电渣重熔过程熔化、流体流动和凝固的二维轴对称数值模拟设计研究

..电渣重熔过程熔化、流体流动和凝固的二维轴对称数值模拟研究李秀杰,张立峰基金项目：中国自然科学基金资助项目〔51274034、51334002和51404019作者简介：基金项目：中国自然科学基金资助项目〔51274034、51334002和51404019作者简介：李秀杰〔1990-,男,硕士；E-mail：lixiujies@163.com；通讯作者：张立峰〔1972-,男,博士,教授；E-mail：zhanglifeng@；〔北京科技大学冶金与生态工程学院,北京100083摘要：本文建立二维轴对称数学模型,使用FLUENT并基于VOF多相流模型对电渣重熔过程的电磁场和流场进行数值模拟。计算结果表明：熔滴在形成、脱离电极和滴落过程中,使系统内的电流重新分布,熔滴受到的电磁力作用也随之发生变化。系统焦耳热、电流密度、电磁力和速度最大值都随熔滴的滴落而逐渐增大,在熔滴脱离电极时达到最大值；相比电极开始熔化时,这些数值变化较大。关键词：电渣重熔；多场耦合；熔化；数值模拟中国分类号：TF744文献标识码：A文章编号：1674-1048〔201504-0359-05Two-dimensionalNumericalSimulationofElectroslagRemeltingProcessLIXiu-jie,ZHANGLi-feng,YULe,RENYing<SchoolofMetallurgicalandEcologicalEngineering,UniversityofScienceandTechnologyBeijing,Beijing100083.>Abstract：Atransient2Daxisymmetricmathematicalmodelwasdevelopedtosimulatethefluidflow,electromagneticfield,heattransferandsolidificationduringelectroslagremelting<ESR>processusingcommercialsoftwareFLUENT,involvingVOFmultiphasemodelanduser-developedsubroutines.Theformation,departureandfallingofdropletsdisturbedthedistributionoftheelectriccurrentdensityandtheelectromagneticforceonthedropletchangedaswell.ThemaximumofJouleheat,currentdensity,electromagneticforceandvelocityincreasedwiththefallingofdroplet,andreachedtheirmaximumvaluewhenthedropletdepartsfromtheelectrode.Keywords:electroslagremelting;multi-fieldscoupling;remelting;numericalsimulation电渣重熔作为一种二次冶金技术,生产出的产品具有高致密度、高洁净度、凝固偏析少、良好的表面质量等优点,通常用来生产一些合金钢和特殊钢。由于电渣生产过程内部参数无法通过直接手段进行测量,但随着计算机技术和数值模拟方法的不断发展和完善,利用数学方法可以非常直观地呈现电渣重熔系统内部电磁场、流场、温度场等物理场的分布规律,这对提高产品质量和优化工艺参数将起到重要作用ADDINEN.CITE<EndNote><Cite><Author>梁强</Author><Year>2012</Year><RecNum>52</RecNum><DisplayText><styleface="superscript">[1]</style></DisplayText><record><rec-number>52</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">52</key></foreign-keys><ref-typename="Thesis">32</ref-type><contributors><authors><author><styleface="normal"font="default"charset="134"size="100%">梁强</style></author></authors></contributors><titles><title><styleface="normal"font="default"size="100%">GH4169</style><styleface="normal"font="default"charset="134"size="100%">电渣重熔过程数值模拟和凝固组织控制</style></title></titles><volume><styleface="normal"font="default"charset="134"size="100%">博士</style></volume><keywords><keyword>ÊýֵģÄâ</keyword><keyword>µçÔüÖØÈÛ</keyword><keyword>Ô²Å÷Á¬Öý</keyword><keyword>ζȳ¡</keyword><keyword>Äý¹Ì</keyword></keywords><dates><year>2012</year></dates><pub-location><styleface="normal"font="default"charset="134"size="100%">北京</style></pub-location><publisher><styleface="normal"font="default"charset="134"size="100%">北京科技大学</style></publisher><work-type><styleface="normal"font="default"charset="134"size="100%">博士学位</style></work-type><urls></urls><remote-database-provider>Cnki</remote-database-provider></record></Cite></EndNote>[\o"梁强,2012#52"1]。早期Szekely等ADDINEN.CITEADDINEN.CITE.DATA[\o"Dilawari,1977#99"2-5]对复杂的电渣重熔系统模型做了一定的简化,通过自编程序计算了系统的电磁场、温度场和流场,但模型过于简化,计算结果只能帮助研究者定性了解整个电渣过程。国内学者魏季和等ADDINEN.CITE<EndNote><Cite><Author>魏季和</Author><Year>1994</Year><RecNum>36</RecNum><DisplayText><styleface="superscript">[6,7]</style></DisplayText><record><rec-number>36</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">36</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>魏季和</author><author>任永莉</author></authors></contributors><titles><title>电渣重熔体系内熔渣流场的数学模拟</title><secondary-title>金属学报</secondary-title></titles><pages>B481-B490</pages><volume>30</volume><number>11</number><dates><year>1994</year></dates><urls></urls></record></Cite><Cite><Author>魏季和</Author><Year>1995</Year><RecNum>35</RecNum><record><rec-number>35</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">35</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>魏季和,</author><author>任永莉</author></authors></contributors><titles><title>电渣重熔体系内磁场的数学模拟</title><secondary-title>金属学报</secondary-title></titles><pages>51-60</pages><volume>31</volume><number>2</number><dates><year>1995</year></dates><urls></urls></record></Cite></EndNote>[\o"魏季和,1994#36"6,\o"魏季和,1995#35"7]基于Maxwell方程组及有关的电磁场理论,在Szekely电磁模型基础上提出了正确的数学模型,研究了无熔滴滴落情况下渣池中的流动情况。近些年来,随着数值模拟技术的发展,一些商用软件如ANSYS、FLUENT等被应用到电渣冶金过程中。李宝宽ADDINEN.CITE<EndNote><Cite><Author>Li</Author><Year>2013</Year><RecNum>111</RecNum><DisplayText><styleface="superscript">[8,9]</style></DisplayText><record><rec-number>111</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">111</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Li,Baokuan</author><author>Wang,Bo</author><author>Tsukihashi,Fumitaka</author></authors></contributors><titles><title>ModelingofElectromagneticFieldandLiquidMetalPoolShapeinanElectroslagRemeltingProcesswithTwoSeries-ConnectedElectrodes</title><secondary-title>MetallurgicalandMaterialsTransactionsB</secondary-title></titles><periodical><full-title>MetallurgicalandMaterialsTransactionsB</full-title></periodical><pages>1122-1132</pages><volume>45</volume><number>3</number><dates><year>2013</year></dates><isbn>1073-5615</isbn><urls></urls></record></Cite><Cite><Author>B.K.Li</Author><Year>2012</Year><RecNum>4</RecNum><record><rec-number>4</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">4</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>B.K.Li,</author><author>F.Wang,</author><author>F.Tsukihashi,</author></authors></contributors><titles><title>Current,MagneticFieldandJouleHeatinginElectroslagRemeltingProcesses</title><secondary-title>ISIJInternational</secondary-title></titles><periodical><full-title>ISIJInternational</full-title></periodical><pages>1289-1295</pages><volume>52</volume><number>7</number><dates><year>2012</year></dates><isbn>0915-1559</isbn><urls></urls></record></Cite></EndNote>[\o"Li,2013#111"8,\o"B.K.Li,2012#4"9]、王芳ADDINEN.CITE<EndNote><Cite><Author>王芳</Author><Year>2011</Year><RecNum>39</RecNum><DisplayText><styleface="superscript">[10]</style></DisplayText><record><rec-number>39</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">39</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>王芳</author><author>李宝宽</author><author>陈瑞</author><author>赵林</author></authors></contributors><titles><title>多电极电渣重熔系统电磁场和焦耳热场研究</title><secondary-title>材料与冶金学报</secondary-title></titles><pages>112-118</pages><volume>10</volume><number>S1</number><dates><year>2011</year></dates><urls></urls></record></Cite></EndNote>[\o"王芳,2011#39"10]利用ANSYS来计算系统的电磁场和温度场分布,并改变电流大小、频率,插入深度等操作参数来研究这些参数对系统的影响,计算结果具有一定的指导意义。余乐ADDINEN.CITE<EndNote><Cite><Author>余乐</Author><Year>2014</Year><RecNum>112</RecNum><DisplayText><styleface="superscript">[11]</style></DisplayText><record><rec-number>112</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">112</key></foreign-keys><ref-typename="ConferencePaper">47</ref-type><contributors><authors><author><styleface="normal"font="default"charset="134"size="100%">余乐</style></author><author><styleface="normal"font="default"charset="134"size="100%">张立峰</style></author><author><styleface="normal"font="default"charset="134"size="100%">任英</style></author><author><styleface="normal"font="default"charset="134"size="100%">王祎</style></author><author><styleface="normal"font="default"charset="134"size="100%">程礼梅</style></author><author><styleface="normal"font="default"charset="134"size="100%">李秀杰</style></author></authors></contributors><titles><title><styleface="normal"font="default"charset="134"size="100%">电渣重熔过程金属熔滴形成及运动的三维数学模型</style></title><secondary-title><styleface="normal"font="default"charset="134"size="100%">全国特种冶金技术学术会议</style></secondary-title></titles><pages>1122-1132</pages><volume>45</volume><number>3</number><dates><year>2014</year></dates><isbn>1073-5615</isbn><urls></urls></record></Cite></EndNote>[\o"余乐,2014#112"11]、贺铸ADDINEN.CITE<EndNote><Cite><Author>王芳</Author><Year>2015</Year><RecNum>122</RecNum><DisplayText><styleface="superscript">[12,13]</style></DisplayText><record><rec-number>122</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">122</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>贺铸刘艳贺李宝宽王芳</author></authors></contributors><titles><title>电极插入深度对电渣重熔过程影响的数模研究</title><secondary-title>东北大学学报<自然科学版></secondary-title></titles><periodical><full-title>东北大学学报<自然科学版></full-title></periodical><pages>218-222</pages><volume>36</volume><number>2</number><dates><year>2015</year></dates><isbn>1005-3026</isbn><urls></urls></record></Cite><Cite><Author>刘艳贺</Author><Year>2013</Year><RecNum>29</RecNum><record><rec-number>29</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">29</key></foreign-keys><ref-typename="ConferenceProceedings">10</ref-type><contributors><authors><author>刘艳贺</author><author>贺铸</author><author>夏添</author><author>王芳</author><author>王强</author><author>李宝宽</author></authors></contributors><titles><title>电渣重熔过程电磁,流动和温度场的数值模拟</title><secondary-title>第十七届<2013年>全国冶金反应工程学学术会议论文集<下册></secondary-title></titles><dates><year>2013</year></dates><urls></urls></record></Cite></EndNote>[\o"王芳,2015#122"12,\o"刘艳贺,2013#29"13]将ANSYS计算的电磁场和焦耳热场以源项的形式加载到FLUENT的动量和能量守恒方程中,研究了重熔过程中熔滴的形成和滴落、系统的温度、流动等情况。RückertADDINEN.CITE<EndNote><Cite><Author>A.Rückert</Author><Year>2007</Year><RecNum>73</RecNum><DisplayText><styleface="superscript">[14,15]</style></DisplayText><record><rec-number>73</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">73</key></foreign-keys><ref-typename="ConferenceProceedings">10</ref-type><contributors><authors><author><styleface="normal"font="default"size="100%">A</style><styleface="normal"font="default"charset="134"size="100%">.</style><styleface="normal"font="default"size="100%">Rückert,</style></author><author>H.Pfeifer,</author></authors></contributors><titles><title>NumericalModellingoftheElectroslagRemeltingProcess</title><secondary-title>16thInternationalMetallurgicalandMaterialsConference</secondary-title></titles><pages><styleface="normal"font="default"size="100%">22</style><styleface="normal"font="default"charset="134"size="100%">-24</style></pages><dates><year>2007</year></dates><urls></urls></record></Cite><Cite><Author>A.Rückert</Author><Year>2009</Year><RecNum>126</RecNum><record><rec-number>126</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">126</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>A.Rückert,</author><author>H.Pfeifer,</author></authors></contributors><titles><title>MathematicalModellingoftheFlowField,TemperatureDistribution,MeltingandSolidificationintheElectroslagRemeltingProcess</title><secondary-title>Magnetohydrodynamics</secondary-title></titles><periodical><full-title>Magnetohydrodynamics</full-title></periodical><pages>527-533</pages><volume>45</volume><number>4</number><dates><year>2009</year></dates><isbn>0024-998X</isbn><urls></urls></record></Cite></EndNote>[\o"A.Rückert,2007#73"14,\o"A.Rückert,2009#126"15]建立二维轴对称模型并假设电磁场不随流动变化,使用FLUENT-MHD模型研究了电渣重熔过程的熔化、流动和凝固情况。以上研究考虑了系统电磁场对流动和温度场的作用,对电渣重熔过程中电磁场、流场、温度场间的相互耦合现象做了一定的探索工作,但是熔滴的形成和滴落改变了系统的相分布,这将直接影响系统的电磁场分布,上述方式却不能体现电磁场随着熔滴瞬态变化,无法将熔滴的运动对电磁场的影响考虑在内。因此,本文旨在建立电磁场瞬态计算模型以实现电磁场和流场的完全耦合。这里借助计算流体力学软件—FLUENT,通过编写UDF求解Maxwell方程组,在每个迭代时间步内重新计算此刻不同相分数分布下的电磁场,并将电磁力以源项形式加载到动量方程中,计算整个电渣重熔体系的流动情况。1数学模型1.1几何模型图1为计算使用的几何模型,本文以电渣炉内自耗电极、渣池、金属熔池为研究对象,建立二维轴对称模型,假定电极熔化端头形状为平底且插入深度为0。表1为模拟所使用的金属和熔渣的材料属性以及相关的模型几何尺寸和供电参数。图1电渣重熔几何模型Fig.1GeometrymodelofESR表1材料物理属性和模型参数Table1Materialsphysicalpropertyandmodelparameter参数数值单位金属密度7160kg·m-3粘度0.0067Pa·s电导率714000Ω-1·m-1磁导率1.257e-6渣密度2570kg·m-3粘度0.042Pa·s电导率251Ω-1·m-1磁导率1.257e-6模型电极半径0.045m渣池半径、高度0.065、0.06m金属锭半径、高度0.065、0.06m操作条件电流大小2800A电流频率50Hz渣-金界面张力0.9N·m-11.2控制方程由于本文采用的是冷态模拟,所以不考虑温度场和凝固过程的计算。因此,描述电渣重熔过程的数值模拟方程包括连续性方程、动量方程、湍流方程、多相流模型和Maxwell方程组。连续性方程ADDINEN.CITE<EndNote><Cite><Author>Wang</Author><Year>2014</Year><RecNum>104</RecNum><DisplayText><styleface="superscript">[16]</style></DisplayText><record><rec-number>104</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">104</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Wang,Qiang</author><author>He,Zhu</author><author>Li,Baokuan</author><author>Tsukihashi,Fumitaka</author></authors></contributors><titles><title>AGeneralCoupledMathematicalModelofElectromagneticPhenomena,Two-PhaseFlow,andHeatTransferinElectroslagRemeltingProcessIncludingConductingintheMold</title><secondary-title>MetallurgicalandMaterialsTransactionsB</secondary-title></titles><periodical><full-title>MetallurgicalandMaterialsTransactionsB</full-title></periodical><pages>2425-2441</pages><volume>45</volume><number>6</number><dates><year>2014</year></dates><isbn>1073-5615</isbn><urls></urls></record></Cite></EndNote>[\o"Wang,2014#104"16]：<1>动量方程ADDINEN.CITE<EndNote><Cite><Author>Wang</Author><Year>2014</Year><RecNum>104</RecNum><DisplayText><styleface="superscript">[16]</style></DisplayText><record><rec-number>104</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">104</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Wang,Qiang</author><author>He,Zhu</author><author>Li,Baokuan</author><author>Tsukihashi,Fumitaka</author></authors></contributors><titles><title>AGeneralCoupledMathematicalModelofElectromagneticPhenomena,Two-PhaseFlow,andHeatTransferinElectroslagRemeltingProcessIncludingConductingintheMold</title><secondary-title>MetallurgicalandMaterialsTransactionsB</secondary-title></titles><periodical><full-title>MetallurgicalandMaterialsTransactionsB</full-title></periodical><pages>2425-2441</pages><volume>45</volume><number>6</number><dates><year>2014</year></dates><isbn>1073-5615</isbn><urls></urls></record></Cite></EndNote>[\o"Wang,2014#104"16]：<2>式中,ρ为流体密度,kg·m-3；t为时间,s；g为重力加速度,m·s-2；为速度,m·s-1；为压力,Pa；μeff为流体有效粘度,Pa·s；为时间平均电磁力,N·m-3。湍流方程ADDINEN.CITE<EndNote><Cite><Author>Wang</Author><Year>2014</Year><RecNum>121</RecNum><DisplayText><styleface="superscript">[17]</style></DisplayText><record><rec-number>121</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">121</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Wang,Shengqian</author><author>Zhang,Lifeng</author><author>Tian,Yue</author><author>Li,Yanlong</author><author>Ling,Haitao</author></authors></contributors><titles><title>SeparationofNon-metallicInclusionsfromMoltenSteelUsingHighFrequencyElectromagneticFields</title><secondary-title>MetallurgicalandMaterialsTransactionsB</secondary-title></titles><periodical><full-title>MetallurgicalandMaterialsTransactionsB</full-title></periodical><pages>1915-1935</pages><volume>45</volume><number>5</number><dates><year>2014</year></dates><isbn>1073-5615</isbn><urls></urls></record></Cite></EndNote>[\o"Wang,2014#121"17]：研究结果表明,电渣重熔过程的流动处于湍流状态ADDINEN.CITE<EndNote><Cite><Author>Weber</Author><Year>2009</Year><RecNum>120</RecNum><DisplayText><styleface="superscript">[18]</style></DisplayText><record><rec-number>120</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">120</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Weber,V</author><author>Jardy,A</author><author>Dussoubs,B</author><author>Ablitzer,D</author><author>Rybéron,S</author><author>Schmitt,V</author><author>Hans,S</author><author>Poisson,H</author></authors></contributors><titles><title>Acomprehensivemodeloftheelectroslagremeltingprocess:descriptionandvalidation</title><secondary-title>MetallurgicalandMaterialsTransactionsB</secondary-title></titles><periodical><full-title>MetallurgicalandMaterialsTransactionsB</full-title></periodical><pages>271-280</pages><volume>40</volume><number>3</number><dates><year>2009</year></dates><isbn>1073-5615</isbn><urls></urls></record></Cite></EndNote>[\o"Weber,2009#120"18]。因此,在计算时湍流模型选择标准k-ε双方程模型,采用标准壁面函数处理近壁区域。湍流方程表示如方程<3>和方程<4>。<3><4>式中,Gk为平均速度梯度产生的湍动能,m2·s-2；Gb为浮力产生的湍动能,m2·s-2；μt为湍流粘度,Pa·s；C1、C2和C3为常数,C1=1.44,C2=1.92；σk,σε为湍流普朗特数,σk=1,σε=1.3。多相流模型ADDINEN.CITE<EndNote><Cite><Author>Wang</Author><Year>2014</Year><RecNum>104</RecNum><DisplayText><styleface="superscript">[16]</style></DisplayText><record><rec-number>104</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">104</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Wang,Qiang</author><author>He,Zhu</author><author>Li,Baokuan</author><author>Tsukihashi,Fumitaka</author></authors></contributors><titles><title>AGeneralCoupledMathematicalModelofElectromagneticPhenomena,Two-PhaseFlow,andHeatTransferinElectroslagRemeltingProcessIncludingConductingintheMold</title><secondary-title>MetallurgicalandMaterialsTransactionsB</secondary-title></titles><periodical><full-title>MetallurgicalandMaterialsTransactionsB</full-title></periodical><pages>2425-2441</pages><volume>45</volume><number>6</number><dates><year>2014</year></dates><isbn>1073-5615</isbn><urls></urls></record></Cite></EndNote>[\o"Wang,2014#104"16]：整个电渣重熔系统包含两相：金属相和渣相。本文选用VOF多相流模型追踪相边界,在此模型中设定计算域内每一单元格中的流体体积分数为α,α满足方程<5>。<5>当控制单元内没有流体时,α=0；当单元内充满流体时,α=1；在两相混合区中,α=0~1。因此,以上各控制方程中局部区域的材料属性值由α决定。电磁场方程ADDINEN.CITEADDINEN.CITE.DATA[\o"Bernardi,2003#110"19-26]：电磁场满足Maxwell方程组<忽略位移电流>：<6><7><8>电磁场本构方程为：<9><10>根据Dilawari和Szekely的研究结果ADDINEN.CITE<EndNote><Cite><Author>Dilawari</Author><Year>1977</Year><RecNum>99</RecNum><DisplayText><styleface="superscript">[2]</style></DisplayText><record><rec-number>99</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">99</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Dilawari,AH</author><author>Szekely,J</author></authors></contributors><titles><title>AmathematicalmodelofslagandmetalflowintheESRprocess</title><secondary-title>MetallurgicalTransactionsB</secondary-title></titles><periodical><full-title>MetallurgicalTransactionsB</full-title></periodical><pages>227-236</pages><volume>8</volume><number>1</number><dates><year>1977</year></dates><isbn>0360-2141</isbn><urls></urls></record></Cite></EndNote>[\o"Dilawari,1977#99"2],在电渣重熔过程中,磁雷诺数较小,磁扩散可以忽略不计,因此电流密度表达式<等式<10>>可简化为：<11>为简化电磁场方程,这里引入矢量磁式<>和标量电势<φ>,则和可写为：<12><13>电流在传导过程中将遵循守恒定律,电流密度守恒方程为：<14>由电磁感应定律可知,当系统内存在变化的磁场时,在导体区域会产生一定大小的感应电流。对于电渣重熔过程,由于从电极顶部通入一定频率的交流电,所以流经整个系统的电流包括2部分：传导电流和感应电流,可以表示为：<15>根据电流密度守恒方程、电流密度表达式和库伦规范,可得电势方程为：<16>对于矢量磁式,利用矢量恒等式和库伦规范,由方程<6>、方程<9>和方程<12>可以得出：<17>由于电磁问题为四维<三个空间分量和一个时间分量>实问题,在数值计算时需要将其降维。因此,引入复数概念,将电磁场四维问题转化为三维空间复问题。在此假设和按正弦规律变化,变化频率为f,为消去方程<17>中的时间因素,可将和表示为：<18><19>将方程<18>、<19>带入方程<17>可得磁矢量式的方程为：<20>将磁矢量式分离成,则方程<17>可离散成：<21><22>通过联立方程<21>、<22>和方程<16>可以求出矢量磁式<>和标量电势<φ>,进而求出、,以及相应的电磁场物理量中的时间平均电磁力Fe和时间平均焦耳热Q：<23><24>式中,为磁场强度,A·m-1；为电场强度,V·m-1；为磁感应强度,T；μ为介质磁导率,F·m-1；μ0为真空磁导率,F·m-1；σ为介质电导率,Ω-1·m-1；为电流密度,A·m-2；为传导电流密度,A·m-2；Jr和Ji分别为电流密度的实部和虚部,A·m-2；ω为角速度,rad·s-1；和分别为磁矢量式的实部和虚部,V·s·m-1。1.3边界条件模拟计算时,忽略电极的运动,设定入口边界为速度入口,大小为0.000676m·s-1,出口边界为outflow,渣池与空气接触面设定为零剪切应力边界条件,其余壁面设定为无滑移边界条件。由于电渣炉的水冷作用,在钢锭表面产生一薄层的渣皮,而且渣的电导率很小,所以认为无电流流过渣皮,也即渣池和金属锭与电渣炉壁间绝缘。根据文献ADDINEN.CITEADDINEN.CITE.DATA[\o"Dilawari,1977#99"2,\o"Wang,2014#104"16,\o"Wang,2015#128"26]：入口和出口处的磁感应强度B连续,壁面处的磁感应强度很小可以忽略。表2ADDINEN.CITE<EndNote><Cite><Author>Wang</Author><Year>2014</Year><RecNum>104</RecNum><DisplayText><styleface="superscript">[16,27]</style></DisplayText><record><rec-number>104</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">104</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Wang,Qiang</author><author>He,Zhu</author><author>Li,Baokuan</author><author>Tsukihashi,Fumitaka</author></authors></contributors><titles><title>AGeneralCoupledMathematicalModelofElectromagneticPhenomena,Two-PhaseFlow,andHeatTransferinElectroslagRemeltingProcessIncludingConductingintheMold</title><secondary-title>MetallurgicalandMaterialsTransactionsB</secondary-title></titles><periodical><full-title>MetallurgicalandMaterialsTransactionsB</full-title></periodical><pages>2425-2441</pages><volume>45</volume><number>6</number><dates><year>2014</year></dates><isbn>1073-5615</isbn><urls></urls></record></Cite><Cite><Author>Kharicha</Author><Year>2014</Year><RecNum>98</RecNum><record><rec-number>98</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">98</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Kharicha,Abdellah</author><author>Ludwig,Andreas</author><author>Wu,Menghuai</author></authors></contributors><titles><title>OnMeltingofElectrodesduringElectro-SlagRemelting</title><secondary-title>ISIJInternational</secondary-title></titles><periodical><full-title>ISIJInternational</full-title></periodical><pages>1621-1628</pages><volume>54</volume><number>7</number><dates><year>2014</year></dates><isbn>0915-1559</isbn><urls></urls></record></Cite></EndNote>[\o"Wang,2014#104"16,\o"Kharicha,2014#98"27]中列出了电磁场的计算边界条件表达式,式中I为电流大小,Rd为金属锭半径。表2电磁场边界条件数值表达式ADDINEN.CITE<EndNote><Cite><Author>Wang</Author><Year>2014</Year><RecNum>104</RecNum><DisplayText><styleface="superscript">[16,27]</style></DisplayText><record><rec-number>104</rec-number><foreign-keys><keyapp="EN"db-id="zwfrdw5vbwpwvdepa9hvwpxp0rzde5zdz22e">104</key></foreign-keys><ref-typename="JournalArticle">17</ref-type><contributors><authors><author>Wang,Qiang</author><author>He,Zhu</au