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.     

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.     

Multilayer differential discrete ordinate method for inhomogeneous participating media

This paper presents a multilayer differential discrete ordinate method to solve the radiative transfer equation for an absorbing, emitting and scattering inhomogeneous plane parallel medium. This method reduces the integro-differential equation into a set of coupled first order ordinary differential equations with two point boundary conditions on using a suitable quadrature scheme. These equations are then solved numerically. Numerical validation of the method for gray medium is done by comparing the results obtained with benchmark cases available in the literature. Validation for a non-gray medium is done by considering a problem concerning radiative transfer from the atmosphere. The brightness temperature at the top of the atmosphere is calculated at various frequencies and validated with those obtained by several other numerical methods.     

Meshless method for solving radiative transfer problems in complex two-dimensional and three-dimensional geometries

A meshless method is presented to solve the radiative transfer equation in complex 2D and 3D geometries. In order to avoid numerical oscillations, the even parity formulation of the discrete ordinates method is used. A moving least squares approximation meshless method is used to solve the second order partial differential equations. Prediction results of radiative heat transfer problems obtained by the proposed method are compared with reference in order to assess the correctness of the present method.     

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.     

Meshless Method for Solving Transient Radiative and Conductive Heat Transfer in Two-Dimensional Complex Geometries

A diffuse approximation meshless method (DAM) is employed to solve the transient radiative and conductive heat transfer problem in a semitransparent medium enclosed in 2-D complex geometries. The computational spatial domain is discretized by a set of nodes scattered in the domain and boundary without information on the relationship between them. The meshless method for radiative transfer equation is based on the even-parity formulation of the discrete ordinates method without any form of upwinding. Results of dimensionless temperature distribution at different dimensionless times are obtained and validated with other benchmark approximate solutions in order to illustrate the performance of the proposed method.     

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.     

Analysis of Conduction and Radiation Heat Transfer in a Differentially Heated 2‐D Square Enclosure

Radiative heat transfer with and without conduction in a differentially heated 2‐D square enclosure is analyzed. The enclosure with diffuse gray boundaries contains radiating and/or conducting gray homogeneous medium. Radiatively, the medium is absorbing, emitting and scattering. On the south boundary, four types of discrete heated regions, viz., the full boundary, the left one‐third, left two third and middle one third, are considered. In the absence of conduction, distributions of heat flux along the south boundary are studied for the effect of extinction coefficient. In the presence of conduction, distributions of radiation, conduction and total heat fluxes along the south boundary are analyzed for the effects of extinction coefficient, scattering albedo, conduction–radiation parameter, and south boundary emissivity. Effects of these parameters on centerline temperature distribution are also studied. To assess the performance of three commonly used radiative transfer methods, in all cases, the radiative transfer equation is solved using the discrete ordinate method (DOM), the conventional discrete ordinate method (CDOM) and the finite volume method (FVM). In the combined mode problem, with volumetric radiative information known from one of the three methods, viz., DOM, CDOM, and FVM, the energy equation is solved using the finite difference method (FDM). In all cases, the results from FDM‐DOM, FDM‐CDOM, and FDM‐FVM are in good agreement. Computationally, all three sets of methods are equally efficient.     

A New and Simple Technique to Normalize the HG Phase Function for Conserving Scattered Energy and Asymmetry Factor

A new, yet simple, technique is formulated for normalizing the Henyey-Greenstein (HG) phase function by ensuring conservation of both scattered energy and asymmetry factor simultaneously, and is analyzed for use in determining accurate radiative transfer predictions in strongly anisotropic scattering media using the discrete ordinates method (DOM). Two recently published simple normalization techniques are able to conserve either scattered energy or asymmetry factor after discretization solely by normalization of the forward-scattering HG phase-function value. However, normalization of only the forward-scattering term cannot conserve two quantities simultaneously. The present technique normalizes both the forward-scattering and backward-scattering terms in order to conserve both scattered energy and asymmetry factor simultaneously and maintain most of the phase-function shape while retaining simplicity and efficiency. Analysis of radiative transfer predictions shows that results generated using the present technique conform accurately to finite-volume method (FVM) and Monte Carlo (MC) predictions, as well as to those generated using the authors' previously developed matrix normalization technique, validating its accuracy.     

A Universal Performance Chart for CPU Cooling Devices

This article deals with performance evaluation of the discrete ordinates method in terms of its capacity to provide accurate results in solving radiation mode problems with different complexities. The problem formulation is done keeping in view the heat transfer process in an evaporator used in a coal-based thermal power station. The hot gas mass in the evaporator is assumed to be a heat source with two different shapes, spherical and conical. The evaporator walls that are covered with a water cooling jacket are modeled with a convective boundary condition. The gas is assumed to be gray and absorbing–emitting. The solution of the radiative transport equation is carried out using the discrete ordinates method. Parametric studies are performed for a wide range of aspect ratio, extinction coefficient, and convective heat transfer coefficient. The code is validated by comparing the result of discrete ordinates method with the exact and the discrete transfer method for nonradiative as well as radiative equilibrium conditions.     

Monte Carlo simulation of transient radiative transfer in a medium with a variable refractive index

In this work, a Monte Carlo method is developed to simulate transient radiative transfer in a refractive planar medium exposed to a collimated pulse irradiation. The time of flight in closed form is derived for a medium with a linearly varying refractive index. The time-resolved reflectance and transmittance of the slab are obtained by tracing photon bundles and calculating the time of flight. There is a very satisfying correspondence between the present results and the discrete ordinates solutions. The magnitude of numerical uncertainty decreases with the increase of bundles. To simulate transient radiative transfer in an optically thick medium, we need a large number of bundles. The reflectance decreases with the increase of the positive gradient of the refractive index considered. The appearance instant of the transmittance peak postpones as the refractive index increases. Influence of the optical thickness, the scattering albedo and the anisotropically scattering coefficient on the time-resolved reflectance and transmittance is also investigated.     

燃烧室内三维温度场的辐射反问题

本文提出了一种在介质辐射特性已知的条件下,由壁面入射辐射热流的测量值反演燃烧室内三维温度场的方法。该方法是在辐射传递方程离散坐标近似的基础上,用求目标函数极小值的共轭梯度法进行反演计算。通过对吸收系数、散射不对称因子、反照率、壁面黑度和燃烧室大小尺寸等参数对反演精度影响的分析,结果表明,即使存在随机测量误差,这些参数对温度场反演精度的影响也不大,本文所提出的方法可较精确地反演燃烧室内三维温度场。     

燃烧产物辐射特性的误差对炉内辐射传热计算精度的影响

本文用离散坐标法对含吸收散射性介质矩形空腔内的３维辐射传递过程进行了模拟，并编写了相应的数值计算程序。利用该程序分析了介质的吸收系数、散射系数、相函数、光谱特性及壁面灰渣沉积层黑度的不确定性对矩形燃烧室内烟气温度及热流计算精度的影响。结果表明计算精度很大程度上取决于燃烧产物辐射特性的取值精度，特别是壁面灰渣沉积层黑度的取值精度。在煤粉燃烧室中，介质的散射不宜忽略。     

3D modelling of radiative heat transfer in circulating fluidized bed combustors: influence of the particulate composition

A three-dimensional model is developed to predict the bed-to-wall radiative heat transfer coefficient in the upper dilute zone of circulating fluidized bed (CFB) combustors. The radiative transfer equation is solved by the discrete ordinates method and Mie scattering theory is applied to calculate the absorption and scattering efficiency factors of particles existing in CFB combustors. Empirical correlations calculate both spacial variation of solid volume fraction and temperature distribution at the wall. The model considers the influences of the particle properties (including particle size distribution, particle optical constants and solid composition) on the radiative heat transfer coefficient. Simulation results show that the particle properties have significant influences on the bed-to-wall radiative heat transfer coefficient in CFB combustors. A very good agreement of predicted results is shown with experimental data.     

Combined conduction and radiation heat transfer in a gray anisotropically scattering planar medium with diffuse-specular boundaries

Coupled conduction and radiation heat transfer in a gray planar nonlinearly anisotropic scattering medium bounded between two plane parallel surfaces reflecting both diffusely and specularly is analyzed. The governing integrodifferential equations are solved by a numerical iterative method consisting of Numerov's method to solve the energy equation and Chandarsekhar's discrete ordinates method in conjunction with the Crank-Nicolson method to solve the radiative transfer equation. Convergence of the solution is enhanced by Ng-acceleration. The numerical algorithm described is found to be fast and reliable. Numerical results based on S₃₂ method indicate that anisotropy plays an important role, and difference between the diffuse and specular reflections is found to be insignificant.     

NATURAL CONVECTION AND RADIATION IN A SQUARE ENCLOSURE

Natural convection of a radiating fluid in a square enclosure is studied numerically. The coupled momentum, energy, and radiative transfer equations are solved by an iterative procedure. The solutions to the equation of radiative transfer are obtained by the discrete ordinates method using S4 and S8 quadratures. The method is based on control volume formulation and is fully compatible with the SIMPLER algorithm used to solve the momentum and energy equations. The effects of optical thickness and scattering on the flow and temperature fields and heat transfer rates are analyzed. The changes in the buoyant flow patterns and temperature distributions due to the presence of radiation in inclined or heat generating enclosures are also studied. Comparative results obtained by the P-I differential approximation are presented.     

A two-step discrete method for reconstruction of temperature distribution in a three-dimensional participating medium

A new two-step discrete method is proposed in this paper for reconstruction of three-dimensional temperature distribution in an absorbing, emitting and isotropically scattering medium. With this new method, the temperature of the wall is also considered. The local radiative source term is reconstructed in the first step through the discrete transfer method from the directional, exit radiation intensities measured by CCD cameras. Then, the temperature of each discrete element is calculated in the second step by subtracting the scattering contribution from the retrieved radiative source term through the discrete ordinate method. The least squares minimum residual algorithm is employed to solve the ill-posed reconstruction equations and the calculation is improved to reduce the computational cost. The performance of the proposed method is examined by numerical test problems with unimodal and bimodal temperature distributions. The ill-conditioning of the reconstruction problem is checked by the Picard condition. The effects of the measurement noise and the radiative properties on the reconstruction accuracy are discussed. The results show that the method proposed in this paper is capable of reconstructing the temperature distribution accurately in large, confined, participating media, even with noisy input data. The computation time reduction of this new method is significant when compared with other methods.     

Integral equation solutions based on exact ray paths for radiative transfer in a participating medium with formulated refractive index

The exact analytical path length of radiation traveling in a slab with formulated variable refractive index is derived. Based on the analytical path lengths, the integral equations in terms of intensity moments for radiative transfer in a participating slab with one of the family of spatially varying refractive indices are developed. We solve the integral equations for radiative transfer in a slab at radiative equilibrium or for radiative transfer in an isothermal slab. The boundaries are assumed to be black for the slab at radiative equilibrium and the index jumps at both boundaries for the isothermal slab are considered. For comparison purpose, we also solve the radiative equilibrium problems by the discrete ordinates method (DOM). The nondimensional emissive power and nondimensional radiative heat flux obtained by solving integral equations show an excellent agreement with those obtained by the DOM. For the slab at radiative equilibrium and with positive gradient of refractive index, the jump of the emissive power at bottom boundary decreases with the increase of optical thickness for the cases with slightly varying refractive index, but the trend may not hold for the cases with significantly varying refractive index. For the non-scattering slab with positive gradient of refractive index and fixed refractive indices at the boundaries, the directional emittances at both boundaries for the case with linear refractive index are smaller than those for the case with a refractive index of slope-increasing profile. Effects of the scattering albedo and the scattering phase function coefficient are investigated too.     

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.     

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.