




版权说明:本文档由用户提供并上传,收益归属内容提供方,若内容存在侵权,请进行举报或认领
文档简介
华南理工大学化工学院,伍钦,,1.3.6MacroscopicMomentumBalances2.LayerflowwithfreesurfaceInoneformoflayerflowtheliquidlayerhasafreesurfaceandflowsundertheforceofgravityoveraninclinedorverticalsurface.华南理工大学化工学院,伍钦,,Ifsuchflowisinsteadystate,withfullydevelopedvelocitygradients,thethicknessofthelayerisconstant.Thereissolittledragatthefreeliquidsurfacethattheshearstresstherecanbeignored.Iftheflowislaminarandtheliquidsurfaceisflatandfreefromripples,thefluidmotioncanbeanalyzedmathematically.华南理工大学化工学院,伍钦,,Alayerofliquidflowinginsteadyflowatconstantrateandthicknessoveraflatplate.Theplateisinclinedatanangleφwiththevertical.Thebreadthofthelayerinthedirectionperpendiculartotheplaneofthefigureisb,andthethicknessofthelayerinthedirectionperpendiculartotheplateisδ华南理工大学化工学院,伍钦,,Theuppersurfaceofthecontrolvolumeisincontactwiththeatmosphere,thetwoendsareplanesperpendiculartotheplateatadistanceL
apart,andthelowersurfaceistheplaneparallelwiththewallatadistancerfromtheuppersurfaceofthelayer.华南理工大学化工学院,伍钦,,Thepossibleforcesactingonthecontrolvolumeinadirectionparalleltotheflowarethepressureforcesontheends,theshearforcesontheupperandlowerfaces,andthecomponentoftheforceofgravityinthedirectionofflow.华南理工大学化工学院,伍钦,,Fromthisequation,notingthatA=bLand华南理工大学化工学院,伍钦,,SincetheflowislaminarRearrangingandintegratingbetweenlimitsgive华南理工大学化工学院,伍钦,,Equation(1.3-40)showsthatinlaminarflowonaplatethevelocitydistributionisparabolic.(1.3-40)华南理工大学化工学院,伍钦,,Industrialprocessesnecessarilyrequiretheflowoffluidsthroughpipes,tubes,andchannelswithanoncircularcrosssection.Thissectiondealswiththesteadyflowofpressiblefluidsthroughclosedpipesandchannels1.4pressibleFlowinPipesandChannels华南理工大学化工学院,伍钦,,FigureFluidelementinsteadyflowthroughpipe.RFlowpττdL-(p+dp)rConsiderthesteadyflowoffluidofconstantdensityinfullydevelopedflowthroughahorizontalpipe.Visualizeadisk-shapedelementoffluid,concentricwiththeaxisofthetube,ofradiusrandlengthdL,asshowninFig.华南理工大学化工学院,伍钦,,
forcesactingontheelement:Forcefromthepressureontheupstreamfacesofthedisk:Abpb=πr2(p+dP)Forcefromthepressureonthedownstreamfacesofthedisk:Aapa=πr2p华南理工大学化工学院,伍钦,,Shearforceactingontherimoftheelement:(2πrdL)τSothetotalforceactingontheelementisSimplifyingthisequationanddividingbyπr2dL(1.4-1)华南理工大学化工学院,伍钦,,Insteadyflow,thepressureatanygivencrosssectionofastreamtubeisconstant,sothatdp/dLisindependentofr,dp/dL=Δp/ΔL,Eq.(1.4-1)thenes1.4-1华南理工大学化工学院,伍钦,,Thismeansthatthemomentumfluxvarieslinearlywiththeradius,asshowninFig.,andthemaximumvalueoccursatr=Ratthewall.华南理工大学化工学院,伍钦,,SubstitutingNewton'slawofviscosity,intoEq.weobtainthefollowingdifferentialequationforthevelocity:华南理工大学化工学院,伍钦,,Integratingusingtheboundaryconditionthatatthewall,u=0atr=R,weobtaintheequationforthevelocitydistribution1.4-15
Thisresultshowsthatthevelocitydistributionisparabolic,asshowninFig华南理工大学化工学院,伍钦,,TheaveragevelocityVforacrosssectionisfoundbysummingupallthevelocitiesoverthecrosssectionanddividingbythecross-sectionalareasubstitutinguintoequationandintegrating华南理工大学化工学院,伍钦,,EquationcalledHagen–Poiseuilleequation,relatesthepressuredropandaveragevelocityforlaminarflowinahorizontalpipe.华南理工大学化工学院,伍钦,,ThemaximumvelocityforapipeisfoundfromEq.(1.4-15)andoccursatr=0:华南理工大学化工学院,伍钦,,wefindthat1.4-16Also,1.4-17
华南理工大学化工学院,伍钦,,ShellMomentumBalanceforFallingFilmWenowuseanapproachsimilartothatusedforlaminarflowinsideapipeforthecaseofflowofafluidasafilminlaminarflowdownundertheforceofgravityoveraverticalsurface.ThecontrolvolumeforthefallingfilmisshowninFig.2.9-3a.华南理工大学化工学院,伍钦,,FluidthicknessisΔxandhasalengthofLintheverticalzdirection.Thisregionissufficientlyfarfromtheentranceandexitregionssothattheflowisnotaffectedbytheseregions.Thismeansthevelocityu(x)doesnotdependonpositionz.华南理工大学化工学院,伍钦,,Theshearstresscanbeignoredatthefreeliquidsurface.Iftheflowislaminarandtheliquidsurfaceisflatandfreefromripples.Thefluidmotioncanbeanalyzedmathematically华南理工大学化工学院,伍钦,,Thepossibleforcesactingonthecontrolvolumeinadirectionparalleltothefloware
thepressureforcesattheends
theshearforcesonthefaces
thecomponentoftheforceofgravityinthedirectionofflow.华南理工大学化工学院,伍钦,,Sincethepressureontheoutersurfaceisatmospheric,thepressuresonthecontrolvolumeattheendsofthevolumeareequaland‘oppositelydirected.Also,byassumption,theshearonthefreeliquidsurfaceoftheelementisneglected.华南理工大学化工学院,伍钦,,Thetwoforcesremainingarethereforetheshearforceontherightsurfaceofthecontrolvolumeandthecomponentofthegravityinthedirectionofflow.Fromthisequation,notingthatA=WLandFg=gρLWx华南理工大学化工学院,伍钦,,SoSincetheflowislaminar,τ=-µdu/dxandRearrangingandintegratingbetweenlimitsgives(theboundaryconditionthatu=0atx=δ)华南理工大学化工学院,伍钦,,andWhereδisthetotalthicknessoftheliquidlayer2.9-25
华南理工大学化工学院,伍钦,,Thismeansthevelocityprofileisparabolic,asshowninFig.2.9-3b.Themaximumvelocityoccursatx=0inEq.(2.9-25)andis2.9-26
华南理工大学化工学院,伍钦,,Theaveragevelocitycanbefound:2.9-27
SubstitutingEq.(2.9-25)into(2.9-27)andintegrating2.9-28
华南理工大学化工学院,伍钦,,
VelocityProfilesinPipesFigureisaplotoftherelativedistancefromthecenterofthepipeversustheratiooflocalvelocitytomaximumvelocityu/umax.Forlaminarflow,thevelocityprofileisatrueparabola.Thevelocityatthewalliszero.华南理工大学化工学院,伍钦,,InmanyengineeringapplicationstheaveragevelocityVinapipeisthemostuseful.Insomecasesonlytheumaxatthecenterpointofthetubeismeasured.TherelationshipbetweenumaxandVcanbeusedtodetermineV.experimentallymeasuredvaluesofV/umaxareplottedasafunctionoftheReynoldsnumbers华南理工大学化工学院,伍钦,,Theaveragevelocityisprecisely0.5timesthemaximumvelocityatthecenterforlaminarflow.Ontheotherhand,forturbulentflow,thecurveissomewhatflattenedinthecenterandtheaveragevelocityisabout0.8timesthemaximum.Thisvalueof0.8variesslightly,dependingupontheReynoldsnumber。华南理工大学化工学院,伍钦,,Theliquidlayerhasafreesurfaceandflowsinlaminarflowundertheforceofgravityoveraverticalsurface,thevelocityprofileis()Averagevelocityis()Problem1华南理工大学化工学院,伍钦,,Theaveragevelocityis()timesthemaximumvelocityatthecenteroftubeforlaminarflow.Theaveragevelocityis()timesthemaximumvelocityatcenterforturbulentflow华南理工大学化工学院,伍钦,,Themaximumvalueofthemomentumfluxoccurs()foralaminarflow.Themaximumvalueofthevelocityoccurs()foralaminarflowinpipe.Inlaminarflowthevelocitydistributionwithr
温馨提示
- 1. 本站所有资源如无特殊说明,都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
- 2. 本站的文档不包含任何第三方提供的附件图纸等,如果需要附件,请联系上传者。文件的所有权益归上传用户所有。
- 3. 本站RAR压缩包中若带图纸,网页内容里面会有图纸预览,若没有图纸预览就没有图纸。
- 4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
- 5. 人人文库网仅提供信息存储空间,仅对用户上传内容的表现方式做保护处理,对用户上传分享的文档内容本身不做任何修改或编辑,并不能对任何下载内容负责。
- 6. 下载文件中如有侵权或不适当内容,请与我们联系,我们立即纠正。
- 7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
最新文档
- 二零二五版产业园区排水管网工程合同
- 高端制造业产品畅销国际路径
- 二零二五年度养老机构整栋租赁合同范本
- 2025版管件维修保养服务合同范本
- 2025版网络安全防护技术合伙开发合同协议书
- 2025版虚拟现实教育平台开发合同
- 二零二五版商业照明灯具租赁与维护服务合同
- 2025版租赁房屋租赁合同租赁用途及限制规范
- 二零二五年度公路货运企业信用评价合同标准范本
- 2025版办公室装修材料运输及保管合同模板下载
- 2023年上海市上海市徐汇区枫林路街道招聘社区工作者真题附带题目详解
- 2025赤峰市松山区中小学教师招聘考试试题及答案
- 赋能智慧矿山的智能决策与优化方案
- 地质调查员职业技能考试题(附答案)
- 初中历史表现性试题情境真实的分析与教学启示
- 2025年中小学班主任基本功大赛笔试试题题库(附答案)
- 2025年上半年广东东莞市寮步镇人民政府招聘工作人员4人易考易错模拟试题(共500题)试卷后附参考答案
- 2025年度链家租赁合同电子版(含房屋设施维修责任)
- 闽教版(2020)小学信息技术四年级上册《综合活动1 规划主题收集资料》说课稿及反思
- 静脉导管常见并发症临床护理实践指南(上)
- 《妇产科学绪论》课件
评论
0/150
提交评论