single as well as concentric cylindrical enclosures are taken up. To check the accuracy of the results obtained from the collapsed dimension method, depending on the availability of results, comparisons are made with the exact method, discrete transfer method, P 3 approximation, Monte Carlo method, and the finite-volume method. The collapsed dimension method has been found to provide very accurate results.

Radiative heat transfer in absorbing–emitting–scattering gray medium inside 1-D gray Cartesian enclosure using the collapsed dimension method

Radiative heat transfer in absorbing–emitting–scattering gray medium inside 1-D gray Cartesian enclosure using the collapsed dimension method

Analysis of conduction–radiation problem in absorbing, emitting and anisotropically scattering media using the collapsed dimension method

Combined conduction–radiation problem is solved using the collapsed dimension method. One-dimensional gray planar absorbing, emitting and anisotropically scattering medium is considered. Non-dimensional medium temperature and heat flux distributions are found for various values of boundary temperature, optical thickness, boundary emissivity and conduction–radiation parameter. Effects of scattering albedo and anisotropy factor are also discussed. For comparison, problems considered are also analyzed with the discrete transfer method and the exact method. Collapsed dimension method is found to give excellent comparison for various radiative parameters considered.

Modified Collapsed Dimension Method for Radiative Heat Transfer Problems

Introduction C OLLAPSED dimensionmethod (CDM) is one of the ray tracing methods used for the solution of radiative transfer problems. Development of this method is partly based on the works of Shih and Chen1 and Shih and Ren2 on the discretized intensity method. Some light on the method was thrown by Blank. Subsequent developmentof the method was made by Blank andMishra.4 Detailed description of this method is available in Ref. 5. In CDM, three-dimensionalradiative information is mapped into a two-dimensionalsolutionplane in terms of effective intensity (EI) andoptical thicknesscoefŽ cient (OTC). Thus, unlikeothermethods, analysis and computations in this method are performed in a twodimensional solution plane instead of three-dimensionalspace. CDMhas beenused for the solutionof radiativetransferproblems with highaccuracy.4;5 Thismethodhasbeenfoundtowork for awide rangeof optical thickness (very low to high optical thickness). Furthermore, CDM has also been found to work well for the conjugate mode heat transfer problems. In CDM, for determinationof heat  ux and temperature information, at each point of interest EIs have to be integrated over planar angle in the solutionplane.Theseangularintegrationsareperformed by dividing the planar angle into intervals of equal sizes. In each subinterval, EI is assumed isotropic. Hence, for higher accuracy, CDM requiresmore EIs and, thus, higher computational time. In the presentwork,with the objectiveofmakingCDMmore economical, the method is modiŽ ed. In the modiŽ ed CDM (MCDM), angular integrations of the EIs are performed differently. The discretizationand integrationsperformed in thismethod originatefrom the concept of the discrete ordinate method (DOM). Here, the planar angle is divided into a Ž nite number of subintervals according to the number of Gaussian quadraturepoints considered.Hence, in MCDM, angular subintervals are unequal and instead of considering average values of interval, the weighted mean corresponding to the Gauss points is considered. With this, the method becomes more realistic, and it derives computational efŽ ciency with a more realistic representationof the radiative transfer process. In the present work, improvements in MCDM over CDM are tested by solving radiative transfer problems in oneand twodimensional Cartesian enclosures with gray and homogeneous, absorbing, emitting, and anisotropically scattering medium. Both radiative and nonradiative equilibrium situations are considered. MCDM and CDM results are compared with results from the exact method and DOM.

TRANSIENT CONDUCTION AND RADIATION HEAT TRANSFER WITH HEAT GENERATION IN A PARTICIPATING MEDIUM USING THE COLLAPSED DIMENSION METHOD

Combined conduction and radiation heat transfer with and without heat generation is investigated. Analysis is carried out for both steady and unsteady situations. One-dimensional gray Cartesian enclosure with an absorbing, emitting, and isotropically scattering medium is considered. Enclosure boundaries are assumed at specified temperatures. The heat generation rate is considered uniform and constant throughout the medium. The Crank?Nicholson scheme is used to solve the transient energy equation. The radiative part of the energy equation is solved using the collapsed dimension method. Transient and steady state temperature and heat flux distributions are found for various radiative parameters. Results are found for situations with and without heat generation. Heat generation is found to have significant bearing on temperature and heat flux. Results are compared with the data reported in the literature. Excellent agreement has been found.

USE OF THE 2-D COLLAPSED DIMENSION METHOD IN GRAY ENCLOSURES WITH ABSORBING-EMITTING-ISOTROPIC SCATTERING MEDIA IN RADIATIVE EQUILIBRIUM

A Cartesian-based method is proposed for combustion-radiation problems in gray enclosures having one ordinate symmetry and containing absorbing-emitting media with isotropic scattering. This method collapses the 3-D radiative information of the problem into its 2-D solution plane and is shown to be applicable for all ranges of optical thicknesses, including the optically very thin case, while maintaining near-analytic accuracy over most of this range. To accomplish this task, the collapsed dimension method makes use of effective intensity rays, each of which contains the information of the conglomerate of all the conventional intensity rays focused on the same point and contained in a plane that is both inclined at the same angle as the ray and is also normal to the solution plane. This novel procedure thus eliminates the use of solid angles and reduces both the complexity and the computational expense involved in the solution of such problems. The method is shown to be applicable to gray wall problems en...