PROBLEMS FOR CHAPTER 1 1-1 - Dr. Syahrullail Samion
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PROBLEMS FOR CHAPTER 1 1-1 A sphere 1.4 cm in diameter is placed in a freestream of 18 m/s at 20°C and 1 atm. Compute the diameter Reynolds number of the sphere if the fluid is (a) air, (b) water and (c) hydrogen. 1-2 A telephone wire 8 mm in diameter is subjected to a crossflow wind and begins to shed vortices. From figure, what wind velocity in m/s will cause the wire to “sing” at middle C (or 256 Hz)? 1-3 If the wire in Problem 1-2 is subjected to a crossflow wind of 12 m/s, use Figure 1-9 to estimate its drag force (in N/m) 1-4 For oil flow in a pipe far downstream of the entrance (Figure 1-10 and 1-11), the axial velocity profile is a function of r only and is given by: = b − b where C is a constant and R is the pipe radius. Suppose the pipe 1 cm in diameter and efg is 30 m/s. Compute the wall shear stress in Pa if =0.3 kg/m.s. 1-5 A tornado may be simulated as two-part circulating flow in cylindrical coordinates, with: j = k =0 l = ≤ l = b > Determine: (a) the vorticity and (b) the strain rates in each part of the flow. 1-7 A two-dimensional unsteady flow has the velocity components: = 1+ = 1 + 2 Find the equation of the streamlines of this flow which pass through the point ( y , y ) at time = 0. 1-8 Using Eq.(1-2) for inviscid flow past a cylinder, consider the flow along the streamline approaching the forward stagnation point , = , . Compute (a) the distribution of strain rates jj and jl along this streamline and (b) the time required for a particle to move from the point 2, to the stagnation point.
Transcript of PROBLEMS FOR CHAPTER 1 1-1 - Dr. Syahrullail Samion
PROBLEMSFORCHAPTER11-1Asphere1.4cmindiameterisplacedinafreestreamof18m/sat20°Cand1atm.ComputethediameterReynoldsnumberofthesphereifthefluidis(a)air,(b)waterand(c)hydrogen.1-2Atelephonewire8mmindiameterissubjectedtoacrossflowwindandbeginstoshedvortices.Fromfigure,whatwindvelocityinm/swillcausethewireto“sing”atmiddleC(or256Hz)?1-3IfthewireinProblem1-2issubjectedtoacrossflowwindof12m/s,useFigure1-9toestimateitsdragforce(inN/m)1-4Foroil flow inapipe fardownstreamof theentrance(Figure1-10and1-11), theaxialvelocityprofileisafunctionofronlyandisgivenby:
𝑢 =𝐶𝜇
𝑅b − 𝑟b
whereCisaconstantandRisthepiperadius.Supposethepipe1cmindiameterand𝑢efgis30m/s.ComputethewallshearstressinPaif𝜇=0.3kg/m.s.1-5Atornadomaybesimulatedastwo-partcirculatingflowincylindricalcoordinates,with:
𝑣j = 𝑣k = 0
𝑣l = 𝑟𝜔𝑖𝑓𝑟 ≤ 𝑅
𝑣l =𝜔𝑅b
𝑟𝑖𝑓𝑟 > 𝑅
Determine:(a)thevorticityand(b)thestrainratesineachpartoftheflow.1-7Atwo-dimensionalunsteadyflowhasthevelocitycomponents:
𝑢 =
𝑥1 + 𝑡
𝑣 =𝑦
1 + 2𝑡
Findtheequationofthestreamlinesofthisflowwhichpassthroughthepoint(𝑥y, 𝑦y)attime𝑡 = 0.1-8UsingEq.(1-2)forinviscidflowpastacylinder,considertheflowalongthestreamlineapproachingthe forwardstagnationpoint 𝑟, 𝜃 = 𝑅, 𝜋 .Compute(a) thedistributionof strainrates𝜖jjand𝜖jlalongthisstreamlineand(b)thetimerequiredforaparticletomovefromthepoint 2𝑅, 𝜋 tothestagnationpoint.
1-9Acommonlyusedequationofstateforwaterisapproximatelyindependentoftemperature:
𝑝𝑝y≈ 𝐴 + 1
𝜌𝜌y
�− 𝐴
where 𝐴 ≈ 3000, 𝑛 ≈ 7, 𝑝y ≈ 1𝑎𝑡𝑚, 𝜌y ≈ 998𝑘𝑔/𝑚� . From this formula, compute (a) thepressure(inatm)requiredtodoublethedensityofwater,(b)thebulkmodulusofwaterat1atm,and(c)thespeedofsoundinwaterat1atm.1-10Asshownbelow,a0.3×0.4(m2)plateslidesdownalong30°inclineonwhichthereisafilmofoil0.5mmthickwithviscosityµ=0.1kg/m.s.Assumingthat theplatedoesnotdeformtheoil film,estimate:(a) Theterminalslidingvelocity(inm/s)(b) Thetimerequiredfortheplatetoacceleratefromrestto99%oftheterminalvelocity.
1-11Estimatetheviscosityofnitrogenat86MPaand49°Candcomparewiththemeasuredvalueof45µPa.s.(86MPaishighpressure,cannotuse“low-density”method)
1-12Estimatethethermalconductivityofairat400°Cand1atmandcomparewiththemeasuredvalueof0.05015W/(m.K).
1-13Itisdesiredtoformagasmixtureof23%CO2,14%O2and63%N2at1atmand20°C.Estimatetheviscosityandthermalconductivityofthismixture.
Theconstituentpropertiesareasfollows:
Constituent Molefraction(𝑥) µ(Pa.s) K(W/m.K)CO2(44) 0.23 1.37×10-5 0.0146O2(32) 0.14 1.92×10-5 0.0244N2(28) 0.63 1.66×10-5 0.0242
1-14Somemeasuredvaluesfortheviscosityofammoniagasareasfollows:
Temp(K) 300 400 500 600 700 800
µ(Pa.s) 1.03×10-5 1.39×10-5 1.76×10-5 2.10×10-5 2.51×10-5 2.88×10-5
1-15AnalyzetheflowbetweentoplatesofFigure1-15byassumingthefluidisadeWaelepower-lawfluidasinEq.1-31a.Compute:(a)Thevelocityprofile𝑢(𝑦)withthepowernasaparameter(b)Thevelocityatthemidpointℎ/2forn=0.5,1.0and2.0
1-16Repeattheanalysisofthevelocityprofilebetweentwoplates(Figure1-15)foraNewtonianfluidbut allow for a slip velocity𝛿𝑢at bothwalls. Compute the shear stress at bothwalls. The slipvelocityis:
𝛿𝑢 ≈ ℓ𝑑𝑢𝑑𝑦
andshearstressattopwallis
𝜏� =𝜇𝑉
ℎ + 2ℓ
1-19Fromthepreviousproblem,ifthetemperature,spheresizeandvelocityremainthesameforairflow,atwhatairpressurewilltheReynoldsnumberbeequalto10,000.1-20Asolidcylinderofmassm, radiusRand lengthL, falls concentrically throughavertical tubeofradiusR+∆R,where∆R≪R.Thetubesisfilledwithgasofviscosityµandmeanfreepathℓ.Neglectfluidforcesonthefrontandbackfacesofthecylinderandconsideronlyshearstressintheannularregion,assumingalinearvelocityprofile.Findananalyticalexpressionfortheterminalvelocityoffall,V,ofthecylinder(a)fornoslip,(b)withslip(Eq.1-91).1-21Oxygenat20°Candapproximately1200Pa(abs)flowsthrougha35µmdiametersmoothcapillarytubeatanaveragevelocityof10cm/s.EstimatetheKnudsennumberoftheflowandwhetherslipflowwillbeimportant.1-22Adiskrotatessteadilyinsideadisk-shapedcontainerfilledwithoilofviscosityµ.Assumelinearvelocityprofileswithnoslipandneglectstressontheouteredgesofthedisk.FindaformulaforthetorqueMrequiredtodrivethedisk.
1-23ShowfromEq.1-86,thatthecoefficientofthermalexpansionofaperfectgasisgivenby𝛽 = �
� .
Usethisapproximationtoestimate𝛽ofammoniagas(NH3)at20°Cand1atmandcomparewiththeacceptedvaluefromadatareference.
1-24Therotating-cylinderviscometershearsthefluidinanarrowclearance∆r,asshown.Assumingalinearvelocitydistributioninthegaps,ifthedrivingtorqueMismeasured,findanexpressionforµby(a)neglectingthebottomfrictionand(b)includingthebottomfriction.
1-25Consider1m3ofafluidat20°Cand1atm.Foranisothermalprocess,calculatethefinaldensityandtheenergy,injoules,requiredtocompressthefluiduntilthepressureis10atm,for(a)airand(b)water.Discussthedifferenceinresults.
1-26Equal layers of two immiscible fluids are being sheared between a moving and a fixed plate.Assuminglinearvelocityprofiles,findanexpressionfortheinterfacevelocityUasafunctionofV,µ1andµ2.
1-27Usetheinviscid-flowsolutionofflowpastacylinder,Eq.1-3,to:
(a) Findthelocationandvalueofthemaximumfluidaccelerationalongthecylindersurface.Isyourresultvalidforgasesandliquids?
(b) Applyyourformulaforamaxtoairflowat10m/spastacylinderofdiameter1cmandexpressyourresultasaratiocomparedtotheaccelerationofgravity.Discusswhatyourresultimpliesabouttheabilityoffluidstowithstandacceleration.
