Two-dimensional radiative equilibrium of a gray medium between concentric cylinders

A method for calculating radiative equilibrium temperature and surface heat-flux distributions is developed for an absorbing-emitting medium between concentric cylinders. The cylinder walls are assumed to be gray diffuse absorbers and emitters and have arbitrary temperature distributions along their peripheries. Heat generation may take place within the medium. As a first approximation, the problem is solved for optically-thick systems by using the differential approximation. To obtain accurate results also for the optically-thin and intermediate regimes, the differential approximation is subsequently improved by a number of geometric parameters, as has been discussed in a previous paper. As examples, two cares are presented in detail: (1) a hollow cylinder with uniform internal heat generation and uniform surface temperature and (2) a hollow cylinder with a cosine temperature distribution imposed on the wall, with no internal heat generation. Comparison with some numerical results generated by Hottel's zonal method shows excellent agreement.     

多维梯度折射率介质内辐射传递的扩散近似

推导多维梯度折射率介质内稳态辐射传递的扩散近似方程.使用有限元法对扩散近似进行离散和求解,利用两个二维半透明介质的稳态辐射传递问题验证该扩散近似的精度及适用性.算例考虑介质为均匀折射率及梯度折射率两种情况.利用扩散近似分别求解辐射平衡时的边界热流、介质内温度场分布,并与辐射传递方程的求解结果进行对比分析.结果表明:介质折射率变化、散射特性、光学厚度及散射反照率均直接影响扩散近似的精度;在光学厚及强散射条件下,该扩散近似可以作为一种快速算法应用于梯度折射率介质稳态辐射传递的求解.     

Analysis of combined conduction and radiation heat transfer in presence of participating medium by the development of hybrid method

The current study addresses the mathematical modeling aspects of coupled conductive and radiative heat transfer in the presence of absorbing, emitting and isotropic scattering gray medium within two-dimensional square enclosure. A blended method where the concepts of modified differential approximation employed by combining discrete ordinate method and spherical harmonics method, has been developed for modeling the radiative transport equation. The gray participating medium is bounded by isothermal walls of two-dimensional enclosure which are considered to be opaque, diffuse and gray. The effect of various influencing parameters i.e., radiation-conduction parameter, surface emissivity, single scattering albedo and optical thickness has been illustrated. The adaptability of the present method has also been addressed.     

Two analytic methods applied to radiative transfer in the linear-anisotropic scattering medium with graded index and diffuse gray boundaries

The curved ray-tracing method is extended to radiative transfer in the graded index medium with diffuse gray boundary conditions instead of black boundary conditions and the pseudo-source adding method is extended to the case of the linear-anisotropic scattering medium with graded index from non-scattering medium. Furthermore, the equivalence of the two methods is verified by formulation derivation. As exact analytical solutions, both the methods have high accuracy and fast computational speed. The predicted temperature distributions and dimensionless radiative heat flux at radiative equilibrium are determined by the proposed methods, and the numerical results are compared with the data in references. The results show that the present methods have a good accuracy. Influences of various combinations of refractive index and boundary emissivities on the temperature distributions and dimensionless radiative heat flux are also investigated.     

Numerical analyses of radiative heat transfer in any arbitrarily-shaped axisymmetric enclosures

A numerical approach for the treatment of radiative heat transfer in any irregularly-shaped axisymmetric enclosure filled with absorbing, emitting and scattering gray media is developed. Radiative transfer equation (RTE) is formulated for a general axisymmetric geometrical configurations, and the discretized equation is conducted using an unstructured meshes, generated by an appropriate computer algorithm, and the control volume finite element method which frequently adopted in CFD problems. A computer procedure has been done to solve the discretized RTE and to examine the accuracy and the computational efficiency of the proposed numerical approach. By using this computer algorithm, five test cases, a cylindrical enclosure with absorbing and emitting medium, a diffuser shaped axisymmetric enclosure, a finite axisymmetric cylindrical enclosure with a curved wall, a furnace with axially varying medium temperature and a rocket nozzle, are treated and the obtained results agree very well with other published works. Furthermore, the developed computer procedure has an accurate CPU time and it can be coupled easily with CFD codes.     

Radiative equilibrium of a gray medium in a rectangular enclosure

Radiative equilibrium temperature distributions are calculated for an absorbing-emitting gray medium enclosed by rectangular walls of different temperatures. A “modified differential approximation” is used to obtain an approximate solution, which compares favorably with direct numerical computations carried out using a zonal method.     

Development of a finite element model for coupled radiative and conductive heat transfer in participating media

Considering the geometrical applicability, a finite element model (FEM) for coupled radiative-conductive heat transfer has been developed which is applicable to enclosures of arbitrary geometry in present research. The present work provides a solution of coupled heat transfer in a rectangular, cylindrical or annulus enclosure with black or gray walls containing an absorbing-emitting-scattering medium. It is also applied to study the influence of conductive/radiation coefficient, albedo and wall emissivity on the temperature distribution in the medium. Compared with the results available in other references, the present FEM has no limitation with respect to geometry and can predict the coupled radiative-conductive heat transfer in participating media accurately.     

倾斜度及温度震荡频率对三维多孔介质方腔自然对流换热的影响

对三维多孔介质倾斜方腔内非稳态自然对流换热进行数值研究.腔体右壁面(X=1)保持恒温T₀,左壁面(X=0)基于温度T₀按正弦规律变化,其他所有壁面保持绝热.采用Brinkman扩展达西模型及SIMPLE算法模拟方腔内的流动.方腔沿y轴转动倾角α1的变化范围为0°~90°,沿x轴转动倾角α2的变化范围为0°~45°,无量纲温度震荡频率f的变化范围为5π~90π.详细研究倾角和温度震荡频率对三维方腔自然对流换热的影响.计算结果表明:当倾角α1=46°,α2=45°及温度震荡频率f=45π时,方腔内的换热最强.     

Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index

The curved ray tracing method (CRT) is extended to radiative transfer in the linear-anisotropic scattering medium with graded index from non-scattering medium. In this paper, the CRT is presented to solve one-dimensional radiative transfer in the linear-anisotropic scattering gray medium with a linear refractive index and two black boundaries. The predicted temperature distributions and radiative heat flux at radiative equilibrium are determined by the proposed method, and numerical results are compared with the data in references. The results show that the CRT has a good accuracy for radiative transfer in the linear-anisotropic scattering medium with graded index and the dimensionless emissive power and dimensionless radiative heat flux depend on the dimensionless refractive index gradient. It can also be seen that the dimensionless refractive index gradient has important effects on the temperature discontinuity at the boundaries.     

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.     

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.     

考虑Soret和Dufour效应的方腔内双扩散自然对流格子Boltzmann模拟

采用格子Boltzmann方法,考虑Soret和Dufour效应,对内置高浓度发热圆的方腔内部双扩散自然对流现象进行数值模拟.高浓度发热圆位于方腔中心,四周壁面均为低温低浓度.在该模型中,用三个独立的LBGK方程分别模拟速度场、温度场和浓度场,并通过Boussinesq近似将它们耦合起来.分析Soret数和Dufour数对方腔内部双扩散自然对流的影响,得到流线图、等温线图、等浓度线图、发热圆表面平均Nusselt数和平均Sherwood数.结果表明：Soret和Dufour效应对方腔内双扩散自然对流影响明显,不能忽略.     

Benchmark solutions of radiative integral transfer equations for three-dimensional emitting–absorbing and scattering media bounded by gray walls

Three-dimensional dimensionless radiative integral transfer equations (RITEs) for a cubic emitting–absorbing and isotropically scattering homogeneous medium of constant properties bounded by gray walls are solved using the method of “product integration”. The resultant system of linear equations for the incident energy is solved iteratively. Evaluation of the accuracy of the numerical solution is achieved by using the computer codes to make predictions for an idealized enclosure in which the exact analytical solution is possible. Comparison of the analytical and numerical values of medium temperatures and heat fluxes has shown that our method is logically correct and has good accuracy. Four benchmark problems for participating medium subjected to various combinations of internally uniform/non-uniform source terms are solved. To make the benchmark problems more complex and more general, different scattering albedo (ω=0.1, 0.5, 0.9), different wall emissivity (ε=0.1, 0.5, 0.9) and different optical thickness (τ=0.1, 1.0, 5.0) are considered. The solutions for the temperature of the medium and the heat flux components are given in tabular form and they may serve as standard values to assess the accuracy of other methods more conveniently.     

Thermal emission and volumetric absorption of a graded index semitransparent medium layer

Both thermal emission and volumetric absorption characteristics of a graded index semitransparent medium layer are investigated numerically. The semitransparent and specular emerging surface of the medium layer is parallel to an opaque and diffuse substrate wall. Monochromatic spectrum or gray medium is considered in the analysis. A pseudo-source adding method is combined with a ray-splitting and -tracing technique to solve the radiative transfer in the medium. As examples, constant and linear refractive index distributions are examined for an isothermal layer and the medium layer with a linear temperature distribution. Directional and hemispherical thermal emissions from the emerging surface as well as the volumetric absorption to the diffuse and parallel incidence of radiation are investigated, respectively. The results show that the refractive index distribution has significant influences on thermal emission and volumetric absorption of a semitransparent medium layer. The optical thickness, temperature distribution and the reflectivity of substrate wall react in combination with the refractive index distribution.     

Upshot of Chemical Species and Nonlinear Thermal Radiation on Oldroyd-B Nanofluid Flow Past a Bi-directional Stretched Surface with Heat Generation/Absorption ina Porous Media

A three-dimensional mathematical model is developed to examine the flow of nonlinear thermal radiation Oldroyd-B nanofluid past a bidirectional linearly stretched surface in a porous medium. The flow is induced by temperature dependent thermal conductivity, chemical reaction and convective heat and mass conditions. Novel characteristics of Brownian motion and thermophoresis are accompanied by magnetohydrodynamic and heat generation/absorption. Self-similar transformations are employed to convert the system of nonlinear partial differential equations to a system of ordinary differential equations with high nonlinearity and are solved by strong analytic technique named as Homotopy Analysis method (HAM). Effects of varied arising parameters on involved distributions are reflected through graphical illustrations. From this study, it is perceived that strong magnetic field hinders the fluid's motion and leads to rise in temperature that eventually lowers heat transfer rate from the surface. Further, decrease in heat transfer rate is also observed for enhanced values of thermal radiation parameter. To validate our results, a comparison with already published paper in limiting case is also given and results are found in excellent oncurrence; hence reliable results are being presented.     

Numerical analysis of nonsteady-state conjugate natural convection between two concentric spheres

Mathematical modeling of the regimes of free convection heat transfer between two concentric spheres is performed. The temperature of the internal sphere is constant during the whole process; the external sphere with a heat-conducting enclosure of finite thickness is affected by the surrounding medium. The boundary-value problem is formulated based on the mass, momentum, and energy conservation laws in dimensionless stream function-velocity vorticity vector-temperature variables in spherical coordinates. The obtained distributions of temperature and streamlines represent the effect of both energy source and environment on the flow regimes and the heat transfer.     

Non-Fourier heat conduction with radiation in an absorbing, emitting, and isotropically scattering medium

This article numerically analyses the combined conductive and radiative heat transfer in an absorbing, emitting, and isotropically scattering medium. The non-Fourier heat conduction equation, which includes the time lag between heat flux and the temperature gradient, is used to model the conductive heat transfer in the medium. It predicts that a temperature disturbance will propagate as a wave at finite speed. The radiative heat transfer is solved using the P₃ approximation method. In addition, the MacCormack's explicit predictor-corrector scheme is used to solve the non-Fourier problem. The effects of radiation including single scattering albedo, conduction-to-radiation parameter, and optical thickness of the medium on the transient and steady state temperature distributions are investigated in detail. Analysis results indicate that the internal radiation in the medium significantly influences the wave nature. The thermal wave nature in the combined non-Fourier heat conduction with radiation is more obvious for large values of conduction-to-radiation parameter, small values of optical thickness and higher scattering medium. The results from non-Fourier-effect equation are also compared to those obtained from the Fourier equation. Non-Fourier effect becomes insignificant as either time increases or the effect of radiation increases.     

Single-phase Stefan problem in selectively absorbing medium

The thermal state of a translucent selectively absorbing medium was studied by the methods of numerical simulation at different values of the optical properties of boundaries and heat transfer from the left surface in approximation of one-phase Stefan problem. The temperature fields and densities of resultant radiation fluxes as well as the thermal state of the left boundary and dynamics of layer reduction in the melting process were analyzed. The processes of phase transition in a flat layer of selective and gray absorbing media and emitting media were compared, and their fundamental differences were shown.     

Three-dimensional non-grey gas radiative transfer analyses using the statistical narrow-band model

Three-dimensional non-grey gas radiation analyses were conducted using the statistical narrow-band model with updated band parameters and three implementation methods: the exact or the correlated formulation, the non-correlated expression, and the grey-band approximation with the absorption coefficient estimated using the local properties. The accuracy of the two approximate narrow-band implementation methods was evaluated for both low-resolution spectral intensity and spectrally integrated radiative source term and wall heat flux in a rectangular enclosure containing (1) isothermal pure water and (2) a CO₂/H₂O/N₂ mixture with a furnace type gas temperature distribution. For spectrally integrated quantities, results of the grey-band approximation are very close to those of the non-correlated formulation and are in qualitative agreement with the results of the correlated formulation. The two approximate methods are capable of predicting qualitatively correct and fairly accurate distributions of low-resolution spectral radiation intensities for the isothermal case. However, they predict less accurate low-resolution spectral intensities for the non-isothermal case.     

Upshot of Chemical Species and Nonlinear Thermal Radiation on Oldroyd-B Nanofluid Flow Past a Bi-directional Stretched Surface with Heat Generation/Absorption in a Porous Media

A three-dimensional mathematical model is developed to examine the flow of nonlinear thermal radiation Oldroyd-B nanofluid past a bidirectional linearly stretched surface in a porous medium. The flow is induced by temperature dependent thermal conductivity, chemical reaction and convective heat and mass conditions. Novel characteristics of Brownian motion and thermophoresis are accompanied by magnetohydrodynamic and heat generation/absorption.Self-similar transformations are employed to convert the system of nonlinear partial differential equations to a system of ordinary differential equations with high nonlinearity and are solved by strong analytic technique named as Homotopy Analysis method(HAM). Effects of varied arising parameters on involved distributions are reflected through graphical illustrations. From this study, it is perceived that strong magnetic field hinders the fluid's motion and leads to rise in temperature that eventually lowers heat transfer rate from the surface. Further, decrease in heat transfer rate is also observed for enhanced values of thermal radiation parameter. To validate our results, a comparison with already published paper in limiting case is also given and results are found in excellent oncurrence; hence reliable results are being presented.