8/7/2019 Computational Fluid Dynamics For Newtonian Fluid Flow Through Concentric Annuli With Center Body Rotation

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COMPUTATIONALFLUID DYNAMICSl!l., FoRNBWTONIANFLUID FLOW THROUGHCONCENTRICANNULI WITHCENTREBODY ROTATION

. Engr.MdMamunurRashidr,Dr.J.A. Nasel' M.Sc n Mech.Engg'96 BUET),M-136'77IEB,M-675BCS,MBSMEAssistant ngineer Mechanical),FCL,Tarakandi, amalpur-2055,angladesh.2MechanicalEngineeringDepartment, angladesh niversityof Engg.& Tech.,Dhaka-000,Bangladesh.

ABSTRACT- Computationalluid dynamicsCFD)hasbeen ery successfuln modelingurbulentgasand iquid flows n many situations.t canprovideanswerso complex low, heat ransferand chemicalreaction usuallycombustion) roblemshat are simplynot solvable y any othermeansexcept esting.Newtonian luid flow phenomenons very important n all the pharmaceuticalndustries nd in manychemicalndustries. computer rogramme apable f predictingNewtonian ehavior. his s because fexperimentalnvestigation f Newtonianluid flow is notonly expensive,aborious nd ime consuming;tis impossiblen manycases. hepowerof prediction nables s o operate xistingequipmentmoresafelyand effrciently.Predictions f the relevantprocess elp us in forecasting nd evencontrollingpotentialdangers.Thesepredictionsoffer economic enefitsand contribute o human well being. The existinganaly'tical nd numerical echniques an only deal with very specific deal cases.Under he situation,areliablecomputer rogramme,which can run on a personal omputer,s very muchdesirable. or thisreason,he aminar low of Newtonian luids throughconcentric nnuli with center ody otationhasbeenstudied umerically. hescope f this studys limited o numerical redictionof axialvelociq'profilesandtangential elocityprofilesat steady tatecondition.A generalcomputerprogram TEACH-T" hasbeenmodified or this purpose. he programwas usedafter suffrcientustification.The computer rogram sused or the predictionof the axial and angential elocities. In the present tudy,confinedlow throughconcentric nnuliwith centre ody otation s examined umerically y solving hemodifiedNavier-Stokesequations. easurementf the axialand angential omponentsf velocity s presentedn non-dimensionalform for a Newtonianluid. The annulargeometry onsists f a rotatingcenterbodywith angularspeed f126 rpm and a radius atioof 0.506.The solutionof governing etof partialdiffertial equationss donebyfinite drfferenceomputation. non-uniform rid arrangement f 52x32with multiple epetitionss used.The governingequations avebeen ntegrated umericallywith the aid of a finite-volumemethod.TheHybrid scheme nd the central differencingwere adopted o properlyaccount or convention-diffirsioneffects,and the coupling of continuity rvith the momentumequationswas treatedwith the SIMPLEalgorithm.The numericalpredictionshavebeenconfirmedby comparing hemwith the experimentallyderivedaxial and angential elocityprofilesobtainedor a Newtonian. or theNewtonianGlucose)luid,the studywascarriedout or Reynolds's umber f 800and1200.Keyword: Center ody.Hydraulicdiameter, onvention-diftrsion frects ndReynolds umber.

TNTRODUCTIONIn the present study, a detailed computationalinvestigationon the Newtonianfluid flow throughconcentric nnuliwith center ody otationwith glucoseas the workingfluid will be carriedout.The geometryand dimensions f theNewtonianluid flow isbased nthe experimental tudiesEscudieret al. (1995).The

presentstudy deals with numerical nvestigationofNewtonian luid flow through concentricannuli withcenterbody rotation. The fluids are dilute solutionofGlucose.The glucose olutions a I : ilw mixtureof aglucose syrup (cerestar)and water. The specificobjectivesof this study are to developa computerprogram or theoreticalnvestigation fcombinedaxialand tangential aminar velocity of concentric nnular

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with center- body rotation, to study the constantrotational speedwith different Reynoldsnumberofconcentricannuli with centerbody rotation low andfinal aftemptwouldbe made o establish eliabilities.suitabilityand assessmentf thequalityof thisprogramthrough comparing he results obtainedwith thoseavailablen the iterature.

PREVIOUSWORKNouri et. al. (1994)presented xperimentalesults orflow of Newtonianand non-Newtonian luids in aconcentric nnuluswith rotationof the innercylinder.Experimentwasconductedor the annularpassagelowwith an outerbmsspipeof nominal nsidediameterDoof 40.3 mm, engthof 2.0 m andan nner stainless teelrod of 20 mm diameter,D-. Theypointedout that themean velocity and the corresponding eynolds hearstresses f Newtonianand non-Newtonianluids weremeasuredn a fully developed oncentric low with adiameleratioof 0.5and nner$linder rotationalpeedof 300 rpm.With theNeqtonian luid in laminar low,the effectsof the inner shaft rotationwere a uniformincreasen the drag co-efficient y about28 percent,flatter and lessskewedaxial meanvelocityand swirlprofile with a narrowboundary lose o the innerwallwith a thickness fabout22 percent fthe gapbetweenthepipes.Escudier t.al . (1995) erformed xperimentsith testsection onsists f five modules achof L027 m lengthand one of 0.64 m which givesan overall engthof5.77 It was ength o hydraulicdiameteratioof I 16.The experimental flow geometry was that of aconcentric moothwalled annuluswith rotatingcenterbodyof radius atio 0.506.The cenlerbody otated t aspeed not exceeding126 rpm. They pointed outincreasing he bulk velocitt' (for constant otationalspeed) roduces progressiveeduction n the levelofthe tangentialvelocity hat is similar for the Glucoseand carboxymethylcelluloseCMC) fluids, exceptanomalous ehavioror CMC at low Reynolds umber.In other aper, scudier t.al .(1995a) emonstratedheapplicabilityor turbulent nnular low in the absencefcenterbody otationusing scalingcriterionproposedyHolt (1991)or drag educingluids n pipe low.

GOVERNING DIFFERENTIALEQUATIONSThis work is concernedwith steady aminar flow inconcentric annuli with center body rotation. The

unity in the presentwork and consistencyndex K,which is also emperaturendependent. he fluid flowin concentric annuli with center body rotation isconsidered nder hefollowingconditions: ) The luidflow in laminar and steady.b) The fluid densityp,consistenryndexK, thermal conductivity , and heatcapacity Co are temperaturendependent.Under theassumptions tatedabove,continuityand momentumequationsor an incompr6essibleluid in cylinder co-ordinate r,0,z) system re:Continuity:Av- Av" V"- - - - - ' - - U t 2 lAr0zrMomentum:

.. OoVz , Ad/, fu,n fuoo oo-l r-^ +t.-*=-:! +--;-: i -r' C r - d d d r)po }po 6o,o 6on 2ot, , -= * l'_- 3 = --:L - -n | - - ,e (3)' 0 r0zAr0zr

,, W, ,,, W, -fur, ,br, ,o"-oeo, r n T r 7 ^ - r ^c r -dc rd rWherehestressensorsregiven y

' . = * ( * ) '\ o z ), , (9?-") '\ c t ' r )

( l )rheologicalequationused n this work is well-knownpower aw, yiz. where ., in the shear stress,n is atemperaturendependent xponent,which is equal o

DISCRETTZEDGOVERNINGDIFFERENTIAL EQUATIONS

In the presentstudy the finite volume approach,as

described y Gosman t al. [989]. is adopted. ypicalhowever,he Newtonianerm,which is included n thepresent tudy,s introducedhrough hesourceerms. nhis approach, he governingdifferential equationsarediscretized y integratinghem over a finite numberofcontrolvolumesor computational ells, nto which the

=K(9!t * avt\Iaz 0r)=-P + zK(Y')"\ar )

(av \ "-P +2K l ^ ' I o "\ d z . ) - z r

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solution dornain is divided. discretized ransport u'illtake he following quasilinear orm.(ao-b)00-Ianu"u+ CWhere he O't6 are coefficicientsnultiplying the valuesof Q at the neighbouringnodessurrounding he centralnode P. The number of neighbour dependson theinterpolationpracticeor differencingschemes sed.Here ?p is the coelTicient f QO givenbvao=Z anu, = Summationover neighbours N, S, E, W)For the presentstudy. he Hybrid Scheme s used.Thename Hvbnd indicates a combination of the CentralDifference Scheme (CDS) and Uprvind DifferenceScheme (lDS) For the range of peclect number(puL/T-) -2 < P"< 2, both the difhrsion and convectiveterm are evaluatedby the CDS. Outside this rangeconvective erms are evaluatedusing the UDS and thediffusion ermsare evaluated sing CDS.Boundaryconditions of the Presentstudy are at inletboundary. la t profile of axial velocity is specified.atoutlet boundary, he gradientsof all variablesare set ozero n the axial direction and at rvall boundaries. uterrvallvelociry s set o a constant alue.

SOLUTIONALGORITHMThe procedure evelopedor the calculationofthe flou,field has beengiven the name SIMPLE, rvhich standsfor Semi- hnplicit Method for Pressure-LinkedEquations. The procedure has been described inPatankar ndSpalding 1972),Caretto t al. (1972). ndPatankar (1975). Operation in the order of theirexecutionareas follorvs:a) Guess he pressureield p*.b) Solve he momentum quationso obtainu ; r,'. an dw'. c) Solve lte p* sqttlion. d) Calculatep by adding$R*. .; Calculateu, v, and rv lrom their starredvaluesusing the velocities +orrection formulas. I Treat thecorrectedpressurep as a nerv guessedpressurep*.return to stepb, and repeat he rvholeprocedure ntil aconverged olution s oblained. Numericalsolutionofthe governingequationfor transportof rnornenturnsobtainedby using the SIMPLE algorithm. The fluid isconsideredNewtonian.The florv geometry s concentricu'ith the inner pipe rotating.More correctorstagesmaybe added following procedure used for the secondcorrector stage.For the florv concentricannuli rvithcenter body rotation in the calculation domain can bedoneby inserting'jnternal' oundary ondition.

RESULTAND DISCUSSIONSThe results f numerical imulation f Newtonianluidflorv through concentricannuli rvith center bodyrotation are presentedand comparedwith theexperimentsf Escudier t al. (1995). he results reobtainedby the numericalmethoddescribedn aboveSolutionalgorithmsection. he solutiondomainwas

bounded by the inlet plane. exit plane. outside solidrvall. nside vall with constant otationalspeed nd axissymmetry. The entire investigationdomain is dividedinto a non-uniform grid anangement of 52x32 rvithmultiple repetition s used.Fine grid spacingwas usednear he solid rvallsand a relativecoursegrid was usedin the florv region. For the presentstudy he followingvalues of parametersare chosen: Porver law indexn=1.00.ConsistencyndexK=0.01N-Vm2. Densi tv= 1134 g /m', OuterRadiusRo =0.0502m. nner adiusRi:0.0254 m, Length X=5.775 m and Rotationalspeedof inner pipeN= 126 pm.

Nurncr ical52r-12)Analyt ical u: n 1969)1 f " ? ' )42x32

(= R f (R , , -R ,)

Fig. I Grid ndependenceest forGlucose f &:800at V Dr =104

,t',/"r./ , \1t '

. g au'd

Ir'tfip]tIIfi1atii,

-a

5. - o

x / Dh=t 0

5

i)

i

.1

1)

t'' \t '\l*,b"ffil|I - Presart relcicn i\-l-l--\

Fig.2 Axialvelocity rofilesor Glucose t&:800Grid ndependenceest s necessaryo testwhetherhepredictedesults re ndependentf grid.At a constantReynoldsnumber,Re=800with 32x22, 42x32 and52x32 grids. The 52x32grid gave reasonably ridpredictionsvhencomparedo the theoretical redicted

xzL

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of Yuan et al . [969] asshown n figurel. Figures an d3 represent he developingaxial velocity profiles.Theprofrles at different non-dimensionalaxial distance sshorvn n such a way so that the gradual changes nprofiles from flat to developedparabolic tlpe can beeasily nspected.The last curve (at length to hydraulic diameter ratio.X/Dh=104) of Figure 3 shows the developedvelocityprofrlecomparedwith experintenlal esultof Escudieretal . (1995). The last curve (a t length to hydraulicdiameter atio,X/Dh:104) of figure 2 is compared vithlaminar Neptonian profile. From Figure 3. themaximum velocity for experimental esult is 1.22 andnumerical solution 1.38. Hence again the differencemay have occurred due to developedof turbulence bythe inner rotating pipe. From figure 2. the maxilnumvelocity for laminar New'tonianprofile is 1.484 andnumerical olution1.49.

xzDi,j

Hence percentageof deviation 0.5%o n maxlmumvelocity profile observed.This indicates he validity ofthe presentmethodology.For both Figures2 and 3 themaxirnumvelocity occurs nearcenterof the annuli forNestonian fluid. Figures 4 and 5 represents thetangential velocity profiles for Neu.tonian luids. Thegradualchange of tangential velocity prohle is shownin concentricannuli rvith centerbodyrotation The lastcurve(at lenglh to hydraulicdiameter atio, )Vdh=104)of Figure 4 is in excellentagreement ith the theoreticaldata. The last curve (at length to hydraulic diameterratio, X/Dh=104) of Figure 5 is comparedwith theexperimental ataof escudier t al. (1995).Also in th ecaseof Newtonian luid it is shown hat as he Reynoldsnumber is increased, the tangential velocity levelswithin the annular gap are progressivelyeduced.Thesamequalified behaviorwas found by Escudieret al .(1995)and Nouri and aw (1994).

1 01 0l l , ' i l' ll*- F::'::i::i:'"'' ' \1 # X/Or=04csdds et at1995)H x / D = 1 0 4e X / D = 5 0.- xDl=rs

9 J X / D = 5(H x/D:=l( - X / D = 0 5

, f l/. ! / J l g ' Jl /| , / /| / a- r . i f ll a I I./.../ I I/ t /t / ',' **j*J-r I, " . r t . , * 1 / l+44

* - ' i ' r t I i) 4 . " 1 a ! *^

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Due to the turbulent diftrsion of fluid at Reynoldsnumber=1200thispredictedesultsshownn Figure5 are not in good agreementwith our numericalpredicton. he reasons that Escudier t al. (1995)mentionedhat n their showed xperimentsurbulentdiftrsion waspresent.Due to this turbulentdiftisionthe luid particlesmove rom highervelocity egion olower velocity region and henceuniform velocityoccur at the central egionof the annuli.But in ournumerical scheme turbulent diffi.rsion was notconsidered.So that for the Newtonian luid flow the tangentialvelocity gradient in the inner layer must besubstantiallyhigher than in the outer layer. Thisexpression as a consequence f the torquebeingconstantwithin the annulargapand he assumptionflaminarsub-layerst each urface. his situationor anon-Newtonian luid is more complex, althoughqualitatively, he same rend evidently exists.Thepresentprediction ailed to reproducehis behavior.As the Reynolds umber s increasedhe tangentialvelocity levels within the annular gap areprogressively educedexcept for non- Newtonianfluid (CMC)at Re 110,which s shownn figure6. InFigure4 Newtonianaminar angentiallow forGlucoseat Re1'nolds umber800 is comparedwith

f- e r -l7 l r , " r i ' IL e = . , . , l - ( t ) 1 - ( D 1 l ' 1 - f 1r ; - r ; l r _ lthe analyical data or following Equation (Yuan S.W.1969) and it is found to be in excellent agreement.This shows the validity of present numericalpredictions.

CONCLUSIONLaminaraxialand angentiallow throughconcentricannuliwith centerbody otationhavebeensimulatedfor Newtonian glucose). The main findings aresummarized elow: a) For Newtonian fluids, theaxial velociff profile at inlet is flat and graduallytransforms o parabolicshape.b) Maximum axialvelocityoccurs t a regionclose o the nner wall. c)The tangential elocitydecreases ith the ncrease fradius.Near he nnerwall it changes harply,whilenear the outer wall it changes slowly. And d)Increasing the Reynolds number for constantrotational speedproducesa progressivelyeducedlevelofthe tansential elocitv.

RECOMMENDATIONSThe same prediction can be carried out for theturbulent cases by incorporating the turbulenttransport quationsor bothNewtonian luid andnon-Newtonian luid. Similar studycan be made oreccentricannularwith centerbodv rotation.Hieher

order schemese.g.LUDS, Quick Scheme) an beused o havebetteraccuracyn this ffi ofprediction.Similar studycan be made or differentsize, ength,diameterand rotationalspeed.Similarprediction anbemadegiving he nnerbody otationwith vibration.ACKNOWLEDGEMENTS

The authorsare grateful o the Ministry of Scienceand technology, angladeshor providing ellowshipduring his research ork.REFERENCES

Escudier,M.P. and Gouldson, W., "Concentricannular low with center ody otationof a Newtonianand shear thinning liquid" Int. JournalofHeat andFluidFlow.Vol.16,No.3, 1995).Gosman,A.D. and Iderials,F.J.K,* TEACH-T:AGeneral computer Program for Two DimensionalTurbulent Recirculating Flows", Department ofMechanicalEngineering,mperialCollege,London,sw7, 1e76).Nouri,J.M. and Whitelaw, .H., " Flowof Newtonianand non-Newtonian luids n a Concentric nnuluswith Rotation of the Inner Cylinder", J. FluidEngineering,ol. 16,pp.82l-827,1994).Nouri,J.M.,Umur,H. andWhitelaw, .H., " FlowofNewtonianand non-Newtonian luids n Concentricand EccentricAnnuli", Journalof Fluid Mechanics,253,6t7-64t,1993).Popovska, . and Wilkinson, W.L., " LaminarHeatMass Transfer to Newtonianand Non-NewtonianFluids in tubes", ChemicalEngineeringScience,32,t154-tt64,197',t).Yuan, S.W., " Foundations f Fluid Mechanics",Prentice-Hallof India Private Limited, New Delhi,(1969).

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