Numerical Solution of the Radiative Transfer Equation in Spherical Shells

A method of solving numerically problems of radiative transferin shells with spherical symmetry is proposed. The differenceequations are derived by discrete ordinate methods and solvedusing algorithms due to Grant & Hunt (1968, 1969). We haveshown that the method is stable, and will yield non-negativesolutions, provided the optical thickness of an elementary shelland the ratio of its thickness to the radius satisfy certainrestrictions. Two simple problems are presented to illustratethe theory.     

Stationary Problem of Radiative Heat Transfer with Cauchy Boundary Conditions

Computational Mathematics and Mathematical Physics - A stationary problem of radiative-conductive heat transfer in a three-dimensional domain is studied in the $${{P}_{1}}$$ -approximation of the...     

Shape Optimal Design Problem with Convective and Radiative Heat Transfer: Analysis and Implementation

We present a study of an optimal design problem for a coupled system, governed by a steady-state potential flow equation and a thermal equation taking into account radiative phenomena with multiple reflections. The state equation is a nonlinear integro-differential system. We seek to minimize a cost function, depending on the temperature, with respect to the domain of the equations. First, we consider an optimal design problem in an abstract framework and, with the help of the classical adjoint state method, give an expression of the cost function differential. Then, we apply this result in the two-dimensional case to the nonlinear integro-differential system considered. We prove the differentiability of the cost function, introduce the adjoint state equation, and give an expression of its exact differential. Then, we discretize the equations by a finite-element method and use a gradient-type algorithm to decrease the cost function. We present numerical results as applied to the automotive industry.     

A Mixed Analytical/Numerical Method for Velocity and Heat Transfer of Laminar Power-Law Fluids

This paper presents a relatively simple numerical method to investigate the flow and heat transfer of laminar power-law fluids over a semi-infinite plate in the presence of viscous dissipation and anisotropy radiation. On one hand, unlike most classical works, the effects of power-law viscosity on velocity and temperature fields are taken into account when both the dynamic viscosity and the thermal diffusivity vary as a power-law function. On the other hand, boundary layer equations are derived by Taylor expansion, and a mixed analytical/numerical method (a pseudo-similarity method) is proposed to effectively solve the boundary layer equations. This method has been justified by comparing its results with those of the original governing equations obtained by a finite element method. These results agree very well especially when the Reynolds number is large. We also observe that the robustness and accuracy of the algorithm are better when thermal boundary layer is thinner than velocity boundary layer.     

纳米流体强化传热问题的高效算法设计及其高精度数值模拟

对周期性温度边界条件下方腔内Copper-水纳米流体的对流传热问题建立了完全四阶高精度紧致差分格式,并探讨了该格式高效算法设计,同时引入伪时间导数项采用Von Neumann方法证明了格式的绝对稳定性.研究了不同Ri数下纳米颗粒体积分数、左右边界温度的正弦振幅比和相位偏差对纳米流体传热效率的影响.数值结果表明,对于给定的Ri数,随着纳米颗粒体积分数的增大和边界温度的正弦振幅比的增加,纳米流体的传热效果会显著增强,而相位偏差只影响方腔右侧的温度分布.     

Theoretical and Numerical Analysis of an Initial-Boundary Value Problem for the Radiative Transfer Equation with Fresnel Matching Conditions

A Cauchy problem for the time-dependent radiative transfer equation in a three-dimensional multicomponent medium with generalized matching conditions describing Fresnel reflection and refraction at the interface of the media is considered. The unique solvability of the problem is proven, a Monte Carlo method for solving the initial-boundary value problem is developed, and computational experiments for different implementations of the algorithm are conducted.     

Mathematical Modelling of Radiative Heat Exchanges in Czochralski Crystal Pulling

Crowley has published a mathematical model of the heat flowin Czochralski crystal pulling, assuming the surfaces of solidcrystal and liquid meniscus to be cooled by radiation to theatmosphere. An enhanced model is described that takes accountof the radiative heat exchanges between the atmosphere and thesurfaces of the crystal, the meniscus and the hot crucible wall. For computational purposes, the modelling of the radiative heatexchanges must be sufficiently simplified to allow frequentre-calculation of the relevant heat fluxes in the course ofa computer simulation of a Czochralski pulling experiment. Radiationoptics suggests a number of simplifications, which are explained.The geometry of the physical system, however, is not one forwhich standard results exist, and the techniques used to overcomethe consequent problems may be of interest in related contexts. The new model is compared numerically with the Crowley modelfor the case of germanium pulling; its sensitivity to variationof the height of the crucible wall, and its responses to variationof some other parameters are also illustrated.     

Modeling of Radiative Heat Conduction on High-Performance Computing Systems

Doklady Mathematics - For problems related to radiative heat conduction, an algorithm is proposed that is well adapted to the architecture of systems with extramassive parallelism. According to the...     

A Hyperbolic Heat Transfer Problem with Phase Changes

There is a large literature on hyperbolic heat-transfer problems,and a few investigators have studied phase-change or melt problemsfor materials governed by such systems. Most formulations ofmelt problems for hyperbolic models insist on continuity ofthe temperature across the melt interface, and a number of investigatorshave observed that this insistence leads to mathematical difficulties.In this paper an alternative model is explored, where continuityof the temperature at the melt interface is not imposed. Instead,we insist that the relaxation process describing the relationbetween heat flux and temperature gradient be interpreted asa conservation equation that must hold across a melt interface.With this formulation, we are led to a solvable problem withdesirable asymptotic properties.     

Variational Principles for the Radiative Transfer Equation of Single Scattering

The equation of radiative transfer for single scattering isrewritten in two alternative ways as a selfadjoint integro-differentialequation. This equation is then expressed in terms of a variationalprinciple, where the corresponding Lagrangian contains integralas well as differential operations. An application is made tothe one-dimensional transfer equation and a conservation lawis established by means of a generalized version of Noether'stheorem. Some consequences of this conservation equation arealso discussed.     

梯度折射率介质内瞬态辐射传递的扩散近似法

基于瞬态时域与频域扩散近似方程求解了一维梯度折射率介质内辐射传递问题.通过计算在不同梯度折射率分布下的穿透率与反射率,对扩散近似的适用性展开研究.结果表明:求解时域穿透率时,扩散近似法始终能够准确计算出平行光的时间延迟,而对漫射部分时间延迟的计算不准确;对时域反射率而言,扩散近似法在反射率出现峰值之前与间断有限元法结果相吻合,随着折射率梯度的增大,扩散近似结果在出现峰值后的时间内偏大,且有明显的时移.对于频域穿透率和反射率计算,扩散近似法精度几乎不受折射率梯度变化的影响,能够较为准确地预测频域穿透率,但是对于频域反射率计算误差较大.     

The Transport Equation of Radiative Transfer in a Three-dimensional Space with Anisotropic Scattering

Present address: Science Research Council, Atlas Computer Laboratory, Chilton, Didcot, Berkshire, England. The exact solution of a radiative transfer problem requiresthe reduction of the integro-differential transport equationto an integral equation. This paper considers the transportequation in a general three-dimensional space, when the scatteringfunction consists of the first three terms of a series of Legendrepolynomials. The resulting integral equations may then be usedfor the solution of Radiative Transfer problems in any space.where the radiation is restricted to normal incidence.     

一种新的变域传热问题的边界条件

回顾古典热传导方程建立的假设条件的基础上,分析了热层材料由于表面烧蚀而引起的传热区域内部的热漏现象.基于能量守恒原理,利用有限元分析法推导出变域热传导方程,并得到了热漏函数的表达式,提出了变域传热问题边界条件的改进形式.为了检验这种边界条件的合理性,利用Crank-Nicholson法对此数学模型进行空间和时间离散化,并进行了数值仿真求解.仿真结果证实,基于边界条件改进形式的数学模型使计算更方便,结果更符合实际,从而为工程应用提供理论分析的依据.同时,该数学模型也为研究动边界发汗冷却控制问题奠定了理论研究的基础.     

圆截面弯管流动的数值研究

本文利用有限差分方法研究稍微有点弯曲的圆截面管道中粘性流体的稳定流动,数值结果表明我们所取的算法是合理的和有效的。