结构化网格中冰水相变传热过程的数值计算
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摘要
结冰和融冰现象与我们的日常生产生活密切相关。从传热学的角度来看,结冰和融冰问题的实质是伴有相变的传热问题。对该现象的数值模拟已经成为近年来数值传热学中的一个热点。
本文基于结构化网格,对冰水相变传热过程进行数值计算。主要研究内容为结构化网格中非稳态导热问题的求解过程,其中包括空间和时间物理量的离散方法,不同边界条件的处理,以及不同源项的情况等。导热控制方程中扩散项的离散采用基元有限容积法,二阶精度;非稳态项的离散,采用全隐格式;源项的离散,将相变过程中的潜热视为移动的热源。
用FORTRAN语言编写模块化计算程序,以此增强其通用性。为了验证此求解过程的正确性,完整性和通用性,对非稳态热传导问题,本文从一维到三维分别给出一个经典的热传导算例,将数值解同数学分析后所得精确解进行比较,得到令人满意的一致性。最后对冰水相变传热过程进行数值计算,这些都为工程计算和应用提供有用的数值计算工具。
The phenomena of icing and melting are related closely with our daily life. From the view of heat transfer, the essence of the phenomena is a heat transfer problem with phase change. The numerical simulation of the phenomena has been a hotspot in the field of numerical heat transfer in the recent years.
The numerical calculations of phase change heat transfer between ice and water were investigated based on structured grids. It included the discretization method of governing equation, the treatment of boundary conditions and source terms. The diffusion term was discretized by cell-based finite volume method (FVM), and the second order accuracy. The unsteady term was discretized by implicit scheme. And, latent heat was calculated as a source term.
Computational program was made using VISUAL FORTRAN, and the codes were divided into different modules according to different purposes. The classic examples of heat transfer problems from one dimension to three dimensions were presented to validate the correction, integrity and universality of the program. The numerical solutions of these examples are agreement when compared with the analytic solutions. All of these prove that the method and the procedure are correct and satisfied. After that, the phase change problems were calculated between ice and water, which supplies useful numerical calculation tools for engineering and application.
本文基于结构化网格,对冰水相变传热过程进行数值计算。主要研究内容为结构化网格中非稳态导热问题的求解过程,其中包括空间和时间物理量的离散方法,不同边界条件的处理,以及不同源项的情况等。导热控制方程中扩散项的离散采用基元有限容积法,二阶精度;非稳态项的离散,采用全隐格式;源项的离散,将相变过程中的潜热视为移动的热源。
用FORTRAN语言编写模块化计算程序,以此增强其通用性。为了验证此求解过程的正确性,完整性和通用性,对非稳态热传导问题,本文从一维到三维分别给出一个经典的热传导算例,将数值解同数学分析后所得精确解进行比较,得到令人满意的一致性。最后对冰水相变传热过程进行数值计算,这些都为工程计算和应用提供有用的数值计算工具。
The phenomena of icing and melting are related closely with our daily life. From the view of heat transfer, the essence of the phenomena is a heat transfer problem with phase change. The numerical simulation of the phenomena has been a hotspot in the field of numerical heat transfer in the recent years.
The numerical calculations of phase change heat transfer between ice and water were investigated based on structured grids. It included the discretization method of governing equation, the treatment of boundary conditions and source terms. The diffusion term was discretized by cell-based finite volume method (FVM), and the second order accuracy. The unsteady term was discretized by implicit scheme. And, latent heat was calculated as a source term.
Computational program was made using VISUAL FORTRAN, and the codes were divided into different modules according to different purposes. The classic examples of heat transfer problems from one dimension to three dimensions were presented to validate the correction, integrity and universality of the program. The numerical solutions of these examples are agreement when compared with the analytic solutions. All of these prove that the method and the procedure are correct and satisfied. After that, the phase change problems were calculated between ice and water, which supplies useful numerical calculation tools for engineering and application.
引文
[1]章熙民,任泽霈.传热学.北京:中国建筑工业出版社,2001
[2]杨世铭,陶文铨.传热学.北京:高等教育出版社,1998
[3]陶文铨.计算传热学的近代进展.北京:科学出版社,2001
[4]Suhas V.Patankar.Computation of Conduction and Duct Flow Heat Transfer.Innovative Research,Inc.7846 Ithaca Lane North Maple Grove,MN55369-8549,USA
[5]陶文铨.数值传热学.西安:西安交通大学出版社,2001
[6]M.N.奥齐西克 著,俞昌铭 译.热传导.高等教育出版社.1984:426-421
[7]Chai,J.C.,Zhang,M.,Moder,J.P.,and Patankar,S.V.Conduction Heat Transfer Calculations Using Structured and Unstructured Grids.ICHMT Symposium CHT'01-Advances in Numerical Heat Transfer.2001(1):519-526
[8]Zhang,M.Modeling of Radiative Heat Transfer and Diffusion Processes Using Unstructured Grid.Ph.D.Tennessee Technological University,USA.2000
[9]Patankar,S.V..Numerical Heat Transfer and Fluid Flow,Hemisphere Publishing,1981
[10]张洪济,热传导.高等教育出版社,1990,458-490
[11]李惜雯.数学物理方法.西安交通大学出版社.2001:178-213
[12]南京工学院数学教研组.数学物理方程和特殊函数.人民教育出版社.1982:101-148
[13]P.Rath,J.C.Chai,Modeling Two-Dimensional Diffusion-Controlled Wet Chemical Etching Using a Total Concentration Approach,International Journal of Heat and Mass Transfer,2005
[14]钱吉裕 李强 宣益民.Lattice-Boltzman方法计算多孔介质内固液相变问题.自然科学进展.2006
[15]王福军.计算流体动力学分析.清华大学出版社.2004
[16]韩占忠,王敬,兰小平.FLUENT流体工程仿真计算实例与应用.北京理工大学出版社.2004
[17]Wen S J,Jeng R H,Chun P K,Lattice Boltzmann Method for the Heat Conduction Problem with Phase Change,Numerical Heat Transfer Part B,2001(39),167-187
[18]周业涛,关振群,顾元宪.求解相变传热问题的等效热容法.化工学报.2004(55)
[19]Crank J.Free and Moving Boundary Problems.Oxford:Clarerdon Press.1984
[20]Voiler V R.Swami Nathan C R.Thomas B G.Fixed Grid Techniques for Phase Change Problems:AReview.J.Num.Meth.Eng.,1990(30):875-898
[21]张寅平,胡汉平,孔祥冬,苏跃红.相变贮能.理论和应用.中国科学技术大学出版社.1996
[22]徐士良.FORTRAN常用算法程序集.清华大学出版社.1992
[23]Bouds S,Davey K,Hinduja S.A Modified Effective Capacitance Method for Solidification Modeling Using Linear TFE.J.Num.Meth.Eng.1996(39):3195-3215
[24]Morgan K,Lewis R W,Zienkiewicz O C.An Improved Algorithm for Heat Conduction Problems with Phase Change.J.Num.Meth.Eng.,1978(2):1191-1195
[25]Commi G.Guidice S D.Finite Element Solution of Non-linear Heat Conduction Problems with Special Reference to Phase Change.J.Num.Meth.Eng.,1974(8):613-624
[26]Glimm J,Graham M J,Grove J.Front Tracking in Two and Three Dimensions.J.Comput.Math.1998(35):1-12
[27]Carslaw,H.S,Jager J C.Conduction of Heat in Solids.Clarendon Press,Oxford.1984:534-548
[28]刘捷,卢文强.非等温相变问题的双倒易边界元法求解.工程热物理学报.2006(27)
[29]Budhia H,Kreith F.Heat Transfer with Melting or Freezing in a Wedge.J.Heat Mass Transfer,1973(16):195-211
[30]Tryggvason G,Bunner B,Esmaeeli A,et al.A Front-Tracking Method for the Computations of Multiphase Flow.J.Comput.Phys.2001(169):708-759
[31]Esmaeeli A,Tryggvason G.A Front Ttracking Method for Computations of Boiling in Complex Geometries.International Journal of Multiphase Flow.2004(30):1037-1050
[32]Hirt C M,ANichols B D.Volume of Fluid(VOF)Method for the Dynamics of Free Boundaries.J.Comput.Phys.1981(39):201-225
[33]Kim S O,No H C.Second-Order Model for Free Surface Convection and Interface Reconstructions.Int.J.Numer.Meth.Fluids.1998(26)1:79-100
[34]高志华,张明义,刘志强,张淑娟.相变温度场中无网格伽辽金法的应用.计算物理.2006(23)
[35]刘儒勋,刘晓平,张磊等.运动界面的追踪和重构方法.应用数学和力学.2004(3)25:279-290
[36]Rathjen K A,Jiji L A.Heat Conduction with Melting or Freezing in a Comer.J.Heat Transfer,1973,93:101-109
[37]Chen S H,Zhang X,Bo J S.L S Mehtod in Body-Fitted Coordinate System for Multi-Dimensional Non-Ideal Detonation Propagation.Chinese Journal of High Pressure Physis.2000(14)4:280-284
[38]Kim,C J and Kaviany,M.Numerical Method for Phase-Change Problems.International Journal of Heat and Mass Transfer.1990,33(12):2721-27
[39]Voller,V R.An Overview of Numerical Methods for Solving Phase-Change Problems.Advanced Numerical Heat Transfer.1997:341-397
[40]Wakamoto,S.,Nakao,K.,Tanaka,N.Study of the Stability of Super cooled Water in an Ice Generator.ASHRAE Trans.1996,102(2):142-150
[41]方贵银.蓄冷空调工程实用新技术.人民邮电出版社.2000年
[42]Inada,T.etc.Active Control of Freezing Temperature of Water by Ultrasonic Vibration.35th National heat transfer symposium of Japan.1998(3):895-896
[43]Hahnain,S.M.Review on Sustainable Thermal Energy Storage Technologies.Heat Storage Materials and Techniques Energy Convers.1998(39)11:1127-1138
[44]Wonjoo Rob and Noboru Kikuchi.Analysis of Stefan Problem with Level Set Method.AIAA 2002-2874
[45]Rieger,H.,and Beer,H.The Melting Process of Ice in Side a Horizontal Cylinder.ASME.1986,:166-182
[46]Chii,D.and Yeh,M.Cool Thermal Discharge from Ice Melting with Heat Fluxes on the Boundary.Energy.1996(2):455-461
[47]孔祥谦.有限单元法在传热学中的应用.科学出版社.1998年:199-216
[48]Harlow F H,Welch J F.Numerical Calculations of Time-dependent Viscous Incompressible Flow of Fluid with Free Surface.Phys Fluids.1965(8):2182-2189
[49]Chen,S.,Merriman,B.,Osher,S.,and Semereka,P.A Simple Level Set Method for Solving Stefan Problems.Journal of Computational Physics,1997(135)1:8-29
[50]Yen,Y.C.,"Free Convection Heat Transfer Characteristics in a Melt Water Layer",ASME.Journal of Heat Transfer,1980(100):550-556
[51]V R.Vo ller,A.D.Brent and C.Prakash.The Modeling of Heat,Mass and Solute Transport in Solidification Systems.International Journal of Heat and Mass Transfer.1989(32):1719-1731
[52]Marilena Giangi,Fulvio Stella,Tomasz A.Kowalewski.Phase Change Problem with Free Convection:Fixed Grid Numerical Simulation.Comput Visial Sci2.1999:123-130.
[53]Wei Shyy,H.S.Udaykuma,and Madhukar M.Rao,etal.Computational Fluid Dynamics with Moving Boundaries.Washington,D.C.,Ta ylor&Francis.1996:163-165.
[54]Einfeldt B.On Godnov-Type Methods for Gas Dynamics.SIMA J.Numer.Anal.1998(25)2:294-318
[55]商立英.非结构化网格上非稳态问题的求解与应用.南京理工大学硕士学位论 文.2006.6
[56]刘晶.结构与非结构网格的生成、转化及应用.南京理工大学硕士学位论文.2006.6
[57]胡亚才,穆弗达赫,李彬轩,冯踏青,屠传经.非稳态导热的混合差分法求解研究,应用力学学报.1998(15)1:100-104.
[58]B.Milne.Adaptive Level Set Methods Interfaces.Technical Report,University of California Berkeley Department of Mathematics.1995
[59]Frink N T,Parikh P.A Fast Unwind Solver for the Euler Equations on Three-Dimensional Unstructured Meshes.AIAA 91-0102
[60]J.Haas,B.Sturtevant.Interaction of Weak Shock Waves with Cylindrical and Spherical Gas in Homogeneities.J.Fluid Mesh.1987(181):41-76
[61]Rudman M.Volume-Tracking Methods for Interfacial Flow Calculations.International Journal Numerical Method of Fluids.1997(24)7:671-691
[62]王春武,赵宁.二维多介质流动问题的界面处理方法.南京航空航天大学学报.2005(37):114-116
[63]Xiaofeng Yang,Ashley J.James,John Lowengrub,Xiaoming Zheng,Vittorio Cristini.An Adaptive Coupled Level-Set/Volume-of-Fluid Interface Capturing Method for Unstructured Triangular Grids.J.Comput.Phys.2006(217):364-394
[64]Wonjoo Roh,Noboru Kikuchi.Analysis of Stefan Problem with Level Set Method.AIAA 2002-2874
[65]Sethian J.A.A Fast Marching Level Set Method for Monotonically Advancing Fronts.Proc.Nat.Acad.Sci.1996(93):1591-1595
[66]Sethian J.A.Level Set and Fast Marching Methods:Evolving Interfaces in Computational Geometry,Fluid Mechanics,Computer Vision and Materials Sciences.Cambridge University Press,Cambridge.UK,1996
[67]Sethian J.A.Evolution,Implementation,and Application of Level Set and Fast Marching Methods for Advancing Fronts.J.Comput.Phys.2001(169):503-555
[68]Sethian J.A.Fast Marching Methods.SIAM Rev.1999(41):199-235
[69]Sethian J.A.A Fast Level Set Method for Propagating Interface.J.Comput.Phys.1995(118):269-277
[70]傅德薰,马延文.计算流体力学.北京:高等教育出版社,2002
[2]杨世铭,陶文铨.传热学.北京:高等教育出版社,1998
[3]陶文铨.计算传热学的近代进展.北京:科学出版社,2001
[4]Suhas V.Patankar.Computation of Conduction and Duct Flow Heat Transfer.Innovative Research,Inc.7846 Ithaca Lane North Maple Grove,MN55369-8549,USA
[5]陶文铨.数值传热学.西安:西安交通大学出版社,2001
[6]M.N.奥齐西克 著,俞昌铭 译.热传导.高等教育出版社.1984:426-421
[7]Chai,J.C.,Zhang,M.,Moder,J.P.,and Patankar,S.V.Conduction Heat Transfer Calculations Using Structured and Unstructured Grids.ICHMT Symposium CHT'01-Advances in Numerical Heat Transfer.2001(1):519-526
[8]Zhang,M.Modeling of Radiative Heat Transfer and Diffusion Processes Using Unstructured Grid.Ph.D.Tennessee Technological University,USA.2000
[9]Patankar,S.V..Numerical Heat Transfer and Fluid Flow,Hemisphere Publishing,1981
[10]张洪济,热传导.高等教育出版社,1990,458-490
[11]李惜雯.数学物理方法.西安交通大学出版社.2001:178-213
[12]南京工学院数学教研组.数学物理方程和特殊函数.人民教育出版社.1982:101-148
[13]P.Rath,J.C.Chai,Modeling Two-Dimensional Diffusion-Controlled Wet Chemical Etching Using a Total Concentration Approach,International Journal of Heat and Mass Transfer,2005
[14]钱吉裕 李强 宣益民.Lattice-Boltzman方法计算多孔介质内固液相变问题.自然科学进展.2006
[15]王福军.计算流体动力学分析.清华大学出版社.2004
[16]韩占忠,王敬,兰小平.FLUENT流体工程仿真计算实例与应用.北京理工大学出版社.2004
[17]Wen S J,Jeng R H,Chun P K,Lattice Boltzmann Method for the Heat Conduction Problem with Phase Change,Numerical Heat Transfer Part B,2001(39),167-187
[18]周业涛,关振群,顾元宪.求解相变传热问题的等效热容法.化工学报.2004(55)
[19]Crank J.Free and Moving Boundary Problems.Oxford:Clarerdon Press.1984
[20]Voiler V R.Swami Nathan C R.Thomas B G.Fixed Grid Techniques for Phase Change Problems:AReview.J.Num.Meth.Eng.,1990(30):875-898
[21]张寅平,胡汉平,孔祥冬,苏跃红.相变贮能.理论和应用.中国科学技术大学出版社.1996
[22]徐士良.FORTRAN常用算法程序集.清华大学出版社.1992
[23]Bouds S,Davey K,Hinduja S.A Modified Effective Capacitance Method for Solidification Modeling Using Linear TFE.J.Num.Meth.Eng.1996(39):3195-3215
[24]Morgan K,Lewis R W,Zienkiewicz O C.An Improved Algorithm for Heat Conduction Problems with Phase Change.J.Num.Meth.Eng.,1978(2):1191-1195
[25]Commi G.Guidice S D.Finite Element Solution of Non-linear Heat Conduction Problems with Special Reference to Phase Change.J.Num.Meth.Eng.,1974(8):613-624
[26]Glimm J,Graham M J,Grove J.Front Tracking in Two and Three Dimensions.J.Comput.Math.1998(35):1-12
[27]Carslaw,H.S,Jager J C.Conduction of Heat in Solids.Clarendon Press,Oxford.1984:534-548
[28]刘捷,卢文强.非等温相变问题的双倒易边界元法求解.工程热物理学报.2006(27)
[29]Budhia H,Kreith F.Heat Transfer with Melting or Freezing in a Wedge.J.Heat Mass Transfer,1973(16):195-211
[30]Tryggvason G,Bunner B,Esmaeeli A,et al.A Front-Tracking Method for the Computations of Multiphase Flow.J.Comput.Phys.2001(169):708-759
[31]Esmaeeli A,Tryggvason G.A Front Ttracking Method for Computations of Boiling in Complex Geometries.International Journal of Multiphase Flow.2004(30):1037-1050
[32]Hirt C M,ANichols B D.Volume of Fluid(VOF)Method for the Dynamics of Free Boundaries.J.Comput.Phys.1981(39):201-225
[33]Kim S O,No H C.Second-Order Model for Free Surface Convection and Interface Reconstructions.Int.J.Numer.Meth.Fluids.1998(26)1:79-100
[34]高志华,张明义,刘志强,张淑娟.相变温度场中无网格伽辽金法的应用.计算物理.2006(23)
[35]刘儒勋,刘晓平,张磊等.运动界面的追踪和重构方法.应用数学和力学.2004(3)25:279-290
[36]Rathjen K A,Jiji L A.Heat Conduction with Melting or Freezing in a Comer.J.Heat Transfer,1973,93:101-109
[37]Chen S H,Zhang X,Bo J S.L S Mehtod in Body-Fitted Coordinate System for Multi-Dimensional Non-Ideal Detonation Propagation.Chinese Journal of High Pressure Physis.2000(14)4:280-284
[38]Kim,C J and Kaviany,M.Numerical Method for Phase-Change Problems.International Journal of Heat and Mass Transfer.1990,33(12):2721-27
[39]Voller,V R.An Overview of Numerical Methods for Solving Phase-Change Problems.Advanced Numerical Heat Transfer.1997:341-397
[40]Wakamoto,S.,Nakao,K.,Tanaka,N.Study of the Stability of Super cooled Water in an Ice Generator.ASHRAE Trans.1996,102(2):142-150
[41]方贵银.蓄冷空调工程实用新技术.人民邮电出版社.2000年
[42]Inada,T.etc.Active Control of Freezing Temperature of Water by Ultrasonic Vibration.35th National heat transfer symposium of Japan.1998(3):895-896
[43]Hahnain,S.M.Review on Sustainable Thermal Energy Storage Technologies.Heat Storage Materials and Techniques Energy Convers.1998(39)11:1127-1138
[44]Wonjoo Rob and Noboru Kikuchi.Analysis of Stefan Problem with Level Set Method.AIAA 2002-2874
[45]Rieger,H.,and Beer,H.The Melting Process of Ice in Side a Horizontal Cylinder.ASME.1986,:166-182
[46]Chii,D.and Yeh,M.Cool Thermal Discharge from Ice Melting with Heat Fluxes on the Boundary.Energy.1996(2):455-461
[47]孔祥谦.有限单元法在传热学中的应用.科学出版社.1998年:199-216
[48]Harlow F H,Welch J F.Numerical Calculations of Time-dependent Viscous Incompressible Flow of Fluid with Free Surface.Phys Fluids.1965(8):2182-2189
[49]Chen,S.,Merriman,B.,Osher,S.,and Semereka,P.A Simple Level Set Method for Solving Stefan Problems.Journal of Computational Physics,1997(135)1:8-29
[50]Yen,Y.C.,"Free Convection Heat Transfer Characteristics in a Melt Water Layer",ASME.Journal of Heat Transfer,1980(100):550-556
[51]V R.Vo ller,A.D.Brent and C.Prakash.The Modeling of Heat,Mass and Solute Transport in Solidification Systems.International Journal of Heat and Mass Transfer.1989(32):1719-1731
[52]Marilena Giangi,Fulvio Stella,Tomasz A.Kowalewski.Phase Change Problem with Free Convection:Fixed Grid Numerical Simulation.Comput Visial Sci2.1999:123-130.
[53]Wei Shyy,H.S.Udaykuma,and Madhukar M.Rao,etal.Computational Fluid Dynamics with Moving Boundaries.Washington,D.C.,Ta ylor&Francis.1996:163-165.
[54]Einfeldt B.On Godnov-Type Methods for Gas Dynamics.SIMA J.Numer.Anal.1998(25)2:294-318
[55]商立英.非结构化网格上非稳态问题的求解与应用.南京理工大学硕士学位论 文.2006.6
[56]刘晶.结构与非结构网格的生成、转化及应用.南京理工大学硕士学位论文.2006.6
[57]胡亚才,穆弗达赫,李彬轩,冯踏青,屠传经.非稳态导热的混合差分法求解研究,应用力学学报.1998(15)1:100-104.
[58]B.Milne.Adaptive Level Set Methods Interfaces.Technical Report,University of California Berkeley Department of Mathematics.1995
[59]Frink N T,Parikh P.A Fast Unwind Solver for the Euler Equations on Three-Dimensional Unstructured Meshes.AIAA 91-0102
[60]J.Haas,B.Sturtevant.Interaction of Weak Shock Waves with Cylindrical and Spherical Gas in Homogeneities.J.Fluid Mesh.1987(181):41-76
[61]Rudman M.Volume-Tracking Methods for Interfacial Flow Calculations.International Journal Numerical Method of Fluids.1997(24)7:671-691
[62]王春武,赵宁.二维多介质流动问题的界面处理方法.南京航空航天大学学报.2005(37):114-116
[63]Xiaofeng Yang,Ashley J.James,John Lowengrub,Xiaoming Zheng,Vittorio Cristini.An Adaptive Coupled Level-Set/Volume-of-Fluid Interface Capturing Method for Unstructured Triangular Grids.J.Comput.Phys.2006(217):364-394
[64]Wonjoo Roh,Noboru Kikuchi.Analysis of Stefan Problem with Level Set Method.AIAA 2002-2874
[65]Sethian J.A.A Fast Marching Level Set Method for Monotonically Advancing Fronts.Proc.Nat.Acad.Sci.1996(93):1591-1595
[66]Sethian J.A.Level Set and Fast Marching Methods:Evolving Interfaces in Computational Geometry,Fluid Mechanics,Computer Vision and Materials Sciences.Cambridge University Press,Cambridge.UK,1996
[67]Sethian J.A.Evolution,Implementation,and Application of Level Set and Fast Marching Methods for Advancing Fronts.J.Comput.Phys.2001(169):503-555
[68]Sethian J.A.Fast Marching Methods.SIAM Rev.1999(41):199-235
[69]Sethian J.A.A Fast Level Set Method for Propagating Interface.J.Comput.Phys.1995(118):269-277
[70]傅德薰,马延文.计算流体力学.北京:高等教育出版社,2002
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