化工原理课程(全英文)教学课件-13

©2015YanweiWangBasicPrinciplesofChemicalEngineeringProcessesLecture13FallSemester,2014YanweiWang(王衍伟)Email:[e-mail]©2015YanweiWang2PreviousLectureDate:April13,2015DragandDragCoefficientsTotalDrag=WallDrag+FormDrag𝑪𝑫=𝝓(𝑹𝒆𝒑)Stokes’Law:𝑭𝑫=𝟑𝝅𝝁𝒅𝒑𝒖GravitationalSettlingEquationofmotion:DragForce,TerminalVelocitySpecialcases:Stokesvs.NewtonCriterionforsettlingregimes(𝑲-parameter)𝑪𝑫=𝑭𝑫𝑨𝒑𝝆𝒖𝟐𝟐𝑹𝒆𝒑=𝒅𝒑𝒖𝝆𝝁𝒎𝒅𝒖𝒅𝒕=𝑭𝒆−𝑭𝒃−𝑭𝑫𝒖𝒕=𝟐𝒈𝝆𝒑−𝝆𝒎𝑨𝒑𝝆𝒑𝑪𝑫𝝆©2015YanweiWang3Today’sTopicFinishingChapter3FallingBallViscometerCentrifugalSettling(pp.131-140)FreeSettlingvs.HinderedSettlingWhenisBrownianmotionimportant?Flowthroughbedsofsolids(pp.128-131)Date:April16,2015©2015YanweiWang4FallingBallViscometer(1/5)Problem3.1,p.157:“FallingBallMethod”isusedtomeasuretheviscosityofliquid.Liquidisplacedintheglasscontainer.Ittakes7.32sforthesteelballwithadiameterof6.35mmtotraveldownward200mmintheliquid.Asknown,thedensityofsteelis7900kg/m3,andthedensityoftheliquidis1300kg/m3.Trytoworkouttheviscosityofliquid.“GravitationalSettling”©2015YanweiWang5FallingBallViscometer(2/5)Solution.SystemSketch𝐻=200mm=0.2m𝑡=7.32s𝑑𝑝=6.35𝑚𝑚=0.00635𝑚𝜌𝑝=7900kgm3𝜌=1300𝑘𝑔𝑚3AssumptionsNeglecttheperiodofinitialacceleration;Use𝑪𝑫datainFig.3.3.ApproachAssumetheparticleReynoldsnumberislow(1),andusetheStokesresult.𝒖𝒕=𝒅𝒑𝟐𝝆𝒑−𝝆𝒈𝟏𝟖𝝁©2015YanweiWang6FallingBallViscometer(3/5)ParticleReynoldsnumber𝑅𝑒𝑝=𝑑𝑝𝑢𝜌𝜇𝑅𝑒𝑝=𝑑𝑝𝑢𝜌𝜇=0.00635m×0.02732ms×1300kgm35.31Pa∙s=0.041Since𝑹𝒆𝒑≪𝟏,theuseoftheStokesresultisjustified.Theperiodofinitialaccelerationisontheorderof𝑹𝒆𝒑𝒅𝒑𝒖𝒕=𝟎.𝟎𝟏𝐬≪𝟕.𝟑𝟐𝐬,soitcanbesafelyneglected.SolutionStep𝑢𝑡=𝑑𝑝2𝜌𝑝−𝜌𝑔18𝜇=𝐻𝑡𝜇=𝑑𝑝2𝜌𝑝−𝜌𝑔18𝐻𝑡Substitutionofknowvaluesof𝒅𝒑,𝝆𝒑,𝝆,𝒈,𝑯,and𝒕gives𝜇=5.31Pa∙s©2015YanweiWang7FallingBallViscometer(4/5)CommentsHowaboutusingtheNewton’sLawinsteadoftheStokes’resulttostartwith?𝑢𝑡=1.75𝑔𝑑𝑝𝜌𝑝−𝜌𝜌Insertingtheknownvariablevaluesgives𝑢𝑡=0.98ms,whereas𝑢𝑡from𝑢𝑡=𝐻𝑡isonly0.027ms.Question#1:Why𝒖𝒕givenbyNewton’slawisindependentofthefluidviscosity?©2015YanweiWang8FallingBallViscometer(5/5)Question#2:Inthefallingballviscometer,thepresenceofcylinderwallwillslowthemotionofafallingballwhenthesizeofballiscomparabletothecylinderdiameter.Doesthiswalleffectartificiallyincreaseordecreasethemeasuredviscosity?Explain.©2015YanweiWang9CentrifugalSettling(1/12)Analogybetweensedimentation¢rifugationQuantityUnitConversionRotationalSpeed,𝒏Revolutionsperminute,rpm1rpm=(𝟐𝝅𝟔𝟎)rad/sAngularVelocity,𝝎Radianspersecond,rad/s𝒓=Radialdistance𝝎=AngularvelocityCentrifugalforce:𝑭𝒄=𝒎𝒓𝝎𝟐Centrifugalacceleration𝒂𝒄=𝑭𝒄𝒎=𝒓𝝎𝟐©2015YanweiWang10CentrifugalSettling(2/12)Exercise:Ifacentrifugerotorisspunat50,000rpm,calculatethecentrifugalforceexperiencedbyaparticleatradius5cmasamultipleofthegravitationalforce(=mg,whereg=9.80665m/s2).©2015YanweiWang11CentrifugalSettling(3/12)Solution.Radialdistance,𝒓=𝟓𝐜𝐦Rotationalspeed,𝒏=𝟓𝟎𝟎,𝟎𝟎𝟎𝐫𝐩𝐦RelativeCentrifugalForce(RCF)IntheTextbook,RCF,𝑲𝒄𝒈,iscalledtheseparationfactorinthecontextofcentrifugalsettling.𝑲𝒄𝒈=𝑭𝒄𝑭𝒈=𝒎𝒓𝝎𝟐𝒎𝒈𝑲𝒄𝒈=𝒓𝝎𝟐𝒈=𝟎.𝟎𝟓𝐦×(𝟓𝟎𝟎𝟎𝟎×𝟐𝝅𝟔𝟎𝐬−𝟏)𝟐𝟗.𝟖𝟎𝟔𝟔𝟓𝐦𝐬𝟐≅𝟏.𝟒×𝟏𝟎𝟓≫𝟏©2015YanweiWangEquationofmotionaccordingtoNewton’s2ndLaw:𝒎𝒅𝒖𝒅𝒕=𝑭𝒄−𝑭𝒃−𝑭𝑫Centrifugalforce:𝑭𝒄=𝒎𝒓𝝎𝟐Buoyantforce:𝑭𝒃=𝝆𝑽𝒓𝝎𝟐=𝝆𝒎𝝆𝒃𝒓𝝎𝟐Dragforce:𝑭𝑫=𝟏𝟐𝑪𝑫𝑨𝒑𝝆𝒖𝟐12CentrifugalSettling(4/12)Buoyantforce©2015YanweiWang13CentrifugalSettling(5/12)SubstitutingalltheforcestoNewton’s2ndLaw:𝒎𝒅𝒖𝒅𝒕=𝒎𝒓𝝎𝟐−𝝆𝒓𝝎𝟐𝒎𝝆𝒑−𝟏𝟐𝑪𝑫𝑨𝒑𝝆𝒖𝟐ComparingittotheEoMinGravitationalSettling:𝒎𝒅𝒖𝒅𝒕=𝒎𝒈−𝝆𝒈𝒎𝝆𝒑−𝟏𝟐𝑪𝑫𝑨𝒑𝝆𝒖𝟐WefindthatthedifferencebetweenCentrifugalsettlingandGravitationalsettlingistheuseofthecentrifugalacceleration𝒓𝝎𝟐inplaceofthegravitationalacceleration,𝒈©2015YanweiWang14CentrifugalSettling(6/12)SubstitutingalltheforcestoNewton’s2ndLaw:𝒎𝒅𝒖𝒅𝒕=𝒎𝒓𝝎𝟐−𝝆𝒓𝝎𝟐𝒎𝝆𝒑−𝟏𝟐𝑪𝑫𝑨𝒑𝝆𝒖𝟐TerminalVelocity(steadystate,𝒅𝒖𝒅𝒕=𝟎)𝒖𝒕=𝟐𝒈𝝆𝒑−𝝆𝒎𝑨𝒑𝝆𝒑𝑪𝑫𝝆Gravitationalsettling𝒖𝒕=𝝎𝟐𝒓𝝆𝒑−𝝆𝒎𝑨𝒑𝝆𝒑𝑪𝑫𝝆Centrifugalsettling©2015YanweiWang15CentrifugalSettling(7/12)ForSphericalParticlesofdiameter𝒅𝒑ProjectedArea,𝑨𝒑=𝝅𝒅𝒑𝟐𝟒;Mass,𝒎=𝝅𝒅𝒑𝟑𝟔𝝆𝒑TerminalVelocity:𝒖𝒕=𝝎𝟐𝒓𝝆𝒑−𝝆𝒎𝑨𝒑𝝆𝒑𝑪𝑫𝝆=𝝎𝟒𝒓𝝆𝒑−𝝆𝒅𝒑𝟑𝑪𝑫𝝆At𝑹𝒆𝒑≤1,UseStokes’Law,𝑪𝑫=𝟐𝟒𝑹𝒆𝒑𝒖𝒕=𝝎𝟒𝒓𝝆𝒑−𝝆𝒅𝒑𝟑𝑪𝑫𝝆→𝒖𝒕=𝒅𝒑𝟐𝝆𝒑−𝝆𝒓𝝎𝟐𝟏𝟖𝝁~𝒅𝒑𝟐Svedbergequation©2015YanweiWang16CentrifugalSettling(8/12)CriterionforSettlingRegimes(𝑹𝒆𝒑=𝒅𝒑𝒖𝝆𝝁)𝑲=𝒅𝒑𝝆𝒑−𝝆𝝆𝒓𝝎𝟐𝝁𝟐𝟏𝟑At𝑹𝒆𝒑≤1,𝑲≤𝟏𝟖𝟏𝟑=2.6(Stokes’Lawrange)At𝟏𝟎𝟑𝑹𝒆𝒑𝟐×𝟏𝟎𝟓(Newton’sLawrange)𝑹𝒆𝒑=𝟏.𝟕𝟓𝑲𝟏.𝟓Thus,if𝟔𝟖.𝟗𝑲𝟐𝟑𝟔𝟎,Newton’sLawApply.For𝟐.𝟔𝑲𝟔𝟖.𝟗or𝑲𝟐𝟑𝟔𝟎,theterminalvelocitymaybeobtainedusingareadoutvalueof𝑪𝑫fromFig.3.3orFig.3.6andatrial-and-errorprocedure.©2015YanweiWang17CentrifugalSettling(9/12)Exercise:Adispersionofoilinwateristobeseparatedusingacentrifuge.Assumethattheoilisdispersedinthef