一维各向同性散射系统内辐射传递分析的ES-DRESOR法
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摘要
同其他传热方式相比,辐射传热包含了发射、吸收、散射、反射等复杂过程,其控制方程为微分积分方程。在辐射换热问题分析以及实际应用中,辐射传递方程的求解和辐射强度的计算至为关键。到目前为止,已经发展了多种方法用来求解辐射传递方程,如离散坐标法、蒙特卡洛法、热流法、区域法、球形谐波法等,这些方法各有优缺点,根据其特点被用于不同的辐射问题求解中。在一些辐射问题的研究中,方向辐射强度信息显得格外重要。基于蒙特卡洛法发展起来的DRESOR (Distributions of Ratio of Energy Scattered by the medium Or Reflected by the boundary surface,被介质散射或者被壁面反射的能量份额分布)法,因能够提供高方向分辨率的辐射强度,而被应用到一系列的辐射传递问题的研究中,如工业炉内燃烧温度场的可视化监控、平行脉冲入射辐射、各向异性散射、梯度折射率介质中等。然而因该方法基于蒙特卡洛法,计算耗时不可避免,同时存在着统计误差,限制了此方法的进一步发展。本文致力于提高DRESOR法计算效率和计算精度的研究,在基本DRESOR法的基础上发展了ES-DRESOR (Equation-solving DRESOR)法。通过求解线性方程组直接得到DRESOR数(DRESOR法求解的核心),无需像先前那样由蒙特卡洛法的能束跟踪得到,以实现计算效率和计算精度上都能有大幅度的提升。本研究对于提高DRESOR法的分析求解能力,拓展其应用范围具有重要意义。具体来说,主要工作如下：在基本DRESOR法的基础上提出了ES-DRESOR。给出了该方法的计算原理,分为透明壁面和漫壁面参与两种情况,对于前者,设定系统内某微元体具有单位黑体发射,其它部分没有发射,这样对DRESOR法给出的辐射强度简化后,可由入射辐射(对辐射强度的空间积分)的两种计算方法(即DRESOR法给出的计算公式和标准计算公式)所得结果相等,建立关于DRESOR数的方程。在漫壁面参与辐射过程后,对DRESOR法计算入射辐射表达式进行了修正,加入壁面的贡献部分。对壁面单元建立方程的原则是被壁面反射的总能量等于其它单元散射或反射到此壁面的总能量；对空间单元建立方程的原则依然是由两种方法计算的入射辐射相等。在一维吸收发射各向同性散射的辐射系统内,对ES-DRESOR求解辐射传递方程进行了研究。数值部分,首先对ES-DRESOR法的可靠性进了验证,包括与文献中的精确解对比,与广泛使用的离散坐标法对比,以及将ES-DRESOR法计算的结果代入到辐射传递方程进行检验。而后将ES-DRESOR法计算出来的DRESOR数与蒙特卡洛法计算的结果进行对比,显示了很好的吻合性,且消除了蒙特卡洛法中因统计误差存在而导致的“抖动”现象。对比ES-DRESOR法同蒙特卡洛法计算出来的辐射强度相对误差,前者的精度高出一个数量级。更为重要的是ES-DRESOR法的计算效率在测试网格数下,与基于蒙特卡洛法的基本DRESOR法相比,有两个数量级的提升。
Compared with other modes of heat transfer, radiative heat transfer contains some more complex process as the emission, absorption, scattering and reflection of the radiative energy. The control equation of the radiative heat transfer is an integral differential equation. How to solve the radiative heat transfer equation (RTE) and calculate the radiative intensity is the key in the analysis of radiative heat transfer problems and practical applications. So far, many methods have been developed to solve the RTE, for example, the Discrete Ordinate method, the Monte Carlo method, the Heat Flux method, the Zone method, the Spherical Harmonics method and others. These methods do have advantages and drawbacks, and are used in different radiative heat transfer problems according to their own characteristics. In some radiative problems, the information of intensity in direction is particularly important. Based on the Monte Carlo method, the DRESOR method (Distributions of Ratio of Energy Scattered by the medium Or Reflected by the boundary surface) can get the radiative intensity with high directional resolution and has been applied to a series of radiative heat transfer problems, as the visual monitoring of the combustion temperature field in the industrial furnaces, the collimated pulse incidence, the an-isotropic scattering media and the gradient index medium and so on. The calculating of the DRESOR values is the kernel and demands the most calculating time of the program. As it is based on the MCM, the time-consuming is unavoidable and the statistical error exits, which limit the development of this method.This paper aims to improve the computational efficiency and accuracy of the DRESOR method. On the basis of the basic DRESOR method, Equation solving DRESOR method, ES-DRESOR method, is devoloped in this paper. In this method, the DRESOR values which is the kernel in the DRESOR method can be got by solving a set of linear equations instead of by the traditional ray-tracing using MCM, so as to have a great enhance on the calculation efficiency and accuracy. This study has an important significance for improving ability of DRESOR method solving radiation problems and expand its application scope. Specifically, the main work is as follows:The principles of ES-DRESOR method are given. The computing principles contain the transparent boundary surfaces and the diffuse boundary surfaces two conditions. For the former, assuming an arbitrary element emission of the system as a blackbody emission while other elements with no emission, which simplifies the intensity description given by the DRESOR. Then equation can be set up by two different calculating description of incident radiation (the standard calculating equation and the equation under the DRESOR method) equals to each other. For the latter, the calculation description of the incident intensity under the DRESOR method needs to be modified, with the contribution of the boundary surface added. The principle of establishing equations for the boundary surface elements is that the total energy reflected by boundary surface equals to the total energy scattered and/or reflected to this boundary surface. As for the spatial elements, the difference with the former one is that the contribution of boundary surface needs to be added.Numerical verification is done on this method in one-dimensional absorbing, emitting, and isotropicall scattering system. First, the reliability of this method is verified including compared with the exact solution in the literature, compared with the widely used DOM, and substituted the ES-DRESOR results into the RTE to test. Then compared the DRESOR values with the MCM, the results show a good agreement of the two method. The shaking phenomena of DRESOR values caused by the statistical error in the MCM is eliminated in the ES-DRESOR method. Compared the relative error of the radiative intensity calculated by the ES and MCM, the accuracy of the former is one order magnitude higher. What is more important is that the efficiency of the ES is two order magnitudes higher than MCM under the same discrete grids.
Compared with other modes of heat transfer, radiative heat transfer contains some more complex process as the emission, absorption, scattering and reflection of the radiative energy. The control equation of the radiative heat transfer is an integral differential equation. How to solve the radiative heat transfer equation (RTE) and calculate the radiative intensity is the key in the analysis of radiative heat transfer problems and practical applications. So far, many methods have been developed to solve the RTE, for example, the Discrete Ordinate method, the Monte Carlo method, the Heat Flux method, the Zone method, the Spherical Harmonics method and others. These methods do have advantages and drawbacks, and are used in different radiative heat transfer problems according to their own characteristics. In some radiative problems, the information of intensity in direction is particularly important. Based on the Monte Carlo method, the DRESOR method (Distributions of Ratio of Energy Scattered by the medium Or Reflected by the boundary surface) can get the radiative intensity with high directional resolution and has been applied to a series of radiative heat transfer problems, as the visual monitoring of the combustion temperature field in the industrial furnaces, the collimated pulse incidence, the an-isotropic scattering media and the gradient index medium and so on. The calculating of the DRESOR values is the kernel and demands the most calculating time of the program. As it is based on the MCM, the time-consuming is unavoidable and the statistical error exits, which limit the development of this method.This paper aims to improve the computational efficiency and accuracy of the DRESOR method. On the basis of the basic DRESOR method, Equation solving DRESOR method, ES-DRESOR method, is devoloped in this paper. In this method, the DRESOR values which is the kernel in the DRESOR method can be got by solving a set of linear equations instead of by the traditional ray-tracing using MCM, so as to have a great enhance on the calculation efficiency and accuracy. This study has an important significance for improving ability of DRESOR method solving radiation problems and expand its application scope. Specifically, the main work is as follows:The principles of ES-DRESOR method are given. The computing principles contain the transparent boundary surfaces and the diffuse boundary surfaces two conditions. For the former, assuming an arbitrary element emission of the system as a blackbody emission while other elements with no emission, which simplifies the intensity description given by the DRESOR. Then equation can be set up by two different calculating description of incident radiation (the standard calculating equation and the equation under the DRESOR method) equals to each other. For the latter, the calculation description of the incident intensity under the DRESOR method needs to be modified, with the contribution of the boundary surface added. The principle of establishing equations for the boundary surface elements is that the total energy reflected by boundary surface equals to the total energy scattered and/or reflected to this boundary surface. As for the spatial elements, the difference with the former one is that the contribution of boundary surface needs to be added.Numerical verification is done on this method in one-dimensional absorbing, emitting, and isotropicall scattering system. First, the reliability of this method is verified including compared with the exact solution in the literature, compared with the widely used DOM, and substituted the ES-DRESOR results into the RTE to test. Then compared the DRESOR values with the MCM, the results show a good agreement of the two method. The shaking phenomena of DRESOR values caused by the statistical error in the MCM is eliminated in the ES-DRESOR method. Compared the relative error of the radiative intensity calculated by the ES and MCM, the accuracy of the former is one order magnitude higher. What is more important is that the efficiency of the ES is two order magnitudes higher than MCM under the same discrete grids.
引文
[1]付茂林,彭志华.大学物理(下册).武汉：华中科技大学出版社,2009.
[2] Siegel, R., Howell, J. R. Thermal radiation heat transfer. (4th). New York:Taylor & Francis,2002.
[3] Modest M F. Radiative heat transfer, second edition. San Diego:Academic Press, 2003.
[4]余其铮.辐射换热原理.哈尔滨：哈尔滨工业大学出版社,2000.
[5]杨世铭,陶文铨.传热学.4版.北京：高等教育出版社,2006.
[6] Kamiuto, K. Kanemaru, K. Approximate Method for Combined Forced-Convection and Radiation Heat Transfer in Absorbing and Emitting Gases Flowing in a Black, plane-parallel Duct, Int. J. Heat Mass Transfer.1997,39:2191-2193.
[7]刘林华.炉膛传热计算方法的发展概况.动力工程,2000,21(1)：523-527.
[8]郑楚光,柳朝晖.弥散介质的光学特性及辐射传热.武汉：华中理工大学出版社,1996.
[9] Carvalho M G, Semiao V S, Coelho P J. Modelling and optimization of the no formation in an industrial glass furnace Journal of Engineering for Industry,1992, 114(4):514-523.
[10] Dombrovsky L A, Sazhin S S, Sazhina E M, et al. Heating and evaporation of semi-transparent diesel fuel droplets in the presence of thermal radiation. Fuel.2001, 80:1535-1544.
[11] Chetverushkin B N, Polyakov S V, Kudryashova T A, et al. Numerical simulation of 2D radiation heat transfer for reentry vehicles. Parallel computational fluid dynamics 2005-Theory and applications.2006,293-299.
[12] Fujita K, Abe T, Suzuki K. Uv radiation impact on heating rate of super-orbital reentry vehicle. International Symposium on Space Technology and Science,21st, 1998, Omiya, Japan,1:963-968.
[13] Coelho P J. Detailed numerical simulation of radiative transfer in a nonluminous turbulent jet diffusion flame. Combustion and Flame.2004,136:481-492.
[14]赵恒凯.辐射散射组合遥感系统关键技术研究：博士学位论文.上海大学,2006.
[15] Kumar S, Mitra K. Microscale aspects of thermal tadiation transport and laser application. Advances in Heat Transfer.1999,33:187-294.
[16] Qui T G, Tien C L. Short pulse laser heating in metals. International Journal of Heat and Mass Transfer.1992,35:719-726.
[17] Bokota A, Iskierka S. Numerical simulation of transient and residual stresses caused by laser hardening of slender elements. Computational Materials Science,1997, 7(4):366-376.
[18] Patterson M S, Wilson B C, Wyman D R. The propagation of optical radiation in tissue Ⅰ. Models of Radiation Transport and Their Application Lasers in Medical Science.1991,6(2):155-168.
[19] Yamada Y. Light-tissue interaction and optical imaging in biomedicine. Annual Review of Heat Transfer 1995,6:1-59.
[20] Okada E, Firbank M, Schweiger M, et al. theoretical and experimental investigation of near-infrared light propagation in a model of the adult head. Applied Optics.1997, 36:21-31.
[21] M. Sakami, K. Mitra and T. Vo-Dinh. Analysis of Short-Pulse Laser Photo Transport through Tissues for Optical Tomography. Opt. Lett.2002,27(5):336-338.
[22] Hebden J C, Veenstra H, Dehghani H, et al. Three-dimensional time-resolved optical tomography of a conical breast phantom. Applied Optics.2001,40:3278-3287.
[23]谈和平,夏新林,刘林华,等.红外辐射特性与传输的数值计算一计算热辐射学.哈尔滨：哈尔滨工业大学出版社,2006.
[24]娄春.煤粉炉内三维温度场及颗粒辐射特性重建：博士学位论文.华中科技大学,2007.
[25]周怀春.炉内火焰可视化检测原理与技术.北京：科学出版社,2005.
[26]范维澄,万跃鹏.流动与燃烧的模型与计算.合肥：中国科学技术大学出版社,1992.
[27] Zhou H C., Sheng F, Han S D, et al. Reconstruction of temperature distribution in a 2-D absorbing-emitting system from radiant energy images. JSME international journal. Series B:Fluids and thermal engineering.2000,43(1):104-109.
[28] Chandrasekhar S. Radiative Transfer, New York:Dove Publications Inc.,1960.
[29] Lathrop K D. Use of discrete-ordinate methods for solution of photon transport problems. Nuclear Science and Engineering,1966,24:381-338.
[30] Love T J, Grosh R J. Radiative heat transfer in absorbing, emitting and scattering media. J. of Heat transfer-Transactions of the ASME,1965,87:161-166.
[31] Fiveland W A. Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures. Journal of Heat Transfer.1984,106(4):699-706.
[32] Fiveland W A. Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media. Journal of Heat Transfer.1987,109(3): 809-812.
[33] Fiveland W A. Three-dimensional radiative heat-transfer solutions by the discrete-ordinates method. Journal of Thermophysics and Heat Transfer.1988,2: 309-316.
[34] Truelove J S. Discrete-rdinate solutions of the radiation transport equation. Journal of Heat Transfer.1987,109(4):1048-1051.
[35] Turelove J S. Three-dimensional radiation in absorbing-emitting-scattering media using discrete-ordinates approximation. Journal of Quantitative Spectroscopy and Radiative Transfer.1988,39(1):27-31.
[36] Thurgood C P, Pollard A, Becker H A. The TN quadrature set for the discrete ordinates method. Journal of Heat Transfer.1995,117(4):1068-1070.
[37]黄志峰.高方向分辨率辐射强度及表面辐射特性计算研究：博士学位论文.华中科技大学,2010.
[38]刘林华,余其铮,阮立明,等.求解辐射传递方程的离散坐标法.计算物理,1998,15(3)：337-343.
[39] Liu L H, Ruan L M, Tan H P. On the discrete ordinates method for radiative heat transfer in anisotropically scattering media. International Journal of Heat and Mass Transfer.2002;45:3259-62.
[40]李本文,姚强,曹新玉,等.一种新的辐射换热离散坐标算法.化工学报,1998,49(3)：288-293.
[41] Li B W, Yao Q, Cao X Y, et al. A new angular quadrature set for discrete ordinates method. Journal of Heat Transfer, Transactions of the ASME,1998,120(2): 514-518.
[42] Li B W, Cen H G, Zhou J H, et al. A finite-volume for predicting a radiant heat transfer in enclosures with participating media. Journal of Heat Transfer, transactions of the ASME,2002,124:482-490.
[43]李宏顺,周怀春,陆继东,郑楚光.用离散坐标法计算炉膛火焰辐射能成像.工程热物理学报,2003,24(5)：843-845.
[44]李宏顺,周怀春,陆继东,郑楚光.炉膛辐射换热计算的一种改进的离散传递法.中国电机工程学报,2003,23(4)：162-166.
[45] Li H S, Flamant G, Lu J D. A new discrete ordinate algorithm for computing radiative transfer in one-dimensional atmospheres. Journal of Quantitative Spectroscopy and Radiative Transfer.2004; 83:407-421.
[46] Huang Z F, Zhou H C, Hsu P F. Improved discrete ordinates method for ray effects mitigation. Journal of Heat Transfer.2011; 133, n 4, p 044502 (5 pp).
[47] Jeans J H. The equations of radiative transfer of energy. Monthly Notices Royal Astronomical Society,1917,78:28-36.
[48] Devaux C, Herman M. Remarks on the use of the method of spherical harmonics in calculations on radiative transfer. Comptes Rendus Hebdomadaires des Seances de l'Academie des Sciences, Serie B (Sciences Physiques).1971;273:849-852.
[49] Menguc M P, Iyer R K. Modeling of radiative transfer using multiple spherical harmonics approximations. Journal of Quantitative Spectroscopy and Radiative Transfer.1988;39:445-461.
[50] Iyer RK, Menguc MP. Quadruple spherical harmonics approximation for radiative transfer in two-dimensional rectangular enclosures. Journal of Thermophysics and Heat Transfer.1989;3:266-273.
[51] Modest M F. The improved differential approximation for radiative transfer in multi-dimensional media. Journal of Heat Transfer, transactions of the ASME,1990, 112:819-821.
[52]王应时,范维澄,周力行,徐旭常.燃烧过程数值模拟.北京：科学出版社,1986.
[53] Hottel H C and Cohen E S. Radiative heat exchange in a gas-filled enclosure: allowance for nonuniformity of gas temperature. AIChE Journal,1958,4:3-14.
[54] Yuen W W. The multiple absorption coefficient zonal method (MACZM), an efficient computational approach for the analysis of radiative heat transfer in multidimensional inhomogeneous nongray media. Numerical Heat Transfer, Part B (Fundamentals).2006;49:89-103.
[55] Mechi R, Farhat H, Said R. Improved zonal method predictions in a rectangular
furnace by smoothing the exchange areas. Turkish Journal of Engineering and Environmental Sciences.2007;31:333-343.
[56] Cui M, Gao X, Chen H. A new inverse approach for the equivalent gray radiative property of a non-gray medium using a modified zonal method and the complex-variable-differentiation method. Journal of Quantitative Spectroscopy and Radiative Transfer.2011;112:1336-1342.
[57] Rath P, Mishra S C, Mahanta P, et al. Discrete transfer method applied to transient radiative transfer problems in participating medium. Numerical Heat Transfer, Part A (Applications).2003;44:183-197.
[58] Mishra S C, Talukdar P, Trimis D, et al. Effect of angular quadrature schemes on the computational efficiency of the discrete transfer method for solving radiative transport problems with participating medium. Numerical Heat Transfer, Part B (Fundamentals).2004;46:463-478.
[59] Mishra SC, Krishna N A. Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium. Journal of Quantitative Spectroscopy and Radiative Transfer.2006; 102:432-440.
[60] Reddy J N Murty V D. Finite element solution of integral equations arising in radiative heat transfer and laminar bounfary-layer theory. Numerical Heat transfer.1978,1:389-401.
[61] Nice M L. Application of finite elements to heat transfer in a participating medium. Numerical properties and Methodologies in Heat transfer.1983:62-71.
[62] Liu L H. Finite element method simulation of radiative heat transfer in absorbing and scattering media. International Journal of Thermophysics and Heat Transfer, 2004,18:555-557.
[63] Liu L H. Finite element solution of radiative transfer across a slab with variable spatial refractive index. International Journal of Heat and Mass Transfer. 2005;48:2260-2265.
[64] Liu L H, Liu L J. Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium. Journal of Quantitative Spectroscopy and Radiative Transfer.2007;105:377-387.
[65]刘林华,赵军明,谈和平.辐射传递方程数值模拟的有限元和谱元法.北京：科学出版社,2008.
[66] Raithby G D, Chui E H. A finite-volume method for predicting a radiant heat transfer in enclosures with participating media. J of heat transfer-transactions of the,ASME,1990,112:415-423.
[67] Chui E H, Hughes P M J, Raithby G D. Implementation of the finite volume method for calculating radiative transfer in a pulverized fuel flame. Combustion science and technology.1993;92:225-242.
[68] Liu J, Shang H M, Chen YS. Parallel simulation of radiative heat transfer using an unstructured finite-volume method. Numerical Heat Transfer, Part B:Fundamentals. 1999;36:115-137.
[69] Baek S W, Kim M Y. Analysis of radiative heating of a rocket plume base with the finite-volume method. International Journal of Heat and Mass Transfer.1997; 40:1501-1508.
[70] 郝金波,董士奎,谈和平.固体火箭发动机尾喷焰红外特性数值模拟.红外与毫米波学报,2003,22(4)：246-250.
[71] Kim S H, Huh K Y. Assessment of the finite-volume method and the discrete ordinate method for radiative heat transfer in a three-dimensional rectangular enclosure. Numerical Heat Transfer, Part B:Fundamentals.1999;35:85-112.
[72] Ko M, Anand N K. Three-dimensional combined convective-radiative heat transfer over a horizontal backward-facing step-A finite-volume method. Numerical Heat Transfer; Part A:Applications.2008;54:109-129.
[73] Murthy J Y, Mathur S R. Finite volume method for radiative heat transfer using unstructured meshes. Journal of Thermophysics and Heat Transfer. 1998;12:313-321.
[74] Murthy J Y, Mathur S R. Radiative heat transfer in axisymmetric geometries using an unstructured finite-volume method. Numerical Heat Transfer, Part B: Fundamentals.1998; 33:397-416.
[75] Kim M Y, Baek S W, Park J H. Unstructured finite-volume method for radiative heat transfer in a complex two-dimensional geometry with obstacles. Numerical Heat Transfer, Part B:Fundamentals.2001; 39:617-635.
[76] Chai J C, Hsu P F, Lam Y C. Three-dimensional transient radiative transfer modeling using the finite-volume method. Journal of Quantitative Spectroscopy and Radiative Transfer.2004;86:299-313.
[77] Zhang H C, Tan H P. Evaluation of numerical scattering in finite volume method for solving radiative transfer equation by a central laser incidence model. Journal of Quantitative Spectroscopy and Radiative Transfer.2009; 110:1965-1977.
[78] Mishra S C, Muthukumaran R, Maruyama S. The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media. International Communications in Heat and Mass Transfer.2011; 38:291-297.
[79] Fleck JA. The calculation of nonlinear radiation transport by a Monte Carlo method: statistical physics. Methods in Computational Physics,1961,1:43-65.
[80] Howell J R and Perlmutter M. Monte Carlo solution of thermal thransfer through radiant media between gray walls. ASME Journal of Heat Transfer,1964, 86(1):116-122.
[81] Howell J R. Application of Monte Carlo to heat transfer problems, in Advances in Heat Transfer, eds. Hartnett JP and Irvine TF, Vol.5, New York:Academic Press, 1968.
[82] Steward F R, Cannon P. The calculation of radiative heat flux in a cylindrical furnace using the monte carlo method. International Journal of Heat and Mass Transfer.1971; 14:245-262.
[83] Modest, M F, Tabanfar S, A Multi-dimensional Differential Approximation for Absorbing/Emitting Anisotropically Scattering Media with Collimated Irradiation, Journal of Quantitative Spectroscopy & Radiative Transfer,1983 29:339-351.
[84] Marchuk G I, Mikhailov G A. Solution of the radiative transfer theory problems by the Monte Carlo method. UCLA International Conference on Radiation and Remote probing of the atmosphere,28-30 Aug 1973. North Hollywood, CA, USA: Western Periodicals; 1974. p.58-73.
[85] Larsen EW, Mercier B. Analysis of a Monte Carlo method for nonlinear radiative transfer. Journal of Computational Physics.1987;71:50-64.
[86] Howell J R. The Monte Carlo method in radiative heat transfer. ASME Proceedings of the 7th AIAA/ASME Joint Thermophisics and Heat Transfer Conference, Volume 1,HTD-Vol.,1998,357(1):1-19.
[87] Wolf S, Henning T, Stecklum B. Multidimensional self-consistent radiative transfer simulations based on the Monte-Carlo method. Astronomy and Astrophysics.1999; 349:839-850.
[88] McClarren R G, Urbatsch T J. A modified implicit Monte Carlo method for time-dependent radiative transfer with adaptive material coupling. Journal of Computational Physics.2009; 228:5669-5686.
[89] Yang W J, Taniguchi H, Kudo K. Radiative Heat Transfer by the Monte Carlo Method, in J. P. Hartnett and F. T. Irvines, eds., Advances in Heat Transfer, San Diego:Academic Press,1995,27:1-215.
[90] Shuai Y, Zhang H C, Tan H P. Radiation symmetry test and uncertainty analysis of Monte Carlo method based on radiative exchange factor. Journal of Quantitative Spectroscopy and Radiative Transfer.2008;109:1281-1296.
[91] Zhou H C, Sheng F, Han S D, et al. Reconstruction of Temperature Distribution in a 2-D Absorbing-Emitting System from Radiant Energy Images, JSME International Journal, Series B,2000,43(1):104-109.
[92] Zhou H C, Han S D, Sheng F, et al. Visualization of three-dimensional temperature distributions in a large-scale furnace via regularized reconstruction from radiative energy images:numerical studies, JQSRT,2002,72:361-383.
[93] Zhou H C, Sheng F, Han S D, et al. A fast algorithm for calculation of radiative energy distributions received by pinhole image-formation process from 2D rectangular enclosure, Numerical Heat Transfer:Applications,2000,38(7):757-773.
[94] Zhou H C, Chen D L, Cheng Q. A new way to calculate radiative intensity and solve radiative transfer equation through using the Monte Carlo method. Journal of Quantitative Spectroscopy and Radiative Transfer,2004.83(3-4):459-481.
[95] Zhou H C, Cheng Q. The DRESOR method for the solution of the radiative transfer equation in gray plane-parallel media. In:Menguc M P and Selcuk N, eds. Proceedings of the Fourth International Symposium on Radaitive Transfer, Istanbul, Turkey. New York:Begell House, Inc.2004.181-190.
[96]程强.求解辐射传递方程的DRESOR法及其应用：博士学位论文.华中科技大学,2007.
[97] Tan H P, Lallemand M. Transeint radiative-conductive heat transfer in flat glasses submitted to temperature, flux and mixed boundary conditions. International Journal, of Heat and Mass transfer,1989,32(5):795-810.
[98] Tan HP, Maestre B, Lallemand M, Transient and steady-state combined heat transfer in semi-transparent materials subjectied to a pulse or a step irradiation, ASME J.
Heat Transfer,1991,113(1):166-173.
[99] Tan HP, Wang PY, Xia XL, Transient coupled radiation and conduction in an absorbing and scattering composite layer, AIAA J. Thermophysics and Heat Transfer,2000,14(1):77-87.
[100] Galinsky VL, Ramanathan V.3D radiative transfer in weakly inhomogeneous medium. Ⅰ. Diffusive approximation. Journal of the Atmospheric Sciences. 1998;55:2946-2959.
[101] Lu X D, Hus P F. Reverse Monte Carlo method for transient radiative transfer in participating media. Transactions of the ASME Journal of Heat Transfer. 2004;126:621-627.
[102] Altac Z. The SKN approximation for solving radiative transfer problems in absorbing, emitting, and isotropically scattering plane-parallel medium:part 1. Transactions of the ASME Journal of Heat Transfer.2002;124:674-684.
[103] Altac Z. The SKN approximation for solving radiative transfer problems in absorbing, emitting, and linearly anisotropically scattering plane-parallel medium: part 2. Transactions of the ASME Journal of Heat Transfer.2002;124:685-695.
[104]赵军明.求解辐射传递方程的谱元法：博士学位论文.哈尔滨工业大学,2007.
[105]谭建宇.求解辐射传递方程的无网格法：博士学位论文.哈尔滨工业大学,2006.
[106] Wawrenczuk A, Kuhnert J, Siedow N. FPM computations of glass cooling with radiation. Computer Methods in Applied Mechanics and Engineering. 2007;196:4656-4671.
[107] Mohamad A A. Local analytical discrete ordinate method for the solution of the radiative transfer equation. International Journal of Heat and Mass Transfer. 1996;39:1859-1864.
[108] Yuen W. The multiple absorption coefficient zonal method (MACZM), an efficient computational approach for the analysis of radiative heat transfer in multidimensional inhomogeneous nongray media. Numerical Heat Transfer, Part B: Fundamentals.2006;49:89-103.
[109] Coelho P J. A modified version of the discrete ordinates method for radiative heat transfer modelling. Computational Mechanics.2004;33:375-388.
[110] Li H S., Werther J. Computation of radiative image formation in isolated source and
collimated irradiation problems. Journal of Quantitative Spectroscopy and Radiative Transfer.2006,97(1):142-159.
[111]娄春,韩曙东,刘浩,等.一种煤粉燃烧火焰辐射成像新模型.工程热物理学报,2002.23(增刊)：93-96.
[112]刘浩,周怀春,娄春,等.燃煤锅炉炉膛断面温度场可视化实验研究.热科学与技术,2003.2(3)：250-254.
[113]吴峰,艾育华,娄春,周怀春.蜡烛火焰三维温度场重建实验研究.工程热物理学报,2003.24(5)：894-896.
[114]曾佳,娄春,程强,等.大型电站锅炉炉内三维燃烧温度场可视化实验研究.工程热物理学报,2004.25(3)：523-526.
[115] Lou C, Zhou H C. Deduction of the two-dimensional distribution of temperature in a cross section of a boiler furnace from images of flame radiation. Combustion and Flame.2005;143:97-105.
[116] Zhou H C, Lou C, Cheng Q, et al. Experimental investigations on visualization of three-dimensional temperature distributions in a large-scale pulverized-coal-fired boiler furnace. Proceedings of the Combustion Institute,2004.30(1):1608-1615
[117]程强,娄春,姜志伟,等.工业炉三维温度场可视化试验研究.全国能源与热工2004年学术年会.
[118] Kudo K, Kuroda A, Eid A, et al. Radiative load problem using the singular value decomposition technique. JSME International Journal, Series B,1996.39(4): 808-814.
[119]周怀春,何诚,程强.平行平板间各向异性散射介质辐射传递方程求解.华中科技大学学报(自然科学版),2006(01)：17-20.
[120] Zhou H C, Cheng Q, Huang, Z F, et al. The influence of anisotropic scattering on the radiative intensity in a gray, plane-parallel medium calculated by the DRESOR method. Journal of Quantitative Spectroscopy and Radiative Transfer.2007,104(1): 99-115.
[121]黄志锋,周怀春,程强,等DRESOR法对含有边界的一维各向异性散射介质辐射传递方程的求解.热科学与技术.2006,5(04)：313-319.
[122]程强,周怀春,黄志锋DRESOR法对瞬态辐射传递问题的研究.工程热物理学报,2006(03)：472-474.
[123]程强,周怀春,黄志锋.任意入射辐射条件下介质特性对瞬态辐射传递的影响.工程热物理学报.2007,(S2)：25-28.
[124]黄志锋,程强,周怀春,等DRESOR法求解超短脉冲在一维非均匀介质中的瞬态辐射传输.工程热物理学报.2008,29(06)：1005-1008.
[125] Cheng Q, Zhou H C, Huang Z F, et al. The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by DRESOR method. Journal of Heat Transfer.2008,130(10):102701-1-102701-15.
[126] Cheng Q, Zhou H C.The DRESOR method for a collimated irradiation on an isotropically scattering layer. Journal of Heat Transfer-Transactions of the ASME, 2007,129(5):634-645.
[127] Cheng Q, Zhou H C, Yu Y L, et al. Highly-directional radiative intensity in a 2-D rectangular enclosure calculated by the DRESOR method. Numerical Heat Transfer. Part B-Fundamentals,2008,54(4):354-367.
[128]程强,周怀春,黄志锋等.DRESOR法对二维辐射传递问题研究.工程热物理学报,2008(10)：1735-1738.
[129] Cheng Q, Luo Z X, Zhou H C. Numerical simulation on radiative heat transfer through a two-dimensional rectangular domain with inhomogeneous, absorbing and isotropic/anisotropic scattering media exposed to collimated irradiation. International Journal of Nonlinear Sciences and Numerical Simulation.2010, 11(11):927-945.
[130] Wang Z C, Cheng Q, Wang G H, et al. The DRESOR method for radiative heat transfer in a one-dimensional medium with variable refractive index. Journal of Quantitative Spectroscopy and Radiative Transfer,2011,11:2835-2845.
[131] Huang Z F, Cheng Q, Zhou H C. The Iterative-DRESOR method to solve radiative transfer in a plane-parallel, anisotropic scattering medium with specular-diffuse boundaries. Journal of Quantitative Spectroscopy and Radiative Transfer.2009, 110(13):1072-1084.
[132] Wells W H, Sidorowich J J, Computational techniques for radiative transfer by spherical harmonics, Journal of Quantitative Spectroscopy and Radiative Transfer, 33(1985)347-363.
[133] Selcuk N, Evaluation of spherical harmonics approximation for radiative transfer in cylindrical furnaces, International Journal of Heat and Mass Transfer,33 (1990)
579-581.
[134] Fiveland W A, Jamaluddin A S. Three-dimensional spectral radiative heat transfer solutions by the discrete-ordinates method, Journal of Thermophysics and Heat Transfer,5(1991)335-339.
[135] Ben Salah M, Askri F, Slimi K, et al. Numerical resolution of the radiative transfer equation in a cylindrical enclosure with the finite-volume method, International Journal of Heat and Mass Transfer,47 (2004) 2501-2509.
[136] Mechi R, Farhat H, Guedri K, et al. Extension of the zonal method to inhomogeneous non-grey semi-transparent medium, Energy,35 (2010) 1-15.
[137] Ho well J R, The Monte Carlo method in radiative heat transfer. Transactions of the ASME. Journal of Heat Transfer,120 (1998) 547-560.
[138] Guo Z X, Kumar S. Radiation element method for transient hyperbolic radiative transfer in plane-parallel inhomogeneous media, Numerical Heat Transfer, Part B (Fundamentals),39 (2001) 371-387.
[139] Altac Z. The SKN approximation for solving radiative transfer problems in absorbing, emitting, and linearly anisotropically scattering plane-parallel medium: part 2, Transactions of the ASME. Journal of Heat Transfer,124 (2002) 685-695.
[140] Tan H P, Yi H L, Wang P Y, et al. Ray tracing method for transient coupled heat transfer in an anisotropic scattering layer, International Journal of Heat and Mass Transfer,47 (2004) 4045-4059.
[141] Busbridge, I. W., Orchard, S. E. Reflection and transmission of light by a thick atmosphere according to phase function:1+xcosq. The Astrophysical Journal.1967, 149:655-664.
[142] Orchard, S. E. Reflection and transmission of light by thick atmospheres of pure scatterers with a phase function:1+w1P1(cosq)+w2P2(cosq). The Astrophysical Journal.1967,149:665-674.
[2] Siegel, R., Howell, J. R. Thermal radiation heat transfer. (4th). New York:Taylor & Francis,2002.
[3] Modest M F. Radiative heat transfer, second edition. San Diego:Academic Press, 2003.
[4]余其铮.辐射换热原理.哈尔滨：哈尔滨工业大学出版社,2000.
[5]杨世铭,陶文铨.传热学.4版.北京：高等教育出版社,2006.
[6] Kamiuto, K. Kanemaru, K. Approximate Method for Combined Forced-Convection and Radiation Heat Transfer in Absorbing and Emitting Gases Flowing in a Black, plane-parallel Duct, Int. J. Heat Mass Transfer.1997,39:2191-2193.
[7]刘林华.炉膛传热计算方法的发展概况.动力工程,2000,21(1)：523-527.
[8]郑楚光,柳朝晖.弥散介质的光学特性及辐射传热.武汉：华中理工大学出版社,1996.
[9] Carvalho M G, Semiao V S, Coelho P J. Modelling and optimization of the no formation in an industrial glass furnace Journal of Engineering for Industry,1992, 114(4):514-523.
[10] Dombrovsky L A, Sazhin S S, Sazhina E M, et al. Heating and evaporation of semi-transparent diesel fuel droplets in the presence of thermal radiation. Fuel.2001, 80:1535-1544.
[11] Chetverushkin B N, Polyakov S V, Kudryashova T A, et al. Numerical simulation of 2D radiation heat transfer for reentry vehicles. Parallel computational fluid dynamics 2005-Theory and applications.2006,293-299.
[12] Fujita K, Abe T, Suzuki K. Uv radiation impact on heating rate of super-orbital reentry vehicle. International Symposium on Space Technology and Science,21st, 1998, Omiya, Japan,1:963-968.
[13] Coelho P J. Detailed numerical simulation of radiative transfer in a nonluminous turbulent jet diffusion flame. Combustion and Flame.2004,136:481-492.
[14]赵恒凯.辐射散射组合遥感系统关键技术研究：博士学位论文.上海大学,2006.
[15] Kumar S, Mitra K. Microscale aspects of thermal tadiation transport and laser application. Advances in Heat Transfer.1999,33:187-294.
[16] Qui T G, Tien C L. Short pulse laser heating in metals. International Journal of Heat and Mass Transfer.1992,35:719-726.
[17] Bokota A, Iskierka S. Numerical simulation of transient and residual stresses caused by laser hardening of slender elements. Computational Materials Science,1997, 7(4):366-376.
[18] Patterson M S, Wilson B C, Wyman D R. The propagation of optical radiation in tissue Ⅰ. Models of Radiation Transport and Their Application Lasers in Medical Science.1991,6(2):155-168.
[19] Yamada Y. Light-tissue interaction and optical imaging in biomedicine. Annual Review of Heat Transfer 1995,6:1-59.
[20] Okada E, Firbank M, Schweiger M, et al. theoretical and experimental investigation of near-infrared light propagation in a model of the adult head. Applied Optics.1997, 36:21-31.
[21] M. Sakami, K. Mitra and T. Vo-Dinh. Analysis of Short-Pulse Laser Photo Transport through Tissues for Optical Tomography. Opt. Lett.2002,27(5):336-338.
[22] Hebden J C, Veenstra H, Dehghani H, et al. Three-dimensional time-resolved optical tomography of a conical breast phantom. Applied Optics.2001,40:3278-3287.
[23]谈和平,夏新林,刘林华,等.红外辐射特性与传输的数值计算一计算热辐射学.哈尔滨：哈尔滨工业大学出版社,2006.
[24]娄春.煤粉炉内三维温度场及颗粒辐射特性重建：博士学位论文.华中科技大学,2007.
[25]周怀春.炉内火焰可视化检测原理与技术.北京：科学出版社,2005.
[26]范维澄,万跃鹏.流动与燃烧的模型与计算.合肥：中国科学技术大学出版社,1992.
[27] Zhou H C., Sheng F, Han S D, et al. Reconstruction of temperature distribution in a 2-D absorbing-emitting system from radiant energy images. JSME international journal. Series B:Fluids and thermal engineering.2000,43(1):104-109.
[28] Chandrasekhar S. Radiative Transfer, New York:Dove Publications Inc.,1960.
[29] Lathrop K D. Use of discrete-ordinate methods for solution of photon transport problems. Nuclear Science and Engineering,1966,24:381-338.
[30] Love T J, Grosh R J. Radiative heat transfer in absorbing, emitting and scattering media. J. of Heat transfer-Transactions of the ASME,1965,87:161-166.
[31] Fiveland W A. Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures. Journal of Heat Transfer.1984,106(4):699-706.
[32] Fiveland W A. Discrete ordinate methods for radiative heat transfer in isotropically and anisotropically scattering media. Journal of Heat Transfer.1987,109(3): 809-812.
[33] Fiveland W A. Three-dimensional radiative heat-transfer solutions by the discrete-ordinates method. Journal of Thermophysics and Heat Transfer.1988,2: 309-316.
[34] Truelove J S. Discrete-rdinate solutions of the radiation transport equation. Journal of Heat Transfer.1987,109(4):1048-1051.
[35] Turelove J S. Three-dimensional radiation in absorbing-emitting-scattering media using discrete-ordinates approximation. Journal of Quantitative Spectroscopy and Radiative Transfer.1988,39(1):27-31.
[36] Thurgood C P, Pollard A, Becker H A. The TN quadrature set for the discrete ordinates method. Journal of Heat Transfer.1995,117(4):1068-1070.
[37]黄志峰.高方向分辨率辐射强度及表面辐射特性计算研究：博士学位论文.华中科技大学,2010.
[38]刘林华,余其铮,阮立明,等.求解辐射传递方程的离散坐标法.计算物理,1998,15(3)：337-343.
[39] Liu L H, Ruan L M, Tan H P. On the discrete ordinates method for radiative heat transfer in anisotropically scattering media. International Journal of Heat and Mass Transfer.2002;45:3259-62.
[40]李本文,姚强,曹新玉,等.一种新的辐射换热离散坐标算法.化工学报,1998,49(3)：288-293.
[41] Li B W, Yao Q, Cao X Y, et al. A new angular quadrature set for discrete ordinates method. Journal of Heat Transfer, Transactions of the ASME,1998,120(2): 514-518.
[42] Li B W, Cen H G, Zhou J H, et al. A finite-volume for predicting a radiant heat transfer in enclosures with participating media. Journal of Heat Transfer, transactions of the ASME,2002,124:482-490.
[43]李宏顺,周怀春,陆继东,郑楚光.用离散坐标法计算炉膛火焰辐射能成像.工程热物理学报,2003,24(5)：843-845.
[44]李宏顺,周怀春,陆继东,郑楚光.炉膛辐射换热计算的一种改进的离散传递法.中国电机工程学报,2003,23(4)：162-166.
[45] Li H S, Flamant G, Lu J D. A new discrete ordinate algorithm for computing radiative transfer in one-dimensional atmospheres. Journal of Quantitative Spectroscopy and Radiative Transfer.2004; 83:407-421.
[46] Huang Z F, Zhou H C, Hsu P F. Improved discrete ordinates method for ray effects mitigation. Journal of Heat Transfer.2011; 133, n 4, p 044502 (5 pp).
[47] Jeans J H. The equations of radiative transfer of energy. Monthly Notices Royal Astronomical Society,1917,78:28-36.
[48] Devaux C, Herman M. Remarks on the use of the method of spherical harmonics in calculations on radiative transfer. Comptes Rendus Hebdomadaires des Seances de l'Academie des Sciences, Serie B (Sciences Physiques).1971;273:849-852.
[49] Menguc M P, Iyer R K. Modeling of radiative transfer using multiple spherical harmonics approximations. Journal of Quantitative Spectroscopy and Radiative Transfer.1988;39:445-461.
[50] Iyer RK, Menguc MP. Quadruple spherical harmonics approximation for radiative transfer in two-dimensional rectangular enclosures. Journal of Thermophysics and Heat Transfer.1989;3:266-273.
[51] Modest M F. The improved differential approximation for radiative transfer in multi-dimensional media. Journal of Heat Transfer, transactions of the ASME,1990, 112:819-821.
[52]王应时,范维澄,周力行,徐旭常.燃烧过程数值模拟.北京：科学出版社,1986.
[53] Hottel H C and Cohen E S. Radiative heat exchange in a gas-filled enclosure: allowance for nonuniformity of gas temperature. AIChE Journal,1958,4:3-14.
[54] Yuen W W. The multiple absorption coefficient zonal method (MACZM), an efficient computational approach for the analysis of radiative heat transfer in multidimensional inhomogeneous nongray media. Numerical Heat Transfer, Part B (Fundamentals).2006;49:89-103.
[55] Mechi R, Farhat H, Said R. Improved zonal method predictions in a rectangular
furnace by smoothing the exchange areas. Turkish Journal of Engineering and Environmental Sciences.2007;31:333-343.
[56] Cui M, Gao X, Chen H. A new inverse approach for the equivalent gray radiative property of a non-gray medium using a modified zonal method and the complex-variable-differentiation method. Journal of Quantitative Spectroscopy and Radiative Transfer.2011;112:1336-1342.
[57] Rath P, Mishra S C, Mahanta P, et al. Discrete transfer method applied to transient radiative transfer problems in participating medium. Numerical Heat Transfer, Part A (Applications).2003;44:183-197.
[58] Mishra S C, Talukdar P, Trimis D, et al. Effect of angular quadrature schemes on the computational efficiency of the discrete transfer method for solving radiative transport problems with participating medium. Numerical Heat Transfer, Part B (Fundamentals).2004;46:463-478.
[59] Mishra SC, Krishna N A. Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium. Journal of Quantitative Spectroscopy and Radiative Transfer.2006; 102:432-440.
[60] Reddy J N Murty V D. Finite element solution of integral equations arising in radiative heat transfer and laminar bounfary-layer theory. Numerical Heat transfer.1978,1:389-401.
[61] Nice M L. Application of finite elements to heat transfer in a participating medium. Numerical properties and Methodologies in Heat transfer.1983:62-71.
[62] Liu L H. Finite element method simulation of radiative heat transfer in absorbing and scattering media. International Journal of Thermophysics and Heat Transfer, 2004,18:555-557.
[63] Liu L H. Finite element solution of radiative transfer across a slab with variable spatial refractive index. International Journal of Heat and Mass Transfer. 2005;48:2260-2265.
[64] Liu L H, Liu L J. Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium. Journal of Quantitative Spectroscopy and Radiative Transfer.2007;105:377-387.
[65]刘林华,赵军明,谈和平.辐射传递方程数值模拟的有限元和谱元法.北京：科学出版社,2008.
[66] Raithby G D, Chui E H. A finite-volume method for predicting a radiant heat transfer in enclosures with participating media. J of heat transfer-transactions of the,ASME,1990,112:415-423.
[67] Chui E H, Hughes P M J, Raithby G D. Implementation of the finite volume method for calculating radiative transfer in a pulverized fuel flame. Combustion science and technology.1993;92:225-242.
[68] Liu J, Shang H M, Chen YS. Parallel simulation of radiative heat transfer using an unstructured finite-volume method. Numerical Heat Transfer, Part B:Fundamentals. 1999;36:115-137.
[69] Baek S W, Kim M Y. Analysis of radiative heating of a rocket plume base with the finite-volume method. International Journal of Heat and Mass Transfer.1997; 40:1501-1508.
[70] 郝金波,董士奎,谈和平.固体火箭发动机尾喷焰红外特性数值模拟.红外与毫米波学报,2003,22(4)：246-250.
[71] Kim S H, Huh K Y. Assessment of the finite-volume method and the discrete ordinate method for radiative heat transfer in a three-dimensional rectangular enclosure. Numerical Heat Transfer, Part B:Fundamentals.1999;35:85-112.
[72] Ko M, Anand N K. Three-dimensional combined convective-radiative heat transfer over a horizontal backward-facing step-A finite-volume method. Numerical Heat Transfer; Part A:Applications.2008;54:109-129.
[73] Murthy J Y, Mathur S R. Finite volume method for radiative heat transfer using unstructured meshes. Journal of Thermophysics and Heat Transfer. 1998;12:313-321.
[74] Murthy J Y, Mathur S R. Radiative heat transfer in axisymmetric geometries using an unstructured finite-volume method. Numerical Heat Transfer, Part B: Fundamentals.1998; 33:397-416.
[75] Kim M Y, Baek S W, Park J H. Unstructured finite-volume method for radiative heat transfer in a complex two-dimensional geometry with obstacles. Numerical Heat Transfer, Part B:Fundamentals.2001; 39:617-635.
[76] Chai J C, Hsu P F, Lam Y C. Three-dimensional transient radiative transfer modeling using the finite-volume method. Journal of Quantitative Spectroscopy and Radiative Transfer.2004;86:299-313.
[77] Zhang H C, Tan H P. Evaluation of numerical scattering in finite volume method for solving radiative transfer equation by a central laser incidence model. Journal of Quantitative Spectroscopy and Radiative Transfer.2009; 110:1965-1977.
[78] Mishra S C, Muthukumaran R, Maruyama S. The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media. International Communications in Heat and Mass Transfer.2011; 38:291-297.
[79] Fleck JA. The calculation of nonlinear radiation transport by a Monte Carlo method: statistical physics. Methods in Computational Physics,1961,1:43-65.
[80] Howell J R and Perlmutter M. Monte Carlo solution of thermal thransfer through radiant media between gray walls. ASME Journal of Heat Transfer,1964, 86(1):116-122.
[81] Howell J R. Application of Monte Carlo to heat transfer problems, in Advances in Heat Transfer, eds. Hartnett JP and Irvine TF, Vol.5, New York:Academic Press, 1968.
[82] Steward F R, Cannon P. The calculation of radiative heat flux in a cylindrical furnace using the monte carlo method. International Journal of Heat and Mass Transfer.1971; 14:245-262.
[83] Modest, M F, Tabanfar S, A Multi-dimensional Differential Approximation for Absorbing/Emitting Anisotropically Scattering Media with Collimated Irradiation, Journal of Quantitative Spectroscopy & Radiative Transfer,1983 29:339-351.
[84] Marchuk G I, Mikhailov G A. Solution of the radiative transfer theory problems by the Monte Carlo method. UCLA International Conference on Radiation and Remote probing of the atmosphere,28-30 Aug 1973. North Hollywood, CA, USA: Western Periodicals; 1974. p.58-73.
[85] Larsen EW, Mercier B. Analysis of a Monte Carlo method for nonlinear radiative transfer. Journal of Computational Physics.1987;71:50-64.
[86] Howell J R. The Monte Carlo method in radiative heat transfer. ASME Proceedings of the 7th AIAA/ASME Joint Thermophisics and Heat Transfer Conference, Volume 1,HTD-Vol.,1998,357(1):1-19.
[87] Wolf S, Henning T, Stecklum B. Multidimensional self-consistent radiative transfer simulations based on the Monte-Carlo method. Astronomy and Astrophysics.1999; 349:839-850.
[88] McClarren R G, Urbatsch T J. A modified implicit Monte Carlo method for time-dependent radiative transfer with adaptive material coupling. Journal of Computational Physics.2009; 228:5669-5686.
[89] Yang W J, Taniguchi H, Kudo K. Radiative Heat Transfer by the Monte Carlo Method, in J. P. Hartnett and F. T. Irvines, eds., Advances in Heat Transfer, San Diego:Academic Press,1995,27:1-215.
[90] Shuai Y, Zhang H C, Tan H P. Radiation symmetry test and uncertainty analysis of Monte Carlo method based on radiative exchange factor. Journal of Quantitative Spectroscopy and Radiative Transfer.2008;109:1281-1296.
[91] Zhou H C, Sheng F, Han S D, et al. Reconstruction of Temperature Distribution in a 2-D Absorbing-Emitting System from Radiant Energy Images, JSME International Journal, Series B,2000,43(1):104-109.
[92] Zhou H C, Han S D, Sheng F, et al. Visualization of three-dimensional temperature distributions in a large-scale furnace via regularized reconstruction from radiative energy images:numerical studies, JQSRT,2002,72:361-383.
[93] Zhou H C, Sheng F, Han S D, et al. A fast algorithm for calculation of radiative energy distributions received by pinhole image-formation process from 2D rectangular enclosure, Numerical Heat Transfer:Applications,2000,38(7):757-773.
[94] Zhou H C, Chen D L, Cheng Q. A new way to calculate radiative intensity and solve radiative transfer equation through using the Monte Carlo method. Journal of Quantitative Spectroscopy and Radiative Transfer,2004.83(3-4):459-481.
[95] Zhou H C, Cheng Q. The DRESOR method for the solution of the radiative transfer equation in gray plane-parallel media. In:Menguc M P and Selcuk N, eds. Proceedings of the Fourth International Symposium on Radaitive Transfer, Istanbul, Turkey. New York:Begell House, Inc.2004.181-190.
[96]程强.求解辐射传递方程的DRESOR法及其应用：博士学位论文.华中科技大学,2007.
[97] Tan H P, Lallemand M. Transeint radiative-conductive heat transfer in flat glasses submitted to temperature, flux and mixed boundary conditions. International Journal, of Heat and Mass transfer,1989,32(5):795-810.
[98] Tan HP, Maestre B, Lallemand M, Transient and steady-state combined heat transfer in semi-transparent materials subjectied to a pulse or a step irradiation, ASME J.
Heat Transfer,1991,113(1):166-173.
[99] Tan HP, Wang PY, Xia XL, Transient coupled radiation and conduction in an absorbing and scattering composite layer, AIAA J. Thermophysics and Heat Transfer,2000,14(1):77-87.
[100] Galinsky VL, Ramanathan V.3D radiative transfer in weakly inhomogeneous medium. Ⅰ. Diffusive approximation. Journal of the Atmospheric Sciences. 1998;55:2946-2959.
[101] Lu X D, Hus P F. Reverse Monte Carlo method for transient radiative transfer in participating media. Transactions of the ASME Journal of Heat Transfer. 2004;126:621-627.
[102] Altac Z. The SKN approximation for solving radiative transfer problems in absorbing, emitting, and isotropically scattering plane-parallel medium:part 1. Transactions of the ASME Journal of Heat Transfer.2002;124:674-684.
[103] Altac Z. The SKN approximation for solving radiative transfer problems in absorbing, emitting, and linearly anisotropically scattering plane-parallel medium: part 2. Transactions of the ASME Journal of Heat Transfer.2002;124:685-695.
[104]赵军明.求解辐射传递方程的谱元法：博士学位论文.哈尔滨工业大学,2007.
[105]谭建宇.求解辐射传递方程的无网格法：博士学位论文.哈尔滨工业大学,2006.
[106] Wawrenczuk A, Kuhnert J, Siedow N. FPM computations of glass cooling with radiation. Computer Methods in Applied Mechanics and Engineering. 2007;196:4656-4671.
[107] Mohamad A A. Local analytical discrete ordinate method for the solution of the radiative transfer equation. International Journal of Heat and Mass Transfer. 1996;39:1859-1864.
[108] Yuen W. The multiple absorption coefficient zonal method (MACZM), an efficient computational approach for the analysis of radiative heat transfer in multidimensional inhomogeneous nongray media. Numerical Heat Transfer, Part B: Fundamentals.2006;49:89-103.
[109] Coelho P J. A modified version of the discrete ordinates method for radiative heat transfer modelling. Computational Mechanics.2004;33:375-388.
[110] Li H S., Werther J. Computation of radiative image formation in isolated source and
collimated irradiation problems. Journal of Quantitative Spectroscopy and Radiative Transfer.2006,97(1):142-159.
[111]娄春,韩曙东,刘浩,等.一种煤粉燃烧火焰辐射成像新模型.工程热物理学报,2002.23(增刊)：93-96.
[112]刘浩,周怀春,娄春,等.燃煤锅炉炉膛断面温度场可视化实验研究.热科学与技术,2003.2(3)：250-254.
[113]吴峰,艾育华,娄春,周怀春.蜡烛火焰三维温度场重建实验研究.工程热物理学报,2003.24(5)：894-896.
[114]曾佳,娄春,程强,等.大型电站锅炉炉内三维燃烧温度场可视化实验研究.工程热物理学报,2004.25(3)：523-526.
[115] Lou C, Zhou H C. Deduction of the two-dimensional distribution of temperature in a cross section of a boiler furnace from images of flame radiation. Combustion and Flame.2005;143:97-105.
[116] Zhou H C, Lou C, Cheng Q, et al. Experimental investigations on visualization of three-dimensional temperature distributions in a large-scale pulverized-coal-fired boiler furnace. Proceedings of the Combustion Institute,2004.30(1):1608-1615
[117]程强,娄春,姜志伟,等.工业炉三维温度场可视化试验研究.全国能源与热工2004年学术年会.
[118] Kudo K, Kuroda A, Eid A, et al. Radiative load problem using the singular value decomposition technique. JSME International Journal, Series B,1996.39(4): 808-814.
[119]周怀春,何诚,程强.平行平板间各向异性散射介质辐射传递方程求解.华中科技大学学报(自然科学版),2006(01)：17-20.
[120] Zhou H C, Cheng Q, Huang, Z F, et al. The influence of anisotropic scattering on the radiative intensity in a gray, plane-parallel medium calculated by the DRESOR method. Journal of Quantitative Spectroscopy and Radiative Transfer.2007,104(1): 99-115.
[121]黄志锋,周怀春,程强,等DRESOR法对含有边界的一维各向异性散射介质辐射传递方程的求解.热科学与技术.2006,5(04)：313-319.
[122]程强,周怀春,黄志锋DRESOR法对瞬态辐射传递问题的研究.工程热物理学报,2006(03)：472-474.
[123]程强,周怀春,黄志锋.任意入射辐射条件下介质特性对瞬态辐射传递的影响.工程热物理学报.2007,(S2)：25-28.
[124]黄志锋,程强,周怀春,等DRESOR法求解超短脉冲在一维非均匀介质中的瞬态辐射传输.工程热物理学报.2008,29(06)：1005-1008.
[125] Cheng Q, Zhou H C, Huang Z F, et al. The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by DRESOR method. Journal of Heat Transfer.2008,130(10):102701-1-102701-15.
[126] Cheng Q, Zhou H C.The DRESOR method for a collimated irradiation on an isotropically scattering layer. Journal of Heat Transfer-Transactions of the ASME, 2007,129(5):634-645.
[127] Cheng Q, Zhou H C, Yu Y L, et al. Highly-directional radiative intensity in a 2-D rectangular enclosure calculated by the DRESOR method. Numerical Heat Transfer. Part B-Fundamentals,2008,54(4):354-367.
[128]程强,周怀春,黄志锋等.DRESOR法对二维辐射传递问题研究.工程热物理学报,2008(10)：1735-1738.
[129] Cheng Q, Luo Z X, Zhou H C. Numerical simulation on radiative heat transfer through a two-dimensional rectangular domain with inhomogeneous, absorbing and isotropic/anisotropic scattering media exposed to collimated irradiation. International Journal of Nonlinear Sciences and Numerical Simulation.2010, 11(11):927-945.
[130] Wang Z C, Cheng Q, Wang G H, et al. The DRESOR method for radiative heat transfer in a one-dimensional medium with variable refractive index. Journal of Quantitative Spectroscopy and Radiative Transfer,2011,11:2835-2845.
[131] Huang Z F, Cheng Q, Zhou H C. The Iterative-DRESOR method to solve radiative transfer in a plane-parallel, anisotropic scattering medium with specular-diffuse boundaries. Journal of Quantitative Spectroscopy and Radiative Transfer.2009, 110(13):1072-1084.
[132] Wells W H, Sidorowich J J, Computational techniques for radiative transfer by spherical harmonics, Journal of Quantitative Spectroscopy and Radiative Transfer, 33(1985)347-363.
[133] Selcuk N, Evaluation of spherical harmonics approximation for radiative transfer in cylindrical furnaces, International Journal of Heat and Mass Transfer,33 (1990)
579-581.
[134] Fiveland W A, Jamaluddin A S. Three-dimensional spectral radiative heat transfer solutions by the discrete-ordinates method, Journal of Thermophysics and Heat Transfer,5(1991)335-339.
[135] Ben Salah M, Askri F, Slimi K, et al. Numerical resolution of the radiative transfer equation in a cylindrical enclosure with the finite-volume method, International Journal of Heat and Mass Transfer,47 (2004) 2501-2509.
[136] Mechi R, Farhat H, Guedri K, et al. Extension of the zonal method to inhomogeneous non-grey semi-transparent medium, Energy,35 (2010) 1-15.
[137] Ho well J R, The Monte Carlo method in radiative heat transfer. Transactions of the ASME. Journal of Heat Transfer,120 (1998) 547-560.
[138] Guo Z X, Kumar S. Radiation element method for transient hyperbolic radiative transfer in plane-parallel inhomogeneous media, Numerical Heat Transfer, Part B (Fundamentals),39 (2001) 371-387.
[139] Altac Z. The SKN approximation for solving radiative transfer problems in absorbing, emitting, and linearly anisotropically scattering plane-parallel medium: part 2, Transactions of the ASME. Journal of Heat Transfer,124 (2002) 685-695.
[140] Tan H P, Yi H L, Wang P Y, et al. Ray tracing method for transient coupled heat transfer in an anisotropic scattering layer, International Journal of Heat and Mass Transfer,47 (2004) 4045-4059.
[141] Busbridge, I. W., Orchard, S. E. Reflection and transmission of light by a thick atmosphere according to phase function:1+xcosq. The Astrophysical Journal.1967, 149:655-664.
[142] Orchard, S. E. Reflection and transmission of light by thick atmospheres of pure scatterers with a phase function:1+w1P1(cosq)+w2P2(cosq). The Astrophysical Journal.1967,149:665-674.